[    {        "title": "1712.07323v1.Unifying_ultrafast_demagnetization_and_intrinsic_Gilbert_damping_in_Co_Ni_bilayers_with_electronic_relaxation_near_the_Fermi_surface.pdf",        "content": " 1 Unifying ultrafast demagnetization and intrinsic Gilbert damping in Co/Ni \nbilayers with electronic relaxation near the Fermi surface  \nWei Zhang, Wei He*, Xiang -Qun Zhang, and Zhao -Hua Cheng*  \nState Key Laboratory of Magnetism and Beijing National Laboratory for \nCondensed Matter Physics, Institute of Physics, Chinese Academy of \nSciences, Beijing 100190, P. R. China  \nJiao Teng  \nDepartment of Materials Physics and Chemistry, University of Sci ence and \nTechnology Beijing, Beijing 100083, P. R. China  \nManfred Fä hnle  \nMax Planck Institute for Intelligent Systems, Heisenbergstra e 3, 70569 \nStuttgart, Germany  \nAbstract  \nThe ability to controllably manipulate the laser -induced ultrafast magnetic \ndynamics  is a prerequisite for future high speed spintronic devices. The optimization \nof devices requires the controllability of the ultrafast demagnetization time, , and \nintrinsic Gilbert damping, . In previous attempts to establish the relationship \nbetween \nM  and \nrint  , the rare -earth doping of a permalloy film with two different \ndemagnetization mechanism is not a suitable candidate. Here, we  choose Co/Ni \nbilayers to  investigate the relations between and  by means of ti me-resolved \nmagneto -optical Kerr effect (TRMOKE) via adjusting the thickness of the Ni layers, \nand obtain an approximately  proportional  relation between these two parameters.   \nM\nintr\nM\nintr 2 The remarkable agreement between TRMOKE experiment and the prediction of \nbreathi ng Fermi -surface model confirms that a large Elliott -Yafet spin -mixing \nparameter \n2b  is relevant to the strong spin -orbital coupling at the Co/Ni interface. \nMore importantly, a proportional relation between \nM  and \nintr  in such metallic \nfilms or heterostructures  with electronic relaxation near Fermi surface suggests the \nlocal spin -flip scattering domains the mechanism of ultrafast demagnetization, \notherwise the spin -current mechanism domains. It is a n effective method to \ndistinguish the dominant contributions to ultrafast magnetic quenching in metallic \nheterostructures by investigating both the ultrafast demagnetization time and Gilbert \ndamping simultaneously. Our work can open a novel avenue to manip ulate the  \nmagnitude and efficiency of Terahertz  emission in metallic heterostructures such as \nthe perpendicular magnetic anisotropic Ta/Pt/ Co/Ni /Pt/Ta  multi layers,  and then it has \nan immediate implication of the design of high frequency spintronic devices . \n \nPACS numbers: 75.78.Jp, 75.40.Gb, 76.50.+g, 78.47.+p  \n*Correspondence and requests for materials  should be addressed to Z.H.C \n(zhcheng@iphy.ac.cn ) or W.H. ( hewei@iphy.ac.cn ) \n   3 Since the pioneering work on ultrafast demagnetization of Ni thin film after \nfemtosecond laser irradiation was demonstrated in 1996 by Beaurepaire et al1, the \nquest for ultrafast modification of the magnetic moments has triggered a new field of \nresearch : Femtomagnetism . It leads to the dawn of a new ear for breaking the ultimate \nphysical limit for the speed of magnetic switching and manipulation, which are \nrelevant to current and future information storage. In the past two decades, the \nultrafast dynamics  in hundreds of femtoseconds have been probed with the \nfemtosecond laser pulse using magneto -optical Kerr1 or Faraday effect2, or other \ntime-resolved techniques such as the high -harmonic generation (HHG) of extreme \nultraviolet(XUV) radiation3, magnetic cir cular dichroism4, or spin resolved two -photo \nphotoemission5.  \nNevertheless, the microscopic mechanism underlying ultrafast quenching of \nmagnetization remains elusive. Various mechanisms including electron -phonon \nmediated spin -flip scattering6-9, electron -electron scattering10,11, electron -magnon \nscattering12,13, direct angular momentum transfer from photon to electron mediated by \nspin-orbit coupling14,15, coherent interaction among spins electrons and photons16, \nwere proposed to explain the ultraf ast spin dynamics.  In addition, since Malinowski17 \net al first proposed that the laser excited spin current transport could increase and \nspeed up the magnetic quenching in metallic heterostructures, the laser -induced \nsuper -diffusive spin current was raise d to play an important role in determining the \nultrafast demagnetization in metallic films or heterostructures18-22. However, the \nrecent demonstration23 shows that the unpolarized hot electrons transport can  4 demagnetize a ferromagnet, indicating the local spin angular momentum dissipation is \nunavoidable even when super -diffusive spin transport domains in the metallic \nheterostructures. Moreover, even in th e similar samples, the local spin -flip scattering \nand nonlocal spin transport mechanism were proposed respectively by different \nexperimental tools19, 24 to explain the ultrafast demagnetization . It is harmful for \nclarifying the underlying ultrafast demagne tization mechanism in such metallic \nheterostructures. Therefore, an effective method to distinguish  the two dominant \ncontribution s to ultrafast  demagnetization in metallic heterostructures is highly \ndesirable19,23,24. Here, we propose that investigating bo th the ultrafast demagnetization \ntime and Gilbert damping25 simultaneously is a candidate method, although the \nrelationship between the two parameters has never been unified successfully so far \nbetween the experiments and theoretical predictions.    \nAn inv erse relation between and was first derived by Koopmans et al. \nfrom a quantum -mechanical calculation on the basis of the Elliot t-Yalfet (EY) \nspin-flip scattering model6. Later, the attempted experiments have ever been carried \nout to demonstrate the predict ion in rare -earth -doped permalloys26,27 and amorphous \nTbFeCo films28. In this case, t he localized 4f electrons  rather than itinerant 5 d6s \nelectrons domain most of the large magnetic moment  in rare -earth elements. Because \nthe 4 f electrons are far from the Fermi level, their ultrafast demagnetization processes \nare medicated by 5 d6s electrons after laser pulse excitation7. The indirect excitation \nleads to the so called type_II ultrafast demagnetization behavior in rare -earth elemen ts, \nwhich is much slower than that of itinerant electrons. Therefore, it is not unexpected \nM\nintr 5 that the ultrafast demagnetization time \nM  of permalloys increases with the doping \ncontents of rare -earth elements increasing. Meanwhile , it happ ened  that the Gilbert \ndamping constant of permalloys is  also increased by doping 4 f elements, which \nmainly comes from the so called “slow relaxing impurities mechanism”29. Therefore, \nby introducing the extra mechanism unavoidablely ，a trivial consequence wa s \nobtained that the ultrafast demagnetization time increases as the Gilbert damping  \n\n increases in rare -earth -doped permalloys26. In hindsight, from this experiment, one \ncan not confirm the relation between ultrafast demagnetization time  and Gilbert \ndamping  \ndue to the defects of the experimental design. A genuine relation between \nultrafast demagnetization time and Gilbert damping should be explored in a clean \nsystem without extra demagnetization mechanism. So far, the explicit relationship \nbetween the two parameters has never been unified successfully between the \nexperiments and theoretical predictions. Our work in Co/Ni bilayers with the electrons \nrelaxing at the Fermi surface can fill in the blank.   \nIn the cas e of pure 3 d itinerant electrons  relaxing near the Fermi surface after the \nlaser excitation , both ultrafast demagnetization  and Gilbert damping are determined \nby spin -flip scattering of itinerant electrons at quasi -particles or impurities . Based on \nthe breathing Fermi -surface model of Gilbert damping and on the EY relation for the \nspin-relaxation time, a proportional relation between and was derived by \nFä hnle et al30,31 for the materials with conductivity -like damping. And an inverse \nrelation was also d erived which is similar with that proposed by B. Koopmans et al \nwhen the resistivity -type damping domains in the materials.  Although the predicted \nM\nM\nM\nintr 6 single numerical values of intr/M are in good agreement with the experimental ones \nfor Fe, Ni, or Co, for a confirmation of the explicit relation between and  one \nhas to vary the values on the two parameters systematically for one system, as we do \nit in our paper by changing the thickness of the films.    \nCo/Ni bilayers with a stack of Ta (3 nm)/Pt (2 nm)/Co ( 0.8 nm)/Ni ( dNi nm)/Pt (1 \nnm)/Ta (3 nm) were grown on glass substrates by DC magnetron sputtering32, 33. The \nthickness of Ni layer changes from dNi = 0.4 nm to dNi = 2.0 nm. T heir static \nproperties have been shown in the Part Ⅰof the Supplementary Materials34. Both\nand for Co/Ni bilayer systems have been achieved by using time -resolved \nmagneto -optical Kerr effect (TRMOKE) technique21, 35. The reasons for selecting the \nCo/Ni bilayers are three -fold. First, Co/Ni bilayers with perpendicular magnetic \nanisotropy  (PMA) are one of candidates for perpendicular magnetic recording (PMR) \nmedia and spintronic devices36-39. Second, the electrons in both Co and Ni are \nitinerant near the Fermi surface and they have the same order of magnitude of \ndemagnetization time7,10. Without rare earth element doping in 3 d metals, one can \nexclude the possibility of an extra slow demagnetization accompanied by doping with \n4f rare-earth metals. Third, both and in Co/Ni bilayers can be tuned by \nchanging the Ni thickness. Therefore, Co/Ni bilayers provide an ideal system to \ninvestigate the relation between and . A nearly p roportional  relationship \nbetween and was evident  in Co/Ni bilayers, suggesting that the \nconductivity -like damping30, 31 plays a dominant role. It is distinct i n physics with \nprevious experiments26 where  the seemingly similar results have been obtained via \nM\nintr\nM\nintr\nM\nintr\nM\nintr\nM\nintr 7 introducing extra slow demagnetization mechanism. Moreover, we discussed the \norigin of Gilbert damping, analyzed its influence on the relation between \nM  and \nintr\n and proposed a new approach to distinguish the intrinsic spin -flip and extrinsic \nspin current mechanism for ultrafast demagnetization in metallic heterostructures. The \nfinding for this unification can provid e the possibility for manipulating the \nlaser -induced ultrafast demagnetization via Gilbert damping in high frequency or \nultrafast spintronic devices  such as the Terahertz emitters .  \nFig. 1(a) shows time -resolved MOKE signals40 for films with various Ni lay er \nthickness measured with an external field  Oe. The quantitative values of \nintrinsic Gilbert damping constant41-44 in Fig. 1(b) can be obtained by eliminating the \nextrinsic contributions (See the Supplementary Materials [34],  PartⅡ for details). It \nwas observed that intr decreases with increasing Ni layer thickness. On the one hand, \nprevious investigations39, 45 have been reported that the large PMA origins from the \nstrong spin -orbit coupling effect at Co/Ni interface. A thickness modification in Co/Ni \nbilayer can change the competition between interface and volume effect, and \nconsequently the PMA. When we plot the intrinsic Gilbert damping constant as a \nfunction of effective anisotropy field  in Fig. 4 in the Part Ⅱ in Supplementary \nMaterial (See the Supplementary Materials [34],  PartⅡ for details), a proportional \nrelation was confirmed in our Co/Ni bilayer system, which demonstrates that \nspin-orbit coupling contributes to both Gilbert damping and PMA  (Also, for the \nachievement of effec tive anisotropy field, please see the Supplementary Materials [34]  \nPartⅡfor details ). On the other hand, the interface between Ni and Pt maybe also \n4000H 8 modified via changing Ni layer thickness. Because the Gilbert damping increases \nlinearly when the Ni layer b ecomes thinner, it seems that the spin current dissipation \nis involved partly. A similar trend was observed in a Pt/CoFeB/Pt system46, in which a \npure non -local spin pumping effect domains the Gilbert damping. Therefore, the total \nGilbert damping equals to  α=𝛼𝑖𝑛𝑡𝑟 +𝛼𝑠𝑝 , in which 𝛼𝑠𝑝  represents the \ncontributions from spin current. Due to the low spin diffusion length of Pt, the \nmagnetization precession in Ni layer entering the Pt layer would be absorbed \ncompletely like in the system of Py/Pt and Py/Pd47 and so on. H owever，we have to \naddress that, i n the case of the variation of ferromagnetic layer thickness, the amount \nof spin current pumped out of ferromagnet is determined entirely by the parameter of \ninterfacial mixing conductance  𝐺𝑒𝑓𝑓𝑚𝑖𝑥 48,49.  It is a constant value once the normal \nmetal thickness is fixed , although the Gilbert damping in thinner magnetic layer is \nenhanced. Therefore, given the spin current contributes partly to the Gilbert damping \nat present, the spin angular momentum transferring from Ni layer to Pt layer would be  \nthe same  for various Ni lay er thickness.       \n  The central strategy  of our study is to establish a direct correlation between \nultrafast demagnetization time and the intrinsic Gilbert damping constant. The \nintrinsic Gilbert damping constant was extracted from magnetization precessi on in \nhundreds of ps timescale. The laser -induced ultrafast demagnetization dynamics has \nbeen measured carefully within time delay of 2.5  ps at a step of 15  fs and low laser \nfluence of 1 was used. Fig. 2 (a) shows the TRMOKE signals of the ultrafast \ndemagn etization evolution after optical excitation. A rapid decrease of magnetization \n2/cmmJ 9 takes place on the sub -picosecond timescale followed by a pronounced recovery. As \ncan be seen in this figure, the ultrafast demagnetization rate is different by changing \nthe Ni  thickness.  \nTo identify the effect of the heat transport across the film thickness on \ndemagnetization time, a numerical simulation50 was carried out to demonstrate that \nthe demagnetization time  variation induced with the thicknesses ranged from 1.2 nm \nto 2.8 nm  is so small that can be ignored  (See the Supplementary Materials [34], Part \nⅢ for details), although a relatively large error of  could be resulted in when the \nsample thickness spans very large. According to the simulation results, the heat \ntransport not only affects  the rate of ultrafast magnetization loss but also the \nmaximum magnetic quenching. So, in experiment we obtain the ultrafast \ndemagnetization time for various samples with almost the sa me maximum quenching \nof 9% to suppress the influence of heat transport7, 21, 51 -54 as well as the non local spin \ncurrent effect17. The temporal evolution of magnetization in sub -picosecond time \nscale was fitted by the analytic solution based on the  phenome nological  three \ntemperature model (3TM)1, 17:  \n \n(1)  \nwhere  presents the convolution product with the Gaussian laser pulse \nprofile, whose full width at half maximum (FWHM) is . A temporal stretching of \nthe laser pulse was introduced by the excited hot ele ctrons55, which is the trigger for \nthe observed ultrafast demagnetization. In the fitting procedure, the demagnetization \nM\n),()()()(\n1)(\n321 1 2\n5.0\n01\nGt\nM EEt\nM EM EtGtAteAAeAA\ntA\nMtMM M \n\n \n\n\n\n\n\n\n\n\n \n),(GtG\nG 10 time  we cared can be influenced by the value of , which is inter -dependence \nwith   within the three temperature model. As is shown in  Table 1  in the \nSupplementary Material34 Part Ⅳ,  was fixed at 330 fs for various samples to \neliminate its relevance with . The time variable in eq. (5) corresponds to  \n, with the free fit parameter characterizing the onset of the \ndemagnetization dynamic s of the actual data trace, which is fixed as 100 fs for various \nsamples.  is a step function,  is the Dirac delta function and  are \nthe fitting constants. The two critical time parameters  are the ultrafast \ndemagnetization time and magnetization recov ery time, respectively. The well fitted \ncurves by 3TM are also shown as the solid lines in Fig. 3(a) from which the ultrafast \ndemagnetization time  and the magnetization recovery time  were evaluated. \nWithin 3TM model, the magnetization recovery process is affected by , \ncharactering the electron -phonon relaxation, and , representing heat transport \ntimescale through the substrates as well as demagnetization time . In the fitting \nprocedure by 3TM model, we assigned a fixed value to  and varies slightly to \nexclude the heat transport effect through thickness. Via changing the single \nparameter ， , we can accurately reproduce the experimental results for various \nsamples. And the heat transport across the thickness domains within 3TM model \ncharacterized by the  parameter of , which is shown in  Table. 1 in  Part Ⅳ of \nSupplementary Material34 as around 2 ps. It is about three times bigger than  \nindicating that we are not mixing the heat transport and the electron -phonon \nrelaxation56. Only in this case, are both th e values of and  genuine. The value \nM\nG\nM\nG\nM\n0 expt tt\n0t\n)(t\n)(t\n3 2 1,,AAA\nE M,\nM\nE\nE\n0\nM\nE\n0\nM\n0\nE\nE\nM 11 of  indicates that the heat was transferred through the substrate in less than 3 ps in \nthis paper, rather than what was observed by F. Busse et al57 where the heat was \ntrapped laterally in the Gaussian profile up to 1  ns. Therefore, the lateral heat \ntransport effect can be ignored, and hencely the modification of precessional \ndynamics here. As illustrated in Fig. 2(b), it can be clearly seen that decreases with \nincreasing dNi.  \n      \nBy replotting Fig. 1(b) and Fig. 2(b), an approximately proportional \nrelationship between and intr was confirmed by our experimental results \n(Fig. 2(c)). This relationship between intr and is consistent well with the \ntheoretical prediction  based on the breathing Fermi -surface model30,31,58 \nfor materials with conductivity -like damping contributions. On the basis of the \nbreathing Fermi -surface model, the Elliott -Yafet spin -mixing parameter 𝑏2 in Co/Ni \nbilayers can be estimated from the theoretical equation30, 31 shown as the red solid l ine \nin Fig. 2(c): \n                                              (2) \nwhere the quantity contains the derivatives of the single -electron energies with respect \nto the orientation e of the magnetization M=Me. p is a material -specific parameter \nwhich should be close to 4. If we use =   from ab initio density \nfunctional electron theory calculation for fcc bulk Ni31, the experimental value of \nElliott -Yafet spin -mixing parameter 𝑏2 = 0.28 can be estimated in Co/Ni bilayers, \nwhich is far larger than that of Co or Ni. The significant enhancement of spin -mixing \n0\nM\nM\nM\nint Mr\n2pbFM\nelM\nelF\nJ231087.1 12 parameters is related to the strong spin -orbital coupling at the Co/Ni interface since b2 \nis proportional to 2 in first -order  perturbation theory, where  is the coefficient of the \nspin-orbit coupling. A detailed ab initio calculation  for Elliott -Yafet spin -mixing \nparameter in Co/Ni bilayers is highly desirable.  For a derivation of eq. (2) it must be \nassumed that the same types of spin-flip scattering processes are relevant for the \nultrafast  demagnetization and for the damping. The assumption does not say anything  \nabout these  detailed  types. It has been shown in Ref. 9 that mere electron -phonon  \nscatterings cannot explain the expe rimentally observed demagnetization  quantitatively. \nIn reality there are also contributions from  electron -electron scatterings11, \nelectron -magnon scatterings12 and from a combination of electron -phonon and \nelectron -magnon scatterings13. Because both for de magnetization and for damping ，\nthe spin  angular momentum  has to be transferred from the electronic spin system to \nthe lattice,  there is no reason why different types of theses spin -flip scatterings  should \nbe relevant for the two situations. Therefore , the Elliott -Yafet  relation, eq. (2) should \nbe applicable for our system. It would not be  valid if non -local spin -diffusion \nprocesses would contribute a lot to  demagnetization. Examples are a superdiffusive \nspin current in the  direction perpendicular to th e film plane, or a lateral diffusion out \nof the spot irradiated by the laser pulse and investigated by the TRMOKE.  However, \nwe definitely found the validity of the Elliott -Yafet relation,  and this shows that \nnonlocal spin -diffusion processes are so small t hat can be neglected in our \nexperiment.  \nDespite this , previous demonstrations17,19-21 show that the ultrafast spin current  13 caused by the transport of spin -majority and spin -minority electrons in the antiparallel \n(AP) state of magnetic multilayers  after the  laser pulse  accelerates the ultrafast \ndemagnetization. Similarly, as is indicated in Fig. 1(b), with the assistance of interface \nbetween FM (Ni) and NM (Pt), the spin current induced by the flow of spin -up and \nspin-down electrons in opposite directions59 may contribute partly to the Gilbert \ndamping in Pt/Co/Ni/Pt mulitilayers.  The femtosecond laser induced spin current lives \nvery shortly which is in sub -picosecond timescale, while the duration of spin current \ntriggered by spin precession is in the timescal e of nanosecond. The difference of the \nduration of the spin current is just related to the timescale of the perturbation of the \nsystem. One has to note that spin currents at the femtosecond time scale gives rise to a \nlowering of the demanetization time17, while spin pumping induced spin current gives \nrise to the enhancement of Gilbert damping and thus a lowering of the relaxation time. \nTherefore, when spin current contributes largely to both ultrafast demagnetization and \nspin precession dynamics, an inverse  relationship between ultrafast demagnetization \ntime and Gilbert damping could be expected. That is, t he more spin current \ntransferred from ferromagnetic layer to normal metal, the faster ultrafast \ndemagnetization should be. Therefore, a t present paper, to  explain the experimental \nresults the local Ellio tt-Yafet scattering theory suffices. And , the non -local spin \ncurrent effect can be ignored, although it contributes partly to the fitted value of \nspin-mixing parameter  𝑏2 . The discussions here inspire us t o continuously  clarify \nthe various relationships between ultrafast demagnetization time and Gilbert dam ping \ncoming from different microscopic mechanisms, which is helpful for understanding  14 the underlying physics of ultrafast spin dynamics as well as the ap plication of ultrafast \nspin current triggered by ultrashort laser60, 61. For instance, recently, the researchers \nare seeking for the potential candidates as the Terahertz waves emitters including the \nmetallic heterostructures. Previous demonstrations show that the magnitude and \nefficiency of Terahertz signals in these multilayers are determined by Gilbert \ndamping60. The investigations of the relationship between Gilbert damping and \nultrafast demagnetization time will open up a new avenue to tailor the Terah ertz \nemission.  \nMeanwhile , the dominant contribution to ultrafast demagnetization in metallic \nheterostructures, either from the localized spin -flip scattering or non -local spin \ntransport, has been a controversial issue for a long time23. Here, a new approa ch, by \nestablishing the relation between the demagnetization time and Gilbert damping, is \nproposed to distinguish the two mechanisms . The proportional relationship indicates \nthe localized spin -flip scattering mechanism domains, otherwise the nonlocal spin \ncurrent domains.          \nIn conclusion, the fast and ultrafast dynamic properties of Ta(3 nm)/Pt(2 \nnm)/Co(0.8 nm)/Ni( dNi nm)/Pt(1 nm)/Ta(3 nm) bilayers with  the electrons  relaxing \nnear the Fermi surface have been investigated by using TRMOKE pump -probe \ntechnique. An genuine  proportional relationship , contrast to previous trivial \nconsequence induced by impurities mechanism,  between ultrafast demagnetization \ntime and Gilbert damping constant is confirmed fr om experimental results. The \nestimated value of spin -mixing parameter on the basis of breathing Fermi -surface  15 model is far larger than that of Co or Ni , which is  originated from the strong \nspin-orbital coupling at the interface. More importantly, distingui shing the dominant \nmechanism underlying ultrafast demagnetization in metallic heterstructures has been a \ntough task for a long time. Here, an effective method by unification of the ultrafast \ndemagnetization time and Gilbert damping is proposed to solve thi s task, namely that, \na proportional relation  between the two parameters indicates the local spin flip \nscattering mechanism domains, otherwise the non local spin current effect domains.   \n   16 Acknowledgments  \nThis work was supported by the National Basic Research Program of China (973 \nprogram, Grant Nos. 2015CB921403 and 2016YFA0300701), the National Natural \nSciences Foundation of China (51427801, 11374350, and 11274361). The authors \nthank Hai -Feng Du, Da -Li Su n and Qing -feng Zhan for critical reading and \nconstructive suggestions for the manuscript.  The authors are indebted to B. Koopmans \nand M. Haag for helpful discussions.  \nAuthor Contributions  \nZ.H.C. supervised project. Z.H.C. and W.Z conceived and designed th e \nexperiments. W.Z. and W.H. performed the polar Kerr loops and TRMOKE \nmeasurement. 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Lett. 101, 072401 (2012).  \n61. T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. \nFreimuth, A. Kronenberg, J. Henrizi, I. Radu, E. Beaurepaire, Y. Mokrousov , P. \nM. Oppeneer , M. Jourdan , G. Jakob , D. Turchinovich , L. M. M. Hayde n , M. \nWolf , M. Mü nzenberg , M. Klä ui , and T. Kampfrath ,  Nat. Photonics. 10, 483 \n(2016).   23  \n \n \nFigure caption:  \nFIG. 1 Spin precession.  (a)TRMOKE signals of Co/Ni bilayers with dNi=0.4-2.0 nm \nin applied field H = 4000 Oe. (b) Intrinsic Gilbert damping constant as a function of \ndNi. \nFIG. 2 Ultrafast demagnetization.  (a) Ultrafast demagnetization curves with various \nNi layer thickness. (b) Ultrafast demagnetization time as a function of Ni layer \nthickness. (c) Ult rafast demagnetization time as a function of Gilbert damping \nconstant. The red full line indicates theoretical fitting .  \n   24  \n \n \n \n \n                  \n \n \n \n \n \n \n \nFig. 1 (Color Online)  Spin precession.  (a)TRMOKE signals of Co/Ni bilayers \nwith dNi=0.4-2.0 nm in applied field H = 4000 Oe. (b) Intrinsic Gilbert damping \nconstant as a function of dNi. \n  \n 25  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.2 (Color Online)  Ultrafast demagnetization.  (a) Ultrafast demagnetization \ncurves with various Ni layer thickness. (b) Ultrafast demagnetization time as a \nfunction of Ni layer thickness. (c) Ultrafast demagnetization time as a function of \nGilbert damping constant. The red full line indicates theoret ical fitting.  \n \n \n 26  \nSupplementary Information  \n \nUnifying ultrafast demagnetization and intrinsic Gilbert damping in Co/Ni \nbilayers with electronic relaxation near the Fermi surface  \n \n \n \nPartⅠ \n \n \nThe measurements of static properties for Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni \n(dNi nm)/Pt (1 nm)/Ta (3 nm) . \n \nFig. 1(a) shows the polar magneto -optical Kerr signal measured at room \ntemperature with maximum applied field of 300 Oe. The static polar Kerr loops of \nCo/Ni bilayers were acquired using a laser diode with a wavel ength of 650 nm.  All \nsamples show very square loops  with a remanence ratio of about 100% , indicating t he \nwell-established perpendicular magnetization anisotropy ( PMA) of the samples. The \nmeasured coercivity Hc decreases with dNi from 103Oe for dNi = 0.4 nm to 37Oe for \ndNi =2.0 nm (Fig. 1(b)). The decrease of coercivity implies that the PMA decreases \nwith the thickness of Ni.  \n \n  27  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.1  Static magnetic properties of of Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni (d Ni \nnm)/Pt (1 nm)/Ta (3 nm) bilayers.  (a) Polar -MOKE loops with various thickness of \nNi layer d Ni. (b) Coercivity Hc and effective anisotropy field as a function of Ni \nlayer thickness d Ni. \n \neff\nKH\n 28  \n \n \n \n \nPartⅡ \n \nThe measurements of spin dynamics for Co/Ni bilayers in ns timescales and the \nanalysis of extrinsic contributions to spin precession  \n \nIn this part, we show the details of spin precession experiment.  For example, Fig. \n2(a) illustrates the scheme for laser -induced magnetization precession. The direction \nof applied field is fixed at .  \nThe typical time -resolved magnetization dynamics with various applied fields for \nTa(3 nm)/Pt(2 nm)/Co(0.8 nm)/Ni(0.8 nm)/Pt(1 nm)/Ta(3 nm) shown in Fig. 2(b) can \nbe best fitted by using the damped harmonic function added to an \nexponential -decaying background1:  \n                (1) \nwhere A and B are the background magnitudes, and  is the background recovery rate. \nC, , f and  are magnetization precession amplitude, relaxation time, frequency and \nphase, respectively. From the f itting curves shown in Fig. 2(b) as the solid lines, the \nvalues of precession frequency f and relaxation time  are extracted. Since the applied \nfields are large enough, we can obtain the Gilbert damping constant using the \nfollowing relationship2  \n                                                (2).  \n80H\n( ) exp( ) exp( )sin(2 )tM t A B t C ft         \n\n1)2( f 29 In the case of films with a relatively low Gilbert damping3-7 as well as thickness \nlarger than the optical penetration depth8, ultrafast laser may generate non -uniform \nspin waves and  affect the relation ship between demagnetization and Gilbert damping \nas extrinsic contributions . In order to check the contribution of non -uniform modes, \nwe performed a fast Fourier transform shown in Fig. 2(c). Only the uniform \nprecession mode was excited at present Co/Ni bi layers with perpendicular magnetic \nanisotropy.     \nBoth  and f as a function of H are plotted in Fig.3. Since the overall damping \nconstant consists of intrinsic damping and extrinsic damping whereby the second one \narises from inhomogeneities in the sample , the Gilbert damping constant decreases \nmonotonously to a constant value as the applied field increases (Fig. 3(a)).  In the low \nexternal fields range, the inhomogeneously distributed anisotropy may lead to higher \n values. Fortunately, the sufficient high  field we used can suppress the extrinsic \ncontributions to the magnetization precession, because for high fields the \nmagnetization dynamics is mainly determined by the external field9. In addition, \nbecause of the interaction between femtosecond laser sourc e and the thin films, the \nlateral heat distribution across the film plane has to be considered as another  \ncandidate contributions to affect the processional dynamics. As is shown by F. Busse \net al6, the heat was trapped as the Gaussian distribution across the film plane of \nCoFeB up to 1 ns due to the use of regenerative amplifier. It can enhance the laser \npower largely while the pump laser spot kept as large as around 90  μm. This \nfacilitates the occurrence of the temperature profile, and consequently the sp in-waves  30 in the range of laser spot size. However, in the absent of regenerative amplifier at \npresent, the laser spot is so small as less than 10 1,10 that one can excite the \nnonequilibrium state of the samples. And the laser fluence used here is around 1\n, which is far weaker than that used in previous report6. Although smaller \nlaser spot seems easier to trigger the nonuniform spin waves, the very low laser power \nwe used here can suppress the influence of lateral heat distribution on the relaxation \ntime o f spin dynamics at present.  M oreover, the absence of non -uniform spin wave \ndemonstrated in Fig. 2(c) in the pump laser spot confirms that the lateral heat \ntransport can be neglected here. In fact, it is found in the main text, within the three \ntemperature  model  (3TM model) describing the ultrafast demagnetization dynamics, \nthat the heat induced by laser pulse mainly transports along the thickness direction to \nsubstrate in less than a few picoseconds. The observation of pronounced \nmagnetization recovery aft er ultrafast demagnetization can exclude the possibility of \nlateral heat trap.     \n   In order to avoid the effect of extrinsic damping constant, the intrinsic \ndamping constants were obtained by fitting the overall damping factor as the function \nof applied  fields with the expression  shown as the red line in Fig. 3(a) : \n                                            (3) \nwhere  and are the intrinsic and extrinsic parts of the damping factor, \nrespectively. The intrinsic part is independent of the external field  or precession \nfrequency, while the extrinsic part is field -dependent.  \nm\n2/cmmJ\n0/\nint 1HH\nrae \nintr\n0/\n1HHe 31 The experimental f-H relation in Fig. 3(b) can be fitted by analytic Kittel formula \nderived from LLG equation2: \n                                                (4) \nwhere , . The \nequilibrium angle of magnetization  was calculated from the relationship\n. The direction of applied field is fixed at . In the \nabove equations, and are the effective perpendicular magnetization \nanisotropy and gyromagnetic ratio, respectively, wher e , . In \nour calculation, the Lande factor  was set to 2.2 as the bulk Co value2. is the \nonly adjustable parameter. The variation of effective field with the thickness of Ni \nlayer was also plotted in Fig. 1(b).  When we plot the intrinsic Gilbert damping \nconstant as a function of effective anisotropy field  in Fig. 4, a proportional relation \nwas confirmed in our Co/Ni bilayer system, which demonstrates that spin -orbit \ncoupling contributes to both Gilbert damping and PMA .   \n \n \n \n \n \n \n \n2 12HH f\n 2\n1 cos ) cos(eff\nK H H H H  \n  2cos ) cos(2eff\nK H H H H  \n\n) sin(22sin    H eff\nKHH\n80H\neff\nKH\n\nseff eff\nKMKH2\n2Bg\nh\ng\neff\nKH 32  \n \n \n \n \n \n \n \n \n \n \nThe numerical simulation for ultrafast demagnetization  \n \n \n \n \n \nFig. 2   (a) Scheme of TRMOKE. (b): TRMOKE signals with various applied \nfield for Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni (0.8 nm)/Pt (1 nm)/Ta (3 nm) \nbilayers. (c): Fast Fourier transform ation s ignals.  \n \n \n \n \n \n \n 33  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3  Gilbert damping and precession frequency. Field dependence of overall \ndamping constant (a) and precession frequency (b) of Co/Ni bilayers with\n \n \nnm dnm dNi Co 8.0 ,8.0  \n 34  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4  Dependence of intrinsic Gilbert damping constant on the effective \nanisotropy field.  \n \n \n \n \n \n \n \n 35  \n \n \n \nPartⅢ \n \nNumerical simulation for the effect of heat transport across the film thickness \non the ultrafast demagnetization time  \n \n \nTo estimate the evolution of heat  transport profile in time, w e carried out a \nnumerical simulation based on M3TM11 model, in which the heat transport12 was \ndominated by electrons and a temperature gradient across the film thickness was \nintroduced.  It is divided in thin slabs in the direct ion normal to the film plane, and the \nslabs is 0.1 nm thick. For each slab, the evolution of the electron and phonon \ntemperatures \neT  and \npT  are determined by a  set of coupled differential equations :13  \n)),)(coth()( 1()()()()),( )(()()),( )(( ))( ()()(\nzTmTzmTzTzRmdtzdmzTzTgdtz dTCzTzTg zTdtzdTzT\nec\ncpp e epp\npe p ep ez ze\ne\n    \n                (5) \n \nWhere \nsMMm ,\n)()(\n0zTzT\npe 4, \n228\nD atB atcBep sf\nEVTkgaR ,with \nat the atomic \nmagnetic moment in units of Bohr magneton \nB , \natV  the atomic volume, and \nDE is \nthe Debye energy.  \neC and \npC  are the heat capacities of the e and p systems \nrespectively. \n)(zTez is the electron temperature gradient normal to the film . \nBk is  36 the Boltzmann constant. \n0k is the material dependent electronic thermal conductivity. \nepg\nis the e -p coupling constant and determines  the decay of the electronic \ntemperature until equilibrium is  reached14. \nsfa represents the spin -flip probability11. \nThe equations of motion for each slab thus describe heating of the electron system by \na Gaussian laser pulse, heat diffusi on by electrons to neighboring slabs, e -p \nequilibration, and finally the evolution of the magnetization due to e -p spin -flip \nscattering. In the simulation, the total magneto -optical signal was obtained by the \ncalculation of \ndzztzm t ) exp(),( )(  .  \nThe electronic system after the action of the laser pulse is in a strongly  \nnon-equilibrium situation. Nevertheless, one can describe the electron  system by use \nof an electron temperature. The reason is that the laser  photons excite electrons, but \nthese excite d electrons thermalize more or  less instantly due to very rapid and \nfrequent electron -electron  scatterings via their Coulomb interactions. This is the \nassumption of the  accepted Elliott -Yafet scenario  which describes the  effect of the \nlaser pulse directly after the action of the laser pulse.   \nFig.4(a) shows the simulated ultrafast demagnetization curves for various film \nthicknesses. We can clearly observe that the evolution of magnetization curves looks \nalmost identical for various film thicknesses , indicating that the effect of heat \ntransport on the demagnetization time can be neglected. Despite this, for the \nremagnetization part, a deviation from the experimental curves occurs. This is mainly \nbecause that the heat diffusion can almost be neglected d uring the ultrafast \ndemagnetization timescale, but starts playing an increasing role from ps timescale  37 onwards. The similar phenomenon was reported previously by B. Koopmans et al. \nFortunately, what we should be focused on here is in the ultrafast demagnet ization \ntimescale, in which the effect of heat transport can be neglected.  In fact, as is shown \nin Fig. 4(b), less than 10 fs variation was induced with the thicknesses ranged from \n1.2 nm to 2.8 nm . The parameters used in the simulation is given in Table.1 .   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  38  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4(a) Dependence of demagnetization as a function of delay time after pulsed laser \nheating at \n0t  (b) Maximum demagnetization and demagnetization time  \nversus the sample thickness.  \n \n \n \n 39  \n \n \n \n \n \nTable 1: Parameters used in the M3TM12,13,15. \n \nParameters  Value  Units  \n\n 5400  \n) /(23KmJ \npC\n \n61033.2  \n) /(3KmJ  \nepg\n \n181005.4  \n) /(3sKmJ  \nDE\n 0.036  \neV  \nat\n 0.62  \ncT\n 630 \nK  \n0\n 90.7 \n) /(smKJ \nsfa\n 0.185  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  40  \n \n \n \n \n \n \nPart  Ⅳ \n \nTable . 1 Values of the main fit parameters of ultrafast demagnetizations curves \nfor various thicknesses of the samples.  \n \n \n \n \n \n \n \n \nReferences :  dNi (nm)       \n0.4 200 860 2.3 330 100 \n0.8 170 860 2.1 330 100 \n1.0 150 860 2.0 330 100 \n1.5 120 860 2.3 330 100 \n2.0 90 860 2.0 330 100 \n)(fsM\n)fsE（\n)(0ps\n)(fsG\n)(0fst 41 1、W. 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P.78  \n \n \n "    },    {        "title": "1712.07534v1.Second_harmonic_magnetic_response_characterizing_magnetite_based_colloid.pdf",        "content": "Second -harmonic magnetic response c haracterizin g magnetite -based  \ncolloid  \nV.A. Ryzhova, I.A. Kiseleva, O.P. Smirnova, Yu.P. Chernenkova, V.V. Deriglazova,*, \nYa.Yu. Marchenkob, L.Y. Yakovlevab, B.P. Nikolaevb, Yu.V. Bogachevc \na Petersburg Nuclear Physics Institute named  by B.P. Konstantinov  of National Research Centre “Kurchatov I n-\nstitute”,  1 Orlova roscha mcr., 188300  Gatchina,  Leningrad Region , Russia  \nb Research Institute of Highly Pure Biopreparations, 7 Pudozhsk aya st, 197110  St-Petersburg , Russia  \nc St-Petersburg State Electrotechnical University LETI, 5 Prof. Popov st, 197376  St-Petersburg, Russia  \nA B S T R A C T  \nNonlinear second -harmonic magnetic response  (M2)  was used to characterize a n aqueous  colloidal solution  of \ndextran -coated  magnetite  (Fe 3O4) nano particles . Data analysis with  the formalism  based on Gilbert -Landau -\nLifshitz equation for stochastic dynamics of superparamagnetic  (SP) particles  ensure d extensive  quantif ying of  \nthe system via  a set of magnetic and magnetodynamic parameters , such as the mean  magnetic moment,  the damp-\ning constant , the longitudinal relaxation time, the magnetic anisotropy  field and  energy , and others . Combined \nwith transmission electron microscopy and dynamic light scattering, M2 technique allow ed obtain ing additional \nparameters, viz., the dextran -coating  thickness  and the interparticle  magnetic  dipolar energy. Aggregated c olloid al \nnanoparticles  were shown to be magn etically correlated  inside the aggregate  due to  magnetic  dipole -dipole (d -d) \ncoupling within  the correlation  radius ~ 50 nm . With  the d-d coupling  account, the volume distribution of  the ag-\ngregates recovered  from M2 measurements  is well consistent with electron microscopy  results . From electron \nmagnetic resonance, abrupt change of SP dynamics with increasing external magnetic field was observed and e x-\nplained. The presented study exemplifies a novel M2 -based  procedure of comprehensive quantitative character i-\nzation  applicable  for a wide variety of  SP systems.  \nKeywords:  Magnetic nano particles , Colloids, Superparamagnetism , Nonlinear magnetic response, Electron ma g-\nnetic resonance  \n1. Introduction  \nMagnetic nanoparticles (MNPs) are widely adopted  in technical, environmental and  biomedical  \nareas  [1-9]. Such MNPs as s uperparamagnetic iron -oxide nanoparticles (SPIONs)  are widely u sed in  \ndisease  diagnostics  as contrast agents in magnetic resonance imaging  and in tumor  treatment  including \nhyperthermia  and drug delivery . Magnetite -based SPIONs are among the most demanded MNPs due to \ntheir biocompatibility and zero coercivity which helps prevent  aggregation in vivo [7, 8]. For many a p-\nplications, SPIONs  are dispersed in  liquid s forming suspensions or colloidal solutions to be inject ed or \nconsumed in some other way . To stabilize the SPIONs in the carrier liquid and to prevent toxicity and \noxidation,  they are coated with specific shells of organic compounds [2, 8], one of which being dextran, \na carbohydrate incorporating the polymers of glucose.  \nHowever, d epending on the solvent, the concentration, the coating fraction  and other factors, the \ndispersed nanoparticles may  aggregat e [7, 10]. This tendency hampers colloidal stability needed in bi o-\nmedicine and modifies relevant magnetic features. Meanwhile, i nformation on  the size distribution of \naggregates  obtained by differ ent techniques  is rather  poor and  contradictory . Thus , electron microscopy  \nneeds dr ying the  suspension  what can modify the size distribution. Another example is d ynamic light \nscattering  (DLS)  which  yield s the hydrodynamic diameter , hard to be related to the  true geometric size. \nMagnetic techniques may somewhat underestimate the aggregate size due to the fact that,  inside an a g-\ngregate, SPIONs are coupled by dipole -dipole interaction resulting in  hardly accountable  finite -radius \n                                                           \n* Corresponding author at: Petersburg Nuclear Physics Institute named  by B.P. Konstantinov of  National Research Centre \n“Kurchatov Institute”, 1 Orlova roscha mcr, 188300 Gatchina, Leningrad Region, Russia.  \nE-mail address:  deriglazov_vv@pnpi.nrcki.ru  (V.V. Deriglazov).  corre lation s of MNP  magnetic moments [ 11]. Also , the study of magnetization dynamics of SPION co n-\ntaining systems face s considerable difficulties in conventional  magnetic measurements. The details of \nself-organization of  the colloidal solution , as well as  its magnetic characteristics , still poorly known , are \ntopics of the present study.  \nTo fabricate  ferro fluid s with definite  attributes , a set of techniques is used for their  characteriz ation . \nThe conventional toolkit includ es steady -field magnetometry  registering also  magnetic hysteresis , ac \nsusceptibility measurements,  X-ray diffraction  (XRD) , and transmission electron microscopy  (TEM) . \nAdditional information  can be obtained  from  magnetorelaxometry  [12], Raman spectroscopy , DLS  also \nknown as photon correlation spectroscopy , magnetoresistivity  measurements  [13] and some others . As \nmagneti c properties  of SPION containing systems , especially, the  magnet ization  dynamic s, are the most \nmeaningful for  certain  applications  (e.g. hyperthermia) , the instruments directly  aiming at  this domain  \nare of  particular significance.  The list of such techniques permanently ex pands indicating topicality of \nthe issue . One of the latest  is the nonlinear  ac Faraday rotation applied to characteriz ation of  magnetite -\nbased  SPION aqueous suspension s [14]. \nHere, we employ a  technique  involving nonlinear magnetic response  on the second harmonic  (M2) \nin the longitudinal geometry of ac - and dc magnetic fields  which  is perfectly well  availa ble to study  SP \nsystems . First, in the megahertz frequency range, particles  with the  relevant  magnetic moment s ~103 – \n106 μB generate an appreciable nonlinear  response with pronounced extrema in small  magnetic fields  of \nthe order  ~10 – 100 Oe. Second, on the same conditions, real and imaginary parts of the response are \ncomparable , hence,  providing , jointly , a great informational content of the measurements. Third , this \ntechnique  is highly  sensitive to  the SP magnet ization dynamic s. \nCapabilities  of the M2 technique w ere repeatedly demonstrated in elaborating a set of condensed \nmatter  [15-17] and biophysical [18] issues . In particular, i ts efficiency  has shown itself in study ing mag-\nneto-electronic phase separation in 3d oxides  [19-21]. As a result, a lot of  novel information  was ob-\ntained on the  emerging system of  ferromagnetic nano clusters  from qualitative  analy sis of  the raw data . \nIn the same way, a biodistribution of magnetite nanoparticles  in tissues  of rodents  in studies of  the brain \ntumor targeting  was evaluated by this technique  [18]. In the present study, this experimental resource  \nwas supplemented with  the recently elaborated rigorous formalism based on Gilbert -Landau -Lifshitz \n(GLL) equation for stochastic dyn amics of SP particles [ 22-24]. Treating M2 experimental data with \nthis formalism enables  to extract a full scope of magnetic and magnetodynamic  parameters characteri z-\ning such systems.  Here, t he GLL data-treatment formalism  was employed to  describe  quantitatively  an \naqueous colloidal solution of magnetite -dextran SPIONs. Such an object may be considered as a trial \nprototype of more complicated magnetic sols relevant for perspective nanomedical applications. \nHowever, the M2 technique alone turned out to be insufficient  for total unambiguous characteriz a-\ntion of  this rather complicated  system. Therefore , conventional nonmagnetic techniques  such as  XRD, \nTEM and DLS have been additionally enabled  to verify and add to M2 data . It concerns , mainly,  geom e-\ntrical parameters , such as  the mean particle volume and the volume distribution width,  which  depend on \nthe nonmagnetic component of the system invisible  by magnetic  probes . Thus, particular  attention was \npaid to mutual consistency of the data obtained by all these techniques.  Jointly, t hey facilitate correct \ninterpretation of M2 data and yield additional information unavailable from the  M2 technique  alone . \nThis approach is intended to be fur ther applied to monitor  functionalized  SPIONs accumulated in \nvarious organs and tissues of experimental animals. It will provide information on the coupling of the \nSPIONs with specific sites of different tissue cells, including tumors, and on aggregation of the SPIONs. \nParticular data obtained  in the present study will be useful  as a reference point . \nEMR measurements were also performed. The resonance spectrum was fitted with the same GLL \nformalism and explained consistently with M2 and TEM observations.  \nIn Section 2, preparation of the colloid al solution  is briefly described and its  attest ation  by XRD is presented. In Section 3, TEM and DLS data are analyz ed and compared.  In the main Section 4, a con-\ncise description of  the M2 technique and the measure ment conditions are offered. The data treatment  \nformalism  is also traced , with the reference on more detailed description. The parameters  characterizing \nthe nano particle  system are analyzed and the structural, magnetic and magneto dynamic properties are \ndiscussed. In Section 5, EMR data are presented  and analyzed involving information obtained from M2  \nand TEM  measurements.  The conclusion is presented  in Section 6. \n2. Sample preparation  and attestation  \nSPIONs were prepared from  solutions of the  iron salts FeSO 4 and FeCl 3 at the ratio of ion conce n-\ntrations Fe+2/Fe+3 = 1/2 by co -precipitation in alkaline media at  the temperature  80°C under  the inert gas \nN2 [25]. To ensure disag gregation, low molecular weight dextran (MW 10 kDa, Sigma) supplemented \nby CsCl was added to the dispersion of magnetic nanoparticles in  the process of sonication during 15 \nmin at the frequency  22.3 kHz. Precipitation was initiated by continual titration with NH 4OH solution \nunder stirring in  the 100 m L reactor. The stock dispersion was collected by  the permanent Nd magnet  \nand then  washed and centrifuged into fractions. A fine fraction of  the SPIONs was treated by dialysis \nand stored at 4 °C before  the experiments. The Fe content was assayed with the  thiocyanate probe by \nmeasuring  the light absorption at  the wavelength  480 nm.  \nThe SPIONs structure and composition were examined by XRD at DRON -3M diffractometer. Fi g. \n1 presents the XRD intensity as a function of the diffraction angle measured at the temperature 290  K. \n30 40 50 60 70050010001500\nFe3O4 \nFe2O3 440\n333\n511\n422\n331400\n222311\n Intensity,  arb.un.\nDiffraction angle,  deg.220\n \nFig. 1. X-ray diffraction intensity vs. diffraction angle for Fe -oxide nanocrystals. Marks at the bottom indicate \nnominal reflections: the upper and the lower sets are for magnetite and hematite, respectively.   \nTo evaluate a size of the nanoparticle crystallinity  region, precise treatment of the XRD pattern was \nperformed with account of the instrumental resolution and a doublet structure of Cu K α line. The diffra c-\ntion peaks broaden, mainly, due to a finite size of the coherent scattering region and internal stress  in the \nsample. Williamson –Hall approach [2 6] clearly differentiates between the size -induced and strain -\ninduced peak broadening by considering  the peak width as a function of angle:  \n𝛽𝑘𝑙cos𝜃=𝑘𝜆\n𝑑+4𝜀sin𝜃 where βhkl is the instrument -corrected breadth (full width at half maximum)  of hkl-reflection located at \nthe angle 2 θ, d is the crystallite size, k ≈ 0.9, λ = 1.54 Ǻ is the wavelength of Cu K α1 radiation and ε is \nthe strain -induced broadening arising from crystal im perfections and distortions. From the meaningful \npeaks of Fig. 1, the mean size of the crystallinity region in Fe 3O4 nanoparticles was found to be 𝑑 =\n8.7(1.3) nm, the strain -induced broadening being small.  \nFor further study, the SPIONs  were coated with dextran and  dispersed in water  to form a colloidal \nsolution  [25].  \n3. Probing  by transmission  electron  microscopy  and dynamic  light  scattering  \n3.1. T ransmission electron microscopy  \nFig. 2 (upper panel)  presents a fragment of the TEM micrograph obtained at the microscope JEM -\n100C (Jeol, Japan) for the freeze -dried colloid on the glass substrate.  \n \n0 50 100 1500102030\n Number of aggregates\nDiameter, nm0 5 10 15 20 25 300200400600\n  Number of particles\nCore diameter,  nm\n \nFig. 2. Upper panel: TEM image of freeze -dried colloid ; lower panel: measured diameter distribution of aggr e-\ngates (histogram)  and its best fit with lognormal distribution ( solid curve) ; inset: the same for distribution of \nmagnetite cores of constituting nano particles. Dashed curve is  lognormal diameter distribution recovered from \nDLS histogram of Fig. 3 (see Subs ection 3 .2). \nNanop articles in the form of granules ~10 nm in diameter make up aggregates of different size and \nirregular form. The micrograph contrast is caused totally by magnetite cores while dextran shells around \nthe cores  are only slightly discernible as thin gaps between dark spots. On the lower panel, a size distr i-\nbution of the aggregates is shown as a function of a diameter of the effective sphere approximating an \naggregate. In the inset, the same distribution is presen ted for magnetite cores of nanoparticles, building \nunits of the aggregates. Both the systems are expectedly well fitted by the lognormal distribution (solid \nlines):  \n𝑓 𝑥 =1\n 2𝜋𝜎𝑥exp −1\n2𝜎2ln2𝑥\n𝑥0                                                                                                                       (1) \nwith the mean 𝑥-value 𝑥 =𝑥0exp(𝜎2/2) and the variance 𝑥02exp 𝜎2(exp 𝜎2−1). The diameter distr i-\nbution parameters are given in Table 1.   \nTable 1  \nParameters of diameter lognormal distribution for nanoparticle magnetite cores (left column) and nanoparticle \naggregates (right column) as obtained from TEM.  \n Particle cores  Aggregates  \nMedian, nm  8.79(5)  35(2)  \nMean diameter, nm  9.46(5)  43(2)  \nStandard deviation  0.383(4)  0.63(4)  \nVariance, nm2 14.1(3)  880(150)  \nWithin the statistical errors, the mean diameters and the variances coincide with these obtained directly \nfrom the histograms.  \nNote , the crystallinity size extracted from XRD is equal, within the measurement accuracy, to the \nmean diameter of the magnetite cores evaluated with TEM. Thus, no non -crystal phase was detected  in \nthe magnetite fraction.  \nThe lognormal distribution over diamete rs with the median 𝐷0 and the standard deviation 𝜎𝐷 means \nalso the lognormal distribution over particle volumes with the median 𝑉0=𝜋𝐷03/6 and the standard \ndeviation 𝜎𝑉=3𝜎𝐷. The mean diameter 𝐷 =𝐷0exp(𝜎𝐷2/2) and the diameter 𝐷  corresponding to  the \nmean volume 𝑉 =𝑉0exp(𝜎𝑉2/2) via 𝑉 =𝜋𝐷 3/6 interrelate as 𝐷 =𝐷 exp 𝜎𝐷2. \nThe particle cores are expected to be in the single -domain state which extends up to the size 128 nm \n[27]. The latter  will be confirmed  by M2 data presented  below.  \n3.2. Dynamic light scattering  \nThe DLS measurements were performed with Zetasizer Nano ZSP (Malvern) equipment for the \nsamples in glass cylindrical cuvettes with the internal diameter 10 mm and the volume 2 mL at the te m-\nperature 298 K. The most part of the collo id, 98.5%, was observed in the aggregated state.  \nThree independent DLS measurements have been performed for this sample. In Fig. 3, a typical hi s-\ntogram of the hydrodynamic diameter distribution is presented. Unlike the microscopy data ( Fig. 2), the \nlognormal distribution (dotted line) insufficiently describes the DLS histogram, with noticeable devi a-\ntions at the histogram edges. The hydrophilic dextran shell of the particles couples with water by hydr o-\ngen bonds creating a solvent layer bound to the aggregate surfaces. A part of water can  also penetrate \ninside an aggregate. Besides, d ue to a strongly irregular form, the aggregates, when moving , capture  \nsome amount of water . So, an effective  thickness of the water layer is not proportional to th e aggregate \nsize resulting in deviation of the measured hydrodynamic diameter from the lognormal  distribution. The \neffective aggregate diameter can be presented as 𝐷=𝐷−2𝛥, where 𝐷 is the hydro dynamic diameter  as seen from DLS and Δ is the effective thickness of the water layer. With the assumption that D con-\nforms  to the lognormal distribution and Δ is independent of D, the fit ting was performed for each of the \nthree measurements. For all of them , the fit curve s perfectly  well lay on the respective  histogram s as \nshown for one of them in Fig. 3 (solid line) . \n0 50 100 15001020\n Number of aggregates,  arb.un.\nHydrodynamic diameter Dh,  nm\n \nFig. 3. Distribution of  aggregates  hydrodynamic diameters  (histogram) and its best fits : with lognormal distrib u-\ntion without water layer account (dotted line) and with water layer account (solid line).  \nThe parameters  averaged over the three measurements with the errors within the reproducibility (the \nconfidence level 0.95)  are presented in Table 2. \nTable 2  \nParameters of diameter distribution for nanoparticle aggregates obtained from DLS: without  account of  water \nlayer (left column) and with  the water layer account (right column).  \n Water layer  \n disregarded  Water layer  \naccounted  \nMedian, nm  50(7) 28(5)  \nMean  hydrodynamic  diameter , nm 52(7)  44(7) \nMean geometrical diameter , nm - 32(5)  \nStandard deviation  0.25(1)  0.51(5)  \nWater layer thickness, nm 0 5.8(8) \nAs seen from Table 2, account of the water layer appreciably reduce s the measured  mean  diameter  \nand increase s the distribution width , approaching them  to the TEM  aggregate  parameters ( Table 1). \nHowever, the DLS  mean  diameter  still somewhat  exceeds the microscopy one. To co mpare  the DLS and \nTEM results in more detail, the lognormal distribution for the effective diameter D was recovered with \nthe fit parameters of the  DLS histogram  and presented  in Fig. 2 by the dashed curve. The curve is seen \nto fairly well match  the TEM histogram peak , thus, demonstrating  a good agreement  of both the me a-surements . Jointly, they may be accepted as a base to verify  the information  obtained  from  the nonlinear \nmagnetic response.  \n4. Nonlinear  magnetic  response  \n4.1. Measurement s \nColloids  with two different concentrations, 2 and 0.02 mM(Fe)/L,  were examin ed with the well d e-\nveloped unconventional technique  [18, 28, 29] exploiting the second harmonic M2 of nonlinear magnetic \nresponse  in parallel ac - and dc magnetic fields , 𝐻 𝑡 =𝐻+0sin𝜔𝑡. The dc field  H was scanned  \nback -and-forth  symmetrically within ±300 Oe with the round -up cycle s 0.125 – 4 s and with high repr e-\nsentativity of 2048 H-points in each scan . The amplitude h0 = 13.8 Oe of the ac field  with the frequency \n𝑓=𝜔/2𝜋= 15.7 MHz ensur ed the condition 𝑀2∝02. This nonrigid  requirement  enable d to directly \nvisualize un distorted H-dependence of the  second -order susceptibility and  somewhat  favore d reliability \nof the data treatment. Both, real and imaginary, components of the signal were simultaneously recorded \nas functions of the dc field at the temperature  region  273 – 297K  with the temperature stabilization ±0.1 \nK. \nIn Fig. 4, the H-field direct and reverse scans for real and imagin ary parts of  the M2 signal from the \nsample with the concentration 0.02 mM(Fe)/L at the temperature 297 K and the round -up cycle 0.125 s \nare presented as a typical example.  \n-300 -200 -100 0 100 200 300-0,20,00,2\n M2,  emu/mole(Fe)\nMagnetic field,  OeIm\nRe\n \nFig. 4.  Real and imaginary parts of nonlinear magnetic response as a function of steady magnetic field: filled ci r-\ncles ( red and magenta ) are direct scan, open circles ( green and cyan ) are reverse scan and solid curves are simu l-\ntaneous best fit.  \nDue to common sym metry requirement, the field dependence of the signal is antisymmetric with respect \nto zero. The two signal components exhibit opposite signs, pronounced extrema, and only slightly di s-cernible hysteresis, all characteristic for SP behavior. Increas e of the round -up cycle up to 4 s leads to \ncomplete disappearance of the hysteresis indicating its dynamical character. The latter is a characteristic \nfeature of single -domain MNPs [ 30]. The spectra for all the temperatures appeared to be quite similar \nmanifes ting the absence of any temperature evolution of the colloidal system in this region. Similarly , \nno distinction was found between the spectra for the two concentrations evidencing the absence of not i-\nceable coupling between components of the SP system in th is concentration range. Increasing the scan \ncycle duration up to 4 s expectedly results only in complete elimination of the hysteresis with no effect \non the other features, thus , evidenc ing a dynamical character of the hysteresis inherent to  single domain \nMNPs  [30]. Anticipating the results, the similarity of all spectra gives rise to one and the same set of \nparameters, within the errors, characterizing the system under study.  \n4.2. Data treatment  \nTreatment of the  obtained  M2 experimental data w as carried out following the  formalism elaborated  \nrecently  by Coffey and colleagues [22 -24]. Real and imaginary components of the measured M2 re-\nsponse were simultaneously fitted with the model function containing  the stationary solution of the \nFokker -Planck equation  for the SP magnetic moment. I n spherical coordinates , it reads  [31]: \n2𝜏𝑁𝜕𝑊\n𝜕𝑡=−1\nsin𝜗 𝜕\n𝜕𝜗(sin𝜗𝐽 𝜗)+𝜕\n𝜕𝜑(𝐽 𝜑)                                                                                                    (2𝑎) \nwith \n𝐽 𝜗=− 𝛽 𝜕ℋ\n𝜕𝜗−1\n𝛼 1\nsin𝜗 𝜕ℋ\n𝜕𝜑 𝑊+𝜕𝑊\n𝜕𝜗  ,                                                                                                    (2𝑏) \n𝐽 𝜑=− 𝛽 1\n𝛼 𝜕ℋ\n𝜕𝜗+1\nsin𝜗 𝜕ℋ\n𝜕𝜑 𝑊+1\nsin𝜗 𝜕𝑊\n𝜕𝜑 .                                                                                          (2𝑐) \nHere, W is the nonequilibrium probability -density function for directions of the particle magnetic m o-\nment,  the dimensionless constant  α is proportional to the damping factor  in the dissipation term of the \nGLL stochastic equation  [31], the time scale  𝜏𝑁∝𝛼+𝛼−1 is the Neél  time in the Gilbert form , and \n𝛽=1/𝑘𝐵𝑇 (𝑘𝐵 is the Boltzmann constant). The magnetic potential ℋ is a sum of the uniaxial  anisotr o-\npy energy and the energy of the magnetic moment in the  external  magnetic field 𝐻(𝑡): \nℋ=−𝐾𝑎𝑉\n𝑚2(𝐦𝐧)2−𝐦𝐇 \nwhere 𝐾𝑎 is the anisotropy constant, 𝑉 and m are the particle volume  and magnetic moment, respectiv e-\nly, and  the unit vector  n is the anisotropy axis direction.  The terms with 1/ α and the rest terms in Eqs. \n(2b) and  (2c) describe precession and thermal diffusion, respectively.  \nAn analytical solution of  the Eq. (2a) for the present case of arbitrary magnetic field direction is a b-\nsent enforcing to solve the problem numerically. By expanding 𝑊(𝑡) in the series on spherical harmo n-\nics \n𝑊 𝑡,𝜗,𝜑 = 𝑐𝑙𝑚 𝑡 𝑌𝑙𝑚 𝜗,𝜑                                                                                                                           (3)\n𝑙𝑚 \nand in  the Fourier series,  the Eq. (2a) is reduced to a linear s et of equations  which, in turn, can be e x-\npressed as a continuous -fraction matrix relation:  𝐒𝑛=−[𝐐𝑛+𝐐+𝐒𝑛+1𝐐𝑛+1]−1                                                                                                                               (4) \nwhere the matrices 𝐐𝑛, 𝐐+, and 𝐐𝑛+1 are composed of the spherical harmonics indices , the direction \ncosines of the ac - and dc magnetic field s, as well as  the parameters entering  Eqs. (2a), (2b) and  (2c) and \nthe magnetic  potential ℋ. As a result, the  (normalized)  k-harmonic of the magnetic moment  in the dire c-\ntion of  the ac field  is expressed as:  \n𝑚𝑘 𝜔 = 4𝜋\n3 𝛾3′𝑐10𝑘 𝜔 + 𝛾1′+𝑖𝛾2′ 𝑐1−1𝑘 𝜔 − 𝛾1′−𝑖𝛾2′ 𝑐11𝑘 𝜔 \n 2                                                        (5) \nwhere 𝛾1,2,3′ are the direction cosines of the ac magnetic field while 𝑐𝑖𝑗𝑘(𝜔) are Fourier transforms of the \nexpansion coefficients 𝑐𝑙𝑚 of Eq. (3) and compose  the column  vector  C proportional to the solution S1 \nof the continuous -fraction Eq. (4). \nThe fit function  is a convolution of 𝑚𝑘(𝜔) for 𝑘=2 with the lognormal  magnetic  moment  distri-\nbution: \n𝑀2(𝐻)=𝑀𝑠 𝑓𝑀𝑚2 𝜔,0,𝐻 𝑑𝑀                                                                                                                      (6) \nwhere 𝑀𝑠 is the saturation magnetization of the SP system.  The distribution  𝑓𝑀 corresponds to the di s-\ntribution over volumes  if all particles are magnetically homogeneous with the same, size independent,  \nmagnetization.  \nThe solution accuracy  is determined by the number of equations in the system, i.e. retained terms in \nthe Fourier and spherical harmonics expansions  which, in turn, specify the matrices sizes . The former \nnumber ±3 and the latter one 7  ensure  a sufficient accuracy.  Computation of t he fit function  (6) at each \nexperimental point impli es multiple implementation of the  procedure for solution of Eq. ( 4) (~ 102 per \nH-field point  per iterate ) while t he CPU time rapidly increases with the matrix size.  Thus, t he total data \ntreatment is essentially time consuming and  feasible only at powerful comput er clusters . The results of \nthis work were obtained using computational resources of Peter the Great Saint -Petersburg Polytechnic \nUniversity  Supercomputing Center ( http://www.spbstu.ru ). \n \n4.3. Result s \n4.3.1.  Effect of magnetic correlations inside aggregates  \nThe fit variable parameters relate to the magneti te fraction of the co lloid and include: (1) the satur a-\ntion magnetization of the SP system  per mole of Fe 𝑀𝑠, (2) the 𝑓𝑀 distribution parameters,  viz., the m e-\ndian magnetic moment  𝑀0 and (3) the distribution width  𝜎𝑀, (4) the mean anisotropy energy  𝐸𝑎, (5) the \nangle  𝛹 between the anisotropy ax is and magnetic field,  and (6) the damping constant  𝛼. Also, the poss-\nible backgrounds linearly depending on the steady field 𝐻 were fitted for real and imaginary parts of the \nsignal.  Some additional quantities can be derived from the fit parameters, viz. (i) the anisotropy field \n𝐻𝑎=𝐸𝑎/𝑀  with the mean magnetic moment 𝑀 =𝑀0exp(𝜎𝑀2/2), (ii) the saturation magnetization  per \nFe ion  𝜇 =𝑀𝑠/𝑁𝐴 where 𝑁𝐴 is the Avogadro number, (iii) the number of Fe ions  corresponding to 𝑀 ,  \n𝑁 =𝑀 / 𝜇 , (iv) the mean  volume 𝑉 =𝑣0𝑁  where 𝑣0≅0.02467  nm3 is the  volume per Fe ion  in ma g-\nnetite  obtained from XRD , (v) the mean diameter 𝐷  corresponding to the mean volume 𝑉 , and the mean \nvalue 𝐷 =𝐷  exp(−𝜎𝑉2/9) of the diameter  lognormal distribution , (vi) the Neél time 𝜏 𝑁=𝑀 𝛽(𝛼+\n𝛼−1)/2𝛾 where 𝛾 is the g yromagnetic ratio and (vii) the zero-field longitudinal relaxation time 𝜏∥. The latter quantity  is proportional to the Neél time , 𝜏∥=𝑄𝜏𝑁. The proportionality  coefficient  𝑄 is a mono-\ntonously increasing  function of 𝛽𝐸𝑎 inferred and tabulated in the studies [32-34]. \nTreatments of all the measured spectra yielded nearly  the similar set s of parameters. Fig. 4 presents \nthe best fit (solid curves)  for the temperature 297 K as  a typical case. The parameter values are pr e-\nsented in Table 3 (left column).   \nThe magnetic moment ~105 μB and the relaxation time ~10-9 s are typical for an SP particle. The \nmean diameter 𝐷  = 36 nm being compared to the microscopy diameters from Table 1 indicates that the \nM2 signal arises from the aggregates rather than from ind ependent  SPIONs. This value  is smaller than \nthe microscopy one (43 nm) as the former comprises  only the magnetically active component of aggr e-\ngates. Thus, 𝐷  is the effective  mean diameter corresponding only to the magnetite fraction.  \nQuestionable, however, is the distribution width 𝜎𝑀. Its value was expected to fit the triple value of \nthe TEM standard deviation for the aggregates diameter distribution presented in Table 1 (right column),  \ni.e. 𝜎𝑉=3𝜎𝐷= 1.9(1). Instead, 𝜎𝑀=0.34 appeared to be not only much less than 𝜎𝑉 but even lower  \nthan that for nanoparticle cores, 𝜎𝑣=3𝜎𝑑=1.15, with 𝜎𝑑 from Table 1 (left column). This discrepancy \nis suggested to arise from the finite distance at which the nanoparticle magnetic moments correlate  in-\nside an aggregate. From Monte -Carlo simulations for growth kinetics of magnetic nanoclusters  with d -d \nTable 3  \nParameters obtained from M2 measurements and identified with aggregates: without account of magnetic correl a-\ntions (left column) and with account of finite -radius magnetic correlations (right column).   \n Infinite correlation \nradius  Finite correlation \nradius  \nSaturation magnetization 𝑀𝑠 , emu/mole(Fe)  257.5(1.6)  702(1)  \nSaturation magnetization per Fe -ion 𝜇  , μB 0.0454(3)  0.127  \nMedian magnetic moment 𝑀0 , μB 45240(160)   \nMean magnetic moment 𝑀  , μB 47950(170)  61200(100)  \nDistribution width 𝜎𝑉 (𝜎𝑀) 0.341(2)  2.11 \nMean anisotropy energy 𝐸𝑎 , K 171(3)  254(22)  \nAnisotropy field 𝐻𝑎 , Oe 53.2(11)  61.8(7)  \nAnisotropy axis direction 𝛹 , deg.  10.71(17)  12.7(1)  \nDamping constant𝛼 0.2246(8)  0.2283(7)  \nNéel  time 𝜏 𝑁 , s 1.442(7) ·10-9 1.81(1) ·10-9 \nLongitudinal relaxation time 𝜏∥, s 1.832(9) ·10-9 2.60(1)·10-9 \nMean volume 𝑉 , nm3 2.565 ·104 3.80 ·104 \nMean diameter 𝐷  , nm 36.6(2)  41.7 \nMean diameter 𝐷  , nm 36.1(2) 25.4 \nMean number of Fe ions 𝑁  1.04 ·106 1.54 ·106 \ncoupling [ 11], magnetic interparticle correlations decay  with increasing the distance and vanish within a \nfinite correlation radius. A rigorous account of magnetic correlations in the nonlinear response is a co m-\nplicated problem. Instead, a cutoff function was introduced for the aggregate magnetic moment imita t-\ning the real correlations:  \n𝑔(𝑟)= 1,          𝑟≤𝑟0= 𝑟𝑐exp(𝜆−1)                                                                                                  \n1−𝜆ln𝑟\n𝑟0,   𝑟0<𝑟<𝑟𝑐                                                                                                                          (7)\n0,          𝑟>𝑟𝑐                                                                                                                                \nwhere 𝑟 is the distance from the center of the aggregate.  Its linear dependence on −ln𝑟 fairly well a p-\nproximates the simulated correlation function for the distances not too close to the correlation radius 𝑟𝑐 [11]. The cutoff function modulates the density of magnetic moment inside the aggregate. Averaging the \nmagnetic moment over the aggr egate volume leads to correction of the former by the factor  \n𝐾(𝑉)=1−𝜆\n3 ln𝑉\n𝑉0+𝑉0\n𝑉−1  \nfor the aggregate volume 𝑉 in the region  𝑉0<𝑉<𝑉𝑐 (2𝑟0<𝐷<2𝑟𝑐), where 𝑉0=4𝜋𝑟03/3 and \n𝑉𝑐=4𝜋𝑟𝑐3/3. For 𝑉<𝑉0 (𝐷<2𝑟0) 𝐾=1, and 𝐾=0 if 𝑉>𝑉𝑐=4𝜋𝑟𝑐3/3 (𝐷>2𝑟𝑐) assuming no \nmagnetic correlations in very large aggregates. With the cutoff function introduced, the magnetic m o-\nment distribution is no more lognormal, whereas the volume distribution is still assumed lognormal.  \nThe results of the data treatm ent with account of the cutoff and with 𝜆 and 𝑟0 also as variable par a-\nmeters are presented in Table 3 (right column). Parameters of the cutoff function are 𝜆=0.283 and \n𝑟0=1.00 nm. The latter value and the correlation radius 𝑟𝑐=34.2 nm should be furth er corrected by a \ncertain factor 𝜉>1 (see Subsection 4.3.3 ). \n The magnetic correlations account is seen to noticeably modify most of the parameters. First of all, \nthe volume distribution width, 𝜎𝑉=2.1, appreciably increased and became quite close to this for aggre-\ngates observed in  microscopy  scans . The mean  magnetic moment 𝑀  raised by 28%. The intensive qua n-\ntity 𝑀𝑠 after correction became almost three times larger, 702 vs. 258 emu/mole(Fe), as the f ormer value \nbelongs only to the magnetically correlated region while the latter one is the mean over -aggregate ma g-\nnetization.  \nThus, m agnetic moments of SPIONs  in aggregates  are, to a great extent, disordered. The  magnetic -\nmoment and  size (or volume) distribution s width of the aggregates evaluated by various techniques \nbased on magnetic measurements c an be appreciably underestimated if finite -radius magnetic correl a-\ntions are disregarded. Besides, considerable magnetic disorder due to anisotropic character of the inte r-\nparticle dipolar forces appreciably diminishes the total aggregate magnetic moment. Being interpreted at \nsimple manner, this could lead to the spurious conclusion on, e.g., a dimer structure of the aggregates.  \n4.3.2.  Quantifying magnetization dynamics  and magnetic anisotropy of aggregates  \nThe damping constant defines  the magnetic moment dissipation rate. It is the most robust fit par a-\nmeter insignificantly correlating with the others and well reproduced even with no correlations account. \nHigh susceptiveness to magneti zation  dynamics is the main virtue of M2 technique. The present small \nvalue 𝛼 ~ 0.2 highlights the precession terms in Eqs. (2b) and  (2c) pointing out an appreciable role of \nprecession in relaxation of the magnetic moment. In the opposite, overdamped, case 𝛼≥1, the relax a-\ntion would have been of purely thermal diffusion type.  \nThe longitudinal relaxation times in the left an d right columns  of Table 3  were  calculated with the \ninterpolated values 𝑄≅ 1.270 and 1.434 corresponding to 𝛽𝐸𝑎≅ 0.5777 and 0.857 [ 34], respectively.  \nThe aggregates were found to be magnetoanisotropic. However, the obtained “easy axis” anisotropy \nenergy 𝐸𝑎=254  K implies a n unexpectedly  small conventionally defined blocking temperature  with \nthe value  𝑇𝐵=𝐸𝑎/25≅10K more inherent to noninteractin g magnetic nanoparticles [ 11]. Notice , the \nsaturation magnetization per Fe ion  𝜇  ~ 10−1μB is one order smaller than the magnetic moment of the \nmagnetite Fe ion 𝜇 ~ 1μB. This finding is indicative of strong disorder of the SPION  magnetic moments \ninside an aggregate. Orientation of SPIONs  anisotropy ax es during formation and growth of the aggre-\ngate is governed  mainly  by d-d coupling between the magnetic moments, the strength and the sign d e-\npending on the mutual position of the int eracting nanoparticles. As a result, anisotropy axes and magne t-\nic moments of nanoparticles inside a large aggregate are oriented almost randomly, as evidenced  also by \nMonte -Carlo simulations [ 11]. In the different approach, randomly positioned  dipoles tend to align due to interaction with the \nmean dipolar field  [35]. However, the position randomness leads to fluctuations inhibiting the ordering. \nAt low densities of dipoles, fluctuations dominate preventing the ordering, whereas at high densitie s, the \nmean field dominates and the ordering is possible. Thus, the partial magnetic ordering is a compromise \nbetween the two opposite tendencies.  \nThe observed relatively small mean magnetic moment of the aggregate s, only twice as large as the \nnanoparticle  core magnetic moment (estimated  in Subsection  5.2), is a result of incomplete mutual co m-\npensation of the moments correlated by dipolar forces sufficiently strong at the measurement temper a-\nture ( also in  Subs ection 5 .2). \nThe angle 𝛹 indicates  the predominant orientation of the aggregates anisotropy axes in the colloid  \nrelative to the applied magnetic field. In the isotropic case, when the aggregates anisotropy axes are \ncompletely disordered, this angle roughly corresponds to the cosine square aver aged over 4 𝜋, in the way \nanisotropy enters the magnetic potential ℋ, viz., 𝛹𝑒𝑓𝑓=cos−1[(cos2𝛹        )1/2]≅55°. The obtained value \n12.7° is too small to agree with this assumption. The small 𝛹 value is suggested to result from the \norienting effect of the external magnetic field which tends to align the aggregates. The mean energy of \nthe aggregate magnetic moment in the steady field, 𝐸 𝐻=𝑀 𝐻, exceeds the thermal energy in the great \npart of the  measured 𝐻-region . Thus, in the field 𝐻=100  Oe, 𝐸 𝐻=410  𝐾. The anisotropy energy \ncompared to the measurement temperature is also large enough to ensure the sufficient coupling b e-\ntween magnetic and rotational degrees of freedom. The moderate Boltzmann factor exp −𝛽𝐸𝑎 ≅0.42 \ncharacterizes the effect of thermal disordering on magnetic moment with respect to the anisotropy axis.  \nTo verify this suggestion, first, the same M2 data were once more treated assuming random distr i-\nbution of  the anisotropy axes, i.e. the function in Eq. (6) was additionally averaged over the axes dire c-\ntions using the Gauss quadrature [3 6]. The resultant fit quality turned out to be noticeably worse, with \n𝜒2 three times greater. Second, an additional M2 measureme nt was performed for the frozen colloid al \nsolution , at the temperature 260 K, after zero -field cooling to retain the orientation disorder. The o b-\ntained value 𝛹≅46° turned out to be fairly  comparable with 𝛹𝑒𝑓𝑓, despite rather an  approximate ch a-\nracter of the 𝛹𝑒𝑓𝑓 estimation. Third, in aqueous colloids , an aggregate experiences Brownian diffusion \nwith the relaxation time 𝜏𝐵=3𝛽𝜂𝑉, where 𝜂=0.87 mPa∙s is the viscosity of water at the measur e-\nment temperature 297  K, and 𝑉=𝜋𝐷3/6≅7.4∙104 nm3 is the hydrodynamic volume estimated with \n𝐷=52 nm from Table 2 (left column). The Brownian relaxation time 𝜏𝐵≅4.8∙10−5 s appeared to be \nmuch less than the round -up cycle of the scan field, 0.125 s, thus, meeting the adiabatic condition. On \nthe contrary, 𝜏𝐵≫1/𝑓≅6.4∙10−8 s and the rotational degrees of freedom are unaffected by the high -\nfrequency  ac magnetic field . Besides, as 𝜏𝐵≫𝜏∥~ 10−9 s, its contribution to the effective relaxation \ntime 𝜏𝑒𝑓𝑓 defined via the relation 𝜏𝑒𝑓𝑓−1=𝜏∥−1+𝜏𝐵−1 is negligibly small.  \nAll estimations  ensure the suggestion on orienting the aggregates by magnetic field in the liquid \ncolloidal solution . The orientation al effect was observed in the similar object  by nonlinear Faraday rot a-\ntion even in smaller magnetic fields, 𝐻≤40 Oe [14]. Orientational mobility of magnetic nanoparticles  \nin suspensions  under magnetic field has been  studied  also by EMR techniques [3 7-39]. \nAs the recovered volume distribution obtained for the aqueous  colloidal solution  satisfactorily fits \nthe distribution observed by TEM on the freeze -dried samples , no noticeable  additional aggregation o c-\ncurs, most probably, when drying the solution  to obtain the TEM specimen.  \n4.3.3.  Distinguishing  between  magnetic and nonmagnetic  components  of the colloid  \nThe recovered volume distributions, all normalized by unity in maxima, are presented in  Fig. 5. The \nnarrow dashed -line peak centered at 104 nm3 corresponds to the treatment without account of correl a-\ntions while the thick -line broad peak centered at 2∙102nm3 is a recovery with the correlations account. The TEM histograms for nanoparticle cores and SPION aggregates ( Fig. 2) recalculated to the volu me \ndistributions are also presented by open and filled circles, respectively, together with their best fits (r e-\nspective solid curves). The cutoff function ( Eq. (7)) monotonously going down with the volume increase \n(red dotted line) is shown, as well. The recovered volume distribution (thick line), being of almost the \nsame width as the aggregates fit, is shifted to lower volumes. This misfit is due to the fact that the a g-\ngregates distribution belongs to the whole colloid magnetite+dextran whereas the recove red distribution \nconcerns only the magnetite fraction. With the known parameters for both the distributions, it is possible \nto estimate the dextran shell thickness of nanoparticles constituting the aggregates.  \n10010110210310410510610-310-210-1100\n dN/dV\nVolume,  nm3\n \nFig. 5. Volume dist ributions, all normalized by unity in maxim a: (i) distribution of SPION s from TEM cent ered \nat 102 nm3 (green open circles) and its best fit (solid line), (ii) distribution of aggregates from TEM cent ered at \n8·102 nm3 (purple filled circles ) and its best fit (solid line), (iii) distribution recovered from M2 data fit (narrow \ndashed peak cent ered at 104 nm3), (iv) distribution recovered from M2 data fit with correction on magnetic corr e-\nlations (see Subsection 4.3.1 ) (black thick curve), (v) cutoff function imitating interparticle magnetic correlations \nwith singularity at 6·105 nm3 corresponding to the correlation radius (red dotted line).  \nFrom the recovered mean volume 𝑉  (Table 3, right column) and the TEM mean nanopar ticle-core \nvolume 𝑣 =688 35 nm3, one obtains the mean number of SPION s in the aggregates, 𝑛𝑐=𝑉 /𝑣 ≅55. \nThe nanoparticle specific volume (the average volume per one nanoparticle in the aggregate) is obtained \nfrom the TEM mean aggregate volume 𝑉𝑐=1.33∙105nm3 as 𝑣𝑐=𝑉𝑐/𝑛𝑐=2420  nm3. After that, the \ndextran shell thickness of the nanoparticle can be determined as  \n𝛿=1\n2 𝑑𝑐−𝑑  . \nHere , 𝑑𝑐 is the mean diameter of SPIONs  related to the mean volume per nanoparticle 𝑣𝑐 as 𝜑𝑣𝑐=\n𝜋𝑑𝑐3/6 where 𝜑 is the occupation factor arising due to the empty space between nanoparticles in the ag-\ngregate. Its value is 𝜑=0.56 and 0.64 for friable and compact irregular packing, respectively, resulting in 𝑑𝑐=13.7 nm for the former and 14.4 nm for the latter case , respectively. The mean core diameter \n𝑑 =11.0 nm relates to the mean volume 𝑣  as 𝑣 =𝜋𝑑 𝑐3/6. As a result, the dextran shell width is est i-\nmated to lie in the reasonable interval 𝛿=1.4−1.7 nm where the lower number corresponds to the \nfriable irregular pa cking and the upper to the compact one.  \nComparing the mean (magnetite only) volume of aggregates 𝑉  obtained from M2 and the mean v o-\nlume  𝑉𝑐 obtained from TEM, one immediately estimates the portion of the volume magnetite occupies in \naggregates,  𝑥=𝑉 /𝑉𝑐≅0.28. As magnetite cores are uniformly spread over the aggregate, the cutoff \nfunction parameters 𝑟0 and 𝑟𝑐 obtained from the M2 data treatment should be rescaled by the factor \n𝜉=1/ 𝑥3≅1.52 yielding the true values 𝑟0≅1.52 nm and 𝑟𝑐≅51.9 nm. The c orrelation radius 𝑟𝑐 \nnoticeably  exceeds the mean aggregate radius 21.5 nm (Table 1, right column) whereas the size 𝑟0 of \nthe complete -correlation area is even smaller than the mean magnetite core radius, 4.7 nm (Table 1, left \ncolumn). The latter is a consequence of strong anisotropy of the d -d coupling and a statistical character \nof the quantity 𝑟0. \n4.3.4.  Complementary  remarks  \nFrom Monte -Carlo simulations [ 11], the growth kinetics of aggregates exhibits the tendency to frac-\ntal formation. The fractal dimension 𝜈 is conventionally defined by the relation 𝑛∝𝐷𝜈 where 𝑛 is the \nnumber of nanoparticles constituting an aggregate of the size 𝐷. From TEM, the aggregate size has the \nlognormal distribution with the width 𝜎𝐷=0.63 (Table 1) while the number of nanoparticles 𝑛=𝑉/𝑣  \nobtained from M2 also has the lognormal distribution with the width 𝜎𝑉=2.1 (Table 3). With Eq. (1), \nthe fractal dimension can be evaluated as 𝜈=𝜎𝑉/𝜎𝐷≅3. Thus, no fractal formation tendency is  ob-\nserved in the colloidal system under study. The aggregates consisting of ~ 10 – 100 nanoparticles are \ncomposed compactly and still too small even for a hint on the fractal structure.  \nThus, the data treatment with  no correction on magnetic correlations , instead of computer time con-\nsuming correlations account , may be implemented  only for rough quantifying . \nNote also that no noticeable difference of M2 parameters was found for colloidal solutions with the \nconcentrations 0.02 and 2 mM(Fe)/L . This finding  indicat es stability of the aggregates  in the concentr a-\ntion range  examined . \n5. Electron  magnetic  resonance  \n5.1. Measurements and data treatment  \nThe measurements  performed  were supplemented with EMR data  obtained from the liquid colloid  \nwith the concentration 4 mM(Fe)/L. The EMR spectra were recorded with the special homemade X -\nrange spectrometer operating at the frequency 𝐹=𝜔/2𝜋=8.54 GHz, which provided high sensitivity \nat registration of wide  resonance  lines  [40]. The spectrometer  was supplied by the cylindrical two -mode \nbalanced cavity with TE 111-type of electromagnetic oscillations. The steady magnetic field 𝐇 was d i-\nrected along the cylinder axis 𝑧. The sample was placed at the bottom of the cavity where it was affected \nby the linearly polarized ac field 𝐡 directed along the 𝑥-axis perpendicular to 𝐇 (excitation xz plane). \nThe detection 𝑦𝑧 plane was perpendicular to the excitation one and , thus,  the detected signal was pr o-\nportional to the  gyrotropic  (off-diagonal ) component of the susceptibility tensor , 𝜒𝑦𝑥 𝜔 , corresponding \nto the 𝑦-component of the induced magnetic moment  𝑀𝑦 𝜔 =𝜒𝑦𝑥 𝜔 𝑥(𝜔). The deep frequency -\nindependent uncoupling between  the excitation and detection modes made  it possible to use a micr o-\nwave source with  the high oscillation power (~1 W) without frequency - or amplitude noise at the input \nof the detector , thus,  provid ing high spectrometer sensitivity.  This facility  has proved its efficiency in a \nnumber of condensed matter studies  [17, 41, 42]. The EMR signal s proportional to t he mixture  of the  dispersion 𝜒𝑦𝑥′- and absorption 𝜒𝑦𝑥′′ parts of the \nmagnetic susceptibility 𝜒=𝜒′−𝑖𝜒′′ were register ed as function s of the magnetic field  ranging from \n260 to 6 400 Oe . \nThe spectrum, measured at the temperature 285 K, is presented in Fig. 6 (circles).  \n0 1000 2000 3000 4000 5000 600005001000\n EMR spectrum,  arb. units\nMagnetic field,  Oe\nFig. 6. EMR spectrum (circles): sharp peak at 3 kOe is nitroxyl  radical  signal used as a calibration witness; solid \ncurves are best fits: for magnetically correlated (H<1250 Oe) and ind ependent  (H>1650 Oe) particles (see Sub-\nsection 5.2 ) with crossover at 1430 Oe; straight dashed line is background Hall signal proportional to magnetic \nfield.  \nThe sharp peak centered at 3 kOe comes from nitrox yl radicals  used as a calibration witness.  The \nmain signal was treated with the s imilar GLL  formalism as  used for treatment of  the M2 data  assuming \nthe SP behavior to persist  also at the EMR  frequency  𝐹. Instead of Eq. (5) valid for the diagonal  re-\nsponse  [43], i.e. parallel to the excitation field, the non -diagonal induced moment implies the form : \n𝑚𝑘 𝜔 = 2𝜋\n3  𝛾2′+𝑖𝛾1′ 𝑐11𝑘 𝜔 − 𝛾2′−𝑖𝛾1′ 𝑐1−1𝑘 𝜔  .                                                                              (8) \nThe function fitting the EMR  response , with 𝑘=1 corresponding to the linear susceptibility, reads:  \n𝑀1(𝐻)=𝑀𝑠 (𝑚1′sinΘ+ 𝑚1′′cosΘ) 𝑓𝑀𝑑𝑀                                                                                                   (9) \nwhere Θ is the angle mixing the real 𝑚1′- and imaginary 𝑚1′′ parts of the induced moment  given by  Eq. \n(8). To fit the spectrum , Eq. (9) was accompanied by the background  Hall signal proportional to the  \nmagnetic field  𝐻 coming from  the cavity  material  of the spectrometer [40]. 5.2. Aggregates vs. independent  nanoparticles  \nAs seen in Fig. 6, two field regions exist where the signal is fitted in different ways. The best-fit \ncurve  well describes the peak and the high -field “tail” of the  signal  in the region 𝐻>𝐻2=1650  Oe, \nwhereas its extrapolation to lower fields  strongly deviates from the measured spectrum. This failure  is \ncaused  by competition between the interparticle d -d coupling and interaction of nano particles with  the \nexternal  magnetic field.  In the low -field region 𝐻<𝐻1=1250  Oe, magnetic moments of SPIONs  con-\nstituting the aggregates are co upled by dipolar forces and the EMR signal is a response of the aggregates  \nwithin the correlation radius  as it occur red in M2 measurements. At elevated magnetic fields , the d -d \ncoupling  is broken  and the signal becomes an additive response of ind ependent  nano particles. In Fig. 6, \nthe signal  in the low er-field r egion  is well described by the  same Eq. (9) with the aggregate parameters  \ndetermined  from  M2 measurements ( Table 3), the only variable parameters being  the normalization fa c-\ntor and the mixing angle.  Both the curves intersect at the crossover field 𝐻𝑐=1430  Oe, while at zero \nfield, they fall to zero. The characteristic  interparticle  dipolar energy reads  \n𝐸𝑑=4𝜋𝑚 2\n𝑣𝑐 . \nRecall  that 𝑣𝑐=2950  nm3 is the mean volume per nano particle  inside an aggregate,  𝑚 =𝜇𝑛  is the \nmean magnetic moment of  nanoparticles where 𝜇 is the Fe -ion magnetic moment averaged over the unit \ncell and 𝑛 =𝑣 /𝑣0≅27900  is the mean number of Fe ions per nano particle. The mean energy of the \nnano particle magnetic moment in the field 𝐻 is, merely , 𝐸𝐻=𝑚 𝐻. The crossover field  correspond s to \nthe condition  𝐸𝑑=𝐸𝐻: \n𝐻𝑐=4𝜋𝑚 \n𝑣𝑐 . \nFrom the resonance  line shape , in addition  limited by the crossover , not all SP parameters  can be r e-\nliably e valua ted. In particular , mainly due to  large  uncertainty in calibration of the signal, the nano par-\nticle magnetic moment is hardly  available  from the  present  EMR data. This forces  to turn to off -site \nmagnetization  measurements . \nThe saturation magnetization of magnetite nanoparticles revealed dependence on size, shape, man u-\nfacturing conditions, and matrix - or coating material, if exists [2 7, 44, 45]. Spherical  nanoparticles ~10 \nnm in diameter were found to become sa turated in strong magnetic fields at the temperature 300 K up to \nthe magnetizations ~60 - 80 emu/g [2 7, 44, 45]. These values correspond to the Fe -ion magnetic m o-\nment 𝜇≅0.8−1.11 𝜇𝐵 resulting in 𝑚 ≅(2.2−3.10)∙104 𝜇𝐵. With these numbers, the crossover \nfield falls into the interval 1120−1490  Oe. Thus, the suggested explanation of the EMR spectrum by \ninterplay of d -d coupling and the external magnetic field seems to be highly credible.  \nWith increasing the field over 𝐻𝑐, the SP dynamics dramatically changes. The damping constant r e-\nduces from 𝛼=0.228 to 0.125 while  the Néel  time 𝜏 𝑁 and the longitudinal relaxation time 𝜏∥ increase \nby five times. This effect, visualized here by EMR, should be  directly  observed in magnetorelaxation \nmeasurements. Its account in practical applications of SPION colloids would be important.  \nThe organic coating may somewhat diminish the magnetization of SPIONs  [27, 45] while the \nmeasured  𝐻𝑐 is quite close to the upper limit corresponding to the magnetic moment of uncoated \nnanoparticles [ 44, 45]. This might imply only a weak effect of the dextran coating on the magnetization \nof SPIONs  under study, if any.  \nBy the way, the crossover field is much greater than the maximal field in M2 measurements, \n𝐻𝑚=300  Oe, and the respective  mean  Zeeman energy fo r ind ependent  nanoparticles, 𝐸𝐻≅500  K, is much less than 𝐸𝑑≅(1.7−3.1)∙103 K. This explains the correlated character of M2 response for the \nwhole array of magnetic moments.  \nThe large dipolar energy much exceeding the measurement temperature ensures also the dipolar -\nglass type of the aggregate magnetic structure.  \nThe large effective value of anisotropy axes dire ctions Ψ≅44° obtained from EMR is indicative of \nthe great extent of disorder of their orientations inside the aggregate s, similarly to the case of frozen col-\nloid in the M2 measurement s mentioned above . The  respective  anisotropy energy  with 𝛽𝐸𝑎≅1.1 seems \nto be large  enough to ensure orienting an isolated nanoparticle by the steady magnetic field. However, \norientation disorder inside the aggregate cannot be eliminated even by rather high EMR magnetic field \nwell exceeding the anisotropy field 𝐻𝑎≅700  Oe, due to chemical bonds between the SPIONs  com-\nbined in aggregates.  \nThe distribution width of the SPIONs  magnetic moments 𝜎𝑚=1.09 is quite close to the expected \nvalue 𝜎𝑣=3𝜎𝑑≅1.15 obtain ed from TEM for nanoparticle cores ( Table 1).  \n6. Conclusions  \nSecond -harmonic magnetic response  to a weak ac field  was employed to study an aqueous colloidal \nsolution of dextran -coated magnetite nanoparticles applicable in biomedicine. Magnetic f ield depen d-\nences of real and imaginary parts of the response  in conjunct ion contain full information on magnetic \nproperties of  the colloid  ferrofluid . Expectedly, t he magnetic  colloid has the tendency to form clusters . \nThe data  processing  based on the  stochastic  Gilbert -Landau -Lifshitz equation  was applied to describe  \nthe system  via a set of parameters,  including  the mean  magnetic moment  of the aggregates , the damping \nconstant , the longitudinal relaxation time, the magnetic anisotropy field and energy, and others.  \nWith M2 technique , magnetic correlations inside the aggregates arising from dipole -dipole coupling \nwere  distinguished . Their a ccount  in the data treatment  enable d to recover  the magnetic correlation r a-\ndius,  the size and volume distributions of aggregates, the concentration  of aggregates  in the solution , the \nmean magnetic moment per nanoparticle  and the energy of dipole -dipole interaction between nanopa r-\nticles . The obtained  parameters  add to and  agree  with the  data obtained  from tr ansmission electron m i-\ncroscopy , dynamic light scattering  and electron magnetic resonance.  In particular, combined  analysis of \nthe M2 and TEM data enabled to distinguish  between magnetic and nonmagnetic components of the co l-\nloid. \nTo recover correctly the magnetic -moment and volume distribution s by techniques based on ma g-\nnetic measurements , account of finite -radius magnetic correlations in aggregates is essentially  require d.  \nThe aggregates were argued  to have dipolar -glass -type structure and  possess  magnet ic anisotrop y. \nThe anisotropy turned out to be strong enough to ensure coupling of magnetic and rotational degrees of \nfreedom resulting in noticeable orienting the aggregates by the steady magnetic field of the order 102 \nOe. \nEMR spectra are well described in the framew ork of superparamagnetic dynamics. Herewith, the \nlower -field part of the signal is generated by magnetically correlated aggregated nanoparticles, similarly \nto the case of nonlinear magnetic response, whereas at higher  fields, the signal is formed by indepe nd-\nently responding  nanoparticles due to break of interparticle dipole -dipole coupling by the external ma g-\nnetic  field.  The break  is accompanied with  abrupt acceleration  of superparamagnetic dynamics in in-\ncreasing magnetic field.  This effect should be accounted for and can be used in practical applications of \nSPION colloids. For instance, probing magnetization dynamics  of SPIONs accumulated  in tissues by \nmagnetorelaxometry  or any other suitable technique  may specify whether the MNPs are in aggregated or \nnonaggregated state.  \nThe present research  exemplifies  application of  the novel M2 -based procedure f or comprehensive \nquantitative characterization of a wide variety of  SP systems , particularly , directed tow ard biomedic al applications . Particular  data obtained in this study  will be employed to examine  the state of function a-\nlized  SPIONs  accumulating  in tissues.  \nAcknowledgments  \nThe authors are grateful  to the management of Peter the Great Saint -Petersburg Polytechnic Unive r-\nsity Supercomputing Cent er for making available the  computational resources , as well as  to A. V. Ar u-\ntyunyan for helpful discussion s and A.  M. 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Mater.  324 (2012 ) \n534-539. https://doi.org/10.1016/j.jmmm. 2011.08.035 . \n "    },    {        "title": "1802.01599v1.Cooper_Pair_Spin_Current_in_a_Strontium_Ruthenate_Heterostructure.pdf",        "content": "Cooper-Pair Spin Current in a Strontium Ruthenate Heterostructure\nSuk Bum Chung,1, 2, 3,\u0003Se Kwon Kim,4,yKi Hoon Lee,2, 3and Yaroslav Tserkovnyak4\n1Department of Physics, University of Seoul, Seoul 02504, Korea\n2Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul National University, Seoul 08826, Korea\n3Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea\n4Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nIt has been recognized that the condensation of spin-triplet Cooper pairs requires not only the\nbroken gauge symmetry but also the spin ordering as well. One consequence of this is the possibility\nof the Cooper-pair spin current analogous to the magnon spin current in magnetic insulators, the\nanalogy also extending to the existence of the Gilbert damping of the collective spin-triplet dynamics.\nThe recently fabricated heterostructure of the thin \flm of the itinerant ferromagnet SrRuO 3on\nthe bulk Sr 2RuO 4, the best-known candidate material for the spin-triplet superconductor, o\u000bers a\npromising platform for generating such spin current. We will show how such heterostructure allows\nus to not only realize the long-range spin valve but also electrically drive the collective spin mode\nof the spin-triplet order parameter. Our proposal represents both a new realization of the spin\nsuper\ruidity and a transport signature of the spin-triplet superconductivity.\n^x^y φ\nJzsp(a) ^d\nφ\nJzsp^n\nJ ↑↑\nJ ↓↓(b) (c)\nFIG. 1. Schematic illustration of the analogy between the\nmagnetic insulator and the spin-triplet superconductor. (a)\nThe planar spiraling of the magnetic order parameter ^ nleads\nto spin current. (b) The same phenomena occurs for that of\nthe spin component ^dof the spin-triplet superconductor order\nparameter, (c) the dual picture of which is the counter\row of\nthe spin up-up and down-down pairs.\nIntroduction : Harnessing spin rather than charge in\nelectronic devices has been a major topic in solid state\nphysics, which not only has been utilized for various\nmemory devices but is also expected to play a key role\nin processing quantum information [1]. In order for vari-\nous spin devices to function robustly, the long-range spin\ntransport needs to be achieved. Metallic wires, however,\ntypically do not transport spins beyond the spin-di\u000busion\nlength due to the single electron spin relaxation [2].\nIn recent years, it has been shown that the exponential\ndamping can be circumvented in the spin transport via\ncollective magnetic excitations. For example, easy-plane\n(ferro- and antiferro-)magnetic insulators, as the U(1)\norder parameter can characterize them, may be consid-\nered analogous to the conventional super\ruid [3{5]. As\nFig. 1 (a) illustrates schematically, the planar spiraling\nof the magnetic order parameter in such magnetic insu-\nlators can give rise to the spin supercurrent, just as the\nphase gradient of the conventional super\ruid gives rise\nto the mass supercurrent; in this sense these magnetic\ninsulators can be regarded as spin super\ruids [6].\nInterestingly, there exists a class of super\ruids and su-\nperconductors which can support both mass and spin su-\npercurrent. Such super\ruids and superconductors wouldneed to involve both spin ordering and gauge symme-\ntry breaking. This occurs in the condensate of both\nthe spin-1 bosons [7] and the spin-triplet Cooper pairs\nof3He atoms [8, 9] or electrons [10, 11]; in the latter\ncase, the dissipationless spin current would be carried by\nthe Cooper pairs. While the vortices with spin supercur-\nrent circulation have been observed in all theses systems\n[12, 13], the bulk spin supercurrent has not been detected\nin the superconductor.\nIn this Letter, we will show how this existence of spin\nsuper\ruidity in the spin-triplet superconductor allows\nnot only the long-range spin current but also electrically\nexciting the spin wave in the bulk. For realizing these\nphenomena, we propose a two-terminal setup with volt-\nage bias between ferromagnetic metal leads in contact\nwith the spin-triplet superconductor. While the static\norder-parameter case [14] can be essentially reduced to\nthe Blonder-Tinkham-Klapwijk type formalism [15] for\nthe interfacial transport, here we need to complement\nit with the appropriate equations of motion for the col-\nlective spin dynamics in the superconductor. Recently,\na thin \flm of the itinerant ferromagnet SrRuO 3has\nbeen epitaxially deposited on the bulk Sr 2RuO 4, the best\nknown candidate material for the spin-triplet supercon-\nductor [16], yielding, due to their structural compatibil-\nity, an atomically smooth and highly conductive interface\n[17] with a strong Andreev conductance [18]. This makes\nSr2RuO 4and SrRuO 3the most suitable candidate mate-\nrials for the bulk and the leads, respectively, of our setup\n[19]. For the remainder of this paper, we will \frst show\nhow the simplest e\u000bective spin Hamiltonian for the spin-\ntriplet superconductor and the resulting spin dynamics\nare analogous to those of the antiferromagnetic insula-\ntor; then, we will discuss the magnetoresistance for the\nDC bias voltage and the coupling between the AC bias\nvoltage and the spin wave.\nGeneral considerations : We \frst point out the close\nanalogy between the spin order parameter of the antifer-arXiv:1802.01599v1  [cond-mat.supr-con]  5 Feb 20182\nromagnet and the spin-triplet superconductor. De\fned\ni(d\u0001\u001b)\u001by=\u0014\u0000dx+idydz\ndzdx+idy\u0015\n\u0011\u0014\u0001\"\"\u0001\"#\n\u0001#\"\u0001##\u0015\n;(1)\nthe d-vector of the spin-triplet pairing, which\nparametrizes the Cooper-pair spin state, be-\nhaves similarly under spin rotations to the N\u0013 eel\norder parameter of an antiferromagnet, i.e.,\n[Si(r);dj(r0)] =i\u0016h\u000fijk\u000e(r\u0000r0)dk(r) and [di;dj] = 0\nfor the condensate spin S(unlike the magnetization,\nneither the N\u0013 eel order parameter nor the d-vector\ngenerate the spin rotation in themselves) [8, 9, 11].\nGiven that the commutation relations establish S\u0002^d\nas the conjugate momentum to din both cases, it is\nnatural that the simplest e\u000bective Hamiltonian for the\nspin-triplet superconductor ^d-vector,\nH=1\n2Z\ndr[A(r^d)2+K^d2\nz+\r2\neS2=\u001f]; (2)\nwhere\reis the electron gyromagnetic ratio, Athe^d-\nvector sti\u000bness, and \u001fthe magnetic susceptibility, should\nbe equivalent to that of the antiferromagnet N\u0013 eel order\nparameter, once we identify the ^d-vector with the N\u0013 eel\norder parameter [4]. In the latter, antiferromagnetic case,\na (xy) planar texture of the orientational order param-\neter^ n!(cos\u001e;sin\u001e;0) is associated with a collective\n(z-polarized) spin current Jz/z\u0001^ n\u0002@i^ n!@i\u001e\row-\ning in theith direction. While this extends directly to\nour spin-triplet case, Eq. (1) gives the intuitive dual pic-\nture of Fig. 1 (c) for the planar spiraling of the d-vector,\ni.e.,^d= (cos\u000b;sin\u000b;0). Namely, as the phase of \u0001 \"\"\n(\u0001##) is given by \u001ec\u0007\u000b(where\u001ecis the overall phase\nof the superconductor), the spiraling of the d-vector on\nthexyplane as shown in Fig. 1 (b), or the gradient of\n\u000b, would imply the counter\row of the spin up-up and\ndown-down pairs. The resultant ( z-polarized) spin cur-\nrent is/\u0000r\u000b. Given the same commutation relation\nand the same e\u000bective Hamiltonian, it is natural that, in\nabsence of dissipation, the equations of motion for these\ntwo cases, the Leggett equations the ^d-vector [8, 9, 20]\nand the Landau-Lifshitz type equation for the N\u0013 eel order\nparameter, are identical.\nWe further argue that both cases have the same phe-\nnomenological form of dissipation as well. For the case of\nthe N\u0013 eel order parameter ^ n, such dissipation, /\u000b(@t^ n)2,\nknown generally as Gilbert damping for collective mag-\nnetic dynamics, has been understood phenomenologically\n[4, 21, 22]. That such dissipation has not been featured in\nthe 3He super\ruid literature can be attributed not to the\nintrinsic nature of the spin-triplet pairing but rather to\nthe very weak relativistic spin-orbit coupling of the 3He\natoms originating solely from the nuclear dipole-dipole\ninteraction [8]. In contrast, electrons in Sr 2RuO 4are\nsubject to the Ru atomic spin-orbit coupling [23] esti-\nmated to be of order 0.1 eV [24]. In this work, we willconsider the decay rate of \u000bn\u0016h\r2\ne=\u001ffor the condensate\nspin, the addition of which makes the Leggett equations\nof motion for spin [25] equivalent to the Landau-Lifshitz-\nGilbert type equations for antiferromagnets:\n@t^d=\u0000^d\u0002\r2\ne\n\u001fS;\n@tS=^d\u0002(Ar2^d\u0000K^dz^ z\u0000\u000bn\u0016h@t^d); (3)\nwhere\u000bis the dimensionless Gilbert damping parame-\nter andnthe Cooper-pair density. This set of equations\nshows how the e\u000bective Hamiltonian of Eq. (2) provides\nthe simplest method for considering the local ^d-vector\ndynamics, including the spin-wave excitation and the col-\nlective dissipation.\nFor the boundary conditions, at the interface between\nthe ferromagnetic lead and the spin-triplet superconduc-\ntor, we consider a two-channel interface conductance due\nto the spins aligned or anti-aligned to the lead magne-\ntization We note, in this regard, that the SrRuO 3thin\n\flm has a very high transport spin polarization, with\na 3-to-1 ratio between the majority and minority spin\nchannels [26{28], while the magnetization gets enhanced\nin the heterostructure [17]. In this Letter, for the sake of\nsimplicity, we shall only consider the case where the lead\nmagnetizations are collinear. Furthermore, the d-vector\nof the bulk spin-triplet superconductor will be taken to be\nperpendicular to the lead magnetization, i.e., the Cooper\npairs are equal-spin paired along the quantization axis\nparallel to the magnetization; it has been claimed for\nthe Sr 2RuO 4superconductor, based on the c-axis NMR\nmeasurement, that its d-vector can be rotated into the\nab-plane by applying magnetic \feld larger than 200 G\n[29], well below the upper critical \feld.\nLong-range spin valve : The simplest physics that can\narise in our two-terminal setup is the spin-valve magne-\ntoresistance due to the relative alignment of the leads.\nWe consider the case where the spin-triplet supercon-\nductor has the easy-plane anisotropy, that is, K > 0\nin Eq. (2), while the lead magnetization is perpendic-\nular to this plane; as already mentioned, the former\ncan be realized for the SrRuO 3/Sr2RuO 4heterostruc-\nture by applying a \u0015200 G \feld along the c-axis. In\nthis case, we can take ^dzto be a small parameter in ^d=\n(q\n1\u0000^d2zcos\u001ez;q\n1\u0000^d2zsin\u001ez;^dz) andjSx;yj\u001cjSzj. In\nsuch a case, [ \u001ez(r);Sz(r0)] =i\u0016h\u000e(r\u0000r0) gives us the con-\njugate pair, leading to the equations of motion\n@t\u001ez=\r2\ne\n\u001fSz; @tSz=Ar2\u001ez\u0000\u000bn\u0016h@t\u001ez;(4)\nwhere the \frst equation is a spin analogue of the Joseph-\nson relation and the second is the spin continuity equa-\ntion with the relaxation term. Note that we measure\nSzwith respect to its equilibrium value. One con\frms\nthe condensate spin imbalance relaxation time to be3\nHa\nVLIVR\n^\n^yz\nx^\nI\nVLVR\nz^\n^y\n^x\nFIG. 2. The setup for the DC voltage bias for the spin valve\n(upper) and the AC bias voltage for the spin-wave detection\n(lower), where ^ x;^ y;^ zcoincide with the crystalline a; b; c -axes,\nrespectively. For the upper \fgure, the lead magnetization is\nalong the c-axis, with the applied magnetic \feld Ha\u0015200 G\nalong the c-axis giving us the easy plane d-vector con\fgura-\ntion on the ab-plane, hence the spiraling in the ab-plane. For\nthe lower \fgure, the lead magnetization is along the a-axis; as\nthe easy-axis d-vector anisotropy favors the alignment along\nthec-axis, in the absence of an applied \fled, the AC bias volt-\nage gives us the low-frequency standing wave of the d-vector\noscillating around the c-axis in the bc-plane.\n\u001f=\u000bn \u0016h\r2\nefrom Eq. (4) through deriving @tSz+r\u0001Jsp\nz=\n\u0000\u000bn\u0016h\r2\neSz=\u001f, where Jsp\nz=\u0000Ar\u001ez. It is also impor-\ntant to note here that the magnitude of the d-vector\nanisotropy Khas no e\u000bect on the in-plane d-vector pre-\ncession, which allows us to ignore the fact that our ap-\nplied \feld gives us the Abrisokov vortices in the spin-\ntriplet superconductor and hence a non-uniform K.\nWe consider the spin-up current and the spin-down\ncurrent to be independent at the interface:\nI\u001b\nL;R=\u0006g\u001b\u001b\nL;R(VL;R\u0000\u0016h@t'\u001b=2e); (5)\nwhereg\u001b\u001b\nL;R's are the conductances for the \u001b-spin,IL;R\nthe\u001b-spin current into (out of) the left (right) lead, and\nVL;Rthe bias voltage of the left (right) lead; this is due\nto the spin-triplet superconductor having the equal spin\npairing axis collinear with the lead magnetization and\ntakingg\"#= 0. From Eq. (1), we see that the overall\n(or charge) phase of the superconductor is given by the\naverage of the spin up-up and the spin down-down con-\ndensate phase, \u001ec=P\n\u001b'\u001b=2, while\u001ezof Eq. (4) is given\nby\u001ez=P\n\u001b\u001b'\u001b=2. We are interested here in the steady-\nstate solution, i.e., @t'\u001b= const, for which we de\fne the\nconstant precession rate of !c\u0011P\n\u001b@t'\u001b=2 for the over-\nall phase\u001ecand \ns\u0011P\n\u001b\u001b@t'\u001b=2 for\u001ez. For such\nsolution, the following continuity conditions can be ap-plied to the charge and spin supercurrents, respectively:\nX\n\u001b(I\u001b\nL\u0000I\u001b\nR)=0;X\n\u001b\u001b(I\u001b\nL\u0000I\u001b\nR)=2\u000bne\nsSL (6)\n(Sis the bulk cross section area and Lthe spacing be-\ntween the two leads), the former from the charge con-\nservation and the latter from applying the steady-state\ncondition on Eq. (4), along with the spin current loss\n/\u000bLin the superconductior.\nThe current through the Sr 2RuO 4bulk can be ob-\ntained from the interface boundary conditions and the\ncontinuity conditions above, with the larger magni-\ntude for the parallel magnetization than the antipar-\nallel magnetization. We de\fne the total conductance\ngL;R\u0011P\n\u001bg\u001b\u001b\nL;R and the conductance polarization\npL;R\u0011P\n\u001b\u001bg\u001b\u001b\nL;R=gL;R, which de\fnes the relevant trans-\nport spin polarization. Applying the continuity condi-\ntions Eq. (6) on the interface boundary conditions Eq. (5)\nand setting VL=\u0000VR=V=2, we obtain\n\u0012\ngL+gRpLgL+pRgR\npLgL+pRgRgL+gR+g\u000b\u0013\u0012\n!c\n\ns\u0013\n=eV\n\u0016h\u0012gL\u0000gR\npLgL\u0000pRgR\u0013\n;\n(7)\nwhereg\u000b\u00114\u000bne2SL\n\u0016h. We can now obtain the dependence\nof the charge current on the conductance polarization:\nIc=X\n\u001bI\u001b=I0\u0014\n1\u0000gLgR(pL\u0000pR)2\n(gL+gR)(gL+gR+g\u000b)\u0000(pLgL+pRgR)2\u0015\n;\n(8)\nwhereI0\u0011gLgRV=(gL+gR). Note that Icis max-\nimized atpL=pR, when the steady-state angle \u001ezre-\nmains static. Di\u000berent spin polarizations at the two ends,\non the other hand, would trigger spin dynamics and re-\nsult in a nonzero dissipation rate of R=1\n2\u000bn\u0016h\n2\ns=\nR0(1\u0000Ic=I0)2=(pL\u0000pR)2per volume of the supercon-\nducting bulk, where R0= 8\u000bn(eV)2=\u0016h. Given that pL;R\nchange sign on the magnetization reversal, the above re-\nsults e\u000bectively give us the spin-valve magnetoresistance\nof our heterostructure, i.e., a larger conductance for the\nparallel magnetizations than for the antiparallel. Any\ne\u000bect that the spin-triplet pairing may have on the mag-\nnetization, hence the conductance polarization, can be\nignored when the Curie temperature of SrRuO 3(\u0018160K)\n[30] is two orders of magnitude higher than the supercon-\nducting critical temperature ( \u00181.5K) Sr 2RuO 4.\nWe emphasize that the above magnetoresistance re-\nsult is obtain solely for the current carried by Cooper\npairs. At a \fnite-temperature, quasiparticle contribu-\ntion would generally result in an exponentially-decaying\nmagnetoresistance, negligible for the lead spacing be-\nyond the spin-di\u000busion length. By contrast, the cur-\nrent of Eq. (8), which is carried by the Cooper pairs,\ngives us the\u00181=Lbehavior for the large spacing limit.\nTherefore, any magnetoresistance beyond the quasipar-\nticle spin-di\u000busion length should arise only below the su-\nperconducting transition at Tc, upon the emergence of4\n0.80.91.|Ic|/I0\n12340.50.60.70.80.91.\nω/ω0|Ic|/I0\nFIG. 3. Charge current versus frequency plotted for ~ g= 0:5,\n~L= 2, \u0000 =!0= 0:1 and ~A= 0:2, with the orange curve\nrepresenting pL=pR=pand the blue pL=\u0000pR=p. Note\nthatp= 0:8 for the top plot and p= 0:2 for the bottom plot.\na Cooper-pair condensate. For our Sr 2RuO 4/ SrRuO 3\nheterostructure, detection of magnetoresistance in the su-\nperconducting state for the lead spacing larger than the\nSr2RuO 4spin-di\u000busion length can be taken as a trans-\nport evidence for the spin-triplet superconductivity. The\nvalue of the spin-di\u000busion length itself can be extracted\nby measuring the exponential decay of the (normal) mag-\nnetoresistance, both above and below the transition.\nElectrically driven spin collective mode : For the case of\nthe easy-axis anisotropy of the d-vector, hence K < 0 in\nEq. (2),the spin collective excitation of the Cooper pairs\n[8, 9, 31, 32] will modify the supercurrent transport under\nthe AC bias voltage. We shall still continue to consider\nthe case where Eq. (5) would be valid, i.e., the equal spin\npairing axis of the spin-triplet superconductor collinear\nto the lead magnetizations. One way to satisfy this con-\ndition would be to have the lead magnetizations collinear\nto thea-axis, with no applied magnetic \feld; that would\nleave thea-axis as the equal spin pairing axis, with the\nd-vector moving on the the bc-plane. The equations of\nmotion, corresponding to spin injection polarized along\nthex-direction, are then modi\fed to\n@t\u001ex=\r2\ne\n\u001fSx; @tSx=Ar2\u001ex\u0000!2\n0\u001f\n\r2ecos\u001exsin\u001ex\u0000\u000b\u0016h@t\u001ex;\n(9)\nwhere\u001exis conjugate to Sxand!2\n0\u0011 jKj\r2\ne=\u001fis\nthe spin-wave energy gap. For the AC voltage bias\nV=V0exp(\u0000i!t), the steady-state solution for thespin phase \u001ex(x;t) =f(x) exp(\u0000i!t) and the charge\nphase\u001ec(x;t) =g(x) exp(\u0000i!t) behave di\u000berently, fo-\ncusing on the frequencies far below the plasma fre-\nquency. Hence the spin equations of motion Eq. (9)\ngives usf(x) =C+cosh\u0014x+C\u0000sinh\u0014x, wherev2\u00142=\n!2\u0000!2\n0\u0000i!\u0000, withv\u0011\rep\nA=\u001f (the^d-vector sti\u000b-\nnessAde\fned in Eq. (2)) being the spin-wave veloc-\nity and \u0000\u0011\u000bn\u0016h\r2\ne=\u001fthe damping rate. By contrast,\nthe charge current Jc(x;t) =\u0000\u001a@x\u001ec, where\u001ais the\n\u001ecsti\u000bness, should be uniform, which means we can set\n\u001ec(x;t) = const:\u0000x(Jc\n0=\u001a) exp(\u0000i!t), with a constant\nJc\n0. By imposing consistency between the current ob-\ntained from the boundary conditions of Eq. (5) and the\ndynamics of Eq. (9), we can solve for Jc\n0andC\u0006; Fig. 3\nshows the numerical results for Ic=Jc\n0Sfor the case of\nbothpL=pRandpL=\u0000pR.\nOur numerical results show that magnetoresistance be-\ncomes signi\fcant at !>\u0018!0, where the collective spin\nmode of the Cooper pairs is activated. For simplicity\nwe have set gL=gR=gand used the dimensionless\nparameters ~ g\u0011g\u0016hv=2eA,~L\u0011!0L=2v, and ~A=A=\u001a.\nFor! <! 0, in addition to barely noticeable magnetore-\nsistance, the charge current amplitude does not oscillate\nwith frequency; it remains close to the DC value I0, which\ncontrasts with the complete transport suppression ob-\ntained for the magnetic insulator [3]. In contrast, for\n! > ! 0, we see an oscillation with the !=! 0period of\nabout\u0019=~L, where the current amplitude maxima for the\nantiparallel lead magnetization occur at the current am-\nplitude minima for the parallel lead magnetization and\nvice versa. As in the ferromagnetic insulator [3], we ex-\npect that for ~L\u001c1 (whileLis still larger than the quasi-\nparticle spin-di\u000busion length), the magnetoresistance of\nEq. (8) is recovered for the static bias, i.e.,!!0.\nWe point out that the detection of the oscillation\nshown in Fig. 3 would determine the yet-unknown energy\nparameters for the spin-triplet pairing of Sr 2RuO 4. From\nthe e\u000bective Hamiltonian of Eq. (2), if we had known\naccurately the \feld Hcalong thec-axis that would ex-\nactly restore the d-vector isotropy, the gap frequency !0\nshould be just the electron Larmor frequency of this \feld\nfrom the spin equations of motion of Eq. (9). However,\nwe know no more than the upper bound Hc<200 G,\nhence only !0< \re\u0002200 G = 3:5 GHz, while the AC\nbias experiment, as shown in in Fig. 3, would allow us to\nde\fnitely identify the spin collective mode gap.\nConclusion and discussion : We have studied the DC\nand AC current transport between the itinerant ferro-\nmagnetic lead with collinear magnetization through the\nspin-triplet superconductor. We showed here that mag-\nnetoresistance can arise for both cases due to the Cooper-\npair spin transport. For the DC bias, the persistence\nof magnetoresistance for the lead spacing larger than\nthe quasiparticle spin-di\u000busion length can be taken as\na transport evidence for the spin-triplet pairing. For\nthe AC bias, the activation of magnetoresistance and5\nfrequency dependent oscillation above the threshold fre-\nquency will allow us to determine the spin anisotropy\nenergy scale. All together, our work shows both a new\nrealization of the spin super\ruidity and a transport sig-\nnature of the spin-triplet superconductivity. The recently\nfabricated SrRuO 3/Sr2RuO 4heterostructure provides a\npromising experimental setup.\nAcknowledgement : We would like to thank Young Jun\nChang, Bongju Kim, Han Gyeol Lee, Seung Ran Lee,\nYoshiteru Maeno, Tae Won Noh, S. Raghu, Manfred\nSigrist and So Takei for sharing their insights. The hos-\npitalities of Natal International Institute for Physics dur-\ning the workshop \\Collective Spin Transport in Electrical\nInsulators\" (S.B.C. and Y.T.) and of Kyoto University\nduring the workshop \\Oxide Superspin 2017\" (S.B.C.),\nwhere parts of this work has been completed, are grate-\nfully acknowledged. This research was supported by IBS-\nR009-Y1 (S.B.C. and K.H.L.), and United States Army\nResearch O\u000ece under Contract No. W911NF-14-1-0016\n(S.K.K. and Y.T.).\n\u0003sbchung0@uos.ac.kr\nyevol@physics.ucla.edu\n[1] S. A. 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B 94, 064508 (2016)."    },    {        "title": "1802.02415v1.Breaking_the_current_density_threshold_in_spin_orbit_torque_magnetic_random_access_memory.pdf",        "content": "arXiv:1802.02415v1  [cond-mat.mes-hall]  7 Feb 2018Breaking the current density threshold in spin-orbit-torq ue magnetic random access\nmemory\nYin Zhang,1,2H. Y. Yuan,3,∗X. S. Wang,4,1and X. R. Wang1,2,†\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2HKUST Shenzhen Research Institute, Shenzhen 518057, China\n3Department of Physics, Southern University of Science and T echnology of China, Shenzhen 518055, China\n4School of Microelectronics and Solid-State Electronics,\nUniversity of Electronic Science and Technology of China, C hengdu, Sichuan 610054, China\n(Dated: March 25, 2022)\nSpin-orbit-torque magnetic random access memory (SOT-MRA M) is a promising technology\nfor the next generation of data storage devices. The main bot tleneck of this technology is the\nhigh reversal current density threshold. This outstanding problem of SOT-MRAM is now solved\nby using a current density of constant magnitude and varying flow direction that reduces the\nreversal current density threshold by a factor of more than t he Gilbert damping coefficient. The\nEuler-Lagrange equation for the fastest magnetization rev ersal path and the optimal current\npulse are derived for an arbitrary magnetic cell. The theore tical limit of minimal reversal current\ndensity and current density for a GHz switching rate of the ne w reversal strategy for CoFeB/Ta\nSOT-MRAMs are respectively of the order of 105A/cm2and 106A/cm2far below 107A/cm2and\n108A/cm2in the conventional strategy. Furthermore, no external mag netic field is needed for a\ndeterministic reversal in the new strategy.\nSubject Areas: Magnetism, Nanophysics, Spintronics\nI. INTRODUCTION\nFast and efficient magnetization reversal is of not only\nfundamentally interesting, but also technologically im-\nportant for high density data storage and massive in-\nformation processing. Magnetization reversal can be in-\nduced by magnetic field [1–3], electric current through\ndirect [4–9] and/or indirect [10–22] spin angular mo-\nmentum transfer from polarized itinerant electrons to\nmagnetization, microwaves [23], laser light [24], and\neven electric fields [25]. While the magnetic field in-\nduced magnetization reversal is a matured technology,\nit suffers from scalability and field localization problems\n[8, 26] for nanoscale devices. Spin transfer torque mag-\nnetic random-access memory is an attractive technol-\nogy in spintronics [26] although Joule heating, device\ndurability and reliability are challenging issues [11, 26].\nIn an spin-orbit-torque magnetic random access mem-\nory (SOT-MRAM) whose central component is a heavy-\nmetal/ferromagnet bilayer, an electric current in the\nheavy-metal layer generates a pure spin current through\nthe spin-Hall effect [10, 11] that flows perpendicularly\ninto the magnetic layer. The spin current, in turn, pro-\nduces spin-orbit torques (SOT) through spin angular\nmomentum transfer [4, 5] and/or Rashba effect [16–22].\nSOT-MRAM is a promising technology because writing\ncharge current does not pass through the memory cells\nso that the cells do not suffer from the Joule heating\nand associated device damaging. In principle, such de-\nvices are infinitely durable due to negligible heating from\n∗[Corresponding author:]yuanhy@sustc.edu.cn\n†[Corresponding author:]phxwan@ust.hkspin current [11]. However, the reversal current density\nthreshold (above 107A/cm2[14, 15] for realistic materi-\nals) in the present SOT-MRAM architecture is too high.\nTo have a reasonable switching rate (order of GHz), the\ncurrent density should be much larger than 108A/cm2\n[14, 15] that is too high for devices. In order to lower the\nminimalreversalcurrentdensityaswellastoswitchmag-\nnetization states at GHz rate at a tolerable current den-\nsity in SOT-MRAM, it is interesting to find new reversal\nschemes (strategies) that can achieveabove goals. In this\npaper, we show that a proper current density pulse of\ntime-dependent flow direction and constant magnitude,\nmuch lower than the conventional threshold, can switch\na SOT-MRAM at GHz rate. Such a time-dependent cur-\nrent pulse can be realized by using two perpendicular\ncurrentspassingthrough the heavy-metallayer. The the-\noretical limit of minimal reversal current density of the\nnew reversal strategy for realistic materials can be of the\norder of 105A/cm2, far below 107A/cm2in the con-\nventional strategy that uses a direct current (DC), both\nbased on macrospin approximation. The validity of the\nmacrospin model is also verified by micromagnetic simu-\nlations.\nII. MACROSPIN MODEL AND RESULTS\nA. Model\nOur new reversal strategy for an SOT-MRAM, whose\ncentral component is a ferromagnetic/heavy-metal bi-\nlayer lying in the xy-plane with initial spin along the\n+z-direction as shown in Fig. 1, uses a current den-\nsityJ=JcosΦˆx+JsinΦˆygenerated from two time-2\nm\nFM \nHM JxJx\nJyJy\nFIG. 1. Schematic illustration of new reversal scheme for\nSOT-MRAMs. Two perpendicular currents flowin the heavy-\nmetal layer of a ferromagnet/heavy-metal bilayer to genera te\na current whose direction can vary in the xy-plane.\ndependent electric currents flowing along the x- and the\ny-directions, where Φ is a time-dependent angle between\nJand the x-axis and Jis a constant total current den-\nsity. The magnetic energy density is ε=−Kcos2θwith\nKbeing the anisotropy coefficient and θbeing the polar\nangle of the magnetization. In the absence of an electric\ncurrent, the system has two stable states m= +ˆzand\nm=−ˆzwheremis the unit direction of magnetization\nM=Mmof magnitude M. The electric current gen-\nerates a transverse spin current perpendicularly flowing\ninto the ferromagnetic layer via the spin-Hall effect [10],\nandthenproducesaneffectiveSOTonthemagnetization[4, 5, 16], i.e.\n/vector τ=−am×(m׈s)+βam׈s, (1)\nwhere the first term on the right-hand-side is the\nSlonczewski-liketorquewhile thesecondtermis thefield-\nlike torque. The spin-polarization direction is ˆ s=ˆJ׈z\n(for other type of spin-Hall effect, see Note [27]) with ˆJ\nbeing the unit vector of current density. a=¯h\n2edθSHJ\nmeasures SOT where ¯ h,e, anddare respectively the\nPlank constant, the electron charge, and the sample\nthickness. θSHis the spin Hall angle which measures\nthe conversion efficiency between the spin current and\ncharge current. βmeasures the field-like torque and can\nbe an arbitrary real number since this torque may also\nbe directly generated from the Rashba effect [16].\nThemagnetizationdynamicsunderanin-planecurrent\ndensityJis governed by the generalized dimensionless\nLandau-Lifshitz-Gilbert (LLG) equation,\n∂m\n∂t=−m×heff+αm×∂m\n∂t+/vector τ, (2)\nwhereαis the Gilbert damping constant that is typically\nmuch smaller than unity. The effective field is heff=\n−∇mεfrom energy density ε. Time, magnetic field and\nenergy density are respectively in units of ( γM)−1,M\nandµ0M2, where γandµ0are respectively the gyro-\nmagnetic ratio and vacuum magnetic permeability. In\nthis unit system, a=¯h\n2edµ0M2θSHJbecomes dimension-\nless.\nThe magnetization mcan be conveniently described\nby a polar angle θand an azimuthal angle φin thexyz-\ncoordinate. In terms of θandφ, the generalized LLG\nequation becomes\n(1+α2)˙θ=−αKsin2θ+a(1−αβ)cosθsin(Φ−φ)+a(α+β)cos(Φ−φ)≡F1, (3a)\n(1+α2)˙φsinθ=Ksin2θ−a(1−αβ)cos(Φ−φ)+a(α+β)cosθsin(Φ−φ)≡F2. (3b)\nB. Derivation of the Euler-Lagrange equation\nThe goal is to reverse the initial state θ= 0 to the\ntarget state θ=πby SOT. There are an infinite number\nof paths that connect the initial state θ= 0 with the\ntarget state θ=π, and each of these paths can be used\nas a magnetization reversal route. For a given reversal\nroute, there are an infinite number of current pulses that\ncan reverse the magnetization. The theoretical limit of\nminimal current density Jcis defined as the smallest val-\nues of minimal reversal current densities of all possible\nreversal routes. Then it comes two interesting and im-\nportant questions: 1) What is Jcabove which there is at\nleast one reversal route that the current density can re-\nverse the magnetization along it? 2) For a given J > Jc,what are the optimal reversal route and the optimal cur-\nrent pulse Φ( t) that can reverse the magnetization at the\nhighest speed?\nDividing Eq. (3b) by Eq. (3a), one can obtain the\nfollowing constraint,\nG≡∂φ\n∂θsinθF1−F2= 0. (4)\nThe magnetization reversal time Tis\nT=/integraldisplayπ\n0dθ\n˙θ=/integraldisplayπ\n01+α2\nF1dθ. (5)\nThe optimization problem here is to find the optimal\nreversal route φ(θ) and the optimal current pulse Φ( t)3\nsuch that Tis minimum under constraint (4). Using the\nLagrange multiplier method, the optimal reversal route\nand the optimal current pulse satisfy the Euler-Lagrange\nequations [28, 29],\n∂F\n∂φ=d\ndθ(∂F\n∂(∂φ/∂θ)),∂F\n∂Φ=d\ndθ(∂F\n∂(∂Φ/∂θ)),(6)\nwhereF= (1 +α2)/F1+λGandλis the Lagrange\nmultipliers which can be determined self-consistently by\nEq. (6) and constrain (4). Given a current density of\nconstant magnitude J, Eq. (6) may or may not have a\nsolution of φ(θ) that continuously passing through θ=\n0 andθ=π. If such a solution exists, then φ(θ) is\nthe optimal path for the fastest magnetization reversal\nand the corresponding solution of Φ( t) is the optimal\ncurrent pulse. The theoretical limit of minimal reversal\ncurrent density is then the smallest current density Jc\nbelow which the optimal reversal path does not exist.\nC. The optimal current pulse and theoretical limit\nof minimal reversal current density\nFrom Eqs. (3a), (3b) and (4) as well as F= (1 +\nα2)/F1+λG, theEuler-Lagrangeequationof (6)becomes\nλd\ndθ(F1sinθ) = 0, (7a)\n1+α2\nF2\n1∂F1\n∂φ−λ∂G\n∂φ=−1+α2\nF2\n1∂F1\n∂Φ+λ∂G\n∂Φ= 0.(7b)\nFrom Eq. (7a), one has λ/ne}ationslash= 0 orλ= 0. Ifλ/ne}ationslash= 0,F1must\nbeF1=C/sinθ(C/ne}ationslash= 0) so that (1+ α2)˙θ=C/sinθ→\n∞asθ→0 orπ. This solution is not physical, and\nshouldbe discarded. Therefore, the only allowedsolution\nmust be λ= 0, and one has ∂F1/∂Φ = 0 according to\nEq. (7b). Interestingly, this is exactly the condition of\nmaximal ˙θ=F1/(1+α2) as Φ varies. Φ satisfies tan(Φ −\nφ) =1−αβ\nα+βcosθ, or\nΦ = tan−1(1−αβ\nα+βcosθ)+φ+π(β <−α) (8a)\nΦ = tan−1(1−αβ\nα+βcosθ)+φ (β >−α).(8b)\nSubstituting Eq. (8) into the LLG equation (3), θ(t)\nandφ(t) are determined by the following equations,\n˙θ=1\n1+α2[aP(θ)−αKsin2θ], (9a)\n˙φ=1\n1+α2[2Kcosθ−a(α+β)(1−αβ)sinθ\nP(θ)],(9b)\nwhereP(θ) =/radicalbig\n(α+β)2+(1−αβ)2cos2θ. To reverse\nmagnetization from θ= 0 toθ=π,amust satisfy a >\nαKsin(2θ)/P(θ) according to Eq. (9a) so that ˙θis no\nnegative for all θ. Obviously, ˙θ= 0 atθ=π/2 whenFIG. 2. The log α-dependence of Jcfor various βare plotted\nas the solid curves for model parameters of M= 3.7×105\nA/m,K= 5.0×103J/m3,θSH= 0.084 and d= 0.6 nm. As\na comparison, Jdc\ncis also plotted as the dashed lines.\nβ=−α. The magnetization reversal is not possible in\nthis case, and β=−αisasingularpoint. The theoretical\nlimit of minimal reversal current density Jcforβ/ne}ationslash=−α\nis\nJc=2αeKd\nθSH¯hQ, (10)\nwhereQ≡max{sin2θ/P(θ)}forθ∈[0,π].\nIn comparison with the current density threshold [13,\n14, 18] (Jdc\nc) in the conventional strategy for β= 0,\nJdc\nc=2eKd\nθSH¯h(1−H√\n2K), (11)\nthe minimal reversal current density is reduced by more\nthan a factor of α. HereH(≃22 Oe in experiments)\nis a small external magnetic needed for a deterministic\nreversal in conventional strategy. Using CoFeB/Ta pa-\nrameters of M= 3.7×105A/m,K= 5.0×103J/m3,\nθSH= 0.084 and d= 0.6 nm [11, 14, 15], Fig. 2 shows\nlogα-dependenceof Jc(solidlines)and Jdc\nc(dashedlines)\nforβ= 0 (black), 0 .3 (red) and −0.3 (blue), respectively.\nBothJdc\ncandJcdepend on β. The lower the damping of\na magnetic material is, the smaller our minimum switch-\ning current density will be. For a magnetic material of\nα= 10−5, the theoretical limit of minimal reversal cur-\nrent density can be five order of magnitude smaller than\nthe value in the conventional strategy.\nFor a given J > Jc, the shortest reversal time is given\nby Eqs. (5) and (9a):\nT=/integraldisplayπ\n01+α2\naP(θ)−αKsin2θdθ. (12)\nThe optimal reversal path is given by φ(θ) =/integraltextθ\n0˙φ\n˙θdθ′\nwhere˙θand˙φare given by Eqs. (9a) and (9b). Eq.\n(9a) gives t(θ) =/integraltextθ\n0(1+α2)/(aP(θ)−αKsin2θ)dθ′and\nthenθ(t) is just θ(t) =t−1(θ). Thus, Φ( θ,φ),φ(θ)\nandθ(t) giveφ(t) =φ(θ(t)) and Φ( t) = Φ(θ(t),φ(t)).4\n(d) (e) mz\nmxmymz\nmy\nmx(a) (b) (c) \n(f) \nmymz\nmx\nFIG. 3. Model parameters of M= 3.7×105A/m,K= 5.0×103J/m3,θSH= 0.084,α= 0.008 and d= 0.6 nm are used to\nmimic CoFeB/Ta bilayer, and β= 0.3 for (a), (c), (d) and (f) while β= 0.1 for (b) and (e). The theoretical limit of minimum\nreversal current density is Jc= 1.56×105A/cm2forβ= 0.1 andJc= 1.28×105A/cm2forβ= 0.3. Optimal current pulses\n((a)-(c)) and fastest reversal routes ((d)-(f)) are for J= 1.92×106A/cm2((a) and (d)), and for J= 9.0×106A/cm2((b),\n(c)), (e) and (f).\nUsing the same parameters as those for Fig. 2 with\nα= 0.008 and various β, Fig. 3 shows the optimal\ncurrent pulses ((a)-(c)) and the corresponding fastest\nmagnetization reversal routes ((d)-(f)) for β= 0.3 and\nJ= 1.92×106A/cm2≈15Jc((a) and (d)), for β= 0.1\nandJ= 9.0×106A/cm2≈58Jc((b) and (e)), and for\nβ= 0.3 andJ= 9.0×106A/cm2≈70Jc((c) and (f)). It\nisknownthatTahaslesseffecton α[11]. Theminimalre-\nversal current density Jcunder the optimal current pulse\nis 1.56×105A/cm2forβ= 0.1 and 1.28×105A/cm2\nforβ= 0.3 which is far below Jdc\nc= 9.6×106A/cm2for\nthe same material parameters [15]. The multiple oscilla-\ntions ofmxandmyreveal that the reversal is a spinning\nprocess and optimal reversal path winds around the two\nstable states many times. Correspondingly, the driving\ncurrent makes also many turns as shown by the multiple\noscillations of JxandJy. The number of spinning turns\ndependsonhowfar JisfromJc. Thecloser JtoJcis, the\nnumber of turns is larger. The number of turns is about\n5 in Figs. 3(a) and 3(d) for J≈15Jcand one turn for\nJ >50Jcas shown in Figs. 3(b), 3(c), 3(e) and 3(f), so\nthat the reversal is almost ballistic. The reversaltime for\nβ= 0.3 andJ= 1.92×106A/cm2is about 10 nanosec-\nonds, for β= 0.1 andJ= 9.0×106A/cm2is about 3.3\nnanoseconds, and for β= 0.3 andJ= 9.0×106A/cm2is\nabout 2.1 nanoseconds. Figure 4 is the reversaltime Tas\na function of current density Junder the optimal current\npulse for the same parameters as those for Fig. 2. TheFIG. 4. Magnetization reversal time Tunder the optimal\ncurrent pulses as a function of Jfor various αandβ.\nreversaltime quickly decreases to nanoseconds as current\ndensity increases. In a real experiment, there are many\nuncertainties so that the current pulse may be different\nfrom the optimal one. To check whether our strategy is\nrobust again small fluctuations, we let the current pulse\nin Fig. 3(c) deviate from its exact value. Numerical\nsimulations show that the magnetization reversal is not\nsignificantly influenced at least when the deviation be-\ntween the real current and optimal current is less than\nfive percents.5\nIII. VERIFICATION OF MACROSPIN MODEL\nBY MICROMAGNETIC SIMULATION\nIn our analysis, the memory cell is treated as a\nmacrospin. A nature question is how good the macrospin\nmodel is for a realistic memory device. To answer this\nquestion, we carried out micromagnetic simulations by\nusing Newton-Raphson algorithm [30] for two memory\ncells of 150 nm ×150nm×0.6 nm (Figs. 5(a), (b), (d) and\n(e)) and 250 nm ×250 nm×0.6 nm (Figs. 5(c) and (f)).\nTo model the possible edge pinning effect due to mag-\nnetic dipole-dipole interaction, we consider square-shape\ndevices instead of cylinder shape device whose edge pin-\nning is negligible. To make a quantitative comparison,\nthe material parameters are the same as those used in\nFig. 3. In our simulations, the unit cell size is 2 nm ×2\nnm×0.6 nm. For a fair comparison, the optimal current\npulses shown in Figs. 3(a) and (c) of respective current\ndensityJ= 1.92×106A/cm2andJ= 9.0×106A/cm2\nwere applied to the memory cell of 150 nm ×150 nm×0.6\nnm. The symbols in Figs. 5(a) and (b) are the time evo-\nlution of averaged magnetization mx,myandmzwhile\nthesolidlinesarethetheoreticalpredictionsofmacrospin\nmodel shown in Figs. 3(d) and (f). The perfect agree-\nments prove the validity of the macrospin approximation\nfor our device of such a size. To further verify that the\nmemory device can be treated as a macrospin, Figs. 5(d)\nand (e) are the spin configurations in the middle of the\nreversal at t= 5.5 ns for Fig. 5(a) and at t= 1.2 ns for\nFig. 5(b). Thefactthatallspinsalignalmostinthesame\ndirection verifies the validity of the macro spin model. In\nreal experiments, non-uniformity of current density is in-\nevitable. To demonstrate the macrospin model is still\nvalid, we let current density linearly varies from 9 .5×106\nA/cm2on the leftmost column of cells to 8 .5×106A/cm2\nonthe rightmostcolumn ofcells. As expected, thereis no\nnoticeable difference with the data shown in Figs. 5(b)\nand (e).\nFor the large memory device of 250 nm ×250 nm×0.6\nnm, the optimal current pulse shown in Fig. 3(c) of cur-\nrent density J= 9.0×106A/cm2was considered. The\ntime evolution of averaged magnetization mx,myand\nmzare plotted in Fig. 5(c), with the symbols for simula-\ntions andsolid linesfor the macrospinmodel. They agree\nvery well although there is a small deviation for device\nof such a large size. Figure 5(f) is the spin configurations\nin the middle of the reversal at t= 1.2 ns for Fig. 5(c).\nThe marcospin model is not too bad although all spins\nare not perfectly aligned in this case.\nIn summary, for a normal SOT-MRAM device of size\nless than 300 nm [11, 15], macrospin model describes\nmagnetization reversal well. However, for a larger sam-\nple size and lower current density ( J <106A/cm2for\nthe same material parameters as those used in Fig. 3),\nonly the spins in sample center can be reversed while the\nspins near sample edges are pinned.(a) (b) (c) \n(d) (e) (f) t = 5.5 ns t = 1.2 ns t = 1.2 ns \nFIG. 5. (a)-(c) Time evolution of the average magnetization :\ncycles for micromagnetic simulations and solid lines are th e-\noretical predictions from macrospin model. (a) and (b) are\nfor the memory cell of 150 nm ×150 nm×0.6 nm and optimal\ncurrent pulse of current density of J= 1.92×106A/cm2and\nJ= 9.0×106A/cm2, respectively. (c) is for the memory\ncell of 250 nm ×250 nm×0.6 nm and optimal current pulse of\ncurrent density of J= 9.0×106A/cm2. (d)-(f) Spin configu-\nrations respectively corresponding to (a)-(c) in the middl e of\nmagnetization reversal at t= 5.5 ns and 1.2 ns. The cell size\nin micromagnetic simulation is 2 nm ×2 nm×0.6 nm.\nIV. DISCUSSION\nObviously, the strategy present here can easily be gen-\neralized to the existing spin-transfer torque MRAM. The\nmathematics involved are very similar, and one expects\na substantial current density reduction is possible there\nif a proper optimal current pulse is used. Of course, how\nto generate such a current pulse should be much more\nchallenge than that for SOT-MRAM where two perpen-\ndicular currents can be used. In the conventional strat-\negy that uses a DC-current, a static magnetic field along\ncurrent flow is required for a deterministic magnetiza-\ntion reversal [13, 14, 18]. Although several field-free de-\nsigns have been proposed [19, 20], an antiferromagnet is\nneeded to create an exchange bias which plays the role\nof an applied magnetic field. As we have shown, such a\nrequirement or complication is not needed in our strat-\negy. Ourstrategydoesnothaveanotherproblemexisting\nin the conventional strategy in which the magnetization\ncan only be placed near θ=π/2 [13, 14, 18] so that\nthe system falls into the target state by itself through\nthe damping. Therefore, one would like to use materials\nwith largerdamping in the conventionalstrategyin order\nto speed up this falling process. In contrast, our strategy\npreferslowdampingmaterials,andreversalisalmostbal-\nlistic when current density is large enough ( >50Jcin the\ncurrent case). To reverse the magnetization from θ=π\ntoθ= 0, one only needs to reverse the current direction\nof the optimal current pulse. One should notice that\nthe Euler-Lagrange equation allows us to easily obtain\nthe optimal reversal current pulse and theoretical limit6\nof the minimal reversal current density for an arbitrary\nmagnetic cell such as in-plane magnetized layer [11] and\nbiaxial anisotropy.\nV. CONCLUSION\nIn conclusion, weinvestigatedthe magnetizationrever-\nsal of SOT-MRAMs, and propose a new reversal strat-\negy whose minimal reversal current density is far be-\nlow the existing current density threshold. For popular\nCoFeB/Ta system, it is possible to use a current densityless than 106A/cm2to reversethe magnetizationat GHz\nrate, in comparison with order of J≈108A/cm2in the\nconventional strategy.\nACKNOWLEDGMENTS\nThis work was supported by the National Natural Sci-\nence Foundation of China (Grant No. 11774296 and\nNo. 61704071) as well as Hong Kong RGC Grants No.\n16300117 and No. 16301816. X.R.W. acknowledges the\nhospitalities of Beijing Normal University and Beijing\nComputationalScience ResearchCenterduringhisvisits.\n[1] C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin,\nD. Weller, E. L. Garwin, H. C. Siegmann, Minimum field\nstrength in precessional magnetization reversal , Science\n285, 864 (1999).\n[2] M. Bauer, J. Fassbender, B. Hillebrands, and R. L.\nStamps, Switching behavior of a Stoner particle beyond\nthe relaxation time limit , Phys. Rev. B 61, 3410 (2000).\n[3] Z. Z. Sun and X. R. 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Nanotech. 11,\n621-625 (2016).\n[15] Jun-Yang Chen, Mahendra DC, Delin Zhang, Zhengyang\nZhao, Mo Li, and Jian-Ping Wang, Field-free spin-\norbit torque switching of composite perpendicular\nCoFeB/Gd/CoFeB layers utilized for three-terminal mag-\nnetic tunnel junctions , Appl. Phys. Lett. 111, 012402\n(2017).\n[16] A. Manchon and S. Zhang, Theory of spin torque due to\nspin-orbit coupling , Phys. Rev. B 79, 094422 (2009).\n[17] Ioan Mihai Miron, Gilles Gaudin, Stphane Auffret,\nBernard Rodmacq, Alain Schuhl, Stefania Pizzini, Jan\nVogel and Pietro Gambardella, Current-driven spin\ntorque induced by the Rashba effect in a ferromagnetic\nmetal layer , Nat. Mat. 9, 230-234 (2010).\n[18] X. Zhang, C. H. Wan, Z. H. Yuan, Q. T. Zhang, H. Wu,\nL. Huang, W. J. Kong, C. Fang, U. Khan, and X. F.\nHan,Electrical control over perpendicular magnetization\nswitching driven by spin-orbit torques , Phys. Rev. B 94,\n174434 (2016).\n[19] A. van den Brink, G. Vermijs, A. Solignac, J. Koo, J. T.\nKohlhepp, H.J.M.SwagtenandB.Koopmans, Field-free\nmagnetization reversal by spin-Hall effect and exchange\nbias, Nat. Commun. 7, 10854 (2016).\n[20] ShunsukeFukami, Chaoliang Zhang, Samik DuttaGupta,\nAleksandr Kurenkov and Hideo Ohno, Magnetization\nswitching by spin-orbit torque in an antiferromagnet-\nferromagnet bilayer system , Nat. Mater. 15, 535-541\n(2016).\n[21] A. Manchon, Spin Hall effect versus Rashba torque: a\nDiffusive Approach , arXiv:1204.4869.\n[22] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY. Mokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G.\nGaudin, and P. Gambardella, Symmetry and magnitude\nof spin-orbit torques in ferromagnetic heterostructures ,\nNat. Nanotech. 8, 587 (2013).\n[23] Z. Z. Sun and X. R. Wang, Magnetization reversal\nthrough synchronization with a microwave , Phys. Rev.\nB74, 132401 (2006).\n[24] J.-Y. Bigot, L. Guidoni, E. Beaurepaire, and P. N. Saeta ,\nFemtosecond Spectrotemporal Magneto-optics , Phys.Rev.\nLett. 93, 077401 (2004).\n[25] Fumihiro Matsukura, Yoshinori Tokura and Hideo Ohno,\nControl of magnetism by electric fields , Nat. Nanotech.\n10, 209-220 (2015).7\n[26] E. Chen, D. Apalkov, Z. Diao, A. Driskill-Smith, D.\nDruist, D. Lottis, V. Nikitin, X. Tang, S. Watts, S.\nWang, S. A. Wolf, A. W. Ghosh, J.W. Lu, S. J. Poon,\nM. Stan, W. H. Butler, S. Gupta, C. K. A. Mewes,\nTim Mewes, and P. B. Visscher, Advances and Future\nProspects of Spin-Transfer Torque Random Access Mem-\nory, IEEE Trans. Magn. 46, 1873 (2010).\n[27] Note: Recently, there are claims that spin polarizatio n ˆs\nhave also a component along ˆJ׈t, whereˆtis the crys-\ntalline direction of the heavy metal, see for example, D.\nMacNeill et al.Nat. Phys. 13, 300 (2016); Alisha M.\nHumphries et al.Nat. Commun. 8, 911 (2017). In thiscase, one needs only to use ˆ s=ˆJ׈z+a1ˆJ׈tin Eq.\n(1), where a1is a model parameter. The rest procedures\nare similar to what was done in the main text.\n[28] X. R. Wang, P. Yan, J. Lu and C. He, Euler equation\nof the optimal trajectory for the fastest magnetization re-\nversal of nanomagnetic structures , Europhys. Lett. 84,\n27008 (2008).\n[29] G. Arfken, Mathematical Methods for Physicists, 3rd ed.\n(Orlando, FL: Academic Press, 1985).\n[30] M. d’Aquino, C. Serpico, G. Miano, I. D. Mayergoyz and\nG. Bertotti, J. Appl. Phys. 97, 10E319 (2005)."    },    {        "title": "1802.03176v2.Monocrystalline_free_standing_3D_yttrium_iron_garnet_magnon_nano_resonators.pdf",        "content": "Monocrystalline free standing 3D yttrium iron garnet magnon\nnano resonators\nF. Heyroth,1C. Hauser,2P. Trempler,2P. Geyer,2F. Syrowatka,1\nR. Dreyer,2S.G. Ebbinghaus,3G. Woltersdorf,2and G. Schmidt2, 1,∗\n1Interdisziplin¨ ares Zentrum f¨ ur Materialwissenschaften,\nMartin-Luther-Universit¨ at Halle-Wittenberg, D-06120 Halle, Germany\n2Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle-Wittenberg, D-06120 Halle, Germany\n3Institut f¨ ur Chemie, Martin Luther Universit¨ at Halle-Wittenberg, D-06120 Halle, Germany\nAbstract\nNano resonators in which mechanical vibrations and spin waves can be coupled are an intriguing\nconcept that can be used in quantum information processing to transfer information between\ndifferent states of excitation. Until now, the fabrication of free standing magnetic nanostructures\nwhich host long lived spin wave excitatons and may be suitable as mechanical resonators seemed\nelusive. We demonstrate the fabrication of free standing monocrystalline yttrium iron garnet\n(YIG) 3D nanoresonators with nearly ideal magnetic properties. The freestanding 3D structures\nare obtained using a complex lithography process including room temperature deposition and lift-\noff of amorphous YIG and subsequent crystallization by annealing. The crystallization nucleates\nfrom the substrate and propagates across the structure even around bends over distances of several\nmicrometers to form e.g. monocrystalline resonators as shown by transmission electron microscopy.\nSpin wave excitations in individual nanostructures are imaged by time resolved scanning Kerr\nmicroscopy. The narrow linewidth of the magnetic excitations indicates a Gilbert damping constant\nof onlyα= 2.6×10−4rivalling the best values obtained for epitaxial YIG thin film material. The\nnew fabrication process represents a leap forward in magnonics and magnon mechanics as it provides\n3D YIG structures of unprecedented quality. At the same time it demonstrates a completely new\nroute towards the fabrication of free standing crystalline nano structures which may be applicable\nalso to other material systems.\nKeyword: Magnonics, 3D nano-fabrication, Magnon resonators, Magnon mechanics, Spin cavit-\nronics, YIG nanostructures\n∗Correspondence to G. Schmidt: georg.schmidt@physik.uni-halle.de\n1arXiv:1802.03176v2  [cond-mat.mes-hall]  5 Apr 2019I. INTRODUCTION\nNanomechanical oscillators are useful tools for quantum information processing. Over\nthe past decade numerous groups have for example demonstrated the conversion of quan-\ntum information from the microwave to the optical regime by means of a micromechanical\nresonator[1–4] . By coupling of electrical excitations in superconducting qubits to mechan-\nical oscillators[5] even readout of quantum information has been demonstrated[6, 7] . The\nnecessary interaction was often obtained by electric fields as in capacitive drum resonators.\nAnother suitable mechanism for information transfer, however, can make use of the cou-\npling of magnetic fields to spin wave modes in a magnon resonator. Indeed the coupling of\na magnon mode in a macroscopic yttrium iron garnet (YIG) sphere to a single qubit has\nalready been demonstrated in 2015[8] . For downscaling and integration, however, smaller\nYIG structures are needed. Taking these results into account it is a promising perspective\nto realize a new transfer mechanism by coupling magnons to mechanical oscillations in a\nnanomechanical resonator via magnetoelastic coupling. Obviously, YIG would be an ideal\ncandidate for these resonators since YIG is the material with the lowest known Gilbert\ndamping[9] and it exhibits extremely long lifetimes for spin waves (magnons) in the µs\nregime. As a single crystalline garnet material with a Young’s modulus of the same order\nof magnitude as that of silicon carbide it is expected to also provide low losses for me-\nchanical waves (phonons) and may yield nanoresonators with high quality factors. Again\nin macroscopic YIG spheres in the sub-mm range the coupling of magnons to phonons\nhas already been demonstrated[10] . However, up to now no method was known to shape\nthree-dimensional nanostructures from monocrystalline YIG. Nanopatterning of thin films\nwith reasonable quality has been demonstrated[11–15] , but no patterning of nano-sized free\nstanding resonators has been put forward. Nevertheless, it would be extremely attractive if\nmicron- or sub-micron sized YIG bridges or cantilevers were available. The mechanical res-\nonance frequencies in such structures may be easily engineered to fall in the range of typical\nmagnon frequencies[16] . As a first step in this direction we have realized the fabrication of\nfreely suspended YIG microbridges with very low damping for spin waves. Although the\nmechanical properties could not yet be investigated in detail, mechanical resonance frequen-\ncies calculated for their dimensions using the elastic properties of YIG fall into the range of\nseveral hundred MHz and may even reach the GHz regime.\n2II. 3D NANO FABRICATION\nFabrication techniques for suspended single crystal nanostructures mostly use subtractive\nprocessing by removing material from a single crystal (bulk or layer). The most straight\nforward method uses focused ion beam (FIB) lithography to directly shape the desired\nstructure from bulk or thin film[17] . Although very flexible in terms of possible geometries\nthis technique suffers from the possible damage to the crystal structure by extended beam\ntails which might be detrimental for the magnetic properties of YIG. Also it requires lateral\naccess for the beam in order to remove the material underneath the suspended structure\npreventing the creation of multiple structures in close vicinity. Alternatively a crystalline\nfilm (resonator material) may be deposited on top of a sacrificial layer. The resonator itself is\nshaped by lithography and dry etching and only becomes free-standing when the underlying\nsacrificial layer is removed by highly selective wet chemical etching[18, 19] . The resulting\ngeometry, however, has several limitations. It is not truly three dimensional but only a\npartly suspended two dimensional structure. Also the suspended resonator must be more\nnarrow than the un-suspended pads to which it is attached. Otherwise the pads are under-\netched during the removal of the sacrificial layer. Unfortunately no sacrificial layers are\nknown for high quality crystalline YIG films which can only be deposited on garnet surfaces\n(especially gallium gadolinium garnet, GGG) and no selective wet etchants are available for\nthese materials.\nOn the other hand nanoscale additive fabrication of polycristalline materials is achieved\nby electron beam lithography, evaporation, and lift-off. A typical example is the fabrication\nof metallic air bridges, well known since more than a decade [20–22] . The process allows for\ndensely packed structures with high flexibility in terms of geometry. However, it requires low\ntemperature deposition of the material because of the limited thermal stability of electron\nbeam resists. This prevents its use for the patterning of monocrystalline materials such as\nYIG, which in most cases need to be deposited at elevated temperatures.\nA new kind of deposition method for thin film YIG has recently been demonstrated.\nAmorphous YIG films are deposited at room temperature on GGG using either pulsed laser\ndeposition[23, 24] or sputtering[25] . In a subsequent annealing step the material adapts\nto the lattice structure of the substrate resulting in thin single-crystalline YIG films. Sur-\nprisingly, the quality of these films in terms of damping surpasses the quality of thin films\n3deposited at high temperature[23–25] . Because deposition is done at room temperature this\ndeposition method is compatible with electron beam lithography. In this way the fabrication\nof laterally nanopatterned YIG with reasonably small Gilbert damping constants has been\ndemonstrated recently[11, 12, 14] . Theoretically, this process also allows the fabrication of\nbeams and bridges when it is adapted to the patterning process used for metal bridges de-\nscribed above. Nevertheless, the higher kinetic energies of the deposited particles in pulsed\nlaser deposition compared to evaporation may necessitate a specially adapted resist profile\nto guarantee a successful lift-off. Further on the recrystallization is more challenging. In\na thin film, crystallization needs to progress only vertically from the substrate to the film\nsurface (with a typical distance of 100 nm or less). In a bridge structure, however, the\ncrystallization starts at the base of the supporting pillars which are in contact with the sub-\nstrate and then needs to progress around bends across the entire span of the bridge in order\nto achieve a monocrystalline structure. Any additional nucleation site for crystallization\nmay disturb the process and introduce an additional grain boundary. As we show in the\nfollowing, it is possible to realize such a 3D lift-off process for YIG with the crystallization\n(which indeed starts at the substrate) extending throughout the complete bridge structure\neven over distances of several micrometers.\nIII. PROCESSING\nFigs. 1a-d schematically show the applied process flow. A thick PMMA layer on a\n<111>oriented GGG substrate is patterned using electron beam lithography at different\nelectron acceleration voltages for the span (low voltage/LV) and pillars (high voltage/HV)\nof the bridges, respectively (Fig. 1a). Further details are provided in the methods section.\nThe resulting structure after development of the e-beam resist is shown in Fig. 1b. It exhibits\nholes down to the substrate for the pillars and a groove for the span of the bridge. At the\nsides the groove has a slight undercut which later facilitates the lift-off process. Onto the\ndeveloped structure the amorphous YIG material is deposited by PLD at room temperature\n(Fig. 1c). Subsequent lift-off and resist removal results in a bridge structure (Fig. 1d) which\nis finally annealed. Fig. 2a shows a scanning electron microscopy (SEM) image of a YIG\nbridge prior to (a) and after annealing (b). The bridge has a nominal span length of 2 µm\nand a YIG layer thickness of approximately 110 nm. The length of the span does not change\n4during the annealing step within the measurement accuracy of the SEM. For the experi-\nment shown here the pillars are not placed at the end of the bridges. This design yields an\noverhang at the end to combine the investigation of short cantilevers fixed on one end only\nwith that of bridge structures which are clamped at both ends. The resulting bridges and\ncantilevers are flat and strain free after the lift-off. Subsequent to annealing the bridge itself\nremains mostly unchanged, however, the overhang is bent upward (Fig. 2b) indicating the\npresence of strain.\nDuring the crystallization at more than than 800◦C the lattice can reorder and a structure\nwith very little or no strain is created. During cool-down, however, the difference in thermal\nexpansion coefficient of YIG and GGG can lead to a small deformation. The YIG now\nexhibits tensile strain. While in a continuous layer on a substrate this strain would lead\nto a change in lattice constant the bridge can now follow the strain by deformation. By\ntilting the feet inward, the length of the span can be decreased while the tilting can of the\nfeet can lead to the small upward bend of the overhang. The thermal expansion coefficients\nfor YIG is smaller than that of GGG by ∼2×10−6K−1. By cooling from 800◦C to room\ntemperature the contraction of the YIG lattice would be approximately 0.1% larger than\nfor GGG. It should be noted that any resulting shortening of the bridge is too small to be\nmeasured with the accuracy of our electron microscope.\nFig. 2c shows a close up view of an annealed YIG bridge with a span of 750 nm also after\nannealing. The deposited YIG has a nominal thickness of 110 nm. The edges of this bridge\nare quite rough and show a lot of residue from the lift-off process. Obviously, these can be\ndetrimental for the quality of mechanical resonances. As we show later, these residues can\nmostly be avoided or removed.\nIV. STRUCTURAL CHARACTERIZATION\nWhile the SEM images show that the molding of the material is successful, the local\ncrystalline quality can only be assessed by transmission electron microscopy (TEM). Atomic\nresolution TEM has been performed on different bridges after annealing (details described\nin the methods section). Fig. 3a shows a cross-sectional view of a small bridge with a span\nof approximately 850 nm and a height between span and substrate of 75 nm. The sample\nwas prepared using a focused ion beam and cut along a {011}plane perpendicular to the\n5surface. The viewing direction of the TEM is along <011>with a small tilt angle.\nThe pillars which are in direct contact with the substrate show an epitaxial monocrys-\ntalline lattice as also observed for large area deposition by Hauser et al. [23] . The transition\nto the span where the material is thinner shows a number of defects likely due to partially\nrelieved shear strain that can be expected in this location.\nThe span of the bridge, however, appears monocrystalline and of perfect crystallinity\nexcept for a single defect in the center (Fig. 3b). This defect is a consequence of the crystal-\nlization process as described below. To investigate possible differences in lattice orientation\nof substrate and bridge FFts of TEM images were taken at different spots of the sample. A\ncomparison of FFTs from the substrate and the bridge shows that except for a minute lattice\nrotation the lattice parameter and orientation are identical for substrate and bridge. This is\nexpected due to the excellent lattice match between YIG and GGG. (mismatch ∼0.06%).\nIn addition FFTs from different points of the bridge are superimposed to see whether the\nlattice orientation varies along the bridge (Fig. 4). A color coded overlay of the FFTs on left,\nright, and center of the span shows that the crystal orientations on both sides are tilted with\nrespect to each other with a tilt angle of about 1◦. A similarly small rotation is observed\nwhen comparing FFTs from bridge and substrate. From these results we can deduce that\ncrystallization starts simultaneously at both pillars, where the material is strained. Thus\nthe two crystallization fronts may be slightly tilted with respect to each other. When they\nmeet at the center of the span the resulting mismatch can only be compensated for by the\nformation of the crystal defect such as a small angle grain boundary observed at the center\nof the bridge. In addition, this mechanism explains the small rotation of left and right hand\npart of the bridge with respect to each other and with respect to the substrate.\nTo investigate the influence of the bridge size on crystallinity also cross-sectional TEM\nimages of longer bridges are studied (Fig. 3c). Even for a length of 2 .8µm a similar quality\nof the span (which is the functional part of the resonator) is obtained.\nV. SPIN DYNAMICS\nBecause of the reduced amount of material it is not possible to measure the saturation\nmagnetization M Sof the bridges directly with magnetometry methods. From previous exper-\niments we know that YIG layers fabricated by room temperature deposition and annealing\n6under similar conditions exhibit M Sup to 27 % below the bulk value of µ0MS≈180mT[26] .\nWe would like to note that the M S-value used for the micromagnetic simulations (132 mT)\nis in excellent agreement with these results.\nIn order to obtain a detailed and accurate measurement of the local dynamic properties\nwe perform time-resolved scanning Kerr microscopy (TR-MOKE) experiments on a 110 nm\nthick YIG-bridge. Using this method it is possible to image directly the different resonant\nmagnon modes in individual bridge structures. To achieve the necessary high frequency\nexcitation of the YIG structures an impedance matched coplanar wavegude (CPW) is de-\nposited by electron beam lithoghraphy and lift-off processes onto the sample. The CPW is\npositioned such that an array of bridges is located in the gap between signal line and ground\nplane (inset of Fig. 8a). The investigated bridge has a width of 600 nm and a span length\nof 3µm. The thickness of the deposited YIG film is 110 nm and the gap under the span is\n100 nm. The sample was deposited using the parameters described in the methods section.\nThe spatially resolved measurements are performed with the external magnetic field ori-\nented along the bridge allowing for the excitation of the backward volume modes (BVM)\nwith k-vectors along the bridge and the Damon Eshbach modes (DEM) with k-vectors at\nan angle of 90◦. Fig. 5 (top row) shows a number of different modes for increasing magnetic\nfield. The fundamental mode with only one antinode is shown in Fig. 5b. Three standing\nBVM with nodes distributed along the bridge are shown in Fig. 5c-e, while a DEM mode\nshows a node extending along the bridge (Fig. 5a). It is clearly visible that the magnons\nare localized in the span of the bridge and no direct coupling to the pillars or beyond is\nobserved.\nWe have also modelled the different magnon modes using MuMax3[27] . Fig. 5\n(bottom row) shows the respective simulations, which are in good agreement with our ex-\nperiments. Like the bridge investigated by TRMOKE the simulated bridge has a width of\n600 nm and a span length of 3 µm. The thickness of the deposited YIG film is 110 nm and\nthe gap under the span is 100 nm. The gyromagnetic ratio γobtained in the simulations is\n178 GHz/T which is close to the value obtained from the MOKE data (171 GHz/T, Fig. 6).\nThe saturation magnetization was fitted to match the spin wave patterns resulting in a value\nofµ0MS≈132 mT which is in good agreement with that of large area films deposited by\nthe same method[23] .\nIn order to obtain a better understanding in terms of the magnetization dynamics in\n7the YIG nano bridges FMR spectra are measured by TRMOKE on a single spot in the\ncenter of the bridge for several frequencies. Such a resonance spectrum is shown in Fig. 8a\nwhere the main resonance peak has a line width of approximately 140 µT at 8 GHz. This\nvalue is among the smallest values reported for PLD grown thin film material so far. Only\nmaterial grown by liquid phase epitaxy exhibits smaller linewidths. From our data we find\na Gilbert damping value of the main resonance of α≈(2.6±0.7)×10−4(Fig. 8b). Also this\nvalue is lower than all values reported for YIG grown by PLD at elevated temperatures. The\ninhomogeneous line width at zero field is µ0∆H0= 75±10µT which is lower than anything\nreported for PLD grown thin film YIG so far. For the given configuration these numbers\ncan also be translated into spin wave life times resulting in 220 ns (3.2 GHz), 160 ns (5.2\nGHz), and 120 ns (8.4 GHz).\nWe also determine the effective saturation magnetization M effwhich also contains any\nanisotropy and the gyromagnetic ratio γthe resonance fields of the main FMR line are\ndetermined as a function of frequency (Fig. 6) and the data is fitted by the Kittel formula:\nω=µ0γ/radicalBig\nHFMR(HFMR+ M eff) (1)\nThe fit yields γ=(180.3±0.6) GHz/T and µ0Meff=(0.125±0.003) T.\nIn addition to the dynamic properties TR-MIKE also allows to investigate the static\nswitching behavior of individual nanobridges. For this we use the method in an off-resonant\nfashion around zero field (Fig. 7). Here the phase and the magnitude of the rf-susceptibility\nare used to detect the switching as first demonstrated in [28] . For the measurement the\nmicrowave frequency is set to 1 GHz. The probing light spot is placed at the center of the\nsame bridge. The magnitude of the response depends on the internal magnetic field and\nis therefore sensitive to the relative alignment of magnetization and applied magnetic field.\nHysteretic behavior is found when magnetization and applied magnetic field are antiparallel.\nFrom this we determine a coercive field of µ0HC≈2 mT for the bridge (lateral dimensions of\nthe span: 600 nm×3µm) when the magnetic field is aligned with the long axis of the bridge\nstructure (easy axis). For the magnetic field aligned along the short axis (hard axis) of the\nbridge we find no hysteretic behavior as expected. This coercive field is considerably larger\nthan for comparable continuous YIG films of the same thickness where we find coercive\nfields of less than 0.1 mT [23] . The enhanced coercive fields in the YIG nano bridges are\nexpected and a consequence of the shape anisotropy and the increased contribution of the\n8domain wall nucleation energy to the magnetization reversal in nanostructures.\nIn the FMR spectrum also a second line is visible which partly overlaps with the main\npeak. Spatially resolved measurements indicate that the two halves of the span which are\nseparated by the central crystalline defect differ in resonance field by approx. 100 µT at 8\nGHz. This can be explained by the rotation of the two sides observed in transmission electron\nmicroscopy. When the field is applied exactly along one half of the bridge, the small tilt of\nthe other half can shift the resonance field in the order of 100 µT at 8 GHz simply because\na very small demagnetizing field is added to the external field. For one degree of tilt this\nmodification can be as large as 0.05% of the resonance field which is sufficient to explain the\nobserved resonance line shift.\nIn addition the spatial resolution of the TR-MOKE also allows to investigate the variation\nof the resonance field between different bridges and between different parts of a single bridge\n(namely span and overhang), respectively. Fig. 9 shows TR-MOKE images of five different\nbridges obtained simultaneously and repeated for two different magnetic fields but at the\nsame excitation frequency. For individual bridges the main resonance (only one antinode)\nappears at fields that vary by almost 0.8 mT, respectively.\nThe resonance in the overhang of a bridge can only be imaged by sweeping the field over\na wider range. Fig.10 shows that the overhang also exhibits a localized resonance. The\nresonance field, however, is offset by approx. 7 mT from the main resonance field of the\ncorresponding span. This shift can be caused by the different strain in the span which is\npinned on both sides and the overhang which is pinned only on one side as well as by the\ndifferent size of the two regions. As the resonance does not extend into the foot of the bridge\nthe k-vector is determined by the length of the area on resonance as we no longer observe a\ntrue uniform mode but a standing spin wave with zero nodes.\nIt should be noted that the fabrication process is not limited to simple bridge geometries\nbut highly flexible and can be extended to more complex structures as shown in the examples\nof Fig. 11 paving the way to a number of applications and experiments. Again, also the\nmagnetic excitations are well defined and can be directly imaged. SEM image and MOKE\ndata in Fig. 11c are obtained from the very same structure. We have also tried to reduce\nedge and surface roughness which may deteriorate the mechanical resonance properties by\nusing an optimized multi-layer resist and a post-annealing wet-etch step. As a result an\nimproved bridge with smoother edges is shown in Fig. 11d.\n9VI. DISCUSSION\nIt is possible to fabricate 3D YIG nanobridges using electron beam lithography, room\ntemperature PLD and lift-off. The structural characterization shows that crystallization\nduring the annealing process progresses throughout the bridge on a length scale of more\nthan oneµm leading to an undisturbed lattice with only very few defects. The span of the\nbridge typically contains a single crystal defect. To the best of our knowledge, until now\nthis kind of long range crystallization process throughout a 3D nanostructure has not been\nreported. The damping does not reach the record values of low temperature grown YIG\nlayers but is still in the range of high quality PLD grown YIG films. The minimum line\nwidth of 140 µT at 8 GHz for a single bridge is well in the range of high quality thin film\nmaterial and various resonant magnon modes can be identified in scanning TR-MOKE. Both\nline width and damping thus rival those obtained for large area thin films deposited at higher\ntemperatures. The mechanical resonances of the YIG bridges have yet to be characterized,\nnevertheless, an estimate of possible resonance frequencies can be given. According to Yang\net al. [29] the resonance frequency of a so called doubly clamped beam which corresponds to\nthe span of our bridge is approximately\nfres≈1.03t\nL2/radicalBigg\nE\nρ(2)\nwith E the Young’s modulus of YIG (2 ×1011Pa)[30] ,ρthe density (5.17 g/cm2)[30] ,t\nthe thickness and Lthe length of the beam. Using these parameters with a thickness of 150\nnm and a length of 1 µm a resonance frequency of 964 MHz is expected while the same beam\nwith a length of 500 nm resonates at 3.86 GHz, which is well in the range of typical magnons\nas measured in our experiments. For further development of the method the next steps will\nbe to fabricate more complex resonators that can not only host magnons with high quality\nfactors but are also suitable for the characterization of mechanical vibrational modes. In\naddition statistics need to be obtained by TR-MOKE on the variation and reproducibility\nof resonance frequencies in nominally identical resonators, which are crucial for applications\nwhere the exact behavior needs to be predictable.\nPossible applications for the nanoresonators can be found in various areas. Spin cavit-\nronics for example investigates strong coupling of cavity resonator modes to magnon modes\nin macroscopic magnetic samples (typically YIG). In these experiments current technology\n10uses large volume YIG samples coupled to macroscopic planar superconducting microwave\nresonators [31] or large cavities [32] . Our YIG nano resonators might be deposited over\nmicron sized superconducting coplanar waveguides allowing for more complex experiments.\nIn spin caloritronics YIG bridges may be used to create large and extremely and well de-\nfined temperature gradients because the span of the bridge is thermally decoupled from the\nsubstrate. It should be noted that a temperature difference of only 1 K over a bridge with\na length of 1 µm corresponds to a temperature gradient of 106K/m. If coupling between\nphonons and magnons in the nanoresonators can be established even an application for read-\nout of qubits or conversion of quantum information between the microwave and the optical\nregime may be possible. Therefore the new technology platform presented here may pave\nthe way for downscaling allowing these schemes to be realized on the micron scale or below\nand facilitating future integration of qubits.\nVII. METHODS\nA. Electron beam lithography\nThe pillars and the span of the bridge are exposed using PMMA as a resist and two\ndifferent respective acceleration voltages. The span is exposed at 2.8 kV while the acceler-\nation voltage for the span is 4.5 kV. For both exposures the area dose is 100 µC/cm2. The\nstructures are developed for 60 s in isopropanol.\nB. Pulsed laser deposition of YIG\nThe YIG is deposited in 0.025 mbar of oxygen from a home-made target. Laser parameters\nare 248 nm wavelength, fluence of 2.5 J cm−2, and a repetition rate of 5 Hz. Annealing is\nperformed in an oxygen atmosphere (99.997%) at ambient pressure and 800◦C for 3 hours.\nC. TEM preparation\nTEM samples from bridges are prepared using a focused gallium ion beam âĂŸFEI\nVERSA 3DâĂŹ dual beam microscope by the classical FIB in-situ lift-out technique as\ndescribed for instance by Bals et al. [33] . Due to the electrically isolating substrate this pro-\n11cedure is extended for the preparation of the sample after thermal treatment by depositing\na thin conductive carbon layer via ion sputtering before transferring the sample to the FIB.\nAs the first step in the preparation procedure inside the FIB a 200 nm thick carbon layer is\ndeposited locally using the electron beam at 5 kV from the top through the bridge to fill the\nspace under the bridge with carbon. The hole under the bridge is filled by locally cracking\nthe organometallic complex gas from the platinum Gas Injection System of the FIB with a\n5 kV electron beam. After lift-out the TEM lamellae are mounted to a grid, thinned down\nto a thickness below 150 nm, and stepwise cleaned on both sides from amorphous material\nby operating the ion beam of the FIB at 5 kV, 2 kV and 1 kV. HRTEM images from these\nsamples are obtained using a JEOL JEM-4010 TEM operated at 400 kV.\nD. TR-MOKE\nFor the time resolved magneto optic Kerr (TR-MOKE) measurements we use a frequency\ndoubled fs-laser operating at 520 nm to illuminate the sample in a scanning optical micro-\nscope with polarization analysis. A detailed description of this method is presented in the\nwork of Farle et al.[34] . In our TR-MOKE measurements the magnetization is excited by\ncontinuous wave microwave magnetic field which is phase synchronized to the optical probe\npulses, i.e. the sampling is stroboscopic. In order to allow for lock-in amplification of the\nmagneto-optical signal the rf-excitation is modulated[35, 36] . The spatial resolution of the\nmeasurements presented in this manuscript is diffraction-limited to about 300 nm.\nE. Micromagnetic simulation\nThe simulations were carried out using MuMax3. The simulated structure is a bridge\nwith a rectangular span of 2700 nm x 600 nm x 110 nm (l x w x h). The pillars are 300 nm\nx 600 nm x 110 nm. 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Heinrich, and C. H. Back, Physical Review Letters 99, 246603\n(2007).\n[36] J. Stigloher, M. Decker, H. S. K¨ orner, K. Tanabe, T. Moriyama, T. Taniguchi, H. Hata,\nM. Madami, G. Gubbiotti, K. Kobayashi, T. Ono, and C. H. Back, Physical Review Letters\n117, 037204 (2016), arXiv:1606.02895 [cond-mat.mes-hall].\n15FIG. 1. Schematic drawing of the patterning process. (a) A resist (green) is exposed with two\ndifferent acceleration voltages. A low voltage exposure is used for the span of the bridge (yellow) and\na high acceleration voltage (red) exposes the pillars down to the substrate. (b) After development\nthe void in the resist has the shape of the bridge and a slight undercut which later facilitates the\nlift-off. (c) The YIG is deposited and the shape of the bridge becomes visible. It is important\nthat the YIG on the resist surface is well separated from the bridge itself. (d) After lift-off a\nfree-standing bridge is obtained.\n16FIG. 2. SEM images of two different bridges. (a) and (b) show a larger bridge before and after\nannealing , respectively. (c) shows a smaller bridge after annealing. The deposited YIG has a\nnominal thickness of 110 nm.\n17FIG. 3. Transmission electron micrographs for bridges with a nominal thickness of the span of\n110 nm. (a) Shows a bridge with a span of approximately 850 nm length and a height of 75\nnm underneath the span. (b) Higher magnification shows single crystalline material with a single\ndefect in the center of the bridge. (c) TEM cross section of a bridge with increased length. Even\nfor a length of 2.8 µm the bridge is free of defects except for the central defect. Above and below\nthe bridge a carbon film is visible which has been deposited using the electron beam during TEM\npreparation to protect the surface of the bridge.\n18FIG. 4. Fast Fourier transforms of different parts of the lattice of a single bridge. (a) shows a TEM\nimage of a bridge with three different square areas color coded in red, green, and blue. (b) shows\nan FFT of the substrate. (c) For the color coded areas the FFT of the lattice is superimposed\nusing the same color code. A zoom into the superposition (frame) shows that a very small rotation\nof the lattice has taken place which is in the range of ∼1◦. All FFTs are obtained from images\nwith the same orientation and magnification.\n19FIG. 5. Time-resolved scanning Kerr microscopy (TR-MOKE) images of standing spin-wave modes\nand simulations. The top row shows TR-MOKE results for the main mode (b), one Damon\nEshbach mode (a), and three different backward volume modes (c-e). Measurement parameters are\n(magnetic field/excitation frequency) 11.96 mT/2 GHz (a), 21.95 mT/2 GHz (b), 25.61 mT/2 GHz\n(c), 89.72 mT/4 GHz (d), and 92.52 mT/4 GHz (e). The modes were imaged at the peak amplitude\nof the respective resonance. The bottom row shows the corresponding simulation results from\nsimulations at fixed respective magnetic fields (see also methods section). Simulation parameters\nare 19.4 mT/2.32 GHz (a), 19.4 mT/2.00 GHz (b), 19.4 mT/1.85 GHz (c), 83.8 mT/3.73 GHz (d),\nand 83.8 mT/3.66 GHz (e). The coordinate system on the left hand side shows the orientation of\nthe external magnetic field H 0.\n20FIG. 6. Resonance frequency plotted as a function of applied magnetic field. The results nicely\nagree except for small deviations at low magnetic fields. The red circles show the measured data\nwhile the blue line is the respective fit using the Kittel formula.\n21FIG. 7. TRMOKE measurement of the static switching behavior. While sweeping the field through\nthe static hysteresis the magnitude of the rf-suceptibility is determined as a function of the applied\nmagnetic field. The hysteretic part of the measurement represents the hysteresis of the static\nswitching of the magnetization.\n22FIG. 8. (a) FMR spectrum obtained by TR-MOKE at the center of a single bridge, excited at 8\nGHz. The red circles show the measured data while the blue line is a fit using three Lorentzian\nline shapes. The arrows are a guide to the eye showing an upper limit for the full width at\nhalf maximum which is 2 µ0∆H. The half width at half maximum µ0∆H is mostly referred to in\nliterature as the line width. The measurement which is performed on a single spot with a diameter\nof approx. 300 nm shows two very sharp lines with a small overlap. The line width µ0∆H is\nsmaller than 140 µT. The insert shows a sketch of an array of bridges located between signal line\nand ground of a CPW. (b) Line width plotted versus frequency. A least mean square fit yields a\nslope corresponding to a Gilbert damping of (2.6 ±0.7)−4.\n23FIG. 9. Two TR-MOKE images obtained at a frequency of 2 GHz showing five adjacent bridges\nat two different magnetic fields, respectively. In both images at least one of the bridges shows an\nintense resonance of the mode with one antinode only. Apparently the resonance field between\nbridges can vary at least by 0.8 mT.\n24FIG. 10. Optical topography image of several bridges (a) and two TRMOKE images of the same\narea acquired at a frequency of 6 GHz at different respective magnetic fields (b-c). In the topogra-\nphy image we can clearly discern the span of the bridge, the base which is slightly darker and the\noverhang at the end. (b) Shows the main resonance of the span with one antinode while in (c) a\nsimilar mode for the overhang of the same bridge is observed. The resonance fields differ by 7 mT.\n25FIG. 11. SEM images of more complex resonators. The process allows to fabricate various shapes\nsuch as open squares (a) or disks and triangles (b). (c) TR-MOKE image of a standing backward\nvolume mode measured on a disk resonator overlayed to an SEM image of the same structure. For\nthese structures the nominal YIG thickness is 210 nm. (d) shows a bridge on which a post-annealing\nwet-etch was applied. The artifacts at the seam of the structure are strongly reduced.\n26"    },    {        "title": "1802.03865v3.Spin_orbit_torque_and_spin_pumping_in_YIG_Pt_with_interfacial_insertion_layers.pdf",        "content": "Spin-orbit torque and spin pumping in YIG/Pt with interfacial insertion layers\nSatoru Emori,1, 2,a)Alexei Matyushov,1, 3Hyung-Min Jeon,4, 5Christopher J. Babroski,1Tianxiang Nan,1, 6Amine\nM. Belkessam,1John G. Jones,4Michael E. McConney,4Gail J. Brown,4Brandon M. Howe,4and Nian X. Sun1\n1)Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115,\nUSA\n2)Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n3)Department of Physics, Northeastern University, Boston, MA 02115, USA\n4)Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, OH 45433,\nUSA\n5)Department of Electrical Engineering, Wright State University, Dayton, OH 45435,\nUSA\n6)Department of Materials Science and Engineering, University of Wisconsin-Madison, WI 53706,\nUSA\n(Dated: April 27, 2018)\nWe experimentally investigate spin-orbit torque and spin pumping in Y 3Fe5O12(YIG)/Pt bilayers with ul-\ntrathin insertion layers at the interface. An insertion layer of Cu suppresses both spin-orbit torque and spin\npumping, whereas an insertion layer of Ni 80Fe20(permalloy, Py) enhances them, in a quantitatively consistent\nmanner with the reciprocity of the two spin transmission processes. However, we observe a large enhance-\nment of Gilbert damping with the insertion of Py that cannot be accounted for solely by spin pumping,\nsuggesting signi\fcant spin-memory loss due to the interfacial magnetic layer. Our \fndings indicate that the\nmagnetization at the YIG-metal interface strongly in\ruences the transmission and depolarization of pure spin\ncurrent.\nThe transmission of pure spin current between a mag-\nnetic insulator and a normal metal is a crucial aspect\nof emerging insulator spintronic devices1,2. Yttrium iron\ngarnet (Y 3Fe5O12, YIG) is an especially promising mag-\nnetic insulator because of its exceptionally low Gilbert\ndamping that allows for e\u000ecient excitation of magne-\ntization dynamics3{5. This magnetic damping can be\nmodi\fed by spin-orbit torque6,7in thin-\flm YIG due\nto absorption of pure spin current8{12, which is gen-\nerated from an electric current in the adjacent metal\n(e.g., Pt) through the spin-Hall e\u000bect13. In the recip-\nrocal process of spin pumping14,15, coherent magnetiza-\ntion dynamics in YIG injects a pure spin current into\nthe metal layer, which can be detected through an en-\nhancement in Gilbert damping16{18or a voltage peak due\nto the inverse spin-Hall e\u000bect19{25. The reciprocity of\nspin-orbit torque and spin pumping is theoretically well\nestablished26. However, while prior reports have shown\nthat various modi\fcations at the YIG-metal interface im-\npact spin pumping (or, more generally, spin transmission\nfrom the YIG to metal layer)17,18,21,22,27{29, how spin-\norbit torque (i.e., spin transmission from the metal to\nthe YIG layer) is a\u000bected by such interfacial modi\fca-\ntions has yet to be reported.\nIn this Letter, we investigate spin-orbit torque and\nspin pumping in the same set of YIG/Pt samples { with\nand without an ultrathin interfacial insertion layer { by\nferromagnetic resonance (FMR) in a microwave cavity.\nThe two spin transmission processes are suppressed with\na nonmagnetic Cu insertion layer and enhanced with a\nmagnetic Ni 80Fe20(permalloy, Py) insertion layer. We\na)Electronic mail: semori@vt.edualso \fnd evidence for substantial spin-memory loss30with\nthe insertion of ultrathin Py. Our \fndings are consistent\nwith the reciprocity of spin-orbit torque and spin pump-\ning, while revealing that the magnetization at the YIG-\nmetal interface has a signi\fcant impact on the transmis-\nsion and scattering of spin current.\nEpitaxial 20-nm thick YIG \flms were grown on\nGd3Ga5O12(111) substrates by pulsed laser deposition\nas reported in Ref. 3. The YIG \flms were transferred\nthrough ambient atmosphere to a separate deposition\nsystem for the growth of the metal overlayers. The YIG\nsamples were sonicated in acetone and ethanol and, after\nintroduction into the deposition chamber, maintained at\n250\u000eC at 50 mTorr O 2for 30 minutes to remove water\nand organics on the surface. The metal overlayers (either\nPt(5 nm), Cu(0.5 nm)/Pt(5 nm), or Py(0.5 nm)/Pt(5\nnm)) were deposited by dc magnetron sputtering at room\ntemperature, base pressure of <\u00182\u000210\u00007Torr, and Ar\nsputtering pressure of 3 mTorr. While the RMS surface\nroughness of the epitaxial YIG \flms is only <\u00180.15 nm\n(consistent with Ref. 3), the nominally 0.5-nm thick Cu\nand Py \\dusting\" layers may not be continuous. Each\nYIG/X/Pt sample (with X = none, Cu, or Py) was pat-\nterned into a 100- \u0016m wide, 1.5-mm long strip by pho-\ntolithography and ion milling. The strip was contacted\nby Cr/Au pads on either end by photolithography, sput-\nter deposition, and lifto\u000b. This sub-mm wide strip ge-\nometry31allows for the use of a cavity electron paramag-\nnetic resonance spectrometer to measure both spin-orbit\ntorque and spin pumping.\nWe \frst demonstrate the transmission of spin current\nfrom the metal layer to the YIG layer through the mea-\nsurement of the damping-like32spin-orbit torque. We\nused a method similar to Refs. 6, 31 where the change inarXiv:1802.03865v3  [cond-mat.mtrl-sci]  29 Apr 20182\n-1.0 0.0 1.00.450.500.55\nJdc (1010 A/m2)\n  W (mT) H||+y\n H||-y\n-1.0 -0.5 0.0 0.5 1.0-0.8-0.40.00.40.8\nYIG/Pt\n  eff (10-4)\nJdc (1010 A/m2)\n-1.0 -0.5 0.0 0.5 1.0-0.8-0.40.00.40.8\nYIG/Cu/Pt\n  eff (10-4)\nJdc (1010 A/m2)\n-1.0 -0.5 0.0 0.5 1.0-0.8-0.40.00.40.8\nYIG/Py/Pt\n  eff (10-4)\nJdc (1010 A/m2)\n238.0 238.5 239.0 239.5H||+y\n  dIFMR/dH (a.u.)\n0H (mT)Jdc= +3 mA\nJdc= -3 mA(a) (b) (c)\n(d) (e) (f)JdchrfH\nSOTPt\nX\nYIGy x\nFigure 1. (a) Schematic of spin-orbit torque (SOT) generated by a dc current Jdcin YIG/(X/)Pt. (b) Jdc-induced modulation\nof FMR spectra in YIG/Pt. The direction of Jdcis as de\fned in (a). (c) Jdc-induced change in FMR linewidth Wwith bias\nmagnetic \feld applied along the + yand -ydirections as de\fned in (a). (d-f) Change in the e\u000bective Gilbert damping parameter\n\u000be\u000bwithJdcfor (d) YIG/Pt, (e) YIG/Cu/Pt, and (f) YIG/Py/Pt. The lines indicate linear \fts to the data.\ndamping is monitored as a function of dc bias current,\nIdc. FMR spectra were measured in a rectangular TE 102\nmicrowave cavity with a nominal excitation power of 10\nmW and several values of Idcin the metallic layer as il-\nlustrated in Fig. 1(a). Each spectrum was \ft with the\nderivative of the sum of symmetric and antisymmetric\nLorentzians (e.g., Fig. 1(b)) to extract the half-width-at-\nhalf-maximum linewidth, W.\nFigure 1(c) shows the variation of WwithIdcunder\nopposite transverse external magnetic \felds, H. The\ndata contain components that are odd and even with\nrespect to Idc, which are due to the spin-orbit torque\nand Joule heating, respectively6,31. The symmetry of the\nspin-Hall spin-orbit torque also gives rise to a component\nofWversusIdcthat is odd with respect to H(Refs. 6{\n8, 31), extracted through \u0001 Wodd(Idc) =fW(Idc;+jHj)\u0000\nW(Idc;\u0000jHj)g=2. We can then obtain the linear change\nin the e\u000bective Gilbert damping parameter due to the\ndc spin-orbit torque, \u0001 \u000be\u000b=j\rj\u0001Wodd=(2\u0019f), where\nj\rj=(2\u0019) = 28 GHz/T and f= 9:55 GHz.\nFrom the linear slope of \u0001 \u000be\u000bover the dc current den-\nsityJdc=Idc=(wtPt) (Fig. 1(d)-(f)), with w= 100\u0016m\nandtPt= 5 nm33, the e\u000bective spin-Hall angle, \u0012e\u000b, can\nbe quanti\fed from7\n\u0012e\u000b=2jej\n~\u0012\nH+Meff\n2\u0013\n\u00160MstYIG\f\f\f\f\u0001\u000be\u000b\nJdc\f\f\f\f;(1)\nwhereMs= 130 kA/m is the saturation magnetization,\nMe\u000b= 190 kA/m is the e\u000bective magnetization including\nthe out-of-plane uniaxial anistropy \feld3, andtYIG= 20\nnm is the thickness of the YIG layer. By \ftting the data\nin Fig. 1(d) with Eq. 1, we arrive at \u0012e\u000b= 0:76\u00060:05%for YIG/Pt.\nWe note that \u0012e\u000bis the product of the intrinsic spin-\nHall angle of Pt, \u0012Pt, and the interfacial spin current\ntransmissivity, T. Assuming that tPtis su\u000eciently larger\nthan the spin di\u000busion length, \u0015Pt, the expression for \u0012e\u000b\nis34{36\n\u0012e\u000b=T\u0012Pt\u00192Ge\u000b\n\"#\u0015Pt\u001aPt\u0012Pt; (2)\nwhereGe\u000b\n\"#is the e\u000bective spin-mixing conductance\n(which includes the spin back\row factor) and \u001aPt\u0019\n4:0\u000210\u00007\n m is the measured resistivity of the Pt layer.\nWith\u0015Pt\u001aPt\u0019(0:6\u00000:8)\u000210\u000015\nm2(Refs. 30, 37, 38),\nwe estimate \u0015Ptto be\u00191.5-2 nm.\nAccording to Eq. 2, the small \u0012e\u000bin our YIG/Pt can\nbe attributed to a reduced T(i.e.,Ge\u000b\n\"#) at the YIG-Pt\ninterface, which may be due to a residual carbon agglom-\neration on the YIG surface39that was not removed by our\ncleaning protocol. In particular, by taking \u0012Pt\u001915\u000030%\nreported from prior spin-orbit torque studies34{36, we ob-\ntain for our YIG/Pt bilayer T\u00190:03\u00000:05, orGe\u000b\n\"#\u0019\n(2\u00005)\u00021013\n\u00001m\u00002, which is an order of magnitude\nlower than the typical values reported for ferromagnetic-\nmetal/normal-metal heterostructures15,34{36, although it\nis comparable to prior reports on YIG/Pt16,20.\nFor YIG/Cu/Pt (Fig. 1(e)), we do not detect a spin-\norbit torque within our experimental resolution, i.e.,\n\u0012e\u000b= 0:01\u00060:10%. Evidently, the Cu dusting layer\nat the YIG-Pt interface suppresses the transmission of\nspin current. By contrast, the Py dusting layer en-\nhances spin transmission from Pt to YIG by \u001940%, with\n\u0012e\u000b= 1:08\u00060:06% derived from the data in Fig. 1(f).\nThe spin-orbit torque experiment thus suggests that the3\ny x\n-3-2-10123-20-1001020\nm0(H-HFMR) (mT)\n  VISH (mV)\n-6-4-20246\nH||-yH||+y YIG/Pt\nVISHW2 (mV mT2)\n-3-2-10123-6-3036\nm0(H-HFMR) (mT)H||-yH||+y YIG/Cu/Pt\n  VISH (mV)\n-0.4-0.20.00.20.4\nVISHW2 (mV mT2)\n-3-2-10123-8-4048\nH||-yH||+y YIG/Py/Pt\n  VISH (mV)\nm0(H-HFMR) (mT)-8-6-4-202468\nVISHW2 (mV mT2)(a) (b) (c) (d)\nhrfH\nFigure 2. (a) Schematic of electrically detected spin pumping in YIG/(X/)Pt. (b-d) Inverse spin-Hall voltage VISHspectra\nmeasured for (b) YIG/Pt, (c) YIG/Cu/Pt, and (d) YIG/Py/Pt. The right vertical axis show VISHscaled by the square of the\nFMR linewidth W, which is proportional to the transmission e\u000eciency of spin current from YIG to Pt.\nnonmagnetic and magnetic insertion layers have opposite\ne\u000bects on spin current transmissivity (Eq. 2).\nIn addition to spin-orbit torque, we show that the mod-\ni\fcation of the YIG-Pt interface equally impacts the re-\nciprocal process of spin pumping. The same sub-mm\nwide YIG/X/Pt strips are measured in the setup identi-\ncal to the spin-orbit torque experiment, except that the\ndc wire leads were connected to a nanovoltmeter, instead\nof a dc current source. As illustrated in Fig. 2(a), FMR\nin the YIG layer pumps a spin current into the Pt layer,\nin which the inverse spin-Hall e\u000bect converts the spin\ncurrent to a charge current that is detected through a\nvoltage peak, VISH, coinciding with FMR. Figure 2(b)-\n(d) shows the VISHspectra obtained at 10 mW of rf ex-\ncitation. The reversal of the voltage polarity with the H\ndirection is consistent with the symmetry of the inverse\nspin-Hall e\u000bect.\nIn the limit of tPtsu\u000eciently larger than \u0015Pt, the re-\nlationship between the peak magnitude of VISHand\u0012e\u000b\nis given by40\njVISHj\u0019h\n2jej\u0012e\u000bfPL\ntPt\u00022; (3)\nwhereL= 1500\u0016m is the length of the sample, P= 1:26\nis the precession ellipticity factor, and \u0002 is the preces-\nsion cone angle. It should be noted that these three\nYIG/X/Pt samples undergo precession at di\u000berent cone\nangles, given by \u0002 = \u00160hrf=W(Refs. 19, 41), since their\nlinewidths Ware di\u000berent. Due to the lack of direct\ncalibration for the microwave \feld amplitude hrfin our\nsetup, the absolute magnitudes of \u0012e\u000bcannot be de-\ntermined accurately from the spin pumping experiment\n(Eq. 3)42.\nNevertheless, we can compare the relative magnitudes\nof\u0012e\u000bamong the three samples. Speci\fcally, we scale\nVISHbyW2(/\u0002\u00002), as shown on the right vertical axis\nof Fig. 2(b)-(d), to quantify the e\u000eciency of spin-current\ntransmission from YIG to Pt. Comparing Fig. 2(c) with\nFig. 2(b), the Cu dusting layer reduces the spin trans-\nmission e\u000eciency ( /VISHW2) by an order of magnitude.\nBy contrast, comparing Fig. 2(d) with Fig. 2(b), the\nPy dusting layer enhnaces the transmission e\u000eciency by\u001940%. This suppression (enhancement) of spin transmis-\nsion with the Cu (Py) insertion layer in the spin pump-\ning experiment quantitatively agrees with the spin-orbit\ntorque experiment, as summarized in Table I. These re-\nsults thus corroborate the reciprocity of the two spin-\ncurrent transmission processes between YIG and Pt.\nWe have revealed that the ultrathin dusting layer of\nnonmagnetic Cu at the YIG-Pt interface suppresses spin\ntransmission, whereas the ferromagnetic Py dusting layer\nenhances it. Our experimental results are qualitatively\nconsistent with the \frst-principles calculations by Jia et\nal.43, which report that the spin-mixing conductance at\nthe YIG-metal interface depends on the interfacial mag-\nnetic moment density. With the ultrathin insertion layer\nof Cu (Py) decreasing (increasing) the interfacial mag-\nnetization, Ge\u000b\n\"#and hence\u0012e\u000bdecrease (increase) as de-\nscribed by Eq. 2. Moreover, the enhancement of spin\ntransmission between YIG and Pt with an ultrathin fer-\nromagnetic insertion layer, quantitatively similar to our\nresults, has been observed in a spin-Seebeck e\u000bect ex-\nperiment by Kikuchi et al.29. We further note that al-\nthough bulk Pt is paramagnetic, it is close to ful\flling the\nStoner criterion such that the direct interface of YIG/Pt\nmay accommodate a higher interfacial magnetic moment\ndensity44,45than YIG/Cu/Pt.\nThe large reduction of spin-orbit torque and spin\npumping with the ultrathin Cu insertion layer may seem\nunexpected, considering that this insertion layer is much\nthinner than the typical spin di\u000busion length of Cu\n(\u0015Cu>100 nm)46. Indeed, prior spin pumping exper-\niments report only a modest decrease (by \u001810%) in spin-\ncurrent transmission between YIG and Pt when the Cu\nspacer thickness is \u00191 nm18,22. However, spin pump-\ning22and spin-Hall magnetoresistance47studies have\nshown that spin transmission decreases by an order-of-\nmagnitude with the insertion of a Cu spacer layer, even\nwhen its thickness (e.g., \u00195 nm) is much smaller than\n\u0015Cu. Other studies also indicate large spin-memory loss\nat the Cu-Pt interface48,49, although we do not observe a\nsigni\fcant increase in spin dissipation (Gilbert damping)\nin YIG/Cu/Pt compared to uncapped YIG, as shown be-\nlow. While further studies are required to understand the\nroles of the Cu spacer layer, one possibility is that spin4\ntransmission is highly sensitive to the nature of the YIG-\nmetal interface, such as the morphology of the ultrathin\nCu layer and the presence of carbon agglomeration39.\nTo gain complementary insight into interfacial spin-\ncurrent transmission, we have examined the enhancement\nof Gilbert damping in YIG/X/Pt strips compared to un-\ncapped YIG \flms. Fig. 3 summarizes the frequency de-\npendence of W, acquired with a broadband FMR setup,\nfrom which the Gilbert damping parameter, \u000b, is quanti-\n\fed. The averaged Gilbert damping parameter for three\nuncapped YIG \flms is \u000b= (4:4\u00060:6)\u000210\u00004, which is\nwithin the range reported by our earlier work3.\nWe observe an increase in \u000bfor each YIG/X/Pt com-\npared to uncapped YIG. Assuming that the damping in-\ncrease is exclusively due to spin pumping, the spin-mixing\nconductance is given by14,15,\nGe\u000b\n\"#=2e2MstYIG\n~2j\rj\u0001\u000b; (4)\nwhere \u0001\u000b(summarized in Table I) is the di\u000berence be-\ntween\u000bof YIG/X/Pt and uncapped YIG. From Eq. 4, we\n\fndGe\u000b\n\"#= (3:3\u00060:5)\u00021013\n\u00001m\u00002for YIG/Pt, which\nis in quantitative agreement with the estimated Ge\u000b\n\"#from\nEq. 2. We also obtain Ge\u000b\n\"#= (0:6\u00060:5)\u00021013\n\u00001m\u00002\nfor YIG/Cu/Pt, which again corroborates the one-order-\nof-magnitude reduction in spin transmission with the ul-\ntrathin Cu insertion layer. Therefore, our experimental\nresults of spin-orbit torque (Fig. 1), electrically detected\nspin pumping (Fig. 2), and Gilbert damping enhance-\nment (Fig. 3) are consistent with each other for YIG/Pt\nand YIG/Cu/Pt.\nThe Gilbert damping enhancement, \u0001 \u000bfor\nYIG/Py/Pt is\u00194 times greater than that for YIG/Pt.\nThis observation is at odds with our \fndings from the\nspin-orbit torque and spin pumping experiments, which\nshow thatGe\u000b\n\"#(i.e., \u0001\u000baccording to Eq. 4) should be\nonly a factor of\u00191.4 greater for YIG/Py/Pt compared\nto YIG/Pt. We thus estimate that only \u001930% of the\ntotal \u0001\u000bis due to spin pumping in YIG/Py/Pt, such\nthat the adjusted value of Ge\u000b\n\"#is\u00195\u00021013\n\u00001m\u00002. The\nremaining\u001970% of \u0001\u000bis likely due to spin-memory loss,\ni.e., spin depolarization by the ultrathin Py layer that\nincreases the Gilbert damping but does not contribute\nto spin-current transmission from YIG to Pt. This large\nspin-memory loss in YIG/Py/Pt is comparable to reports\non ferromagnetic-metal/Pt heterostructures30,49,50.\nIn summary, we have measured the transmission of\nspin current between YIG and Pt thin \flms, separated by\nan interfacial dusting layer of nonmagnetic Cu or mag-\nnetic Py, through FMR-based spin-orbit torque and spin\npumping experiments. Spin transmission decreases by an\norder of magnitude when ultrathin Cu is inserted at the\nYIG-Pt interface and increases by \u001940 % with the inser-\ntion of ultrathin Py. The quantitatively consistent results\nfrom the spin-orbit torque and spin pumping experiments\ncon\frm the reciprocity of these two processes. However,\nwith the Py insertion layer, the Gilbert damping param-\n0 5 10 15051015YIG/Py/Pt\nYIG/Pt\nYIG/Cu/Pt\nYIGW (mT)\nf (GHz)Figure 3. Frequency dependence of half-width-at-half-\nmaximum FMR linewidth, W.\nTable I. Essential extracted parameters - \u0012SOT\ne\u000b: e\u000bective spin-\nHall angle from the spin-orbit torque experiment; VISHW2:\ne\u000eciency of spin transmission from the electrically detected\nspin pumping experiment; Ge\u000b\n\"#: e\u000bective spin-mixing conduc-\ntance from the enhancement in Gilbert damping (YIG/Py/Pt\nadjusted to account for spin-memory loss); \u0001 \u000b: total en-\nhancement of the Gilbert damping parameter.\nYIG/Pt YIG/Cu/Pt YIG/Py/Pt\n\u0012SOT\ne\u000b(%) 0 :76\u00060:05 0:01\u00060:10 1:08\u00060:06\nVISHW2(\u0016V mT2) 4:0\u00060:2 0:35\u00060:02 5:6\u00060:4\nGe\u000b\n\"#(1013\n\u00001m\u00002) 3:3\u00060:5 0:6\u00060:5 \u00195\n\u0001\u000b(10\u00004) 4 :8\u00060:7 0:9\u00060:7 21 \u00061\neter is much larger than expected from spin pumping,\nsuggesting substantial spin-memory loss in YIG/Py/Pt.\nOur \fndings shed light on the roles of interfacial magneti-\nzation in the transmission and depolarization of spin cur-\nrent between a magnetic insulator and a normal metal.\nAcknowledgments: This work is funded by NSF\nERC TANMS 1160504, AFRL through contract FA8650-\n14-C-5706, and by the W.M. 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However, because of the\nlarge quadratic change of HFMR withIdc(\u00190.3 mT/mA2) due\nto thermal e\u000bects, it is di\u000ecult to reliably extract the linear Idc-\ninduced shift of HFMR. The upper bound of the linear HFMR\nshift is 0.01 mT per mA of Idc, about a factor of \u00191.5 greater\nthan the expected Oersted \feld of \u00160jIdcj=(2w) = 0:0063 mT per\nmA, in the direction of the Oersted \feld for all samples.\n33For simplicity, we assume that all of the dc bias current \rows\nthrough the Pt layer. This simplifying assumption is supported\nby the resistance ( R\u00191200 \n) of YIG/Pt, YIG/Cu/Pt, and\nYIG/Py/Pt being within a few % of each other.\n34T. Nan, S. Emori, C. T. Boone, X. Wang, T. M. Oxholm, J. G.\nJones, B. M. Howe, G. J. Brown, and N. X. Sun, Phys. Rev. B\n91, 214416 (2015).\n35C.-F. Pai, Y. Ou, L. H. Vilela-Le~ ao, D. C. Ralph, and R. A.\nBuhrman, Phys. Rev. B 92, 064426 (2015).\n36W. Zhang, W. Han, X. Jiang, S.-H. Yang, and S. S. P. Parkin,\nNat. Phys. 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Lett. 110, 147207\n(2013).\n45T. Kikkawa, M. Suzuki, J. Okabayashi, K.-i. Uchida, D. Kikuchi,\nZ. Qiu, and E. Saitoh, Phys. Rev. B 95, 214416 (2017).\n46T. Kimura, J. Hamrle, and Y. Otani, Phys. Rev. B 72, 014461\n(2005).\n47H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kaji-\nwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi,\nR. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh,\nPhys. Rev. Lett. 110, 206601 (2013).\n48J. Bass and W. P. Pratt, J. Phys. Condens. Matter 19, 183201\n(2007).\n49K. Dolui and B. K. Nikoli\u0013 c, Phys. Rev. B 96, 220403 (2017).\n50A. J. Berger, E. R. J. Edwards, H. T. Nembach, O. Karis,\nM. Weiler, and T. J. Silva, (2017), arXiv:1711.07654."    },    {        "title": "1802.05548v1.Damping_s_effect_on_the_magnetodynamics_of_spin_Hall_nano_oscillators.pdf",        "content": "Damping's e\u000bect on the magnetodynamics of spin Hall nano-oscillators\nYuli Yin,1, 2,\u0003Philipp D urrenfeld,2Mykola Dvornik,2Martina\nAhlberg,2Afshin Houshang,2Ya Zhai,1and Johan \u0017Akerman2, 3\n1Department of Physics, Southeast University, 211189 Nanjing, China\n2Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden\n3Department of Materials and Nano Physics, School of Information and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden\n(Dated: July 27, 2021)\nWe study the impact of spin wave damping ( \u000b) on the auto-oscillation properties of nano-\nconstriction based spin Hall nano-oscillators (SHNOs). The SHNOs are based on a 5 nm Pt layer\ninterfaced to a 5 nm Py 100\u0000x\u0000yPtxAgymagnetic layer, where the Pt and Ag contents are co-varied\nto keep the saturation magnetization constant (within 10 %), while \u000bvaries close to a factor of\nthree. We systematically investigate the in\ruence of the Gilbert damping on the magnetodynamics\nof these SHNOs by means of electrical microwave measurements. Under the condition of a constant\n\feld, the threshold current scales with the damping in the magnetic layer. The threshold current as\na function of \feld shows a parabolic-like behavior, which we attribute to the evolution of the spatial\npro\fle of the auto-oscillation mode. The signal linewidth is smaller for the high-damping materials\nin low magnetic \felds, although the lowest observed linewidth was measured for the alloy with least\ndamping.\nPACS numbers: 75.70.-i, 76.50.+g, 75.78.-n\nINTRODUCTION\nSpin Hall nano-oscillators (SHNO) are spintronic de-\nvices in which magnetization oscillations are induced by\npure spin currents [1]. These pure spin currents can be\nexperimentally realized via the spin Hall e\u000bect (SHE)\nin an adjacent heavy metal layer [2{4] or by non-local\nspin injection [5, 6]. SHNOs, which use the SHE in\na heavy metal layer, have been fabricated in a vari-\nety of device layouts, which all utilize the focusing of\ncharge current into a region with a lateral size of tens\nto hundreds of nanometers. This focusing is commonly\ndone via a nano-gap between two highly conductive elec-\ntrodes [3, 7, 8], with a nanoconstriction [9{13], or with\na nanowire [14, 15]. Most recently, nanoconstriction-\nSHNOs have attracted large interest, due to their rel-\native ease of fabrication, their direct optical access to\nthe magnetization oscillation area, and their potential for\nlarge scale and large distance synchronization of multiple\nSHNOs [16, 17].\nNanoconstriction-SHNOs consist of a bilayer of a fer-\nromagnetic free layer and a SHE inducing heavy metal\nlayer. Since the SHE and the concomitant spin accumu-\nlation at the bilayer interface are only in\ruenced by the\ncurrent density in the heavy metal layer, magnetization\noscillations of the device under a constant current can\nbe directly linked to the magnetodynamic properties of\nthe magnetic free layer. Until now, the variety of materi-\nals from which SHNOs has been fabricated is limited to a\nfew standards like permalloy (Py, Ni 80Fe20), (Co,Fe)B, or\nyttrium iron garnet (YIG). However, these materials are\ndi\u000berent from each other in every one of the key magneto-\ndynamic parameters, such as magnetization ( M), Gilbertdamping (\u000b), or exchange constant ( A).\nIn a recent study, we have shown how the magneto-\ndynamic properties of Py can be engineered by alloying\nwith the noble metals Pt, Au, and Ag [18]. While alloy-\ning with Pt leads to a large increase in damping but only\na small decrease in magnetization, alloying with Ag has\nonly a weak e\u000bect on the damping but reduces the mag-\nnetization relatively strongly. Co-alloying with both ele-\nments Pt and Ag thus results in Py 100\u0000x\u0000yPtxAgy\flms,\nwhoseMand\u000bcan be tuned independently, e.g. the\nmagnetization can be kept constant, while the damping\nis strongly increased with increasing Pt concentration.\nHere, we employ a series of alloyed Py 100\u0000x\u0000yPtxAgy\nthin \flms in nanoconstriction-SHNOs, where we vary the\ne\u000bective damping of the free layer by a factor of three,\nwhile we keep the magnetization of the \flms constant.\nBased on these \flms, we fabricate geometrically identical\nnanoconstriction-SHNOs and compare their microwave\nauto-oscillation characteristics. This allows us to directly\nanalyze the in\ruence of one single magnetodynamic prop-\nerty, namely the Gilbert damping, on the spectral char-\nacteristics, i.e. the onset current ( Ith\nDC), the output power\n(P), and the linewidth (\u0001 f).\nSPIN HALL NANO-OSCILLATOR DEVICES\nBilayers of 5 nm Py 100\u0000x\u0000yPtxAgyand 5 nm Pt were\nsputter-deposited onto sapphire substrates in a high-\nvacuum chamber with a base pressure of less than\n3\u000210\u00008Torr. The deposition was carried out with\n3 mTorr argon gas at a \row rate of 30 sccm. The alloyed\nlayers were co-sputtered from up to 3 targets, and the\nPy target power was kept constant at 350 W, while thearXiv:1802.05548v1  [cond-mat.mes-hall]  15 Feb 20182\n150 nm\n200 nm\n(a) (c)\n(b)\nHIPCurrent\nFIG. 1. (a) Schematic representation of the sputtered bilayer\nstructure. (b) SEM micrograph of a nanoconstriction-SHNO\nshowing the relative orientations of current and \feld. (c) Op-\ntical micrograph showing the microwave wave guide used for\ncontacting the SHNOs.\nnoble metal sputtering powers and the sputtering time\nwas adjusted for composition and thickness, respectively.\nThe top Pt layer was magnetron sputtered with a dc\npower of 50 W. The alloy compositions are Py 84Ag16\n(S01), Py 77:5Pt10Ag12:5(S02), Py 75Pt15Ag10(S03), and\nPy73Pt19Ag8(S04), chosen to result in a constant satura-\ntion magnetization throughout the series of SHNOs [18].\nDevices for electrical measurements were fabricated\nfrom these bilayers by electron beam lithography and ar-\ngon ion beam etching, using the negative resist as an etch-\ning mask. Nanoconstrictions were formed by two sym-\nmetrical indentations with a 50 nm tip radius into 4 µm\nwide stripes, see Fig. 1(b). The width of the nanocon-\nstrictions is 150 nm. Finally, 1 µm thick copper waveg-\nuides with a 150 µm pitch were fabricated by optical\nlithography and lift-o\u000b, see Fig. 1(c).\nFILM CHARACTERIZATION\nCharacterization of the extended bilayer samples was\nperformed by ferromagnetic resonance (FMR), and two-\npoint anisotropic magneotresistance (AMR) measure-\nments. The FMR was carried out with in-plane applied\n\felds using a NanOsc Instruments PhaseFMR with a\n200µm wide coplanar waveguide (CPW). An asymmet-\nric Lorentzian was \ft to the absorption peaks. The fre-\nquency dependence of the determined resonance \felds\nand linewidths was subsequently used to extract the ef-\nfective magnetization ( \u00160Me\u000b) and the damping param-\n0.0\n0.2\n0.4\n0.6\n0.8\nμ0M eff\nα\n0.02\n0.03\n0.04\n0.05\n0.06\nα  \n0\n5\n10\n15\n20\n0.3\n0.4\n0.5\nAMR (%)\nxPt(%)φ (°)\n0\n90\n180\n270\n360\n0.0\n0.1\n0.2\n0.3\n0.4\nMR (%)\nS01μ0Meff(T) μ0Meff = 0.617 TFIG. 2. (a) Magnetization and damping of the alloyed \flms\nin the bilayer as measured by CPW-based FMR. (b) AMR of\nthe extended layer structure. The inset shows the angular-\ndependent relative resistance of the Py 84Ag16/Pt (S01) bi-\nlayer, together with a \ft to a cos2-function.\neter (\u000b), respectively [18]. Figure 2(a) shows the two\nparameters, \u00160Me\u000band\u000b, as a function of Pt concen-\ntration. The magnetization is constant throughout the\nsample series ( \u00160Me\u000b= 0.617(34) T), while the damp-\ning increases linearly from 0 :023(1) to 0 :058(3) as the\nPt concentration increases from 0 (Py 84Ag16) to 19 %\n(Py73Pt19Ag8). The small layer thickness compared to\nthe \flms in Ref. 18 results in a slightly lower magnetiza-\ntion, whereas the damping is enhanced as a consequence\nof spin pumping into the adjacent Pt layer [19{21].\nThe AMR was determined by probing the resistance\nof 4µm wide stripes in a rotating 90 mT in-plane mag-\nnetic \feld. A representative AMR measurement is pre-\nsented in the inset of Fig. 2(b), together with a \ft of a\ncos2-function to the data. The angle '= 0\u000edenotes a\nperpendicular orientation between current and \feld, and\nthe AMR (Fig. 2(b)) is calculated by the di\u000berence in\nresistance at perpendicular and parallel alignments via\nAMR =Rk\u0000R?\nR?. The AMR is below 1 %, which is a re-\nsult of the majority of the current \rowing through the\nnonmagnetic platinum layer, which has a higher conduc-\ntivity than the Py alloys. The AMR reduces by \u001930 %\nacross the samples series, but the absolute resistance of\nthe bilayers decreases by less than 5 %. The AMR magni-\ntude is therefore most likely governed by the alloy compo-\nsition, since the amount of current in the magnetic layer\ndoes not change signi\fcantly.3\nf(GHz)\nCurrent (mA)\n2.2 2.4 2.6 2.8 3.0 3.2 3.45.96.06.1\n2468S01, H = 500 mT\n5.85 5.95 6.05 6.150246\nf(GHz)PSD (nV2/Hz)\nFIG. 3. Power spectral density (PSD) of the Py 84Ag16/Pt\n(S01) SHNO as a function of current in an external \feld of\n\u00160Hext= 0:5 T, tilted 80\u000eOOP. The inset shows the PSD at\nIDC= 3:26 mA and the solid line is a Lorentzian \ft resulting\nin \u0001f= 5:98 MHz and P= 1:02 pW.\nMICROWAVE EMISSION MEASUREMENTS\nAND DISCUSSION\nThe microwave measurements were conducted with the\ndevices placed in a magnetic \feld oriented at an out-of-\nplane (OOP) angle of 80\u000efrom the \flm plane, and an\nin-plane angle of '= 0\u000e. The in-plane component of the\nmagnetic \feld ( HIP\next) was thus perpendicular to the cur-\nrent \row direction, as sketched in Fig. 1(b). The relative\norientation of the current and HIP\nextyields a spin-torque\ncaused by the spin current from the Pt layer, which re-\nduces the damping in the Py layer and leads to auto-\noscillations for su\u000eciently large positive applied dc cur-\nrents (IDC) [22]. The current was applied to the samples\nvia the dc port of a bias-tee and the resulting microwave\nsignals from the devices were extracted from the rf port of\nthe bias-tee. The microwave signals were then ampli\fed\nby a broadband (0 :1 to 40 GHz) low-noise ampli\fer with\na gain of +32 dB before being recorded by a spectrum an-\nalyzer (Rohde&Schwarz FSV-40) with a resolution band-\nwidth of 500 kHz. All measurements were carried out at\nroom temperature.\nA typical microwave measurement of a Py 84Ag16/Pt\n(S01) device in a constant \feld of \u00160Hext= 0:5 T and\na varying current is displayed in Fig. 3. The peak fre-\nquency \frst decreases slightly after the oscillation onset\natIth\nDC= 2:26 mA, then reaches a minimum at \u00182:6 mA,\nand \fnally increases up to the maximum applied current\nof 3:4 mA. A Lorentzian peak function \fts well to the\nauto-oscillation signal, see inset of Fig. 3, allowing for de-\ntermination of the full-width at half-maximum linewidth\n(\u0001f) and the integrated output power ( P). Besides the\n10\n100\nΔf(MHz)\n5.9\n6.0\n6.1\nS01\nS02\nS03\nS04\nf(GHz)\n2.2\n2.4\n2.6\n2.8\n3.0\n3.2\n3.4\n0.1\n1\nPower (pW)\nI (mA)(a)\n(b)\n(c)FIG. 4. (a) Frequency, (b) linewidth, and (c) integrated power\nof the microwave auto-oscillations as a function of current for\nfour di\u000berent SHNOs with increasing damping. The applied\n\feld is\u00160Hext= 0:5 T, tilted 80\u000eOOP.\nhighly coherent auto-oscillation mode, no other modes\nare excited under these \feld conditions.\nFigure 4 shows the determined auto-oscillation char-\nacteristics of SHNOs with di\u000berent alloy composition\nand damping. The measurements were again made in\na constant \feld of 0 :5 T. The oscillation frequencies in\nFig. 4(a) lie around 6.0 \u00060.1 GHz for all samples, and the\ncurrent-frequency dependence is virtually identical above\nthe individual threshold currents. However, the current\nrange where fdecreases with IDCis missing for the S04\nsample, which suggests that the threshold current is un-\nderestimated for this device. The comparable frequencies\nof all samples con\frm that the saturation magnetization\nis constant throughout the alloy series. Furthermore, the\nquantitatively similar current tunability implies that the\nincreased damping does not change the fundamental na-\nture of the excited auto-oscillation mode.\nThe linewidth of the SHNOs decreases rapidly after the\nauto-oscillation onset and then levels o\u000b for higher IDC\nvalues, as shown in Fig. 4(b). This behavior is consistent\nwith previous studies on nanoconstriction-SHNOs made4\n300\n400\n500\n600\n700\n800\n2.0\n2.4\n2.8\n3.2\nS02\nS01\nS03\nS04\nIth\nDC(mA)\nField (mT)\nFIG. 5. Threshold current ( Ith\nDC) as a function of external\nmagnetic \feld for the four devices of this study.\nof permalloy \flms [11, 17]. The low-damping device S01\nreaches its minimum level at \u0001 f\u001811 MHz, while the\nSHNOs with higher damping materials all have a simi-\nlar minimum linewidth of \u0001 f\u00185 MHz. The linewidth\nis inversely proportional to the mode volume [23], and\nthe decrease in \u0001 fcan therefore be attributed to a spa-\ntial growth of the auto-oscillation region as the damping\nincreases. Nevertheless, the active area of the device is\ncon\fned to the nanoconstriction, which limits the reduc-\ntion in linewidth.\nThe output power of the four nanoconstriction-SHNOs\nis shown as a function of IDCin Fig. 4(c). The power\ngrows almost exponentially with increasing current for\nall samples. However, Pdrops dramatically as the Pt\nconcentration increases. The AMR also decreases in the\nhigher damping samples, but the reduction is too small\nto fully account for the drop in power. Together with the\ntrend in linewidth, the evolution of the power contradicts\nthe general assumption \u0001 f/\u000b=P [23{25]. This equa-\ntion is only valid in the vicinity of the threshold current\nand a direct comparison to the data is problematic, due\nto the experimental di\u000eculties of determining Ith\nDC. Still,\nthe direct relation between the intrinsic oscillator power\nand the electrically measured power is put into question\ndue to the remarkable decrease in the measured P. A\nnumber of factors could in\ruence the signal strength, e.g.\nrecti\fcation, spin-pumping, and the inverse spin-Hall ef-\nfect.\nThe onset current for auto-oscillations was determined\nby current scans for external \felds ranging between 0 :3 T\nand 0:8 T, and the results are shown in Fig. 5. The\n\feld dependence of Ith\nDCis parabola-like for all samples.\nThis kind of behavior has been predicted in a numerical\nstudy by Dvornik et al. [13]. The non-monotonic behav-\nior of threshold current as a function of applied \feld is\na result of a re-localization of the auto-oscillation mode\nand a corresponding change in the spin-transfer-torque\n(STT) e\u000eciency. In weak oblique magnetic \felds, the\nmode is of edge type and samples a signi\fcant portion of\nthe pure spin current, which is largest at the nanocon-striction edges due to the inhomogeneous current den-\nsity. When the \feld strength increases, the mode shows\nan even stronger localization towards the region of the\nhigher current density. Thereby, the STT e\u000eciency in-\ncreases and the threshold current drops. When the \feld\nstrength increases further, the mode detaches from the\nedges and eventually transforms to the bulk type. As\nthis transformation gradually reduces the spatial corre-\nlation between the spin current density and the location\nof the mode, the STT e\u000eciency drops and the threshold\ncurrent increases. The lower \feld tunability of Ith\nDCof\nthe high damping samples imply an initially larger mode\nvolume, which also was suggested by the evolution of the\nlinewidth.\nThe \feld and current range with detectable auto-\noscillations is strongly dependent on \u000b. The threshold\ncurrent should increase linearly with damping [13] and\nthe minimum Ith\nDCindeed scales with \u000b. The enhance-\nment is smaller than predicted (a factor of three), which\nindicates that the increase in damping is accompanied\nwith a higher STT e\u000eciency. A possible reason for the\nimproved e\u000eciency is a larger SHE through a more trans-\nparent interface for alloyed \flms. The origin of the ob-\nserved damping dependence of the threshold \feld is un-\nclear at this stage, calling for a closer inspection of the\nimpact of the applied \feld on the spectral characteristics.\nThus, a further investigation of our devices is targeted\ntowards the microwave emission as a function of \feld\nwith a constant IDC= 3.2 mA, i.e. above or at the pre-\nviously measured auto-oscillation threshold for all \felds.\nWhile the peak frequencies are virtually identical for all\nthe samples, see Fig. 6(a), the varied damping manifests\nin a clear pattern in Pand \u0001f. The microwave power,\nshown in Fig. 6(c), \frstly increases for all samples with\nincreasing \feld, peaks for an intermediate \feld, and \f-\nnally drops relatively sharp until a point where no more\noscillations are detectable. An opposite behavior can be\nseen for \u0001f, which shows a minimum for intermediate\n\felds. The \feld at which the SHNOs emit their maxi-\nmum output power decreases monotonically from 0.64 T\nto 0.4 T with increasing damping. The same trend is\nvisible for the point of minimum linewidth, which de-\ncreases with increased damping from 0.71 T to 0.49 T,\nand is therefore at a typically \u00180.1 T larger \feld than the\nrespective maximum power. The lowest overall linewidth\ncan be achieved for the lowest damping SHNO (S01) at\nhigh \felds, where only this device still shows a detectable\nsignal, i.e., \u0001 f= 1:2 MHz at\u00160Hext= 0:71 T. How-\never, at low applied \felds \u00160Hext\u00140:48 T a clear trend\nis noticeable towards smaller linewidths for the alloyed\npermalloy \flms with larger damping.\nIn light of this inverse trend, we can argue that auto-\noscillations in nanoconstriction-SHNO should also be de-\nscribed in the framework of non-linear auto-oscillators,\nalthough the study in Ref. 13 has shown that oscilla-\ntions in nanoconstriction-SHNOs emerge from a local-5\nΔf(MHz)\n(b)\n(c)\n4\n6\n8\n10\nS01\nS02S03\nS04\n1\n10\n100\n300\n400\n500\n600\n700\n800\n0.1\n1\nPower (pW)\nField (mT)(a)f(GHz)\nFIG. 6. (a) Frequency, (b) linewidth, and (c) integrated power\nof the auto-oscillations as a function of the applied external\nmagnetic \feld at a constant drive current IDC= 3:2 mA.\nized linear mode. The generation linewidth of nanocon-\ntact spin torque oscillators, which are a prime example\nof non-linear auto-oscillators, has been studied analyti-\ncally [23, 26] and experimentally [27]. The linewidth as\na function of current and magnetic \feld angle was shown\nto follow the expression:\n\u0001f=\u00000\n2\u0019\u0012kBT\nE0\u0013\"\n1 +\u0012N\n\u0000e\u000b\u00132#\n; (1)\nwherekB,T, andE0(IDC=Ith\nDC) are the Boltzmann con-\nstant, temperature and the average oscillator energy, re-\nspectively. Nis the nonlinear frequency shift, a material\nproperty that is determined by the internal magnetic \feld\nand the magnetization [28]. \u0000 e\u000bis the e\u000bective nonlinear\ndamping rate and \u0000 0is the positive damping rate, and\nboth have an explicit linear dependence on the Gilbert\ndamping\u000b[23]. Assuming everything else equal amongst\nour devices, a decrease of the linewidth with \u000bcan be\nthus expected, when the second term in the brackets in\nEq. 1 dominates. This is likely for low to intermediate\felds, since Ncan be calculated to take up the largest\nvalues under these conditions [28], which are thus in ac-\ncordance with the range of \felds, where we observe the\ndiscussed linewidth vs. damping behavior in our devices.\nCONCLUSIONS\nIn conclusion, we have fabricated a series of sam-\nples where the magnetization is constant, while the\nspin wave damping is varied by a factor of three. We\nhave shown that the damping of the magnetic layer in\nnanoconstriction-SHNOs has an important in\ruence on\nall the spectral characteristics of the devices. The re-\nsults of our study will encourage the application of tai-\nlored materials in SHNOs and can be used for a further\nunderstanding of the magnetodynamics in nanodevices,\ne.g. the coupling mechanisms in mutually synchronized\nSHNOs.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from the China\nScholarship Council (CSC), the G oran Gustafsson\nFoundation, the Swedish Research Council (VR), the\nKnut and Alice Wallenberg Foundation (KAW), and\nthe Swedish Foundation for Strategic Research (SSF).\nThis work was also supported by the European Re-\nsearch Council (ERC) under the European Communitys\nSeventh Framework Programme (FP/2007-2013)/ERC\nGrant 307144 \\MUSTANG\".\n\u0003yuri@seu.edu.cn\n[1] T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli,\nA. Houshang, A. A. Awad, P. D urrenfeld, B. G. Malm,\nA. Rusu, and J. \u0017Akerman, \\Spin-Torque and Spin-Hall\nNano-Oscillators,\" Proc. IEEE 104, 1919 (2016).\n[2] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. 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B 76, 024437 (2007)."    },    {        "title": "1802.07195v1.Ultrafast_magnetization_dynamics_in_pure_and_doped_Heusler_and_inverse_Heusler_alloys.pdf",        "content": "Ultrafast magnetization dynamics in pure and doped Heusler and inverse Heusler\nalloys\nR. Chimata,1, 2E. K. Delczeg-Czirjak,2J. Chico,3M. Pereiro,2B. Sanyal,2O. Eriksson,2, 4and D. Thonig2\n1Argonne National Laboratory, Lemont, IL 60439, United States\n2Department of Physics and Astronomy, Materials Theory, University Uppsala, SE-75120 Uppsala, Sweden\n3Peter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, D-52425 J ulich, Germany\n4School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\n(Dated: February 21, 2018)\nBy using a multiscale approach based on \frst-principles density functional theory combined with\natomistic spin dynamics, we investigate the electronic structure and magnetization dynamics of an\ninverse Heusler and a Heusler compound and their alloys, i. e. Mn 2\u0000xZxCoAl and Mn 2\u0000xZxVAl,\nwhere Z= Mo, W, Os and Ru, respectively. A signature of the ferrimagnetic ordering of Mn 2CoAl\nand Mn 2VAl Heusler alloys is re\rected in the calculated Heisenberg exchange constants. They decay\nvery rapidly with the interatomic distance and have short range, which is a consequence of the\nexistence of the \fnite gap in the minority spin band. The calculated Gilbert damping parameter\nof both Mn 2CoAl and Mn 2VAl is high compared to other half-metals, but interestingly in the\nparticular case of the inverse Mn 2CoAl alloys and due to the spin-gapless semiconducting property,\nthe damping parameters decrease with the doping concentration in clear contradiction to the general\ntrend. Atomistic spin dynamics simulations predict ultrafast magnetisation switching in Mn 2CoAl\nand Mn 2VAl under the in\ruence of an external magnetic \feld, starting from a threshold \feld of\n2 T. Our overall \fnding extends with Heusler and inverse Heusler alloys, the class of materials that\nexhibits laser induced magnetic switching.\nI. INTRODUCTION\nThe \feld of the ultrafast magnetization dynamics has\nbecome one of the most important topics in magnetism,\nstarting from the pioneering experiment on ferromag-\nnetic nickel from Beaurepaire et al.1in 1996. Since\nthen, numerous experiments were carried out on 3 d\n(Fe2, Co3, Ni4,5), 4f(Tb and Gd6) ferromagnets, as\nwell as on several alloys (GdFeCo7{13, TbCo14, CoPt15)\nand half metallic systems (CrO 216, Co 2Cr0:6Fe0:4Al17,\nCo2FeSi, Co 2MnGe, Co 2FeAl18, and Co 2FexMn1\u0000xSi19\nand Co 2MnSi19,20) aiming to \fnd faster ways of manip-\nulating spins in a controllable way, opening a new \feld\nin the advanced information/data storage and data pro-\ncessing technologies.\nExperimental observations revealed that the charac-\nteristic demagnetization times of 3 delements are within\nthe 100 fs time scale, much faster than that of the 4 f-\nferromagnets, which show more complex behavior in-\nvolving a two-step demagnetization process in 10 ps time\nscale. Surprisingly, recent pump-probe experiments on\nhalf-metallic Heusler alloys measured distinguished and\ntypically larger all-optical switching times when com-\npared to 3d-ferromagnets8,19. In these materials, one of\nthe spin channels is completely or partially unoccupied\naround the Fermi energy, consecutively the magneto op-\ntical excitations from one channel to another channel are\nforbidden.\nAttempts to understand the momentum transfer be-\ntween the electrons, spins and phonons after a short\nlaser pulse have opened a new debate in the \feld. Sev-\neral quantitative models had been proposed to describe\nthe mechanism of the ultrafast demagnetization suchas the microscopic three-temperature model21, stochas-\ntic atomistic descriptions22, models using the stochas-\ntic Landau-Lifshitz-Bloch equation23,24and models sug-\ngesting the presence of di\u000busive or superdi\u000busive spin\ncurrents6,25{28. The \frst three models relate the spin-\nscattering to the Gilbert damping parameter, \u000b, that de-\nscribes the energy dissipation in a magnetic system via\nelementary spin-\rip processes29,30. Here, we combine the\nab initio description of the magnetic exchange interaction\nand Gilbert damping31{33parameter with the Landau-\nLifshitz-Gilbert equation to investigate the demagneti-\nzation process in half-metallic ferrimagnetic Heusler and\ninverse Heusler alloys.\na0\nAl\nV\nMn\na0\nAl\nCo\nMna) b)\nFIG. 1. (Color online) Schematic crystal structures of a)\nthe Heusler alloy Mn 2VAl and b) the inverse Heusler alloy\nMn2CoAl. Di\u000berent atom types are represented by di\u000berent\ncolours. Solid and dashed lines indicate the bond between the\natoms and are added to guide the eye. The lattice constant\na0is also indicated.arXiv:1802.07195v1  [cond-mat.mtrl-sci]  20 Feb 20182\nHeusler and inverse Heusler alloys are de\fned as\nternary intermetallic compounds with a composition of\nX2YT(cf. Fig. 1). Heusler alloys crystallize in the L2 1\nstructure (space group Fm \u00163m, 225), with the 4 a(0, 0, 0),\n4b(1\n2,1\n2,1\n2) and 8c(1\n4,1\n4,1\n4) Wycko\u000b positions. XandY\nare transition metals occupying the 8 cand 4apositions,\nrespectively, and Tis a main group III, IV or V element\nsitting in the 4 bposition. Inverse Heusler alloys adopt the\nHg2CuTi prototype structure (space group F \u001643m, 216),\nwith 4a(0,0,0), 4b(1\n2,1\n2,1\n2), 4c(1\n4,1\n4,1\n4) and 4d(3\n4,3\n4,\n3\n4) positions. In this case, XandYare transition metals,\nXoccupying the 4 aand 4dpositions while Yis the 4cpo-\nsition. The main element T sits in the 4 bposition. Both\nstructures may be regarded as a cubic unit cell, which\nconsists of four interpenetrating fcc sublattices. There\nare four atoms in the diagonal of the cube following the\nX-Y-X-Tsequence for Heusler alloys and X-X-Y-Tfor\nthe inverse Heuslers.\nHere, we study the demagnetization dynamics of a\nHeusler and an inverse Heusler compound and their al-\nloys, i.e. Mn 2\u0000xZxVAl and Mn 2\u0000xZxCoAl, where Z=\nMo, W, Os and Ru. Mn 2VAl is a well known half-metallic\nferrimagnetic Heusler compound34{39where the minority\nspin channel is the conducting one40. Mn 2CoAl adopts\nthe inverse Heusler structure41and it is predicted41and\ncon\frmed42to be a spin gapless magnetic semiconductor.\nThese peculiarities of the band structure are re\rected in\nthe Gilbert damping parameter and a\u000bect the magneti-\nsation dynamics under the in\ruence of a laser pulse, as\nwill be described below.\nThe paper is divided as follows: In Section II we intro-\nduce our numerical technique to study materials proper-\nties and magnetization dynamics in Heusler alloys. Elec-\ntronic and magnetic properties of the parent Heusler al-\nloys Mn 2CoAl and Mn 2VAl as well as doping of these\nmaterials with Os, Ru, W, and Mo is discussed in Sec-\ntion III A. Demagnetisation studies of these alloys caused\nby a femtosecond laser are described in Section III E. Fi-\nnally, the article concludes in Section IV with an outlook.\nII. METHODS\nA. Electronic structure calculation\nThe electronic and magnetic properties of the studied\nmaterials are obtained from \frst principle calculations\nby applying full-relativistic multiple scattering theory\nas formulated in the Korringa-Kohn-Rostocker (KKR)\napproach43. This method is implemented in the SPR-\nKKR package44,45. Solving the Dirac equation, relativis-\ntic e\u000bects are fully accounted for, especially the spin-orbit\ninteraction which is essential for heavy elements such as\nthe here considered dopants Os, W, Ru, and Mo. The\npotentials are treated by the atomic sphere approxima-\ntion (ASA) and obtained by self-consistently solving the\nKohn-Sham density functional theory (DFT) equation\nwithin the local density (LDA) or generalized gradientapproximation (PBE) as devised by Perdew, Burke and\nErnzerhof46,47. Note that we applied the PBE functional\nif not further speci\fed. The irreducible Brillouin zone is\nsampled by\u0019500 k-points. To describe substitutional\ndisorder in the sub-lattices of the alloys we make use\nof the coherent potential approximation (CPA)48. The\nspin-polarized scalar relativistic full-potential (SR-FP)\nmode49of the KKR approach is used to calculate the to-\ntal energies as a function of volume [ E(V)], which gives\nan estimate of the lattice constant a0.\nB. Calculation of Heisenberg exchange and Gilbert\ndamping\nThe angular momentum transfer in terms of Heisen-\nberg exchange interactions Jijand energy dissipation\nrelated to the Gilbert damping parameter \u000bis deter-\nmined by an ab-initio method with the aim to ad-\ndress the magnetic ground state and also the dynami-\ncal properties by using the Landau-Lifshitz-Gilbert equa-\ntion. The interatomic exchange interactions, Jij, were\ncalculated via the Liechtenstein-Katsnelson-Antropov-\nGubanov (LKAG) formalism50\nJij=1\n\u0019Z\"F\n\u00001Im Tr\u0010\n\u0001i\u001c\"\nij\u0001j\u001c#\nji\u0011\nd\": (1)\nwhere \u0001i=t\u00001\ni;\"\u0000t\u00001\ni;#is the spin-resolved di\u000berence of the\nsingle-site scattering matrix tiat siteiand\u001cijis the scat-\ntering path operator, describing the propagation of the\nelectrons between two sites iandj. The Fermi energy\nis denoted by \"F. Note that in CPA, the multiple scat-\ntering matrix is replaced by the scattering properties of\nthe e\u000bective medium ^ \u001ci\u0016;j\u0017 =Xi\u0016\u001cCPA\nijXj\u0017constructed\nfrom a defect of type \u0016;\u0017at sitei;j, respectively. The\ndefects are taken into account by the defect matrix Xi\u0016.\nFrom the calculated exchange interactions, it is possible\nto obtain the spin wave sti\u000bness, D, which is expressed\nas:51\nD= lim\n\u0011!02\n3X\nje\u0000\u0011jr0jj\na0J0jjrijj2(2)\nby using super cell calculation with random con\fgura-\ntions of the dopants in 12 ensembles and starting from\na reference site i= 0. The distance between site iand\njis given by rijand the parameter \u0011is introduced to\nguarantee convergence within a Pade interpolation ap-\nproximation.\nThe Gilbert damping parameter is identi\fed on the\nbasis of the linear response theory33by means of the\nmultiple scattering formalism52. The diagonal elements\n\u0016=x;y;Zof the Gilbert damping tensor can be written\nas33:\n\u000b\u0016\u0016=g\n\u0019mtotX\njTr\nT\u0016\n0~\u001c0jT\u0016\nj~\u001cj0\u000b\nc; (3)3\nwhere the e\u000bective g-factor g= 2(1 +morb=mspin) and\ntotal magnetic moment mtot=mspin+morbare given by\nthe spin and orbital moments, mspinandmorb, respec-\ntively, ascribed to a unit cell. Equation (3) gives \u000b\u0016\u0016for\nthe atomic cell at lattice site 0 and implies a summation\nover contributions from all sites indexed by j, including\nj= 0. Moreover, ~ \u001cijis related to the imaginary part of\nthe multiple scattering operator that is evaluated only at\nthe Fermi energy \"F. Finally,T\u0016\nirepresents the matrix\nelements of the torque operator ^T\u0016=\f\u001b\u0016Bxc(r). The\nnotationh:::icrepresents the con\fgurational average, in-\ncluding vertex corrections33derived by Butler53and ac-\ncounting for \fnite temperature using the alloy analogy\nmodel within CPA54.\nC. Atomistic spin dynamics\nThe evolution of atomistic spins in a thermal bath is\ndescribed by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion55,56, where the dynamics of a magnetic moment is\nexpressed in terms of precession and damping:\ndmi(t)\ndt=\u0000\r\n(1 +\u000b2)\u0012\nmi(t)\u0002Bi(t)\n+\u000b\nmimi(t)\u0002(mi(t)\u0002Bi(t))\u0013\n:(4)\nHere\ris the gyromagnetic ratio, \u000brepresents the di-\nmensionless Gilbert damping constant, and mi=miei\nis an individual atomic moment on site i. The e\u000bective\nmagnetic \feld is given by Bi=\u0000@H\n@mi+bi, whereH=\n\u0000P\ni6=jJijei\u0001ejandbiis an stochastic \feld. The latter\ndescribes white noise ( hbi(t)\u0001bj(t0)i= 2D\u000eij\u000e(t\u0000t0)),\nwhere the \ructuation width is D=\u000bkBTs=\rm. Thus, the\nspin temperature Tsdirectly passes into LLG equation\nvia the stochastic magnetic \feld biand is obtained from\nsolving the two-temperature (2T) model57. The analyti-\ncal expression of this two temperature model reads,\nTs=T0+ (5)\n(TP\u0000T0)\u0002(1\u0000exp(\u0000t=\u001cinitial ))\u0002exp(\u0000t=\u001cfinal)+\n(TF\u0000T0)\u0002(1\u0000exp(\u0000t=\u001cfinal))\nwhereT0is the initial temperature of the system, TP\nis the peak temperature after the laser pulse is applied\nandTFis the \fnal temperature. Both the initial and\n\fnal temperature are set to 300 K, where the peak tem-\nperature is a parameter in the simulations. The time-\ndependent parameters \u001cinitial and\u001c\fnalare exponential\nparameters, \fxed by \u001cinitial = 10 fs and \u001c\fnal= 20 ps from\nRef. [58]. Note that both relaxation times are materials\nspeci\fc and kBis the Boltzmann constant.III. RESULTS AND DISCUSSION\nThis current section is divided in \fve parts. In the \frst\nand second part we discuss the electronic structure and\nthe magnetic moments, respectively, of pure and doped\nHeusler and inverse Heusler materials based on DFT-\noptimized lattice constants. The third part deals with\nthe Heisenberg interaction, spin wave sti\u000bness, as well\nas the ordering temperature. The Gilbert damping is\ndiscussed in part four. The last part focuses on the de-\nmagnetisation and reliable switching in Heusler materials\nbased on the LLG equation.\nA. Electronic structure calculations\nLattice parameters are estimated from total energy cal-\nculations, compared to Refs. [38] and [59], and listed in\nTable I. For undoped Mn 2CoAl and Mn 2VAl, we im-\nproved the theoretically predicted values used in Ref. [59]\nby 10% and they are closer to the experimentally mea-\nsured lattice constant. The improvement comes from tak-\ning into account the full-potential, which is known to im-\nprove lattice constants60. By doping Mn with 4d and\n5d metals Mo, Ru, W, and Os, we observe an expected\nincrease of the lattice constant with the concentration\nof the dopants, since the atomic radius of the dopant is\nlarger than the one of Mn. For Mn 2VAl, the increase\nof the lattice constant is substantially bigger ( \u00191% for\nx= 1% doping) than for Mn 2CoAl (\u00190:1% forx= 1%\ndoping).\nThermal switching within our classical atomistic model\nis completely determined by the Heisenberg exchange\nand the Gilbert damping of the system61, which are in\nturn identi\fed by the scattering-path matrices and the\nsingle-site scattering matrices of the Kohn-Sham prob-\nlem in Eqs. (1) and (3). Hence, we \frst have to ad-\ndress the electronic structure by means of the density of\nstates (DOS; Fig. 2). The here studied inverse Heusler\nMn2CoAl is known to be a spin gapless semiconductor,\nwhere an almost zero-width energy gap at the Fermi level\nexists in the majority-spin channel (the majority states\nare plotted with positive values and the minority spin\nstates with negative values) but a regular energy gap oc-\ncurs in the minority spin-channel (see inset in the bottom\npanel of Fig. 2). This was already reported, for example,\nin Ref. [59]. The density of states and, consequently,\nthe gap are sensitive to the applied exchange correlation\nfunctional. Using local density approximation (LDA),\nstates are shifted up in energy (not shown here) com-\npared to the PBE by about 10 meV and, consequently,\nno gap at the Fermi energy is observed. Note that the\no\u000bset of the energy from the real axis in Fig. 2 (the\nspectral width of the electron bands) is small and about\n1 meV, which causes sharp features in the DOS. A \fnite\nspectral width also gives rise to an overlap of the states\naround the Fermi level and `hide' the zero-width energy\ngap; a \fnite density of states at \"Fis observed. Bands4\n-120-80-4004080120Density of states (1/Ryd)\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1\nE−EF(Ryd)-120-80-4004080120\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-120-80-4004080120\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-120-80-4004080120\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-120-80-4004080120\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1Mn 2CoAlDOS\n-0.02 0.0 0.02\nE−EF(Ryd)-0.02 0.0 0.02 -0.02 0.0 0.02 -0.02 0.0 0.02 -0.02 0.0 0.02\n-80-4004080Density of states (1/Ryd)\n-80-4004080\n-80-4004080\n-80-4004080\n-80-4004080W\nRu\nMo\nOs\nMn 2VAlDOS\nE−EF(Ryd)\nFIG. 2. (Color online) Density of states (DOS) for Mn 2CoAl\n(lower panel) and Mn 2VAl (upper panel) without doping(gray\nbackground) and with doping of W (red lines), Ru (blue lines),\nMo (green lines), and Os (orange lines). Positive (negative)\nDOS values correspond to the majority-(minority-) spin elec-\ntrons and are indicated by bold up-(down-) arrays. The inset\nis a magni\fcation of the DOS around the Fermi level.\nthat cross the Fermi level, are mainly allocated to Mn1\nand Co (band structure is not shown here, but it can be\nfound elsewhere41). Note that the superscripts 1 and 2\nbetween the two Mn atoms. In contrast to Ref. [59], the\nFermi energy is not located at the centre of the minority\nband gap, which will a\u000bect the coupling between the col-\nlective and single-electron excitations, i.e. the exchange\ninteractions.\nThe chemical compound Mn 2VAl, however, is half-\nmetallic (cf. Fig. 2) with a gap in the majority spin-\nchannel. The width of the majority band gap (0 :7 eV) is\nbigger than the minority spin gap in Mn 2CoAl (0:4 eV),\nwhich signi\fcantly a\u000bects the magnetic properties. In\nthe minority spin channel and at the Fermi energy Mn\nprojected states cause a strong peak in the DOS that hy-bridize with V atoms. States above the Fermi energy are\ndominated by the d-states of V atoms.\nThe spin-gapless semiconducting or half-metallic be-\nhaviour in Mn 2\u0000xZxCoAl and Mn 2\u0000xZxVAl is destroyed\nby replacing some of the Mn atoms with heavy metals, Z\n= Mo, W, Os, Ru of a given concentration x= 0:05 and\n0:1. Comparing total energies (not shown here) allows us\nto conclude that for the inverse Heusler Mn 2\u0000xZxCoAl\ndoping at both Mn-sites (Mn1-Mn2) has the lowest en-\nergy. We obtained a maximal energy di\u000berence of \u0001 E\u0019\n40 meV when doping at Mn1-Y, Mn2-Y, orYwith 1%\nof the dopants W, Ru, Mo, Os. There is no major vari-\nation found in \u0001 Ebetween the di\u000berent dopands. Note\nthat we used here the same lattice constant as shown in\nTable I, but in principle it will vary when doping at Mn1-\nMn2, Mn1-Y, Mn2-Y, orY. However, Mn 2\u0000xZxVAl has\nthe lowest energy when doping only the V atom, but to\ntreat both material on the same footing, we consider also\nMn2\u0000xZxVAl to be doped at the Mn1-Mn2atoms.\nIn the case of Mn 2CoAl, W and Mo generate states\nat the spin-gap majority states at the Fermi level, where\non the other hand the gap in the minority spin chan-\nnel survives. In terms of a rigid band model, W- and\nMo-doping decreases the Fermi energy, which relocates\nthe DOS to higher energies. The dopants Os and Ru\nhave one electron more than Mn in the valence band\nand, consequently, a\u000bect the density of states in the op-\nposite way: Minority states are added and become occu-\npied. The Fermi energy increases, which shifts the den-\nsity of states to smaller energies. For Mn 2VAl, doping\nwith Ru and Os preserves the half-metallic behaviour; it\nadd states below the Fermi energy and typically at the\nenergy\"=\u00000:025 Ryd. Doping with Mo and W reduces\nthe width of the band-gap and shifts it above the Fermi\nenergy. Related to the alloying, the density of states\nsmears out in the whole energy range.\nB. Magnetic moments\nThe exchange splitting in the DOS and, consequently,\nthe total magnetic moment is a\u000bected by doping (see\nFig. 3). Both Heusler materials are ferrimagnetic.\nAn antiferromagnetic coupling between the Mn atoms\nwas observed for the inverse Heusler alloy Mn 2CoAl\n(cf. Table I), caused by the inequivalence of the two\nMn atoms. These results are in good agreement with\nexperiments41,42and existing theoretical predictions59,62.\nAccording to the Bethe-Slater curve63, transition-metal\natoms such as Mn tend to have an antiferromagnetic\nspin moment when they are close to each other. In\nMn2\u0000xZxVAl, the Mn atoms are equivalent and, thus,\nhave the same magnetic moment that couple ferromag-\nnetically. The V atom, however, is antiferromagnetic\nwith respect to the Mn atoms and has a strong induced\nmagnetic moment of 0 :91\u0016B. Opposite to the total mag-\nnetic moment, the size of the element resolved magnetic\nmoments is sensitive to the lattice constant of the system5\n1.71.81.92.02.12.2m(µB)\n0.0 0.05 0.1\nxW\nRu\nMo\nOsMn 2−2xZ2xVAl\nMn 2−2xZ2xCoAl\nFIG. 3. (Color online) Total magnetic moments of\nMn2\u0000xZxCoAl (triangles) and Mn 2\u0000xZxVAl (circles) as a\nfunction of dopant concentration x. The symbol Z represents\nMo (green lines and symboles), Os (orange lines and symbols),\nRu (blue lines and symbols), and W (red lines and symbols).\nand moments can vary up to 13 %, which was also found\nin Ref. 59.\nThe size but not the sign of the elemental magnetic\nmoments changes by doping the Heusler materials with\n4d and 5d heavy metals, and, thus, also the total mag-\nnetic moment. Typically, the induced magnetic moments\nof dopants are parallel to the magnetic moment of Mn\natoms and they become larger if the magnetic moment\nof the Mn atom is smaller. In the case of Mn 2CoAl, the\ndopants W, Ru, Mo, and Os cause a decay of the to-\ntal magnetic moment of about 0 :1\u00000:2\u0016Bforx= 1%,\nwhile in the case of Mn 2VAl, only the dopants Ru and\nMo decrease the magnetic moment. This is caused by a\nsigni\fcant change of the Mn magnetic moments of about\n\u0001m\u00190:1\u00000:2\u0016B, but also for Co atoms the moment\nvariation is about \u0001 m\u00190:2\u0016B.\nC. Heisenberg exchange parameter and Curie\ntemperatures\nBased on our electronic structure analysis in the Sec-\ntion III B, we calculated the Heisenberg exchange param-\neterJij(see Fig. 4). The already revealed ferrimag-\nnetic behaviour is re\rected also in the exchange con-\nstantsJ. The magnetic exchange parameters decay very\nrapidly with the interatomic distance, rij, which is as-\ncribed to the existence of the \fnite spin gap in the\nminority-channel51,64. Our results for Mn 2CoAl are sim-\nilar to the ones already reported in Refs. [62,59]. Note\nthe factor of 2 in Ref. [59] may be caused by a di\u000berent\ndouble-counting convention of the Heisenberg Hamilto-\nnian. For the compound Mn 2CoAl, the antiferromagnetic\ninteraction between Mn1and Mn2dominates the ferri-\nmagnetism, whereas the Mn2-Co interatomic exchange\ninteraction is ferromagnetic. In Mn 2VAl, the situation is\nthe opposite: the Mn to V interaction is dominating and\nantiferromagnetic, where only the Mn1-Mn2contributes\n-2.5-2.0-1.5-1.0-0.50.00.51.01.5Jij(mRyd)\n0.4 0.8 1.2 1.6 2.0\nrij·a−1\n0Mn1-Mn1\nMn1-Mn2\nMn1-Co\nMn2-Mn2\nMn2-Co\n-0.8-0.6-0.4-0.20.00.20.4Jij(mRyd)\n0.4 0.8 1.2 1.6 2.0\nrij·a−1\n0Mn 1-Mn 1\nMn 1-Mn 2\nMn 1-V\nMn 2-Mn 2\nMn 2-VFIG. 4. (Color online) Intersublattice Heisenberg exchange\nparameter as a function of renormalized interatomic distance\nfor a) Mn 2VAl and b) Mn 2CoAl. Di\u000berent colours represents\nthe coupling between Mn1-Mn1(red dotes), Mn1-Mn2(blue\ndotes), Mn1-Co or Mn1-V (green dotes), Mn2-Mn2(orange\ndotes) and Mn2-Co or Mn2-V (cyan dotes).\nwith a ferromagnetic coupling but with half the strength\nof the Mn-V interaction. The coupling between equiv-\nalent Mn atoms in Mn 2VAl (Mn1-Mn1and Mn2-Mn2)\nis small and negligible. The calculated interactions de-\npend to some extent on the details of the calculations.\nIn particular, the JMn-CoandJMn-Vinteractions depend\nstrongly on the applied exchange-correlation functional,\nbut also on the lattice constant of the system. Notice that\nforJMn-CoandJMn-Vin LDA we obtain twice the size of\ntheJ's from PBE (not shown here). The other couplings\n(e.g.JMn-Al,JCo-Al,JV-Al) turned out to be negligible,\nprimarily caused by a vanishing magnetic moment on the\nAl atom.\nAs shown in Fig. 5, doping with 4d and 5d elements\nreduces nearest-neighbour interactions and the correla-\ntion length between magnetic moments, which is a direct\nconsequence of the disorder and the coherent potential\napproximation65. Nearest neighbour interactions are af-\nfected mostly by the doping. In general, the exchange6\nCompound a0(\u0017A)m[Mn1]m[Z1]m[Mn2]m[Z2]m[Y]\nMn2CoAl 5 :73 [ 59] \u00001:64 2 :77 0 :93\nMn1:8W0:2CoAl \u00001:52\u00000:52 2:75 0:26 0:78\nMn1:8Ru0:2CoAl \u00001:62\u00000:10 2:76 0:06 0:91\nMn1:8Mo0:2CoAl \u00001:53\u00000:56 2:75 0:33 0:78\nMn1:8Os0:2CoAl \u00001:56\u00000:12 2:76 0:18 0:92\nMn2VAl 5 :69 [ 38] 1 :32 1 :32 \u00000:66\nMn1:8W0:2VAl 1 :32 0:19 1:32 0:19\u00000:57\nMn1:8Ru0:2VAl 1 :31 0:08 1:31 0:08\u00000:61\nMn1:8Mo0:2VAl 1 :36 0:25 1:36 0:25\u00000:58\nMn1:8Os0:2VAl 1 :31 0:09 1:31 0:09\u00000:60\nMn2CoAl 5:79 [5:84 exp] \u00001:81 2 :91 0 :96\nMn1:8W0:2CoAl 5 :79 \u00001:37\u00000:46 2:62 0:22 0:77\nMn1:8Ru0:2CoAl 5 :79 \u00001:81\u00000:10 2:90 0:07 0:96\nMn1:8Mo0:2CoAl 5 :79 \u00001:71\u00000:60 2:89 0:39 0:82\nMn1:8Os0:2CoAl 5 :80 \u00001:80\u00000:12 2:92 0:19 0:98\nMn2VAl 5:84 [5:88 exp] 1:47 1 :47 \u00000:91\nMn1:8W0:2VAl 5 :92 1:73 0:37 1:73 0:37\u00000:99\nMn1:8Ru0:2VAl 5 :86 1:50 0:05 1:50 0:05\u00000:89\nMn1:8Mo0:2VAl 5 :91 1:72 0:45 1:72 0:45\u00000:98\nMn1:8Os0:2VAl 5 :92 1:52 0:06 1:52 0:06\u00000:91\nTABLE I. Lattice constant and atom resolved magnetic moments (in \u0016B) of the host Mn 2CoAl and Mn 2VAl . The upper panel\nshows results for a \fxed lattice constant obtained from literature, where the lower panel is for lattice constants calculated from\ntotal energy minimization. The superscripts 1 and 2 distinguish between the two Mn atoms. The symbol Yrepresents either\nCo or V. The magnetic moment of Al is negligibly small.\ncouplings diminish with doping concentration xup to\n0:6 mRyd for W and x= 0:1. For Os and Ru doping,\nthere is a slight increase of the exchange coupling (about\n0:03 mRyd).\nWith knowledge about the trends in the exchange cou-\nplingsfJg, one can estimate the spin-wave sti\u000bness D\nand the phase transition temperature from both mean\n\feld theory via kBTMF\nC =3=2P\njJ0jor from Monte\nCarlo simulations. The results are shown in Fig. 6. The\nspin-wave sti\u000bness (upper panel in Fig. 6) for Mn 2\u0000xCoAl\nis in good agreement with Ref. [59], while for Mn 2\u0000xVAl\nwe reproduce the spin wave sti\u000bness constant Dalready\nreported in Ref. [66] ( D= 324 meV \u0017A2), but not the ex-\nperimentally measured sti\u000bness67(D= 534 meV \u0017A2).\nFor the Co based Heusler compounds we obtain a hard-\nening of the spin-waves after an initial softening, where\nfor the V based Heusler compound, only hardening of\nthe spin-waves with doping is observed. The phase tran-\nsition temperature TC, which turns out to be inversely\nproportional to D, decreases with doping concentration\nxfor two reasons, namely: i)reduction of the magnetic\nmoment due to doping and, consequently, stronger \ruc-tuations at a given temperature as well as ii)reduction of\ncorrelation. The critical temperature TCis obtained from\nMonte Carlo simulations on the Metropolis algorithm68,\nfrom Binder's fourth cumulant68for di\u000berent simulated\nsystem sizes but also from the spin susceptibility \u001f. Note\nthat the \frst method could fail for antiferro- and ferri-\nmagnets. Thus, we obtain a systematic error of about\n\u00065 K.\nOur simulations of ordering temperature (680 K for\nMn2CoAl and 475 K for Mn 2VAl) underestimate the\ntransition temperature observed from experiment (720 K\nfor Mn 2CoAl42and 768 K for Mn 2VAl). This discrep-\nancy that is most notable for Mn 2\u0000xVAl was reported\nearlier66and could have multiple reasons. First, magnetic\nproperties in Heusler alloys are sensitive to the intersti-\ntial region spanned by the mu\u000en tin potential. Thus,\nfull-potential simulations are required as it was shown in\nRefs. [41, 42, and 69]. Also the results depend crucially on\nthe choice of the exchange-correlation functional and on\nelectron correlations e.g. addressed by including a Hub-\nbardU66. Second, the Heisenberg exchange is calculated\nfor a collinear ferrimagnetic state but when the magnetic7\n-2.0-1.5-1.0-0.50.00.51.0Jij(mRyd)\n0.4 0.8 1.2 1.6\nrij·a−1\n0W\nMn1-Mn1\nMn1-Mn2\nMn1-Co\nMn2-Mn2\nMn2-Co\n0.4 0.8 1.2 1.6 2.0\nrij·a−1\n0Ru-2.0-1.5-1.0-0.50.00.51.0Jij(mRyd)Mo Os\n-1.0-0.8-0.6-0.4-0.20.00.2Jij(mRyd)\n0.4 0.8 1.2 1.6\nrij·a−1\n0W\nMn 1-Mn 1,Mn 2-Mn 2\nMn 1-Mn 2\nMn 1-V,Mn 2-V\nMn 1-X1\nMn 1-X2\n0.4 0.8 1.2 1.6 2.0\nrij·a−1\n0Ru-1.0-0.8-0.6-0.4-0.20.00.20.4Jij(mRyd)Mo Os\nFIG. 5. (Color online) Intersublattice Heisenberg exchange\nparameter as a function of renormalized interatomic distance\nfor a) Mn 2\u0000xZxVAl and b) Mn 2\u0000xZxCoAl, where the di\u000ber-\nent subpanels show the dopants W (bottom left), Ru (bottom\nright), Mo (top left), and Os (top right). Di\u000berent colours\nrepresents the coupling between Mn1-Mn1(red dotes), Mn1-\nMn2(blue dotes), Mn1-Co or Mn1-V (green dotes), Mn2-Mn2\n(orange dotes) and Mn2-Co or Mn2-V (cyan dotes).\ndisorder is taken into account in the electronic structure,\nusually the exchange interaction is biased65. Based on\nthe alloy analogy model54, we modelled also the temper-\nature stability of the magnetic properties (magnetic mo-\nments and magnetic exchange) coming from electronic\nstructure by the partial disordered local moment (DLM)\n600700800900100011001200TMF\nC(K)\n0.0 0.05 0.1\nxW\nRu\nMo\nOsMn 2−2xX2xVAl\nMn 2−2xX2xCoAl450500550600650TC(K)250300350400450D(meV ˚A2)FIG. 6. (Color online) Spin wave sti\u000bness D, critical tem-\nperaturesTC, and mean \feld critical temperatures TMF\nCof\nMn2\u0000xZxCoAl (triangles) and Mn 2\u0000xZxVAl (circles) as a\nfunction of dopand concentration x. Dopands are Mn (black\ncircles), Os (red squares), Ru (green diamonds), and W (or-\nange triangles).\napproximation within the Ising model65. DLM approach\nis believed to accurately describe `spin temperature' in\nthe electronic structure70. However, it turned out that\nthe disordered local moment theory can not be applied\nto both Heusler and inverse Heusler for similar reasons as\nfor Ni71: the magnetic moments in Al and Co/V disap-\npear. For Mn 2CoAl, our simulations show furthermore\nthat the magnetic moment of the Mn2atom is zero in\nthe paramagnetic phase and, consequently, the magnetic\nexchange and the phase transition temperature are zero.\nThis result is independent of the doping with 4d and 5d\nelements. These results indicate the inconsistency of the\nDLM model for Heusler materials. It is still an open\nquestion, if results get improved by applying relativis-\ntic DLM theory72. Third, we consider only a simpli\fed\napproach for electron correlation in the LDA and GGA\ndensity functional. However, it is known66that improved\nmodels for electron correlation have the trend to increase\nslightly the phase transition temperature.\nD. Gilbert damping\nPrevious studies61have shown that Gilbert damping is\na crucial parameter in the ultrafast switching procedure\nand, thus, call for ab-initio footing. Figure 7 shows the8\nGilbert damping \u000bas a function xatT= 300 K. Note\nthat for these calculations both lattice and magnetic \ruc-\ntuations terms are considered, where the magnetic \ructu-\nations are assumed from a linear correlation between the\nmagnetization and the temperature. This could result in\nerrors, in particular at high temperatures.\n012345α·10−3\n0.00.40.81.21.6\nn(EF) (eV)\n0.0 0.05 0.1\nxMn 2−2xX2xCoAl2345678910α·10−3\n2.83.23.64.04.4\nn(EF) (eV)Mn 2−2xX2xVAl\nFIG. 7. (Color online) Gilbert damping parameters \u000b\n(solid lines) and density of states at the Fermi level n(EF)\n(dotted lines) of Mn 2\u0000xZxCoAl (triangular symboles) and\nMn2\u0000xZxVAl (circle symboles) vs dopand concentration x.\nDopands are W (red color), Ru (blue color), Mo (green color),\nand Os (orange color).\nThe Gilbert damping of both undoped Heusler materi-\nals (Mn 2CoAl:\u000b= 0:0030, Mn 2VAl:\u000b= 0:0029) is high\ncompared to other half-metals reported, e.g., in Ref. [66]\nor low-damping alloys like Fe 0:75Co0:2573. The trends of\nthe Gilbert damping parameters with dopant concentra-\ntion are di\u000berent for Heusler and inverse Heusler materi-\nals. In Mn 2\u0000xZxVAl, doping leads to an increase of the\ndamping with x, except for the case of Ru. The slope\nof\u000bversus concentration xfollows the general increase\nof the total density of states at the Fermi level as it is\nproposed in Refs. [33, 73, and 74], but not linear to it.\nThis non-linearity was already observed for Heusler ma-\nterials in Ref. [66] or doped permalloy with the heavy\n4d and 5d elements used here75. The observed damping\n\u000bis di\u000berent from zero, however, small. This is in line\nwith the theory proposed in Ref. [74], in which damping\nis proportional to the product of the spin-polarised DOS\nand, consequently \u000b\u00190. The increase of damping can be\nalso understood in terms of the Kambersk\u0013 y model76,77:\nAlloying broadens the electron bands and more spin-\rip\ntransitions between the electron states occur. This is\ntrue only, if interband transitions are already dominat-\ning. In the inverse Heusler material Mn 2CoAl we even\n\fnd a decrease with x. This is due to the spin-gaplesssemiconducting behaviour (cf. Fig. 2): Only a low num-\nber of states exist at the Fermi energy, making interband\ntransitions unlikely. The damping is dominated by intra-\nband transitions, that tend to decrease with very small\nx. With increasing x, however, states appear within the\ngap and interband transition are preferred. Thus, a small\nincrease with even higher concentration is expected and\nobserved. However, not only the number of states at the\nFermi energy and the spectral width of the states con-\ntribute to the damping, but also the spin-orbit coupling\n(SOC), the Land\u0013 e factor, and the saturation magnetiza-\ntion a\u000bect the damping parameter. Since we dope with\nrather heavy elements W, Mo, Ru, and Os, spin orbit\ncoupling strongly contributes to the variation of damp-\ning with concentration x: the higher the `mass' of the\ndopant atom (W and Os compared to Ru and Mo) is,\nthe higher is the damping parameter.\nAfter we addressed all relevant parameters for the sim-\nulation based on the Landau-Lifshitz-Gilbert equation,\nwe are able to perform ultrafast switching calculations.\nE. Ultrafast switching\n-1.0-0.50.00.51.0M/M s\n0 5 10 15 20\nt (ps)V\nMn1\nMn2-1.0-0.50.00.51.0M/M sCo\nMn1\nMn2\nFIG. 8. (Color online) Ultrafast switching behaviour of\nMn2CoAl (upper panel) and Mn 2VAl (lower panel). The de-\nmagnetization is shown element resolved (blue and green lines\n- Mn atoms, red line - Co/ V atom). The peak temperature is\n600 K for Mn 2VAl and 900 K for Mn 2CoAl. The external mag-\nnetic \feld is B= 2:5 T and damping parameter is \u000b= 0:009.\nThe arrow indicates the crossing point at where the switching\ntakes place.9\n7007508008509009501000TP(K)\n0 2 4 6 8 10\nB(T)α= 0.006\n0 2 4 6 8 10\nB(T)α= 0.009\n0.02.04.06.08.0\nt(ps)\n450500550600650700750TP(K)\n0 2 4 6 8 10\nB(T)α= 0.006\n0 2 4 6 8 10\nB(T)α= 0.009\n0.02.04.06.08.0\nt(ps)b)a)\nFIG. 9. (Color online) Thermal switching phase diagram for\ndi\u000berent damping parameter (0.006 and 0.009) in a) Mn 2CoAl\nand b) Mn 2VAl. The peak temperature is represented versus\nthe strength of the external magnetic \feld. The colour scale\n(fast switching - blue colour, slow switching - red colour) rep-\nresents the time in units of ps where the switching (indicated\nby an arrow in Fig. (8)) takes place. No switching is repre-\nsented by the black background.\nIn order to study the ultrafast switching process in\nHeusler alloys we combined the two temperature model\nwith an atomistic spin dynamics code78. Here, we consid-\nered a very long thermal pulse of 20 ps with di\u000berent peak\ntemperatures TP. Typical timescales of the ultrafast de-\nmagnetization and remagnetization process for Mn 2VAl\nand Mn 2CoAl are in the orders of picoseconds (1 \u00005 ps)\n(see Fig. 8). The time scales are mainly dictated by the\nGilbert damping \u000b, which is varied in our studies between\n0:003, 0:006, and 0:009, but can depend on the Heisen-\nberg exchange12. As demonstrated above, these damping\nvalues are achievable by doping the `pure' Heusler mate-\nrials. There is only a slight shift observable in the demag-\nnetization time of each individual element in Mn 2CoAl,\nwhere for Mn 2VAl, it is not. After demagnetization, the\nHeusler material undergoes reliable switching only when\nan external magnetic \feld induced by the pump-pulse is\npresent. Thus, three parameters | damping, peak tem-\nperature and pulse induced external magnetic \feld |\nspan a phase space for observing reliable switching, as\nshown in Fig. 9.\nWe did not observe any magnetic switching for both\nHeusler materials with \u000b= 0:003 (data not shown here).\nTypically for certain threshold peak temperatures TP\nabove the magnetic phase transition temperature ( TC=\n700 K for Mn 2CoAl andTC= 475 K for Mn 2VAl) switch-\ning occurs. The peak temperature can be tuned by the\nlaser intensity and the pulse duration. The presence of\nan e\u000bective magnetic \feld during pumping is discussedin literature79,80. It was argued that the electric \feld of\nthe pump pulse induces a strong material speci\fc mag-\nnetic \feld of 10\u0000100 T. Even below but above certain\nminimum magnetic \feld of 1 \u00002 T, we observed reliable\nswitching. This threshold magnetic \feld as well as the\nswitching time (indicated by reduced contrast in Fig. 9)\ndecreases with increasing damping. The time when the\nswitching occurs (crossing point in Fig. 8 and colour scale\nin Fig. 9) typically passes a maximum at certain and de-\ncreases for larger peak temperatures. However, there is\nalso a minimum switching time of around 2 \u00003 ps, con-\ntrolled by the demagnetization rate. Note that due to the\ndi\u000berent spin polarization and resulting di\u000berent atomic\nmagnetic moments and magnetic states, an asymmetry\nin the phase diagram between Mn 2CoAl and Mn 2VAl oc-\ncurs.\nNevertheless, our approach has certain limitations. For\ninstance, we explicitly neglect the electronic motion and\ne\u000bects like super di\u000busion or spin-\rip scattering, as dis-\ncussed in Ref.25. We also assume the damping to be `spin-\nand phonon-temperature' independent. This is a rough\napproximation, in particular, due to the important role of\nphonons in the demagnetization process (e.g. Ref. [81])\nand for energy dissipation in magnetic systems33. Fur-\nthermore, we neglect the change of the magnetic ex-\nchange interaction with temperature, although magnetic\nmoments of Co and V atoms vanish in the DLM approx-\nimation. This behaviour in the disordered local moment\ntheory is well studied71and occurs also for Ni atoms. But\nwe have shown elsewhere58but also others82{84, that our\nmethodology is applicable for demagnetization in bulk\nbcc Fe and hcp Co compounds and, likely, for the Heusler\nmaterials studied here. We also neglect possible struc-\ntural phase transition to A2 or B2 disorder during de-\nmagnetization.\nIV. CONCLUSION\nWe have demonstrated thermal switching in Heusler\nand inverse Heusler materials making use of magnetic\n\feld pulse induced by the pump-pulse. We found a sensi-\ntive dependence of the possible switching and the switch-\ning time on the magnetic \feld pulse strength, the peak\ntemperature in the e\u000bective two-temperature model as\nwell as intrinsic materials properties, say the Heisenberg\nexchange and the Gilbert damping parameter. We have\nshown that the latter can be tuned by doping heavy ele-\nments, say W, Mo, Ru, Os, to both, higher and lower\ndamping values, especially in the case of spin-gapless\nsemiconductor. This calls for further investigations on\nother spin-gapless semiconductor59, aiming for tuning the\nGilbert damping to very low values, which may enable in-\nteresting spintronic and magnonic applications85. Within\nour methodology, we could reproduce exchange parame-\nter and, consequently, phase transition temperatures re-\nported in literature59. Our overall \fnding extends with\nHeusler and inverse Heusler alloys the class of materials10\nthat exhibits laser induced magnetic switching and calls\nfor future theoretical and experimental studies.V. ACKNOWLEDGEMENT\nWe acknowledge \fnancial support from the Swedish\nResearch Council. O.E. and and E.K.D.-Cz. ac-\nknowledged support from KAW (projects 2013.0020\nand 2012.0031) as well as acknowledges eSSENCE and\nSTandUP. The calculations were performed at NSC\n(Link oping University, Sweden) under a SNAC project.\n1E. Beaurepaire, J. C. Merle, A. Daunois, and J. Y. Bigot,\nPhys. Rev. 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Although the effects of the renormalization for\nnonchiral ferromagnets are usually minor and hardly detectable, we show that the renormalization does play a\ncrucial role for chiral magnets. Here the renormalization is chiral and as such we predict experimentally identi-\nfiableeffectsonthephenomenologyofmagnetizationdynamics. Inparticular,ourtheoryfortheself-consistent\nmagnetization dynamics of chiral magnets allows for a concise interpretation of domain wall creep motion. We\nalso argue that the conventional creep theory of the domain wall motion, which assumes Markovian dynamics,\nneedscriticalreexaminationsincethegyromagneticratiomakesthemotionnon-Markovian. Thenon-Markovian\nnature of the domain wall dynamics is experimentally checkable by the chirality of the renormalization.\nRenormalization is a useful concept to understand interac-\ntion effects between a physical system and its environment.\nIn metallic ferromagnets, magnetic moments experience such\nrenormalization due to their coupling to conduction electrons\nthrough exchange interactions. Spin magnetohydrodynamic\ntheory [1–3] examines the renormalization of dynamical pa-\nrameters in the Landau-Lifshitz-Gilbert (LLG) equation as\nfollows. Magnetization dynamics exerts a spin motive force\n(SMF) [4, 5] on conduction electrons, and the resulting spin\ncurrentgeneratesspin-transfertorque(STT)[6–8]thataffects\nthemagnetizationdynamics itself. Thisself-feedbackofmag-\nnetizationdynamics[9]renormalizestheGilbertdampingand\nthe gyromagnetic ratio. However, its consequences rarely go\nbeyond quantitative corrections in nonchiral systems [10–14]\nand are commonly ignored.\nChiralmagnetsareferromagnetsthatpreferaparticularchi-\nrality of magnetic texture due to spin-orbit coupling (SOC)\nand broken inversion symmetry. Examples include ferro-\nmagnets in contact with heavy metals, such as Pt [15] and\nthose with noncentrosymmetric crystal structures [16]. Mag-\nnetization dynamics in chiral magnets are usually described\nby generalizing the conventional LLG equation to include\nthe chiral counterpart of the exchange interaction called the\nDzyaloshinskii-Moriya interaction (DMI) [17–19] and that of\nSTTcalledspin-orbittorque(SOT)[20–23]. Thisdescription\nis incomplete, however, since it ignores the renormalization\nby the self-feedback of magnetization dynamics. Although\nthe renormalization in chiral magnets has been demonstrated\ntheoretically for a few specific models [24–27], most experi-\nmentalanalysesofchiralmagnetsdonottakeintoaccountthe\nrenormalization effect.\nInthiswork,wedemonstratethattherenormalizationinchi-\nralmagnetsshouldbechiralregardlessofmicroscopicdetails\nand these effects should be nonnegligible in chiral magnets\nwith large SOT observed in many experiments [21–23, 28–30]. Unlike in nonchiral systems, the chiral renormalization\ngenerates experimentally identifiable effects by altering the\nphenomenology of magnetization dynamics. This provides a\nuseful tool to experimentally access underlying physics. We\nillustratethiswiththefield-drivenmagneticdomainwall(DW)\nmotion with a controllable chirality by an external magnetic\nfield [31, 32]. We find that not only is the steady state DW\nvelocity chiral due to the chiral damping [25], but also the\neffective mass of the DW [33] is chiral due to the chiral gy-\nromagnetic ratio. The chiral gyromagnetic ratio also signifi-\ncantly affects the DW creep motion, which is one of the tech-\nniques to measure the strength of the DMI [32]. We argue\nthat the chiral gyromagnetic ratio is the main mechanism for\nthe non-energetic chiral DW creep velocity [34], contrary to\nthe previous attribution to the chiral damping [25, 34]. We\nalso highlight the importance of the tilting angle excitation\nand its delayed feedback to the DW motion. This has been\nignoredinthetraditionalcreeptheory[35,36]foralongtime,\nsince its effects merely alter the velocity prefactor which is\nindistinguishable from other contributions, such as the impu-\nrity correlation length [37]. However, in chiral magnets, it is\ndistinguishablebymeasuringtheDWvelocityasa functionof\nchirality (not a single value).\nTo get deep insight into the chiral renormalization, we\nadopt the self-feedback mechanism of magnetization dynam-\nics through conduction electrons and develop a general, con-\ncise, and unified theory for chiral magnets. There are several\nprevious reports on the anisotropic or chiral renormalization\nof the magnetic damping [24–26, 38] and the gyromagnetic\nratio[27,38,39]intheRashbamodel[40]. Tounifyandgen-\neralize the previous works, we start from the general Onsager\nreciprocityrelationandpredictallthecoreresultsoftheprevi-\nous reports. Our theory can be generalized to situations with\nanyphenomenologicalspintorqueexpression,whichcaneven\nbedeterminedbysymmetryanalysisandexperimentswithoutarXiv:1803.00017v2  [cond-mat.mes-hall]  14 Mar 20182\nMagnetization under\nchiral self-feedback Effective  equation of \nmotion for magnetization SOT \nchiral \nSMF\nLLG ( γ, α)\nchiral LLG ( ζγ , G )(a) (b) \nFIG.1. (a)Magnetizationdynamicsdescribedbytheunrenormalized\nLLG equation. The dynamics of magnetization and that of electrons\narecoupledtoeachotherbytheexchangeinteraction. (b)Aftertracing\nout the electron degree of freedom, the gyromagnetic ratio ( \u0010\r) and\nthe magnetic damping ( G) are chirally renormalized [Eq. (1)].\nknowing its microscopic mechanism. We provide a tabular\npicture(SeeTableIbelow)forphysicalunderstandingofeach\ncontribution to the chiral renormalization. Furthermore, one\ncan utilize the generality of the Onsager relation to include\nmagnon excitations [26], thermal spin torques [41], and even\nmechanical vibrations [42] in our theory.\nToexaminetheconsequencesofthechiralrenormalization,\nwestartfromthefollowingrenormalizedLLGequation,which\nwe derive in the later part of this paper,\n(\u0010\r)\u00001\u0001@tm=\u0000m\u0002He\u000b+\r\u00001m\u0002G\u0001@tm+\r\u00001Text;(1)\nwhere mis the unit vector along magnetization, \ris the un-\nrenormalized gyromagnetic ratio, He\u000bis the effective mag-\nneticfield,and Textreferstospintorqueinducedbyanexternal\ncurrent.\u0010andG,whicharegenerallytensorsandfunctionsof\nmand its gradients, address respectively the renormalization\nofthegyromagneticratioandthemagneticdamping,depicted\nin Fig. 1. If the renormalization is neglected, Eq. (1) reduces\nto the conventional LLG equation with \u0010= 1andG=\u000b,\nwhere\u000bistheunrenormalizedGilbertdamping. Otherwise \u0010\nandGare dependent on the chirality of magnetic texture. At\nthe end of this paper, we show that the chiral renormalization\nis completely fixed once the expressions of STT and SOT are\ngiven.\nWe first examine implications of the chiral renormaliza-\ntion on a few exemplary types of field-driven DW dynamics\n(Fig.2). Westartfrom He\u000b=H0+Hext+Hth,where H0is\nthe energetic contribution (without an external field), Hext=\n(Hx;0;Hz)is the external field, and Hthis a thermal fluctu-\nation field. We use the DW profile m(x) = (sin\u001esech[(x\u0000\nX)=\u0015];cos\u001esech[(x\u0000X)=\u0015];tanh[(x\u0000X)=\u0015])whereX,\n\u001eand\u0015aretheposition,thetiltingangle,andthewidthofthe\nDW, respectively. Taking Xand\u001eas the collective coordi-\nxyz\nφφ\nHxHz v(   )FIG. 2. Chiral dynamics of a DW between domains with m=\u0007^z\n(red and blue respectively). The DW chirality is characterized by\nthe DW tilting angle \u001e[the positivity (negativity) of \u001ecorresponds\nto the left-handed (right-handed) chirality], and can be controlled\nby an in-plane field ( Hx). The DW motion is driven by an applied\nfield (Hz). Measuring the DW velocity as a function of \u001e(orHx),\nthe difference between v(\u001e)andv(\u0000\u001e)gives the information of the\nchiral renormalization.\nnates, Eq. (1) gives\n\u000bX\ne\u000b\n\u0015dX\ndt+1\n\u0010e\u000bd\u001e\ndt=FX+\u0018X; (2a)\n\u00001\n\u0010e\u000bdX\ndt+\u000b\u001e\ne\u000b\u0015d\u001e\ndt=F\u001e+\u0018\u001e; (2b)\nwhereFX=\u001e= (\r=2)R\n(H0+Hext)\u0001(@X=\u001em)dxrefertothe\nforce onXand\u001e.\u0018X=\u001e= (\r=2)R\nHth\u0001(@X=\u001em)dxis the\nthermal force on Xand\u001e.\nTheeffectivedamping \u000bX=\u001e\ne\u000bandthegyromagneticratio \u0010e\u000b\nare given by\n\u000bX\ne\u000b=\u0015\n2Z\n(@Xm\u0001G\u0001@Xm)dx; (3a)\n\u000b\u001e\ne\u000b=1\n2\u0015Z\n(@\u001em\u0001G\u0001@\u001em)dx; (3b)\n\u0010\u00001\ne\u000b=1\n2Z\u0002\n(m\u0002@\u001em)\u0001\u0010\u00001\u0001@Xm\u0003\ndx:(3c)\nNote that without the chiral renormalization, Eq. (2) reduces\ntotheThieleequations[43]with \u000bX=\u001e\ne\u000b=\u000band\u0010e\u000b= 1. We\nemphasizethat \u000bX=\u001e\ne\u000band\u0010e\u000bdependonthetiltingangle \u001eand\nthus on the chirality of the DW. Figure 3 shows the \u001edepen-\ndenciesoftheseparameters. Theasymmetricdependenceson\n\u001econfirmtheirchiraldependences. Notethat,evenforpurely\nfield-drivenDWmotion,thechiraldependencesoftheparam-\netersaredeterminedbytheexpressionof current-induced spin\ntorque.\nWe first consider the steady-state dynamics of DW in the\nflow regime, where the effects of the pinning and the thermal\nforces are negligible. Then, translational symmetry along\nXguarantees the absence of contribution from H0toFX,\nthus only the external field contribution survives in the right-\nhand side of Eq. (2a), FX+\u0018X\u0019\u0000\rHz. In a steady state\n(d\u001e=dt = 0), Eq. (2a) gives the DW velocity as\nv\row=\u0000\r\u0015\n\u000bX\ne\u000bHz; (4)3\nα\u0001\u0002\u0002\u0001ϕα\u0001\u0002\u0002\u0001\u0001\nα\u0001\u0002\u0002ϕϕα\u0001\u0002\u0002ϕ\u0001\nζ\u0001\u0002\u0002\u0003ϕζ\u0001\u0002\u0002\u0003\u0001\n\u0001\u0002\u0003 \u0001\u0002\u0004 \u0001\u0002\u0001 \u0001\u0002\u0004 \u0001\u0002\u0003\u0001\u0002\u0005\u0001\u0002\u0004\u0001\u0002\u0006\u0001\u0002\u0001\u0001\u0002\u0006\u0001\u0002\u0004\u0001\u0002\u0005\nϕ π\u0007\b\tRighthandedchirality Left handedchirality\nFIG. 3. The effective dynamical parameters, \u000bX\ne\u000b(the red, solid\ncurve),\u000b\u001e\ne\u000b(the red, dashed curve), and \u0010\u00001\ne\u000b(the blue curve), as a\nfunction of the DW tilting angle \u001e. We take the phenomenological\nexpression of spin torque in magnetic bilayers [21–23, 30], which is\na typical example with large SOT: T= (\r~=2eMs)f(js\u0001r)m\u0000\n\f1m\u0002(js\u0001r)m+kSO(^z\u0002js)\u0002m\u0000\f2kSOm\u0002[(^z\u0002js)\u0002\nm]g, where each term refers to the adiabatic STT [44], nonadiabatic\nSTT[45,46],fieldlikeSOT[47,48],anddampinglikeSOT[30,49–\n51], induced by the spin current js. Here,Ms= 1000 emu =cm3is\nthesaturationmagnetization, e>0isthe(negative)electroncharge,\n^zistheinterfacenormaldirection, kSO= 1:3 (nm)\u00001characterizes\nthestrengthoftheSOT.Wetake \f1= 0:05,\f2= 5,\u0015= 8 nm,and\nthe electrical conductivity \u001b\u00001\n0= 6\u0016\ncm. The parameters are on\nthe order of the typical values for Pt/Co systems [28, 45, 52].\nwhichisinverselyproportionaltothechiraldamping \u000bX\ne\u000beval-\nuatedatthesteady-statetiltingangle \u001eeqforwhichd\u001e=dt = 0.\nAs\u001eeqcanbemodulatedby Hx,themeasurementof v\rowasa\nfunction ofHxprovides a direct test of the chiral dependence\non\u000bX\ne\u000b.\nAs an experimental method to probe the chiral dependence\nof\u0010e\u000b, we propose the measurement of the DW mass, called\nthe Döring mass [33]. It can be performed by examining\nthe response of DW under a potential trap to an oscillating\nfieldHz[53]. Unlike v\row,\u001eis not stationary for this case,\nand dynamics of it is coupled to that of X. Such coupled\ndynamicsof \u001eandXmakes\u0010e\u000brelevant. IntheSupplemental\nMaterial[54],weintegrateoutthecoupledequations[Eq.(2)]\nto obtain the effective Döring mass,\nmDW=1\n\u00102\ne\u000b2MsS\n\rjF0\n\u001e(\u001eeq)j; (5)\nwhereSis the cross-sectional area of the DW. Here, \u0010e\u000brep-\nresents a measurement of its value for \u001e=\u001eeq, which can be\nvariedbyHx.mDWprovidesanexperimentalwaytomeasure\nthe chiral dependence of \u0010e\u000b.\nIn the creep regime of the DW where the driving field is\nmuch weaker than the DW pinning effects, the implication of\nthechiralrenormalizationgobeyondmerelychiralcorrections\ntotheDWvelocity. TherecentcontroversiesonthechiralDW\ncreep speed vcreepmeasured from various experiments [32,\n34, 55, 56] require more theoretical examinations. Typically,vcreepis believed to follow the Arrhenius-like law vcreep =\nv0exp(\u0000\u0014H\u0000\u0016\nz=kBT)[35, 36], where v0is a prefactor, \u0016is\nthe creep exponent typically chosen to be 1/4 [57], and \u0014is\na parameter proportional to the DW energy density. Based\non the observation that the DMI affects \u0014, an experiment [32]\nattributedthechiraldependenceof vcreeptotheDMI.However\nlater experiments [34, 55, 56] found features that cannot be\nexplained by the DMI. In particular, Ref. [34] claimed that\nthe chiral dependence of vcreepis an indication of the chiral\ndamping [25], based on the observation v0/(\u000bX\ne\u000b)\u00001. On\nthe other hand, our analysis shows that the explanation of the\nchirality dependence may demand more fundamental change\nto the creep law, which assumes the dynamics of \u001eto be\nessentially decoupled from that of Xand thus irrelevant for\nvcreep. As a previous experiment on the DW creep motion in\na diluted semiconductor [58] argued the coupled dynamics of\n\u001eandXto be important, it is not a prioriclear whether the\nassumptionofdecoupling Xand\u001eholdsinthecreepregime.\nWe consider the coupling between the dynamics of Xand\n\u001easfollows. Afterthedynamicsof Xexcites\u001e,thedynamics\nof\u001eresults in a feedback to Xwith a delay time \u001c. Since the\ndynamics at a time tis affected by its velocity at past t\u0000\u001c,\nit is non-Markovian. The traditional creep theory takes the\nMarkovian limit ( \u001c!0), thus\u001e=\u001eeqat any instantaneous\ntime,decoupledfromthedynamicsof X. Toshowthecrucial\nroleofafinitefeedbacktime \u001c,wecalculatetheescaperateof\nthe DW over a barrier, which is known to be proportional to\nv0[37] and apply the Kramer’s theory [59] for barrier escape\nand its variations for non-Markovian systems [60, 61]. After\nsomealgebraintheSupplementalMaterial[54],Eq.(2)gives\nv0/\u001a(\u000bX\ne\u000b)\u00001\u001c\u00170\u001c\u00102\ne\u000b\u000bX\ne\u000b\u000b\u001e\ne\u000b(Markovian );\n\u0010e\u000b\u001c\u00170&\u00102\ne\u000b\u000bX\ne\u000b\u000b\u001e\ne\u000b(non-Markovian );\n(6)\nwhere\u00170is called the reactive frequency [61] and is on the\norder of 2\u0019times the attempt frequency ( \u00191 GHz[37]). We\nemphasizethatthetworegimesshowverydifferentbehaviorin\nthesenseofunderlyingphysicsaswellasphenomenology. The\nvalidityoftheMarkovianassumptiondependsonthetimescale\nof\u001ccomparedto \u00102\ne\u000b\u000bX\ne\u000b\u000b\u001e\ne\u000b. Sincethedampingissmall,itis\nnotguaranteedforoursituationtobeintheMarkovianregime.\nIndeed,wedemonstrateintheSupplementalMaterial[54]that\nthesecondregime(non-Markovian)inEq.(6)ismorerelevant\nwithrealisticvalues,thusthechiralityof v0mainlyoriginates\nfrom the gyromagnetic ratio, not the damping [34]. One can\nmeasure the chiral dependence of \u000bX\ne\u000band\u0010e\u000bfrom the flow\nregime[Eqs.(4)and(5)]andcomparetheirchiraldependences\ntothecreepregimetoobservethenon-Markoviannatureofthe\nDWdynamics. Thisadvantageoriginatesfromthepossibility\nthatonecanmeasuretheDWvelocityasa functionofchirality,\nin contrast to nonchiral magnets where one measures the DW\nvelocity as a single value.\nSo far, we present the role of the chiral renormalization for\ngiven renormalized tensors Gand\u0010. To provide underlying\nphysical insight into it, we present a analytic derivation of\nEq. (1) in general situations. We start from the LLG equation4\n\r\u00001@tm=\u0000m\u0002He\u000b+\r\u00001\u000bm\u0002@tm+\r\u00001Tandreferto\nthe scenario illustrated in Fig. 1. Note that There includes a\ncontribution from an internally generated SMF ( Tint) as well\nas that from an external current [ Textin Eq. (1)]. We write\ndown the spin torque in a general current-linear form T=\n\u0000(\r~=2eMs)m\u0002P\nuAu(m)js;u,whereurunsoverx;y;z.\nHere the spin current jsis split into an internally generated\nSMF [4, 5] js;intand the external current js;ext. The former\nisproportionalto @tm,thusitrenormalizesthegyromagnetic\nratio and the damping. The latter generates Textin Eq. (1).\nTheexpressionof js;intisgivenbytheOnsagerreciprocityof\nSTT and SMF [62]: js;int;u=\u0000(\u001b0~=2e)Au(\u0000m)\u0001@tm,\nwhere\u001b0is the charge conductivity [63]. Substituting this to\nTint= (\r~=2eMs)m\u0002P\nuAu(m)js;int;ugivestheeffective\nLLGequation \r\u00001@tm=\u0000m\u0002He\u000b+\r\u00001m\u0002A\u0001@tm+\n\r\u00001Text, whereA=\u000b+\u0011P\nuAu(m)\nAu(\u0000m),\u0011=\n\r~2\u001b0=4e2Msand\nisthedirecttensorproduct. Asaresult,\nTintis taken care of by renormalizing \u000bintoAin the LLG\nequation.\nTherenormalizeddampingandgyromagneticratioaregiven\nby separating different contributions of Awith different time\nreversal properties. A damping contribution is required to\nbe dissipative (odd in time reversal), whereas a gyromagnetic\nterm should be reactive (even in time reversal). Separating\nthese gives Eq. (1) where G= (A+AT)=2and\u0010\u00001=\n1\u0000m\u0002(A\u0000AT)=2. Theparticularchoicefortheadiabatic\nSTTandthenonadiabaticSTT Au(m) =m\u0002@um+\f@um\nreproduces the renormalized LLG equation for nonchiral sys-\ntems [1–3]. When one uses Au(m)for a particular chiral\nsystem, Eq. (1) produces the effective LLG equation for it, as\nreported by a numerical study for a one-dimensional Rashba\nmodel [27].\nIn chiral magnets, it is known that spin torque includes two\nmore contributions called fieldlike SOT [47, 48] and damp-\ninglikeSOT[30,49–51]. Thecharacterizationoffieldlikeand\ndampinglikeSOTisregardlessofthechoiceofSOC,sinceitis\ndetermined by the time reversal characteristic. Since Au(m)\nconsistsoffourcontributions,thereare16contributionsinthe\nfeedback tensor \u0001A=\u0011P\nuAu(m)\nAu(\u0000m)for each\nu. We tabulate all terms of \u0001Ain Table I. The contributions\nSTT:Ax(m)\nSMF:\nAx(\u0000m)Adiabatic\nm\u0002@xmNonadiabatic\n\f1@xmFLT\nkSOm\u0002(^y\u0002m)DLT\n\f2kSO^y\u0002m\nm\u0002@xmG\u0010\u00001G\u0010\u00001\n\u0000\f1@xm\u0010\u00001G\u0010\u00001G\nkSOm\u0002(^y\u0002m)G\u0010\u00001G\u0010\u00001\n\u0000\f2kSO^y\u0002m\u0010\u00001G\u0010\u00001G\nTABLE I. Example characterization of contributions in Ax(m)\nAx(\u0000m). Counting orders of gradients and mgives the conven-\ntional (white), chiral (lighter gray), or anisotropic (darker gray) con-\ntributions to the gyromagnetic ratio ( \u0010\u00001) or the damping (G). The\nform of the fieldlike SOT (FLT) and dampinglike SOT (DLT) are\ntaken from magnetic bilayers [30, 47–51] for illustration, but the\ncharacterization procedure works generally.withthewhitebackgroundarezerothorderinSOCbutsecond\norder in gradient and are the conventional nonlocal contribu-\ntions [3, 65]. Those with the lighter gray background are first\norder in gradient and chiral [27]. Those with the darker gray\ncolor are zeroth order in gradient and anisotropic [66]. In\nthisway,ourtheoryprovidesaunifiedpictureontheprevious\nworks. Whether a term contributes to \u0010\u00001orGis deter-\nmined by the order in m. After a direct product of STT and\nSMF, a term even (odd) in mgivesG(\u0010\u00001), since it gives\na time irreversible (reversible) contribution appearing in the\nLLGequationas m\u0002A\u0001@tm. Thesameanalysiswithsimple\norder countings works for any Au(m). It holds even if our\ntheory is generalized to other physics, such as magnons [26],\nthermal effects [41], and mechanical effects [42].\nAsanexampleofapplicationsofTableI,weadoptthespin-\nHall-effectdrivenSOT[21,67,68],wherealargedampinglike\nSOTarises. FromTableI,onecanimmediatelyfigureoutthat\nthecombinationofthedampinglikeSOTandtheconventional\nSMF(themosttoprightcell)givesachiralgyromagneticratio\ncontribution. As another example, one notices that the com-\nbination of the dampinglike SOT and its Onsager counterpart\n(the fourth term in the SMF) gives an anisotropic damping\ncontribution. Note that the Onsager counter part of the spin-\nHall-effect driven SOT is the inverse spin Hall effect driven\nby spin pumping current generated by the magnetization dy-\nnamics. In this way, Table I provides useful insight for each\ncontribution.\nTable I also allows for making the general conclusion that\nthe magnitude of the chiral gyromagnetic ratio is determined\nbythesizeofthedampinglikeSOT( \f2)andthatofthenona-\ndiabatic STT ( \f1). This is an important observation since\nmany experiments on magnetic bilayers and topological in-\nsulators [21–23, 30] shows a large dampinglike SOT. This\nconclusionisregardlessofthemicroscopicdetailsoftheSOT,\nbecauseadampinglikecontributionissolelydeterminedbyits\ntime-reversal property.\nTo summarize, we demonstrate that the chiralities of the\ngyromagnetic ratio and Gilbert damping have significant im-\nplicationswhichgofurtherbeyondmerelythechangeinmag-\nnetization dynamics. The chirality plays an important role in\ninvestigating underlying physics because physical quantities,\nwhich were formerly treated as constants, can now be con-\ntrolled through their chiral dependence. An example is the\nnon-Markovian character of the DW creep motion, which is\ndifficult to be verified in nonchiral systems. From the non-\nMarkovian nature of the DW creep motion, we conclude that\nthe non-energetic origin of chiral DW creep originates from\nthe chiral gyromagnetic ratio rather than the chiral damping.\nWealsoprovideageneral,concise,andunifiedtheoryoftheir\nchiralities, which provide useful insight on the self-feedback\nof magnetization.\nWe acknowledge M. D. Stiles, Y. Tserkovnyak, A. Thiav-\nille, S.-W. Lee, V. Amin, and D.-S. Han for fruitful discus-\nsion. This work is supported by the Alexander von Humboldt\nFoundation, the ERC Synergy Grant SC2 (No. 610115), the\nTransregionalCollaborativeResearchCenter(SFB/TRR)1735\nSPIN+X, and the German Research Foundation (DFG) (No.\nEV 196/2-1 and No. SI 1720/2-1). 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Phys. 87, 1213 (2015).Supplementary Materials for\n“Roles of chiral renormalization of magnetization dynamic s in chiral magnets\"\nKyoung-Whan Kim,1Hyun-Woo Lee,2Kyung-Jin Lee,3, 4Karin Everschor-Sitte,1Olena Gomonay,1, 5and Jairo Sinova1, 6\n1Institut für Physik, Johannes Gutenberg Universität Mainz, Mainz 5512 8, Germany\n2PCTP and Department of Physics, Pohang University of Science and Te chnology, Pohang 37673, Korea\n3Department of Materials Science and Engineering, Korea University, S eoul 02841, Korea\n4KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea\n5National Technical University of Ukraine “KPI\", Kyiv 03056, Ukraine\n6Institute of Physics, Academy of Sciences of the Czech Republic, Cuk rovarnická 10, 162 53 Praha 6 Czech Republic\nI. THE NON-MARKOVIAN NATURE OF THE DW DYNAMICS\nA. Integrating out φ\nIn the linear response regime, we may take Fφ≈ −|F′\nφ(φeq)|(φ−φeq)and the dynamical coefficients ζeffandαX/φ\neffto be\nevaluated at φ=φeq. Without loss of generality, we assume the initial conditio nX(0) = 0 andφ(0) = φeq. We then define the\nLaplace transforms L[f(t)](s) =/integraltext∞\n0e−stf(t)dt. We denote L[X] =QandL[φ−φeq] =P. Then the Laplace transform of\nEq. (2) in the main text is\nsαX\neff\nλQ+s\nζeffP=L[FX] +L[ξX], (S1a)\n−s\nζeffQ+sαφ\neffλP=−|F′\nφ(φeq)|P+L[ξφ], (S1b)\nEliminating Pin Eq. (S1) gives\n1\nζ2\neffs2\n|F′\nφ(φeq)|+sαφ\neffλQ+sαX\neff\nλQ=−γHz\ns+L[Fpin] +/parenleftBigg\nL[ξX]−s\nζeffL[ξφ]\nb+sαφ\neffλ/parenrightBigg\n, (S2)\nwhich is an equation of Xonly. Taking the inverse Laplace transform, we obtain the fo llowing non-Markovian equation:\n1\nλ/integraldisplayt\n0f(t−u)X′(u)du=FX+˜ξX(t). (S3)\nHere f(t)is a feedback function from φ, whose explicit form is\nf(t) =L−1/bracketleftBigg\nαX\neff+1\nζ2\neffαφ\neffsτ\n1 +sτ/bracketrightBigg\n=/parenleftBigg\nαX\neff+1\nζ2\neffαφ\neff/parenrightBigg\nδ(t)−1\nζ2\neffαφ\neffτe−t/τΘ(t), (S4)\nandτ=αφ\neffλ/|F′\nφ(φeq)|is the relaxation time of φdegree of freedom. The correlation relation for the effectiv e thermal\nfluctuation field ˜ξX(t)is given by the fluctuation-dissipation theorem ∝angbracketleft˜ξX(t)˜ξX(t′)∝angbracketright ∝Tf(|t−t′|)where Tis the temperature.\nThe noise is ‘colored’ in the sense that it is no longer a white random noise.\nB. Order-of-magnitude estimation of τ\nTo estimate the order of magnitude of τ, we use the fact that the magntude of |Fφ|is determined by the DMI or the hard axis\nanisotropy: |F′\nφ(φeq)| ≈γλ(π/2)×(2H⊥orDλ−1). We take the DMI field Dλ−1being 30 mT [1] for a rough estimation.\nThen, |F′\nφ(φeq)|/λ≈γ×30 mT ≈5 GHz , so that τ=αφ\neffλ/|F′\nφ(φeq)| ≈αφ\neff×0.2 ns, which is small compared to the time\nscale of the dynamics of X.2\nC. First order approximation - chiral mass correction\nSince τis small, compared to the times scale of the dynamics of X, we may expand f(t)byτ, in the sense of the gradient\nexpansion in time space. Then, f(t)≈ L[αX\neff+ (1/ζ2\neffαφ\neff)sτ] =αX\neffδ(t) + (τ/ζ2\neffαφ\neff)δ′(t). Putting this into Eq. (S3) gives\nτ\nζ2\neffαφ\neff1\nλd2X\ndt+αX\neff\nλdX\ndt=FX+˜ξX(t), (S5)\nwhere the first term represents a massive term. To obtain the D W mass, we need to find the factor which makes FXhave the\ndimension of force. Note that the force generated by pushing the DW is calculated by Ms/integraltext\nHeff·∂Xmd3x= (2MsS/γ)FX.\nTherefore, the mass is defined by multiplying the factor 2MsS/γ,\nmDW=1\nζ2\neff2MsSτ\nγαφ\neffλ, (S6)\nwhich is equivalent to Eq. (5) in the main text.\nD. Higher order contributions - chiral creep\nTo calculate v0, one needs to solve a barrier escape problem. For an energy ba rrierEb, Kramer [2] derived the thermal escapes\nrate\nΓ =ν\n2π/radicalBigg\n|F′(Xm)|\n|F′(XM)|e−Eb/kBT, (S7)\nwhere F′(Xm)andF′(XM)are the derivatives of the force (second derivatives of the p inning energy landscape) evaluated at\nthe potential well and the saddle point respectively. νis called the reactive frequency [3] which we calculate belo w. Then, v0\nis proportional to Γ. According to the Kramer’s theory, ν∝1/αX\nefffor a high damping and Markovian limit, which was also\nconfirmed by the functional renormalization group techniqu e [4].\nHowever, we generalize this result to a non-Markovian situa tion [Eq. (S3)]. To do this, we apply the theory of escape rate for\na non-Markovian equation of motion [3, 5], based on which, th e reactive frequency νcorresponding to Eq. (S3) is given by the\npositive root of the following algebraic equation:\n1\nλνL[f(t)](ν) =|F′(XM)|, (S8)\nwhose exact solution can be calculated from Eq. (S4). As a res ult,\nν=2ν0\n(1−τν0) +/radicalBig\n(1 +τν0)2+ 4τν0/ζ2\neffαX\neffαφ\neff≈\n\nν0∝1\nαX\neffν0τ≪ζ2\neffαX\neffαφ\neff,\nζeff/radicalBigg\nν0αX\neffαφ\neff\nτ∝ζeffν0τ/greaterorsimilarζ2\neffαX\neffαφ\neff,(S9)\nwhere ν0=λ|F′(XM)|/αX\neffis the reactive frequency for τ= 0. In the second limit, we assume that the damping parameters\nare small, thus the last term in the denominator in Eq. (S9) do minates the other terms in the denominator. The two limits sh ows\ncompletely different dependences of νon the dynamical parameters. Therefore, it is important to d etermine the relevant regime.\nAssuming Fpinis random, |F′(XM)| ≈ |F′(Xm)|in Eq. (S7) gives ν0/2πto be the typical attempt frequency ≈1 GHz [6].\nFrom τ≈αφ\neff×0.2 ns estimated above, we obtain τν0≈αφ\neffwhich is an order of magnitude larger than ζ2\neffαX\neffαφ\neff, thus the\nsecond regime in Eq. (S9) is more relevant, contrary to the tr aditional creep theory just taking τ= 0.\n[1] S.-G. Je, D.-H. Kim, S.-C. You, B.-C. Min, K.-J. Lee, and S.- B. Choe, Phys. Rev. B 88, 214401 (2013).\n[2] H. A. Kramers, Physica (Amsterdam) 7, 284 (1940).\n[3] E. Pollak, H. Grabert, and P. Hänggi, J. Chem. Phys. 91, 4073 (1989).\n[4] P. Chauve, T. Giamarchi, and P. Le Doussal, Phys. Rev. B 62, 6241 (2000)\n[5] R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 (1980).\n[6] K. Gorchon, S. Bustingorry, J. Ferré, V. Jeudy, A. B. Kolton, a nd T. Giamarchi, Phys. Rev. Lett. 113, 027205 (2014)."    },    {        "title": "1803.01280v2.Optimization_of_Time_Resolved_Magneto_optical_Kerr_Effect_Signals_for_Magnetization_Dynamics_Measurements.pdf",        "content": "Optimization of Time -Resolv ed Magneto -optical Kerr Effect S ignals for \nMagnetization Dynamics Measurements  \nDustin M. Lattery1, Delin Zhang2, Jie Zhu1, Paul Crowell3, Jian-Ping Wang2 and Xiaojia Wang1* \n1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN \n55455, USA   \n3School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA  \n*Corresponding authors: wang4940@umn.edu \n \nAbstract: Recently magnetic storage and magnetic memory have shifted towards the use of \nmagnetic thin films with perpendicular magnetic anisotropy (PMA). Understanding the magnetic \ndamping in these ma terials is crucial, but normal Ferromagnetic Resonance (FMR) measurements \nface some limitations. The desire to quantify the damping in materials with PMA has resulted in \nthe adoption of Time -Resolved Magneto -optical Kerr Effect (TR -MOKE) measurements. In t his \npaper, we discuss the angle and field dependent signals in TR -MOKE, and utilize a numerical \nalgorithm based on the Landau -Lifshitz -Gilbert (LLG) equation to provide information on the \noptimal conditions to run TR -MOKE measu rements . \n \nI. INTRODUCTION  \nSpintronics utilizing perpendicular magneti c anisotropy (PMA) are very promising for the \nadvancement of computer memory, logic, and storage. Due to the time scale of magnetic switching \nin these devices (~ 1 ns), it is crucial to understand the ultrafast dy namic magnetization, which \nbehave according to the Landau -Lifshitz -Gilbe rt (LLG) equation. The application of this equation to understand magnetization dynamics requires knowledge of the magnetic anisotropy and the \nGilbert damping (α). While anisotropy can  be determined through magnetostatic measurements, \nextracting α requires measurements that can capture the dynamic magnetization at time scales \nfaster than magnetic switching. To date, the most common method to do this is through frequency \ndomain measureme nts of ferr omagnetic resonance ( FMR ). By measuring the resonance frequency \nand linewidth as a function of field, FMR can probe both the magnetic anisotropy a nd Gilbert \ndamping . As spintronic  applications  begin to use materials  with large PMA, the use  of another \ntechnique, time -resolved magneto -optical Kerr effect (TR -MOKE), has increased. This technique \n(which is essentially a time -domain FMR measurement technique) is able to measure at higher \nresonance frequencies and external fields, which allows ex tremely hard mag netic materials to be \nmeasured . \nThere are many papers discussing TR -MOKE measurements for measuring the Gilbert \nDamping. Most of these papers utilize similar polar MOKE measurement techniques, but there is \noften a large variation in both  the Hext range for measurements and in the angle of external field. \nWhile some papers utilize in -plane external field because of its well -understood frequency \ndependence, others choose to apply the field at a chosen angle away from the surface normal . It \nhas been theorized and shown in measurements that the process of applying the field at some angle \nbetween 0 and 90°  is beneficial to increase the TR -MOKE signal amplitude, but the explanations \nas to why this occurs are lacking . In this paper, we aim to discuss why the signal depends on the \nangle of external field and calculate the optimal angle for conducting TR -MOKE measurements \nof damping on magnetic materials with PMA.  \n \n II. FINITE DIFFERENCE METHOD LANDAU -LIFSHITZ -GILBERT EQUATIONS  \nSimulations in this  work utilize a finite difference approach to solve the LLG equation \n(Eq. 1) with an explicit solution for the magnetization vector ( M) as a function of time following \nthe forward Euler method.  \neff\nsdd\ndt M dt    MMM H M\n          (1) \nwhere M is the magneti zation vector with a magnitude of Ms (the saturation magnetization), γ is \nthe gyromagnetic ratio, Heff is the effective magnetic field, and α is the Gilbert damping parameter. \nThe vector Heff is determined by taking the gradient of the magnetic free energy  density ( F) with \nrespect to the magnetization direction (\neff FM H ). The scalar quantity F is the summation of \ncontributions from Zeeman energy (from the external magnetic field, Hext), perpendicular uniaxial \nmagnetic anisotropy ( Ku), and  the demagnetizing field (assuming the sample is a magnetic thin \nfilm).  \nWhile Eqn. 1 is often used to describe magneto -dynamics due to the use of α, it is not \nconducive to numerical solutions of this ordinary differential equation. To simplify the \ndevelopment of computational algorithms, it is preferential to utilize the Landau -Lifshitz equation \n(Eq. 2).  \n  eff eff 2\ns'd\ndt M    MM H M M H\n.             (2) \nThe coefficients in Eq. 2 can be related to the previously defined constants in Eqs. 3 and 4 [1]. \n2'\n1\n\n\n           (3) \ns'M \n           (4) In equilibrium, M is parallel to Heff, and so the magnetization does not precess . If the \nmagnetization is removed from the equilibrium direction, it will begin precessing around the \nequilibrium direction, finally damping towards equilibrium at a rate determined by the magnitude \nof α (shown in Fig. 1) .  \n \nFigure 1 . A three -dimensional representation of the magnetization vector ( M) precessing around the equilibrium \ndirection ( θ) displayed on the surface of a sphere of radius Ms. The equilibrium direction  is controlled by the magnitude \nand direction ( θH) of the external magnetic field vec tor (Hext). The change in the z-component of magnetization (Δ Mz) \nis proportional to the TR -MOKE signal.  \n \nTo initiate precession, a thermal  demagnetization process  is applied , emulating  TR-MOKE \nmeasurements. For TR -MOKE measurements, a “pump” laser pulse increases the temperature at \nan ultrafast time scale, causing a thermal demagnetization (a decrease in Ms caused by temperature) \n[2, 3] . This thermal demagnetization temporarily moves the equilibrium direction causing the \nmagnetization to begin precession, which is continued even when Ms has recovered to its original \nstate.  Here, the demagnetization process is treated as a step decrease in Ms that lasts for 2.5 ps \nbefore an instant recovery to the initial value.  All signal analysis discussed in this work  is following \nthe recovery of Ms. \nFor polar MOKE measurements, the projected magnetization in the z -direction ( Mz, \nthrough -plane magnetization ) is proportional to the Kerr rotation  [4]. The projection of Mz in time \nduring precession will appear is a decaying sinusoid  (\n sin exp /zM t t t      ), which is \nalso captured by TR -MOKE measurements. The amplitude of the precession will greatly depend \non the applied field magnitude and angle, which is also carried into TR -MOKE signal. By \nanalyzing the precession as a function of field and angle, the precession amplitude (delta Mz) can \nbe extracted. Figure 2 shows the process of extracting the amplitude as a function of angle for two \ndifferent regions of magnetic field. Tracking this signal amplitude as a function of θH, reveals  that \nthe precession (and thus the signal) will be maximized for a certain θH as shown in Fig. 2(b). \nMaximizing the oscillation implies that it will be beneficial to maximize the “magnetic torque” \nterm (M × Heff, which prefers a large angle between M and Heff), but it also important to factor in \nthat TR -MOKE measures the projection of the magnetization along the z-direction (which prefers \nθ = 90°). Because of this, the value of θH,MAX  requires weighing inputs from both the magnetic \ntorque and the z-direction projection of  magnetization.  \n \nFigure 2. For specific conditions, the LLG simulation will produce a time -dependent magnetization vector. The \ndifference between the maximum and mini mum of the z-component of magnetization in time (Δ Mz) provides \ninformation about the strength of the TR -MOKE signal. These simulations are conducted for a range of θH resulting \nin the curves in (b). The trend of signal with increasing θH also depends on th e magnitude of the external field relative \nto Hk,eff, as shown by the black ( Hk,eff < Hext) and red ( Hk,eff > Hext) lines.  \nDepending on whether the field ratio ( Hext/Hk,eff) the angular dependence on magnitude \nwill drastically change. For Hext<Hk,eff, the magnetization will be in equilibrium between the \nperpendicular direction and the in -plane direction (0  ≤ θ ≤ 90°). Maximizing the magnetic torque \nand projection in the z -direction in these cases will cause Hext applied in -plane (θH = 90°) to be the \noptimal setup [shown by the red line in Fig. 4(b)]. Once Hext exceeds Hkeff, the Stoner -Wolfarth \nminimum energy model predicts that the magnetization will approach the direction of external \nfield, but never align (except along θH = 0 or 90 °). Because these two directions will have no \nmagnetic torque, there should be no magnetic precession, and thus there will be amplitude minima \nat these extremes. Between these two angles, the two effects for optimizing signal will complete, \nleading to a n amplit ude maximum at an angle that depends on the size of the ratio.  \nFigure 3 shows a contour plot of the dependence on  signal amplitude as a function of both \nthe magnitude of Hext and θH. The highest amplitude of precession will occur near Hk,eff when the \nfield is applied in the film plane. If the external field is greater than Hk,eff, it is beneficial to conduct \nthe measurement at an angle that is out of the film plane. To reveal this trend, the dotted red line \nin Fig. 3 indicates the angle of maximum signal ( θH,MAX ) as a function at specified field ratios. \nNote that the curve does not follow the gradient of signal vs. θH and Hext. This is due to the \ndefinition of θH,MAX  as the value of θH that maximizes signal for a given Hext , instead of a \nmaximization of signal with both parameters.  Based on these results, measurement conditions can \nbe tuned to maximize the signal based on the field ratio. For example, if the maximum strength of \nthe magnetic field is only 2 Hk,eff, then it would be beneficial to set θH > 60°. Furthermore, \nmeasurements conducted at a constant field and a varied magnetic field angle, should not \nnecessarily conduct the measurement at the highest possible Hext if the goal is to maximize SNR.   \nFigure 3. A contour plot of the relative signal size  as a function of field ratio ( Hext/Hk,eff) and θH where a value of “1” \nindicates the maximum possible signal. The dotted line shows the θH where the signal is maximized at a specific field \nratio.  \n \nFor field -swept measurements, (where the angle is held constant and the field is swept) \nFig. 4 should provide a simple guide for maximizing signals  (a summary of θH,MAX  in Fig. 3). To \nfurther assist in the design of TR -MOKE signals to maximize SNR, we suggest a simplified \nestimation for the determination of th e amplitude of TR -MOKE signal. Equation 5 predicts the \nprecession amplitude based on the equilibrium direction ( θ, from Fig. 1) and the external field \nangle. The magnitude of Hext is integrated into Eq. 5 through the θ through Eq. 6 whic h provides \nthe mini mum energy condition.  \nH\nssin sinzM\nM  \n      (5) \n  ext H k,eff2 sin sin 2HH  \n         (6) \nThis simplified expression is based on the product of the two components for signal \nmaximization previously discussed: the projection of the magnetization in the z -direction , \n sin , \nand the magnetic torque, \nH sin . While the simplified expression presented in Eq. 2 cannot \ncapture all the details of a more complex LLG simulation, it is more than accurate enough for an \ninitial estimate of θH,MAX , as shown by the comparison in Fig. 4.  \n \nFigure 4. The trend of θH,MAX  at a given field ratio. The open circles indicate results from the LLG simulation discussed \nin Section I, while the red curve is the simplified model from Eq. 5.  \n \nIII. COMPARING SIMULATION RESULTS TO TR -MOKE MEASUREMENTS  \nTo verify the precited results for the m aximum TR -MOKE signal amplitude, a series of \nmeasurements were conducted on a 300 °C post -annealed W/CoFeB/MgO film (see our previous \npublication for more information ). After conducting measurements, the thermal background was \nsubtracted leaving purely the decaying sinusoidal term. The oscillation amplitude from \nmeasurement was calculated as shown in Fig. 2a. Results from four values of Hext and six value s \nof θH are summarized in Fig. 5.  \n \nFigure 5. Normalized TR -MOKE oscillation amplitudes directly for a W/CoFeB/MgO  when Hext is 4, 6, 8, and \n10 kOe. The open red circles show the measurement data (a line between points is provided to guide the eye) while \nthe black curves indicate the results from the LLG simulations for a material with Hk,eff ≈ 6 kOe.  \n  \nComparisons between the trends predicted simula tions and measurement results show \nremarkable agreement. As expected, the signal amplitude decreases with increasing angle for \nHext < Hk,eff (Hk,eff ≈ 6 kOe ) and decreases with increasing angle for Hext > Hk,eff. These \nmeasurements can even capture the predicted peak of amplitude at nearly the same θH for fields \nnear Hk,eff. For the 6 kOe measurements, there is a slight deviation in the amount of decay in signal \nstrength for decreasing θH (simulations predict a s lower decrease). This is most likely due to an \ninhomogeneous broadening effect (i.e. the Hk,eff in the sample has a distribution of values) leading \nto a deviation from theory near Hk,eff. While the θH in the setup used in this experiment was limited, \nthese  results verify that the excellent agreement between simulation and measurement.  \n \nIV.  CONCLUSION  \nIn conclusion, we utilized a numerical approach to calculate the dynamic  response of \nmagnetization to a demagnetization process. We find that the size of the magnetic precession, and \nthus the size of the TR -MOKE signal depends on the angle and amplitude of the external field \n(relative to Hk,eff). To verify the results of these  simulations, we conducted measurements on a \nW/CoFeB/MgO sample with perpendicular magnetic anisotropy. The results of the measurements \nshow that the magnitude of the TR -MOKE signal shows good agreement with our prediction. \nThese results should assist to m aximize the SNR in TR-MOKE measurements.  \n \nACKNOWLEDGEMENTS  \nThis work is supported by C -SPIN  (award #: 2013 -MA-2381) , one of six centers of STARnet, a \nSemiconductor Research Corporation progra m, sponsored by MARCO and DARPA.  \n \nREFERENCES  \n[1] Iida, S., 1963, \"The difference between gilbert's and landau -lifshitz's equations,\" Journal of \nPhysics and Chemistry of Solids, 24(5), pp. 625 -630. \n[2] van Kampen, M., Jozsa, C., Ko hlhepp, J. T., LeClair, P., Lagae, L., de Jonge, W. J. M., and \nKoopmans, B., 2002, \"All -Optical Probe of Coherent Spin Waves,\" Physical Review Letters, \n88(22), p. 227201.  \n[3] Zhu, J., Wu, X., Lattery, D. M., Zheng, W., and Wang, X., 2017, \"The Ultrafast Laser Pump -\nProbe Technique for Thermal Characterization of Materials With Micro/Nanostructures,\" \nNanoscale and Microscale Thermophysical Engineering, 21(3), pp. 177 -198. \n[4] You, C. -Y., and Shin, S. -C., 1998, \"Generalized analytic formulae for magneto -optical Kerr \neffects,\" Journal of Applied Physics, 84(1), pp. 541 -546. \n "    },    {        "title": "1803.10064v2.Dynamics_of_a_Magnetic_Needle_Magnetometer__Sensitivity_to_Landau_Lifshitz_Gilbert_Damping.pdf",        "content": "arXiv:1803.10064v2  [physics.gen-ph]  19 Oct 2018Dynamics of a Magnetic Needle Magnetometer: Sensitivity to\nLandau–Lifshitz–Gilbert Damping\nY. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nAn analysis of a single-domain magnetic needle (MN) in the pr esence of an external magnetic\nfieldBis carried out with the aim of achieving a high precision magn etometer. We determine the\nuncertainty ∆ Bof such a device due to Gilbert dissipation and the associate d internal magnetic\nfield fluctuations that give rise to diffusion of the MN axis dir ectionnand the needle orbital angular\nmomentum. The levitation of the MN in a magnetic trap and its s tability are also analyzed.\nA rigid single-domain magnet with large total spin,\ne.g.,S≃1012/planckover2pi1, can be used as a magnetic needle magne-\ntometer (MNM). Recently Kimball, Sushkov and Budker\n[1] predicted that the sensitivity of a precessing MNM\ncan surpass that of present state-of-the-art magnetome-\nters by orders of magnitude. This prediction motivates\nour present study of MNM dynamics in the presence of\nan external magnetic field B. Such analysis requires in-\nclusion of dissipation of spin components perpendicular\nto the easy magnetization axis (Gilbert damping). It is\ndue to interactions of the spin with internal degrees of\nfreedom such as lattice vibrations (phonons), spin waves\n(magnons), thermal electric currents, etc. [2, 3]. Once\nthere is dissipation, fluctuations are also present [6], and\nresult in a source of uncertainty that can affect the ac-\ncuracy of the magnetometer. Here we determine the un-\ncertainty in the measurement of the magnetic field by a\nMNM. We also analyze a related problem concerning the\ndynamics of the needle’s levitation in an inhomogeneous\nmagnetic field, e.g., a Ioffe-Pritchard trap [8].\nThe Hamiltonian for a MN, treated asa symmetric top\nwith body-fixed moments of inertia IX=IY≡ I ∝negationslash=IZ,\nsubject to a uniform magnetic field Bis,\nH=1\n2IˆL2+(1\n2IZ−1\n2I)ˆL2\nZ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nHR−(ω0//planckover2pi1)(ˆS·ˆn)2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHA−ˆµ·B/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHB,\n(1)\nwhere a hat denotes quantum operator. In the rotational\nHamiltonain HR,ˆLis the orbital angular momentum op-\nerator and ˆLZ=ˆL·ˆZis its component along the body-\nfixed symmetry axis. ˆSis the needle spin angular mo-\nmentum operator, and ˆnis the operator for nthat is the\nunit vector in the direction of the easy magnetization\naxis. The frequency appearing in the anisotropy Hamil-\ntonianHA[4] isω0= 2γ2KS/V, whereKis the strengthofthe anisotropy, Vis the needle volume, and γ=gµB//planckover2pi1\nis the gyromagnetic ratio, in which µBis the Bohr mag-\nnetron, and gis theg-factor (taken to be a scalar for\nsimplicity). In the expression for the Zeeman Hamilto-\nnianHB,ˆµ=gµBˆSis the magnetic moment operator.\nThe Heisenberg equations of motion are\n˙ˆS=−gµBB׈S+2ω0\n/planckover2pi1(ˆS׈n)(ˆS·ˆn),(2)\n˙ˆL=-2ω0\n/planckover2pi1(ˆS׈n)(S·ˆn), (3)\n˙ˆJ=−gµBB׈S, (4)\n˙ˆn=I−1\n/planckover2pi1[ˆL׈n+i/planckover2pi1ˆn], (5)\nwhereˆJ=ˆL+ˆSis the total angularmomentum operator\nandIis the moment of inertia tensor.\nThe dynamics of a MN can be treated semiclassically\nbecause Sis very large. A mean–field approximation\n[9–11] is obtained by taking quantum expectation values\nof the operator equations and assuming that for a given\noperator ˆA, the inequality/radicalBig\n∝angbracketleftˆA2∝angbracketright−∝angbracketleftˆA∝angbracketright2≪ |∝angbracketleftˆA∝angbracketright|holds,\n(an assumption warranted for large S). Hence, the ex-\npectation values of a product of operators on the RHS\nof Eqs. (2)-(5) can be replaced by a product of expecta-\ntion values. The semiclassical equations are equivalent\nto those obtained in a classical Lagrangian formulation.\nDissipation is accounted for by adding the Gilbert term\n[2, 4]−αS×(˙S//planckover2pi1−Ω×S//planckover2pi1) to the RHS of the expecta-\ntion value of Eq. (2) and subtracting it from the RHS of\nEq. (3). Here αis the dimensionless friction parameter,\nand the term Ω×Stransforms from body fixed to space\nfixed frames. Note that Gilbert damping is due to inter-\nnalforces, hence Jis not affected and Eq. (4) remains\nintact.2\nIt is useful to recast the semiclassical dynamical equa-\ntions of motion in reduced units by defining dimension-\nless vectors: the unit spin m≡S/S, the orbital angu-\nlar momentum ℓ≡L/S, the total angular momentum,\nj=m+ℓand the unit vector in the direction of the\nmagnetic field b=B/B:\n˙m=ωBm×b+ω0(m×n)(m·n)−αm×(˙m−Ω×m),(6)\n˙ℓ=−ω0(m×n)(m·n)+αm×(˙m−Ω×m),(7)\n˙n=Ω×n, (8)\n˙j=ωBm×b, (9)\nwhere the angular velocity vector Ωis given by\nΩ= (ω3−ω1)(ℓ·n)n+ω1ℓ\n= (ω3−ω1)[(j−m)·n]n+ω1(j−m).(10)\nHereωB=γ|B|is the Larmor frequency, ω1=S/IX,\nandω3=S/IZ. Similar equations were obtained in\nRef. [5], albeit assuming that the deviations of n(t) and\nm(t) frombare small. We show below that the dynam-\nics can be more complicated than simply precession of\nthe needle about the magnetic field, particularly at high\nmagnetic fields where nutation can be significant.\nFor the numerical solutions presented below we are\nguided by Ref. 1, which uses parameters for bulk cobalt,\nand take ω1= 100 s−1,ω3= 7000 s−1, anisotropy fre-\nquencyω0= 108s−1, Gilbert constant α= 0.01, tem-\nperature T= 300 K, and N=S//planckover2pi1= 1012. First, we elu-\ncidate the effects of Gilbert dissipation, and consider the\nshorttimebehaviorin aweakmagneticfield, ωB= 1s−1.\nThe initial spin direction is intentionally chosen notto be\nalong the easy magnetic axis; n(0) = (1 /2,1/√\n2,1/2),\nm(0) = (1 /√\n2,1/√\n2,0),ℓ(0) = (0 ,0,0). Figure 1(a)\nshows the fast spin dissipation as it aligns with the easy\naxis of the needle, i.e., m(t)→n(t) after a short time,\nand Fig. 1(b) shows relaxation of the oscillations in ℓ(t),\nwhileℓx(t) andℓy(t) approach finite values. Figure 1(c)\nshowsthe innerproduct m·n, which clearlytendstounity\nonthe timescaleofthe figure. Increasing αleadsto faster\ndissipation of m(t), but the short-time saturation values\nof bothm(t) andℓ(t) are almost independent of α.\nWe consider now the long time dynamics (still in\nthe weak field regime) and take the initial value of the\nspin to coincide with the easy magnetization axis, e.g.,\nm(0) =n(0) = (1 /√\n2,1/√\n2,0), with all other param-\neters unchanged. The spin versus time is plotted in\nFig. 2(a). The unit vectors m(t) andn(t) are almost\nidentical, andsincetheir z-componentisnearlyzero,they\nmove together in the x-yplane. In this weak field case,\nthe nutation is small, and the fast small-oscillations due\nto nutation are barely visible. The orbital angular mo-\nmentum dynamics is plotted in Fig. 2(b) [note the differ-\nenttimescalein (a)and(b)] andshowsthat ℓ(t) oscillates\nwith a frequency equal to that of the fast tiny-oscillation\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0002\n-\u0001\u0002\u0003\u0001\u0002\u0003\u0004\u0002\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003\u0007\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0001\u0001\u0002\u0007\u0001\u0001\u0002\u0007\u0003\u0004\u0002\u0001\u0001{\u0002\u0002\u0003}\nFIG. 1: (color online) (a) The normalized spin vector mver-\nsus time for the low-field case at short times (5 orders of\nmagnitude shorter than in Fig. 2) when the initial spin is\nnot along the fast axis. (b) The reduced orbital angular mo-\nmentum vector ℓ(t). (c) The inner product m(t)·n(t) (the\nprojection of the spin on the fast magnetic axis of the needle .\nofm(t) [the oscillation amplitude is 0 .02|m(t)|]. Fig-\nure 2(c) shows a parametric plot of m(t) versus time.\nThe nutation is clearly very small; the dynamics of m(t)\nconsists almost entirely of precession at frequency ωB.\nFigure 3 shows the dynamics at high magnetic field\n(ωB= 105s−1) with all the other parametersunchanged.\nFigure 3(a) shows mversus time, and now the nutation\nis clearly significant. For the high magnetic field case,\nm(t) is also almost numerically equal to n(t).ℓ(t) is\nplotted in Fig. 3(b). Its amplitude is very large, ℓ(t)≈\n40m(t). However, its oscillation frequency is comparable\nwith that of m(t). In contrast with the results in Fig. 2,\nhere, in addition to precession of the needle, significant\nnutation is present, as shown clearly in the parametric\nplot of the needle spin vector m(t) in Fig. 3(c).\nWe now determine the uncertainty of the MNM due to\ninternal magnetic field fluctuations related to the Gilbert\ndamping. A stochastic force ξ(t), whose strength is de-\ntermined by the fluctuation–dissipation theorem [6], is3\n\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0003\u0004\u0005\u0006\u0002\n-\u0005\u0007\u0006-\u0006\u0007\b\u0006\u0007\b\u0005\u0007\u0006{\u0001\u0001\t\u0001\u0002\t\u0001\u0003\t\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\nFIG. 2: (color online) Dynamics for thelow-field case ( ωB= 1\ns−1), over relatively long timescales relative to those in Fig. 1.\n(a)mversus time in units of seconds (note that nis indistin-\nguishable from mon the scale of the figure). (b) ℓ(t) (note\nthat it stays small compared to S). (c) Parametric plot of the\nneedle spin vector m(t) showing that nutation is almost im-\nperceptible for small fields [contrast this with the large fie ld\nresult in Fig. 3(c)]; only precession is important.\nadded to Eq. (6), in direct analogy with the treatment of\nBrownianmotionwherebothdissipationandastochastic\nforce are included [12]:\n˙m=m×(ωBb+ξ)+ω0(m×n)(m·n)\n−αm×(˙m−Ω×m). (11)\nξ(t) is internal to the needle and therefore it does not\naffect the total angular momentum jdirectly, i.e., ξ(t)\ndoes not appear in Eq. (9) [since the term −m×ξis also\nadded to the RHS of (7)]. However, as shown below, ξ(t)\naffectsℓas well as m, causing them to wobble stochas-\ntically. This, in turn, makes jstochastic as well via the\nZeeman torque [see Eq. (9)].\nThe fluctuation-dissipation theorem [6] implies\n∝angbracketleftξαξβ∝angbracketrightω≡/integraldisplay\ndt∝angbracketleftξα(t)ξβ(0)∝angbracketrighteiωt\n=δαβαωcoth(/planckover2pi1ω/2kBT)\nN≈δαβ2αkBT\n/planckover2pi1N,(12)\u0001\u0001\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0007\u0002\u0001-\u0001\u0002\b\u0001\u0002\b\u0007\u0002\u0001{\u0001\u0001\t\u0001\u0002\t\u0001\u0003}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0004\u0001-\u0003\u0001\u0003\u0001\u0004\u0001{\u0001\u0001\n\b\u0001\u0002\n\u0000\u0001\u0003}\nFIG. 3: (color online) High-field case ( ωB= 105s−1). (a)\nm(t) [which is almost numerically equal to n(t)]. (b)ℓ(t)\n(note the ordinate axis scale is [ −40,40]). (c) Parametric plot\nof the needle spin vector m(t) showing that strong nutation\noccurs for large fields in addition to precession.\nwhereN=S//planckover2pi1, and the last approximation is ob-\ntained under the assumption that /planckover2pi1ω≪kBT. Note that\nEq. (11) should be solved together with Eqs. (8) and (9).\nThe presence of the anisotropy term in Eq. (11) makes\nnumerical solution difficult for large ω0. Hence, we con-\nsider a perturbative expansion in powers of λ≡ω1/ω0:\nm(t) =n0(t)+λδm(t)+...,n(t) =n0(t)+λδn(t)+...,\nj(t) =j0(t) +λδj(t) +.... Sinceω0is the largest fre-\nquency in the problem, the inequalities αω0≫ωB,ω1,ω3\nhold. Moreover, the Gilbert constant αis large enough\nto effectively pin m(t) ton(t) [hencej(t) =ℓ(t)+m(t)≈\nℓ(t)+n(t)]. Therefore, anadiabaticapproximationtothe\nset of dynamical stochastic equations can be obtained.\nThe zero order term in λreads:\n˙j0=ωBn0×b,˙n0=ω1j0×n0,(13)4\nwhereΩwas approximated by Ω0= (ω3−ω1)(j0·n0−\n1)n0+ω1(j0−n0) in Eqs. (8) and (10) in obtaining (13)\n[7]. The solution to Eqs. (13) [for times beyond which\nGilbert dissipation is significant so m(t)≈n(t)] is very\nclose to that obtained from Eqs. (6)-(8).\nExpanding Eq. (11) in powers of λand keeping only\nthe first order terms (the zeroth order term on the LHS\nvanishes since m0=n0), we get: ω1(δm−δn)×n0=\n˙n0−ωBn0×b+αn0×(˙n0−Ω0×n0)−n0×ξ. Taking\nEq. (13) into account and introducing the notation δη≡\nδm−δn, we obtain\nδη×n0=j0×n0−(ωB/ω1)n0×b−(1/ω1)n0×ξ,(14)\nand from Eqs. (8) and (9) we find\nd\ndtδj=ωB(δn+δη)×b, (15)\nd\ndtδn=ω1(j0−n0)×δn+ω1(δj−δn−δη)×n0\n=ω1j0×δn+ω1(δj−δη)×n0. (16)\nTo first order in λ,δn⊥n0(sincenmust be a unit\nvector), and δm⊥n0, henceδη⊥n0. Therefore, δη×\nb= [j0−(j0·n0)n0]×b+(ωB/ω1)[b−(b·n0)n0]×b+\nω−1\n1[ξ−(ξ·n0)n0]×bon the RHS of Eq. (15) and\nd\ndtδj=ωBδn×b+ωB[j0−(j0·n0)n0]×b\n−ω2\nB\nω1(b·n0)n0×b+ωB\nω1[ξ−(ξ·n0)n0]×b.(17)\nEquations (13), (16) and (17) form a closed system of\nstochastic differential equations [upon using Eq. (14) to\nsubstitute for δη×n0on the RHS of Eq. (16)]. With\nthe largest frequency ω0eliminated, a stable numerical\nsolution is obtained. Moreover, for small magnetic field\n(whereωBis the smallest frequency in the system), an\nanalytic solution of these equations is achievable. To ob-\ntain an analytic solution to Eqs. (13), let us transform\nto the frame rotating around Bwith frequency ωBto\nget equations of the formd\ndτv=d\ndtv+ωBb×v(which\ndefinesτ):\nd\ndτn0=−ω1n0×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n,(18)\nd\ndτj0=ωBb×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n.(19)\nIf the initial condition is n0(0)−j0(0)+(ωB/ω1)b= 0,\nthen, in the rotating frame j0(τ) andn0(τ) are constant\nvectors. Note that this initial condition is only slightly\ndifferent from the “ordinary” initial condition n0(0) =\nj0(0)since( ωB/ω1)≪1forsmallmagneticfields. Hence,\nin the rotating frame,\nd\ndτδn=ω1n0×(δn−δj+δη),(20)d\ndτδj=−ωBb×(δn−δj+δη).(21)\nWith the special initial conditionbeing satisfied, Eq. (14)\nbecomes δη×n0=−(1/ω1)n0×ξ, and Eqs. (20)-(21)\nbecome a set of first order differential equations with\ntime-independent coefficients. Their solution for initial\nconditions, δn(t= 0) = 0, δj(t= 0) = 0 is,\n/parenleftbiggδn(t)\nδj(t)/parenrightbigg\n=t/integraldisplay\n0dt1exp[C(t−t1)]C/parenleftbiggδη(t1)\n0/parenrightbigg\n,(22)\nwhere the constant matrix C=/parenleftbiggA−A\n−B B/parenrightbigg\nhas di-\nmension 6 ×6 and the 3 ×3 matrices AandBare given by\nAij=−ω1ǫijknk\n0,Bij=−ωBǫijkbk. Without loss of gen-\neralitywecanchoose n0=ˆzandb=ωB(cosθˆz+sinθˆx),\nwhereθis the angle between the easy magnetization\naxis and the magnetic field. In this basis, ∝angbracketleftδηxδηx∝angbracketrightω=\n∝angbracketleftδηyδηy∝angbracketrightω≈ω−2\n0∝angbracketleftξxξx∝angbracketrightω=ω−2\n0∝angbracketleftξyξy∝angbracketrightω=Sa(ω), and\n∝angbracketleftδηzδηz∝angbracketrightω= 0. Here ∝angbracketleftxx∝angbracketrightω≡/integraltext\ndteiωt∝angbracketleftx(t)x(0)∝angbracketrightand [see\nEq. (12)] Sa(ω) =αωcoth(/planckover2pi1ω/2kBT)\nω2\n0N≈2αkBT\nN/planckover2pi1ω2\n0.\nWe are particularly interested in the quantities\n∝angbracketleftδn2\ny(t)∝angbracketright ≡ ∝angbracketleftδny(t)δny(t)∝angbracketrightand∝angbracketleftδj2\ny(t)∝angbracketright ≡ ∝angbracketleftδjy(t)δjy(t)∝angbracketright\nbecause, in the basis chosen above, the y-axis is the di-\nrection of precession of n0aroundb. Using Eq. (22) we\nobtain∝angbracketleftδn2\ny(t)∝angbracketright ≈tω2\n1Sa(ω∼ω1). Assuming the pre-\ncession of nis measured, [or equivalently, the precession\nofm, since they differ only for short timescales of or-\nder (αω0)−1], the uncertainty in the precession angle is\n∝angbracketleft(∆ϕ)2∝angbracketright ≈tω2\n1Sa(ω∼ω1). We thus arrive at our central\nresult: the precision with which the precession frequency\ncan be measured is, ∆ ωB=√\n/angbracketleft(∆ϕ)2/angbracketright\nt≈ω1\nω0/radicalBig\n2αkBT\n/planckover2pi1N1√\nt.\nEquivalently, the magnetic field precision is,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω1\nω0/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt.(23)\nFor the parameters used in this paper we find ∆ B≈\n5×10−18√\nt[s]Tesla (independent of ωB). This result should\nbe compared with the scaling ∆ B∝t−3/2obtained in\nRef. 1. Therein, the initial uncertainty of the spin di-\nrection relative to the needle axis was estimated from\nthe fluctuation-dissipation relation and the deterministic\nprecession resulted in the t−3/2scaling of the precession\nangle uncertainty (in addition this angle was assumed to\nbe small). In contrast, we consider the uncertainty ac-\nquired due to Gilbert dissipation duringthe precession,\nallowing the precession angle to be large. Thus, the stan-\ndard1/√\ntdiffusion scalingis obtained and dominates for\ntimes that are even much longer than those considered\nin Ref. 1.\nIntheSupplementalMaterial[13]wediscussthreerele-\nvant related issues. (a) The time at which diffusion stops\nbecause equipartition is reached (we estimate the time5\nwhen the energy stored in stochastic orbital motion be-\ncomes of order kBT). (b) The uncertainty of the mag-\nnetic field for experiments in which the fast precession of\nnaroundjis averaged out in the measurement, and the\ndiffusion of jdetermines ∆ B. (c) We consider the related\nproblem of the dynamics and stability of a rotating MN\nin an inhomogeneous field (e.g., levitron dynamics in a\nIoffe-Pritchard trap [14, 15]).\nIn conclusion, we show that ∆ Bdue to Gilbert damp-\ning is very small; external noise sources, as discussed in\nRef. [1], will dominate over the Gilbert noise for weak\nmagnetic fields. A closed system of stochastic differen-\ntial equations, (13), (16) and (17), can be used to model\nthe dynamics and estimate ∆ Bfor large magnetic fields.\nA rotating MN in a magnetic trap can experience levi-\ntation, although the motion does not converge to a fixed\npoint or a limit cycle; an adiabatic–invariant stability\nanalysis confirms stability [13].\nThis work was supported in part by grants from the\nDFG through the DIP program (FO703/2-1). Useful\ndiscussions with Professor Dmitry Budker are gratefully\nacknowledged. A. S. was supported by DFG Research\nGrant No. SH 81/3-1.\n[1] D. F. J. Kimball, A. O. Sushkov, and D. Budker, Phys.\nRev. Lett. 116, 190801 (2016).\n[2] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004)\n[3] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet. 8153\n(1935). In L. D. Landau, Collected Papers. Ed. by D. ter\nHaar, (Gordon and Breach, New York, 1967), p. 101.\n[4] W. F. Brown Jr., Phys. Rev. 130, 1677 (1963).\n[5] H. Keshtgar, et al., Phys. Rev. B 95, 134447 (2017).\n[6] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).\n[7] We note in passing that Eqs. (13) are equivalent to the\nequations of motion of a symmetric top in a gravita-\ntional field when the top is anchored at a point a=an\non its axis a distance afrom the center of mass. The\nequations of motion are: dL/dt=T, where Land\nT=an×(−mgz) are taken with respect to the fixedpoint, and dn/dt=Ω×n. The angular velocity is given\nbyΩ=I−1\n1[L−(L·n)n] +I−1\n3(L·n)n, where the mo-\nments of inertia ( I1,I1,I3) are calculated relative to the\nfixed point. Introducing a characteristic scale L0so that\nL=L0j(jis not a unit vector and its length is not\nconserved) we obtain Eqs. (13) with ωB=mga/L 0and\nω1=L0/I1. Here, the analog of the magnetic field is the\ngravitational field and the analog of bisz.\n[8] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein.\n[9] O. Zobay and B. M. Garraway, Phys. Rev. A 61, 033603\n(2000); J. Liu, L. Fu, B.-Y. Ou, S.-G. Chen, D.-I. Choi,\nB. Wu, and Q. Niu, Phys. Rev. A 66, 023404 (2002).\n[10] Y. B. Band, I. Tikhonenkov, E. Pazyy, M. Fleischhauer,\nand A. Vardi, J. of Modern Optics 54, 697-706 (2007).\n[11] Y. B. Band, Phys. Rev. E 88, 022127 (2013); Y. B. Band\nand Y. Ben-Shimol, Phys. Rev. E 88, 042149 (2013).\n[12] H. P. Breuer and F. Petruccione, The Theory of\nOpen Quantum Systems (Oxford University, Cambridge,\n2002); M. Schlosshauer, Decoherence and the Quantum-\nto-Classical Transition (Springer, Berlin, 2007).\n[13] See Supplemental Material at\nhttp://link.aps.org/supplemental/10.1103/PhysRevLet t.121.160801\nwhich contains a discussion of the three issues enumer-\nated in the text, and which includes Refs. 16-20.\n[14] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[15] A movie showing the dynamics of a Levitron can be seen\nathttps://www.youtube.com/watch?v=wyTAPW_dMfo .\n[16] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a\nMagnetic Needle Magnetometer: Sensitivity to Landau–\nLifshitz–Gilbert Damping”, Phys. Rev. Lett. (to be pub-\nlished).\n[17] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[18] C. C.Rusconi, V.P¨ ochhacker, K.Kustura, J.I.Ciracan d\nO. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017);\nC. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac\nand O. Romero-Isart, Phys. Rev B 96, 134419 (2017); C.\nC. Rusconi and O. Romero-Isart, Phys. Rev B 93, 054427\n(2016).\n[19] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[20] D. R. Merkin, Introduction to the Theory of Stability ,\n(Springer–Verlag, New York, 1997); F. Verhulst, Non-\nlinear Differential Equations and Dynamical Systems ,\n(Springer–Verlag, Berlin, 1990).arXiv:1803.10064v2  [physics.gen-ph]  19 Oct 2018Supplemental Material for “Dynamics of a Magnetic Needle Ma gnetometer:\nSensitivity to Landau–Lifshitz–Gilbert Damping”\nY. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nIn this supplemental material we expand the discussion of the main t ext [1] and address the following three issues.\n(a) The time τeat which the diffusion of the magnetic needle axis direction nand the magnetic needle orbital\nangular momentum ℓstops because equipartition is reached, i.e., we estimate the time req uired for the energy stored\nin stochastic orbital motion to become of order kBT. (b) The uncertainty ∆ Bof the magnetic field for experiments\nin which the fast precession of naroundjis averaged out in the measurement process and the uncertainty ∆ Bis\ndetermined by the diffusion of j. (c) The dynamics of a magnetic needle in an inhomogeneous field, e.g., levitron\ndynamics of a rotating magnetic needle in a Ioffe-Pritchard trap [2], s ee Refs. [3–5].\n(a):τecan be estimated by noting that the diffusion determined in [1] stops once equipartition is reached. The\nenergy ∆ Estored in stochastic orbital motion is given by\n∆E∼/planckover2pi1ω1N/angbracketleftδℓ2/angbracketright, (1)\nwhere where N=S//planckover2pi1(note that δj−δn=δℓ). By requiring ∆ E∼kBTwe can estimate that the diffusion given\nby Eqs. (20-21) of [1] stops when τe∼ω2\n0/(αω3\n1) (this result can also be obtained by expanding Eq. (11) further in\npowers of λ≡ω1/ω0). For the parameters used in [1] this is an extremely long time ( τe∼1012s∼5 years). Hence,\nwe conclude that the diffusion of Eqs. (20-21) and the error estima tes given for ∆ Bin Ref. [1] are relevant for all\nreasonable times.\n(b): In [1] we calculate ∆ Bassuming the experimental measurement follows the temporal dyn amics of nandj.\nAn alternative assumption is that the precession of naroundjis averaged out by the measurement process and one\nmeasures the diffusion of j. For the latter we obtain the leading term\n/angbracketleftδj2\ny(t)/angbracketright ≈tω2\nBcos2θSa(ω∼ω1), (2)\nwhereSa(ω) is given in Eq. (23) of [1]. At θ=π/2 the leading contribution obtained in Eq. (2) vanishes and the\nremaining sub-leading term is\n/angbracketleftδn2\ny(t)/angbracketright ≈t2ω4\nB\nω2\n1Sa(ω∼ω1), (3)\nhence for θ/negationslash=π/2 we obtain\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBωB\nω0cosθ/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt, (4)\nwhereas at θ=π/2,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω2\nB\nω0ω1/radicalbigg\n4αkBT\n/planckover2pi1N1√\nt. (5)\nTakingωB= 1s−1we obtain ∆ B≈cosθ×5×10−23√\nt[s]Tesla for θ/negationslash=π/2, and ∆ B≈7×10−25√\nt[s]Tesla for θ=π/2.2\n(c): A rotating magnet can be levitated in an inhomogeneous magnet ic field [3–5]. This is possible despite Earn-\nshaw’s theorem [6] from which one can conclude that levitation of a non-rotating ferromagnetin a static magnetic field\nis not possible. Two important factors regarding magnetic levitation are the forces on the magnet and its stability\n(ensuring that it does not spontaneously slide or flip into a configura tion without lift). The dynamics of a magnetic\nneedle in an inhomogeneous magnetic field can be modelled using Eqs. (6 ), (7) and (8) of [1] augmented by the\nequations of motion for the center of mass (CM) degrees of freed om of the needle,\n˙p=∇(µ·B(r)), (6)\n˙r=p/m , (7)\nwhererandpare the needle CM position and momentum vectors. Our numerical re sults show levitation of the\nmagnetic needle when the initial rotational angular momentum vecto r of the needle is sufficiently large and points\nin the direction of magnetic field at the center of the trap. We shall s ee that the dynamical variables do not evolve\nto a fixed point or a simple cyclic orbit. Moreover, a linear stability analy sis yields a 15 ×15 Jacobian matrix with\neigenvalues having a positive real part, so the system is unstable. However, a stability analysis of the system using\nthe adiabatic invariant |µ||B|[3] does yield a stable fixed point (contrary to the full numerical re sults which show a\nmore complicated levitation dynamics).\nFigure 1 shows the dynamics of the system over time in the trap. We u se the same magnetic needle parameters\nused in Fig. 2 of [1] and a Ioffe-Prichard magnetic field [2]\nB(r) =ex/parenleftbigg\nB′x−B′′\n2xz/parenrightbigg\n+ey/parenleftbigg\nB′y−B′′\n2zy/parenrightbigg\n+ez/parenleftbigg\nB0+B′′\n2(z2−x2+y2\n2)/parenrightbigg\n, (8)\nwith field bias B0, gradient B′, and curvature B′′parameters chosen so that the Zeeman energy and its variation ov er\nthe trajectory of the needle in the trap are substantial (as is clea r from the results shown in the figure). We start\nthe dynamics with initial conditions: r(0) = (0,0,0),p(0) = (0,0,0),m(0) = (0,0.0011/2,−(1−0.001)1/2) (almost\nalong the −zdirection), n(0) =m(0),ℓ(0) = (0 ,0,0.001) [this is large orbital angular momentum since ℓis the\norbital angular momentum divided by S]. Figure 1(a) shows the needle CM position r(t) versus time. Fast and slow\noscillations are seen in the xandymotion, whereas z(t) remains very close to zero. Figure 1(b) shows oscillations of\nthe CM momentum p(t) with time. px(t) andpy(t) oscillate with time, and pz(t) remains zero. Figure 1(c) plots the\nspinm(t) versus time. Initially, m(0) points almost in the −zdirection, and the tip of the needle n(t) =m(t) carries\nout nearly circular motion in the nx-nyplane. Figure 1(d) plots the orbital angular momentum ℓ(t). The components\nℓx(t) andℓy(t) undergo a complicated oscillatory motion in the ℓx(t)-ℓy(t) plane but ℓz(t)≈ℓz(0). Figure 1(e) is a\nparametric plot of m(t); the motion consists of almost concentric rings that are slightly dis placed one from the other.\nThe full dynamics show levitation but they do not converge to a fixed point or a limit cycle.\nQuite generally, for a system of dynamical equations, ˙ yi(t) =fi(y1,...,y n),i= 1,...n, a linear stability analysis\nrequires calculating the eigenvalues of the Jacobian matrix evaluate d at the equilibrium point y∗wheref(y∗) =0,\nJij=/parenleftBig\n∂fi\n∂yj/parenrightBig\ny∗[7]. The system is unstable against fluctuations if any of the eigenvalu es ofJijhave a positive real\npart. Equations (6), (7) and (8) of [1] together with Eqs. (6) and (7) above have a Jacobian matrix with eigenvalues\nwhose real part are positive, so the linear stability test fails. Howev er, if the Zeeman force −∇HZin Eq. (6) is\nreplaced by the gradient of the adiabatic invariant, µ·∇|B(r)|, none of the eigenvalues of the Jacobian matrix have\na positive real part and the system is linearly stable, i.e., the stability a nalysis using the adiabatic-invariant predicts\nstability. Note that substituting the adiabatic invariant for the Zee man energy in the full equations of motion yields\nr(t) andp(t) vectors that are constant with time and n(t),m(t) andℓ(t) are similar to the results obtained with\nthe full equations of motion (but the parametric plot of m(t) is a perfectly circular orbit). Thus, adiabatic–invariant\nstability analysis of a rotating magnetic needle in a magnetic trap confi rms stability of its levitation as obtained in\nthe numerical solution of the dynamical equations.3\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0004-\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0004\n\u0001{\u0001\u0001\u0002\u0001\u0003}\u0002\u0003\u0004\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0003\u0001-\u0001\u0005\u0001\u0002\u0001\u0005\u0001\u0001\u0001\u0005\u0001\u0002\u0001\u0005\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0003\u0005\u0001-\u0001\u0005\u0006-\u0001\u0005\u0007-\u0001\u0005\b-\u0001\u0005\u0004\u0001\u0005\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0001\u0006-\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0006\u0001\u0005\u0001\u0001\u0001\u0007\u0001\u0005\u0001\u0001\u0001\b\u0001\u0005\u0001\u0001\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\nFIG. 1: (color online) Dynamics of a needle in a Ioffe-Pritcha rd magnetic field. (a) rversus time, (b) pversus time, (c) m\nversus time (note that n(t) is indistinguishable from m(t) on the scale of the figure). (d) ℓversus time (note that |ℓ(t)|is small\ncompared to Sbut rotational angular momentum L(t) =Sℓ(t) is large since S= 1012). (e) Parametric plot of the needle spin\nvectorm(t) (nutation is very small for this case of small magnetic field ).4\n[1] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a Magnet ic Needle Magnetometer: Sensitivity to Landau–Lifshitz–\nGilbert Damping”, Phys. Rev. Lett. (to be published).\n[2] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87, 3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[3] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[4] A movie a a Levitron can be seen at https://www.youtube.com/watch?v=wyTAPW_dMfo .\n[5] C. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017); C. C.\nRusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romer o-Isart, Phys. Rev B 96, 134419 (2017); C. C. Rusconi and\nO. Romero-Isart, Phys. Rev B 93, 054427 (2016).\n[6] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[7] D. R. Merkin, Introduction to the Theory of Stability , (Springer–Verlag, New York, 1997); F. Verhulst, Nonlinear Differential\nEquations and Dynamical Systems , (Springer–Verlag, Berlin, 1990)."    },    {        "title": "1803.10925v1.Giant_resonant_nonlinear_damping_in_nanoscale_ferromagnets.pdf",        "content": "Giant resonant nonlinear damping in nanoscale ferromagnets\nI. Barsukov,1,\u0003H. K. Lee,1A. A. Jara,1Y.-J. Chen,1A. M. Gon\u0018 calves,1\nC. Sha,1J. A. Katine,2R. E. Arias,3B. A. Ivanov,4, 5and I. N. Krivorotov1\n1Physics and Astronomy, University of California, Irvine, CA 92697, USA\n2Western Digital, 5600 Great Oaks Parkway, San Jose, CA 95119, USA\n3Departamento de F\u0013 \u0010sica, CEDENNA, FCFM, Universidad de Chile, Santiago, Chile\n4Institute of Magnetism, National Academy of Sciences of Ukraine, Vernadsky av. 36 B, Kyiv, 03142, Ukraine\n5National University of Science and Technology MISiS, Moscow, 119049, Russian Federation\nMagnetic damping is a key metric for emerging technologies based on magnetic nanoparticles,\nsuch as spin torque memory and high-resolution biomagnetic imaging. Despite its importance,\nunderstanding of magnetic dissipation in nanoscale ferromagnets remains elusive, and the damping\nis often treated as a phenomenological constant. Here we report the discovery of a giant frequency-\ndependent nonlinear damping that strongly alters the response of a nanoscale ferromagnet to spin\ntorque and microwave magnetic \feld. This novel damping mechanism originates from three-magnon\nscattering that is strongly enhanced by geometric con\fnement of magnons in the nanomagnet. We\nshow that the giant nonlinear damping can invert the e\u000bect of spin torque on a nanomagnet leading\nto a surprising current-induced enhancement of damping by an antidamping torque. Our work\nadvances understanding of magnetic dynamics in nanoscale ferromagnets and spin torque devices.\nI. INTRODUCTION\nNanoscale magnetic particles are the core components\nof several emerging technologies such as nonvolatile spin\ntorque memory [1], spin torque oscillators [2{7], targeted\ndrug delivery, and high-resolution biomagnetic imaging\n[8{11]. Control of magnetic damping holds the key to\nimproving the performance of many nanomagnet-based\npractical applications. In biomagnetic characterization\ntechniques such as magnetic resonance imaging [12], re-\nlaxometry [13], and magnetic particle imaging [14, 15],\nmagnetic damping a\u000bects nanoparticles relaxation times\nand image resolution. In spin torque memory and oscil-\nlators, magnetic damping determines the electrical cur-\nrent necessary for magnetic switching [1] and generation\nof auto-oscillations [16] and thereby determines energy-\ne\u000eciency of these technologies. The performance of\nnanomagnet-based microwave detectors is also directly\na\u000bected by the damping [17{19]. Despite its impor-\ntance across multiple disciplines, magnetic damping in\nnanoparticles is poorly understood and is usually mod-\neled as a phenomenological constant [6, 16].\nIn this article, we experimentally demonstrate that a\nferromagnetic nanoparticle can exhibit dynamics quali-\ntatively di\u000berent from those predicted by the constant\ndamping model. We show that nonlinear contributions\nto the damping can be unusually strong and the damp-\ning parameter itself can exhibit resonant frequency de-\npendence. Our work demonstrates that nonlinear damp-\ning in nanomagnets is qualitatively di\u000berent from that in\nbulk ferromagnets and requires a new theoretical frame-\nwork for its description. We show both experimentally\nand theoretically that such resonant nonlinear damping\noriginates from multi-magnon scattering in a magnetic\n\u0003igorb@ucr.edusystem with a discrete spectrum of magnons induced by\ngeometric con\fnement.\nWe also discover that the resonant nonlinear damping\ndramatically alters the response of a nanomagnet to spin\ntorque. Spin torque arising from injection of spin cur-\nrents polarized opposite to the direction of magnetization\nacts as negative damping [2]. We \fnd, however, that the\ne\u000bect of such antidamping spin torque is reversed, lead-\ning to an enhanced dissipation due to the nonlinear res-\nonant scattering. This counterintuitive behavior should\nhave signi\fcant impact on the operation of spin torque\nbased memory [1], oscillators [2{7] and microwave detec-\ntors [17{19].\nII. RESULTS\nA. Spin wave spectroscopy\nWe study nonlinear spin wave dynamics in nanoscale\nelliptical magnetic tunnel junctions (MTJs) that consist\nof a CoFeB free layer (FL), an MgO tunnel barrier, and a\nsynthetic antiferromagnet (SAF) pinned layer [20]. Spec-\ntral properties of the FL spin wave modes are studied in a\nvariety of MTJs with both in-plane and perpendicular-to-\nplane equilibrium orientations of the FL and SAF magne-\ntization. We observe strong resonant nonlinear damping\nin both the in-plane and the perpendicular MTJs, which\npoints to the universality of the e\u000bect.\nWe employ spin torque ferromagnetic resonance (ST-\nFMR) to measure magnetic damping of the FL spin wave\nmodes. In this technique, a microwave drive current\nIacsin(2\u0019ft) applied to the MTJ excites oscillations of\nmagnetization at the drive frequency f. The resulting\nmagnetoresistance oscillations Racsin(2\u0019ft+\u001e) generate\na direct voltage Vmix. Peaks in ST-FMR spectra Vmix(f)\narise from resonant excitation of spin wave eigenmodes\nof the MTJ [21{28]. To improve signal-to-noise ratio,arXiv:1803.10925v1  [cond-mat.mes-hall]  29 Mar 20182\n0 0.5 1 1.5 2H1H2\n36912\nField (kOe)0 0.5 1 1.5 2H2H1\n00.20.4\nField (kOe)Linewidth (GHz)Experiment\nSimulationFrequency (GHz)\n+1\n1a b\n<Vmix>~\nFIG. 1. Spin wave spectra in a nanoscale MTJ. (a) Normalized ST-FMR spectra h~Vmix(f)iof spin wave eigenmodes in a\nperpendicular MTJ device (Sample 1) measured as a function of out-of-plane magnetic \feld. Resonance peaks arising from\nthree low frequency modes of the MTJ free layer j0i,j1i, and j2iare observed. (b) Spectral linewidth of the quasi-uniform\nj0ispin wave mode as a function of out-of-plane magnetic \feld. Strong linewidth enhancement is observed in the resonant\nthree-magnon regime at H1andH2.\nthe magnitude of external magnetic \feld Happlied par-\nallel to the free layer magnetization is modulated, and\na \feld-derivative signal ~Vmix(f) = dVmix(f)=dHis mea-\nsured via lock-in detection technique [20]. Vmix(f) can\nthen be obtained via numerical integration (Supplemen-\ntal Material).\nFigure 1(a) shows ST-FMR spectra ~Vmix(f) measured\nas a function of out-of-plane magnetic \feld Hfor an el-\nliptical 52 nm\u000262 nm perpendicular MTJ device (Sam-\nple 1). Three spin wave eigenmodes with nearly linear\nfrequency-\feld relation fn(H) are clearly visible in the\nspectra. Micromagnetic simulations (Supplemental Ma-\nterial) reveal that these modes are three lowest frequency\nspin wave eigenmodes of the FL (Supplemental Material).\nThe lowest frequency (quasi-uniform) mode j0iis node-\nless and has spatially uniform phase. Each of the two\nhigher-order modes jni(n= 1;2) has a single node at\nthe FL center that is either perpendicular ( n= 1) or\nparallel (n= 2) to the ellipse long axis.\nThe spectral linewidth of the resonances in Fig. 1(a)\ncan be used for evaluation of the mode damping. The\nquasi-uniform mode j0iresonance visibly broadens at\ntwo magnetic \feld values: H1= 0:74 kOe (4 GHz) and\nH2= 1:34 kOe (6 GHz). Near H1, the modej1iresonance\nalso broadens and exhibits splitting, same behavior is ob-\nserved for the mode j2iatH2. At these \felds, the higher-\norder mode frequency is twice that of the quasi-uniform\nmodefn= 2f0. This shows that three-magnon con\ru-\nence [29{33] is the mechanism of the quasi-uniform mode\ndamping increase: two magnons of the quasi-uniform\nmodej0imerge into a single magnon of the higher-ordermodejni.\nThe most striking feature of the quasi-uniform mode\nresonance near H1is its split-peak shape with a local min-\nimum at the resonance frequency. Such a lineshape can-\nnot be \ft by the standard Lorentzian curve with symmet-\nric and antisymmetric components [20]. We therefore use\na double-peak \ftting function (Supplemental Material)\nto quantify the e\u000bective linewidth \u0001 f0of the resonance\npro\fle. For applied \felds su\u000eciently far from H1, the\nST-FMR curve recovers its single-peak shape and \u0001 f0\nis determined as half width of the standard Lorentzian\n\ftting function [20]. Figure 1(b) shows \u0001 f0as a function\nofHand demonstrates a large increase of the linewidth\nnear the \felds of the resonant three-magnon regime H1\nandH2. The stepwise increase of \u0001 f0nearH1is a result\nof the ST-FMR curve transition between the split-peak\nand single-peak shapes. For \felds near H2, the resonance\npro\fle broadens but does not develop a visible split-peak\nlineshape. As a result, \u0001 f0(H) is a smooth function in\nthe vicinity of H2.\nB. E\u000bect of spin torque\nIn MTJs, direct bias current Idcapplied across the\njunction exerts spin torque on the FL magnetization, act-\ning as antidamping for Idc>0 and as positive damping\nforIdc<0 [22, 34]. The antidamping spin torque in-\ncreases the amplitude of the FL spin wave modes [22, 35]\nand decreases their spectral linewidth [36]. We can em-\nploy spin torque from Idcto control the amplitude of spin3\nFIG. 2. E\u000bect of spin torque on spin wave resonance lineshape. (a)-(b) Spin wave resonance lineshapes in the nonresonant\nregime at H > H 1for di\u000berent values of direct bias current Idc. (c)-(d) Spin wave resonance lineshapes in the resonant three-\nmagnon regime at H=H1. (a), (c) Measured ST-FMR spectra (Sample 2). (b), (d) Solutions of Eqs. (3) and (4). Identical\nbias current values Idc(displayed in (a) are used in (a)-(d).\nwave eigenmodes excited in ST-FMR measurements, and\nthereby study the crossover between linear and nonlinear\nregimes of spin wave resonance.\nFigure 2 shows the dependence of ST-FMR resonance\ncurve of thej0imodeVmix(f) onIdcfor a 50 nm\u0002110 nm\nelliptical in-plane MTJ (Sample 2). For in-plane mag-\nnetic \feld values far from the three-magnon resonance\n\feldsHn, the amplitude of ST-FMR resonance curve\nVmix(f) shown in Fig. 2(a) monotonically increases with\nincreasing antidamping spin torque, as expected. At\nH=H1, the antidamping spin torque has a radically\ndi\u000berent and rather surprising e\u000bect on the resonance\ncurve. As illustrated in Fig. 2(c), increasing antidamp-\ning spin torque \frst broadens the resonance at H=H1\nand then transforms a single-peak resonance lineshape\ninto a split-peak lineshape with a local minimum at the\nresonance frequency f0. The data in Fig. 2 demonstrate\nthat the unusual split-peak lineshape of the resonance is\nonly observed when (i) the three-magnon scattering of\nthe quasi-uniform mode is allowed by the conservation of\nenergy and (ii) the amplitude of the mode is su\u000eciently\nhigh, con\frming that the observed e\u000bect is resonant and\nnonlinear in nature.\nFig. 2(c) reveals that antidamping spin torque can in-\ncrease the spectral linewidth and the e\u000bective damping\nof the quasi-uniform spin mode if the mode undergoes\nresonant three-magnon scattering. Figure 3 further illus-\ntrates this counterintuitive e\u000bect. It shows the linewidth\nof the quasi-uniform mode of a 50 nm \u0002110 nm elliptical\nin-plane MTJ (Sample 3) measured as a function of bias\ncurrent. In Fig. 3, blue symbols show the linewidth mea-\nsured at an in-plane magnetic \feld su\u000eciently far fromthe three-magnon resonance \felds Hn. At this \feld, the\nexpected quasi-linear dependence of the linewidth on Idc\nis observed for currents well below the critical current\nfor the excitation of auto-oscillatory magnetic dynamics.\nNear the critical current, the linewidth increases due to\na combination of the fold-over e\u000bect [37{39] and ther-\nmally activated switching between the large- and small-\namplitude oscillatory states of the fold-over regime [22].\nThe red symbols in Fig. 3 show the linewidth measured\nin the resonant three-magnon regime at H=H1. In con-\ntrast to the nonresonant regime, the linewidth increases\nwith increasingjIdcjfor both current polarities. Fur-\nthermore, the maximum linewidth is measured for the\nantidamping current polarity.\nIII. THEORETICAL MODEL\nNonlinear interactions among spin wave eigenmodes\nof a ferromagnet give rise to a number of spectacu-\nlar magneto-dynamic phenomena such as Suhl instabil-\nity of the uniform precession of magnetization [40, 41],\nspin wave self-focusing [42] and magnetic soliton forma-\ntion [43{45]. In bulk ferromagnets, nonlinear interac-\ntions generally couple each spin wave eigenmode to a\ncontinuum of other modes via energy- and momentum-\nconserving multi-magnon scattering [40]. This kinemat-\nically allowed scattering limits the achievable amplitude\nof spin wave modes and leads to broadening of the spin\nwave resonance. These processes lead to a resonance\nbroadening [40, 46{48] and cannot explain the observed\nsplit-peak lineshape of the resonance. In nanoscale ferro-4\nmagnets, geometric con\fnement discretizes the spin wave\nspectrum and thereby generally eliminates the kinemati-\ncally allowed multi-magnon scattering. This suppression\nof nonlinear scattering enables persistent excitation of\nspin waves with very large amplitudes [49] as observed in\nnanomagnet-based spin torque oscillators [2, 50]. Tun-\nability of the spin wave spectrum by external magnetic\n\feld, however, can lead to a resonant restoration of the\nenergy-conserving scattering [31]. The description of\nnonlinear spin wave resonance in the nanoscale ferromag-\nnet geometry therefore requires a new theoretical frame-\nwork. To derive the theory of resonant nonlinear damp-\ning in a nanomagnet, we start with a model Hamilto-\nnian that explicitly takes into account resonant nonlinear\nscattering between the quasi-uniform mode and a higher-\norder spin wave mode (in reduced units with ~\u00111):\nH=!0aya+!nbyb+\t0\n2ayayaa+\tn\n2bybybb (1)\n+( naaby+ \u0003\nnayayb)\n+\u0010\b\nexp(\u0000i!t)ay+ exp(i!t)a\t\nwhereay,aandby,bare the magnon creation and an-\nnihilation operators for the quasi-uniform mode j0iwith\nfrequency!0and for the higher-order spin wave mode\njnimode with frequency !n, respectively. The non-\nlinear mode coupling term proportional to the coupling\nstrength parameter  ndescribes the annihilation of two\nj0imagnons and creation of one jnimagnon, as well as\nthe inverse process. The Hamiltonian is written in the\nresonant approximation, where small nonresonant terms\nsuch asaab,aaayare neglected. The terms proportional\nto \t 0and \t ndescribe the intrinsic nonlinear frequency\nshifts [51] of the modes j0iandjni. The last term de-\nscribes the excitation of the quasi-uniform mode by an\nexternal ac drive with the amplitude \u0010and frequency !.\nWe further de\fne classically a dissipation function Q,\nwhere\u000b0and\u000bnare the intrinsic linear damping param-\neters of the modes j0iandjni[52{54]:\nQ=day\ndtda\ndt(\u000b0+\u00110aya) +dby\ndtdb\ndt(\u000bn+\u0011nbyb) (2)\nFor generality, Eq. (2) includes intrinsic nonlinear\ndamping [16] of the modes j0iandjnidescribed by the\nnonlinearity parameters \u00110and\u0011n. However, our analy-\nsis below shows that the split-peak resonance lineshape\nis predicted by our theory even if \u00110and\u0011nare set equal\nto zero.\nEquations describing the nonlinear dynamics of the\ntwo coupled spin wave modes of the system follow from\nEq. (1) and Eq. (2):\nida\ndt=@H\n@ay+@Q\n@(day=dt)(3)\nidb\ndt=@H\n@by+@Q\n@(dby=dt)(4)\nIt can be shown (Supplemental Material) that these\nequations have a periodic solution a= \u0016aexp (\u0000i!t) and\n100 50 0 50 10000.10.20.30.40.5\nI (A)f0 (GHz)Resonant regime\nNonresonant regimeFIG. 3. E\u000bect of spin torque on linewidth. Linewidth of the\nquasi-uniform spin wave mode as a function of the applied\ndirect bias current (Sample 3): blue symbols { in the non-\nresonant regime H6=H1and red symbols { in the resonant\nthree-magnon regime H=H1. Lines are numerical \fts using\nEqs. (3) and (4).\nb=\u0016bexp (\u0000i2!t), where \u0016a,\u0016bare the complex spin wave\nmode amplitudes. For such periodic solution, Eqs. (3)\nand (4) are reduced to a set of two nonlinear algebraic\nequations for absolute values of the spin wave mode am-\nplitudesj\u0016ajandj\u0016bj, which can be solved numerically.\nSince the ST-FMR signal is proportional to j\u0016aj2(Supple-\nmental Material), the calculated j\u0016aj2(!) function can be\ndirectly compared to the measured ST-FMR resonance\nlineshape.\nWe employ the solution of Eqs. (3) and (4) to \ft the\n\feld dependence of the quasi-uniform mode linewidth in\nFig. 1(b). In this \ftting procedure, the resonance line-\nshapej\u0016aj2(!) is calculated, and its spectral linewidth\n\u0001!0is found numerically. The resonance frequencies !0\nand!nare directly determined from the ST-FMR data\nin Fig. 1(a). The intrinsic damping parameters \u000b0and\n\u000bnnearH1andH2are found from linear interpolations\nof the ST-FMR linewidths \u0001 f0and \u0001fnmeasured at\n\felds far from H1andH2. We \fnd that \u0001 !0weakly\ndepends on the nonlinearity parameters \t and \u0011, and\nthus these parameters are set to zero (Supplemental Ma-\nterial). We also \fnd that the calculated linewidth \u0001 !0\ndepends on the product of the drive amplitude \u0010and\nmode coupling strength  n, but is nearly insensitive to\nthe individual values of \u0010and nas long as\u0010\u0001 n= const\n(Supplemental Material). Therefore, we use \u0010\u0001 nas a\nsingle \ftting parameter in this \ftting procedure. Solid\nline in Fig. 1(b) shows the calculated \feld dependence\nof the quasi-uniform mode linewidth on magnetic \feld.\nThe agreement of this single-parameter \ft with the ex-\nperiment is excellent.\nFigures 2(b) and 2(d) illustrate that Eqs. (3) and\n(4) not only describe the \feld dependence of ST-FMR\nlinewidth but also qualitatively reproduce the spectral5\nlineshapes of the measured ST-FMR resonances as well\nas the e\u000bect of the antidamping spin torque on the line-\nshapes. Fig. 2(b) shows the dependence of the calculated\nlineshapej\u0016aj2(!) on antidamping spin torque for a mag-\nnetic \feldHfar from the three-magnon resonance \felds\nHn. At this nonresonant \feld, increasing antidamping\nspin torque induces the fold-over of the resonance curve\n[37] without resonance peak splitting. The dependence of\nj\u0016aj2(!) on antidamping spin torque for H=H1is shown\nin Fig. 2(d). At this \feld, the resonance peak in j\u0016aj2(!)\n\frst broadens with increasing antidamping spin torque\nand then splits, in qualitative agreement with the ex-\nperimental ST-FMR data in Fig. 2(c). Our calculations\n(Supplemental Material) reveal that while the nonlinear-\nity parameters \t 0,\u00110, \tnand\u0011nhave little e\u000bect on\nthe linewidth \u0001 !0, they modify the lineshape of the res-\nonance. Given that the nonlinearity parameter values\nare not well known for the systems studied here, we do\nnot attempt to quantitatively \ft the measured ST-FMR\nlineshapes.\nEquations (3) and (4) also quantitatively explain\nthe observed dependence of the quasi-uniform mode\nlinewidth \u0001 !0on direct bias current Idc. Assuming an-\ntidamping spin torque linear in bias current [36, 55, 56]:\n\u000b0!\u000b0(1\u0000Idc=Ij0i\nc),\u000bn!\u000bn(1\u0000Idc=Ijni\nc), where\nIjni\nc>Ij0i\ncare the critical currents, we \ft the measured\nbias dependence of ST-FMR linewidth in Fig. 3 by solv-\ning Eqs. (3) and (4). The solid lines in Fig. 3 are the best\nnumerical \fts, where \u0010\u0001 nandIcare used as indepen-\ndent \ftting parameters. The rest of the parameters in\nEqs. (3) and (4) are directly determined from the experi-\nment following the procedure used for \ftting the data in\nFig. 1(b). Theoretical curves in Fig. 3 capture the main\nfeature of the data at the three-magnon resonance \feld\nH1{ increase of the linewidth with increasing antidamp-\ning spin torque.\nIV. DISCUSSION\nFurther insight into the mechanisms of the nonlinear\nspin wave resonance peak splitting and broadening by an-\ntidamping spin torque can be gained by neglecting the in-\ntrinsic nonlinearities \t nand\u0011nof the higher-order mode\njni. Setting \t n= 0 and\u0011n= 0 in Eqs. (3) and (4) allows\nus to reduce the equation of motion for the quasi-uniform\nmode amplitudej\u0016ajto the standard equation for a single-\nmode damped driven oscillator (Supplemental Material)\nwhere a constant damping parameter \u000b0is replaced by\nan e\u000bective frequency-dependent nonlinear damping pa-\nrameter\u000be\u000b\n0:\n\u000be\u000b\n0=\u000b0+\u0014\n\u00110+4\u000bn 2\nn\n(2!\u0000!n)2+ 4\u000b2n!2\u0015\nj\u0016aj2(5)and the resonance frequency is replaced by an e\u000bective\nresonance frequency:\n!e\u000b\n0=!0+\u0014\n\t0+2j nj2(2!\u0000!n)\n(2!\u0000!n)2+ 4\u000b2n!2\u0015\nj\u0016aj2(6)\nEquation (5) clearly shows that the damping parame-\nter of the quasi-uniform mode itself becomes a resonant\nfunction of the drive frequency with a maximum at half\nthe frequency of the higher order mode ( !=1\n2!n). The\namplitude and the width of this resonance in \u000be\u000b\n0(!) are\ndetermined by the intrinsic damping parameter \u000bnof\nthe higher-order mode jni. If\u000bnis su\u000eciently small,\nthe quasi-uniform mode damping is strongly enhanced\nat!=1\n2!n, which leads to a decrease of the quasi-\nuniform mode amplitude at this drive frequency. If the\ndrive frequency is shifted away from1\n2!nto either higher\nor lower values, the damping decreases, which can re-\nsult in an increase of the quasi-uniform mode amplitude\nj\u0016aj. Therefore, the amplitude of the quasi-uniform mode\nj\u0016aj(!) can exhibit a local minimum at !=1\n2!n. Due to\nits nonlinear origin, the tendency to form a local min-\nimum inj\u0016aj(!) at1\n2!nis enhanced with increasing j\u0016aj.\nSincej\u0016ajis large near the resonance frequency !0, tun-\ning!0to be equal to1\n2!ngreatly ampli\fes the e\u000bect of\nlocal minimum formation in j\u0016aj(!). This qualitative ar-\ngument based on Equation (5) explains the data in Fig. 2\n{ the split-peak nonlinear resonance of the quasi-uniform\nmode is only observed when external magnetic \feld tunes\nthe spin wave eigenmode frequencies to the three-magnon\nresonance condition !0=1\n2!n.\nEquation (6) reveals that the nonlinear frequency shift\nof the quasi-uniform mode is also a resonant function of\nthe drive frequency. In contrast to the nonlinear damping\nresonance described by Equation (5), the frequency shift\nresonance is an antisymmetric function of !\u00001\n2!n. The\nnonlinear shift is negative for ! <1\n2!nand thus causes\na fold-over towards lower frequencies while it is positive\nfor!>1\n2!ncausing fold-over towards higher frequencies.\nAt the center of the resonance pro\fle, the three-magnon\nprocess induces no frequency shift. This double-sided\nfold-over also contributes to the formation of the split-\npeak lineshape of the resonance shown in Figs. 2(c) and\n2(d) and to the linewidth broadening. As with the non-\nlinear damping resonance, the antisymmetric nonlinear\nfrequency shift and the double-sided fold-over become\ngreatly ampli\fed when the spin wave mode frequencies\nare tuned near the three-magnon resonance !0=1\n2!n.\nEquations (5) and (6) also shed light on the origin\nof the quasi-uniform mode line broadening by the an-\ntidamping spin torque. The antidamping spin torque in-\ncreases the quasi-uniform mode amplitude j\u0016ajvia transfer\nof angular momentum from spin current to the mode [57].\nSince the nonlinear damping and the nonlinear frequency\nshift are both proportional to j\u0016aj2and both contribute to\nthe line broadening, the antidamping spin torque can in-\ndeed give rise to the line broadening. Equation (5) reveals\ntwo competing e\u000bects of the antidamping spin torque on\nthe quasi-uniform mode damping parameter \u000be\u000b\n0: spin6\ntorque from Idcdecreases the linear component of the\ndamping parameter \u000b0!\u000b0(1\u0000Idc=Ij0i\nc) and increases\nthe nonlinear component via increased j\u0016aj2. Whether the\nantidamping spin torque decreases or increases the spec-\ntral linewidth of the mode depends on the system param-\neters. Our numerical solution of Eqs. (3) and (4) shown\nin Fig. 3 clearly demonstrates that the antidamping spin\ntorque can strongly increase the linewidth of the quasi-\nuniform mode when the three-magnon resonance condi-\ntion!0=1\n2!nis satis\fed. Furthermore, we \fnd that\nthe three-magnon process exhibits no threshold behav-\nior upon increasing amplitude (Supplemental Material)\nor decreasing intrinsic damping.\nThe key requirement for observation of the resonant\nnonlinear damping is the discreteness of the magnon\nspectrum imposed by geometric con\fnement in the\nnanoscale ferromagnet. The split-peak nonlinear reso-\nnance discovered in this work cannot be realized in bulk\nferromagnets because the three-magnon resonance con-\ndition in bulk is not only valid at the uniform mode\nfrequency!0=1\n2!nbut instead in a broad frequency\nrange. Owing to the magnon spectrum continuity in\nbulk, shifting the excitation frequency away from !0does\nnot suppress the three-magnon scattering of the uniform\nmode { it simply shifts it from one group of magnons to\nanother [29, 40]. Therefore, the amplitude of the uni-\nform mode does not increase when the drive frequency is\nshifted away from !0and the split-peak resonance is not\nrealized.\nWe expect that the resonant nonlinear damping dis-\ncovered in this work will have strong impact on the\nperformance of spin torque devices such as spin torque\nmagnetic memory, spin torque nanooscillators and spin\ntorque microwave detectors. Since all these devices rely\non large-amplitude oscillations of magnetization driven\nby spin torque, the amplitude limiting resulting from the\nresonant nonlinear damping is expected to have detri-\nmental e\u000bect on the device performance.V. CONCLUSIONS\nIn conclusion, our measurements demonstrate that\nmagnetic damping of spin wave modes in a nanoscale\nferromagnet has a strong nonlinear component of reso-\nnant character that appears at a discrete set of magnetic\n\felds corresponding to resonant three-magnon scattering.\nThis strong resonant nonlinearity can give rise to unusual\nspin wave resonance pro\fle with a local minimum at the\nresonance frequency in sharp contrast to the properties\nof the linear and nonlinear spin wave resonances in bulk\nferromagnets. The resonant nonlinearity has a profound\ne\u000bect on the response of the nanomagnet to spin torque.\nAntidamping spin torque, that reduces the quasi-uniform\nspin wave mode damping at magnetic \felds far from the\nresonant three-magnon regime, can strongly enhance the\ndamping in the resonant regime. This inversion of the\ne\u000bect of spin torque on magnetization dynamics by the\nresonant nonlinearity is expected to have signi\fcant im-\npact on the performance of nanoscale spin torque devices\nsuch as magnetic memory and spin torque oscillators.\nACKNOWLEDGMENTS\nThis work was supported by the National Science\nFoundation through Grants No. DMR-1610146, No.\nEFMA-1641989 and No. ECCS-1708885. We also ac-\nknowledge support by the Army Research O\u000ece through\nGrant No. W911NF-16-1-0472 and Defense Threat Re-\nduction Agency through Grant No. HDTRA1-16-1-0025.\nA. M. G. thanks CAPES Foundation, Ministry of Educa-\ntion of Brazil for \fnancial support. R.E.A acknowledges\nFinanciamiento Basal para Centros Cienti\fcos y Tec-\nnologicos de Excelencia under project FB 0807 (Chile),\nand Grant ICM P10-061-F by Fondo de Innovacion para\nla Competitividad-MINECON. 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Krivorotov1\n1Physics and Astronomy, University of California, Irvine, CA 92697, USA\n2Western Digital, 5600 Great Oaks Parkway, San Jose, CA 95119, USA\n3Departamento de F´ ısica, CEDENNA, FCFM, Universidad de Chile, Santiago, Chile\n4Institute of Magnetism, National Academy of Sciences of Ukraine, Vernadsky av. 36 B, Kyiv, 03142, Ukraine\n5National University of Science and Technology MISiS, Moscow, 119049, Russian Federation\nI. METHODS\nA. Linewidth evaluation\nAll measurements presented were carried out with magnetic field applied along the easy axis of the MTJ devices so\nthat the magnetic moments of the free and pinned layers are collinear to each other. In this geometry, the ST-FMR sig-\nnals are dominated by photo-resistance contribution and are proportional to the square of the transverse component of\nthe dynamic magnetization magnetization [1], which allows us to directly compare calculated |a|2(ω) resonance curves\nto measured ST-FMR resonance curves ˜Vmix(f) and toVmix(f) approximated by numerical integration/integraltext˜Vmix(f)df.\nWhenVmix(f) and|a|2(ω) are single-peak curves, they are fit to a sum of symmetric and antisymmetric Lorentzian\ncurves with identical central frequencies and linewidth parameters as described in Ref. [2], and the spectral linewidth\nis determined as half-width at the half-maximum of the symmetric Lorentzian curve.\nIn order to quantify the linewidth of the split-peak resonance profile, we introduce a fitting function that is a sum\nof two Lorentzian curves with different central frequencies separated by δf. The half width of the resonance profile\n∆f0is then defined as the average of the half widths of the two Lorentzians plus δf/2.\nSupplemental Figure 1. Spatial profiles of spin wave eigenmodes. Normalized amplitude and phase of the three lowest frequency\nspin wave eigenmodes of the MTJ free layer, given by micromagnetic simulations.\nB. Micromagnetic simulations\nMicromagnetic simulations were performed using OOMMF software [3, 4]. To account for all magnetic interactions\nin the MTJ, a three dimensional model was employed with three ferromagnetic layers: free, SAF top and SAF bottom.\nWe use material parameters obtained from the measurements and/or their accepted literature values (see Ref. [2] for2\nthe MTJ structure and fabrication details). Magnetization dynamics is excited by a combined pulse of spin torque\nand Oersted field, resulting from a sinc-shaped spatially uniform current pulse. The spatial profile of the Oersted\nfield corresponds to that of a long wire with elliptical cross section. The direction of the spin torque vector acting\non the free layer is determined by the magnetization orientation of the SAF top layer. The spectrum of spin wave\neigenmodes is obtained via fast Fourier transform (FFT) of the time dependent components of the layers’ magnetic\nmoment. Spatial mapping of the resulting Fourier amplitude and phase at a given frequency provides the mode\nprofiles (Supplmental Fig. 1). The observed excitations are confirmed to be spin wave modes localized to the free\nlayer. SAF modes are found at much higher frequencies than the free layer modes, and their frequencies are found to\nbe incommensurable to the free layer quasi-uniform mode frequency [5].\nII. SOLUTION OF THE EQUATIONS OF MOTION\nThe Hamiltonian equations of motion describing the coupled dissipative dynamics of the quasi-uniform ( a) and the\nhigher-order ( b) spin wave modes are:\nida\ndt=∂H\n∂a†+∂Q\n∂(da†/dt)(1)\nidb\ndt=∂H\n∂b†+∂Q\n∂(db†/dt)(2)\nwhereHis the Hamiltonian of the system and Qis the dissipation function, given by:\nH=ω0a†a+ωnb†b+1\n2Ψ0a†a†aa+1\n2Ψnb†b†bb+ (ψ∗\nnaab†+ψna†a†b) +ζ{exp(−iωt)a†+ exp(iωt)a} (3)\nQ=da†\ndtda\ndt(α0+η0a†a) +db†\ndtdb\ndt(αn+ηnb†b) (4)\nBy using Eq. (3) and Eq. (4) in Eq. (1) and Eq. (2), the Hamiltonian equations can be written as:\nida\ndt−(α0+η0a†a)da\ndt=ω0a+ 2ψna†b+ Ψ 0a†aa+ζexp(−iωt) (5)\nidb\ndt−(αn+ηnb†b)db\ndt=ωnb+ψ∗\nnaa+ Ψnb†bb (6)\nUsing a periodic ansatz a= ¯aexp(−iωt) andb=¯bexp(−2iωt) in Eq. (5) and Eq. (6), where ¯ aand¯bare complex\namplitudes, reduces the Hamiltonian equations to a set of two algebraic equation for the complex amplitudes:\n/parenleftbig\nω−ω0−Ψ0|¯a|2+i(α0+η0|¯a|2)ω/parenrightbig\n¯a−2ψn¯a∗¯b=ζ (7)\n/parenleftbig\n2ω−ωn−Ψn|¯b|2+ 2i(αn+ηn|¯b|2)ω/parenrightbig¯b=ψ∗\nn¯a2(8)\nWe solve Eq. (8) for ¯band multiply the numerator and denominator of this expression by the complex conjugate of\nthe denominator:\n¯b=ψ∗\nn¯a2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n−i2(αn+ηn|¯b|2)ω\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(9)\nthen we multiply Eq. (9) by2ψn¯a∗\n¯aand evaluate the real and imaginary parts.\n/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=|ψn|2|¯a|22/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(10)\n/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=|ψn|2|¯a|2−4(αn+ηn|¯b|2)ω\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(11)\nBy taking the modulus of Eq. (8), we obtain:\n|¯a|2=|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2 (12)3\nUsing Eq. (12) in Eqs. (10-11), we derive:\n/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(13)\n/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=−4(αn+ηn|¯b|2)ω|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(14)\nTaking the modulus squared of Eq. (7):\n/braceleftBigg/parenleftbigg\nω−ω0−Ψ0|¯a|2−/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg/parenrightbigg2\n+/parenleftbigg\n(α0+η0|¯a|2)ω−/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg/parenrightbigg2/bracerightBigg\n|¯a|2=ζ2(15)\nand using Equations (12)–(14) in Eq. (15) gives us an algebraic equation for the absolute value of the higher order\nmode amplitude|¯b|:\n\n\n\nω−ω0−Ψ0|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2−2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2\n2\n+\n\n/parenleftbigg\nα0+η0|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2/parenrightbigg\nω−−4(αn+ηn|¯b|2)ω|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2\n2\n\n×\n|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2=ζ2\n(16)\nAfter numerically solving Eq. (16) for |¯b|, and using it in Eq. (12), we can calculate the amplitude of the quasi-uniform\nmode|¯a|.\nIII. EFFECTS OF THE DRIVE AMPLITUDE AND INTRINSIC NONLINEARITITES\nTo understand the impact of the intrinsic nonlinearity parameters (Ψ 0, Ψn,η0,ηn) on the quasi-uniform spin wave\nmode resonance, we plot the numerical solution of Eq. (16) in Supplemental Figure 2. Each panel of this figure shows\na reference lineshape of the resonance calculated with all intrinsic nonlinearity parameters set to zero (red curve) and\na lineshape calculated with one of the intrinsic nonlinearity parameter different from zero (blue curve). This figure\nreveals that increasing η0decreases the mode amplitude and slightly increases the linewidth. Increasing ηndecreases\nthe degree of the double-peak lineshape splitting. Increasing Ψ nincreases the lineshape asymmetry. Increasing Ψ 0\nincreases lineshape asymmetry and induces fold-over.\nSupplemental Figure 3 shows the linewidth as a function of the drive amplitude for three scenarios, where the\nintrinsic nonlinearities Ψ 0, Ψn,η0,ηnare set to zero for simplicity. If the coupling parameter is zero, ψn= 0, the\nlinewidth does not depend on the drive amplitude, as expected for a single-mode linear oscillator. The second case\ndemonstrates that the linewidth remains constant when the product ψn·ζis constant. For a constant non-zero\ncoupling parameter, the linewidth shows an increase with the drive amplitude. This observation allows us to employ a\nsingle fitting parameter ( ψn·ζ) to fit the data in Fig. 1b. This conjecture can be confirmed analytically by introducing\na normalized spin wave amplitude ˆ a=ψn¯a, which allows us to rewrite Eq. (16) omitting all intrinsic nonlinearities\ninto the following form:\nω/bracketleftbigg\n1 +iα0+i4αn|ˆa|2\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\nˆa−ω0ˆa−2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2|ˆa|2ˆa=ψnζ (17)\nThis equation describes an effective single-mode nonlinear oscillator with renormalized excitation amplitude ψnζ.4\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nη0\n2 2.5 3 3.500.010.02\nFrequency (GHz)|a|20\nηn\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nΨ0\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nΨn|a|2|a|2|a|2a b\nc d\nSupplemental Figure 2. Effect of intrinsic nonlinearities on the quasi-uniform spin wave resonance lineshape. Spectral lineshape\nof the quasi-uniform spin wave mode resonance |¯a|2(ω) at the three-magnon resonance condition 2 ω0=ωncalculated by\nnumerically solving Eq. (16). The red curve is a reference lineshape calculated with all intrinsic nonlinearity parameters\n(η0,ηn,Ψ0,Ψn) set to zero. The blue lineshape in each panel is calculated with one of the intrinsic nonlinearity parameters set\nto a non-zero value: (a) η0= 1.325·10−24J, (b)ηn= 3.313·10−24J, (c) Ψ 0= 1.325·10−24J, (d) Ψ n= 1.325·10−23J. Other\nparameters employed in the calculation are: ω0= 2π·2.63 GHz,ωn= 2π·5.26 GHz;α0= 0.02662,αn= 0.03042 atIdc= 0;\nψn·ζ=h2·0.006 GHz2, wherehis the Planck constant.\n0 0.05 0.1 0.15 0.20.150.20.25\nζ h-1 (GHz)∆f0 (GHz)\n(i)(ii)(iii)\nSupplemental Figure 3. Effect of the drive amplitude on linewidth in the resonant three-magnon regime. Calculated linewidth\nof the quasi-uniform spin wave mode as a function of the drive amplitude ζfor different values of the mode coupling parameter\nψn. (i) Green: ψn= 0, (ii) red: variable ψnwith a constraint ψn·ζ=h2·0.006 GHz2, and (iii) blue: ψn=h·0.1 GHz. All\nintrinsic nonlinearity parameters: Ψ 0, Ψn,η0andηnare set to zero. his the Planck constant. Other parameters employed in\nthe calculation are: ω0= 2π·2.63 GHz,ωn= 2π·5.26 GHz;α0= 0.02662 andαn= 0.03042 atIdc= 0.5\nIV. EFFECTIVE SINGLE-MODE NONLINEAR OSCILLATOR APPROXIMATION\nIf we neglect intrinsic nonlinearities Ψ nandηnof the higher order spin wave mode, Eq. (16) can be reduced to a\ncubic equation for ¯ aand solved analytically. This approximation allows us to obtain several important qualitative\ninsights into the properties of the resonant nonlinear damping of the quasi-uniform mode. By setting Ψ n= 0 and\nηn= 0 in Eq. (8), we obtain an exact solution for ¯b:\n¯b=ψ∗\nn¯a2\n2ω(1 +iαn)−ωn(18)\nUsing this result, we reduce Eq. (16) to a cubic algebraic equation for ¯ a:\nω/bracketleftbigg\n1 +i(α0+η0|¯a|2) +i4|ψn|2αn|¯a|2\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n¯a−ω0¯a−/bracketleftbigg\nΨ0+2|ψn|2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2¯a=ζ (19)\nThis equation describes the amplitude ¯ aof an effective single-mode nonlinear oscillator.\nIt is evident from Eq. (19) that the frequency of the quasi-uniform mode experiences a nonlinear shift:\nωeff\n0=ω0+/bracketleftbigg\nΨ0+2|ψn|2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2(20)\nThe nonlinear frequency shift has a well-pronounced antisymmetric resonant character near the resonance frequency\nωn/2, that arises from the resonant three-magnon scattering.\nFurther, it is clear from Eq. (19) that the effective damping of the quasi-uniform mode also acquires a term arising\nfrom the three-magnon interaction:\nαeff\n0=α0+/bracketleftbigg\nη0+4|ψn|2αn\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2(21)\nThe last term describes a resonant enhancement of the nonlinear damping by three-magnon scattering near the\nresonance frequency ωn/2. Strikingly, the magnitude of the resonant damping enhancement at ωn/2 increases when\nthe intrinsic damping of the higher order mode αndecreases. In the limit αn→0, the effective damping becomes\nαeff\n0→α0+/bracketleftbigg\nη0+2π|ψn|2\nωδ(2ω−ωn)/bracketrightbigg\n|¯a|2(22)\nwhereδis Dirac’s delta function. Equation (21) suggests that the effective damping of the quasi-uniform mode αeff\n0\ncan increase with increasing antidamping spin torque applied to the nanomagnet. Indeed, the antidamping spin torque\ntends to increase the amplitude [6] of the quasi-uniform mode |¯a|and decrease the intrinsic damping parameter of the\nhigher order mode αn→αn(1−Idc/I|n/angbracketright\nc), both enhancing the nonlinear damping term in Eq. (19). For a sufficiently\nlarge mode coupling parameter ψn, the enhancement of the nonlinear damping term by the antidamping spin torque\ncan exceed the reduction of the linear damping parameter α0→α0(1−Idc/I|0/angbracketright\nc) by the torque, leading to an increase\nofαeff\n0byIdc>0 and broadening of the quasi-uniform mode resonance by the antidamping spin torque. This scenario\nis indeed realized in the MTJ devices studied here as demonstrated by the data and calculations in Fig. 3.\nV. MODE COUPLING PARAMETER\nIn this Supplementary Note, we discuss how the coupling parameter between the spin wave modes, ψnin Eq. (3),\ncan be calculated. We consider a very thin, magnetically soft ferromagnetic disk with elliptical cross section, that\nis magnetized in-plane. Within a classical micromagnetic model, we include Zeeman, dipolar and exchange terms in\nthe free energy. An applied field Halong thexdirection (long axis of the ellipse) magnetizes the sample to a nearly\nuniform state. Through a classical Holstein-Primakoff transformation [7] we introduce variables c(/vector x,t) andc∗(/vector x,t) to\ndescribe the magnetization such that the magnetization magnitude is conserved:\nmx= 1−cc∗, m +=c√2−cc∗, m−=c∗√2−cc∗, (23)\nwhere/vector m=/vectorM/Ms, andm±≡mz±imy. Approximating the exchange energy to the fourth order in candc∗, the\nnormalized free energy of the disk, U≡E/4πM2\ns, is given by\nU/similarequal−hx/integraldisplay\n(1−cc∗) dV+ (lex)2/integraldisplay/bracketleftbigg\n/vector∇c·/vector∇c∗+1\n4c2(/vector∇c∗)2+1\n4c∗2(/vector∇c)2/bracketrightbigg\ndV−1\n2/integraldisplay\ndV/vectorhD(/vector m)·/vector m, (24)6\nwithhx≡H/4πMs,lex≡/radicalbig\nA/2πM2sis the exchange length, and /vectorhD(/vector m) =/vectorHD(/vector m)/4πMsis the normalized\ndemagnetizing field. The Landau-Lifshitz equations of motion in the new variables are: i˙c=δU/δc∗,i˙c∗=−δU/δc\nwitht/prime= 4πMs|γ|t.\nAssuming the normal modes involved in three magnon scattering dominate the magnetization dynamics, the free\nenergy in Eq. (24) can be written in terms of amplitudes of these modes, by expressing cin terms of aandb:\nc(/vector x,t)/similarequala(t)f(/vector x) +a∗(t)g(/vector x) +b(t)p(/vector x) +b∗(t)q(/vector x) (25)\nThe functions f,g,p,q can be determined from calculating the linear modes of oscillation of the sample. The terms of\nthe free energy proportional to aab∗anda∗a∗bdescribe the three-magnon process and the magnitude of these terms\ngives the coupling parameter ψn.\nIf the magnetization state is approximated as exactly uniform, the dipolar energy for a very thin film may be\napproximated as UD=m2\nz/2 = (c+c∗)2(1−cc∗/2), and in this case all three-magnon terms are zero. However, when\nthe effects due to the sample edges (such as spatial inhomogeneity of the demagnetization field and edge roughness)\nare taken into account, the equilibrium magnetization configuration is generally nonuniform. In this case, there are\nnon-zero three-magnon terms in the free energy expression. An explicit calculation of the corresponding overlap\nintegrals is necessary for a quantitative prediction of ψn. Refs. [8, 9] show such extensive calculations for circular disks\nand include explicit expressions for the exchange and dipolar energies.\n[1] Michael Harder, Yongsheng Gui, and Can-Ming Hu, “Electrical detection of magnetization dynamics via spin rectification\neffects,” Phys. Rep. 661, 1–59 (2016).\n[2] A. M. Gon¸ calves, I. Barsukov, Y.-J. Chen, L. Yang, J. A. Katine, and I. N. Krivorotov, “Spin torque ferromagnetic\nresonance with magnetic field modulation,” Appl. Phys. Lett. 103, 172406 (2013).\n[3] M. J. Donahue and D. G. Porter, OOMMF User’s Guide (National Institute of Standards and Technology, Gaithersburg,\nMD, 1999).\n[4] Robert D. McMichael and Mark D. Stiles, “Magnetic normal modes of nanoelements,” J. Appl. Phys. 97, 10J901 (2005).\n[5] P. S. Keatley, V. V. Kruglyak, A. Neudert, R. J. Hicken, V. D. Poimanov, J. R. Childress, and J. A. Katine, “Resonant\nenhancement of damping within the free layer of a microscale magnetic tunnel valve,” J. Appl. Phys. 117, 17B301 (2015).\n[6] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, “Spin-transfer-driven\nferromagnetic resonance of individual nanomagnets,” Phys. Rev. Lett. 96, 227601 (2006).\n[7] T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58,\n1098–1113 (1940).\n[8] D. Mancilla-Almonacid and R. E. Arias, “Instabilities of spin torque driven auto-oscillations of a ferromagnetic disk mag-\nnetized in plane,” Phys. Rev. B 93, 224416 (2016).\n[9] D. Mancilla-Almonacid and R. E. Arias, “Spin-wave modes in ferromagnetic nanodisks, their excitation via alternating\ncurrents and fields, and auto-oscillations,” Phys. Rev. B 95, 214424 (2017)."    },    {        "title": "1804.00554v1.Anisotropic_Gilbert_damping_in_perovskite_La___0_7__Sr___0_3__MnO___3___thin_film.pdf",        "content": "Anisotropic Gilbert damping in perovskite La 0:7Sr0:3MnO 3thin film\nQing Qin,1Shikun He*,2Haijun Wu,1Ping Yang,1, 3Liang Liu,1\nWendong Song,2Stephen John Pennycook,1and Jingsheng Chen*1\n1Department of Materials Science and Engineering,\nNational University of Singapore, Singapore 117575\n2Data Storage Institute, Agency for Science, Technology and Research (A*STAR),\n2 Fusionopolis Way 08-01 Innovis, Singapore 138634\u0003\n3Singapore Synchrotron Light Source (SSLS),\nNational University of Singapore, 5 Research Link, Singapore 117603\n1arXiv:1804.00554v1  [cond-mat.mtrl-sci]  2 Apr 2018Abstract\nThe viscous Gilbert damping parameter governing magnetization dynamics is of primary importance for\nvarious spintronics applications. Although, the damping constant is believed to be anisotropic by theories.\nIt is commonly treated as a scalar due to lack of experimental evidence. Here, we present an elaborate\nangle dependent broadband ferromagnetic resonance study of high quality epitaxial La 0:7Sr0:3MnO 3films.\nExtrinsic effects are suppressed and we show convincing evidence of anisotropic damping with twofold\nsymmetry at room temperature. The observed anisotropic relaxation is attributed to the magnetization\norientation dependence of the band structure. In addition, we demonstrated that such anisotropy can\nbe tailored by manipulating the stain. This work provides new insights to understand the mechanism of\nmagnetization relaxation.\nA. INTRODUCTION\nThe magnetization relaxation process determines the speed of magnetization relaxation and the\nenergy required for current-induced magnetization reversal [1–6]. Understanding the mechanism\nand controlling of magnetization relaxation [7–12], including intrinsic Gilbert damping and extrinsic\neffects, pave the way for ultra-low power and high performance spintronic devices based on spin\ntransferandspinorbittorques[13–15]. IthasbeendemonstratedthatGilbertdampingconstant( \u000b)\ncanbetunedeffectivelybyengineeringthedensityofstatesandspinorbitcoupling(SOC)[9,16–18].\nIn addition, magnetization relaxations subjected to finite size and interfacial effects have also been\nextensively investigated [8, 19, 20]. However, it is still an open question that if magnetic damping\nis anisotropic. In principle, \u000bis magnetization orientation dependent and should be a 3 \u00023 tensor\nin the phenomenological Gilbert equation [21, 22], yet it is often treated as a scalar (isotropic). In\nthe case of polycrystalline thin films prepared by sputtering, such treatment is reasonable due to\nthe smearing of long range structural order. Whereas for single crystal thin films, it is still difficult\nto draw a conclusion due to the lack of convincing experimental evidence. From the view of theo-\nries, the Gilbert damping is determined by two scattering processes, the interband resistivity-like\nscattering and the intraband conductivity-like scattering [12]. Both terms vary with temperature\nthrough their dependence on electron relaxation time. The interband scattering which dominates\ndamping in most ferromagnets becomes isotropic at room temperature [23]. Therefore, anisotropic\nlinewidth in 3d magnetic metals was only observed at low temperature[24]. From the aspect of\nexperimental technique, Seib et al. have predicted that the precession trajectory of magnetization\nin a ferromagnetic resonance (FMR) measurement (standard technique for measuring damping)\nmay partially average out the anisotropy [25]. Hence, detecting the anisotropy in Gilbert damping\nis extremely difficult. Furthermore, the existence of several angle dependent extrinsic contributions\nto damping in most materials further hinders the determination of a possible weak anisotropic\ndamping [11, 26–28]. We note that in a ferromagnet with nearly half-metallic band structure, the\nisotropic interband term is suppressed [29] and the damping can be dominated by the anisotropic\nintraband contribution[23]. Recent reports have claimed the observation of anisotropic damping in\nhalf-metallic Heusler alloy[30, 31]. However, unavoidable chemical disorder [32, 33]of Heusler alloy\nintroduces extrinsic effects such as spin wave scattering hence complicates the verification procedure\nof such anisotropy.\n\u0003heshikun@gmail.com msecj@nus.edu.sg\n2La0:7Sr0:3MnO 3(LSMO) is an oxide perovskite material exhibited half-metallic band structure\nand ultra-low damping at room temperature [34, 35]. In this work, we studied the magnetiza-\ntion relaxation of LSMO films deposited on NdGaO 3(NGO) (110) substrates using angle-resolved\nbroadband ferromagnetic resonance. The purpose of choosing NGO (110) substrates is to utilize its\nnon-equalaandbaxis value. Such asymmetry will potentially lead to non-spherical Fermi surface.\nTwo types of high quality samples with different static magnetic anisotropies were investigated. The\nnormal LSMO film (hereafter denoted as S-LSMO) exhibited weak uniaxial magnetic anisotropy\nwhereas the other with modulated strain relaxation mode (hereafter denoted as W-LSMO) have\nboth uniaxial and cubic anisotropy fields. The angle dependence of the in-plane intrinsic Gilbert\ndampingshowedtwo-foldsymmetryinbothtypeofsamples. Strikingly, theorientationofminimum\ndamping differs 90 degree. This work provided strong evidence of anisotropic nature of magneti-\nzation relaxation and demonstrated the tuning of anisotropy in damping through stress relaxation\nengineering.\nB. RESULTS\nEpitaxial growth of LSMO\nPulsed laser deposition (PLD) was used to deposit LSMO thin films with a thickness of 25nm\non (110) NGO substrates. The energy and repetition frequency of KrF laser (248nm) were 225mJ\nand 2Hz, respectively. During deposition, the substrate temperature was fixed at 950\u000eC. The\noxygen pressure was 225mTorr for S-LSMO and 200mTorr for W-LSMOAfter deposition, S-LSMO\nwas cooled down to room temperature at 10K/min under the oxygen pressure of 1 Torr, whereas\nW-LSMO at 5K/min under the oxygen pressure of 100 Torr in order to promote the modification\nof strain hence micro-structurestructure.\nCrystalline quality analysis\nThe crystallographic structures of the films were characterized by synchrotron high resolution\nX-ray diffraction. Reciprocal space maps (RSMs) taken at room temperature around {013} pc(here\nthe subscript pc stands for pseudocubic) reflections confirm the epitaxial growth of LSMO layers\non the NGO substrate as shown in Fig. 1 (a). The vertical alignment of LSMO and NGO reciprocal\nlatticepointclearlyshowsthattheLSMOfilmiscompletelystrainedontheNGOsubstrate. Lattice\nmismatch along [100] pcand [010] pcare 1.03% and 0.8%, respectively. Considering the position of\nthe LSMO reciprocal lattice point in the {013} pcmappings, equal Lvalues of (103) pcand (-103) pc\nindicates the perpendicular relation between vector aandcin the lattice, whereas different L values\nfor (013) pcand (0-13) pcshows that the angle between bandcis not equal to 90Âř. Thus, the LSMO\nis monoclinic phase which is consistent with previous reports [36]. The good crystalline quality was\nfurther verified by aberration-corrected scanning transmission electron microscopy (AC-STEM).\nFig. 1 (b, c) are the simultaneously acquired high angle annular dark field (HAADF) and annular\nbright field (ABF) images of S-LSMO along [100]pc direction, while Fig. 1 (d, e) are for [010] pc\ndirection. The measurement directions can be differentiated from the diffraction of NGO substrate:\n31/2[010] superlattices for [100] pcdirection (inset of Fig. 1(c)) and 1/2[101] superlattices for [100] pc\ndirection (inset of Fig. 1(e)). High quality single crystalline films are essential for the present\npurposes because high density of defects will result in spin wave scattering [26].\nMagnetic anisotropy fields\nThe magnetic dynamic properties were investigated by a home-built angle-resolved broadband\nFMR with magnetic field up to 1.5T. All measurements were performed at room temperature.\nShown in Fig. 2(a) is the color-coded plot of the transmission coefficient S21 of the S-LSMO sample\nmeasured at 10GHz. 'His the in-plane azimuth angle of the external magnetic field counted\nfrom [010] pcdirection (Fig. 2(b)). This relative orientation was controlled by a sample mounting\nmanipulator with a precision of less than 0.1\u000e. The olive shape of the color region indicates the\nexistence of anisotropy field, whereas the very narrow field region of resonances is an evidence of low\ndamping. Three line cuts at 'H=0, 45 and 90 degrees are plotted in Fig. 2(c), showing the variation\nof both FMR resonance field ( Hres) and line shape with 'H. All curves are well fitted hence both\ntheHresand resonance linewidth \u0001H are determined. The 'Hdependence of H resat two selected\nfrequencies (20 and 40 GHz) are shown in Fig. 2(d) for S-LSMO. The angle dependencies of the\nresonance field Hres('H)is calculated starting from the total energy [37]:\nE=\u0000MH [cos\u0012Hcos\u0012M+ sin\u0012Hsin\u0012Mcos('M\u0000'H)] + 2\u0019M2cos2\u0012M\u00001\n2MH 2?cos2\u0012M\n\u00001\n4MH 4?cos4\u0012M\u00001\n2MH 2ksin2\u0012Mcos2('M\u0000\u001e2IP)\u00001\n4MH 4k3+cos 4('M\u0000\u001e4IP)\n4sin4\u0012M(1)\nwhere\u0012Mand'Mare the polar angle and the azimuth angle of the magnetization ( M),H2?,\nH4?,H2k,H4kare the uniaxial and cubic out-fo-plane and in-plane anisotropy fields. The easy axes\nof in-plane anisotropies are along \u001e2IPand\u001e4IP, respectively. According to Smit-Beljers equation\nthe resonance condition for \u0012M=\u0019/2 is [38]:\n2\u0019f=\r\nMsin\u0012p\nE\u0012\u0012E'' (2)\nHere,E\u0012\u0012=Hrescos('M\u0000'H) + 4\u0019Me\u000b\u0000H2kcos2('M\u0000\u001e2IP) +H4k(3 + cos 4('M\u0000\u001e4IP)=4)and\nE''=Hrescos('M\u0000'H)+H2kcos 2('M\u0000\u001e2IP)+H4kcos 4('M\u0000\u001e4IP)aresecondpartialderivatives\nof the total energy with respect to the polar and azimuth angles. \r=1.76\u0002107s\u00001G\u00001denotes the\ngyromagnetic ratio, 4\u0019Me\u000b= 4\u0019M\u0000H2?is the effective magnetization. The resonance field of\nS-LSMO shows pronounced minimum at 'H=n\u0001\u0019, indicating the existence of uniaxial magnetic\nanisotropy with easy axis along \u001e2IP= 0or [010] pcdirection. Cubic anisotropy is negligible hence\nH4k=0. Such uniaxial anisotropy observed in S-LSMO is consistent with previous reports [39],\nwhich is attributed to anisotropic strain produced by the NGO(110) substrate [40–42]. Compared\nto the resonance fields in our measurement, the magnetic anisotropy fields are orders of magnitude\nsmaller. Therefore, the calculated difference between 'Hand'Mare always smaller than 1\u000eand\n'='H='Mis assumed in the following discussion.\n4Magnetization orientation dependence of Gilbert damping\nIn order to study the symmetry of magnetization relaxation of the sample. The FMR linewidth\n\u0001Hfor a matrix of parameter list (72 field orientations and 36 frequency values) are extracted.\nThe results are shown by 3-D plots in Fig. 3(a) . Here, zaxis is \u0001Handx,yaxes aref\u0001cos'\nandf\u0001sin', respectively. The figure clearly shows that the linewidth depends on magnetization\norientation. At a given frequency, the linewidth is maximum (minimum) at '= 0('=\u0019=2) for\nS-LSMO. Fig. 3(c) shows the \u0001Hversus frequency for three field orientations. The FMR linewidth\ndue to intrinsic magnetic damping scales linearly with frequency \u0001HGL= 4\u0019\u000bf=\r cos ('M\u0000'H)\naccording to Laudau-Lifshitz-Gilbert phenomenological theory [43, 44]. However, a weak non-\nlinearity in the low frequency range can be identified. In general, extrinsic linewidth contributions\nsuch as inhomogeneity and magnon scattering will broaden the FMR spectrum hence result in\nadditionallinewidthcontributionsscalesnon-linearlywithfrequency[9,11]. Theinterfacialmagnon\nscattering is suppressed due to relative large film thickness (25 nm) and the bulk magnon scattering\ncontribution to the linewidth is negligible in our samples with very good atomic order. However, the\nstatic magnetic properties of the thin film may vary slightly in the millimeter scale. Since the FMR\nsignal is an averaged response detected by the coplanar waveguide (5mm long), a superposition of\nlocation resonance modes broadens the FMR spectrum. Such well-known contribution to linewidth,\ndefined as \u0001Hinhom, are generally treated as a constant [9, 44, 45]. However, it is frequency\ndependent for in-plane configuration and need to be treated carefully for samples with ultra-low\ndamping. Here, we fit the data with \u0001H= \u0001HGL+ \u0001Hinhom, taking into account the frequency\nand orientation dependence of \u0001Hinhom. As can be seen from Fig. 3(c), the data are well reproduced\nfor every field orientations. Hence, the magnetization orientation dependence of intrinsic damping\nconstant is determined and plotted in Fig. 3(e). Remarkably, the damping constant shows two-fold\nsymmetry. The lowest damping of S-LSMO with in-plane magnetization, observed at '= 0and\n'=\u0019, is(8:4\u00060:3)\u000210\u00004and comparable to the value measured under a perpendicular field\n(Tbl. I). The maximum damping at '=\u0019=2and'= 3\u0019=2is about 25% higher.\nSince the magnetization damping and resonance field of the S-LSMO sample exhibited identical\nsymmetry (Fig. 2 (d) and Fig. 2(e)), it seems that the observed anisotropic damping is directly\nrelatedtocrystallineanisotropy. Therefore, wepreparedtheW-LSMOsamplewithslightlydifferent\nstructureandhencemodifiedstaticmagneticanisotropyproperties. TheW-LSMOsampleexhibited\n1D long range atomic wave-like modulation [36] (twining domain motif) along [100] pcaxis near the\ninterface between substrate and film. Due to different strain relaxation mechanism as compared to\nS-LSMO, the 'Hdependence of Hresfor the W-LSMO have additional features and can only be\nreproduced by including both H2k(13:9\u00060:9Oe) andH4k(11:8\u00061:2Oe) terms. The easy axis of\nthe uniaxial anisotropy ( \u001e2IP=0 ) is the same as S-LSMO whereas the additional cubic anisotropy\nis minimum at \u001e4IP=45Âř. The magnetization orientation dependence of the FMR linewidth for\nW-LSMO is significantly different (Fig. 3(b)) as compared to S-LSMO. Such change in trend can be\nclearlyidentifiedfromthefrequencydependenceoflinewidthforselectedmagnetizationorientations\nshown in Fig. 3(d). Magnetization damping values are extracted using the same procedure as S-\nLSMObecausethespinwavecontributionisexcluded. Thedampingconstantagainshowedtwo-fold\nin-plane symmetry. However, in contrast to S-LSMO, the maximum damping value of W-LSMO is\nobserved at '= 0and'=\u0019.\n54\u0019Meff(T)H2k(Oe)H4k(Oe) \u000b? \u000b('= 0)\u000b('=\u0019=2)\nS-LSMO 0.3280 \u00060.0011 37\u000640 (8:6\u00060:5)\u000210\u00004(8:4\u00060:3)\u000210\u00004(11\u00060:6)\u000210\u00004\nW-LSMO 0.3620 \u00060.002513.9\u00060:911.8\u00061:2(4:7\u00060:7)\u000210\u00004(6:5\u00060:3)\u000210\u00004(5:3\u00060:3)\u000210\u00004\nTable I. Summary of the parameters for S-LSMO and W-LSMO samples.\nC. DISCUSSION\nAnisotropy in linewidth at low temperatures have been reported decades ago, however, data in\nmost early publications were taken at a fixed frequency in a cavity-based FMR [24, 46]. Due to\nlack of frequency dependence information, it is not clear if the anisotropy in linewidth is due to\nintrinsic damping or extrinsic effects [47–49]. In this study, besides wide range of frequencies, we\nalso adopted samples with effective anisotropy orders of magnitude smaller than the external field.\nTherefore, the field dragging effect and mosaicity broadening, both of which are anisotropic in natur\ne[50], are negligibly small and the Gilbert damping constant is determined reliably. Furthermore,\nthemechanisminthissimplesystemisdifferentfrompreviousreportsrelatedtointerfacialexchange\ncoupling and spin pumping[51, 52]. Since both S-LSMO and W-LSMO exhibited in-plane uniaxial\nmagnetic anisotropy, the opposite trends observed in these two samples exclude the existence of a\ndirect link between anisotropic damping and effective field. Both magnetic anisotropy and damping\nare related to the band structure but in quite different ways. According to perturbation theory,\nthe magnetic anisotropy energy is determined by the matrix elements of the spin-orbit interaction\nbetween occupied states. Hence, the contributions from all the filled bands must be considered to\ncalculate the absolute value of magnetic anisotropy. On the other hand, the magnetic damping is\nrelated to the density of states at the Fermi level.\nThe damping term in the Landau-Lifshitz-Gilbert equation of motion is\u000b\njMj\u0000\nM\u0002dM\ndt\u0001\n, there-\nfore, anisotropy in damping can have two origins, one related to the equilibrium orientation of\nmagnetization M(orientation anisotropy) and the other depends on the instantaneous change in\nmagnetization dM=dt(rotational anisotropy). In FMR experiments the magnetization vector ro-\ntates around its equilibrium position, therefore, the rotational anisotropy may be smeared out [25].\nThe orientation anisotropy is described by both interband and intraband scattering process. Ac-\ncording to Gilmore et al.[23], the latter is isotropic at sufficiently high scattering rates at room\ntemperature. We suspect that the anisotropic damping in LSMO is due to its half-metallic band\nstructure. As a result of high spin polarization, interband scattering is suppressed and the room\ntemperature damping is dominated by intraband scattering. The intraband contribution to damp-\ning exhibit anisotropy for all scattering rates [23] which agree well with our experiments. The\nsuppression of interband scattering is evidenced by the ultra-low damping in the order of 10\u00004.\nNotably, the absolute value of the observed anisotropy, 2.6 \u000210\u00004for S-LSMO and 1.2 \u000210\u00004for\nW-LSMO, is so small that could not be identified reliably for a material with typical damping\nvalues between 5 \u000210\u00003to 2\u000210\u00002.\nIn a microscopic picture, the Gilbert damping is proportional to the square of SOC constant ( \u0018)\nand density of states at the Fermi level, \u000b\u0018\u00182D(EF). The shape of the Fermi surface depends on\nthe orientation of the magnetization due to SOC. 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Back, Nature Physics (2018).\n9Figure 1. Structure characterization of S-LSMO sample. (a) and (b) XRD profiles around S-LSMO\n(00L) reflections (L=1,2,3,4) with the incident beam aligned along the [100] pcand [010] pc, respectively.\n(b) and (c) STEM-HAADF/ABF lattice images of S-LSMO along [100] pcdirection. (d) and (e) STEM-\nHAADF/ABF images of S-LSMO along [010] pcdirection. the insets are the intensity profile and FFT\nimage; The red dashed line indicates the interface.\n10(b)\n(c)\n10GHz(a)\n30\n21060\n24090\n270120\n300150\n330180 010GHz\n242022002420Hres(Oe)\n5.625.70\n12.6512.712.75\nHres(kOe)(d)\n40GHz\n20GHzS-LSMO\n0 90 180 270 360\n0\n0 fit\n45\n45 fit\n90\n90 fitS21 (a.u.)\n2200 2300 2400\nHres(Oe)Figure 2. Magnetic anisotropy characterization. (a) The 2D polar color plot of the FMR spectra of\nS-LSMO. The frequency is 10GHz. (b) Schematics of the FMR setup and the definition field orientation.\n(c) FMR spectra for 'H=0, 45 and 90 degrees for S-LSMO. (d) Field orientation ( 'H) dependence of the\nresonance fields ( Hres) of the S-LSMO sample at f=20 and 40GHz. The solid lines in (c) and (d) are\ncalculated values.\n11404040\n0 060\n-40\n40 4020\n0 030\n-40\nf (GHz) f (GHz)0 10 20 30 40204060ΔH (Oe)\n102030ΔH (Oe)\n0 10 20 30 40\n810120\nfit\n45\n90fit\nfit0\nfit\n45\n90fit\nfit(a) (b)\n(c) (d)\n(e) (f)ΔH (Oe)\nΔH (Oe)\nf (GHz) f (GHz)α (10-4)\n0 360 270 180 90\nφ567\n0 90 180 270 360α (10-4)\nφf (GHz) f (GHz)Figure 3. Anisotropic linewidth and damping: (a)-(b) 3-D plot of frequency and in-plane field ori-\nentation dependence of FMR linewidth. (c)-(d) frequency dependence of FMR linewidth for seleted field\norientations. Solid symbols are experimental data and the lines are calculated value. (e)-(f) Damping\nconstant as a function of '. (a),(c), (e) are for S-LSMO and (b),(d), (f) are for W-LSMO.\n12"    },    {        "title": "1804.03174v1.Damping_and_clustering_into_crowded_environment_of_catalytic_chemical_oscillators.pdf",        "content": "arXiv:1804.03174v1  [physics.chem-ph]  9 Apr 2018Damping and clustering into crowded environment of catalyt ic chemical oscillators\nCarlos Echeverria,∗Jos´ e L. Herrera∗,†and Kay Tucci∗‡\n∗CeSiMo, Facultad de Ingenier´ ıa, Universidad de Los Andes, M´ erida 5101, Venezuela.\n†ICTP South American Institute for Fundamental Research, IF T-UNESP, S˜ ao Paulo, SP Brazil 01440-070.\n‡SUMA, Facultad de Ciencias, Universidad de Los Andes, M´ eri da 5101, Venezuela.\nOrlando Alvarez-Llamoza§\nGrupo de Investigaci´ on de Simulaci´ on, Modelado,\nAn´ alisis y Accesibilidad. Universidad Cat´ olica de Cuenc a, Cuenca 010105, Ecuador.\nMiguel Morales¶\nUnidad Acad´ emica de Ingenier´ ıa en Nanotecnolog´ ıa,\nUniversidad Polit´ ecnica de Sinaloa, Mazatl´ an, Sinaloa 8 2199, M´ exico.\n(Dated: September 25, 2018)\nAsystemformed byacrowdedenvironmentofcatalytic obstac les andcomplex oscillatory chemical\nreactions is inquired. The obstacles are static spheres of e qual radius, which are placed in a random\nway. The chemical reactions are carried out in a fluid followi ng a multiparticle collision scheme\nwhere the mass, energy and local momentum are conserved. Fir stly, it is explored how the presence\nof catalytic obstacles changes the oscillatory dynamics fr om a limit cycle to a fix point reached\nafter a damping. The damping is characterized by the decay co nstant, which grows linearly with\nvolume fraction for low values of the mesoscale collision ti me and the catalytic reaction constant.\nAdditionally, it is shown that, although the distribution o f obstacles is random, there are regions\nin the system where the catalytic chemical reactions are fav ored. This entails that in average the\nradius of gyrations of catalytic chemical reaction does not match with the radius of gyration of\nobstacles, that is, clusters of reactions emerge on the cata lytic obstacles, even when the diffusion is\nsignificant.\nKeywords: Selkov Reaction, Reactive Multiparticle Collis ion, Damping in Chemical Reaction, Crowded\nEnvironment, Clustering effects\nI. INTRODUCTION\nIn chemical oscillations, as in other chemical and bi-\nological processes, reaction and diffusion are two of the\nmost basic transport mechanisms underlying their de-\nscription. Several numerical and experimental studies\nhave been performed in order to understand these trans-\nport mechanisms in homogeneousmedia. Additionally, it\nis known that when the environment is crowded with ob-\nstacles the transport processes can be significantly mod-\nified [1, 2]. An important system with a crowded en-\nvironment is a biological cell, where the volume is oc-\ncupied by structural elements such as microtubules and\nfilaments,variousorganellesandavarietyofothermacro-\nmolecular species [3]. For example, within bacterial cells\nmacromolecules account for more than 40% of their vol-\nume [4]. Under these conditions the fluid and the chemi-\ncal reactions occurring in it may behave differently from\nhow they do in solute solutions [5–7]. Understanding\nsuch changes and differences is of great importance in\nprocesses as essential as protein folding [8–13], protein-\n∗cecheve@ula.ve\n†jdiestra@ictp-saifr.org\n‡kay@ula.ve\n§llamoza@gmail.com\n¶mmorales@upsin.edu.mxprotein binding [6, 14], gene regulation [15–19] and en-\nzyme activity [20–24], among others.\nIn a catalytic crowded system, the reaction dynam-\nics of reactive particles introduces new features to the\nreaction-diffusion kinetics. In particular, the rate con-\nstantsand the diffusion coefficient depend on the fraction\nof volume that is occupied by reactive particles in non-\ntrivial ways. There are experiments where phenomena\nlike synchronization, quorum sensing [25, 26] and emer-\ngence of chimeras [27, 28] are modulated by the fraction\nof the volume occupied by reactive obstacles. The theo-\nretical treatments of this problem requires that the long-\nrange nature of the diffusive coupling among the reactive\nobstacles to be properly taken into account [2, 29–31].\nIn this paper we explore the behavior of complex os-\ncillating chemical reactions in a catalytic crowded en-\nvironment with hydrodynamic coupling. We consider a\nsimple model where hundreds of thousands of small par-\nticles undergo motion among a random distribution of\nstationary catalytic spherical obstacles and compute the\ndependence of the system’s attractor and the decay con-\nstant of oscillations with respect to the volume fraction\nof obstacles and the viscosity of the fluid. Moreover, by\nmeans of the radius of gyration of the obstacles on which\nthe catalytic reactionstake place, westudy the clustering\nof the chemical reactions, which could be a signature of\nsome emerging properties induced by the crowding.\nDue to the intrinsic difficulties to trackanalyticallythe2\nevolution ofthe system for arbitraryvalues ofthe volume\nfractionandviscosity, ourresultsareobtainedfromsimu-\nlations adapting the chemical reactions to the Multipar-\nticle Collision technique (MPC) [32–35]. Although our\nmodelissimple, itcapturessomeofthefeaturesofcrowd-\ning effects on oscillatory chemical reactions and sets a\nstarting point for the construction of more detailed mod-\nels in crowded environments. Also, an emergent cluster-\ning phenomenon is observed in which the mean distance\nbetween reactionson catalyticobstaclesissmallerthan it\nshould be, that is, the reactions are closer than expected.\nIn Section II we give the details of the model, de-\nscribing how the Selkov oscillatory chemical reaction is\nimmersed into hydrodynamic fluid with obstacles [36]\nthrough a multiparticle collision approach [37]. The re-\nsults are presented in Section III. The conclusions of the\nstudy are given in Section IV.\nII. THE MODEL\nA. Reactive multiparticle collision dynamics\nUnlike most studies in this area, which focus on the re-\nactive catalytic event on the obstacles when the particles\nsimply diffuse between them, here we study a situation in\nwhich, in addition to the reactive events on the catalyst\nsurfaces, there are complex reactions in the fluid.\nThe simulations were carried out on a three-\ndimensional cubic system with volume Vthat contains\na large number of particles, undergoing a reactive dy-\nnamics in a field of catalytic obstacles. More specifically,\nthe system contains N=/summationtexts\nℓ=1Nℓpoint particles with\nmassm, where the sub-index ℓindicates to which of the\nsspecies the Nℓparticles belong.\nAdditionally, thevolumecontains NOnon-overlapping,\nidentical and immobile catalytic spheric obstacles of ra-\ndioσ. The volume fraction occupied by the obstacles is\nφ=NOVO/V, whereVO= 4πσ3/3 is the volume of\neach sphere, being Vf=V(1−φ) the remaining volume\nfree of obstacles. FIG. 1 shows a typical configuration\nof the system, where it can be appreciated the catalytic\nobstacles and the reactive particles of the fluid. Periodic\nboundary conditionswere employedin the Npoint parti-\ncles displacement, while the NOobstacles are completely\ninside the simulation box.\nTo adapt the chemical reactions to Multiparticle Col-\nlision (MPC) [32–35], the simulation time unit is set as\nthe time between collisions where reactive events could\noccur [38, 39]. In MPC the particles have continuous\npositions and velocities with free stream between multi-\nparticle collision events that occur at discrete times τ.\nTo carry out collisions, the volume Vis partitioned into\nNcubic cells of volume V, where V=N × V. Each\ncell is labeled with a index ξ. There are Nξ\nℓparticles of\nspeciesℓin cellξ, and the total number of particles in\nthat cell is Nξ=/summationtext\nℓNξ\nℓ. With a single species ( ℓ= 1)\nthe multiparticles collisions are carried out as follows: at\nFIG. 1. Snapshot of a typical configuration of the system.\nThe big gray spheres are the catalytic obstacles and the smal l\nballs represent the reactive particles X(yellow) and Y(blue)\nof the fluid.\nevery time step τ, a random rotational operator ˆ ωξis as-\nsigned to each cell. The velocity of the center of mass in\nthe cellξisVξ=N−1\nξ/summationtextNξ\ni=1vi, whereviis the velocity\nof the particle ibefore the collision. After the collision\nis performed, the velocity of particle iwill be given by\nv′\ni=Vξ+ ˆωξ(vi+Vξ).\nThe step-collision rule can be generalized to multicom-\nponent species in the system [40]. If ℓi∈ {1,2,...,s}\ndenotes the species label of particle i, then we may write\ncollision rule as\nv′\ni=Vξ+ ˆωξ(Vℓi\nξ−Vξ)+ ˆωℓi\nξˆωξ(vi−Vℓi\nξ),(1)\nwhereVℓi\nξis the velocityofthe centerofmassofparticles\nof species ℓiin cellξand ˆωℓi\nξis the rotational operator\nthat only acts on the corresponding subset of particles ℓi\ninξ. Bothrotationaloperators, ˆ ωξand ˆωℓi\nξ, arerandomly\nchosenateachcollisionstep. This collisionruleconserves\nmass, momentum and energy, and preserves phase space\nvolumes.\nIn the bulk solutions, we assume that molecules may\nalso undergo chemical reactions of the form\nRµ:s/summationdisplay\nℓ=1νµ\nℓXℓkµ−→s/summationdisplay\nℓ=1¯νµ\nℓXℓ, (2)\nwhereνµ\nℓand ¯νµ\nℓare the stoichiometric coefficients for\nreaction Rµ,Xℓis the density of chemical species ℓand\nkµis the velocity constant of the reaction.\nIn reactive multiparticle collision dynamics, reactive\ncollisionsoccuratdiscretetime intervals τRwhichismul-\ntiple ofthe time step τ. Theprobabilitythat the reaction3\nRµoccurs before any other event in the cell ξin the in-\ntervalτRis given by\nPξ\nµ(Nξ,τR) =aξ\nµ\naξ(1−e−aξτR), (3)\nwhereNξis the vector of species populations in the cell\nandaξ=/summationtext\nµaξ\nµ.\nWhen there are reactive collisions the probability that\na reaction Rµwill occur in a cell ξwith a free volume Vξ\nf\nduring the interval ( t,t+dt) is given by [38]\naξ\nµ=kµ(Vξ\nf)hξ\nµ, (4)\nwhere the notation kµ(Vξ\nf) indicates that the rate con-\nstants have been scaled to take into account the free vol-\nume of the cell Vξ\nf, andhξ\nµis a combinatorial factor that\naccounts for the number of different ways the reaction\ncan occur in the cell, given by\nhξ\nµ=s/productdisplay\nℓ=1Nξ\nℓ!\n(Nξ\nℓ−νµ\nℓ)!. (5)\nIn the presence of catalytic obstacles, emerges a set of\nchemical reactions of the form\nRµ:Xℓ+Ckµ−→Xℓ′+C, (6)\nthat take place on the surface of the obstacles and con-\nverts species ℓintoℓ′. In such reactions the rate constant\nis given by\nkµ=PR/parenleftbigg8πkBT\nm/parenrightbigg1/2φ\nVO, (7)\nwherePRis the reaction probability, and the other terms\nare related to the cross section [40]. Note that when the\nvolume fraction occupied by catalytic obstacles is large,\neffects that modify the mass-action chemical rate laws\nand rate constants emerge.\nB. Selkov reaction with catalytic obstacles\nSpecifically, we consider a system comprising a solu-\ntion of reactive species ℓ={A,B,X,Y }that follow the\nreversible version of the Selkov reaction [41]\nR1,R2: Ak1⇄\nk−1X ,\nR3,R4:X+2Yk2⇄\nk−23X , (8)\nR5,R6: Yk3⇄\nk−3B ,\nwhereAandBdenote species with constant concentra-\ntions that act as feeds which maintain the system out ofequilibrium. This is a very simplified model of the phos-\nphofructokinasereactionscheme portion ofthe glycolytic\ncycle that contributes to the oscillations seen in this sys-\ntem [41]. Selkov reaction is a convenient test case for our\nstudy because it is a complex chemical reaction that is\nreal. Also, it is very simple, non-linear, with limit cycles\nand whose mean field reaction dynamics of the reversible\nversion shows oscillatory and steady-state behaviors and\nwith a well known phase diagram [42].\nAs particles in their free streaming movement can col-\nlide with the obstacles and undergo a bounce-back col-\nlisions, where particles of the species Xchange to the\nspeciesYwith probability PC, in the model, addition-\nally to the reactions in Eq. (8), we include the following\nreaction\nR7:X+CkC−→C+Y . (9)\nThechemicalratelawcorrespondingtoEqs.(8-9)isgiven\nby\ndCX\ndt=k1−k−1CX−k2CXC2\nY+k−2C3\nY\n−kCCXCO, (10)\ndCY\ndt=k2CXC2\nY−k−2C3\nY−k3CY+k−3\n+kCCXCO, (11)\nwhere the constant concentrations of the feed species A\nandBhave been incorporated in the k1andk−3rate\nconstants.\nIII. RESULTS\nSimulations are performed using the multiparticle col-\nlision (MPC) approach. We have set the volume of cells,\nV= 1, the rotational operations ˆ ωξare taken from the\nset{±π/2}aboutrandomlychosenaxes, the massofpar-\nticles of the reactive species {A,B,X,Y }ism= 1, and\nthe radius of the catalytic spheres is σ= 2.5.\nFor different values of the fraction of volume occu-\npied by the obstacles φ, the system volume Vis adjusted\nto keep the particles density constant in the remaining\nvolume free of obstacles, n=N/Vf= 11, varying the\nnumber of particles Nin the system as little as possible.\nThe initial concentrations of XandYareCX= 3.0 and\nCY= 0.8 for all cases.\nThe temperature in reduced units ( m,V,τ) is set as\nT= 5/12; hence, particles move a fraction of the length\nof the cell on average, which introduces the impossibility\nto maintain Galilean invariance [43]. This is corrected\napplying a shifting to this invariance. The rate constants\nin the Selkov reaction, which yield to oscillatory dynam-\nics, arek1= 0.0009485, k−1= 0.0001,k2= 0.0004,\nk−2= 0.0004,k3= 0.001, and k−3= 0.0001265.\nThe rate constant kCthat characterizes the reac-\ntion on the catalytic sphere can be written as [44]\nkC=pCkmkD/(km+kD), where kmis an intrinsic4\nrate constant and kD= 4πDσis the Smoluchowski dif-\nfusion rate constant where, Dis the diffusion coefficient\nthatcanbecomputedformultiparticlecollisiondynamics\n[34, 45] as D=D0(1−φ)/(1 +φ/2), in a first approx-\nimation, where D0≈0.479τis the diffusion coefficient\nin a system without obstacles. The intrinsic rate con-\nstant, computed approximately from collision theory, is\nkm=σ2/radicalbig\n8πkBT/m. Table I shows these reaction rates\nfor two different values of τ. These values of the rates\nmean that as the time elapses the reaction will be in-\ncreasingly controlled by the diffusion mechanism.\nTABLE I. Approximate values of the intrinsic reaction rate\nkm, the Smoluchowski diffusion rate , kD, and the catalytic\nreaction rate, kC, for two different values of the simulation\nstep,τ, for the simulation setup.\nτ k m kD kC\n1.0 20 .225 15 .053 8 .630\n0.5 20 .225 7 .527 5 .485\nA. Reactions in bulk solution\nThe full reaction-diffusion dynamics in the presence of\nan arbitrary number of catalytic obstacles can be simu-\nlated using reactive multiparticle collision dynamics. As\ndescribed in [38], for long time scales in a well mixed sys-\ntem, this mesoscopic dynamics reduces to the mean-field,\nmass-action equations of chemical kinetics.\nThe simulation results for the globally averaged con-\ncentrations XandYare compared with the mean-\nfield concentrations as trajectories in the phase-space\n(CX(t),CY(t)), as shown in FIG. 2 for two different val-\nues of time step τ= 1.0 (top) and τ= 0.5 (bottom).\nIt is noticeable that as φincreases, the limit cycle in the\nphase-spacereducesuntil reachingafixed point. Thisbe-\nhavior appears for both values of τ; however, being more\ndiscernible for τ= 1.0. We observe that for φ= 0.1\nandφ= 0.2 the steady state is a limit cycle, which size\ndepends on τ, being larger for smaller values of τ. Addi-\ntionally, the limit cycle vanishes for φ= 0.3 andτ= 1.0,\nindicating that the diffusion is determinant in the steady\nstate.\nAsφincreases, the mean-field approximation fails to\ndescribe the behavior of the system, because the effects\nof diffusion (in the cyclic limit) are small. To observe the\neffects of diffusion in this regime we calculate the instan-\ntaneous difference of concentration Xfor two values of\nτ, ∆CX=C(τ=1.0)\nX(t)−C(τ=0.5)\nX(t).\nFIG. 3 showsthe evolution of ∆ CXfor two values of φ.\nFor the mean field approximation (top) ∆ CXincreases\nwith time for both values of φ, taking larger values for\nlarger values of the density φ. On the other hand, the\nbehavior of ∆ CXfor the simulations (bottom) only in-\ncreases for small values of density, φ= 0.1, while for\nlarger values, φ= 0.3, the difference ∆ CXfalls into con-\n❂ ✵ \u0000\n✷✶✿ ✺✁✂✄ ☎✆\n✝ ✞ ✟\n✻ ✠ ✹ ✸ ✡\n☛☞✌ ✍✎✏✑ ✒✓\n✔ ✕ ✖\n✗ ✘ ✙\n✚ ✛ ✜ ✢ ✣\n❂ ✵ \u0000\n✷✶✿ ✺✁✂✄ ☎✆\n✝ ✞ ✟\n✻ ✠ ✹ ✸ ✡\n☛☞✌ ✍✎✏✑ ✒✓\n✔ ✕ ✖\n✗ ✘ ✙\n✚ ✛ ✜ ✢ ✣\nFIG. 2. Trajectories in the phase-space, ( CX(t),CY(t)), for\nPC= 4×10−5and various values of φ.τ= 1.0 (top) and\nτ= 0.5 (bottom). The solid blue and black lines are the\nsimulation and mean-field results, respectively. Dashed li ne\nrepresent the attractor of the mean-field model. Simulation\ntime ist= 105iterations.\nstant oscillations. We attribute this behavior to the pres-\nence of two different steady states in the system, a fixed\npoint and a limit cycle.\nB. Behavior of transients\nAnother characteristic of the dynamics that could be\nappreciated in FIG. 2 is that for a given value of φ, the\namplitude of oscillations start to converge either to a cy-\ncle limit or a fixed point. This behavior resembles that\nof a mass connected to a spring in the presence of a fric-\ntional force, and can be described by\nCX(t)∼cos(ωt+d)exp(−γt), (12)\nwhere the frequency ( ω), the phase shift ( d), and the de-\ncay constant γare fitting parameters. In a mechanical\nsystem,γrepresents the friction; however, a more accu-5\n✶✵\u0000✁✂✷✄✸\n☎✆ ✝✞✟✠ ✽✡☛☞ ✌ ✻✍✎✏✑ ✹✒✓ ✔✕ ✖✗✘✙ ✚ ✛\n✜✿ ✢✣✤✥✦ ✧★✩✪ ✫\nFIG. 3. Evolution of the difference between X concentrations\nfor two different values of the time step, τ= 1.0 andτ= 0.5.\nThe evolution of ∆ CXis calculated for φ= 0.1 (black lines)\nandφ= 0.3 (blue lines) with the same parameters used in\nFIG. 2. Top: Results obtained using the mean field theory.\nBottom: Results obtained from the simulations of the MPC\nmodel.\nrate interpretation of γfor our system is that it’s related\nwith the transport properties of media and the value of\nPC. To verify this relationship, FIG. 4 shows how γde-\npends on the density of catalytic spheres φ, that is, on\nthe reaction surface. Note that in all cases the depen-\ndency is linear but the slope changes for the different\nvalues of PCandτ, meaning that γeffectively depends\nontheprobability PCanddiffusioncoefficientinasystem\nwithout obstacles D0, according to the linear expression\nγ∼mτ\nPCφ.\nIn this way, table II shows the ratios among the dif-\nferent values of mτ\nPCand compare them with the ratios\nbetween values of τandPC.\nTABLE II. Ratios between the values of τ,PCandmτ\nPC. To\nsimplify the notation all values of PCare shown multiplied by\n105.\nτRatio PCRatio mτ\nPCRatio\n1.0/0.5 = 2.0 10 /10 = 1.0/parenleftbig\nm1.0\n10/m0.5\n10/parenrightbig\n≈1.66\n1.0/0.5 = 2.0 04 /04 = 1.0/parenleftbig\nm1.0\n04/m0.5\n04/parenrightbig\n≈2.01\n1.0/1.0 = 1.0 10 /04 = 2.5/parenleftbig\nm1.0\n10/m1.0\n04/parenrightbig\n≈2.10\n0.5/0.5 = 1.0 10 /04 = 2.5/parenleftbig\nm0.5\n10/m0.5\n04/parenrightbig\n≈2.56\nIn the table we observe that the better correlations oc-\ncur forPC= 4×10−5whenmτ\nPCratio is 2.01 and τ\nratio is equal to 2; and for τ= 0.5 whenmτ\nPCratio is\n2.56 and PCratiois 2.5. In both cases, the values of mτ\nPCthat are closer to τandPCare smaller than these pa-\nrameters. In other words, when the velocities of catalytic\nreactions and diffusion are slow, there is a linear effect\non the damping, while for the opposite case, the effect is\nsmaller than linear. Although the relationship between\n❂ ✶ ✵\n\u0000✿ ✷✁✂ ✾✄☎ ✻✆✝ ✸✞\n✟ ✠ ✺✡☛ ☞ ✌✍ ✎ ✏ ✑✒ ✓ ✔✕ ✖✗ ✘ ✙ ✚ ✛ ✜ ✢✣ ✤\n✥✦ ✧★✩ ✹✪✫ ✬✭\nFIG. 4. Decay constant, γ, as function of catalytic spheres\ndensity, φ, for two values of reaction probability, PC= 4×\n10−5(circles blue points) and PC= 1×10−4(squares black\npoints); andtwo valuesof time step, τ= 1.0 (top)and τ= 0.5\n(bottom). Error bars are the standard deviations computed\nover 8 realizations.\nmτ\nPC,τandPCis not trivial, we can see that the change\nin the slope of the damping is linear respect to variations\nofτandPCwhen the reaction is sufficiently slow, this is\nPC= 4×105, or when the diffusion is sufficiently slow,\nthis isτ= 0.5.\nWe know that when the number of catalytic spheres\nincreases the surface of reaction grows, then the value\nkCbecomes greater. This effect could be suppressed\nsetting the reaction probability on catalytic obstacles\n/tildewiderPC=P0\nCN0\nC/NO(φ), where P0\nCandN0\nOare the reac-\ntion probability and the volume fraction occupied by the\ncatalytic spheres with which the system has the desired\nreaction rate. FIG. 5 shows the decay constant γas a\nfunction of the obstacle volume fraction φusing/tildewiderPCas\nreaction probability on catalytic obstacles. Note that\ndespite compensating the increase of the reaction surface\nwith the decrease of /tildewiderPC, the value of the damping factor\nchanges, increasing linearly with an approximate slope\nequal to 0 .85. This shows us that the effect of the spa-\ntial distribution of the catalytic spheres is significant on\noscillatory chemical reaction.\nC. Reactions on catalytic sphere\nReactions on the surface depend on the concentration\nof the species Xand on its transport properties. Conse-\nquently, if CX(t)oscillatesthe numberofreactionsonthe\ncatalytic spheres will oscillate as well. Despite this oscil-\nlations, the cooperative effects of reactions over spheres\ncan be analyzed observing the radius of gyration of the\nspheres where the reactions take place during each τR6\n✵✿ ✸ \u0000 ✁ ✷✺ ✂ ✄ ☎ ✆✝ ✶✞ ✟✠ ✡ ☛☞ ✌ ✍ ✎\n✏✑ ✒✓✔✕ ✖✗✘ ✙✚✛✜ ✢✣✤ ✥✦✧\nFIG. 5. Decay constant γas function of the volume fraction\nof obstacles φfor reaction probability /tildewiderPC=P0\nCN0\nO/NO(φ).\nPoints represent the mean values averaged over 8 realizatio n\nwithP0\nC= 6×10−4,N0\nO= 10 and τ= 0.5. Error bars\nshow the standard deviations around the mean. The line is\nthe best fit of a linear function among of points with a slope\napproximately of 0.85.\ninterval time given by\nρR\ng=/radicaltp/radicalvertex/radicalvertex/radicalbt1\nNRNR/summationdisplay\ni=1(ri−rP)2, (13)\nwhereNRis the number of reaction events that occurred\non the catalytic spheres in an interval τR,riis the posi-\ntion of the sphere where the i-th event took place, and\nrPis the center of mass of all catalytic spheres involved\nin the reaction during the time interval. In a random\nprocess, reactions can occur on the surface of any of the\ncatalytic obstacles with the same probability and in such\ncase the difference between the radius of gyration of all\nobstacles ( ρO\ng) and the average value over all time inter-\nvals of the radii of gyration of obstacles where chemical\nreactions take place ( ρRg) should be equal to zero, that\nis ∆ρg=ρO\ng−ρRg≈0. However, if the distribution of\nobstacles and the diffusion process favor the triggering of\nreactions on the obstacles in some regions of the volume\nV, there should exist a difference between these two radii\nof gyration, that is, ∆ ρg/ne}ationslash= 0.\nFIG. 6 evidences that ∆ ρgis greater than zero for all\nvalues of φand for both values of the time step, τ= 1.0\n(black squares) and τ= 0.5 (blue circles); i.e. in all\nstudied cases ρRgis less than expected, that is, reactions\non the catalytic spheres are forming clusters. It can also\nbe appreciated that for low values of the volume frac-\ntion of catalytic obstacles ( φ≤0.1) the values of ∆ ρg\nare independent of the values of τconsidered, which im-\nplies that ρRgdoes not depend significantly of the diffu-\nsion constant in this range of values of φ. Nonetheless,\nfor greater values of φthe difference ∆ ρgonly increaseswhen the diffusion is low ( τ= 0.5), while for higher dif-\nfusion (τ= 1.0) its value reaches an apparent constant\nbehavior.\n✵✿ ✸ \u0000✁ ✷✺ ✂✄ ☎ ✆✝ ✶ ✞ ✟✠ ✡ ☛☞ ✌ ✍ ✎\n✹✏ ✽✑✒ ✓✔✕✖ ✻✗✘ ✙\nFIG. 6. Effect of the volume fraction occupies bythe catalyti c\nspheres, φ, on the difference between the radius of gyration of\nthe obstacles and the average values over all time intervals of\ntheradiiofgyrationofobstacles wherechemicalreactions take\nplace, ∆ ρg. Simulations are done with pC= 10−4,τ= 1.0\n(black squares) and τ= 0.5 (blue circles). Error bars shows\nthe standard deviations over 8 realizations.\nAs an example of a system where these properties can\nbe determining, thereis an experiment published byTay-\nlor et al. [25] where they studied large populations of\ndiscrete chemical oscillators that presents synchronized\noscillatory behavior. The experiment was carried out\nwithφ≈0.05 in an agitated medium, in other words,\na medium where the diffusion coefficient D0is large. In\nour model, these parameter values correspond to a sys-\ntem with ρRgless than expected and where clustering of\nthe reactions does not depend significantly on the diffu-\nsion coefficient. The clustering of the catalytic reactions\nobserved in the experiment [25] as well as in our model,\nsuggests that the system’s behavior reported by Taylor\net al. could be in part due to phenomena that is also\npresent in our model.\nTo understand the nature of the clustering, we calcu-\nlate the number of reactions that occur during the time\nintervalτR, and denote it by nR. FIG 7 shows the av-\nerage distribution ( H) of the first three values of nRas\na function of their respective ρR\ng. To create the distri-\nbutions, we counted, for each ρR\ng, how many times each\nnRoccurred during a simulation, averaging these results\nover 8 realizations. Note that the results of the simula-\ntions, which were carried out in systems with a volume\nfraction of obstacles φ= 0.3 and a probability of reac-\ntionPC= 10−4, show that most groups of reactions take\nplace in obstacles which radii of gyration are less than\nthe radius of gyration of all obstacles, ρO\ng, indicated in\nthe FIG 7 with vertical lines. This behavior is observed\nfor both time steps, τ= 1.0 (left) and τ= 0.5 (right).7\n✷✵ ✶✻ \u0000 ✁ ✽ ✹ ✂\n✄✺☎✆✝✞✟✠✡☛☞✌✍✎✏✑✒✓✔✕\n✷✵ ✶ ✻ \u0000✁ ✽ ✹ ✂\n✼✄ ☎✆✝ ✞✺✟ ✠✡☛ ☞✸✌ ✍✎✏ ✑✒✓ ✔✕\nFIG. 7. Average distribution of the the number of reactions t hat occur during an interval τRwith the same radius of gyration\nof the obstacles where the reactions take place, nR, as a function of their respective radii of gyration, ρR\ng. Averages were made\nover 8 simulations with φ= 0.3,PC= 10−4andτ= 1.0 (left), and τ= 0.5 (right). Solid, dashed and dotted lines represent\nthe distributions of nR= 2,3 and 4 respectively. The vertical line indicates the mean va lue of the radius of gyration of all\nobstacles, ρO\ng.\nNotice that with greater diffusion, the maximum val-\nues of the distributions are closer to the value ρO\ngand the\nnumber of reactions is considerably larger. Additionally,\nit canbeobservedthat the structuresformedbythe reac-\ntions depend on the diffusion mechanism when the close-\nness among reactions is favored by a greater diffusion\nrate. In both cases it is also appreciated, as expected,\nthat asnRincreases, the corresponding average radius of\ngyration approaches to ρO\ng. Finally, for sufficiently large\nvalues of diffusion, once more we can see that the obsta-\ncles, where small groups of catalytic reactions take place,\nhave an average radius of gyration ( ρRg) which is clearly\nsmaller than the radius of gyration of all the obstacles\npresent in the system ( ρO\ng), in other words, in the system\nemerge clusters of catalytic reactions.\nIV. CONCLUSION\nWe show that, as previously reported [46], the effect of\nthe introduction of catalytic obstacles in a system with a\ncomplex oscillating chemical reaction is to reduce the os-\ncillations in it, leading the system to a fixed point where\nthe limit cycles are shifted. Additionally, we fit these os-\ncillations to a damped periodic function, finding that its\ndecay constant γscales linearly with the volume fraction\noccupied by the obstacles ( φ) after a certain amount of\nobstacles is introduced; meaning that there exists a crit-\nical damping value for φ. Our results show that when\nthe time step ( τ) and the probability of reaction on thecatalytic spheres ( PC) take low values, the slope of the\ndamping ( mτ\nPC) is linear with respect to them. Nonethe-\nless, when either the diffusion effects or the rate of cat-\nalytic reactions begin to be significant, the crowding of\nobstacles in the system becomes a crucial factor, even\nwhen the reaction probability p0\nCis adjusted to keep con-\nstantthe quantityofreactionsthat occuron the catalytic\nsurface of the spheres per time unit.\nFurthermore we find that, as a result of the presence\nof catalytic obstacles in the system, on average the ra-\ndius of gyration of obstacles where chemical reactions\ntake place is smaller than expected ( ρR\ng<ρOg); that is,\na clustering effect of the catalytic reactions emerges in\nthe system. In addition, there is a range of values of\nthe volume fraction occupied by the obstacles ( φ≤0.1)\nwhere the average radius of gyration ρRgdoes not seem\nto depend significantly on the constant of diffusion D0.\nIn other words, the model of oscillatory chemical reac-\ntions presented in this document is able to show and\nmeasure how changes in diffusive transport properties\nhave an important role in the distribution of catalytic\nreactions in a crowed environment, where hundreds of\nthousands of particles are involved, which could explain\nthe emergence of phenomena such as the quorum sensing\nand the chimeric patterns, observed in experiments with\nthis kind of systems [25, 47].\nAcknowledgements Jos´ e L. Herrera Diestra is sup-\nportedbythe S˜ aoPauloResearchFoundation (FAPESP)\nunder grants 2016/01343-7 and 2017/00344-2\n[1] R. Kapral and K. Showalter, Chemical waves and pat-\nterns, Vol. 10 (Springer Science & Business Media, 2012).[2] C. Echeveria, K. Tucci, and R. 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Oscillations were classi\fed\naccording to their velocity amplitude: 106 small-amplitude oscillations (SAOs), with velocities <\n10 km s\u00001, and 90 large-amplitude oscillations (LAOs), with velocities >10 km s\u00001. Both SAOs and\nLAOs are common, with one event of each class every two days on the visible side of the Sun. For\nnearly half of the events we identi\fed their apparent trigger. The period distribution has a mean\nvalue of 58\u000615 min for both types of oscillations. The distribution of the damping time per period\npeaks at\u001c=P = 1:75 and 1:25 for SAOs and LAOs respectively. We con\frmed that LAO damping\nrates depend nonlinearly on the oscillation velocity. The angle between the direction of motion and\nthe \flament spine has a distribution centered at 27\u000efor all \flament types. This angle agrees with the\nobserved direction of \flament-channel magnetic \felds, indicating that most of the catalogued events\nare longitudinal (i.e., undergo \feld-aligned motions). We applied seismology to determine the average\nradius of curvature in the magnetic dips, R\u001989 Mm, and the average minimum magnetic-\feld\nstrength,B\u001916 G. The catalog is available to the community online, and is intended to be expanded\nto cover at least 1 solar cycle.\n1.INTRODUCTION\nFilament oscillations were \frst observed visually\n(Greaves, Newton, & Jackson, reported by Dyson 1930;\nNewton 1935; Bruzek 1951), followed by photographic\nobservations that revealed a signi\fcant relationship with\n\rares (Dodson (1949); Bruzek & Becker (1957) and\nBecker (1958)). Moreton & Ramsey (1960) con\frmed\nthat wave disturbances initiated during the impulsive\nphase of \rares were responsible for triggering prominence\noscillations both near and far from the \rare. Ramsey &\nSmith (1966) determined periods and damping times for\nseveral oscillating \flaments, but did not \fnd any correla-\ntion between the period or damping time and the dimen-\nsions of the \flament, the distance to the associated \rare,\nor its size. In these early observations, some events were\ncalled \\winking \flaments\" because these \flaments were\nvisible inH\u000bwhen they were at rest, but disappeared\nwhile oscillating. Because these observations were made\nwith narrow-band H\u000b\flters, Doppler-shifted absorption\nfrom prominence material traveling at su\u000eciently large\nline-of-sight (LOS) velocities ( >23 km s\u00001) fell outside\nthe 0.5 \u0017A bandpass of the \flter and thus became invisible\ninH\u000b.\nNowadays, thanks to both space- and ground-based\ninstruments, observations of large-amplitude \flament os-\ncillations (LAOs: v>10 km s\u00001) have become common.\n1Instituto de Astrof\u0013 \u0010sica de Canarias, E-38200 La Laguna,\nTenerife, Spain\n2Departamento de Astrof\u0013 \u0010sica, Universidad de La Laguna, E-\n38206 La Laguna, Tenerife, Spain\n3NASA Goddard Space Flight Center, Greenbelt, MD 20771,\nUSA\n4Departament de F\u0013 \u0010sica, Universitat de les Illes Balears\n(UIB), E-07122 Palma de Mallorca, Spain\n5Institute of Applied Computing & Community Code (IAC3),\nUIB, Spain\n6Catholic University of America, Washington, DC 20064, US(The terms \\large\" and \\small\" amplitude are de\fned\nlater in this Section.) The exciters identi\fed thus far in-\nclude Moreton or EIT waves (Eto et al. 2002; Okamoto\net al. 2004; Gilbert et al. 2008; Asai et al. 2012), EUV\nwaves (Liu et al. 2012; Shen et al. 2014a; Xue et al. 2014;\nTakahashi et al. 2015), shock waves (Shen et al. 2014b),\nnearby jets, sub\rares and \rares (Jing et al. 2003, 2006;\nVr\u0014 snak et al. 2007; Li & Zhang 2012), and the eruption\nof the \flament (Isobe & Tripathi 2006; Isobe et al. 2007;\nPouget 2007; Chen et al. 2008; Foullon et al. 2009; Boc-\nchialini et al. 2011).\nMany of the observed \rare-induced LAOs in \flaments\nexhibit motions in di\u000berent directions relative to the ax-\nial magnetic \feld (polarization). For instance, the mate-\nrial can undergo vertical (Eto et al. 2002; Okamoto et al.\n2004; Shen et al. 2014a), horizontal (Kleczek & Kupe-\nrus 1969; Hershaw et al. 2011; Gosain & Foullon 2012;\nLiu et al. 2012; Shen et al. 2014b), or longitudinal (\feld-\naligned) (Jing et al. 2003, 2006; Vr\u0014 snak et al. 2007; Li &\nZhang 2012; Zhang et al. 2012; Luna et al. 2014; Shen\net al. 2014b) motions. Oscillations with a mixed charac-\nter (Gilbert et al. 2008) have also been observed.\nThe \frst theoretical models proposed to explain the\nexcitation, restoring forces, and damping mechanisms of\nlarge-amplitude longitudinal oscillations were purely an-\nalytical (Hyder 1966; Kleczek & Kuperus 1969). One-\ndimensional, hydrodynamic, numerical models have been\nemployed successfully to describe longitudinal oscilla-\ntions (Vr\u0014 snak et al. 2007; Luna & Karpen 2012; Luna\net al. 2012; Zhang et al. 2012, 2013; Ruderman & Luna\n2016; Zhou et al. 2017), while 2D and 3D MHD models\nhave described more completely the features of observed\nlongitudinal and transverse oscillations (Terradas et al.\n2013; Terradas et al. 2015, 2016; Luna et al. 2016).\nSpectroscopic techniques have revealed oscillations\nwith much smaller peak velocities than those of LAOs,arXiv:1804.03743v1  [astro-ph.SR]  10 Apr 20182 Luna et al.\nwith amplitudes from the noise level of 0 :1 km s\u00001to\n10 km s\u00001. Harvey (1969) \frst measured oscillatory pe-\nriods between 1 and 17 min, while later observations\nyielded characteristic periods ranging from a few to 90\nmin. Although the triggering mechanisms of these small-\namplitude oscillations (SAOs) have not been clearly iden-\nti\fed, they are generally believed to be excited by the\nperiodic motions of \flament magnetic \felds driven by\nphotospheric or chromospheric oscillations (see review by\nArregui et al. 2012).\nA variety of approaches has been used to categorize\nand understand \flament oscillations. The simplest are\nbased on a single property, such as the peak velocity\n(e.g., Arregui et al. 2012), the nature of the trigger (e.g.,\nOliver 1999; Oliver & Ballester 2002), or the period (e.g.,\nArregui et al. 2012). The apparent tendency of periods\nto group below 10 min, in the range 10 - 40 min, or 40 -\n90 min (Arregui et al. 2012) led to classi\fcations denoted\nas short-, intermediate- and long-period oscillations, re-\nspectively. Very short-periods below 1 min (Balthasar\net al. 1993), very long-periods above 5 hours (Foullon\net al. 2004; Pouget et al. 2006), and even periods longer\nthan 20 hours (Efremov et al. 2016) have been reported.\nClassi\fcation based only on the period does not re\rect\nthe nature, origin, or exciter of the oscillations, however.\nMore complex schemes have proven to be di\u000ecult to em-\nploy consistently (e.g., Vr\u0014 snak 1993).\nBecause oscillation velocities have been measured from\nthe observable threshold to 100 km s\u00001, the velocity\namplitude alone is not the most de\fnitive criterion by\nwhich oscillation events can be categorized. In spite of\nthese limitations, a widely accepted, velocity-based di-\nvision between small-amplitude and large-amplitude os-\ncillations has proven to be both convenient and physi-\ncally justi\fable. We can relate the observed oscillation\namplitudes to their linear or nonlinear character by con-\nsidering the characteristic Alfv\u0013 en and sound speeds in\nprominences, which are of the order of 100 km s\u00001and\n10 km s\u00001, respectively. Therefore, oscillations with ve-\nlocity amplitudes above 10 km s\u00001exceed the local sound\nspeed, and hence can be considered nonlinear oscilla-\ntions, while smaller velocity amplitudes would be lin-\near. In general, small-amplitude oscillations (SAOs) ex-\nhibit amplitudes below 10 km s\u00001, are not related to\n\rare activity, are local, and can be appropriately ana-\nlyzed or modeled using methods of linear perturbations.\nLAOs are usually associated with energetic events, are of\nglobal character, and as the velocity amplitude is \u001510-\n20 km s\u00001, require a nonlinear approach. As we demon-\nstrate in the present work, however, exceptions to these\n\\rules\" exist.\nProminence seismology aims to determine physical pa-\nrameters that are di\u000ecult to measure by direct means in\nthese magnetized plasma structures. This remote diag-\nnostics method combines observations of oscillations and\nwaves in these structures with theoretical results from\nthe analysis of oscillatory properties of given prominence\nmodels, as \frst suggested by Tandberg-Hanssen (1995).\nThe \frst seismological determinations of magnetic \feld\nstrength in winking \flaments used a simple model of lon-\ngitudinal motions based on a harmonic oscillator (Hyder\n1966; Kleczek & Kuperus 1969). Vr\u0014 snak et al. (2007) an-\nalyzed large-amplitude longitudinal oscillations (LALOs)in a prominence to infer the Alfv\u0013 en speed; assuming the\nmass density of the prominence plasma, they also deter-\nmined the azimuthal and axial magnetic \feld strengths.\nOur theoretical investigation of oscillations in simulated\nprominence threads strengthened the foundations of the\ndamped harmonic oscillator model for LALOs, providing\na basis for applications to observations (Luna & Karpen\n2012; Luna et al. 2012). Subsequent seismological anal-\nyses of LALOs in prominences have derived the radius\nof curvature of dipped \feld lines supporting prominence\nthreads, the minimum magnetic \feld strength, the en-\nergy injected by the triggering jet, and the mass accre-\ntion rate according to the thermal nonequilibrium model\n(Li & Zhang 2012; Bi et al. 2014; Luna et al. 2014, 2016;\nZhang et al. 2017b). Using the same seismological tech-\nniques, we determined the curvature radius of the mag-\nnetic \feld dips and the minimum \feld strength from the\nlargest prominence oscillation ever reported in the lit-\nerature; these results were validated by reconstructing\nthe \flament magnetic \feld from the photospheric \feld in\ncombination with the \rux-rope insertion method (Luna\net al. 2017).\nTo interpret observed prominence LAOs directed\ntransverse to the magnetic \feld, an MHD approach is\nrequired. Some observations of oscillatory behavior have\nbeen interpreted and analyzed as global or standing kink\nmodes (e.g., Hershaw et al. 2011; Liu et al. 2012; Xue\net al. 2014). A theoretical analysis predicted a linear re-\nlationship between the damping time ( \u001c) and the period\n(P) that could be compatible with resonant absorption\nas the damping mechanism (Ruderman & Roberts 2002;\nOfman & Aschwanden 2002; Arregui et al. 2008b). How-\never, this interpretation must be considered with care\nbecause the use of scaling laws to discriminate between\ndamping mechanisms is questionable, at least for res-\nonant absorption (Arregui et al. 2008a). Much work\nremains before the physical models of both longitudi-\nnal and transverse LAOs are su\u000eciently detailed and\ncomprehensive to adequately link theory and simulations\nwith observed prominence motions.\nTo date, all studies of oscillating prominences have\nbeen focused on one or, at most, a few episodes. In\norder to understand this phenomenon thoroughly and\nderive key physical characteristics via seismology of all\ntypes of prominences over the solar cycle, we have be-\ngun to compile a systematic, large data set of oscillation\nevents. Thus far we have identi\fed and analyzed 196\nevents during several months close to the maximum of\nsolar cycle 24, using GONG H\u000bdata. We found that\nLAOs are very common on the Sun (one event every\ntwo days on the visible hemisphere), and that the fre-\nquency of SAOs is similar to that of LAOs, yielding one\nSAO or LAO per day. Our large sample of prominence\noscillations has enabled the \frst statistically signi\fcant\nstudy of \flament oscillations and their pertinent prop-\nerties, including their apparent triggers, damping times,\nperiods, \flament type, \flament dimensions, peak veloci-\nties, directionality with respect to the \flament spine, and\nmaximum displacements. With the information in this\ncatalog, one can derive minimum \feld strength and other\nunobservable characteristics through seismology, and be-\ngin to explore the implications of longitudinal and trans-\nverse oscillations for prominence stability, evolution, and\neruption. We have made the catalog available to theGONG Catalog of Solar Filament Oscillations 3\ncommunity at the following URL: http://www.iac.es/\ngaleria/mluna/pages/gong-catalogue-of-laos.php\nThis paper presents both individual examples of in-\nterest and statistical analyses that explore potential re-\nlationships among the derived parameters. In x2 the\nGONG data used in the catalog are described, while in\nx3 the GONG catalog of prominence oscillations is in-\ntroduced.x4 presents the method used to detect oscil-\nlations and select events for the catalog. The criteria\nused to classify prominence types are introduced in x5.\nx6 explains how we identi\fed the triggering mechanism\nand derived the \flament parameters. x7 andx8 discuss\nthe time-distance approach and analysis methods used\nto characterize the oscillations, respectively. x9 describes\nselected events in detail, while in x10 we present the re-\nsults of our statistical study of \flament oscillations. A\nseismological analysis of selected events comprises x11,\nand the results are summarized in x12. A full list of\nevents and their oscillation parameters is in Appendix\nA. We describe our new method for constructing time-\ndistance diagrams with data from curved slits in Ap-\npendix B.\n2.DESCRIPTION OF THE NSO GONG NETWORK DATA\nNowadays, it is possible to monitor the full Sun nearly\ncontinuously with the space-based Solar Dynamics Ob-\nservatory (SDO; Lemen et al. 2012) or the ground-based\nnetwork of telescopes of the Global Oscillation Network\nGroup (GONG) ( http://gong2.nso.edu ). Continuous\ncoverage of the full Sun is needed for a complete study of\n\flament oscillation events. SDO o\u000bers the best spatial\nresolution and temporal cadence, and the observations\nare independent of the local conditions of the Earth's\natmosphere, in contrast to the GONG telescopes. How-\never, the \flaments and their periodic movements are not\neasy to detect in SDO data. In some situations the os-\ncillation is clear in the GONG H \u000bdata, but it is not\npossible to see the \flament in absorption in the SDO\nEUV images because of foreground emission. In addi-\ntion, the structures seen by SDO are complex and very\ndynamic, making the detection of periodic movements\nvery di\u000ecult. Therefore we use GONG data to perform\nour survey of \flament oscillations. The GONG network\ntelescopes o\u000ber su\u000eciently good spatial resolution and\ntemporal cadence to detect prominence oscillations with\nperiods of a few tens of minutes.\nThe GONG H \u000bimages allow us to identify \flaments\neasily and to follow their motions. We interpreted the\n\flament motions as displacements of the prominence\nplasma in the plane of the sky. However, H\u000bintensity\ndepends on LOS velocities. It is worth to mention that\nexists the possibility that this e\u000bect may produce a dis-\nappearance of parts of the \flament giving the impression\nthat the remaining visible \flament is moving. With the\nH\u000bGONG data we can study the massive set of oscilla-\ntions observed since August 2010, the date when the net-\nwork started to operate. Here we focus on an analysis of\nGONG data from several months close to the maximum\nof solar cycle 24, from 1 January 2014 to 30 June 2014.\nCycle 24 started in 2008 and reached minimum in early\n2010, with a double-peaked maximum in 2013 and 2014.\nThe GONG network telescopes are of identical design\nand construction and are placed around the world at\nthe following locations: Learmonth (L), Udaipur (U), ElTeide (T), Cerro Tololo (C), Big Bear (B) and Mauna\nLoa (M). The telescope locations were selected to fol-\nlow the diurnal motion of the Sun in the sky, in order to\ncollectively ensure full-day coverage (Harvey et al. 1996).\nEach telescope takes data daily, weather permitting, with\nsome temporal overlap of coverage between telescopes.\nThe temporal cadence of the GONG data is 1 min with\na pixel size of\u00181 arcsec. For each data sequence of each\ntelescope we compensate the solar di\u000berential rotation\nusing the drot map.pro solarsoft routine. The reference\ntime to de-rotate the images is the central time of each\ntemporal data sequence for each telescope and day.\n3.GONG CATALOG OF PROMINENCE OSCILLATIONS\nThe objective of our GONG catalog is to completely\ndescribe the oscillations detected in solar \flaments be-\ntween 1 January 2014 and 30 June 2014. The catalog\ncontains information about the properties of the oscil-\nlating \flaments, the apparent triggers of the oscillation,\nand the oscillation parameters. With this information we\nconstruct a comprehensive global picture of the \flament\noscillations close to the maximum of solar cycle 24.\nIn the following sections we describe the methods we\nused to construct the catalog ( x4 tox8). The full results\nof the survey are shown in Tables 1 to 8 in Appendix A.\nThe \frst group (Tables 1 to 4) displays data describing\nthe observations and the \flaments. The \frst column cor-\nresponds to the number of the oscillation event, ordered\nin time starting 1 January 2014. The second column\nlists the telescope where the event is detected (L, U, T,\nC, B, M). The third column lists the central time of the\ntemporal sequence associated with each telescope used\nto analyze the event (see x2). The fourth column shows\nthe averaged position of the \flaments at the reference\ntime (see detailed description in x9). The \ffth column\nindicates the \flament type (AR, IT, QS) described in\nx5. The sixth and seventh columns contain the length,\nL, and width, Wof the \flament measured as described\ninx6. In the eighth column we indicate the possible\ntriggering agent described in x6. The last column shows\nwhether the \flament erupted in the temporal sequence\nanalyzed, indicated by a Y (Yes).\nThe second group, Tables 5 to 8, shows the oscillation\nparameters resulting from the \ftting method described\nin detail inx8 with Equation (2). The columns indicate\n(1) the event number; (2) the initial time of the sequence\nused for the \ft; (3) the angle \u000bbetween the oscillation\ndirection and the \flament spine; (4) the period ( P); (5)\ndamping time ( \u001c); (6) damping time per period ( \u001c=P);\n(7) maximum displacement ( A); and (8) velocity ampli-\ntude (V).\nInx5 tox8 we will use Event 1 from Table 5 as our\nreference event to describe the methodology. Although\nthe \fgures are speci\fc to this event, the results and ex-\nplanations are valid for all events listed in the Tables.\n4.EVENT SELECTION\nOur \frst action was to detect the \flaments that may\noscillate by visual inspection of the GONG H \u000bdata\n(http://gong2.nso.edu ), in which the \flaments are\nseen as dark absorption structures (see Figure 1). The\noscillations were identi\fed as periodic displacements of\na part of the \flament. The GONG webpage shows very\ngood quality movies with full cadence for all six network4 Luna et al.\ntelescopes. We analyzed daily observations from each\ntelescope, and selected data that showed a clear or sus-\npected oscillatory event for in-depth analysis. In this\ninitial inspection we identi\fed 408 potential cases. We\ninitially identi\fed each event to be associated with one\nday and one telescope. For cases where the oscillation\ncontinued at the end of the observing period of the se-\nlected telescope, we did not utilize the subsequent tele-\nscope observations in order to extend the oscillation data.\nIn addition, we checked the data carefully to avoid double\ncounting the same oscillation observed by two telescopes\nwith overlapping data. For cases in which a second os-\ncillation appeared during a given observing period, we\nde\fned a new event with the same location and tele-\nscope as the preceding event and we marked it with an\nasterisk next to the event number.\nOnce we identi\fed the \flaments that might oscillate,\nwe downloaded the reduced H \u000bdata in the form of FITS\n\fles from the GONG server. We de-rotated the images\nin order to compensate for solar rotation and to study\nthe proper motion of the \flaments over the solar surface.\nAll images were de-rotated using a reference time that\ncorresponds to the central time of the temporal sequence\nas described inx2. This de-rotation algorithm only works\non the solar disk, so we discarded events in prominences\nseen at the limb and focused exclusively on \flaments seen\nin absorption on the disk. The coordinates are given in\nthe usual Heliocentric-Cartesian coordinates (Thompson\n2006).\n5.FILAMENT CLASSIFICATION\nIn the catalog we assigned \flament types exclusively\nbased on GONG H \u000bdata, according to the position\nscheme of Engvold (2014), as active region (AR), inter-\nmediate (IT), or quiescent (QS). In Figure 1 we have\nmarked the three types by colored arrows (AR - red, IT\n- green, QS - blue). AR \flaments are located close to\nsunspots and plages with a prominent spine and few or\nno barbs. ITs have one end close to an active region (AR)\nand the other end far from an AR; they exhibit both a\nspine and barbs. The QSes are far from any AR or plage\nregion, with no clear spine. For \flaments whose type was\ndi\u000ecult to determine, we used SDO HMI magnetograms\nto distinguish whether the \flament is close to a strong\nmagnetic \feld or a quiet region. The catalog includes 45\nAR, 99 IT and 52 QS \flaments.\nFollowing this classi\fcation we identify our reference\ncase 1 as intermediate or IT (see Table 1), because the\n\flament has one end located in plage associated with the\nactive region NOAA 11938 and the other end in a quiet\nregion (see Fig. 2).\n6.TRIGGERING AND FILAMENT PARAMETERS\nWe constructed a movie with the FITS data showing\nthe region of interest surrounding each \flament, enabling\nus to identify the most likely triggering agent and to\nstudy the \flament motion. For more than half of the\nevents we could not identify what triggered the oscilla-\ntions, so we left column 4 empty in Tables 1 to 4. Those\ncases for which we found a trigger were marked FLARE\nwhen a sudden H \u000bbrightening was detected nearby just\nbefore the oscillation onset; prominence eruption (PE)\nwhen a nearby \flament erupted before oscillation onset;\nand JET when the trigger was a jet of plasma that hit\nFigure 1. GONG H\u000bimage from the Learmonth telescope illus-\ntrating the 3 types of \flaments, which are seen as dark structures\nagainst the bright chromosphere. The red arrows point to active\nregion (AR) \flaments; green arrows point to intermediate \flaments\n(IT) between ARs; blue arrows point to quiescent (QS) \flaments.\nthe \flament. In one case, 91, we clearly observed a More-\nton Wave (MW) emanating from a \rare and hitting the\n\flament, triggering its oscillation as described in x9.3.\nIn Figure 2 the sequence of events is shown for our\nreference case 1. The trigger was identi\fed as a \rar-\ning region located south of the \flament. In order to\nparametrize the \rare position, we averaged the measured\npositions of several bright regions in the \rare; this aver-\naged position is marked by a red dot in Figure 2(a). This\npanel also shows the equilibrium position of the \flament\nbefore the trigger perturbed the \flament. Panels (b) and\n(c) show the di\u000berence images at the given times with the\nimage shown in (a) subtracted, thus visualizing the dis-\nplacements with respect to the equilibrium con\fguration.\nThe initial motion was in the northwest direction (Fig.\n2(b)), then the motion was reversed to travel toward the\nsoutheast (Fig. 2(c)).\nBecause the \flaments were very dynamic and their\nshapes changed considerably during the observation in-\ntervals, we \frst generated an average image of the region\nof interest, as shown in Figure 3 for case 1. This image\nwas constructed by averaging 10 equally spaced images\nfrom the data sequence of the day and telescope selected\n(i.e., Cerro Tololo or C in this case). From this averaged\nimage we determined the position of the spine following\nthe dark \flament (thick white line in Fig. 3), the length\nof the spine, L, and the average width of the \flament,\nW. We de\fned the width of the \flament at 5 equidistant\npositions along the spine as the length of the 5 segments\nplotted in the \fgure as thin lines. The average width\nand length characterize the size of the \flament, to be\nused later in our statistical study. For this example the\nlength isL= 269 Mm and width W= 15 Mm (see event\n1 in Table 1). Using the positions of the \flament spineGONG Catalog of Solar Filament Oscillations 5\nFigure 2. Temporal sequence of the triggering and oscillations in event 1. (a) H \u000bimage showing the dark \flament in its equilibrium\nposition (12:55 UT, before oscillation onset), outlined by a white contour. The brightening associated with the triggering \rare and the\naveraged \rare position (red dot) are also visible. (b) Base di\u000berence H \u000bimage (13:53 UT - 12:55 UT) showing the northward displacement\nof the \flament. The dark and white regions correspond to negative and positive di\u000berences, respectively. The contour of the equilibrium\n\flament is overplotted in white. (c) Base di\u000berence H \u000bimage (14:33 UT - 12:55 UT) showing the southern displacement of the central\npart of the \flament. In (b) and (c) the orange contour marks the slit used to track the \flament motion.\nwe also obtained the averaged position of the \flament on\nthe solar disk, marked with a cross in the \fgure.\nFigure 3. H\u000bimage of event 1 averaged over 10 equally spaced\ntimes from the full observing period, showing the spine position\n(white line), the spine length ( L), and the average \flament width\n(W). The 5 thin line segments were used to calculate the average\nwidth of the dark band. The orange contour corresponds to the\nslit used to follow the motion and to construct the time-distance\ndiagrams.\u000bmeasures the angle between the direction of motion,\ni.e. the slit, and the \flament spine. In this example \u000b= 16\u000e,\nL= 260 Mm, and W= 15 Mm. The red dot marks the averaged\n\rare position; the white cross is the averaged \flament position on\nthe solar disk.\n7.TIME-DISTANCE DIAGRAMS AND DIRECTION OF THE\nMOTION\nWe used the time-distance approach to analyze the \fla-\nment oscillations. Because many oscillations reported in\nthis work did not follow straight trajectories, but rather\nmoved along curved paths, we could not apply the tech-\nnique described by Luna et al. (2014) based on straight\nslits. In addition, due to the relatively low resolution\nof the GONG data, we needed to generate time-distance\ndiagrams with minimal reduction of the e\u000bective resolu-\ntion in curved slits. To de\fne the curved slit in the H \u000bimages for each event, we tracked the path of the oscilla-\ntions by visual inspection. In order to generate the slit,\nwe \frst traced the motion of the \flament segment with\nthe clearest and largest displacement. The slit, of length\nland width wpixels, was placed lengthwise along the\ncurved path of the motion as described in Appendix B.\nWith the technique shown in the Appendix, we averaged\nthe intensity over the transverse pixels, w, resulting in\nan intensity distribution along l. The time-distance dia-\ngrams display this intensity along the slit as a function of\ntime. Figure 2 shows that the slit matches the trajectory\nof the cool plasma for case 1. The vertical coordinate in\nthe time-distance diagrams (e.g. Fig. 2(a)) corresponds\nto the distance along the slit in Mm, with the origin set\nto coincide approximately with the equilibrium position\nof the \flament. This distance is measured in the plane of\nthe sky inx\u0000ycoordinates (i.e., Heliocentric-Cartesian\ncoordinates); thus the displacements are projections of\nthe actual motions onto the plane of the sky. Similarly,\nthe velocities measured in the time-distance diagrams are\nalso projections, so the actual values are probably larger.\nThe angle \u000bbetween the direction of the oscillatory\nmotion and the \flament spine was measured at the in-\ntersection of the spine curve and the slit (orange curve\nin Fig. 3). This angle (dotted arc) characterizes the po-\nlarization of the oscillations in terms of longitudinal or\ntransverse movements. In this example \u000b= 16\u000e(see Ta-\nble 5), so the oscillation is longitudinal. In case 1, we\nfound that the triggering location was aligned with the\nslit used to track the motion (Fig. 3), suggesting that\nthe perturbation from the \rare followed the same direc-\ntion as the slit to reach the \flament. The angle \u000bis a\nprojection onto the sky plane of the actual angle. The\ndi\u000berence between these angles depends on the \flament's\nposition and orientation on the solar disk.\nUsing the technique explained in Appendix B, we con-\nstructed time-distance diagrams for each event. In the\nresulting time-distance diagram for our reference case 1\n(Fig. 4(a)), the \flament appears as a dark band sur-\nrounded by bright emission from the adjacent chromo-\nspheric plasma, and the \flament oscillations are clearly\nvisible.\nThe slit was traced visually following the motion of the6 Luna et al.\nFigure 4. Oscillation diagnostics of event 1. (a) Time-distance\nH\u000bdiagram. The dark band is the \flament seen in absorption, sur-\nrounded by bright emission from the adjacent chromosphere. The\ntwo dashed lines mark the 1 \u001blevel as discussed in x8.1. (b) Trian-\ngles show the central position of the \flament, s0(t), as a function\nof timetas determined from the Gaussian \ft along the slit using\nEq.(1). The 1 \u001bregion is delimited by two thick dashed lines. The\nthick solid line is the best \ft to the triangles using Eq. (2). The\nblue dashed line is the same \ftted function extrapolated to times\noutside the temporal range used to construct the \ft. (c) The veloc-\nity as a function of time, computed as the time derivative of s0(t).\nThe velocity obtained with the \ftted function (2) is overplotted as\na solid curve.\ncool plasma in the H \u000bimages, introducing a subjective\nfactor in the determination of the slit path. In addi-\ntion, the relatively low spatial resolution of GONG data\ncould produce a misalignment of the slit with the actual\ntrajectory of the cool plasma, reducing the measured dis-\nplacements over the slit and yielding another source of\nerror in the measurements of the displacements. We have\nnot quanti\fed explicitly the error introduced by the mis-\nalignment, however, because we have overestimated the\nerrors in the displacements as discussed in x8.1. Addi-\ntionally, the misalignment introduces an error in \u000b, but it\nis unclear how to assess the uncertainties in this param-\neter. We are currently developing automated techniques\nto track the motion of the \flament, which will enable\nus to quantify and reduce the errors in the displacement\nand\u000b.\n8.OSCILLATION ANALYSIS\nFigure 5. H\u000bintensity along the slit for case 1 at 16:06 UT\n(thin line), corresponding to the blue vertical line in Fig. 4(a).\nAn asterisk marks the intensity minimum. The thick line is the\nGaussian \ft to the observed intensity pro\fle using Eq. (1).\nTo determine the central position of each \flament as\na function of time, we plotted the intensity along the\nslit and \ftted a Gaussian function to the intensity for\neach image of the observing sequence. Because the ab-\nsorption of the \flament depends on the column depth of\ncool plasma along the line of sight (LOS), we assumed\nthat the intensity minimum corresponds to the central\nposition in the direction along the slit. This Gaussian\nfunction also enabled us to determine the uncertainties\nof the oscillatory parameters ( x8.1). We used the gauss-\n\ft.pro IDL routine with a functional form\nI(s) =g0e\u00001\n2\u0010s\u0000s0\ng1\u00112\n+g2+g3s+g4s2; (1)\nwheresis the coordinate along the slit, g0<0 is the in-\ntensity amplitude, s0is the central position of the Gaus-\nsian,g1=\u001bGis the standard deviation, and the remain-\ning terms are the background chromospheric emission.\nFigure 5 shows the intensity along the slit at \u001816:06\nUT (blue vertical line in Figure 4(a)) in the reference\ncase, with the position of the local minimum indicated\nby an asterisk. To avoid errors due to noisy data, we\nconsistently used the central position of the Gaussian\nfunction,s0(t), to track the \flament motion (see Figure\n4(b) for case 1). The measured velocity for all events\nwas derived from the observations by computing the nu-\nmerical derivative of s0(t). The function used to \ft the\noscillation is an exponentially decaying sinusoid, plus a\nthird-order polynomial function to de-trend the proper\nmotions of the \flament:\ny(t) =A0e\u0000A1(t\u0000t0)cos [A2(t\u0000t0) +A3] +\nA4+A5(t\u0000t0) +A6(t\u0000t0)2+A7(t\u0000t0)3;(2)\nwhereAiare the coe\u000ecients of the \ft. Sometimes the\nbeginning of the oscillation is not well described by Equa-\ntion (2), so we performed the \ft in a selected time in-\nterval when the oscillation is clear in the time-distance\ndiagram for each event. In Equation (2), t0is the ini-\ntial time of the \ftted function (column 2, Tables 5 to 8).\nThe \frst few coe\u000ecients of the \ft are associated with the\noscillation in the following way: A0is the \ftted displace-\nment amplitude; A1= 1=\u001c, where\u001cis the damping time;\nA2= 2\u0019=P, wherePis the period; and A3is the initialGONG Catalog of Solar Filament Oscillations 7\nphase. The remaining terms are the coe\u000ecients of the\npolynomial function that \fts the background motion of\nthe \flament. This trend function \flters out motions as-\nsociated with long-period oscillations. Very long periods\nhave been observed in a few oscillating \flaments (Foullon\net al. 2004, 2009; Efremov et al. 2016), but these motions\nare not clear in our data. Hence we focus our attention\non more rapid oscillations.\nFigure 4(c) shows the measured velocity (diamonds)\nand the best \ft obtained from Equation 2(solid curve)\nfor case 1. The \flament remained essentially station-\nary before 13:00 UT, then the velocity increased slowly\nbetween 13:00 and 13:30 UT In the following \u001930 min-\nutes the velocity suddenly jumped up to 60 km s\u00001, re-\nturned to zero at the time of maximum displacement (see\nFig. 4(b)), and increased again in the opposite direction\nto\u000055 km s\u00001. In this phase the acceleration reached\n140 m s\u00002. The velocity does not resemble a sinusoidal\noscillation until after \u001814:00 UT. The measured velocity\nand the \ftted function agree very well.\n8.1. Errors\nSeveral sources of uncertainty in the measured oscil-\nlation parameters are attributable to the relatively low\nspatial resolution of the GONG images, including jitter,\nthe uncertainties in the exposure time, and the above-\nmentioned misalignment of the slit with the cool plasma\ntrajectories. We assumed that the uncertainty in the\noscillation parameters comes mainly from the errors in\ndetermining the position of the \flament along the slit.\nIn Figure 6 we have plotted a time-distance diagram in\nan interval where no oscillation was evident. The aster-\nisks mark the central positions of the Gaussian \ft, s(t).\nWe clearly see a noisy signal in the \fgure, where the\n1\u001bstandard deviation (thin straight lines) around the\nmean value is \u001bnoise = 1:4 Mm. The \fgure also shows\nthe width of the Gaussian \ft, \u001bG, to the intensity (Eq.\n1) as dashed lines, which coincide with the borders of the\ndark band. We consider \u001bGto overestimate the uncer-\ntainty of the central position because this uncertainty is\ncomparable to the \flament width. In fact, \u001bG&3\u001bnoise\nin all analyzed events. Thus, we decided somewhat arbi-\ntrarily to set the uncertainty of the \flament position as\n\u001b= 0:5\u001bG&\u001bnoise(indicated by two solid curves in Fig-\nure 6). This overestimates the error because \u001b > \u001b noise\nand the uncertainty in the position is larger than the\nvariations seen in the \fgure. This error \u001bwill be con-\nsidered the only source of uncertainty in the estimated\noscillation parameters.\nIn Table 5 we have tabulated the best-\ft parameters\nfrom Equation 2 for all oscillation events. Event 1 has a\nperiod ofP= 76\u00061 min, damping time \u001c= 121\u000615\nmin, maximum displacement A= 23\u00062 Mm, and peak\nvelocity amplitude V= 26\u00064 km s\u00001. The ratio\n\u001c=P = 1:6\u00060:2, indicating that the damping was strong\nand very e\u000ecient. Because the velocity amplitude was\nlarger than 10 km s\u00001, event 1 is a LAO. Large velocities\nexceeding 40 km s\u00001occurred early in this event, but this\nphase did not produce a \ft compatible with subsequent\nmotions (Fig. 4(c)). The best \ft derived from Equa-\ntion (2) has a much lower peak value of 26 km s\u00001and\nagrees well with most of the subsequent velocity oscil-\nlations. At the end of the \ftted range, around 18:30\nFigure 6. Time-distance diagram of an interval with no clear\noscillations. The asterisks mark the central positions of the Gaus-\nsian \ft of Eq. (1), g1(t). The standard deviation of these points is\n\u001bnoise; the two thin horizontal lines correspond to \u0006\u001bnoise with re-\nspect to the mean value of these positions. The two dashed curves\nare atg1\u0006\u001bG, containing the dark region in the diagram. We use\n\u001b= 0:5\u001bGas the uncertainty in the \flament position. Here the\nuncertainty region g1\u0006\u001bis delimited by two solid lines.\nUT, the \frst oscillation ceased and a new one appeared.\nThis new oscillation, case 2*, had a di\u000berent phase and\namplitude than case 1, which can be seen by comparing\nthe observed triangles with the blue dashed line from the\n\ft of case 1 in Figure 4(b). Cases 1 and 2* have com-\nparable periods, indicating that they are characteristic\noscillations of the \flament. The direction of the motion\nis o\u000bset by 16\u000efrom the \flament spine (see Fig. 3(b) and\nTable 5), suggesting that the oscillation is longitudinal as\nwell as large-amplitude (i.e., a LALO). This angle is com-\nparable to the typical angle between the magnetic \feld\nand the \flament spine, according to direct measurements\n(Leroy et al. 1983, 1984; E. Tandberg-Hanssen 1995; Tru-\njillo Bueno et al. 2002; Casini et al. 2003; L\u0013 opez Ariste\net al. 2006), suggesting that the oscillation is aligned with\nthe \flament magnetic \feld.\n9.SELECTED CASE STUDIES\nIn the previous Sections we used case 1 as a represen-\ntative oscillation example from our event catalog. Here\nwe describe selected additional events from the catalog\nto illustrate the intriguing variety of behaviors and oscil-\nlation characteristics encountered in our survey.\n9.1. Event 58: LALO triggered by a two-ribbon \rare\nEvent 58 of the survey (Table 5) is a LAO with a peak\nvelocity of 14 km s\u00001, triggered by a two-ribbon \rare that\nstraddled the AR \flament (see Figure 7(a)). This \rare\nproduced an oscillation in which the cool plasma was\ndisplaced initially in the west-east direction from its pre-\n\rare equilibrium position (white contour in Figure 7(a)).\nThen the motion was reversed, so the \flament reached\nmaximum western elongation (Fig. 7(b)), followed by\nanother reversal that produced a peak eastward displace-\nment of smaller amplitude (Fig. 7(c)). The Figure also\ndemonstrates that the slit used to track the motion and\nconstruct the time-distance diagram closely follows the\ntrajectory of the cool \flament plasma. Very clear oscil-\nlations in the H \u000bintensity are evident in Figure 8(a),\ntriggered around 11:50 UT and lasting for more than 78 Luna et al.\nperiods from\u001812:00 to 18:00 UT. Before \rare onset the\ndark \flament band is almost horizontal, so the \flament\nwas almost at rest. In Figure 8(b), the central position\ns(t) and its best \ft from Eq. 2 agree well until \u001816:00\nUT. Thereafter the \ft is more damped than s(t) and a\nslight phase di\u000berence is also evident. After 19:00 UT\nthe measured s(t) is very noisy because the quality of\nthe H\u000bimages was reduced.\nThe \ftted velocity matches the measured velocity well\n(Fig. 8(c)). There were no large velocities during the\ninitial phase associated with the triggering, in contrast\nwith case 1 (Fig. 4(c)). Although the initial velocity\nexceeded 10 km s\u00001, indicating that this event is a LAO,\nin less than a period the velocity fell below this thresh-\nold. The direction of motion was o\u000bset by 32\u000efrom the\n\flament spine, identifying this event as a possible LALO.\n9.2. Event 63: LALO in a large quiescent \flament\nIn case 63 the oscillation occurred in a very large, frag-\nmented quiescent \flament (QS) in the southern hemi-\nsphere. The oscillation appeared in the southern seg-\nment, possibly triggered by the \rare centered at the red\ndot in Figure 9(a). The H \u000b\rare brightened slightly\nbefore the oscillation began at \u001818:26 UT. Part of the\nprominence plasma moved toward the northwest or west\nalong a curved trajectory. Di\u000berence images in Figures\n9(b) and (c) reveal the maximum elongation at two times\nafter the initiation: a dark region along the slits ap-\npeared, \frst on one side and then on the other side of\nthe pre-oscillation position of the \flament (Fig. 9(c)).\nSeveral threads of cool plasma moved in a complex way\nbefore oscillation onset at 20:26 UT (Fig. 10(a)). There-\nafter the oscillation was very clear but only was observed\nfor 2 cycles. Apparently the whole \flament was displaced\nto the southeast before the oscillation, possibly indicat-\ning that the structure of the \flament changed during the\npre-\rare phase. In Figure 10(a) the positional uncer-\ntainty,\u001b, is indicated by two dashed lines. This uncer-\ntainty is very small where the dark band is very narrow.\nInitially some threads moved towards the centroid of the\noscillation, but our tracking code ignored these outliers\nand only followed the motion of the main body, the dark-\nest band at s\u001810 Mm. After 20:39 UT the entire \fla-\nment oscillated, and Equation (2) provided a good \ft to\ns(t) (Fig. 10(b)). The measured velocity amplitude, V,\nreached very large values >50 km s\u00001(Fig. 10(c)) .\nThe results of the \ft are shown in Table 5. The oscil-\nlation had a velocity amplitude of V= 48:5\u00062:4 km s\u00001,\nP= 103\u00061 min, a damping time \u001c= 175\u000612 min, and\n\u001c=P = 1:7\u00060:1 indicating very strong damping. The\nmotion was only \u000b= 2\u000emisaligned with the \flament\nspine, indicating that this event was probably a LALO.\n9.3. Event 91: LALO triggered by a Moreton wave\nEvent 91 occurred in an intermediate \flament (IT) lo-\ncated close to AR NOAA 12017. Around 17:41 UT a\nmajor \rare occurred in the AR followed by a Moreton\nwave, which is visible in Figure 11. This is the only\nevent in the survey for which we could identify a More-\nton wave connected to the \flament oscillation. The wave\nemanated from the \rare region (Fig. 11(a)), hit the \fl-\nament (Fig. 11(b)) then continued to propagate north-\nward (Fig. 11(c)). Once the wave encountered the \fla-ment (Fig. 12(b)), the \flament started to oscillate per-\npendicular to the propagation direction of the wave front.\nAt 18:53 UT (Fig. 12(c)), the motion was reversed and\nthe oscillatory motion was fully established. The time-\ndistance diagram in Figure 13(a) reveals signi\fcant mo-\ntions of the \flament before and after \rare onset. Around\noscillation onset, at 17:47 UT, a white vertical region ap-\npears in Figure 13(a), signalling the arrival of the More-\nton wave at the \flament throughout the slit. The More-\nton wave initially produced complex \flament dynamics,\nbest seen in a movie of the event (not shown here). The\noscillation became distinct around 18:00 UT, with a max-\nimum southeastward displacement of 12 Mm from the\nequilibrium position around 18:22 UT (Fig 13(b)) and a\nsmaller opposing displacement at 18:53 UT (Fig 13(b)).\nInterestingly, when the \flament moved in the northwest\ndirection, it was darker than when the motion was in\nthe opposite direction (Fig. 13(a)). The central posi-\ntion of the dark band oscillated clearly (Fig. 13(b)), and\nwas \ftted very well with Equation (2) after 18:23 UT.\nHowever, between the triggering at 17:47 UT and 18:23\nUT the motion does not \ft the sinusoidal function (blue\ndashed line), probably due to the very complex motions\nprior to and during the passage of the wave through the\n\flament.\nBefore the oscillations were triggered at 17:47 UT the\n\flament reached velocities above 5 km s\u00001, but this mo-\ntion was not periodic (Fig. 13(c)). The oscillation veloc-\nity peaked at 18:00 UT; after \u001818:23 UT the measured\nvelocity was very well \ftted by the derivative of Equation\n(2), yielding a peak amplitude V= 19:3\u00062:3 km s\u00001, a\npeak displacement A= 11\u00061 Mm, period P= 58\u00061\nmin, damping time \u001c= 108\u000612 min, and \u001c=P =\n1:9\u00060:2. The angle between the motion and the \flament\nspine was\u000b= 26\u000e, again consistent with the typical di-\nrect measurements of the orientation of the prominence\nmagnetic \feld relative to the spine (Leroy et al. 1983,\n1984; E. Tandberg-Hanssen 1995; Trujillo Bueno et al.\n2002; Casini et al. 2003; L\u0013 opez Ariste et al. 2006). There-\nfore the motion may be aligned with the local magnetic\n\feld and the event is probably a LALO.\n9.4. Event 31: SALO with a very small velocity\namplitude\nIn the survey we also detected oscillations with very\nsmall amplitudes of only a few km s\u00001, below the LAO\nlower limit of 10 km s\u00001. Event 31 has the smallest ve-\nlocity amplitude of the survey: V= 1:6\u00069 km s\u00001. The\n\flament is an IT and the oscillation was triggered by\na nearby \rare. In this example the displacements were\ntoo small to be visible in a \fgure similar to Figure 2.\nTherefore we refer the reader instead to the movie of\nthis event at the URL: http://www.iac.es/galeria/\nmluna/pages/gong-catalogue-of-laos.php .\nAlthough the displacements are very small in this\nevent, the oscillatory pattern is very clear (Fig. 14(a)).\nBefore 17:00 UT the \flament appears to be moving, but\nclear oscillations started at 17:17 UT and ended at 21:06\nUT. The displacement appears more or less constant\nthroughout the event (Fig. 14(b)). However, the extrap-\nolated oscillation some time before and after the \ftted\noscillation (blue dashed line in Figure 14(b)) does not fol-\nlow the \flament motion, and the oscillation apparently\nended without strong damping. In general, the veloc-GONG Catalog of Solar Filament Oscillations 9\nFigure 7. Temporal sequence of the triggering and oscillations in event 58. Panels and annotations are as in Fig. 2. (a) Pre-oscillation\nH\u000bimage at 11:34 UT. The white contour outlines the equilibrium \flament position at \rare onset (11:24 UT). The triggering two-ribbon\n\rare is not visible in this initial frame. The red dot marks the average position of the \raring region (evident in (b)). (b) Base di\u000berence\nH\u000bimage (12:10 UT - 11:34 UT). The two \rare ribbons appear as bright patches north and south of the \flament. (c) Base di\u000berence H \u000b\nimage (12:33 UT - 11:34 UT).\nFigure 8. Oscillation diagnostics of event 58. Panels and anno-\ntations are as in Fig. 4\nity has a large uncertainty for small-amplitude events.\nIn this case the velocity error is 9 km s\u00001, much greater\nthan the velocity V= 1:6 km s\u00001. Figure 14 exhibits\ndistinct oscillatory motions, so we are probably overesti-\nmating the positional errors (see x8.1). The displacement\nwas very small ( A= 1\u00061 Mm), the oscillation period\nP= 77\u00064 min, and the damping time \u001c= 662\u00061546min, so the damping was very weak with a large uncer-\ntainty (see Table 5). The angle between the slit and the\nspine\u000b= 26\u000e, suggesting that the oscillation was longi-\ntudinal.\n9.5. Events 145 and 146*: double event\nIn these events, two clear oscillations occurred in the\nsame \flament within one observing interval; therefore\nwe labeled them as separate events. In our example the\n\flament is of the IT type with a curved initial structure\n(Fig. 15(a)). Both oscillations involve motions mainly\ntransverse to the \flament spine, with \u000b= 50\u000e. A nearby\n\rare, marked by the red dot in the \fgure, is the most\nlikely trigger. At onset the entire \flament was displaced\nlaterally toward the northeast (Fig. 15(b)), then after a\nhalf period the \flament moved to the other side of its\nequilibrium position (Fig. 15(c)). The slit (red outline)\nis over the region of the \flament that oscillated with the\nlargest displacement, which protruded from the rest of\nthe structure.\nThere were many data gaps, which appear as white\nvertical bands at the beginning of the time-distance dia-\ngram (Fig. 16(a)). The \frst event, 145, oscillated most\nvisibly between 13:31 and 15:33 UT. The oscillation pe-\nriod wasP= 39\u00062 min,V= 14:4\u00066:1 km s\u00001and\n\u001c= 48\u000613 min (see Table 7), indicating very strong\ndamping with \u001c=P = 1:2\u00060:4. Event 146*, which started\nat 17:38 UT and ended at 19:20 UT, had P= 36\u00062 min,\nV= 3:4\u00069:4 and\u001c= 119\u0006146 min with \u001c=P = 3:3\u00064.\nFigure 16(a) leaves the impression that both events were\npart of a long oscillation starting at 13:31 UT and con-\ntinuing throughout the temporal sequence. However, the\nvelocity evolution demonstrates that case 145 ended at\n15:33 UT (Fig. 16(c)), followed by an interval of complex\noscillation. Event 146* started at 17:38 UT with a very\nclear oscillation that was out of phase with previous mo-\ntions. We conclude that repetitive triggering occurred in\nthis \flament.\nIt is interesting that both events had similar periods,\nsuggesting that the \flament oscillated with a character-\nistic frequency of the system (e.g., Hyder (1966)). How-\never, the damping times were very di\u000berent, indicating\nthat either the damping mechanisms were di\u000berent or\nthe damping e\u000eciency changed between events. The \frst10 Luna et al.\nFigure 9. Temporal sequence of the triggering and oscillations in event 63. Panels and annotations are as in Figure 2. Here panels (b)\nand (c) show a smaller region centered at the \flament. (a) Pre-oscillation H \u000bimage at 20:26 UT. (b) Base di\u000berence H \u000bimage (21:04 UT\n- 20:26 UT), showing the initial northwestward displacement of the \flament along the slit. (c) Base di\u000berence H \u000bimage (22:04 UT - 20:26\nUT) showing the subsequent southeastward displacement.\nFigure 10. Oscillation diagnostics of event 63. Panels and anno-\ntations are as in Fig. 4\ncase had a peak velocity \u001814 km s\u00001, much larger than\nthe second event with V\u00183 km s\u00001. Inx10.3 we show\nthat the damping time decreases with Vfor the entire\nset of events. This nonlinear e\u000bect might explain the\ndi\u000berent damping times for cases 145 and 146*. Both\noscillations are very clear in Figure 16, and both \fts ac-cording to Equation (2) are good. Although the second\noscillation continued after 19:20 UT, the extrapolated\n\ftted function (blue dashed line in Fig. 16(b)) does not\nfollow the center of the dark band (triangles) well dur-\ning this interval, probably because the dark band is very\nlight and the signal-to-noise ratio is very low in the time-\ndistance diagram. Event 145 is a transverse LAO with\nV= 14:4 km s\u00001, and 146* is a transverse SAO with\nV= 3:4 km s\u00001(see Figure 16(c)).\n9.6. Event 151 and 152*: double event and ampli\fed\noscillation\nUnlike the double event described in the previous sec-\ntion, the cases presented here exhibit very di\u000berent oscil-\nlation periods: P= 52\u00062 min for case 151 and P= 66\u00063\nmin for case 152*. Figure 17 shows the \frst stages of\nevent 151; the initial motion is southward, followed by a\nreversal toward the north. The direction of motion for\nevent 152* is similar to 151. Both events were apparently\ntriggered by nearby \raring (red dot in Figure 17(a)).\nEvent 151 occurred between 00:00 UT and 3:33 UT\n(Fig. 18(a)). The best-\ft solution to the central position\ntracks the oscillation well, except for a small discrepancy\nin the last period (Fig. 18(b)). The measured and \ftted\nvelocities plotted in panel (c) also agree well. The oscil-\nlation parameters are P= 52\u00062 min,\u001c= 89\u000623 min,\nand\u001c=P = 1:7\u00060:5 (strong damping). This event is a\nSAO withV= 6:8\u00065:1 km s\u00001. In both events \u000b= 36\u000e,\nsuggesting longitudinal oscillations.\nFigure 18 also shows the unusual oscillation of event\n152*, which started at 5:37 UT, increased in amplitude,\nand ended at 8:53 UT. The best \ft agrees well with\ns(t) between 5:37 and 8:15 UT but not after this in-\nterval, suggesting that the plasma motion of this ampli-\n\fed oscillation is more complex than Equation (2) (Fig.\n18(b)). The oscillation parameters are P= 66\u00063 min,\n\u001c=\u0000163\u0006105 min, and \u001c=P =\u00002:5\u00062, indicating\nvery strong ampli\fcation. The large error in the damp-\ning time is probably overestimated, because the \ftted\nfunction agrees very well with the oscillation. The \ftted\nmaximum velocity amplitude is V= 4:0\u00067:8 km s\u00001, but\nthe measured velocity reached a maximum of 14 km s\u00001\nat 08:40 UT. Later H \u000bdata reveals that the oscillation\nceased and the \flament became stationary again. Simi-\nlar behavior was found by Molowny-Horas et al. (1999)GONG Catalog of Solar Filament Oscillations 11\nfor an ampli\fed \flament oscillation.\nIn order to amplify the oscillation the cool plasma must\ngain energy. Recently Zhou et al. (2017) and Zhang et al.\n(2017b) found LALOs with an ampli\fed oscillation fol-\nlowed by a damped phase, which they explain as a beat-\ning phenomenon between two interacting oscillators (see,\nFigure 11. Running di\u000berence H \u000bimages showing the propagat-\ning Moreton wave at selected times. In all panels the \flament equi-\nlibrium position is outlined by a white contour, the slit is marked\nby a red arc, and the averaged \rare position is marked by a red\ndot. (a) 17:46 UT, shortly after the wave was generated at the\n\raring region. (b) 17:48 UT, when the wave (white patch) reached\nthe \flament. (c) 17:51 UT, as the wave (white arc) continued to\nexpand and travel northward.e.g, Luna et al. 2006; Luna et al. 2008). In this scenario\nthe oscillations of two regions of the \flament are cou-\npled: an active oscillator that transfers energy to the\nother part of the \flament (the passive oscillator). The\npassive oscillator gains energy with time so its oscillation\nis ampli\fed, while the active portion loses energy, reduc-\ning its amplitude. The section of the \flament oscillating\nin event 152* should be the passive oscillator because it is\ngaining energy. However, the active oscillator should be\nthe other region of the \flament, but it does not oscillate\nwith a larger amplitude. Therefore this hypothesis does\nnot explain events 151 and 152*. Ballester et al. (2016)\nfound that cooling the prominence plasma could amplify\nits oscillations, but we can't test this hypothesis for lack\nof relevant temperature diagnostics. Alternatively, repet-\nitive nearby \rares could produce both oscillation events\nand possibly amplify the second. The H \u000bdata shows\n\raring activity close to the \flament (Fig. 17(a)). In\nthis situation the ampli\fed oscillation would be driven\nby external forcing, which could explain why the period\nis di\u000berent from that of the non-ampli\fed previous event.\n9.7. Events 107 and 108*: double event with ampli\fed\noscillation and eruption\nThis double event is similar to that discussed in x9.5\n| a damped oscillation (case 107) immediately followed\nby an ampli\fed oscillation (case 108*) | but with a \fnal\neruption. Both events occurred in the same IT \flament,\nand both events were triggered by \raring in an active\nregion north of the \flament (Fig. 19(a)). For the \frst\neventP= 50\u00061 min,V= 6:6\u00062:2 km s\u00001, and\u001c=P =\n3:1\u00060:7; for the second event P= 40\u00063 min,V=\n5:6\u00069:3 km s\u00001, and\u001c=P =\u00002:4\u00062:0. Both events are\nSAOs with \u000b= 20\u000e, denoting longitudinal polarization.\nAs in events 151 and 152*, the periods are di\u000berent.\nSimultaneous with the oscillation onset for event 107, a\nwhite spot appeared north of the \flament (marked with a\nwhite arrow in Figure 19(a)) and continued almost to the\nend of event 107. Because the slit in Figure 19(b,c) passes\nover the white spot, we can also see this brightening in\nthe resulting time-distance diagram (Fig. 20(a)). Base\ndi\u000berence images show the maximum elongation of the\ncool plasma in event 107 (Fig. 19(b)) and for event 108*\n(Fig. 19(c)). At the end of event 108* the prominence\nerupts (\u001823:20 UT, not shown). The white spot that\nappeared north of the \flament appears as a bright region\nat the top of the dark band in Figure 20(a). This bright\nemission apparently followed the motion of the threads\nfrom 18:20 to 21:20 UT (end of event 107).\nAs with events 145 and 146*, the oscillation seems to\nbe continuous between 18:20 UT and 23:26 UT. However,\nthe event 107 and 108* oscillations di\u000ber signi\fcantly in\nphase, period, and damping time. Figure 20(b) shows\nthis discrepancy clearly: one \ft (Eq. (2)) is very good in\nthe \frst event and another is good in the second, except\nat the end of the event when the motion was obviously\na\u000bected by the eruption. The di\u000berences between two\nevents are equally evident in the measured velocities (Fig.\n20(c)). Before 18:20 UT the motions were small and\ndisorganized, while after this time the damped oscillation\nis very clear. At 23:26 UT the period changed and the\noscillation started to grow, ending in an eruption.\nThe main di\u000berence between events 107-108* and the\nampli\fed oscillation of x9.6 is that the \flament erupts.12 Luna et al.\nFigure 12. Temporal sequence of the triggering and oscillations in event 91. Panels and annotations are as in Figure 2. (a) H \u000bimage\nof the \raring region and the \flament at oscillation onset (17:47 UT). The red dot indicates the approximate position of the \rare that\nproduced the Moreton wave that triggered the oscillation. (b) and (c) Running di\u000berence H \u000bimages of a smaller region centered on the\n\flament. The \flament was displaced initially to the southeast (b), then toward the northwest (c).\nFigure 13. Oscillation diagnostics of event 91. Panels and anno-\ntations are as in Fig. 4\nHowever, similar explanations might apply for the am-\npli\fcation. The potential relationship between the am-\npli\fed oscillation and the eruption is an intriguing topic\nfor further study.\n10. STATISTICS\nFigure 14. Oscillation diagnostics of event 31. Panels and anno-\ntations are as in Fig. 4.\nThe catalog consists of 196 oscillation events which we\nfound by analyzing six months of GONG data in cycle 24\n(see Tables 1 to 8). In about 43% of the cases we identi-\n\fed the apparent trigger of the oscillation: 72 events were\ntriggered by \rares, 11 by prominence eruptions, 1 by a\njet, and 1 by a Moreton wave. However, in 111 casesGONG Catalog of Solar Filament Oscillations 13\nFigure 15. Temporal sequence of the triggering and oscillations in event 145. Panels and annotations are as in Fig. 2. In (a) and (b) the\nred dot indicates the approximate position of the \rare that probably triggered the oscillation.\nFigure 16. Oscillation diagnostics of events 145 and 146*. Panels\nare as in Fig. 4 .\nthe triggering agent was not identi\fed. In 9 cases the\n\flament erupted during the temporal range analyzed.\nAs discussed in x1 we classi\fed the oscillations ac-\ncording to their maximum velocity amplitude as SAOs\n(V < 10 km s\u00001) and LAOs ( V > 10 km s\u00001). Of the\n196 oscillation events there are 106 SAOs and 90 LAOs.\nOver the six months of the survey this averages to one\noscillation event per day on the visible solar disk. The\noccurrence rate of one LAO event every two days implies\nthat LAOs are a common phenomena on the Sun, in con-\ntrast to previous statements that LAOs are scarce (e.g.,Tripathi et al. 2009). We also found a similar rate for\nSAOs.\nThe data presented in the catalog enabled us to search\nfor possible dependencies between pairs of \flament and\noscillation parameters: the velocity amplitude ( V), oscil-\nlation period ( P), damping time ( \u001c), damping time per\nperiod (\u001c=P), displacement ( A), and angle between the\nproper motion and the \flament spine ( \u000b). We also com-\nputed the Pearson correlation matrix (Neter et al. 1993)\nusing the IDL subroutine correlate.pro . The matrix ele-\nments are the correlations between pairs of parameters,\nand range from -1 to 1. A linear correlation between two\nparameters yields an associated matrix element close to\n1 (or -1). Although we found that the values of the ma-\ntrix are small, in general, we will discuss those pairs of\nparameters whose correlations or lack thereof are inter-\nesting. Figures 21 to 23 show scatter plots of some pairs\nof these parameters. In Figure 21 the scatter plots of\nthe period, P, vs the other parameters are displayed in\n6 panels (a-f). Figure 22 shows the damping parame-\nters,\u001cor\u001c=P, vsvand\u000b(panels a-e). Figure 23 plots\nseveral parameters vs solar latitude of the \flament. In\nthese scatter plots, the LAOs and SAOs are plotted with\ncircles and squares, respectively.\n10.1. Velocity Amplitude, V\nIn the survey we found velocity amplitudes from a\nfew km s\u00001to 55 km s\u00001(see Tables 5 to 8). His-\ntograms of the velocity distribution for all events and\nthe distribution according to \flament type are plotted\nin Figure 24(a) and (d), respectively. The vertical dot-\nted line separates LAOs ( V > 10 km s\u00001) from SAOs\n(V < 10 km s\u00001). The total number of LAO events de-\ncreases with the velocity amplitude, as expected: more\nenergetic events are less frequent than less energetic ones.\nThe velocity ranges for all \flament types (AR - red, IT -\ngreen and QS - blue) are similar (Figure 24(d)), indicat-\ning that all types of \flaments can support both SAOs and\nLAOs. The velocity distribution for each \flament type\nfollows the same trend as the total distribution except for\nAR \flaments. The apparent rollover in the AR \flament\ndistribution below 5 km s\u00001probably re\rects the di\u000e-\nculty in detecting small \flaments and small-amplitude\nevents by eye, implying that we have underestimated the\nnumber of SAOs.\nThe histograms also do not distinguish two separate14 Luna et al.\nFigure 17. Temporal sequence of the triggering and oscillations in events 151 and 152*. Panels and annotations are as in Fig. 2. In (c)\nthe white arrow points to the part of the prominence that oscillates.\nFigure 18. Oscillation diagnostics of events 151 and 152*. Panels\nare as in Fig. 4 .\npopulations associated with large- and small-amplitude\noscillations, regardless of the choice of LAO threshold\n(i.e., 10 km s\u00001or 20 km s\u00001). Additionally, 32 of the\n106 SAOs were clearly triggered by an identi\fed ener-\ngetic disturbance. These contradict the idea that the\nLAOs and SAOs have a di\u000berent nature and they are trig-\ngered by di\u000berent mechanisms (Oliver & Ballester 2002;\nArregui et al. 2012).\nTheP\u0000Vscatter plot (Figure 21(a)) and the small\ncorrelation P\u0000Vvalue reveal no dependence of the ve-\nlocity on the period, neither for all events nor for dif-ferent \flament types. In contrast, the V\u0000\u000bscatter\nplot (Fig. 22(a)) shows a clear pattern: the Vrange de-\ncreases with \u000b, and theVvalues drop sharply for events\nwith\u000bbeyond 40\u000e. This tendency leads to no LAOs for\n\u000b>65\u000e. The two populations can be also distinguished\nin theA\u0000\u000bscatter plot of Figure 22(b) as we will dis-\ncuss inx10.4. The evident correlation between velocity\namplitude and damping time will be discussed in x10.3.\n10.2. Period,P\nThe period re\rects the restoring force and the under-\nlying physics of the oscillation. The period values range\nfrom 30 to 110 min for the total population, with a mean\nvalue of 58 min, a standard deviation of 15 min, and a\nclear peak centered at \u001858 min (Figure 24(b)). The pe-\nriod distributions for LAOs (striped) and SAOs (shaded)\nhave mean values and standard deviations comparable to\nthose of the Pdistribution for all events. This indicates\nthat SAOs and LAOs are not two distinct populations of\nevents with respect to their periods.\nThe period distributions for the three \flament types\ndo not di\u000ber signi\fcantly from each other or from the\ntotal distribution (Figure 24(e)). For IT \flaments the\nmean period is 56 min \u000614 min; the distribution for AR\n\flaments peaks at 57 \u000616 min; the mean period for QS\n\flaments is 62\u000617 min with long-period tail extending\nto 110 min. If LAOs were nonlinear, as discussed in x1,\nthe period could depend on VorA. However, Figures\n21(a) and 21(b), together with the negligible P\u0000Vand\nP\u0000Acorrelation elements, demonstrate that Pdoes not\ndepend on either VorAfor the catalogued events.\nMany theoretical models of MHD modes in \flaments\npredict a relationship between the oscillation period and\nthe \flament length or width (see review by Arregui et al.\n2012). To test this hypothesis, we plotted the oscillation\nperiod as a function of length Land widthWin Figures\n21(c) and 21(d), respectively. We found no correlation\nbetweenPandLfor all types. Although the period is not\ncorrelated with Wfor AR and IT \flaments, QS \flament\nperiods tend to increase with W. The correlation element\nis relatively large, 0.74, and the linear P\u0000Wrelationship\nis\nPQS= 23:4\u00060:4 + (2:31\u00060:02)WQS; (3)\nwherePQSis in minutes and the errors in the period have\nbeen considered. The general tendency is for wider QS\n\flaments to oscillate with longer periods than narrowerGONG Catalog of Solar Filament Oscillations 15\nFigure 19. Temporal sequence of the triggering and oscillations in events 107 and 108*. Panels and annotations are as in Fig. 2.\nFigure 20. Oscillation diagnostics of events 107 and 108*. Panels\nand annotations are as in Fig. 4 .\nprominences.\nFigure 21(e) shows that, for angles \u000b<70\u000e, the range\nof possible periods generally decreases with \u000b. For\u000b <\n20\u000ethe periods occupy the range from 30 to 110 min,\nwhereas for 20\u000e<\u000b< 40\u000ethe periods range from 30 to\n95 min and for 40\u000e<\u000b< 70\u000ethe range is from 30 to 80\nmin. Only a few cases have \u000b>70\u000e, and some of them\ndo not follow this trend. Pdecreases gradually with \u000b,\nso there is a no clear drop in Pfor\u000b>40\u000eas we found\nforV(seex10.1).\nThe decrease of Pwith\u000b, in conjunction with the\nsharp decrease in Vat\u000b > 40\u000e, suggests a connec-tion with the polarization of the oscillations. Theoretical\nmodeling predicts that oscillations along the \feld have\nlonger periods than transverse oscillations. Wang et al.\n(2016) and Zhang et al. (2017a) observed simultaneous\nlongitudinal and transverse oscillations in a prominence,\nand con\frmed that the transverse oscillation period was\nshorter than the longitudinal period. At this point we are\ntempted to de\fne longitudinal and transverse oscillations\naccording to the V\u0000\u000bresults: longitudinal for \u000b<40\u000e\nand transverse for \u000b>40\u000e. High-resolution observations\nreveal that on-disk \flaments are composed of many nar-\nrow, \feld-aligned threads oriented at a shallow angle to\nthe spine (e.g. Lin et al. 2005), and often are composed\nof segments spaced along a common PIL. That is, the\nspine is not necessarily a coherent, magnetically contin-\nuous structure. Transverse oscillations involve coherent\nmovement of the whole magnetic structure or magnet-\nically linked portions thereof, whereas longitudinal os-\ncillations involve individual thread motions at an angle\nwith the spine. In almost all catalog events, only an small\nfraction of the \flament oscillates, suggesting that the lo-\ncal, rather than global, magnetic \feld is engaged. Fur-\nther study of individual, well-observed events is needed\nto resolve whether \u000bis a reliable marker of the boundary\nbetween transverse and longitudinal events.\nInx9.5 we reported two consecutive oscillations in the\nsame \flament during the same data sequence. The pe-\nriods of both events, PandP\u0003, agreed, suggesting that\nthe common period is the characteristic period of oscilla-\ntion of the structure (Ramsey & Smith 1966). However,\nin the cases described in x9.6 andx9.7,PandP\u0003are\nclearly di\u000berent. In Figure 25 the scatter plot of P\u0003\nvsPis shown for all the double events in the catalog.\nFor several cases the ellipse is inside or close to the re-\ngion ofP\u0003\u0018P\u00065 min (region between the two dotted-\nlines). For these cases, we can reasonably consider the\noscillation as a characteristic of the system. The shaded\nellipses correspond to the double cases with ampli\fed\noscillations, which in some events were probably associ-\nated with \raring activity near the \flament (see, e.g., x9.6\nandx9.7). Thus, these oscillations were probably forced\nand are not characteristic motions. However, more cases\nexhibit signi\fcant di\u000berences between P\u0003andPnot as-\nsociated with oscillation ampli\fcation. In almost all of\nthese cases we found that the substantial period di\u000ber-\nences were associated with recon\fguration of the \flament\nstructure. For example, in cases 174-175* and 186-187*16 Luna et al.\nFigure 21. Scatter plots of period, P, vs: (a) velocity amplitude, V. (b) displacement amplitude, A. (c) Length of the spine, L. (d)\nWidth of the spine, W. (e) Angle between the direction of motion and the spine, \u000b. (f) Damping time per period, j\u001c=Pj. The square\nsymbols are for SAO events ( V < 10 km s\u00001) and circles are for LAOs ( V\u001510 km s\u00001). For greater clarity the error bars are not plotted,\nbut can be found in Tables 1-8. The colors represent the \flament type: active region (AR, red), intermediate (IT, green) and quiescent\n(QS, dark blue). The big black diamonds indicate events with negative values of \u001c=P.\nthe \flament structures change with time, judging from\nthe observed \rows along the slit and movements of the\nequilibrium position of the \flament.\n10.3. Damping,\u001cand\u001c=P\nj\u001c=Pjmeasures the number of oscillations within the\ncharacteristic damping time. The absolute value of\n\u001c=P is considered because \u001cis negative when an os-\ncillation is ampli\fed with time, as discussed in xx9.6\nand 9.7. A large value of j\u001c=Pjindicates weak damp-\ning, while a small ratio indicates strong damping. The\nj\u001c=Pjhistogram (Fig. 24(c)) for all events extends from\n0.6 to 2711 (not shown in the histogram), and peaks\natj\u001c=Pj= 1:25. Most events are strongly damped\n(j\u001c=Pj<3), and a signi\fcant number are very strongly\ndamped (j\u001c=Pj<1). A value ofj\u001c=Pj\u001510 essentially\nsigni\fes an undamped oscillation. In contrast with the V\nandPdistributions considered above, the j\u001c=Pjdistribu-\ntions for SAOs and LAOs clearly di\u000ber: the SAO distri-\nbution is wide, with a peak close to 1.75, while the LAO\ndistribution is narrower with a peak near 1.25 and scat-\ntered points at larger values of j\u001c=Pj. The LAO events\n(V > 10 km s\u00001) are mainly below j\u001c=Pj= 3 while SAOs\ncover a larger range. The distributions for the 3 \flament\ntypes appear similar (Fig. 24(f)) to the total j\u001c=Pjdis-\ntribution.\nFigure 22(c) shows that larger velocity amplitudes are\npositively correlated with stronger damping, which in-\ndicates that the higher-speed oscillations are likely to\nbe nonlinear. The sharp transition in the j\u001cjrange at\nV= 10 km s\u00001divides LAOs from SAOs in Figure 22(a),\nre\recting a distinct boundary between linear and nonlin-\near oscillations. The scatter plot j\u001c=Pj-Vis not shown\nbut resembles that of Figure 22(a) with the same trend:the damping time j\u001c=Pjdecreases as Vincreases.\nZhang et al. (2013) found a nonlinear relationship be-\ntween\u001candVin their simulations of prominence mass\nformation: \u001c\u0018V\u00000:3. This scaling law (solid black line\nin Figure 22(c)) is roughly consistent with observed and\nderived values from our events, suggesting that LAOs\nmay be damped through radiative cooling. In their\nmodel, each \rux tube supporting a cool thread has two\ncoronal segments that connect the thread with the chro-\nmosphere at both footpoints. The oscillations alternately\ncompress and rarefy both segments, heating or cooling\nthe coronal plasma. The combined density and temper-\nature increases raise the radiative losses, thus damping\nthe oscillations. The Zhang et al. (2013) model predicted\nthat this e\u000bect could yield a temperature variation of\nseveral hundred thousand Kelvins, which should be ob-\nservable in some EUV lines. An alternative mechanism\nthat can explain strong damping is the mass accretion as-\nsociated with thermal nonequilibrium (Luna & Karpen\n2012; Ruderman & Luna 2016), when evaporated chro-\nmospheric plasma continually condenses onto the promi-\nnence threads. In this model the damping is not related\ndirectly to the oscillation velocity. However, events with\nlargerVare associated with violent events, which could\nproduce increased evaporation and consequently stronger\ndamping. A combination of mass accretion and radiative\ndamping is also possible.\nj\u001c=Pjand dimensions LandWare uncorrelated (the\ncorresponding correlation elements are close to zero), im-\nplying that the damping is not related to the promi-\nnence size. The building blocks of prominences are cool,\nelongated threads aligned with the magnetic \feld, so\nthe damping process is probably associated with the lo-\ncal magnetic or plasma characteristics and not with theGONG Catalog of Solar Filament Oscillations 17\nFigure 22. Scatter plots of (a) damping time, \u001c, vsV. (b)j\u001c=Pj\nvs\u000b. (c)Vvs\u000b. (d)Avs\u000b. Symbols and colors are as in Fig.\n21.\nglobal dimensions of the \flament. Similarly j\u001c=Pjis un-\ncorrelated with PorA.\nFigure 23. Scatter plots of latitude vs: (a) P. (b)\u000b. (c)A. (d)\nV. Symbols and colors are as in Fig. 21.\nThe\u001c=P-\u000bscatter plot (Fig. 22(d)) shows a decreased\nrange of\u001c=P for\u000b>40\u000e. This behavior is similar to the\nV-\u000b(x10.1) and the A-\u000bscatter plots, as we will discuss18 Luna et al.\nFigure 24. Histograms of the number of events binned by V(\frst column), P(second column), and j\u001c=Pj(third column). In the top\nrow the shaded and striped areas represent SAO and LAO events, respectively, for three properties: V(a),P(b) andj\u001c=Pj(c). In (a)\nthe vertical dashed line indicates the separation between SAOs and LAOs at V= 10 km s\u00001. In (b) and (c) the curve with a white area\nunderneath is the histogram of the total number of events. In the bottom row, histograms of (d) V, (e)P, and (f)j\u001c=Pj, divided according\nto the three types of \flaments: active region (AR, red), intermediate (IT, green) and quiescent (QS, blue) are shown.\nFigure 25. Scatter plot of the two periods in double events in\nthe same \flament. Pis the \frst oscillation period and P\u0003is the\nsubsequent one. The data are shown as ellipses where the vertical\nsemi-axis is the error bar for P\u0003and the horizontal semi-axis is the\nerror bar for P. Shaded ellipses are for double events including one\nampli\fed oscillation, with the relevant event numbers written on\nthe side of each ellipse.\ninx10.4.\nEvents 6;7;38;65;108;134;152;156;and 171 were char-\nacterized by ampli\fed oscillations ( \u001c <0). In Figures 21to 23 these cases are marked by symbols surrounded by a\nbig diamond. Cases 6*, 7, 65, 171 are similar to 108* and\n152*: an ampli\fed oscillation prior to a \flament recon-\n\fguration or eruption. Case 38 is less clear but probably\nis associated with recon\fguration. The ampli\fcation in\ncases 134 and 156 is not evident in the time-distance\ndiagrams, and might be associated with \flament proper\nmotions. These ampli\fed oscillations are very interesting\nand deserve to be studied in greater depth.\n10.4. Displacement, A\nThe maximum displacement of the \flament mass with\nrespect to the equilibrium position during the \ftted os-\ncillation,A, was derived from Equation (2):\nA=MAX (jA0e\u0000A1(t\u0000t0)cos [A2(t\u0000t0) +A3]j):(4)\nThe distributions of Afor SAOs and LAOs di\u000ber sub-\nstantially (Figure 26(a)). For SAOs, the distribution is\nconcentrated at the origin with a large peak in the range\n0-5 Mm, many fewer events between 5-10 Mm, and no\nevents with A > 10 Mm. In contrast, LAO displace-\nments cover a larger range ( A= 0-50 Mm), with a peak\nat 7.5 Mm. The Adistributions for the three \flament\ntypes are similar, with a maximum in the range 0-5 Mm\nand a decreasing number of events for increasing A(Fig.\n26(d)).\nIn theP\u0000Ascatter plot (Fig. 21(b)), SAOs are\nconcentrated at A < 10 Mm while LAOs extend up to\nA= 46 Mm. Note that no events have large Aand\nlowP. Because the velocity amplitude is approximately\nV\u0018A=P, the region of large Aand lowPcorrespondsGONG Catalog of Solar Filament Oscillations 19\nFigure 26. Histograms of the number of events binned by A(\frst column), \u000b(second column), and latitude (third column). Panels and\nannotations are as in Fig. 24.\nto very large Vvalues where no events were found in\nour survey. Recently, we discovered an oscillation event\nwith the largest velocity amplitude reported thus far\n(100 km s\u00001) and a displacement of more than 50 Mm\nLuna et al. (2017), which would \ft in the empty region\nof Figure 21(b).\nIn theA\u0000\u000bscatter plot (Fig. 22(b)) we see that\nthe range of displacements is reduced when \u000bincreases.\nSimilar to the V\u0000\u000bor\u001c=P-\u000bplots,Adrops signi\fcantly\nfor\u000b > 40\u000eand there are no LAOs for \u000b > 65\u000e. This\nsuggests that the oscillation or excitation mechanisms\ndi\u000ber on either side of \u000b= 40\u000e, as discussed in xx10.1\nand 10.3. Figure 23(c) shows that Ais independent of\nthe \flament latitude.\n10.5. Direction of motion \u000b\nThe parameter \u000bis the angle between the direction of\nthe oscillation and the \flament spine ( x9). Within the\ncatalog we found oscillations in any direction from 0\u000e\nto 90\u000e(Fig. 26(b)). The total distribution has a peak\nclose to 18\u000eand a mean value of 27\u000e\u000618\u000e. For LAOs,\nthe maximum is \u001828\u000ewith a mean of 25\u000e\u000614\u000e, while\nfor SAOs the peak also is close to 18\u000eand the mean\nvalue is 29\u000e\u000621\u000e. The number of events decreases for\n\u000b>40\u000eand only SAOs have \u000b>65\u000e, as we found for the\nV\u0000\u000b,j\u001c=Pj-\u000bandA\u0000\u000bscatter plots (xx10.1, 10.3, and\n10.4). Therefore we de\fne two populations of oscillations\nwith respect to \u000b: 163 events with \u000b<40\u000eand 33 with\n\u000b>40\u000e.\nFigure 26(b) shows that LAOs and SAOs have sim-\nilar\u000bdistributions. The mean values are consistent\nwith direct measurements of the angle between the \fl-\nament magnetic \feld and its spine ( \u000b\u001825\u000eon average;\nLeroy et al. 1983, 1984; E. Tandberg-Hanssen 1995; Tru-jillo Bueno et al. 2002; Casini et al. 2003; L\u0013 opez Ariste\net al. 2006). This suggests that most of the oscillations\nin the catalogued events are aligned with the magnetic\n\feld (longitudinal).\nThe\u000bdistribution is clearest for IT events (Fig. 26(e)):\nthe oscillations are aimed in all directions, but the peak\ncoincides with the mean at 25\u000e\u000613\u000e. The\u000bdistribution\nfor AR events has a peak at 37.5\u000ewith a mean value of\n25\u000e\u000614\u000e. Interestingly, there are no oscillations in AR\n\flaments with \u000b>45\u000e, indicating that the motions are\nmainly longitudinal. For QS \flaments the distribution\nhas a maximum around 18\u000eand a mean value at 22\u000e\u0006\n20\u000e. The QS\u000b-distribution covers the entire domain, but\nthe oscillation o\u000bsets are mainly below 40\u000e. In summary,\nthe mean\u000bvalues for all \flament types agree with the\nobserved magnetic-\feld orientation relative to the spine,\nimplying longitudinal polarization, particularly for IT \fl-\naments.\n10.6. Latitude\nProminence oscillations may re\rect the global struc-\nture of the supporting \flament channels, which is intrin-\nsically tied to the large-scale solar magnetic \feld. Figure\n23 displays several oscillation properties | V,P,A, and\n\u000b| as functions of solar latitude in Stonyhurst Helio-\ngraphic Coordinates (Thompson 2006). V,P, and\u000bgen-\nerally display larger ranges of values at speci\fc latitudes.\nFor IT \flaments, these oscillation properties largely oc-\ncupy the region between \u000025\u000eand 0\u000elatitudes (see Fig-\nures 23(a), (b), and (d)). In contrast, AR events exhibit\nlarger ranges of these properties around two latitudes,\n\u000015\u000eand 15\u000e. ForAthis trend is less evident (Figure\n23(c)), but a 2D histogram (not shown) reveals the same\ntrend.20 Luna et al.\nThe latitude distribution (Figure 26(c)) shows that all\nsurvey events were located between 50\u000eand\u000050\u000e, typi-\ncal for solar maximum. However, a substantial fraction of\nevents accumulated around \u000015\u000e, in the southern hemi-\nsphere, regardless of oscillation type (SAO or LAO). In\nFigure 26(f) the latitude distributions for the three \fla-\nment types are shown. The distribution peaks at \u000015\u000e\nand 15\u000efor AR \flaments, at \u000015\u000efor IT \flaments, and\nat\u000025\u000eand 5\u000efor QS \flaments. It is evident From Fig-\nures 26(c) and 26(f) that the regions of a large number\nof events coincide with the regions of large dispersion of\noscillation parameters of Figure 23. This suggests that\nin those latitudes there are more \flaments and more ac-\ntivity triggering oscillations. In this sense, the existence\nof these latitudes is not necessarily showing a latitudinal\ndependence of oscillation parameters or intrinsic charac-\nteristics of the \flaments.\nBashkirtsev & Mashnich (1993) found a smooth, sinu-\nsoidal latitudinal dependence for 30 SAO events observed\nover more than 8 years, with periods of 80 min at \u000020\u000e\nand 20\u000elatitudes and 40 min at 0\u000e. We have not found a\nclear relationship between the periods or other properties\nand the \flament latitude. Their study covered almost a\nsolar cycle, so their latitudinal dependence could be re-\nlated to the well-known migration of \flaments from the\npoles toward the equator during the cycle. To determine\nwhether this potentially profound relationship is solid,\nour catalog would have to be expanded signi\fcantly to\ninclude oscillation events throughout at least 1 solar cy-\ncle.\n11. SEISMOLOGY\nProminence seismology combines observations and the-\noretical modeling to infer hard-to-measure parameters\nsuch as the magnetic \feld (see x1). There are essen-\ntially three driving mechanisms for prominence oscilla-\ntions: gravitational force, pressure imbalance, and mag-\nnetic Lorentz force.\nLongitudinal oscillations are driven by a combination\nof gravity projected along the \feld (pendulum model,\nLuna & Karpen 2012) and gas pressure gradients (slow\nmodes, Joarder & Roberts 1992). In the pendulum\nmodel, the period depends exclusively on the radius of\ncurvature of the dips supporting the cool prominence\nplasma,R. Luna et al. (2012) and Zhang et al. (2013)\ndetermined that gas pressure gradients contribute negli-\ngibly to the restoring force when the radius of curvature\nis much smaller than a limit de\fned by the prominence\ncharacteristics ( R\u001cRlim), whereRlimis\nRlim= 1=4Lt(Lf\u0000Lt)\u0014g=c2\nsc: (5)\nHere\u0014is the temperature contrast between the cool and\nadjacent hot plasmas, Lfis the \feld line length, Ltis the\nthread length, gis the solar gravitational constant, and\ncscis the coronal sound speed. In that case the period is\nP= 2\u0019s\nR\ng: (6)\nAssuming that the magnetic tension in the dipped\npart of the tubes must be larger than the weight of the\nthreads, the minimum magnetic-\feld strength, B, de-\npends on the particle number density of the prominencethread,n, and the period P. In the absence of direct\ndensity measurements, Luna et al. (2014) adopted the\nrange of typical values n= 1010\u00001011cm\u00003as the main\nsource of uncertainty and determined that\nB(G)\u0015(0:28\u00060:15)P(min): (7)\nFor transverse horizontal oscillations, Kleczek & Kupe-\nrus (1969) assumed that the \flament was supported by\na single line-tied magnetic \rux tube, and that the restor-\ning force was supplied by magnetic tension. We assume\nagain thatntakes typical prominence values, and using\ntheir Eq. (9), we \fnd\nB(G) = (5:5\u00063)L(Mm)\nP(min); (8)\nwhereLis the length of the \flament. The uncertainty\nin the numerical coe\u000ecient is associated with the uncer-\ntainty inn.\nWithout additional data analysis and \feld extrapola-\ntion (e.g., Luna et al. 2017), it is di\u000ecult to establish\nwhich catalog events are oscillations parallel or perpen-\ndicular to the magnetic \feld. However, our statistical\nanalysis revealed a clear distinction between oscillations\nwith\u000b < 40\u000eand those with \u000b > 40\u000e(x10). Although\nthe two populations are not necessarily uniquely associ-\nated with di\u000berent oscillation polarizations, for seismol-\nogy purposes we applied the longitudinal model to the\noscillations with \u000b < 40\u000eand the transverse model to\nthe\u000b>40\u000ecases. This is also justi\fed because the two\nmodels predict approximately the same Bfor a given\nevent. We determined BandRfrom Equations (6) and\n(7) for the events with \u000b < 40\u000e(Figure 27(a)). The\nshaded area covers the uncertainties in B. The magnetic\n\feld ranges from 9 to 48 G, andRfrom 25 to 300 Mm.\nThe mean values are B= 16 G and R= 89 Mm. The\nobtained values are consistent with the rare direct mea-\nsurements of prominence magnetic \felds (see review by\nMackay et al. 2010).\nThe magnetic \feld plotted in Figure 27(a) is a lower\nlimit, so we expect larger values to occur. In particular\nthe \feld could be signi\fcantly underestimated for small\nradii of curvature, R. The reason is that the magnetic\ntension is proportional to B2=Rand the weight of the\nprominence is proportional to ng. Thus, assuming sim-\nilarn, theBnecessary to balance the gravity is smaller\nfor smaller Rthan for larger R.\nIn order to check the validity of the pendulum model,\nwe computed Equation 5 and compared it with Rfor all\ncatalog cases. Because we do not have direct measure-\nments ofLfandLt, we usedLandW, the length of the\nspine and width of the \flament. Wis probably compa-\nrable to the thread lengths, but Lis a lower limit on the\nlength of the sheared \feld lines in the \flament channel\nfor\u000b>0.cscis typically\u0018200 km s\u00001and the typical\ntemperature contrast is \u0014= 100. The resulting Rlimis\nlargely greater than Rlim, demonstrating the applicabil-\nity of the pendulum model to the catalog events.\nFigure 27(b) shows the inferred magnetic \feld as a\nfunction of \u000b. The pendulum model (Eq. (7)) is used\nfor events with \u000b<40\u000e, and transverse model (Eq. (8))\nfor\u000b >40\u000e. For longitudinal oscillations ( \u000b <40\u000e) the\nBrange generally decreases with \u000b, reminiscent of the\nbehavior of P. The same trend applies to the transverseGONG Catalog of Solar Filament Oscillations 21\nFigure 27. Seismology diagnostics for longitudinal and transverse\noscillations. (a) The lower limit on Bas a function of Rfor lon-\ngitudinal oscillations. The shaded area corresponds to the uncer-\ntainty range. (b) The estimated magnetic \feld strength for events\nwith longitudinal and transverse oscillations, from Equations (7)\nand (8) respectively. The vertical dot-dashed line indicates the as-\nsumed separation between longitudinal and transverse oscillations.\nSymbols and colors are as in the scatter plots.\noscillations ( \u000b>40\u000e), although some events reach large\nBvalues (38 G). For transverse oscillations the Bval-\nues are consistent with direct measurements (see, e.g.,\nHarvey 1969). Our AR events are all longitudinal, while\nIT and QS events occupy both categories. It is interest-\ning to note that the minimum \feld strengths do not dif-\nfer signi\fcantly among the \flament types, although AR\n\flaments are embedded in higher \feld-strength regions.\nThis lower limit is consistent with direct measurements\nin AR \flaments (Kuckein et al. 2009; Sasso et al. 2010;\nKuckein et al. 2012; Sasso et al. 2014) showing strong\n\felds of up to several hundred Gauss.\n12. SUMMARY AND CONCLUSIONS\nIn this work we have surveyed prominence oscilla-\ntions detected through visual inspection of the GONG\nnetwork H\u000bdata during January - June 2014, provid-\ning an extensive sample of events close to solar maxi-\nmum of cycle 24. We have catalogued a large variety\nof oscillations including strongly damped motions, un-\ndamped oscillations, and ampli\fed oscillations, enabling\nthe \frst statistically signi\fcant study of \flament oscilla-\ntions and their pertinent properties. The \flament and\noscillation parameters are described in the text and Ta-bles; additional information and animations can be found\nin the online catalog: http://www.iac.es/galeria/\nmluna/pages/gong-catalogue-of-laos.php .\nWe have found 196 oscillation events, including 106\nSAOs and 90 LAOs. In 85 cases we have identi\fed the\ntriggering agents of the oscillations as \rares, prominence\neruptions, a jet, and a Moreton wave. For the remaining\n111 events the triggering agent is not identi\fed. The\noccurrence rate of one LAO event every two days implies\nthat LAOs are common phenomena on the Sun, as are\nSAOs.\nWe have parametrized the oscillations by \ftting an ex-\nponentially decaying sinusoid, and statistically the dis-\ntributions and correlations of key physical parameters.\nThe \ftted velocity amplitudes, V, are in the range\n1:6\u000055 km s\u00001, and show a clear tendency to occur\nless frequently with increased V. This indicates that the\nLAOs are less common than SAOs, particularly since we\nprobably underestimated the number of SAOs approach-\ning the small-amplitude limit. The Vrange decreases\nwith\u000b, dropping sharply for events beyond 40\u000e, and\nthere are no LAOs for \u000b>65\u000e.\nThe oscillation periods, P, range from 32 to 110 min.\nSurprisingly, the periods of both LAOs and SAOs have\nwell-de\fned distributions centered at P= 58\u000615 min.\nThis indicates that LAOs and SAOs are not two dis-\ntinct populations of events with respect to their periods.\nFor all three \flament types the mean oscillation period\nis around 1 hour. The Prange decreases with the an-\ngle between the oscillation displacement and the \flament\nspine,\u000b. In general, we have not found strong correla-\ntions between Pand other oscillation parameters.\nThe damping time per period, \u001c=P covers a large\nrange, including some cases with negative values (am-\npli\fcation). The \u001c=P distribution for LAOs peaks at\n1.25, and most of the events exhibit very strong damp-\ning. For SAOs, the range of observed \u001c=Pvalues is wider,\npeaking at 1.75. The three \flament types behave simi-\nlarly. For LAOs \u001cand\u001c=P decrease with V, regardless\nof \flament type, con\frming that LAOs involve nonlin-\near motions with velocity-dependent damping. This is\na very interesting result because the kinetic energy in-\nvolved in large-amplitude oscillations is enormous, due\nto the combination of large thread masses and large ve-\nlocities. Therefore the physical mechanism must be ef-\n\fcient enough to damp the substantial motion in a few\noscillations. Our earlier theoretical studies showed that\nreasonable rates of mass accretion could explain the ob-\nserved damping rates. On the other hand, the observed\nrelation between \u001candVis consistent with the Zhang\net al. (2013) scaling law, \u001c\u0018V\u00000:3, which suggests that\nthe damping is associated with radiative cooling. More\nobservational and theoretical work needs to be done to\nunderstand the damping process more thoroughly.\nFor the catalog events, the direction of the motion with\nrespect to the \flament spine, \u000b, covers all possible angles\nbetween 0\u000eand 90\u000e, and the\u000bdistributions for LAOs\nand SAOs exhibit no clear peak. However, the mean \u000b\nvalue is the same for all three \flament types: 27\u000e, which\nagrees with previous direct measurements of \u000b\u001825\u000eon\naverage (Leroy et al. 1983, 1984; E. Tandberg-Hanssen\n1995; Trujillo Bueno et al. 2002; Casini et al. 2003; L\u0013 opez\nAriste et al. 2006). Thus, most of the oscillation displace-\nments are probably aligned with the \flament magnetic22 Luna et al.\n\felds.\nWe have not found evidence of any relationships be-\ntween the oscillation parameters and the solar latitude,\nin contrast to the \fndings of Bashkirtsev & Mashnich\n(1993). However, their study covered almost a solar cy-\ncle, and their latitudinal dependence could be associated\nwith the well-known migration of \flaments from poles to\nequator. To determine whether this profound relation-\nship is solid, our catalog must be expanded to include\nevents throughout at least 1 solar cycle .\nWe have applied seismological techniques to the en-\ntire catalog. For the longitudinally oscillating cases,\nwe determined the radius of curvature of the magnetic\ndips hosting the prominence, R, and the minimum \feld\nstrength,B, required to support the mass against grav-\nity.R= 25\u0000300 Mm and B= 2\u000038 G with mean values\nofR= 89 Mm and B= 17 G. For transverse oscillations,\nthe magnetic \feld strength derived from the magnetic\nrestoring force yields a wider range of B= 2\u000038 G but\na similar mean value.\nMost of the oscillations are longitudinal, with the mo-\ntion directed along the local magnetic \feld. Surprisingly,\nthe period distributions for both SAOs and LAOs have\na strong peak centered at 58 min, which implies that\nmost solar \flaments share a common structure. Namely,\ntheir structure is composed of dipped \rux tubes with a\nradius of curvature of \u001890 Mm and an angle between\nthe threads and the spine of \u001830\u000e. The magnetic-\feld\nstrength is probably larger than the minimum estimate\nof 16 G. We also found that many SAOs are initiated by\nenergetic disturbances, which contradicts the idea that\nSAOs are exclusively driven by photospheric or chromo-\nspheric waves. On the other hand, Ning et al. (2009) and\nHillier et al. (2013) studied numerous oscillations in small\nprominence features, and found velocities in general be-\nlow 10 km s\u00001and periods of the order of minutes. These\nlocalized versions of SAOs are more consistent with wave\ndriving than our SAOs, which a\u000bect large portions or the\nentire \flament.In future research we will extend the catalog to events\nnear the solar minimum of the same cycle 24, to augment\nour statistics and explore the possibility that oscillation\nparameters and \flament properties evolve during the so-\nlar cycle. We invite the community to utilize this cata-\nlog for other research projects and to aid in expanding\nits contents, in order to advance our understanding of\nthe fundamental structure and evolution of solar promi-\nnences.\nThe Global Oscillation Network Group (GONG) Pro-\ngram is managed by the NSO and operated by AURA,\nInc. under a cooperative agreement with the NSF.\nThe data are acquired by instruments operated by the\nBig Bear Solar Observatory, High Altitude Observa-\ntory, Learmonth Solar Observatory, Udaipur Solar Ob-\nservatory, Instituto de Astrof\u0013 \u0010sica de Canarias, and\nCerro Tololo Interamerican Observatory. The opera-\ntion of Big Bear Solar Observatory is supported by\nNJIT, US NSF AGS-1250818, and NASA NNX13AG14G\ngrants. This paper made use of the IAC Supercom-\nputing facility HTCondor ( http://research.cs.wisc.\nedu/htcondor/ ), partly \fnanced by the Ministry of\nEconomy and Competitiveness with FEDER funds, code\nIACA13-3E-2493. This research also made use of NASA\nAstrophysics Data System.\nThis work was initiated during International Space\nScience Institute (ISSI) team 314 meetings in Bern led\nby M. Luna on \\Large-Amplitude Oscillations in So-\nlar Prominences\". M. Luna acknowledges the support\nby the Spanish Ministry of Economy and Competitive-\nness through project AYA2014-55078-P. H. Gilbert, J.\nKarpen, T. Kucera and K. Muglach acknowledge sup-\nport by the NASA Heliophysics Guest Investigator pro-\ngram. J. Terradas and J. L. Ballester want to thank the\n\fnancial support from MINECO AYA2014-54485-P and\nFEDER Funds, and the Conselleria d'Innovaci\u0013 o, Recerca\ni Turisme del Govern Balear to IAC3.GONG Catalog of Solar Filament Oscillations 23\nAPPENDIX\nEVENT CATALOG\nEvent # Time Tel. Pos. (x,y) Type L (Mm) W (Mm) Trigger Erupts\n1 1-Jan-2014 16:50 C -18, -76 IP 269 15 FLARE\n2* 1-Jan-2014 16:50 C -18, -76 IP 269 15 FLARE\n3 4-Jan-2014 22:04 B -251, 302 IP 172 7 FLARE\n4 5-Jan-2014 17:03 C -399, 48 IP 87 14 FLARE\n5 5-Jan-2014 17:03 C -308, -116 AR 240 13\n6* 5-Jan-2014 17:03 C -308, -116 AR 240 13\n7 5-Jan-2014 17:03 C 730, -430 QS 139 10 PE\n8* 5-Jan-2014 17:03 C 730, -430 QS 139 10 PE\n9 5-Jan-2014 22:11 B 806, -79 IP 206 11\n10 6-Jan-2014 08:59 U 84, 492 QS 44 9 FLARE\n11 6-Jan-2014 07:11 U -132, -261 IP 185 8\n12 6-Jan-2014 16:56 C -268, -466 QS 102 8\n13 6-Jan-2014 16:56 C -184, 57 AR 78 12\n14* 6-Jan-2014 16:56 C -184, 57 AR 78 12\n15 6-Jan-2014 08:59 U 57, 304 IP 119 8 FLARE\n16 7-Jan-2014 09:13 U -335, 78 IP 387 13 FLARE\n17 7-Jan-2014 09:13 U -325, -282 IP 190 7 FLARE\n18 7-Jan-2014 09:13 U 143, -370 IP 445 8\n19 7-Jan-2014 16:58 C -100, -501 QS 143 12 PE\n20 8-Jan-2014 08:59 U -139, -316 IP 313 14 PE\n21 8-Jan-2014 08:59 U 8, -494 IP 187 10\n22* 8-Jan-2014 08:59 U 8, -494 IP 187 10\n23 8-Jan-2014 08:59 U 260, -367 IP 543 9 FLARE\n24 9-Jan-2014 17:10 C 155, -306 IP 287 20 FLARE\n25 9-Jan-2014 17:10 C 242, -505 IP 303 12\n26 10-Jan-2014 14:03 T 433, 80 IP 54 9\n27 11-Jan-2014 16:43 C 815, -393 IP 221 9 PE\n28 15-Jan-2014 20:00 B 502, -217 AR 125 6\n29 16-Jan-2014 07:37 U 268, 125 AR 80 15\n30 16-Jan-2014 19:59 B -624, -298 IP 228 9\n31 24-Jan-2014 17:00 C -311, -385 IP 281 10 FLARE\n32 24-Jan-2014 19:11 M 284, -151 IP 394 14\n33 25-Jan-2014 07:40 U -184, -376 IP 251 8\n34 25-Jan-2014 16:41 C -110, -129 AR 163 10 FLARE\n35* 25-Jan-2014 16:41 C -110, -129 AR 163 10 FLARE\n36 25-Jan-2014 16:41 C 508, -95 IP 527 15 FLARE\n37 26-Jan-2014 08:27 U 850, 99 IP 81 8\n38 27-Jan-2014 16:41 C 335, -142 AR 255 9\n39* 27-Jan-2014 16:41 C 335, -142 AR 255 9\n40 27-Jan-2014 16:41 C 449, -402 IP 209 17\n41 28-Jan-2014 20:23 B 635, -420 IP 247 9\n42 29-Jan-2014 07:23 U 445, -304 IP 82 11\n43 29-Jan-2014 16:49 C 556, -428 IP 251 9\n44 29-Jan-2014 19:59 B -822, 11 IP 182 11 FLARE\n45 30-Jan-2014 07:23 U 702, 33 IP 195 9 PE\n46 31-Jan-2014 15:20 C -557, 16 IP 201 15\n47 1-Feb-2014 07:17 U -404, 58 AR 317 9 FLARE\n48 5-Feb-2014 17:07 C -812, -129 IP 206 10 FLARE Y\n49 5-Feb-2014 17:07 C -212, -269 IP 205 10 FLARE\n50 6-Feb-2014 17:05 C -626, -433 IP 363 13 FLARE\n51 6-Feb-2014 17:05 C -388, -266 IP 231 10 FLARE\n52 7-Feb-2014 13:24 T -506, 243 IP 111 6\n53 8-Feb-2014 20:10 B 428, -271 AR 211 9 FLARE\n54 8-Feb-2014 13:24 T -193, 114 IP 84 16\n55 8-Feb-2014 13:24 T -122, -162 AR 181 8\n56 9-Feb-2014 16:52 C -82, -93 IP 162 11 FLARE\n57 9-Feb-2014 16:52 C -192, -119 IP 109 8 FLARE\n58 9-Feb-2014 16:52 C -390, -197 AR 232 7 FLARE\n59 11-Feb-2014 17:39 C 517, -203 AR 101 9 FLARE\n60 12-Feb-2014 13:28 T 368, 297 AR 184 8 FLARE\nTable 1\nTable of the observation details and parameters of the \flamet for events 1 to 60. In \frst column the asterisk indicates that the oscillation\nis in the same time-distance diagram than in previous case.24 Luna et al.\nEvent # Time Tel. Pos. (x,y) Type L (Mm) W (Mm) Trigger Erupts\n61 12-Feb-2014 19:42 B 571, -105 IP 160 14 FLARE\n62 13-Feb-2014 17:05 C 601, 238 AR 236 6 FLARE\n63 13-Feb-2014 20:04 B 708, -382 QS 721 33 FLARE\n64 14-Feb-2014 13:27 T 723, 248 AR 295 9 FLARE\n65 14-Feb-2014 19:34 B -592, -233 IP 805 8 FLARE\n66 16-Feb-2014 19:51 B -17, -93 AR 132 7 FLARE\n67 17-Feb-2014 07:28 U -110, -202 IP 898 14 FLARE\n68 17-Feb-2014 18:52 B 3, -199 IP 1051 13\n69 19-Feb-2014 19:47 B 380, -215 IP 913 13 FLARE\n70 22-Feb-2014 07:28 U -16, -378 IP 359 9\n71 23-Feb-2014 16:55 C -448, 277 AR 120 12 FLARE\n72 24-Feb-2014 16:59 C 44, -311 IP 219 22\n73 24-Feb-2014 16:59 C 330, -267 IP 200 10\n74 25-Feb-2014 16:38 C 548, -213 IP 447 7\n75 25-Feb-2014 16:38 C 366, -172 AR 114 8\n76 25-Feb-2014 16:38 C 526, 144 QS 46 12 Y\n77 27-Feb-2014 13:22 T -1, 164 IP 209 9 FLARE\n78 27-Feb-2014 16:57 C -203, 350 IP 147 9\n79* 27-Feb-2014 16:57 C -203, 350 IP 147 9\n80 28-Feb-2014 13:22 T 125, -328 QS 813 24\n81 7-Mar-2014 17:16 C 590, -74 IP 437 12 FLARE\n82 12-Mar-2014 07:29 U -567, -237 IP 210 9 FLARE\n83 14-Mar-2014 16:58 C -812, 265 AR 192 8 FLARE\n84 16-Mar-2014 07:22 U -539, -213 IP 831 13 PE\n85 20-Mar-2014 16:59 C -315, -4 IP 242 9\n86 21-Mar-2014 19:21 B 521, 347 AR 335 11 PE\n87 23-Mar-2014 16:54 C 398, 0 IP 320 12 JET\n88 24-Mar-2014 18:57 B 586, -13 IP 152 9\n89* 24-Mar-2014 18:57 B 586, -13 IP 152 9\n90 28-Mar-2014 18:09 B 683, 41 IP 79 18\n91 29-Mar-2014 16:57 C 329, 372 IP 120 9 MW\n92 29-Mar-2014 16:57 C 99, -201 AR 217 6 FLARE\n93 30-Mar-2014 07:10 U 83, -405 IP 95 10\n94 31-Mar-2014 18:49 B 379, -390 IP 289 14 FLARE\n95* 31-Mar-2014 18:49 B 379, -390 IP 289 14 FLARE\n96 7-Apr-2014 18:33 B 522, 416 AR 72 6 FLARE\n97 9-Apr-2014 12:42 T 469, 216 IP 139 9 FLARE\n98 9-Apr-2014 18:30 M 520, 200 AR 111 8 Y\n99 10-Apr-2014 05:07 L 653, -431 QS 458 18\n100 14-Apr-2014 16:44 C 146, -110 IP 248 10 Y\n101 17-Apr-2014 05:01 L -648, -161 IP 531 9 FLARE\n102 17-Apr-2014 05:01 L 297, 447 QS 210 20\n103 17-Apr-2014 16:52 C 5, -324 IP 177 8 PE\n104 17-Apr-2014 16:52 C 731, -150 QS 141 21\n105 19-Apr-2014 04:48 L 602, 466 QS 240 17\n106 19-Apr-2014 16:47 C -185, -125 IP 691 11 FLARE\n107 21-Apr-2014 18:20 B 701, -371 IP 149 7 FLARE Y\n108* 21-Apr-2014 18:20 B 701, -371 IP 149 7 FLARE Y\n109 22-Apr-2014 16:41 C 61, 296 IP 229 6\n110 23-Apr-2014 16:49 C 264, 294 IP 270 7 FLARE\n111 24-Apr-2014 07:05 U -262, -263 AR 197 9\n112 24-Apr-2014 07:05 U 205, 148 IP 289 11\n113 25-Apr-2014 07:10 U 424, 132 IP 264 14\n114 26-Apr-2014 13:27 T 465, -370 IP 800 14 PE\n115 1-May-2014 18:11 B -380, 231 IP 290 8 FLARE\n116 1-May-2014 18:11 B 753, -385 IP 428 11 FLARE\n117 1-May-2014 07:10 U -625, -379 IP 137 12\n118 2-May-2014 07:04 U -296, 190 AR 212 9 FLARE\n119* 2-May-2014 07:04 U -296, 190 AR 212 9 FLARE\n120 2-May-2014 13:42 T 36, -342 IP 173 12\nTable 2\nSame as Table 1 for events 61 to 120.GONG Catalog of Solar Filament Oscillations 25\nEvent # Time Tel. Pos. (x,y) Type L (Mm) W (Mm) Trigger Erupts\n121 5-May-2014 17:22 B 92, -391 IP 291 8\n122 10-May-2014 06:31 U -499, 417 IP 254 14\n123 10-May-2014 13:07 T -181, -581 QS 71 19 PE\n124 11-May-2014 18:06 B -610, -486 QS 395 13 FLARE\n125 12-May-2014 04:11 L -142, 443 QS 117 15\n126 12-May-2014 04:11 L 77, -22 IP 192 12 FLARE\n127 12-May-2014 04:11 L -494, -634 QS 330 10\n128 12-May-2014 07:21 U 281, -102 IP 96 9\n129 12-May-2014 18:01 M -346, 361 IP 164 14\n130 13-May-2014 18:07 B 323, -26 IP 115 9\n131 13-May-2014 18:07 B -86, -464 IP 222 18\n132 14-May-2014 07:22 U 413, -626 QS 149 35\n133 15-May-2014 16:43 C 285, 142 IP 330 25\n134 16-May-2014 06:47 U 354, 197 QS 246 33\n135 16-May-2014 13:07 T -514, 355 IP 142 9 FLARE\n136 17-May-2014 04:54 L 521, -70 IP 202 12\n137 17-May-2014 13:07 T 649, 135 QS 175 14\n138 18-May-2014 19:54 B 455, -96 IP 133 13\n139 23-May-2014 12:58 T -101, 240 IP 228 11\n140 23-May-2014 13:07 T 269, -302 IP 292 9\n141 23-May-2014 19:27 B 658, 146 QS 164 18\n142 23-May-2014 18:10 B -577, -192 IP 329 9 FLARE\n143 26-May-2014 03:39 L 243, -432 IP 603 11 FLARE\n144 26-May-2014 17:08 C 743, -32 QS 97 18\n145 26-May-2014 16:45 C -348, -260 IP 143 10 FLARE\n146* 26-May-2014 16:45 C -348, -260 IP 143 10 FLARE\n147 27-May-2014 04:26 L 280, -233 AR 121 8\n148 27-May-2014 11:24 T 552, -404 IP 796 9\n149 28-May-2014 04:21 L -26, -352 AR 219 7 FLARE\n150 29-May-2014 04:37 L 184, -346 AR 183 8\n151 30-May-2014 05:31 L 371, -353 AR 235 11 FLARE\n152* 30-May-2014 05:31 L 371, -353 AR 235 11 FLARE\n153 30-May-2014 11:42 T 765, -153 IP 152 8 FLARE\n154* 30-May-2014 11:42 T 765, -153 IP 152 8 FLARE\n155 1-Jun-2014 04:36 L 692, -354 AR 123 7 FLARE Y\n156 2-Jun-2014 13:25 T 291, 166 IP 239 19\n157 2-Jun-2014 17:45 B -203, 59 QS 124 19\n158 3-Jun-2014 04:32 L -85, 54 QS 166 15\n159 4-Jun-2014 17:42 B -628, -55 IP 339 10\n160 5-Jun-2014 13:09 T 97, -236 IP 256 9\n161 5-Jun-2014 19:42 M -174, 404 QS 98 12\n162 6-Jun-2014 13:14 T 323, -235 IP 104 12 Y\n163 7-Jun-2014 04:45 L 23, -204 IP 108 15\n164 7-Jun-2014 13:09 T -592, -239 IP 206 9\n165* 7-Jun-2014 13:09 T -592, -239 IP 206 9\n166 8-Jun-2014 13:09 T 3, 205 AR 290 11\n167 8-Jun-2014 17:56 B -345, -224 IP 355 13\n168 9-Jun-2014 13:11 T 242, 212 AR 229 11\n169* 9-Jun-2014 13:11 T 242, 212 AR 229 11\n170 9-Jun-2014 13:11 T 379, 6 IP 326 17\n171 9-Jun-2014 13:11 T 45, -304 AR 166 13\n172* 9-Jun-2014 13:11 T 45, -304 AR 166 13\n173 10-Jun-2014 18:52 M 499, 220 AR 73 17 FLARE Y\n174 12-Jun-2014 04:33 L -328, -575 QS 99 12\n175* 12-Jun-2014 04:33 L -328, -575 QS 99 12\n176 12-Jun-2014 04:46 U 289, -344 IP 176 13\n177* 12-Jun-2014 04:46 U 289, -344 IP 176 13\n178 12-Jun-2014 04:33 L 438, -202 IP 166 11\n179 12-Jun-2014 12:51 T 469, 490 QS 196 13\n180* 12-Jun-2014 12:51 T 469, 490 QS 196 13\nTable 3\nSame as Table 1 for events 121 to 180.26 Luna et al.\nEvent # Time Tel. Pos. (x,y) Type L (Mm) W (Mm) Trigger Erupts\n181 13-Jun-2014 04:30 L -163, -574 QS 125 16\n182 13-Jun-2014 13:11 T 599, -297 IP 606 19\n183 14-Jun-2014 04:40 L 672, -294 IP 288 13\n184 14-Jun-2014 13:10 T 745, 55 IP 23 7\n185 14-Jun-2014 13:10 T -527, 490 QS 58 10 FLARE\n186 15-Jun-2014 17:51 M -345, 512 IP 86 16 FLARE\n187* 15-Jun-2014 17:51 M -345, 512 IP 86 16 FLARE\n188 16-Jun-2014 13:11 T -836, 5 IP 189 15\n189 16-Jun-2014 13:11 T -214, 521 IP 61 14\n190* 16-Jun-2014 13:11 T -214, 521 IP 61 14\n191 17-Jun-2014 17:56 B -334, -510 QS 116 12\n192 17-Jun-2014 13:12 T -757, -7 IP 301 13 FLARE\n193 17-Jun-2014 13:12 T -39, 520 IP 119 12\n194 17-Jun-2014 13:12 T 695, 385 IP 107 7 FLARE\n195 19-Jun-2014 18:44 M 241, -231 IP 131 13\n196 29-Jun-2014 16:50 C 130, -200 IP 691 21\nTable 4\nSame as Table 1 for events 181 to 195.GONG Catalog of Solar Filament Oscillations 27\n# Time Ini. \u000b(\u000e)P(min.)\u001c(min.) \u001c=P A (Mm) V (km s\u00001)\n1 1-Jan-2014 13:50 16 76\u00062 121\u000615 1.6\u00060.2 22\u00062 26.5\u00063.6\n2* 1-Jan-2014 18:56 16 71\u00063 173\u0006107 2.4\u00062. 8\u00063 10.7\u00067.0\n3 4-Jan-2014 20:30 26 57\u00062 105\u000633 1.8\u00060.6 4\u00061 8.0\u00063.5\n4 5-Jan-2014 11:43 9 63\u00064 | | 5\u00063 9.4\u00069.2\n5 5-Jan-2014 12:01 32 52\u00061 | | 1\u00061 2.7\u00066.0\n6* 5-Jan-2014 17:05 32 76\u00063 -115\u000638 -1.5\u00060.6 3\u00061 3.5\u00066.0\n7 5-Jan-2014 10:44 0 63\u00063 -390\u0006545 -6.2\u00069. 2\u00061 3.9\u00066.7\n8* 5-Jan-2014 21:31 0 43\u00063 160\u0006329 3.7\u00068. 5\u00063 10.7\u000611.1\n9 5-Jan-2014 20:13 37 44\u00062 38\u00069 0.9\u00060.2 5\u00062 11.6\u00066.2\n10 6-Jan-2014 06:58 22 68\u00061 617\u0006367 9.0\u00065. 8\u00061 12.4\u00061.9\n11 6-Jan-2014 08:02 23 63\u00062 94\u000623 1.5\u00060.4 4\u00061 5.5\u00063.8\n12 6-Jan-2014 11:07 38 40\u00063 158\u0006141 4.0\u00064. 2\u00061 6.5\u00067.9\n13 6-Jan-2014 12:27 16 61\u00061 118\u000618 1.9\u00060.3 10\u00061 15.4\u00062.1\n14* 6-Jan-2014 16:26 16 56\u00066 475\u00061313 8.5\u000620 3\u00062 5.6\u000611.5\n15 6-Jan-2014 06:45 1 64\u00061 | | 8\u00061 12.8\u00061.7\n16 7-Jan-2014 05:58 14 65\u00062 302\u0006158 4.7\u00063. 4\u00061 6.2\u00063.6\n17 7-Jan-2014 04:04 3 46\u00061 96\u000624 2.1\u00060.5 5\u00061 11.0\u00064.3\n18 7-Jan-2014 04:45 23 50\u00061 46\u00065 0.9\u00060.1 15\u00062 25.5\u00064.7\n19 7-Jan-2014 14:16 26 44\u00061 83\u00068 1.9\u00060.2 14\u00062 31.2\u00064.5\n20 8-Jan-2014 05:47 30 50\u00062 67\u000615 1.4\u00060.3 15\u00063 36.5\u00068.8\n21 8-Jan-2014 04:49 2 43\u00062 100\u000645 2.3\u00061. 4\u00061 9.0\u00064.7\n22* 8-Jan-2014 08:18 2 35\u00062 | | 1\u00061 2.5\u00069.3\n23 8-Jan-2014 03:28 24 59\u00061 102\u000614 1.7\u00060.2 7\u00061 14.7\u00062.9\n24 9-Jan-2014 16:56 40 51\u00062 97\u000629 1.9\u00060.6 5\u00062 9.3\u00065.7\n25 9-Jan-2014 12:31 44 46\u00063 73\u000633 1.6\u00060.8 4\u00062 10.8\u00067.5\n26 10-Jan-2014 11:23 45 43\u00062 80\u000626 1.9\u00060.7 3\u00061 10.2\u00065.5\n27 11-Jan-2014 12:37 24 66\u00063 114\u000651 1.7\u00060.8 2\u00061 2.7\u00067.5\n28 15-Jan-2014 17:38 35 39\u00061 78\u000613 2.0\u00060.3 7\u00062 16.2\u00065.8\n29 16-Jan-2014 08:29 13 53\u00062 232\u0006168 4.4\u00063. 2\u00061 3.1\u00065.4\n30 16-Jan-2014 20:46 12 57\u00062 58\u000610 1.0\u00060.2 9\u00062 18.8\u00066.1\n31 24-Jan-2014 17:17 26 77\u00064 663\u00061546 8.6\u000620 1\u00061 1.6\u00069.5\n32 24-Jan-2014 21:00 21 94\u00062 | | 17\u00062 20.4\u00063.5\n33 25-Jan-2014 07:16 34 72\u00062 94\u000620 1.3\u00060.3 4\u00061 4.4\u00063.0\n34 25-Jan-2014 12:19 37 34\u00063 103\u0006185 3.0\u00066. 1\u00061 3.1\u000612.9\n35* 25-Jan-2014 14:47 37 50\u00062 46\u00066 0.9\u00060.1 9\u00062 25.2\u00066.5\n36 25-Jan-2014 16:06 15 87\u00061 185\u000626 2.1\u00060.3 23\u00062 28.6\u00063.1\n37 26-Jan-2014 08:09 36 74\u00061 343\u0006117 4.6\u00062. 5\u00061 7.8\u00062.7\n38 27-Jan-2014 14:28 27 45\u00063 -189\u0006216 -4.2\u00065. 2\u00061 4.5\u000610.7\n39* 27-Jan-2014 20:02 27 55\u00062 172\u0006112 3.1\u00062. 4\u00061 7.4\u00065.2\n40 27-Jan-2014 16:04 39 52\u00061 327\u0006154 6.3\u00063. 2\u00061 4.4\u00063.8\n41 28-Jan-2014 20:14 29 59\u00063 101\u000635 1.7\u00060.6 6\u00062 11.0\u00065.1\n42 29-Jan-2014 08:23 7 59\u00062 131\u000634 2.2\u00060.6 4\u00061 8.2\u00063.2\n43 29-Jan-2014 19:36 35 52\u00062 424\u0006617 8.2\u000610 2\u00061 4.7\u00068.1\n44 29-Jan-2014 18:35 26 69\u00062 | | 4\u00061 6.5\u00064.2\n45 30-Jan-2014 05:20 28 62\u00061 117\u000614 1.9\u00060.2 6\u00061 11.7\u00062.6\n46 31-Jan-2014 13:40 28 44\u00061 72\u000610 1.6\u00060.2 10\u00062 21.0\u00063.7\n47 1-Feb-2014 07:55 39 59\u000610 159\u0006264 2.7\u00065. 1\u00061 2.4\u000615.4\n48 5-Feb-2014 13:53 26 52\u00062 150\u000658 2.9\u00061. 4\u00062 7.7\u00066.4\n49 5-Feb-2014 20:03 44 54\u00063 | | 2\u00061 3.7\u00068.7\n50 6-Feb-2014 16:45 30 68\u00062 78\u000613 1.2\u00060.2 11\u00062 15.6\u00064.5\n51 6-Feb-2014 17:26 34 60\u00062 105\u000620 1.8\u00060.4 9\u00062 12.2\u00064.2\n52 7-Feb-2014 09:44 30 67\u00062 140\u000629 2.1\u00060.5 8\u00062 13.0\u00063.5\n53 8-Feb-2014 21:28 64 47\u00061 61\u00065 1.3\u00060.01 13\u00061 22.9\u00062.5\n54 8-Feb-2014 10:57 68 36\u00061 194\u0006140 5.3\u00064. 1\u00061 2.8\u00065.9\n55 8-Feb-2014 10:34 25 36\u00061 | | 2\u00061 5.5\u00067.0\n56 9-Feb-2014 14:10 35 63\u00061 78\u00068 1.2\u00060.1 22\u00063 46.2\u00065.0\n57 9-Feb-2014 15:04 42 37\u00061 | | 1\u00061 3.9\u00067.0\n58 9-Feb-2014 11:48 32 47\u00061 82\u000620 1.8\u00060.4 5\u00062 14.0\u00065.0\n59 11-Feb-2014 17:13 35 57\u00061 72\u00066 1.3\u00060.01 14\u00062 25.7\u00063.3\n60 12-Feb-2014 14:19 18 55\u00061 283\u0006170 5.1\u00063. 3\u00061 6.3\u00064.7\nTable 5\nTable of oscillation best \ft parameters for 1 to 60. In \frst column the asterisk indicates that the oscillation is in the same time-distance\ndiagram than in previous case.28 Luna et al.\n# Time Ini. \u000b(\u000e)P(min.)\u001c(min.) \u001c=P A (Mm) V (km s\u00001)\n61 12-Feb-2014 17:45 24 73\u00063 260\u0006144 3.6\u00062. 4\u00061 4.6\u00063.9\n62 13-Feb-2014 18:05 19 48\u00061 118\u000624 2.5\u00060.5 10\u00062 20.9\u00064.2\n63 13-Feb-2014 20:39 2 103\u00061 175\u000612 1.7\u00060.1 47\u00062 48.5\u00062.4\n64 14-Feb-2014 08:38 21 56\u00061 79\u00066 1.4\u00060.1 31\u00062 52.1\u00063.7\n65 14-Feb-2014 20:06 60 63\u00063 -40\u00067 -0.6\u00060.1 10\u00061 17.0\u000611.5\n66 16-Feb-2014 17:26 37 40\u00062 188\u0006255 4.7\u00067. 2\u00062 6.7\u000611.9\n67 17-Feb-2014 02:42 12 92\u00063 122\u000618 1.3\u00060.2 9\u00061 14.9\u00063.1\n68 17-Feb-2014 19:59 25 39\u00063 117\u0006149 3.0\u00064. 3\u00062 8.1\u000611.6\n69 19-Feb-2014 15:38 16 76\u00062 88\u000613 1.2\u00060.2 20\u00063 28.7\u00066.1\n70 22-Feb-2014 09:04 23 51\u00062 51\u000613 1.0\u00060.3 10\u00063 25.7\u00068.2\n71 23-Feb-2014 16:04 26 34\u00061 42\u00068 1.2\u00060.2 11\u00062 34.7\u00066.1\n72 24-Feb-2014 15:03 22 73\u00061 416\u0006127 5.7\u00062. 7\u00061 10.5\u00062.9\n73 24-Feb-2014 19:13 20 54\u00062 155\u000681 2.9\u00062. 5\u00062 9.3\u00067.9\n74 25-Feb-2014 15:23 50 65\u00062 102\u000621 1.6\u00060.3 5\u00061 9.8\u00063.5\n75 25-Feb-2014 15:28 16 65\u00063 80\u000627 1.2\u00060.5 4\u00062 8.9\u00066.5\n76 25-Feb-2014 12:37 15 78\u00062 | | 3\u00061 4.8\u00063.3\n77 27-Feb-2014 09:48 7 57\u00061 75\u00068 1.3\u00060.2 29\u00063 54.6\u00066.3\n78 27-Feb-2014 15:59 34 53\u00062 136\u000690 2.6\u00062. 4\u00062 9.6\u00066.2\n79* 27-Feb-2014 19:41 34 51\u00061 87\u000620 1.7\u00060.4 7\u00062 18.9\u00065.9\n80 28-Feb-2014 12:08 10 109\u00062 312\u0006106 2.9\u00061. 9\u00062 9.6\u00062.8\n81 7-Mar-2014 16:44 21 49\u00061 423\u0006345 8.6\u00067. 4\u00061 9.5\u00063.7\n82 12-Mar-2014 09:02 1 105\u000610 115\u000652 1.1\u00060.6 12\u00063 20.5\u000611.1\n83 14-Mar-2014 13:31 35 70\u00062 89\u000614 1.3\u00060.2 8\u00062 14.4\u00063.0\n84 16-Mar-2014 03:31 37 46\u00061 74\u000615 1.6\u00060.4 11\u00062 26.4\u00065.9\n85 20-Mar-2014 16:02 19 93\u00062 73\u00068 0.8\u00060.09 16\u00062 21.8\u00063.8\n86 21-Mar-2014 17:23 7 61\u00062 89\u000625 1.5\u00060.4 13\u00062 19.6\u00065.1\n87 23-Mar-2014 16:05 22 76\u00061 71\u00065 0.9\u00060.07 16\u00062 29.6\u00063.5\n88 24-Mar-2014 19:17 29 44\u00062 50\u00068 1.2\u00060.2 7\u00061 16.0\u00064.0\n89* 24-Mar-2014 20:57 29 38\u00062 251\u0006269 6.6\u00067. 1\u00061 3.8\u00066.5\n90 28-Mar-2014 17:52 0 81\u00063 135\u000632 1.7\u00060.4 8\u00062 10.8\u00063.9\n91 29-Mar-2014 18:23 26 58\u00061 108\u000612 1.9\u00060.2 11\u00061 19.3\u00062.3\n92 29-Mar-2014 15:50 12 87\u00062 139\u000620 1.6\u00060.3 31\u00063 38.8\u00064.4\n93 30-Mar-2014 03:53 10 59\u00065 | | 8\u00062 15.7\u00069.2\n94 31-Mar-2014 16:11 30 51\u00062 43\u000611 0.8\u00060.2 6\u00062 18.9\u00067.0\n95* 31-Mar-2014 19:38 30 58\u00061 325\u000662 5.6\u00061. 17\u00062 32.5\u00062.5\n96 7-Apr-2014 21:46 19 41\u00061 70\u00069 1.7\u00060.2 10\u00061 24.2\u00062.8\n97 9-Apr-2014 11:21 33 56\u00061 71\u000611 1.3\u00060.2 10\u00062 24.9\u00064.5\n98 9-Apr-2014 18:19 29 51\u00064 103\u0006110 2.0\u00062. 3\u00062 6.8\u000610.6\n99 10-Apr-2014 06:14 20 52\u00061 94\u000613 1.8\u00060.3 8\u00061 13.7\u00062.9\n100 14-Apr-2014 12:16 7 50\u00061 53\u00064 1.0\u00060.09 20\u00062 32.4\u00062.7\n101 17-Apr-2014 05:20 25 52\u00061 83\u00068 1.6\u00060.2 9\u00061 22.8\u00062.8\n102 17-Apr-2014 02:21 3 92\u00062 179\u000635 1.9\u00060.4 8\u00061 9.7\u00062.4\n103 17-Apr-2014 17:41 39 56\u00061 61\u00067 1.1\u00060.1 11\u00062 25.5\u00063.7\n104 17-Apr-2014 18:38 8 73\u00064 | | 1\u00061 2.1\u00067.4\n105 19-Apr-2014 02:57 0 59\u00062 118\u000634 2.0\u00060.6 9\u00062 12.9\u00064.4\n106 19-Apr-2014 12:44 26 65\u00061 114\u000620 1.8\u00060.3 8\u00062 15.0\u00064.3\n107 21-Apr-2014 18:20 20 50\u00061 155\u000634 3.1\u00060.7 4\u00061 6.6\u00062.2\n108* 21-Apr-2014 21:29 20 40\u00063 -97\u000670 -2.4\u00062. 2\u00061 5.6\u00069.3\n109 22-Apr-2014 12:17 11 61\u00061 110\u000619 1.8\u00060.3 5\u00061 8.5\u00063.4\n110 23-Apr-2014 14:32 26 51\u00064 45\u000615 0.9\u00060.3 7\u00062 10.5\u00067.4\n111 24-Apr-2014 04:04 16 43\u00061 232\u000642 5.3\u00060.1 8\u00061 18.5\u00063.3\n112 24-Apr-2014 10:53 21 41\u00062 37\u00066 0.9\u00060.1 13\u00062 27.7\u00066.2\n113 25-Apr-2014 03:17 16 58\u00062 276\u0006189 4.7\u00063. 2\u00061 3.0\u00066.4\n114 26-Apr-2014 12:40 54 47\u00063 55\u000621 1.2\u00060.5 6\u00062 11.0\u00068.7\n115 1-May-2014 17:40 8 70\u00061 78\u00064 1.1\u00060.06 25\u00062 48.4\u00062.8\n116 1-May-2014 16:17 23 52\u00061 218\u0006103 4.2\u00062. 3\u00061 6.4\u00064.9\n117 1-May-2014 03:08 81 52\u00062 125\u000644 2.4\u00060.9 2\u00061 3.3\u00064.8\n118 2-May-2014 02:17 8 76\u00061 377\u0006144 5.0\u00062. 6\u00061 7.0\u00062.4\n119* 2-May-2014 09:08 8 72\u00065 110\u000689 1.5\u00061. 5\u00063 9.3\u00069.8\n120 2-May-2014 11:04 48 50\u00064 51\u000619 1.0\u00060.4 4\u00062 6.1\u00068.6\nTable 6\nSame as Table 5 for events 61 to 120.GONG Catalog of Solar Filament Oscillations 29\n# Time Ini. \u000b(\u000e)P(min.)\u001c(min.) \u001c=P A (Mm) V (km s\u00001)\n121 5-May-2014 16:13 15 35\u00063 26\u00068 0.7\u00060.3 6\u00062 14.2\u00069.1\n122 10-May-2014 08:01 18 81\u00063 198\u0006104 2.4\u00061. 3\u00061 3.7\u00065.8\n123 10-May-2014 09:14 72 69\u00063 119\u000658 1.7\u00060.9 3\u00062 4.6\u00066.5\n124 11-May-2014 18:47 9 64\u00063 46\u00068 0.7\u00060.2 18\u00063 24.0\u00065.8\n125 12-May-2014 02:40 18 65\u00066 86\u000652 1.3\u00060.9 4\u00062 5.3\u00069.1\n126 12-May-2014 01:19 31 88\u00062 212\u000655 2.4\u00060.7 8\u00062 12.3\u00063.3\n127 12-May-2014 06:20 21 32\u00062 34\u00069 1.1\u00060.3 9\u00063 25.0\u00069.1\n128 12-May-2014 08:22 7 55\u00062 123\u000635 2.2\u00060.7 5\u00061 8.4\u00064.0\n129 12-May-2014 01:26 15 43\u00062 50\u000616 1.2\u00060.4 3\u00061 8.0\u00065.8\n130 13-May-2014 18:26 16 50\u00063 72\u000625 1.4\u00060.5 4\u00061 6.8\u00065.6\n131 13-May-2014 14:46 74 77\u00062 185\u000683 2.4\u00061. 2\u00061 2.4\u00063.5\n132 14-May-2014 07:13 87 99\u00065 227\u0006138 2.3\u00061. 4\u00062 3.6\u00065.6\n133 15-May-2014 18:25 44 76\u00062 87\u000612 1.1\u00060.2 8\u00061 12.9\u00062.7\n134 16-May-2014 02:25 35 85\u00065 -224\u0006195 -2.6\u00062. 2\u00061 4.5\u00069.0\n135 16-May-2014 11:49 7 50\u00061 | | 11\u00061 24.7\u00063.2\n136 17-May-2014 02:37 2 59\u00063 218\u0006219 3.7\u00064. 3\u00062 5.1\u00067.4\n137 17-May-2014 10:35 20 58\u00062 78\u000612 1.4\u00060.2 19\u00062 29.5\u00065.1\n138 18-May-2014 18:22 38 57\u00061 49\u00066 0.9\u00060.1 9\u00062 23.5\u00064.3\n139 23-May-2014 15:06 48 56\u00062 179\u0006111 3.2\u00062. 2\u00061 4.1\u00065.4\n140 23-May-2014 14:58 18 48\u00062 82\u000620 1.7\u00060.4 8\u00062 13.8\u00065.2\n141 23-May-2014 14:14 4 48\u00063 213\u0006216 4.4\u00065. 2\u00061 4.0\u00068.5\n142 23-May-2014 18:01 37 46\u00062 32\u00067 0.7\u00060.2 6\u00062 20.1\u00067.2\n143 26-May-2014 02:38 13 62\u00062 53\u00066 0.9\u00060.01 15\u00062 21.2\u00062.8\n144 26-May-2014 17:52 62 53\u00062 180\u0006145 3.4\u00063. 3\u00061 6.9\u00066.2\n145 26-May-2014 13:31 50 39\u00062 48\u000613 1.2\u00060.4 5\u00062 14.4\u00066.1\n146* 26-May-2014 17:38 50 36\u00062 119\u0006146 3.3\u00064. 1\u00061 3.4\u00069.4\n147 27-May-2014 03:16 1 70\u00062 444\u0006385 6.3\u00066. 3\u00061 4.5\u00065.4\n148 27-May-2014 09:16 36 34\u00063 40\u000622 1.2\u00060.7 2\u00061 4.8\u000610.0\n149 28-May-2014 00:34 42 75\u00066 467\u00061263 6.2\u000620 2\u00062 3.5\u00069.2\n150 29-May-2014 01:16 27 42\u00061 86\u000613 2.1\u00060.3 8\u00061 21.2\u00064.3\n151 30-May-2014 00:01 36 52\u00062 89\u000623 1.7\u00060.5 4\u00061 6.8\u00065.1\n152* 30-May-2014 05:37 36 66\u00063 -163\u0006105 -2.5\u00062. 3\u00061 4.0\u00067.8\n153 30-May-2014 11:43 36 45\u000612 | | 1\u00063 4.1\u000630.4\n154* 30-May-2014 13:42 36 37\u00063 49\u000625 1.3\u00060.7 7\u00064 17.2\u000615.4\n155 1-Jun-2014 02:06 32 32\u000613 101\u0006859 3.1\u000630 0\u00062 1.7\u000652.6\n156 2-Jun-2014 07:05 51 73\u00063 -291\u0006236 -4.0\u00063. 2\u00061 3.6\u00065.9\n157 2-Jun-2014 18:26 14 84\u00062 | | 2\u00061 2.4\u00063.9\n158 3-Jun-2014 02:33 19 74\u00067 557\u00062260 7.5\u000630 1\u00061 1.2\u000613.5\n159 4-Jun-2014 17:30 33 79\u00062 155\u000629 2.0\u00060.4 6\u00061 6.6\u00062.6\n160 5-Jun-2014 10:20 50 60\u00061 272\u000673 4.6\u00061. 2\u00061 3.6\u00063.2\n161 5-Jun-2014 19:57 16 49\u00062 51\u000613 1.1\u00060.3 5\u00062 10.3\u00066.1\n162 6-Jun-2014 10:46 2 48\u00063 | | 2\u00061 3.9\u00067.0\n163 7-Jun-2014 00:36 64 45\u00063 33\u00069 0.7\u00060.2 9\u00063 16.2\u000613.1\n164 7-Jun-2014 07:26 42 52\u00063 106\u000662 2.1\u00061. 3\u00061 5.9\u00066.4\n165* 7-Jun-2014 11:26 42 58\u00062 123\u000639 2.1\u00060.7 5\u00062 9.1\u00064.9\n166 8-Jun-2014 10:17 13 56\u00065 46\u000620 0.8\u00060.4 5\u00064 11.3\u000611.9\n167 8-Jun-2014 16:55 29 55\u00063 193\u0006167 3.5\u00063. 2\u00061 3.6\u00068.0\n168 9-Jun-2014 07:32 34 47\u00062 64\u000616 1.3\u00060.4 6\u00062 11.7\u00065.2\n169* 9-Jun-2014 11:55 34 40\u00061 84\u000621 2.1\u00060.5 6\u00061 12.4\u00063.5\n170 9-Jun-2014 14:46 40 58\u00064 | | 1\u00061 2.2\u000617.5\n171 9-Jun-2014 09:34 1 62\u00063 -202\u0006226 -3.2\u00064. 1\u00061 2.6\u00069.3\n172* 9-Jun-2014 13:59 1 57\u00062 280\u0006189 4.9\u00063. 3\u00061 5.5\u00065.3\n173 10-Jun-2014 18:19 35 36\u00061 84\u000620 2.4\u00060.6 6\u00061 17.3\u00064.8\n174 12-Jun-2014 00:01 20 63\u00061 275\u0006118 4.4\u00062. 5\u00061 9.9\u00063.3\n175* 12-Jun-2014 04:31 20 45\u00061 207\u0006134 4.6\u00063. 1\u00061 2.8\u00065.1\n176 12-Jun-2014 01:50 21 67\u00063 | | 3\u00061 5.1\u00065.7\n177* 12-Jun-2014 05:37 21 54\u00063 56\u000635 1.0\u00060.7 13\u00065 15.1\u00069.5\n178 12-Jun-2014 00:30 47 67\u00062 118\u000631 1.8\u00060.5 4\u00061 5.3\u00065.1\n179 12-Jun-2014 13:11 13 52\u00061 57\u00068 1.1\u00060.2 10\u00062 22.1\u00065.0\n180* 12-Jun-2014 14:10 13 51\u00061 142\u000648 2.8\u00060.1 4\u00061 8.3\u00064.1\nTable 7\nSame as Table 5 for events 121 to 180.30 Luna et al.\n# Time Ini. \u000b(\u000e)P(min.)\u001c(min.)\u001c=P A (Mm) V (km s\u00001)\n181 13-Jun-2014 05:38 28 60\u00065 | | 1\u00061 2.3\u000612.9\n182 13-Jun-2014 16:37 2 51\u00063 245\u0006353 4.8\u00067. 1\u00061 3.2\u00068.5\n183 14-Jun-2014 01:44 1 58\u00062 493\u0006673 8.4\u000610 1\u00061 1.7\u00067.4\n184 14-Jun-2014 11:38 81 61\u00064 249\u0006313 4.1\u00065. 1\u00061 2.3\u00068.5\n185 14-Jun-2014 09:46 17 45\u00063 | | 2\u00062 6.3\u000610.0\n186 15-Jun-2014 02:37 9 44\u00066 | | 1\u00061 1.4\u000619.3\n187* 15-Jun-2014 21:04 9 54\u00061 118\u000622 2.2\u00060.4 7\u00061 11.1\u00062.6\n188 16-Jun-2014 11:43 57 63\u00068 237\u0006356 3.7\u00066. 2\u00061 3.9\u000612.6\n189 16-Jun-2014 08:50 88 38\u00069 125\u0006501 3.3\u000610 0\u00061 1.0\u000628.2\n190* 16-Jun-2014 15:59 88 41\u00063 84\u000667 2.1\u00062. 2\u00061 3.7\u000610.1\n191 17-Jun-2014 21:14 20 67\u00066 39\u00069 0.6\u00060.2 4\u00062 4.7\u00067.4\n192 17-Jun-2014 15:50 35 57\u00061 162\u000651 2.9\u00060.9 6\u00062 15.0\u00064.6\n193 17-Jun-2014 09:50 65 42\u00061 246\u0006166 5.8\u00064. 1\u00061 2.8\u00064.5\n194 17-Jun-2014 12:36 27 51\u00062 68\u000616 1.3\u00060.3 4\u00061 9.0\u00064.0\n195 19-Jun-2014 18:10 14 47\u00063 62\u000621 1.3\u00060.5 3\u00061 7.3\u00065.1\n196 29-Jun-2014 16:21 50 56\u00064 61\u000628 1.1\u00060.6 5\u00063 9.4\u000610.6\nTable 8\nSame as Table 5 for events 181 to 195.\nTIME-DISTANCE DIAGRAMS IN CURVED SLITS\nThe GONG network telescopes o\u000ber fairly good spatial resolution of around 1 arcsec per pixel. However, the seeing\nconditions at the network telescope locations often limit the quality of the images, yielding poor e\u000bective spatial\nresolution greater than 1 arcsec. As we discussed in x2, it was necessary to follow the motion of the large-amplitude\ndisplacements with curved paths in order to accurately track the entire motion of the \flament.\nA time-distance diagram is constructed to follow the motion along the path de\fned by the arti\fcial slit. In many\ncases (e.g., Luna et al. 2014), straight slits consisting of rectangles of length land width win pixels are placed\nlengthwise along the path of the motion studied. In order to increase the signal-to-noise ratio, the intensity is averaged\nalong the width w, which essentially projects the intensity onto the axis of the slit. The resulting intensity along the\nslit as a function of time is the time-distance diagram. Using a curved slit is theoretically similar to using a straight\nslit. In a curved slit the projection is de\fned along the normal lines to the curved slit axis. Thus, for each pixel, there\nis a normal line intersecting the slit axis at ( xq,yq) and the distance between the pixel and the slit is d(i.e. between\nthe pixel and ( xq,yq)). A pixel belongs to the slit if d\u0014w=2.\nFigure 28. (a) H\u000bimage of case 1. The \flament is located in the center of the image. The white curve is the axis of the slit, S. (b)\nIsocontours of Dist(i;j) andInt(i;j) (Equation B1) for the image in (a). (c) Close-up view of a region of the slit showing the bins used to\nconstruct the time-distance diagram. All positions inside the bins have B(i;j) =q=constant (Equation B3). The grey gradient highlights\nthe di\u000berent bins, with white corresponding to q= 1 and black to q=Npix.\nIn general an image is described by the 2D function I(x;y), whereIis the intensity in the \flter considered (H \u000bin\nour situation) and xandyare the coordinates of each position in the image. We assume, without loss of generality,\nthat the origin of the coordinates ( x;y) = (0;0) is at the left-bottom boundary of the image. These coordinates take\nentire values of the resolution \u000eof the image, then x=i\u000eandy=j\u000ewhereiandjde\fne the position within the\nimage in pixels. Alternatively, the image can be described in pixels I(i;j).\nWe \frst de\fne a su\u000eciently smooth curve, S, that represents the axis of the slit, by clicking repetitively on the\nimage along the path of the oscillatory motion and \ftting these points with a polynomial function of 4th degree. The\nwhite line in Figure 28(a) shows the curve Sobtained for event 1 of the catalog. We divide this curve into segmentsGONG Catalog of Solar Filament Oscillations 31\nof length\u000ein order to pixelate the curve as the image. The coordinate along the slit axis is then s=\u000eqwhereqis a\none dimensional array with Npix=l=\u000eelements.\nThe time-distance diagram consists of I(t;iq;jq) =hI(t;A)iwhere (iq;jq) is the position of the q-segment of the axis\nof the slit, tis time,Ais an area surrounding ( iq;jq), and theh:::imeans the average of the intensity over A. The\nmain di\u000eculty is how A is de\fned. Some authors just de\fnes a square area centered at ( iq;jq). However, this mixes\nthe intensities from points that are not projected perpendicularly to the slit, and some pixels are projected twice in\nconsecutive segments of the slit. We will de\fne A as the area enclosed by the normal lines between both ends of slit\nsegment,qand within the slit, d\u0014w=2.\nFor this end, we de\fne two matrices, Dist(i;j) is the distance from any point ( i;j) to the closest point along the\nslit, andInd(i;j) is the index of that point along the slit. These are\nDist(i;j) =MIN\u00121\n\u000eq\n(xi\u0000xq)2+ (yj\u0000yq)2\u0013\n(B1)\nInd(i;j) =qmin: (B2)\nwhere the MIN is the minimum over the qindex. We construct these arrays by computing the distancep\n(xi\u0000xq)2+ (yj\u0000yq)2between each pixel of the image, ( i;j), and all the positions over the slit, q. This is equivalent\nto computing the distance, d, between the pixel and the slit axis. However, this way is much more computationally\ne\u000bective. To calculate Ind(i;j) we then \fnd the value of qminthat minimizes the distancep\n(xi\u0000xq)2+ (yj\u0000yq)2.\nThis is the position over the slit where the intensity of the image pixel will be projected. We repeat this process for\nall image pixels and obtain the arrays de\fned by Equations (B1) and (B2). Thus, Dist(i;j) is the distance measured\nin pixels from ( i;j) to the curve S. The closed thick lines in Figure 28(b) are the isolines of the Dist function over the\nimage. We see that each isoline represents the positions of the pixels that are equidistant to the curve segment S, i.e.\nthe slit axis. In the example of Figure 28, the slits has w= 6, which corresponds to the area inside the most internal\nisoline with a distance to Sof 3 pixels. In general we de\fne the slit as the set of pixels ( i;j) that ful\fll the condition\nDist(i;j)\u0014w=2. Thus, to select the pixels inside the slit, we de\fne a masking function\nMask (i;j) =\u001a1;ifDist(i;j)\u0014w=2\n0;ifDist(i;j)>w= 2:(B3)\nThe thin straight lines in Figure 28(b) plot are the isolines of Indmatrix. This isolines coincide with the normal lines\nto the slit axis. We de\fne a new function\nB(i;j) =Mask (i;j)\u0002Ind(i;j): (B4)\nThe values in this array are zero outside the slit and range from 1 to Npixin the slit. In this way we have binned the\nregions of the image that are going to be averaged over the q-position of the slit. We clearly see these bins of constant\nB(i;j) in Figure 28(c), as well as the bins de\fned by the area inside the region formed by the isolines of Dist andInt.\nThen the intensity over the slit, I(q), is the average of the intensity over the bin where B(i;j) =q, that is\nI(q) =hI(B(i;j) =q)i: (B5)\nThis technique can also be used for straight slits to reduce the computational time, because the images do not need\nto be rotated in order to align the x- ory-axis with the direction of the slit. 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Bauer1, 3\n1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Institute for Advanced Studies in Basic Science, 45195 Zanjan, Iran\n3Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\n(Dated: July 11, 2018)\nWe theoretically investigate pumping of phonons by the dynamics of a magnetic film into a non-\nmagnetic contact. The enhanced damping due to the loss of energy and angular momentum shows\ninterferencepatternsasafunctionofresonancefrequencyandmagneticfilmthicknessthatcannotbe\ndescribed by viscous (“Gilbert”) damping. The phonon pumping depends on magnetization direction\nas well as geometrical and material parameters and is observable, e.g., in thin films of yttrium iron\ngarnet on a thick dielectric substrate.\nThe dynamics of ferromagnetic heterostructures is at\nthe root of devices for information and communication\ntechnologies [1–5]. When a normal metal contact is at-\ntached to a ferromagnet, the magnetization dynamics\ndrives a spin current through the interface. This effect\nis known as spin pumping and can strongly enhance the\n(Gilbert) viscous damping in ultra-thin magnetic films\n[6–8]. Spin pumping and its (Onsager) reciprocal, the\nspin transfer torque [9, 10], are crucial in spintronics, as\nthey allow electric control and detection of magnetiza-\ntion dynamics. When a magnet is connected to a non-\nmagnetic insulator instead of a metal, angular momen-\ntum cannot leave the magnet in the form of electronic or\nmagnonic spin currents, but they can do so in the form\nof phonons. Half a century ago it was reported [11, 12]\nand explained [13–16] that magnetization dynamics can\ngenerate phonons by magnetostriction. More recently,\nthe inverse effect of magnetization dynamics excited by\nsurface acoustic waves (SAWs) has been studied [17–20]\nand found to generate spin currents in proximity normal\nmetals [21, 22]. The emission and detection of SAWs was\ncombined in one and the same device [23, 24], and adia-\nbatic transformation between magnons and phonons was\nobserved in inhomogeneous magnetic fields [25]. The an-\ngular momentum of phonons [26, 27] has recently come\ninto focus again in the context of the Einstein-de Haas\neffect [28] and spin-phonon interactions in general [29].\nThe interpretation of the phonon angular momentum in\ntermsoforbitalandspincontributions[29]hasbeenchal-\nlenged [30], a discussion that bears similarities with the\ninterpretation of the photon angular momentum [31]. In\nour opinion this distinction is rather semantic since not\nrequired to arrive at concrete results. A recent quantum\ntheory of the dynamics of a magnetic impurity [32] pre-\ndicts a broadening of the electron spin resonance and a\nrenormalized g-factor by coupling to an elastic contin-\nuum via the spin-orbit interaction, which appears to be\nrelated to the enhanced damping and effective gyromag-\nnetic ratio discussed here.\nA phonon current generated by magnetization dynam-\nics generates damping by carrying away angular momen-\ntum and energy from the ferromagnet. While the phonon\nphonon sinkzmagnet\nnon-magnet0\nphononsmHFigure 1. Magnetic film (shaded) with magnetization mat-\ntached to a semi-infinite elastic material, which serves as an\nideal phonon sink.\ncontribution to the bulk Gilbert damping has been stud-\nied theoretically [33–38], the damping enhancement by\ninterfaces to non-magnetic substrates or overlayers has\nto our knowledge not been addressed before. Here we\npresent a theory of the coupled lattice and magnetiza-\ntion dynamics of a ferromagnetic film attached to a half-\ninfinite non-magnet, which serves as an ideal phonon\nsink. We predict, for instance, significantly enhanced\ndamping when an yttrium iron garnet (YIG) film is\ngrown on a thick gadolinium gallium garnet (GGG) sub-\nstrate.\nWe consider an easy-axis magnetic film with static ex-\nternal magnetic field and equilibrium magnetization ei-\nther normal (see Fig. 1) or parallel to the plane. The\nmagnet is connected to a semi-infinite elastic material.\nMagnetization and lattice are coupled by the magne-\ntocrystalline anisotropy and the magnetoelastic interac-\ntion, giving rise to coupled field equations of motion in\nthe magnet [39–42]. By matching these with the lattice\ndynamics in the non-magnet by proper boundary con-\nditions, we predict the dynamics of the heterostructure\nas a function of geometrical and constitutive parameters.\nWe find that magnetization dynamics induced, e.g., by\nferromagnetic resonance (FMR) excites the lattice in the\nattachednon-magnet. Inanalogywiththeelectroniccase\nwecallthiseffect“phononpumping” thataffectsthemag-\nnetization dynamics. We consider only equilibrium mag-\nnetizations that are normal or parallel to the interface,\nin which the pumped phonons are pure shear waves that\ncarry angular momentum. We note that for general mag-arXiv:1804.07080v2  [cond-mat.mes-hall]  16 Jul 20182\nnetization directions both shear and pressure waves are\nemitted, however.\nWe consider a magnetic film (metallic or insulating)\nthat extends from z=\u0000dtoz= 0. It is subject to suffi-\nciently high magnetic fields H0such that magnetization\nis uniform, i.e. M(r) =M:For in-plane magnetizations,\nH0> Ms, where the magnetization Msgoverns the de-\nmagnetizing field [43]. The energy of the magnet|non-\nmagnet bilayer can be written\nE=ET+Eel+EZ+ED+E0\nK+Eme;(1)\nwhich are integrals over the energy densities \"X(r). The\ndifferent contributions are explained in the following.\nThe kinetic energy density of the elastic motion reads\n\"T(r) =(\n1\n2\u001a_u2(r); z> 0\n1\n2~\u001a_u2(r);\u0000d<z< 0; (2)\nand the elastic energy density [44]\n\"el=(\n1\n2\u0015(P\n\u000bX\u000b\u000b(r))2+\u0016P\n\u000b\fX2\n\u000b\f(r); z> 0\n1\n2~\u0015(P\n\u000bX\u000b\u000b(r))2+ ~\u0016P\n\u000b\fX2\n\u000b\f(r);\u0000d<z< 0;\n(3)\nwhere\u000b;\f2fx;y;zg,\u0015and\u0016are the Lamé parameters\nand\u001athe mass density of the non-magnet. The tilded\nparameters are those of the magnet. The strain tensor\nX\u000b\fis defined in terms of the displacement fields u\u000b(r),\nX\u000b\f(r) =1\n2\u0012@u\u000b(r)\n@r\f+@u\f(r)\n@r\u000b\u0013\n: (4)\nEZ=\u0000\u00160VM\u0001Hextis the Zeeman energy for Hext=\nH0+h(t), where h(t)is time-dependent. ED=\n1\n2\u00160VMTDMis the magnetostatic energy with shape-\ndependent demagnetization tensor DandVthe volume\nof the magnet. For a thin film with zaxis along the sur-\nface normal n0,Dzz= 1while the other components van-\nish.E0\nK=K1V(m\u0002n0)2is the uniaxial magnetocrys-\ntalline anisotropy in the absence of lattice deformations,\nwhere m=M=MsandK1is the anisotropy constant.\nThe magnetoelastic energy Emecouples the magnetiza-\ntion to the lattice, as discussed in the following.\nThe magnetoelastic energy density can be expanded as\n\"me(r) =1\nM2sX\n\u000b;\fM\u000b(r)M\f(r)\n\u0002[B\u000b\fX\u000b\f(r) +C\u000b\f\n\u000b\f(r)]:(5)\nFor an isotropic medium the magnetoelastic constants\nB\u000b\fread [45]\nB\u000b\f=\u000e\u000b\fBk+ (1\u0000\u000e\u000b\f)B?: (6)\nRotational deformations as expressed by the tensor\n\n\u000b\f(r) =1\n2\u0012@u\u000b(r)\n@r\f\u0000@u\f(r)\n@r\u000b\u0013\n(7)are often disregarded [39–42, 46], but lead to a position\ndependence of the easy axis n(r)from the equilibrium\nvalue n0=ezand an anisotropy energy density [29, 47,\n48]\n\"K(r) =K1\nM2s[M\u0002n(r)]2: (8)\nTo first order in the small deformation\n\u000en(r) =n(r)\u0000n0=0\n@\nxz(r)\n\nyz(r)\n01\nA; (9)\n\"K(r) =\"0\nK+ 2K1(n0\u0000mzm)\u0001\u000en(r):(10)\nFrom \n\u000b\f=\u0000\n\f\u000bit follows that (for non-chiral crystal\nstructures) C\u000b\f=\u0000C\f\u000b. For the uniaxial anisotropy\nconsidered here Cxz=Cyz=\u0000K1. The magnetoelastic\ncoupling due to the magnetocrystalline anisotropy thus\ncontributes [47]\n\"K\nme(r) =\u00002K1\nM2sMz(r) [Mx(r)\nxz(r) +My(r)\nyz(r)]:\n(11)\nPureYIGismagneticallyverysoft, sothemagnetoelastic\nconstants are much larger than the anisotropy constant\n[49, 50]\nBk= 3:48\u0002105Jm\u00003; B?= 6:96\u0002105Jm\u00003;\nK1=\u00006:10\u0002102J m\u00003; (12)\nbut this ratio can be very different for other magnets.\nWe find below that for the Kittel mode dynamics both\ncoupling processes cannot be distinguished, even though\nthey can characteristically affect the magnon-phonon\ncoupling for finite wave numbers.\nThe magnetization dynamics within the magnetic film\nis described by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion [51, 52]\n_m=\u0000\r\u00160m\u0002He\u000b+\u001c(\u000b)\nm; (13)\nwhere\u0000\ris the gyromagnetic ratio, the effective mag-\nnetic field which includes the magnetoelastic coupling\nHe\u000b=\u0000rmE=(\u00160VMs); (14)\nand the Gilbert damping torque [52]\n\u001c(\u000b)\nm=\u000bm\u0002_m: (15)\nThe equation of motion of the elastic continuum reads\n[44]\nu(r;t) =c2\nt4u(r;t) + (c2\nl\u0000c2\nt)r[r\u0001u(r;t)];(16)\nwith longitudinal and transverse sound velocities\ncl=s\n\u0015+ 2\u0016\n\u001a; ct=r\u0016\n\u001a; (17)3\nwhere elastic constants and mass density of non-magnet\nand magnet can differ.\nA uniform precession of the magnetization interacts\nwith the lattice deformation at the surfaces of the mag-\nnetic film [13, 14] and at defects in the bulk. The present\ntheorythenholdswhenthethicknessofthemagneticfilm\nd\u001cp\nA, whereAis the cross section area. The Kittel\nmode induces lattice distortions that are uniform in the\nfilm planeu\u000b(r) =u\u000b(z)[14]. The elastic energy density\nis then affected by shear waves only:\n\"el(z) =(\n\u0016\n2\u0000\nu02\nx(z) +u02\ny(z)\u0001\n; z> 0\n~\u0016\n2\u0000\nu02\nx(z) +u02\ny(z)\u0001\n;\u0000d<z< 0;(18)\nwhereu0\n\u000b(z) =@u\u000b(z)=@z. The magnetic field\nHext=\u0000hx(t); hy(t); H 0\u0001Twith monochromatic drive\nhx;y(t) = Re\u0000\nhx;ye\u0000i!t\u0001\nand static component H0along\nthezaxis. At the FMR frequency !?=!H+!Awith\n!H=\r\u00160H0and!A=\r(2K1=Ms\u0000\r\u0016Ms). The equi-\nlibrium magnetization is perpendicular for !?>0. The\nmagnetoelastic energy derived above then simplifies to\nEz\nme=(B?\u0000K1)A\nMsX\n\u000b=x;yM\u000b[u\u000b(0)\u0000u\u000b(\u0000d)];(19)\nwhichresultsinsurfaceshearforces F\u0006(0) =\u0000F\u0006(\u0000d) =\n\u0000(B?\u0000K1)Am\u0006, withF\u0006=Fx\u0006iFy. These forces\ngenerate a stress or transverse momentum current in the\nzdirection (see Supplemental Material)\nj\u0006(z) =\u0000\u0016(z)u0\n\u0006(z); (20)\nwith\u0016(z) =\u0016forz >0and\u0016(z) = ~\u0016for\u0000d < z < 0,\nandu\u0006=ux\u0006iuy, which is related to the transverse mo-\nmentump\u0006(z) =\u001a( _ux(z)\u0006i_uy(z))by Newton’s equa-\ntion:\n_p\u0006(z) =\u0000@\n@zj\u0006(z): (21)\nThe boundary conditions require momentum conserva-\ntion and elastic continuity at the interfaces,\nj\u0006(\u0000d) = (B?\u0000K1)m\u0006;(22)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(23)\nu\u0006(0+) =u\u0006(0\u0000): (24)\nWe treat the magnetoelastic coupling as a small pertur-\nbation and therefore we approximate the magnetization\nm\u0006entering the above boundary conditions as indepen-\ndent of the lattice displacement u\u0006. The loss of angular\nmomentum (see Supplemental Material) affects the mag-\nnetization dynamics in the LLG equation in the form of a\ntorque, which we derive from the magnetoelastic energy\n(19),\n_m\u0006jme=\u0006i!c\nd[u\u0006(0)\u0000u\u0006(\u0000d)]\n=\u0006i!cRe(v)m\u0006\u0007!cIm(v)m\u0006;(25)where!c=\r(B?\u0000K1)=Ms(for YIG:!c= 8:76\u0002\n1011s\u00001) andv= [u\u0006(0)\u0000u\u0006(\u0000d)]=(dm\u0006). We can\ndistinguish an effective field\nHme=!c\n\r\u00160Re(v)ez; (26)\nand a damping coefficient\n\u000b(?)\nme=\u0000!c\n!Imv: (27)\nThe latter can be compared with the Gilbert damping\nconstant\u000bthat enters the linearized equation of motion\nas\n_m\u0006j\u000b=\u0006i\u000b_m\u0006=\u0006\u000b!m\u0006: (28)\nWith the ansatz\nu\u0006(z;t) =(\nC\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d<z< 0;\n(29)\nwe obtain\nv=Ms!c\n!\rd~\u001a~ct2h\ncos(~kd)\u00001i\n\u0000i\u001act\n~\u001a~ctsin(~kd)\nsin(~kd) +i\u001act\n~\u001a~ctcos(~kd);(30)\nand the damping coefficient for perpendicular magneti-\nzation\n\u000b(?)\nme=\u0010!c\n!\u00112Ms\n\rd~\u001a~ct\u001act\n~\u001a~ct4 sin4\u0010~kd\n2\u0011\nsin2(~kd) +\u0010\n\u001act\n~\u001a~ct\u00112\ncos2(~kd);\n(31)\nwhere!=ctk= ~ct~k. The oscillatory behavior of the\ndamping\u000b(?)\nmecomes from the interference of the elastic\nwaves that are generated at the top and bottom surfaces\nof the magnetic film. When they constructively (destruc-\ntively) interfere at the FMR frequency, the damping is\nenhanced(suppressed), becausethemagnon-phononcou-\npling and phonon emission are large (small).\nWhen\u001act\u001c~\u001a~ct(soft substrate) or when acoustic\nimpedances are matched ( \u001act= ~\u001a~ct), damping at the\nresonance ~kd= (2n+ 1)\u0019withn2N0[14] simplifies to\n\u000b(?)\nme!\u0010!c\n!\u001124Ms\n\rd\u001act: (32)\nWhen\u001act\u001d~\u001a~ct(hard substrate), the magnet is acousti-\ncally pinned at the interface and the acoustic resonances\nare at ~kd= (2n+ 1)\u0019=2[14] with\n\u000b(?)\nme!\u0010!c\n!\u00112Ms\n\rd~\u001a~ct\u001act\n~\u001a~ct: (33)\nIn contrast to Gilbert damping, \u000b(?)\nmedepends on the\nfrequency and vanishes in the limits !!0and!!1.\nTherefore, it does not obey the LLG phenomenology4\nand in the non-linear regime does not simply enhance\n\u000bin Eq. (15). The magnetization damping \u000b0in bulk\nmagnetic insulators, on the other hand, is usually of\nthe Gilbert type. It is caused by phonons as well, but\nnot necessarily the magnetoelastic coupling. A theory\nof Gilbert damping [38] assumes a bottleneck process\nby sound wave attenuation, which appears realistic for\nmagnets with high acoustic quality such as YIG. In the\npresent phonon pumping model, energy and angular mo-\nmentum is lost by the emission of sound waves into an\nattached perfect phononwaveguide, so thepumpingpro-\ncess dominates. Such a scenario could also dominate the\ndamping in magnets in which the magnetic quality is rel-\natively higher than the acoustic one.\nWhen the field is rotated to Hext =\u0000\nhx(t); H 0; hz(t)\u0001T, the equilibrium magnetization\nis in the in-plane ydirection and the magnetoelastic\nenergy couples only to the strain uy,\nEy\nme=(B?\u0000K1)A\nMsMz[uy(0)\u0000uy(\u0000d)]:(34)\nThe FMR frequency for in-plane magnetization !k=\n!Hp\n1\u0000!A=!Hwith!A< !H. The magnetoelastic\ncoupling generates again only transverse sound waves.\nThe linearized LLG equation including the phononic\ntorques reads now\n_mx= (!H+!me)mz\u0000\r\u00160hz\u0000!Amz\n+ (\u000b+\u000bme) _mz; (35)\n_mz=\u0000!Hmx+\r\u00160hx\u0000\u000b_mx; (36)\nwhere\u000bmeis given by Eq. (27) and !me=\r\u00160Hmewith\neffective field Hme=Hme\u0001ezgiven by Eq. (26). Both\nHmeand\u000bmecontribute only to _mx. The phonon pump-\ning is always less efficient for the in-plane configuration:\n\u000b(k)\nme=1\n1 + (!k=!H)2\u000b(?)\nme: (37)\nAs an example, we insert parameters for a thin YIG\nfilm on a semi-infinite gadolinium gallium garnet (GGG)\nsubstrate at room temperature. We have chosen YIG\nbecause of its low intrinsic damping and high quality in-\nterface to the GGG substrate. Substantially larger mag-\nnetoelastic coupling in other materials should be offset\nagainst generally larger bulk damping. For GGG, \u001a=\n7:07\u0002103kg m\u00003,cl= 6411 m s\u00001, andct= 3568 m s\u00001\n[53]. For YIG, Ms= 1:4\u0002105A m\u00001,\r= 1:76\u0002\n1011s\u00001T\u00001,~\u001a= 5170 kg m\u00003,~cl= 7209 m s\u00001,~ct=\n3843 m s\u00001, and!c= 8:76\u00021011s\u00001[49, 50]. The ratio\nof the acoustic impedances ~\u001a~ct=\u001act= 0:79. The damp-\ning enhancement \u000b(?)\nmeis shown in Fig. 2 over a range\nof FMR frequencies and film thicknesses. The FMR fre-\nquencies!?=!H+!Aand!k=!Hp\n1\u0000!A=!Hfor\nthe normal and in-plane configurations are tunable by\nthe static magnetic field component H0via!H=\r\u00160H0.\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n<10−510−410−3\nα(⊥)\nmeFigure 2. Damping enhancement \u000b(?)\nmeby phonon pumping\nin a YIG film on a semi-infinite GGG substrate, as given by\nEq. (31).\nThe damping enhancement peaks at acoustic resonance\nfrequencies \u0017\u0019n~ct=(2d). The counter-intuitive result\nthat the damping increases for thicker films can be un-\nderstood by the competition between the magnetoelastic\neffect that increases with thickness at the resonances and\nwins against the increase in total magnetization. How-\never, with increasing thickness the resonance frequencies\ndecrease below a minimum value at which FMR can be\nexcited. For a fixed FMR frequency \u000bme!0ford!1:\nFor comparison, the Gilbert damping in nanometer thin\nYIG films is of the order \u000b\u001810\u00004[54] which is larger\nthan corresponding values for single crystals. We con-\ncludethattheenhanceddampingisatleastpartlycaused\nby interaction with the substrate and not by a reduced\ncrystal quality.\nThe resonances in the figures are very broad because\nthe\u001act\u0019~\u001a~ctimplies very strong coupling of the discrete\nphonons in the thin magnetic layer with the phonon con-\ntinuum in the substrate. When an acoustic mismatch\nis introduced, the broad peaks increasingly sharpen, re-\nflecting the increased lifetime of the magnon polaron res-\nonances in the magnet.\nThe frequency dependent effective magnetic field H(?)\nme\nis shown in Fig. 3. The frequency dependence of H(?)\nme\nimplies a weak frequency dependence of the effective gy-\nromagnetic ratio\n\r(?)\ne\u000b=\r \n1 +\r\u00160H(?)\nme\n!!\n: (38)\nIn the limit of vanishing film thickness, \u00160H(?)\nme!\n\u0000(B?\u0000K1)2=(Ms~\u0016).\nWe assumed that the non-magnet is an ideal phonon\nsink, which means that injected sound waves do not\nreturn. In the opposite limit in which the phonons\ncannot escape, i.e. when the substrate is a thin film\nwith high acoustic quality, the additional damping van-\nishes. This can be interpreted in terms of a phonon5\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n−60−45−30−15015304560\nµ0H(⊥)\nme[µT]\nFigure3. Effectivefield H(?)\nmegeneratedbythemagnetoelastic\ngeneration of phonons in a YIG film on a semi-infinite GGG\nsubstrate, as given by Eq. (26).\naccumulation that, when allowed to thermalize, gener-\nates a phonon chemical potential and/or non-equilibrium\ntemperature. The non-equilibrium thermodynamics of\nphonons in magnetic nanostructures is subject of our on-\ngoing research.\nThe damping enhancement by phonons may be com-\npared with that from electronic spin pumping [6–8],\n\u000bsp=\r~\n4\u0019dMsh\ne2g; (39)\nwhich is inversely proportional to the thickness dof the\nmagnetic film and does not depend on the FMR fre-\nquency, i.e. obeys the LLG phenomenology. Here, g\nis the spin mixing conductance per unit area at the in-\nterface. While phonons can be pumped into any elastic\nmaterial, spin pumping requires an electrically conduct-\ning contact. With a typical value of hg=e2\u00181018m\u00002\nthe damping enhancement of YIG on platinum is \u000bsp\u0018\n10\u00002nm=d:The physics is quite different, however, since\n\u000bsp;in contrast to \u000bme, does not require coherence over\nthe interface.\nIn conclusion, the pumping of phonons by magnetic\nanisotropy and magnetostriction causes a frequency-\ndependent contributions to the damping and effective\nfield of the magnetization dynamics. The generation\nof phonons by magnetic precession can cause significant\ndamping in a magnetic film when grown on an insulating,\nnon-magnetic substrate and partly explains the increased\ndampinginvariablyobservedforthinnerfilms. Theimpli-\ncations of further reaching ramifications, such as phonon-\ninduced dynamic exchange interactions, phonon accumu-\nlations and phonon spin Seebeck effect require additional\nresearch.\nThis work is financially supported by the Nederlandse\nOrganisatievoorWetenschappelijkOnderzoek(NWO)as\nwell as by the Grant-in-Aid for Scientific Research on In-\nnovative Area, ”Nano Spin Conversion Science” (Grant\nNo. 26103006). H. K. acknowledges support from theIran Science Elites Federation. We acknowledge use-\nful discussions with Yaroslav Blanter, Rembert Duine,\nAkashdeep Kamra, Eiji Saitoh, and Sanchar Sharma.\n[1] S. Bader and S. Parkin, Annu. Rev. Condens. Matter\nPhys. 1, 71 (2010).\n[2] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010).\n[3] F. Pulizzi, Nat. Mater. 11, 367 (2012).\n[4] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nat. Phys. 11, 453 (2015).\n[5] S. A. Nikitov, D. V. Kalyabin, I. V. Lisenkov, A. 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Bodzenta, Phys. Status Solidi B\n146, 467 (1988).\n[54] H. Chang, P. Li, W. Zhang, T. Liu, A. Hoffmann,\nL. Deng, and M. Wu, IEEE Magn. Lett. 5, 1 (2014).\n[55] E. B. Magrab, Vibrations of Elastic Systems (Springer,\nDordrecht, 2012).1\nSupplemental Material\nInSecs.IandIIofthisSupplementwederivetheangu-\nlar and transverse momentum of transverse elastic waves\nand the corresponding momentum currents. In Sec. III\nwe give results for a magnet sandwiched between two\nnon-magnets and in Sec. IV we present a theory for the\nmagnetizationdampingenhancementfrompumpingflex-\nural waves into a thin beam.\nI. ANGULAR MOMENTUM\nThe magnetization M=Msmof an uniformly mag-\nnetized magnet with saturation magnetization Msand\nvolumeVis associated with the angular momentum\nS=\u0000MsV\n\rm; (S1)\nwhere\u0000\ris the gyromagnetic ratio. The angular mo-\nmentum density relative to the origin of an elastic body\nwith displacement field u(r;t)and constant mass density\n\u001areads\nl(t) =\u001a(r+u(r;t))\u0002_u(r;t): (S2)\nWith uniaxial anisotropy axis along z, FMR generates\nthe transverse elastic wave\nu(z;t) = Re2\n40\n@ux\nuy\n01\nAeikz\u0000i!t3\n5; (S3)\nwith dispersion relation !=ctk. Defining the time av-\nerage\nhf(t)i= lim\nT!11\nTZT\n0dtf(t); (S4)\nhli= (0;0;hlzi)can be expressed as\nhlzi=\u001ahux_uy\u0000_uxuyi\n=\u0000\u001a!\n4\u0010\nju+j2\u0000ju\u0000j2\u0011\n; (S5)\nwhereu\u0006=ux\u0006iuyand where we used\n\nRe(ae\u0000i!t)Re(be\u0000i!t)\u000b\n=1\n2Re (a\u0003b):(S6)\nThe non-magnet harbors a constant phonon angular mo-\nmentum density in the z-direction, which implies pres-\nence of a phonon angular momentum current Ahjz\nliat\nthe interface to the magnet with area Athat is ab-\nsorbed at the phonon sink. In our model the angular\nmomentum loss rate of the magnet by phonon pumping\nh_Szjmei=\u0000Ahjz\nliand\nD\n_Sz\f\f\f\nmeE\n=hfSz;Ez\nmegi=MsV\u000b(?)\nme!\n4\r\u0010\njm+j2\u0000jm\u0000j2\u0011\n(S7)wheref;gis the Poisson bracket and Ez\nmethe magnetoe-\nlastic energy Eq. (19) in the main text. From Eq. (31)\nand\nu\u0006=Cm\u0006; (S8)\nwith\nC=(B?\u0000K1)h\ncos(~kd)\u00001i\nik\u0016cos(~kd) +~k~\u0016sin(~kd); (S9)\nwe obtain the relation\nhjz\nli=cthlzi; (S10)\nwhichagreeswiththesimplephysicalpictureofanelastic\nwave carrying away its angular momentum density hlzi\nwith transverse sound velocity ct.\nII. TRANSVERSE MOMENTUM\nFor the transverse elastic wave (S3) in a magnet ex-\ntending from z=z0toz=z1(withz1> z0), the time\nderivative of the transverse momentum P\u0006=Px\u0006iPy\nreads\n_P\u0006=\u001aZ\nVd3ru\u0006(z;t)\n=\u0016A\u0002\nu0\n\u0006(z1;t)\u0000u0\n\u0006(z0;t)\u0003\n:(S11)\nThe change of momentum can be interpreted as a trans-\nverse momentum current density j\u0006(z0) =\u0000\u0016u0\n\u0006(z0)\nflowing into the magnet at z0and a current j\u0006(z1) =\n\u0000\u0016u0\n\u0006(z1)flowing out at z1. The momentum current\nis related to the transverse momentum density p\u0006(z) =\n\u001a_u\u0006(z)by\n_p\u0006(z) =\u0000@\n@zj\u0006(z); (S12)\nwhich confirms that\nj\u0006(z;t) =\u0000\u0016u0\n\u0006(z;t): (S13)\nThe instantaneous conservation of transverse momentum\nisaboundaryconditionsattheinterface. Itstimeaverage\nhj\u0006i= 0, but the associated angular momentum along z\nis finite, as shown above.\nIII. SANDWICHED MAGNET\nWhen a non-magnetic material is attached at both\nsides of the magnet and elastic waves leave the magnet\natz= 0andz=\u0000d, the boundary condition are\nj\u0006(\u0000d\u0000)\u0000j\u0006(\u0000d+) = (B?\u0000K1)m\u0006;(S14)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(S15)\nu\u0006(0+) =u\u0006(0\u0000); (S16)\nu\u0006(\u0000d+) =u\u0006(\u0000d\u0000); (S17)2\nwithd\u0006=d\u00060+. Since the Hamiltonian is piece-wise\nconstant\nu\u0006(z;t) =8\n><\n>:C\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d<z< 0\nF\u0006e\u0000ikz\u0000i!t: z< \u0000d;\n(S18)\nUsing the boundary conditions\nv=u\u0006(0)\u0000u\u0006(\u0000d)\ndm\u0006=Ms!c\n!\rd~\u001a~ct2\ni\u001act\n~\u001a~ct\u0000cot(~kd\n2);(S19)\nleading to the damping coefficient\n\u000b(?)\nme=\u0010!c\n!\u00112Ms\n\rd~\u001a~ct2\n\u001act\n~\u001a~ct+~\u001a~ct\n\u001actcot2\u0010~kd\n2\u0011:(S20)\nWhen ~\u001a~ct=\u001act,\n\u000b(?)\nme=\u0010!c\n!\u001122Ms\n\rd\u001actsin2 ~kd\n2!\n;(S21)\nwhich differs from the sin4\u0010\n~kd=2\u0011\ndependence obtained\nfor the magnet|non-magnet bilayer. This result can be\nexplained by the phonon angular momentum leaking at\ntwo interfaces that should enhance the damping for thin\nmagneticfilms. However, thephononpumpingisacoher-\nent process that couples both interfaces, so the damping\nis not increased simply by a factor of 2 as in case of in-\ncoherent spin pumping of a magnetic film sandwiched by\nmetals. The position of the resonances, ~kd= (2n+1)\u0019=2\nwithn2N0, are independent of the ratio \u001act=~\u001a~ctwith\n\u000bme=\u0010!c\n!\u001122Ms\n\rd\u001act: (S22)\n\u000b(?)\nmeand the effective magnetic field for YIG sandwiched\nbetween two infinitely thick GGG layers are shown in\nFigs. S1 and S2.\nIV. FLEXURAL (BENDING) WAVES IN THIN\nBEAMS\nIn the main text we focus on the generation of trans-\nverseorlongitudinalsoundwaves. Infree-standingstruc-\ntured samples such as cantilevers, additional modes be-\ncome important that can be excited by magnetization\ndynamics as well. This can be illustrated by a thin\ncylindrical elastic beam (see Fig. S3) with cross section\nareaA=\u0019r2, in which flexural waves are generated by\nthe magnet of volume V=Adattached to the top of\nthe beam. The elastic energy according to the Euler-\nBernoulli beam theory [44]\nEel=ZL\n0dz\u00141\n2\u001aA_u2(z;t) +1\n2EYI?u002(z;t)\u0015\n;(S23)\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n<10−510−410−3\nα(⊥)\nmeFigure S1. Phonon pumping-enhanced \u000b(?)\nmein a YIG film\nsandwiched between two infinitely thick GGG layers.\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n−40−30−20−10010203040\nµ0H(⊥)\nme[µT]\nFigure S2. Phonon pumping effective field H(?)\nmein a YIG film\nsandwiched between two infinitely thick GGG layers.\nleads to the equation of motion for the flexural waves [44]\n\u001aAu\u0006(z;t) +EYI?u(4)\n\u0006(z;t) = 0;(S24)\nwhereI?=R\ndAx2=\u0019r4=4and elastic modulus EY=\n\u0016(3\u0015+ 2\u0016)=(\u0015+\u0016). The dispersion relation of the flex-\nural waves is quadratic,\n!=s\nEYI?\n\u001aAk2: (S25)\nWhen the dimensions of the magnet are much smaller\nthan the wavelength of the elastic waves, the magne-\ntoelastic coupling is suppressed and magnetization and\nlattice are coupled exclusively by the magnetocrystalline\nand, in contrast to the bulk magnet, also the shape\nanisotropies,\nEA=EK+ED; (S26)\nwith\nEK=K1V(m\u0002n)2; (S27)\nED=1\n2\u00160VMTDM: (S28)3\nm\nH\nz 0magnet non-magnet\nflexural wave\nFigure S3. Thin film magnet (shaded) with magnetization m\nattached to a thin semi-infinite elastic beam.\nFor a thin magnetic film Dzz=D3= 1. When the\nmagnet become very thick ( d\u001dr), i.e. a needle with\nits point forming the contact, and a coordinate system\nwithzaxis along the surface normal n,Dxx=Dyy=\nD?=\u00001=2. All otherD\u000b\fvanish. In contrast to the\nextended bilayer treated in the main text nis now a\ndynamic variable with n\u0006=\u0000u0\n\u0006(0;t). The mechanical\ntorque exerted by the magnet on the elastic beam reads\n\u001c=_L=VMs\n\r_m+_J; (S29)\nwhere _J=\u00160VMsm\u0002Hextis the torque exerted by the\nexternal magnetic field on the total angular momentum.\nFor a magnet with equilibrium magnetization mkn0\n\u001c\u0006=\u0006if(m\u0006\u0000n\u0006); (S30)\nwheref=VMs!A=\rand\n!A=(\n2\rK1=Ms\u0000\r\u00160Ms;thin film\n2\rK1=Ms+1\n2\r\u00160Ms;needle:(S31)\nIn order to compute the angular momentum current\npumped into the attached non-magnet, _L\u0006=\u001c\u0006, we\nhave to specify four boundary conditions. Two are pro-\nvided by the assumption that the beam is infinitely long\nso that the are no reflections. The absence of shear forces\nat the boundary is expressed by u000(0;t) = 0, while the\nbending by the torque follows from the principle of least\naction [55]\nu00\n\u0006(0;t) =\u0006i\u001c\u0006\nEYI?: (S32)The general solution for the differential equation can be\nwritten\nu\u0006(z;t) =e\u0000i!t\u0000\nA\u0006eikz+B\u0006e\u0000kz\u0001\n;(S33)\nbecause there are no back-reflections in the semi-infinite\nbeam. We find\nn\u0006=wm\u0006\nwith\nw=\u00002f\nEYI?k\u0012\n1 +i\u00002f\nEYI?k\u0013\u00001\n;(S34)\nand the following source term in the LLG equation,\n_m\u0006jan=\u0007i!An\u0006\n=\u0006i!ARe(w)m\u0006\u0007!AIm(w)m\u0006;(S35)\nThe first term on the right-hand-side is a field-like torque\nequivalent to the effective field\n\u00160Han=!A\n\rRe(w)ez; (S36)\nand the second one a damping-like torque with damping\ncoefficient\n\u000ban=\u0000!A\n!Imw: (S37)\nSince for weak magnetoelastic coupling we expect \u000ban\u001c\n1and thereforejwj \u001c 1, which is equivalent to\n2f=(EYI?k)\u001c1, we may approximate\nw\u0019f(1\u0000i)\nEYI?k; (S38)\n\u000ban\u0019VMs!2\nA\n!k\rI?EY; (S39)\n\u00160Han\u0019\u0000VMs!2\nA\n\r2EYI?kez: (S40)\nThe damping enhancement scales as\n\u000ban/V\nA2!3\n2: (S41)\nFor a needle-shaped YIG magnet attached to GGG with\nEY= 2:5\u00021011Paand!A= 1:4\u00021010s\u00001\n\u000ban\u00198:6\u000210\u00006d=nm\n(\u0017=GHz)3\n2(r=nm)2;(S42)\n\u00160jHanj\u00193:1\u000210\u00007d=nm\n(\u0017=GHz)1\n2(r=nm)2T;(S43)\nwhich are very small numbers even at nanoscale dimen-\nsions."    },    {        "title": "1804.07674v1.A_Weakly_Nonlinear_Model_for_the_Damping_of_Resonantly_Forced_Density_Waves_in_Dense_Planetary_Rings.pdf",        "content": "Draft version September 5, 2018\nTypeset using L ATEX default style in AASTeX62\nA WEAKLY NONLINEAR MODEL FOR THE DAMPING OF RESONANTLY FORCED DENSITY WAVES IN\nDENSE PLANETARY RINGS\nMarius Lehmann,1Jurgen Schmidt,1and Heikki Salo1\n1Astronomy Research Unit, University of Oulu, Finland\n(Received April 23, 2016; Revised July 9, 2016; Accepted July 14, 2016; Published September 23, 2016)\nSubmitted to The Astrophysical Journal\nABSTRACT\nIn this paper we address the stability of resonantly forced density waves in dense planetary rings.\nAlready by Goldreich and Tremaine (1978) it has been argued that density waves might be unstable,\ndepending on the relationship between the ring's viscosity and the surface mass density. In the recent\npaper Schmidt et al. (2016) we have pointed out that when - within a \ruid description of the ring\ndynamics - the criterion for viscous overstability is satis\fed, forced spiral density waves become unstable\nas well. In this case, linear theory fails to describe the damping, but nonlinearity of the underlying\nequations guarantees a \fnite amplitude and eventually a damping of the wave. We apply the multiple\nscale formalism to derive a weakly nonlinear damping relation from a hydrodynamical model. This\nrelation describes the resonant excitation and nonlinear viscous damping of spiral density waves in a\nvertically integrated \ruid disk with density dependent transport coe\u000ecients. The model consistently\npredicts density waves to be (linearly) unstable in a ring region where the conditions for viscous\noverstability are met. Su\u000eciently far away from the Lindblad resonance, the surface mass density\nperturbation is predicted to saturate to a constant value due to nonlinear viscous damping. The\nwave's damping lengths of the model depend on certain input parameters, such as the distance to the\nthreshold for viscous overstability in parameter space and the ground state surface mass density.\nKeywords: planets and satellites: rings, Hydrodynamics, Instabilities\n1.INTRODUCTION\nDensity wave theory, originally proposed to explain the spiral structure of galactic disks, has been applied to wave\ntrains excited at resonances in Saturn's rings, notably the inner Lindblad resonances (ILR) of various Saturnian\nsatellites.\nThe linear \ruid description of density waves is well developed (Goldreich and Tremaine (1979); Shu (1984)) and it\nhas been applied to derive estimates on the local surface mass density of Saturn's rings (Esposito et al. (1983); Lissauer\net al. (1985); Nicholson et al. (1990); Rosen et al. (1991a); Rosen et al. (1991b); Spilker et al. (2004); Tiscareno et al.\n(2007)). Also, estimates for the ring viscosity have been obtained from the observed damping of density waves. These\nare based on a linear damping relation with the assumption of a constant viscosity. However, already Goldreich and\nTremaine (1978) pointed out that linear density waves might also be unstable, depending on the behavior of the\nviscosity as a function of the local surface mass density of the rings. If a density wave is unstable, a linear description\nis not applicable.\nTraditionally, the theory of the excitation and damping of \\nonlinear density waves\" in Saturn's rings is based on\na description of the ring matter as Lagrangian \ruid elements (streamlines) that interact with each other in terms of\nperturbations of their orbital elements (Borderies et al. (1983a), Borderies et al. (1983b), Borderies et al. (1985), Shu\net al. (1985a), Shu et al. (1985b), Longaretti and Borderies (1986), Borderies et al. (1986), Borderies and Longaretti\nCorresponding author: Marius Lehmann\nmarius.lehmann@oulu.\farXiv:1804.07674v1  [astro-ph.EP]  20 Apr 20182 Lehmann et al.\n(1987), Borderies et al. (1989)). In this formalism the degree of nonlinearity (of a wave) is a direct measure of the\nrelative spacing between adjacent streamlines. Borderies et al. (1985) applied this streamline formalism to describe the\npropagation and damping of spiral density waves as self-excited features in \ruid disks. They computed the pressure\ntensor from a \ruid model describing a collection of densely packed inelastic spheres (Ha\u000b (1983)). The pressure tensor\ndetermines the damping behavior of the waves and they concluded that (free) density waves should be unstable in\nclosely packed rings and stable in dilute rings. Borderies et al. (1986) derived radial pro\fles of tightly wound nonlinear\ndensity waves by computing for di\u000berent viscosity models the isothermal steady state structure of a ring patch in\nthe presence of a density wave. They used a damping relation for waves derived from the conservation of angular\nmomentum luminosity and the nonlinear dispersion relation for tightly wound density waves, as derived with the\nstreamline formalism. Shu et al. (1985a) and Shu et al. (1985b) adopted a similar approach. They also took into\naccount the local heating of the ring due to the density wave itself in their calculation of the pressure tensor from\nthe second-order moment of the Boltzmann equation. They used a Krook ansatz to account for the e\u000bect of particle\ncollisions. Borderies and Longaretti (1987) applied the streamline formalism to a pro\fle of the Mimas 5:3 density\nwave, obtained by the Voyager Photo Polarimeter Subsystem, to retrieve the radial dependences of all the functions\ninvolved in this formalism. The resulting theoretical surface density pro\fle \fts the Voyager data quite well. Later,\nRappaport et al. (2009) applied an improved inversion procedure to derive essentially the same functions, but with\nhigher accuracy, since their procedure is based on a combination of multiple optical depth pro\fles obtained from data\nfrom the Cassini Radio Science Subsystem (RSS). Conclusions of Borderies et al. (1986) and Shu et al. (1985b) were\nthat their nonlinear density wave models can successfully describe many features of observed strong density waves in\nSaturn's rings, but also that they underestimate the wave damping in dense ring regions.\nRecently, the applicability of the linear hydrodynamic damping relation for density waves has been revisited by\nSchmidt et al. (2016) in relation to the strong variety of damping lengths among the \frst order Janus waves, as seen in\nCassini data. In that paper it has been shown that, in a \ruid description of dense rings, a linear density wave is unstable\nif the criterion for the onset of viscous overstability is ful\flled. The viscous overstability is an intrinsic instability of\nthe ring \row which leads to the formation of axi-symmetric wave trains with typical wavelengths \u0015\u0018100m\u0000200m\n(Schmit and Tscharnuter (1995); Schmit and Tscharnuter (1999); Schmidt et al. (2001); Salo et al. (2001); Schmidt\nand Salo (2003); Latter and Ogilvie (2009); Latter and Ogilvie (2010)). The occurrence of this periodic micro structure\nin the dense Saturnian rings has been con\frmed in Cassini RSS and Ultra Violet Imaging Spectrograph (UVIS) data\n(Thomson et al. (2007); (Colwell et al. (2007))). This leads to the conclusion that in some regions of Saturn's rings\ndensity waves might su\u000ber from linear instability and one would expect that these waves show a qualitatively di\u000berent\ndamping behavior than the one predicted by linear theory, even for weakly forced waves.\nIt is the main goal of this paper to approach the problem of an unstable density wave according to the scenario\npictured above in terms of a \ruid model. The wave pattern, which forms at resonance is also responsive to the\nlinear instability of the ring \row. The latter is controlled by a threshold parameter, which is in this case the density\ndependence of the viscosity, a criterion directly related to the viscous overstability. Not too far away from the instability\nthreshold, one expects that the wave pattern obeys a cubic amplitude equation (Cross and Hohenberg (1993)). By\nperforming a multiple scale expansion (Kevorkian and Cole (1996); Cross and Hohenberg (1993)) of the hydrodynamical\nequations we derive a weakly nonlinear model, which describes the damping of resonantly forced spiral density waves\nin a vertically integrated \ruid disk with density dependent transport coe\u000ecients in terms of such a cubic amplitude\nequation.\nIn the idealized model, a density wave subject to viscous overstability is predicted to travel inde\fnitely with an\nasymptotically constant amplitude, limited by nonlinearity. This behavior is also predicted by the streamline model\nof Borderies et al. (1986). In our new model this saturation results from the nonlinear terms in the hydrodynamic\nbalance equations and is not captured by the linearized theory. Nonlinear terms in\ruence the density wave damping as\nsoon as the perturbation of surface density attains a signi\fcant fraction of its background value, and their magnitude\nis a measure for the nonlinearity of the wave.\nThe paper is organized as follows. In Section 2 we brie\ry review important aspects of the linear theory of density\nwaves, particularly the damping by viscosity. Section 3 provides detailed information on the numerical values used for\nthe parameters of our hydrodynamical model. The model equations for the description of a dense ring are presented\nin Section 4.1. In Section 4.2 we use the results of the linear theory in combination with symmetry arguments to place\nrestrictions on the shape of the amplitude equation. These considerations motivate the de\fnition of parameters which\nwill then be used for a rigorous derivation of the nonlinear amplitude equation in terms of a multiple scale expansionDamping of Nonlinear Density Waves in Dense Planetary Rings 3\n(Sections 4.3-4.6). Further, in Sections 4.6 and 4.7 we discuss some general implications of the resulting amplitude\nequation. In section 5 we include the e\u000bects of satellite forcing in the model and derive a forced nonlinear amplitude\nequation. In Section 6 we compare our model with the nonlinear streamline model of Borderies et al. (1986). Section\n7 provides a summary of the derived results and addresses remaining open questions.\n2.LINEAR VISCOUS THEORY\nIn the linear hydrodynamic theory (Goldreich and Tremaine (1978); Shu (1984)), a trailing density wave in the\nvicinity of an inner Lindblad resonance is described in terms of the perturbed surface mass density\n\u001b(r;\u0012;t ) =\u001b0+ Re[A(r)\u0001exp(\niZr\nk(s) ds)\n\u0001expfi(m\u0012\u0000!t)g]; (1)\nwhere Re[ ] denotes the real part and where \u001b0is the unperturbed surface mass density. Here we adopt a cylindrical\ncoordinate system ( r,\u0012,z), restricted to the plane z= 0. We consider waves which are excited by single Fourier modes\nof an external satellite potential, which can be written in the form\n\u001es(r;\u0012;t ) =^\u001es(r) expfi(m\u0012\u0000!t)g; (2)\nwhere the radial function ^\u001es(r) varies slowly with radius r(see Goldreich and Tremaine (1980) for more details). In\nthis paper we neglect higher moments of Saturn's mass distribution, so that the orbital frequencies of ring particles are\nidentical to their corresponding epicyclic frequencies and frequencies of vertical oscillations, i.e. \n ( r) =\u0014(r) =\u0016(r)\nwith\n\n (r) =r\nGMp\nr3(3)\nwhereMpdenotes Saturn's mass and Gis the gravitational constant. Then the relationship between the Lindblad\nresonance radius rLand the forcing frequency !is\n!= (m\u00001) \n (rL): (4)\nFor simplicity we assume that the satellite is on an uninclined, circular orbit so that vertical and corotation resonances\ncan be ignored. Then, the forcing frequency is simply\n!=m\ns; (5)\nwheremis the same positive integer as in (1) and \n sdenotes the satellite's orbital frequency. Since the satellite\npotential (2) varies slowly in r-direction one typically evaluates the corresponding forcing terms in the evolutionary\nequations (Section 5) at the resonance r=rL. For all times tthe perturbation (1) is a tightly wound m-armed spiral\nwave which satis\fes the WKB-approximation, such that jkAj\u001d@rA, meaning that its phase varies much faster with\nradial distance than its amplitude. The excitation of the density wave takes place within a small region around the\nILR of approximately one wavelength.\nThe complex wavenumber of the density wave in (1) reads (Goldreich and Tremaine (1978); Shu (1984))\nk=Dx\n2\u0019G\u001b 0+i\nLD2x2\n(2\u0019G\u001b 0)3\u00110\n\u001b0\u00147\n3+\r\u0015\n(6)\nwith\nx=r\u0000rL\nrL(7)\nbeing the scaled distance from the resonance radius rL. Further,D= 3 (m\u00001) \n2\nL(Cuzzi et al. (1984)) with \n L=\n\n (rL) being the orbital frequency at resonance. In the expression for kthe real part yields the linear wavelength\ndispersion of the wave, while the imaginary part accounts for the e\u000bects of viscosity on the wave amplitude. \u00110is the\nconstant dynamic shear viscosity and the constant \rdenotes the ratio of the bulk and shear viscosities. If one allows\nfor a density dependence of the dynamic viscosity (Schmit and Tscharnuter (1995))\n\u0011=\u00170\u001b0\u0012\u001b\n\u001b0\u0013\f+1\n; (8)4 Lehmann et al.\nwhere\u00170denotes the ground state value of the kinematic shear viscosity \u0017=\u0011=\u001b, one obtains (Schmidt et al. (2016))\npossible values of \u001babout the groundstate value one obtains (Schmidt et al. (2016))\nk=Dx\n2\u0019G\u001b 0+i\nLD2x2\n(2\u0019G\u001b 0)37\n3\u00170\u0014\n1\u00009\n7(\f+ 1) +3\n7\r\u0015\n: (9)\nClearly, expression (6) is recovered in the limit \f!\u00001, corresponding to a constant dynamic viscosity (8). From (9)\none obtains an exponentially growing wave amplitude with increasing distance from the resonance if\n\f >1\n3\u0012\n\r\u00002\n3\u0013\n\u0011\fc: (10)\nThis is the hydrodynamic criterion for the onset of viscous overstability (Schmit and Tscharnuter (1995); Schmidt\net al. (2001)) in the long wavelength limit. Estimates for the bulk viscosity of a dense ring from N-body simulations\n(Salo et al. (2001)) yield \r\u00182\u00004. This suggests \fc\u00191 in dense rings. For optical depths \u001c\u00151,\fwas found\nanalytically (Araki and Tremaine (1986)) and by molecular dynamics simulations (Wisdom and Tremaine (1988); Salo\n(1991)) to be larger than one. Moreover, in local N-body simulations, including gravitational inter-particle forces, it\nwas shown that in the presence of self-gravity wakes the kinematic shear viscosity behaves as (Daisaka et al. (2001))\n\u0017'CG2\u001b2\n\n3; (11)\nwhich would imply \f= 2. In this formula Gis the gravitational constant and Cdenotes a dimensionless constant that\ndepends on particle size and bulk density.\nSchmidt and Salo (2003), which are listed in Table 1. vertical oscillations component of self-gravity would do. (Salo\n(1992); Richardson (1994); Daisaka and Ida (1999); Ohtsuki and Emori (2000)), model with the respect to temperature\nand surface density, respectively. of Schmidt and Salo (2003) by the corrective factor\n3.HYDRODYNAMIC PARAMETERS\nFor the numerical solution of the hydrodynamic equations (14), we must specify several quantities, like the pressure\nand the shear viscosity, along with their dependence on the surface mass density of the ring. To this end we will use\nparameters derived from N-body simulations (Salo et al. (2001)), as they were used by Schmidt and Salo (2003) to\ndescribe the nonlinear evolution of overstable modes in N-body simulations of a dense ring in terms of a hydrodynamic\nmodel. The speci\fc numbers used in our integrations are listed in Table 1.\nThe hydrodynamic parameters from Salo et al. (2001) were determined from simulations without direct particle-\nparticle self-gravity. Instead, e\u000bects of self-gravity were mimicked by using an arti\fcially increased frequency of vertical\noscillations of the ring particles, a treatment that was introduced by Wisdom and Tremaine (1988). This modi\fcation\nincreases the collision frequency between particles in a qualitatively similar manner as the vertical component of self-\ngravity would do, which tends to promote overstability. This treatment of self-gravity, however, misses the e\u000bect of self-\ngravity wakes, which form in the system as a result of gravitational instability (Salo (1992); Richardson (1994); Daisaka\nand Ida (1999); Ohtsuki and Emori (2000)). Observational evidence for the presence of self-gravity wakes is found in\nlarge parts of Saturn's rings (Colwell et al. 2006; French et al. 2007; Colwell et al. 2007). Nevertheless, the hydrodynamic\ncoe\u000ecients determined by Salo et al. (2001) are well suited as parameters for the numerical study performed in this\npaper, because our model, as well as any other theory for spiral density waves presented in the literature so far (see\nSection 1 for references), does not yet take into account the e\u000bect of self-gravity wakes on the evolution of the density\nwave. How the presence of such micro structure, like self-gravity wakes or overstable waves, a\u000bects the density waves\nremains a challenge for future modeling.\nOne problem with the hydrodynamic parameters given by Salo et al. (2001) is that they depend on the velocity\ndispersion of the ring particle ensemble, in addition to their dependence on the surface mass density. In contrast,\nour hydrodynamic model (Section 4.1) is isothermal, i.e. it assumes a constant velocity dispersion everywhere in the\nperturbed ring. But it was shown (Salo et al. 2001; Schmidt et al. 2001) that thermal modes play a stabilizing role for\nthe development of overstability in a planetary ring (see also Spahn et al. (2000)), such that the stability boundary for\noverstability is shifted to higher optical depths when compared to an isothermal treatment. It was noted by Schmidt\nand Salo (2003) that the e\u000bects of the thermal modes on the oscillation frequency and on the growth rate of a linearDamping of Nonlinear Density Waves in Dense Planetary Rings 5\noverstable wave can be incorporated easily into a purely isothermal model if one renormalizes two hydrodynamic\nparameters appropriately. These parameters are the derivative of pressure with respect to surface mass density ( p\u001b)\nand the ratio of the bulk and shear viscosities ( \rin this paper). The renormalization uses the non-isothermal transport\ncoe\u000ecients determined by Salo et al. (2001) and the linear non-isothermal mode analysis by Schmidt et al. (2001).\nSpeci\fcally, it is achieved by absorbing in equation (24) of Schmidt et al. (2001) the quantity F2into an e\u000bective p\u001band\nby absorbing the quantity F3into an e\u000bective value for the constant ratio \r. This method led to a quantitative match\nof an isothermal model for the nonlinear evolution of viscously overstable modes with N-body simulations (Schmidt\nand Salo (2003)). We will use the same method to \fx the parameters of our isothermal model for density waves.\nThe speci\fc parameter sets used in this paper are listed in Table 1. These correspond directly to the numbers given in\nTable (1) of Schmidt and Salo (2003), taking into account the di\u000berent scalings that were applied to non-dimensionalize\nthe parameters (see captions of the tables). The parameter p\u001bin Table 1 corresponds to the e\u000bective quantity in the\ntable by Schmidt and Salo (2003). Similarly, our parameter \r, describing the ratio of bulk to shear viscosity, relates\nto the parameter \u000be\u000bin Schmidt and Salo (2003) through \r=\u000be\u000b\u00004=3.\nTable 1 :Values for Parameters and their Scaling\nQuantity Scaling (Typical) Value\n\u000f=2\u0019G\u001b 0\nrLD(dimensionless parameter) 10\u00008\u000010\u00009\nG(gravitational constant) 6:67\u000110\u000011m3kg\u00001s\u00002\nrL(resonance radius) 108m\n\nL(orbital frequency at resonance) 2\u000110\u00004s\u00001\n\u001b0(ground state surface density)\nt(time) \n\u00001\nL\nk(wavenumber) \u000f\u00001r\u00001\nL\nu,v(planar velocity components ) \u000frL\nL\n\u001e,\u001es(gravitational potentials) \u000f2r2\nL\n2\nL\np(scalar pressure) \u001b0\u000f2r2\nL\n2\nL\n\u001b(surface mass density) \u001b0\n\n (orbital frequency) \nL\n!(forcing frequency) \nL\nD= 3 (m\u00001) \n2\nL \n2\nL\nHydrodynamic parameters (from Schmidt and Salo (2003))\n\u001c(optical depth) 1.0 (\u001c10)1.4 (\u001c14)1.5 (\u001c15)2.0 (\u001c20)\n\u00170[10\u00004m2s\u00001] \u000f2r2\nL\nL 4:43 6:06 6:47 8:93\n\fc 1.23 0.97 0.93 0.92\n\f 0.85 1.03 1.06 1.16\n\r 4.37 3.59 3.47 3.42\n\u000e\u0017 0:56i 0.25 0.37 0.51\np\u001b[10\u00006m2s\u00002] \u000f2r2\nL\n2\nL 0:52 0:63 0:67 1:00\nNote: the quantity \u000e\u0017is imaginary for \u001c= 1:0 which follows from Eq. (22). Further, \fcis de\fned in Eq. (10). For\nexplanations of the hydrodynamic parameters see Sections 2, 3, 4.1 and 4.3.6 Lehmann et al.\n4.NONLINEAR DAMPING OF FREE DENSITY WAVES\nIn the following we study the in\ruence of nonlinearities in the hydrodynamic equations (14) on the propagation of\ndensity waves in a ring region which is described by the viscosity model (8) and that may exhibit viscous overstability.\nWe formally restrict our considerations to the weakly nonlinear regime. Strictly, this means that we are su\u000eciently\nclose to the threshold for the instability (10) [i.e. j\f\u0000\fc\n\fcj\u001c1] and that the density wave has a small initial amplitude\nat the resonance location1. It is then appropriate (Cross and Hohenberg (1993)) to calculate the nonlinear pattern in\nterms of a multiple scale expansion about the marginally unstable (or marginally stable) wave of the linear theory [cf.\n(1)]:\n\u001b=\u001b0+ Re[A(\u0018)\u0001exp(\niZx\nk(s) ds)\n\u0001expfi(m\u0012\u0000!t)g] + hh: (12)\nThe amplitudeAwill now depend on a \\slow\" radial length scale \u0018(formally much larger than one wavelength) and\nis governed by a Landau-type nonlinear amplitude equation. The wave will accordingly develop nonlinear properties,\nincluding the excitation of its higher harmonics (hh). The amplitude equation which will be derived below is a nonlinear\ngeneralization of the linear damping relation in the case of density dependent viscosities (Eq. 9).\n4.1. Hydrodynamic Equations\nWe use the cylindrical coordinate system ( x;\u0012;z ) in the plane z= 0 with the dimensionless distance xas de\fned in\nSection 2. We scale length with rL\u000f, where the small dimensionless parameter\n\u000f=2\u0019G\u001b 0\nrLD(13)\ndescribes the strength of self-gravity as compared with the gravity of the central planet. Expressed in terms of the\nToomre critical wavelength \u0015crthis length scale yields typical values rL\u000f\u0018\u0015cr=6\u0019\u00182 m. Further, time is scaled with\n1=\nLand surface density with its ground state value \u001b0. The scaled z-integrated nonlinear isothermal \ruid equations\nin the plane ( z= 0) then read (Stewart et al. (1984); Schmidt et al. (2009))\n@t\u001b=\u0000\n(x)@\u0012\u001b\u0000\u000f(\u001b@xu+u@x\u001b);\n@tu=\u0000\n(x)@\u0012u+ 2\n(x)v\u0000\u000fu@xu\n+\u00170\u000f2\u00124\n3+\r\u0013\n(1 +\f)\u001b\f\u00001@x\u001b@xu\n+\u00170\u000f2\u00124\n3+\r\u0013\n\u001b\f@2\nxu\u0000\u000fp\u001b@x\u001b\n\u001b\u0000\u000f@x\u001e\u0000\u000f@x\u001es;\n@tv=\u0000\n(x)@\u0012v\u00001\n2\n (x)u\u0000\u000fu@xv\n+\u00170\u000f(1 +\f)\u001b\f\u00001@x\u001b\u0012\n\u000f@xv\u00003\n(x)\n2\u0013\n+\u00170\u000f2\u001b\f@2\nxv\u0000\u000f@\u0012\u001es:(14)\nThe symbols \u001b,uandvdenote the surface mass density, the radial and the tangential velocities, respectively. Note\nthatvdoes not include the Keplerian ground state velocity \n r[cf. (3)] and that we neglect curvature terms, since\nwe will focus on the description of tightly wrapped waves whose wavelengths ful\fll \u0015\u001cr. We use the viscosity\nprescription (8). In the equation for the radial velocity uthe derivative of the scalar pressure pis approximated as\ndp\ndx=p\u001b@\u001b\n@x(15)\nwith\np\u001b\u0011\u0014@p\n@\u001b\u0015\n0(16)\n1such that the density perturbations are much smaller than the equilibrium value: j\u001b(r)\u0000\u001b0\n\u001b0j\u001c1.Damping of Nonlinear Density Waves in Dense Planetary Rings 7\nwhere the subscript \\0\" denotes that the derivative has to be taken at the ground state. The quantity p\u001bis scaled with\n(\u000frL\nL)2. Using values for p\u001bfrom simulations (listed in Table 1) this linearized treatment of the equation of state\nretains e\u000bects of non-local pressure. Further, the quantities \u001eand\u001esare the self-gravity potential and the satellite\npotential, respectively. From here on, all parameters and quantities are scaled as denoted in Table 1.\nEquations (14) are vertically averaged. This restricts their applicability to phenomena which occur on radial length\nscales much greater than the vertical extent of the disk. For density waves this condition is ful\flled by a large\nmargin. However, it is expected that in regions of high compression, such as the peaks of density waves, vertical\nsplashing of the ring material occurs, similar as in overstable oscillations (Salo et al. (2001)). In these regions the\nisothermal approximation is violated. Qualitatively, one would expect an increased velocity dispersion in regions of\nhigher compression, such that, to \frst order, the increased energy in the random motions gets balanced by an enhanced\nfrequency of inelastic particle collisions. Keplerian. The e\u000bect of the satellite forcing terms in Eqs. (14) will be studied\nin Section 5. However, in the free wave analysis which follows below we exclude these terms. We restrict our analysis\ntolong trailing density waves near an inner Lindblad resonance such thatx\u001c1.\n4.2. Nonlinear Amplitude Equation\nOur aim is to derive a complex nonlinear amplitude equation for the radial steady state pro\fle of a density wave,\npropagating away from an ILR. Before we proceed with a rigorous derivation in Section 4.3, we can already place\ncertain restrictions on the shape of this equation by means of physical arguments and by using the results of linear\ntheory (Section 2).\nFirst of all, since the wave amplitude Ais time independent in a stationary state, time derivatives shall not appear.\nFurther, the equation should be invariant upon multiplying Aby an arbitrary phase factor. This can be seen by applying\nthe multiplication A!A\u0001 exp (i\b) to (12) which describes the density wave state in the lowest approximation of the\norder parameter expansion which follows below. One sees that it is always possible to absorb the phase factor exp ( i\b)\nin the phase i!t, which corresponds to a translation in time and the amplitude equation must be invariant upon this\ntranslation. Therefore, the simplest possible nonlinear amplitude equation has the form\ndA\ndx=g(x)A\u0000l(x)AjAj2; (17)\nwhere nonlinear e\u000bects are described through the cubic term and where the radial dependence of the amplitude is\nwritten in terms of the regular length scale x. The applied sign convention resembles the one which is commonly used\nwhen formulating the complex Ginzburg-Landau equation (Aranson and Kramer (2002)). In general the functions g(x)\nandl(x) are complex and their dependence on xre\rects the fact that the considered system is notinvariant upon\ntranslation in x-direction, since, to lowest order in x, the wavenumber of the density wave depends linearly on x[cf.\n(9)].\nIn the linear limit , obtained for su\u000eciently small values of A, the nonlinear term in (17) is negligible and the equation\nshould reproduce the linear damping relation described by the imaginary part kiof the complex wavenumber (9). In\nthis limit, integration of (17) yields\nA(x) = exp\u0012Zx\ng(s) ds\u0013\n: (18)\nOn the other hand, if we insert the imaginary part kiof the wavenumber (9) in (1) and apply the scaling discussed in\nSection 4.1, we obtain\nA(x) = exp\u0012Zx3\u00170s2\n\u000fD(\f\u0000\fc) ds\u0013\n; (19)\nwhere the quantities \u00170andDare scaled. Comparison of (18) with (19) leads to the condition\nRe[g(x)] =3\u00170x2\n\u000fD(\f\u0000\fc): (20)\nThis can be rewritten in the form\ngr(x) =\u00170x2(3\r\u00002)\n3\u000fD\u0012\f\u0000\fc\n\fc\u0013\n(21)\nwhere we de\fned gr(x) = Re[g(x)]. The last expression clearly displays the role of the viscous parameter \fas a\nthreshold parameter for the linear instability of a density wave as it occurs for \f >\fc. In the same manner \fis used\nas threshold parameter in the linear theory of viscous overstability (Schmit and Tscharnuter (1995)).8 Lehmann et al.\nIn the case \f >\fcthe wave amplitude (19) grows exponentially. This means that the linear description fails and the\ncubic term in (17) is necessary to provide a damping of the wave. In the following sections we perform a multi-scale\nexpansion of the hydrodynamic equations (14) in order to derive Eq. (17), where the linear coe\u000ecient g(x) will be\nidentical to (21), consistent with the linear theory.\n4.3. Multiple Scale Expansion\nIn order to perform the multiple scale expansion, we de\fne as control parameter for the bifurcation from the ground\nstate\n\u000e\u0017=s\n\f\u0000\fc\n\fc; (22)\nsuch that the bifurcation occurs at \u000e\u0017= 0. The subscript \u0017is used to distinguish this expansion parameter from\nthe small parameter \u000eswhich will be introduced in Section 5 to describe forced density waves. From this de\fnition\ndirectly follows that the linear coe\u000ecient (21), and consequently the imaginary part of the wavenumber (9), are both\nproportional to \u000e2\n\u0017such that these quantities change their signs at the bifurcation which marks the threshold for linear\ninstability.\nThe state variables are expanded as a series in powers of j\u000e\u0017j:\n\u001e=j\u000e\u0017j\u001e1+j\u000e\u0017j2\u001e2+j\u000e\u0017j3\u001e3+\u0001\u0001\u0001; (23a)\nu=j\u000e\u0017ju1+j\u000e\u0017j2u2+j\u000e\u0017j3u3+\u0001\u0001\u0001; (23b)\nv=j\u000e\u0017jv1+j\u000e\u0017j2v2+j\u000e\u0017j3v3+\u0001\u0001\u0001; (23c)\n\u001b= 1 +j\u000e\u0017j\u001b1+j\u000e\u0017j2\u001b2+j\u000e\u0017j3\u001b3+\u0001\u0001\u0001; (23d)\n\f=\fc+j\u000e\u0017j\f1+j\u000e\u0017j2\f2+\u0001\u0001\u0001: (23e)\nFurther, we introduce a \\slow radial length scale\" \u0018by\n@x!@x+j\u000e\u0017j2@\u0018: (24)\nWe use the absolute value j\u000e\u0017jfor the expansion. This is done as to avoid a negative slow length scale which would\notherwise occur if \f < \fcsince for this case \u000e2\n\u0017<0. By using the absolute value we further ensure that all terms\nin the expressions (23a)-(23e) are real-valued. The necessary expansion of \farises from the choice of the expansion\nparameter (22). The consistency of the expansion requires that \fnally\n\f=\fc\u0000\n1 +\u000e2\n\u0017\u0001\n; (25)\nsuch that\fis a real quantity and can take values greater or smaller than \fc. With (23e) this implies the conditions\n\f1= 0; (26a)\n\f2=\fcsgn\u0000\n\u000e2\n\u0017\u0001\n; (26b)\nwhere sgn\u0000\n\u000e2\n\u0017\u0001\ndenotes the sign of \u000e2\n\u0017.\nThe two length scales xand\u0018are well separated for small values of j\u000e\u0017j, i.e. near the threshold for instability.\nThe \\fast\" scale xdescribes the radial oscillation of the density wave, de\fned by the real wavenumber, which will be\nwritten in the following as k(x). In contrast, the variation of the wave amplitude occurs on the \\slow\" scale \u0018. For\nthe vector of state in the plane ( z= 0) we adopt the short notation\n\t=X\nij\u000e\u0017ji\ti(x;\u0012;t;\u0018 ) (27)\nwith\n\ti(x;\u0012;t;\u0018 ) =0\nB@\u001ei(x;\u0012;t;\u0018 )\nui(x;\u0012;t;\u0018 )\nvi(x;\u0012;t;\u0018 )1\nCA; (28)\nwhere the self-gravity potential is used in place of the surface mass density \u001bto describe the hydrodynamic state. This\nmeans that the surface density in Eqs. (14) has to be replaced by a solution of Poisson's equation which is derived inDamping of Nonlinear Density Waves in Dense Planetary Rings 9\nAppendix B. Inserting (24), (25) and (27) in the nonlinear evolution equations (14), and sorting by orders in j\u000e\u0017jone\nobtains the following hierarchy of balances when requiring that all orders of j\u000e\u0017jvanish separately:\nO\u0000\nj\u000e\u0017j1\u0001\n: ^L\t1= 0;\nO\u0000\nj\u000e\u0017j2\u0001\n: ^L\t2=N2(\t1;\t1);\nO\u0000\nj\u000e\u0017j3\u0001\n: ^L\t3=N3(\t1;\t2) +@\u0018(^M\u0001\t1):(29)\nIn the following it is shown that the desired amplitude equation (17) can be obtained from the O\u0000\nj\u000e\u0017j3\u0001\nequations.\nTherefore we truncate the expansion at this order. In the above equations the same linear operator ^Lappears on the\nleft hand side at each order of \u000e\u0017.^Litself is of orderj\u000e\u0017j0. The vectorial terms Niare the nonlinear terms at order\nj\u000e\u0017jiand are provided in Appendix A. With this expansion we have split the problem in an iterative hierarchy of linear\ninhomogeneous equations where the inhomogeneities at a given order are functions of the solutions of the lower order\nequations. ^Mdenotes a 3\u00023-matrix containing complex constants of order j\u000e\u0017j0. The operator ^Land its adjoint ^Ly\nare given by (30) and (31), respectively. Note that the terms d \u001b=d\u001ewill in general be di\u000berent for each order in j\u000e\u0017j\n(cf. Appendix B).\n^L=0\nBBBBBBB@d\u001b\nd\u001e(@t+ \n@\u0012) \u000f@x 0\n\u000f\u0010\n1 +p\u001bd\u001b\nd\u001e\u0011\n@x@t+ \n@\u0012\u0000\u00004\n3+\r\u0001\n\u00170\u000f2@2\nx\u00002\n3\n2\n\u00170(1 +\fc)d\u001b\nd\u001e\u000f@x1\n2\n @t+ \n@\u0012\u0000\u00170\u000f2@2\nx1\nCCCCCCCA: (30)\n^Ly=0\nBBBBBB@\u0000d\u001b\nd\u001e(@t+ \n@\u0012)\u0000\u0010\n1 +p\u001bd\u001b\nd\u001e\u0011\n\u000f@x\u00003\n2\n\u00170(1 +\fc)d\u001b\nd\u001e\u000f@x\n\u0000\u000f@x\u0000@t\u0000\n@\u0012\u0000\u00004\n3+\r\u0001\n\u00170\u000f2@2\nx1\n2\n0 \u00002\n\u0000@t\u0000\n@\u0012\u0000\u00170\u000f2@2\nx1\nCCCCCCA: (31)\nTo solve (29) we have to apply appropriate solvability conditions2. Therefore a scalar product has to be de\fned in the\nspace of the vectors (28) [whose components are of the form (12)]. If ^Lyis the adjoint operator of ^Land\t0adone of\nits null solutions, i.e.\n^Ly\t0ad= 0; (32)\nthen taking the scalar product of the expanded Eqs. (29) with \t0adleads to the solvability conditions\nO\u0000\nj\u000e\u0017j2\u0001\n:h\t0adjN2(\t1;\t1)i= 0;\nO\u0000\nj\u000e\u0017j3\u0001\n:h\t0adjN3(\t1;\t2) +@\u0018(^M\t1)i= 0:(33)\nThe solvability condition at O\u0000\nj\u000e\u0017j3\u0001\nyields the desired di\u000berential equation (17) for the complex amplitude Awhich\nis the \fnal goal of the weakly nonlinear analysis. As a scalar product we can use\nh\tkj\tli\u00111\n2\u0019Z2\u0019\n0d\u0012[\u001e\u0003\nk\u001el+u\u0003\nkul+v\u0003\nkvl]; (34)\nwhere a star denotes complex conjugate. This choice is appropriate since all \felds (27) can be decomposed in exponen-\ntial phase factors exp\b\nj\u0002R\nik\n\u000fdx+im\u0012\u0000i!t\u0003\t\nwithj=\u00061;\u00062;\u0001\u0001\u0001such that expressions with di\u000berent exponential\nphase factors are orthogonal in terms of (34).\n2This is necessary to avoid resonant driving of the linear equations (Cross and Hohenberg (1993)).10 Lehmann et al.\n4.4. Linear Stability Problem\nIn this section we solve the order j\u000e\u0017j1equations (29). For the order j\u000e\u0017j1vector of state we assume [cf. (12)]\n\t1=A(\u0018)A\t1(x) exp\u001aZ\nik\n\u000fdx+im\u0012\u0000i!t\u001b\n+c:c:(35)\nwith the slowly varying amplitude A(\u0018) and where c:c:denotes complex conjugate. Note that since ^Ldoes not act\non the slow length scale \u0018, the amplitudeA(\u0018) will be carried along as pre-factor until we arrive at the order j\u000e\u0017j3\nequations in Section 4.6. The system of partial di\u000berential equations ^L\t1= 0 with the di\u000berential operator ^L, together\nwith the ansatz (35) yield the set of algebraic equations ^L1A\t1(x) = 0 with the complex matrix\n^L1\u00110\nBBBBBBB@\u0000i(!\u0000m\n) \u0000iD 0\nik\u0010\n1\u0000p\u001bk\nD\u0011\n\u0000i(!\u0000m\n) +k2\u00004\n3+\r\u0001\n\u00170\u00002\n\u00003\nik2(1 +\fc)\u00170\n2D1\n2\n\u0000i(!\u0000m\n) +k2\u001701\nCCCCCCCA(36)\nand its null space A\t1(x). Below we will see that the x-dependency of A\t1(x) vanishes in the leading order approx-\nimation. For a null space A\t1(x) to exist, the determinant of ^L1must vanish, i.e.\nDet^L1= (!\u0000m\n)\u0012D\nD\u0000k\u0013\n+i\u0010\n\u0000(!\u0000m\n)2(7 + 3\r) + 9 (\fc+ 1) \n2\u0011\n\u00170+ 3p\u001b(!\u0000m\n)\n3Dk2\n\u0000i\u00170k3+\u00170(3ip\u001b+ (4 + 3\r) (!\u0000m\n)\u00170)\n3Dk4= 0:(37)\nHere we used the de\fnition\n\n2\u0000(!\u0000m\n)2=D (38)\nwithD(rL) = 0 [cf. (4)] and rL[dD=dr]rL=D. The condition of marginal stability a\u000bords that the wavenumber is\nreal, i.e. Im [ k(x)] = 0. Therefore, we can solve Eq. (37) separately for its real and imaginary parts. The imaginary\npart yields the exact curve of marginal stability:\n\fc(k) =\u00001 +(7 + 3\r) (!\u0000m\n)2\n9\n2+D\n3\n2k\u0000p\u001b\n3\n2k2: (39)\nThe condition that the real part of the determinant is zero yields the dispersion relation:\nD=Dk\u0000p\u001bk2\u0000(4 + 3\r)\u00172\n0\n3k4: (40)\nIf we assume small distances from the Lindblad resonance x\u001c1:\nD=Dx+O\u0000\nx2\u0001\n: (41)\nThus, solving equation (40) perturbatively with the series expansion k(x)\u0011k0+k1x+k2x2+\u0001\u0001\u0001yields the (scaled)\nwavenumber of marginally stable long trailing density waves [cf. (9)]\nk=x+O\u0000\nx2\u0001\n: (42)\nWe also need to consider the leading order expressions of the frequencies:\n\n = 1 +O(x); (43a)\n!= (m\u00001) +O(x): (43b)Damping of Nonlinear Density Waves in Dense Planetary Rings 11\nWith the above approximations, the curve of marginal stability reduces to relation\n\fc=1\n3\u0012\n\r\u00002\n3\u0013\n+O(x): (44)\nThe condition \f >\fcis the condition for viscous overstability in the hydrodynamic approximation [cf. (10)].\nThe \fnal goal of the nonlinear analysis is the nonlinear amplitude equation (17) with the proper coe\u000ecient functions\ng(x) andl(x). If we carry out our calculations with the relations (39) and (40), such that the null spaces of ^L\nand ^Lyexist and can be calculated exactly, we can apply the approximations (42), (43a, b) at the end and expand\ncorresponding terms to leading order in x. Then, the linear coe\u000ecient g(x) in (17) should reduce to the negative of\nthe imaginary part of (9) from linear theory.\nThe expressions arising in course of the analysis will contain a vast number of terms. We will present here only\nterms to leading order in xfor the sake of brevity and clarity. For the evolution of the amplitude of the waves these\nterms provide an excellent approximation.\nUsing (39) and (40), as well as the approximations (42) and (43a, b), the marginal null vector of ^L1in the vicinity\nof an ILR ( x\u001c1) is given by\nA\t1(x) =0\nBB@\u00002iD\n\u00002i\n11\nCCA; (45)\nshowing no x-dependency. In order to proceed we also need to compute the adjoint null space, i.e. the null space of\n(31). With the ansatz\n\t0ad=A\t0ad(x) exp\u001aZ\nik\n\u000fdx+im\u0012\u0000i!t\u001b\n(46)\nEquation (32) reads ^Ly\n1A\t0ad(x) = 0 with the matrix\n^Ly\n1\u00110\nBBBBBBB@i(!\u0000m\n)\u0000ik\u0010\n1\u0000p\u001bk\nD\u00113\nik2(1 +\fc)\u00170\n2D\niDi(!\u0000m\n) +k2\u00004\n3+\r\u0001\n\u001701\n2\n0 \u00002\n i(!\u0000m\n) +k2\u001701\nCCCCCCCA(47)\nbeing the adjoint of (36). With relations (39) and (40), the determinant of ^Ly\n1vanishes as well and its expanded null\nspace reads\nA\t0ad(x) =0\nBB@\u00001\n2Dx\n\u0000i\n2\n11\nCCA: (48)\n4.5. Second Order Solution\nWith the solution for \t1given by (35), we can proceed to the computation of the second order inhomogeneity\nN2(\t1;\t1) (cf. Appendix A), the second order solvability condition and the second order vector of state \t2. The12 Lehmann et al.\nsecond order inhomogeneity, expanded to leading order in x, reads\nN2(\t1;\t1) =0\nBB@\u00008ix2\n4ix\n\u00002x1\nCCA\u0001exp\u001a\n2i\u0014Zk\n\u000fdx\u0000!t+m\u0012\u0015\u001b\nA(\u0018)2\n+0\nBB@0\n0\n3\f1\u00170x21\nCCA\u0001exp\u001aZ\nik\n\u000fdx\u0000i!t+im\u0012\u001b\nA(\u0018)\n+0\nBB@0\n\u00008\u00004\n3+\r\u0001\n\u00170x3\n4x1\nCCA\u0001jA(\u0018)j2+c:c: :(49)\nTo obtain a solution for \t2from theO\u0000\nj\u000e\u0017j2\u0001\nequation in (29), the corresponding solvability condition (33) must be\nful\flled. Considering the de\fnition of the scalar product (34) and the adjoint null solution (46), it is clear that after\nevaluating the scalar product, the only remaining terms are those proportional to A(\u0018) in (49). Thus, the solvability\ncondition reads\n3\f1\u00170x2= 0; (50)\nfrom which follows:\n\f1= 0: (51)\nThis is consistent with (26a). The same relation follows from the exact second order solvability condition, which is\nalso proportional to \f1and which will not be displayed here, as explained above. With (51) being satis\fed, we are\nnow able to obtain a particular solution for \t2.\nConsidering (49) and (51) the particular solution will take the form\n\t2;p=\nA(\u0018)2A\t2a(x) exp\u001a\n2i\u0014Zk\n\u000fdx\u0000!t+m\u0012\u0015\u001b\n+jA(\u0018)j2A\t2b(x)\n+c:c: :(52)\nExpression (52) contains six unknown quantities which can be computed by evaluating the three O\u0000\nj\u000e\u0017j2\u0001\nequa-\ntions (29) for the oscillatory terms ( \u0018A(\u0018)2exp\b\n2i\u0002Rk\n\u000fdx\u0000!t+m\u0012\u0003\t\n) and the non-oscillatory ( \u0018jA (\u0018)j2) terms\nseparately. As a result, to leading order in xit is found\nA\t2a(x) =0\nBB@4Dx\n4x\n2ix1\nCCA;\nA\t2b(x) =0\nBB@0\n8x\n4\u00004\n3+\r\u0001\n\u00170x31\nCCA:(53)\nAgain, higher order corrections in x\u001c1 are omitted here.\nOne notes that the phase shifts between the components of (53) are the same as for the components of (45). With theDamping of Nonlinear Density Waves in Dense Planetary Rings 13\nparticular solution (53), the full solution of ^L\t2=N2(\t1;\t1) is given by\n\t2=\nA(\u0018)2A\t2a(x) exp\u001a\n2i\u0014Zk\n\u000fdx\u0000!t+m\u0012\u0015\u001b\n+jA(\u0018)j2A\t2b(x)\n+CA(\u0018)A\t1(x) exp\u001aZ\nik\n\u000fdx+im\u0012\u0000i!t\u001b\n+c:c:(54)\nwith an arbitrary constant C. The solution contains the second harmonics as well as non-wave contributions for the\nvelocitiesu,v. It turns out that the contribution proportional to C, which contains the null solution (45), does not\ncontribute to the amplitude equation which will be derived in the following section. Therefore we may choose C= 0.\n4.6. Third Order Solvability Condition and Amplitude Equation\nWith the solutions for \t1and\t2, we are able to compute the third order inhomogeneity N3(\t1;\t2) (cf. Appendix\nA). Before proceeding we note that it is not necessary to compute the third order vector of state \t3, which would\nrequire a complete solution of the third order equations. We merely need to evaluate the third order solvability\ncondition\nh\t0adjN3(\t1;\t2) +@\u0018(^M\t1)i= 0; (55)\nwhich already yields the desired equation of the form (17) for A(\u0018), i.e. explicit expressions for the coe\u000ecient functions\ng(x) andl(x). The result is\ndA\nd\u0018=\u000e2\n\u0017[gr(x) +igi(x)]A\u0000[lr(x) +ili(x)]AjAj2(56)\nwith real-valued coe\u000ecient functions gr(x),gi(x),lr(x) andli(x) denoting the real and imaginary parts of the\ncoe\u000ecients, respectively. Note that the consistency of the multiple scale expansion requires the condition (26b), which\nwas used to arrive at (56), as well as the identity sgn\u0000\n\u000e2\n\u0017\u0001\nj\u000e2\n\u0017j=\u000e2\n\u0017. The coe\u000ecient functions to leading order in xare\ngiven by\ngr(x) =(3\r\u00002)\u00170\n3D\u000fx2\u0011^grx2; (57a)\ngi(x) =(3\r\u00002)\u00172\n0\n3D\u000fx4\u0011^gix4; (57b)\nlr(x) =\u0000 \n4\n\u000f\u00004\u0000\n589 + 204\r+ 9\r2\u0001\n\u00170\n81D\u000f!\nx4\u0011^lrx4; (57c)\nli(x) =4x3\n\u000f\u0011^lix3; (57d)\nwhere the second equations de\fne the constants ^ gr, ^gi,^lrand^li. Returning to un-scaled units j\u000e\u0017j2@\u0018=@xand\nAj\u000e\u0017j=~A, we \fnally write\nd~A\ndx=\u000e2\n\u0017\u0002\n^grx2+i^gix4\u0003~A\u0000h\n^lrx4+i^lix3i\n~Aj~Aj2: (58)\nWe see that the real part of the linear coe\u000ecient in this equation is indeed equivalent to (21). Dropping the tildes and\nwriting for the complex amplitude A(x)\u0011jAj (x) expfi\u0002 (x)g, we arrive at\ndjAj\ndx=\u000e2\n\u0017^grx2jAj\u0000 ^lrx4jAj3; (59a)\nd\u0002\ndx=\u000e2\n\u0017^gix4\u0000^lix3jAj2: (59b)14 Lehmann et al.\nAn exact solution of (59a) is derived in Appendix C. It reads in terms of the initial amplitude jA0j\njAj(x) =jA0jexp\b1\n3\u000e2\n\u0017^grx3\t\nq\n2^lrjA0j2Rx\n0t4exp\b2\n3\u000e2\u0017^grt3\t\ndt+ 1: (60)\nThe precise behavior of the solution is fairly complicated due to the x-dependency of the coe\u000ecients. In the case of a\nviscously overstable ring ( \u000e2\n\u0017>0) the solution (60) \frst grows exponentially for x&0:\njAj(x)\u0019jA 0jexp\u001a1\n3\u000e2\n\u0017^grx3\u001b\n: (61)\nForx&x\u0003, wherex\u0003denotes a critical distance which depends on the used model parameters, the amplitude closely\nfollows the power law\njAj(x)\u0019s\n\u000e2\u0017^gr\n^lr1\nx: (62)\nValues ofx\u0003lie in the range 10\u00003\u001810\u00002for realistic model parameters, as shown below.\nWe can relate initial values for the wave amplitude A0to the linear (inviscid) satellite torque Tsby using Eq. (30)\nin Goldreich and Tremaine (1979)\nTs=\u0000mrL[4D(\u000frL\nL)2A0]2\n4G: (63)\nThis expression is valid to leading order in xand we also used (45). It results from the angular momentum luminosity\ncarried by a long trailing wave with constant amplitude A0in the linear inviscid theory . This is also the value of\nthe accumulated linear satellite torque, since in the linear inviscid approximation, the wave transports away all the\nangular momentum which is excited at the resonance by the satellite.\nFurther, with (60) we are able to solve for the nonlinear phase shift given by Eq. (59b):\n\u0002 (x) = \u0002 0\u0000^liZx\n0t3jAj2(t) dt (64)\nwith \u0002 0= \u0002 (0). Thus, nonlinearity introduces a phase shift in addition to the radial phase functionRk\n\u000fdx=x2\n2\u000fof\nthe long trailing density wave [cf. Eq. (35)]. With\n\t1=jAj(x)A\t1(x) exp\u001a\ni\u0002 (x) +Z\nik\n\u000fdx+im\u0012\u0000i!t\u001b\n+c:c:\nwe can de\fne the nonlinear wavenumber\nknl=k+\u000fd\u0002\ndx: (65)\nFigure 1 shows example plots for knlfor di\u000berent satellite torque values. The plots show that local nonlinear e\u000bects of\nself-gravity, expressed through the coe\u000ecient li(x) (57d), give rise to a reduction of the local wavenumber. The waves\nin this plot are linearly unstable and their amplitudes follow the relation (62) for larger distances from resonance (here\nr\u0000rL&500km). Therefore their wave numbers depart from the linear limit with growing distance from resonance.\nLet us now consider the \fnal amplitude equation\ndjAj\ndx=\u000e2\n\u0017(3\r\u00002)\u00170\n3D\u000fx2jAj\n+ \n4\n\u000f\u00004\u0000\n589 + 204\r+ 9\r2\u0001\n\u00170\n81D\u000f!\nx4jAj3:(66)\nFor a viscously overstable ring , the linear instability of the density wave manifests through the \frst term with \u000e2\n\u0017>0,\ncorresponding to \f >\fc[cf. (22)]. The case of \\linear viscous damping\" is described by values \u000e2\n\u0017<0, correspondingDamping of Nonlinear Density Waves in Dense Planetary Rings 15\n \n0 50 100 150 200\nr-rL [km]012345knl [1/km](σ/σ0)max =  1.53\n(σ/σ0)max =  2.22\n(σ/σ0)max =  3.16\nFigure 1 : Nonlinear dispersion relations (65) for density waves with increasing torque values at resonance Ts=\n\u00006:7\u0001108kg m2s\u00002(red);\u00001:1\u00011010kg m2s\u00002(green );\u00004:3\u00011010kg m2s\u00002(blue). We used the \u001c15-parameters (from\nTable 1) with rL= 108m,\u001b0= 350 kg m\u00002andm= 4. The torques Tsfollow from chosen initial amplitudes A0\nthrough (63). The dashed line is the linear dispersion relation for reference. Also indicated are the maximal values\nof the density contrast for these nonlinear cases. The wave assumes these values at those radial distances where the\ndeviations from the linear dispersion relation are highest.\nto\f <\fc. Nonlinear damping, described by the cubic term /jAj3, is dominated physically by viscous terms as well.\nThe \frst term in the bracket is purely self-gravitational and positive. Hence, in the limit of small viscosities, this term\nwould, theoretically, cause a nonlinear instability for large values3ofx. However, for realistic values of the viscosity\nthe self-gravity term is negligible compared to the viscous contribution, which has negative sign. Note that in Eq.\n(66) the relative magnitude of the nonlinear term /jAj3(strongly) depends on the distance to the Lindblad resonance\n(/x4), and not only on the magnitude of Aitself. This x-dependency, and also those of the other coe\u000ecients arise\nfrom the radial dependence of the scaled wavenumber k(x) =x.\nIn conclusion, Eq. (66) describes the damping of a density wave under the in\ruence of density dependent viscosities\nin the weakly nonlinear regime. It is a generalization of the linear viscous damping relation with constant viscosities\n(6). A consequence of Eq. (62) is that if the condition for viscous overstability ( \u000e2\n\u0017>0) is ful\flled, the surface density\nperturbation of the weakly nonlinear model does not decay to zero but rather saturates to a \fnite value at large\ndistance from the resonance. Namely, the WKB-solution (B5) of Poisson's equation and the \frst component of the\nnull vector (45) yield the following expression for the \frst order density perturbation:\n\u001b1=\u00004xjAj(x) sin\u0012x2\n2\u000f+ \u0002 (x) +m\u0012\u0000!t\u0013\n: (67)\nWith the asymptotic solution for jAjfor largexgiven by (62), it follows that the amplitude of the unscaled surface\ndensity perturbation \u001b(x) saturates to a constant value\n\u001b(x!1 )\u00194\u001b0q\n\u000e2\u0017^gr=^lr: (68)\nThe saturation to this value occurs for smaller xif\fincreases.\n3which would require the derivation of a stabilizing quintic term in the amplitude equation.16 Lehmann et al.\nIn Figure 2 numerical solutions of the amplitude equation (66) are plotted for di\u000berent parameter sets. These\nparameter sets were obtained in N-body simulations [Salo et al. (2001), Schmidt and Salo (2003)] and are listed in\nTable 1. One notes that the values of \u000e\u0017(also provided in Table 1) corresponding to these parameters are signi\fcant\nfractions of unity. Thus, we expect our model to be qualitatively correct for these parameters, though not quantitatively.\nTurning to the discussion of Figure 2, the dashed green curve represents the linear viscous damping relation, which\nresults from the limit of a vanishing density dependence of viscosities. This is an unrealistic assumption for dense\nplanetary rings, as outlined in Section 2. In the nonlinear cases, the amplitude decays substantially slower. The\nN-body parameter sets with optical depths \u001c= 1:4,\u001c= 1:5 and\u001c= 2 and corresponding viscous parameters \f= 1:03,\n\f= 1:06 and\f= 1:16, ful\fll the (hydrodynamic) condition for viscous overstability. For these cases we observe a\nturnover to a power law damping relation (62). This turnover occurs for smaller x, the larger the value of \f, or,\nequivalently, of \u001c. The parameter set with \u001c= 1:0 (and\f= 0:85) does not exhibit overstability, since \f < \fc.\nFor this linearly stable case, the damping behavior follows the nonlinear solutions with \f > \fcfor a certain range,\nbut eventually turns into an exponential decay. The reason for this nonlinear behavior for small xis that the initial\namplitudejA0jis chosen fairly high, such that the wave becomes already nonlinear within a few wavelengths from\nresonance.\n \n0.000 0.001 0.010\nx0.11.010.0100.0|A|(x)\nτ10τ14τ15τ20\nNonlin. ( β = 1.16)\nNonlin. ( β = 1.06)\nNonlin. ( β = 1.03)\nNonlin. ( β = 0.85)\nLin. ( η = const)\nPowerlaw (x-1)\nFigure 2 : Numerical solutions of the nonlinear amplitude equation (66) for the parameter sets listed in Table 1 with\nm= 4. Further, we used rL= 108m and\u001b0= 350 kg m\u00002. The green dashed line shows the linear viscous damping\nrelation in the limit of constant viscosity ( \u00170and\rfrom the\u001c20-parameter set). Note that for the \u001c10-parameters\nthe condition for viscous overstability is not ful\flled and eventually the amplitude damps exponentially. The initial\namplitudejA0j= 100 atx= 0 corresponds to a satellite torque Ts=\u00009:54\u0001108kg m2s\u00002.\nThe amplitudejAjis related to the nonlinearity parameter qof a streamline model by\nq= 4xjAj (69)\nwhich follows from equations (7) and (8a) in Longaretti and Borderies (1986) and Eq. (67) of this paper. In the\nstreamline model, perturbed ring matter is described in terms of eccentric streamlines and qmeasures the radial\ndisplacement of adjacent streamlines, relative to their unperturbed distance. Generally we have 0 \u0014q < 1. The\nunperturbed state corresponds to q= 0, whereas in a strongly nonlinear wave one \fnds q.1. For a detailed\ndescription of the streamline formalism we refer to the papers listed in Section 1.\nRelation (69) is, strictly speaking, valid only in the weakly nonlinear limit. In this limit qis equal to the amplitude of\nthe \frst order density perturbation (67). In Figure 3 we present the radial pro\fles of qcorresponding to the amplitude\nsolutions in Figure 2. One notes a saturation of qfor those parameters that ful\fll the condition for viscous overstability.Damping of Nonlinear Density Waves in Dense Planetary Rings 17\nFurther, Fig. 4 shows the hydrodynamic \feld quantities corresponding to the case \f= 1:06 in Figure 2. The surface\n \n0.000 0.001 0.010\nx0.00.20.40.60.81.0q(x)\nτ10τ14τ15τ20Nonlin. ( β = 1.16)\nNonlin. ( β = 1.06)\nNonlin. ( β = 1.03)\nNonlin. ( β = 0.85)\nLin. ( η = const)\nFigure 3 : Radial pro\fles of the nonlinearity parameter qcorresponding to the amplitude solutions in Fig. 2.\ndensity oscillations persist inde\fnitely. In the same \fgure, fSGis the scaled self-gravity force per unit mass\nfSG(x) =\u0000@\u001e\n@r\n=iD\n\u000f\u001b(x) +c:c:(70)\nwhere we used the solution of the Poisson Eq. (B8). Recall that Dis scaled with \n2\nL. The plots in Figure 4 are the\nsecond order representation of the vector of state (27), where we used (35), (42), (45), (53) and (54). As a result,\nthe density pro\fles are not entirely smooth, particularly in the density minima, lacking corrections by the orders\n>2. Nevertheless, the second order contribution in (27) correctly leads to a \rattening of the density minima and a\nsharpening of the maxima. This is in contrast to the density wave pro\fles that result from `streamline models' which\nare based on a Lagrangian equation of continuity. However, the wave amplitude which is computed from Eq. (58), is\nnot a\u000bected by the restriction on second order harmonics. In Section 6 we compare our weakly nonlinear model with\nthe streamline model applied in Borderies et al. (1986).\n4.7. Non-WKB E\u000bects of Self-Gravity\nThe model derived in the previous sections includes self-gravity e\u000bects that go beyond the WKB-approximation. In\nAppendix B we solve the Poisson equation while including the e\u000bects of the slow length scale, i.e. the slow change of\namplitude. The solution (B9c) includes the term\nis\u000f\nD@\u001e1\n@\u0018(71)\nwhich describes the slow amplitude-related change of the self-gravity potential (in the lowest approximation). In\ncontrast, the WKB-result involves only derivatives of the rapidly varying phase of the potential. We generally observe\nthat this correction to the WKB-order gives rise to shorter damping lengths of density wave pro\fles. An example is\nshown in Figure 5.18 Lehmann et al.\n \n    \n 0.61.01.41.8σ\n \n    \n -6000600u\n \n    \n -3000300v\n \n0 100 200 300\nr-rL [km]-404fSG\nFigure 4 : Scaled hydrodynamic quantities to order j\u000e\u0017j2related to the density wave corresponding to the case\n(\u001c= 1:5;\f= 1:06) in Figure 2. Also shown is fSG, the scaled self-gravity force, de\fned in (70).\n.\n5.NONLINEAR DAMPING OF FORCED DENSITY WAVES\nIn this section we include the forcing by an external satellite in our analysis of Eqs. (14) and derive a nonlinear\namplitude equation describing the propagation of forced density waves subject to viscous stress.\nThe wave is excited by one particular Fourier mode of the potential of an orbiting satellite. We neglect orbital\neccentricity and inclination of the satellite. Thus, we consider only \frst order resonances of the type m:m\u00001. This\nrestriction is made for the sake of simplicity of the calculations and does not occlude any physical aspects that are\ninvestigated here. We adopt again the scalings of quantities and parameters as given in Table (1).\nWe can apply the multiple scale analysis presented in Section 4 to resonantly forced density waves with slight\nmodi\fcations in the derivation. The scaled resonant mode of the forcing satellite potential reads\n\u001es(x;\u0012;t ) =\u0000GMs\na\u000f2r2\nL\n2\nLbm\n12(x) exp (im\u0012\u0000i!t)\n+c:c:(72)\nwhereais the satellite's semi major axis and Msits mass. Further, bm\n12(x) is a Laplace-coe\u000ecient:\nbm\n12(x) =2\n\u0019Z\u0019\n0d\tcos (m\t)q\n1 +\u001a(x)2\u00002\u001a(x) cos (\t)(73)\nwith\n\u001a=r\na;andr=rL(1 +x): (74)\nNote that we evaluate the satellite forcing terms in Eqs. (14) at the resonance location with \u001a(x= 0) =rL=a=\n[(m\u00001)=m]2=3(Goldreich and Tremaine (1978)).\nThe de\fnition of the expansion parameter now reads\n\u000es=\u0014GMs\na\u000fr2\nL\n2\nL\u00151=3\n(75)Damping of Nonlinear Density Waves in Dense Planetary Rings 19\n \n0 100 200 300\nr-rL [km]0.51.01.52.0σWKB\nNon-WKB\nFigure 5 : Illustration of the self-gravity correction due to amplitude change on the density wave pro\fle. The\nparameters are the \u001c15-parameters with m= 4 andTs=\u00009:54\u0001108kg m2s\u00002. The wave pro\fle labeled \\Non-WKB\"\nhas been computed by using the coe\u000ecient functions (57a-d). For the pro\fle labeled \\WKB\" we re-derived the\ncoe\u000ecient functions from a multiple scale expansion without the corrective term (71). Further, we used rL= 108m\nand\u001b0= 350 kg m\u00002. Note that small cusps in the density minima result from the omission of higher-order terms in\nour theory (see the discussion at the end of Section 4.6).\nwhich describes the ratio of satellite forcing to the self-gravity force. This parameter is very similar to the forcing\nparameter fin Eq. (36) in Shu et al. (1985a). Strictly, it must have values much smaller than unity to ensure\nthe validity of the weakly nonlinear analysis. Using the values rL= 96248 km (Hedman and Nicholson (2016)),\nMS= 1:898\u00011018kg (Jacobson et al. (2008)) and m= 2, corresponding to the (strongly nonlinear) Janus 2:1 density\nwave, as well as \u001b0= 600 kg m\u00002, one obtains \u000es= 0:47. The linear (inviscid) satellite torque value corresponding to\nthis forcing strength is TS=\u00003:61\u00011011kg m2s\u00002, obtained with Eq. (89) below. For such large value of \u000eswe do\nnot expect our model to produce quantitatively correct results. For illustrative examples which follow below we will\nuse the above values for rLand\u001b0in combination with various viscosity parameters and di\u000berent values of \u000es.\nWe now expand the vector of state in powers of the parameter \u000esas we did in Eqs. (27), (28) with the parameter\n\u000e\u0017. Since\u000es>0 always, we do not need to work with its absolute value. The forcing terms will appear now at O\u0000\n\u000e3\ns\u0001\nof the expansion. This is desirable since we want to obtain an equation describing the damping of forced waves, as we\nalready know that nonlinear viscous damping occurs at this order. We again expand the viscous parameter [cf. (8),\n(23e)]\n\f=\fc+\u000es\f1+\u000e2\ns\f2+\u0001\u0001\u0001: (76)\nAs in Section 4.5, the second order solvability condition yields \f1= 0. However, important to note is that in the\ncurrent situation \f2is not restricted (by the de\fnition of the expansion parameter \u000es) to yield\f2=\fc, as it was\nthe case in Section 4.3. But the contribution \u000e2\ns\f2should strictly be much smaller than \fcfor the consistency of the\nexpansion (76). We can de\fne \f2such that\n\u000e2\ns\f2\u0011\u000e2\n\u0017\fc (77)\nwhere the free parameter \u000e\u0017is identical to (22) and controls the distance of the system to the threshold for viscous\noverstability. With these de\fnitions the derivations of the vector of state components at O(\u000es) andO\u0000\n\u000e2\ns\u0001\nyield the\nsame results as in Sections 4.4 and 4.5. However, at O\u0000\n\u000e3\ns\u0001\n, we obtain a modi\fed inhomogeneity\nNf\n3=N3(\t1;\t2) +f (78)20 Lehmann et al.\nwhere\nf=0\nBB@0\n[@xbm\n12(x)]rLcos (m\u0012\u0000!t)\n\u0000m[bm\n12(x)]rLsin (m\u0012\u0000!t)1\nCCA(79)\ndescribes the satellite forcing. From the inhomogeneity (78) we arrive at a modi\fed third order solvability condition,\nwhich now includes the additional term\nF(x) =1\n2\u0019Z2\u0019\n0d\u0012exp\u001a\n\u0000Z\nik\n\u000fdx+im\u0012\u0000i!t\u001b\nhA\t0ad(x)jfi: (80)\nwith the adjoint null vector A\t0ad(x) given by (48) and where the scalar product h\u0001\u0001\u0001j\u0001\u0001\u0001i is de\fned through (34).\nThus, compared to the free wave analysis in Section 4, the nonlinear amplitude equation is changed according to\ndA\nd\u0018!dA\nd\u0018\u0000F(x). Evaluating the integral over \u0012in (80) yields\nF(x) =[!@xbm\n12(x)\u0000m\n (2bm\n12(x) +@xbm\n12(x))]rL\n4i\n\u0001exp\u001a\n\u0000Z\nik\n\u000fdx\u001b\n:(81)\nReturning to ordinary length scale @\u0018!(1=\u000e2\ns)@x,A! (1=\u000es)~A, dropping the tildes and using the relations (41),\n(42), (43a) and (43b) for a long trailing density wave in the vicinity of an ILR, the nonlinear amplitude equation now\nreads\ndA\ndx= [fr(x) +ifi(x)] +\u000e2\n\u0017[gr(x) +igi(x)]A\n\u0000[lr(x) +ili(x)]jAj2A;(82)\nwhere the leading orders of the coe\u000ecient functions are given by\nfr(x) =\u0000@x\u001es(rL) + 2m\u001es(rL)\n4Dsin\u0012x2\n2\u000f\u0013\n; (83a)\nfi(x) =\u0000@x\u001es(rL) + 2m\u001es(rL)\n4Dcos\u0012x2\n2\u000f\u0013\n; (83b)\ngr(x) =(3\r\u00002)\u00170\n3D\u000fx2; (83c)\ngi(x) =(3\r\u00002)\u00172\n0\n3D\u000fx4; (83d)\nlr(x) =\u0000 \n4\n\u000f\u00004\u0000\n589 + 204\r+ 9\r2\u0001\n\u00170\n81D\u000f!\nx4; (83e)\nli(x) =4x3\n\u000f: (83f)\nClearly, the functions gr,gi,lrandliare equivalent to (57a)-(57d). For convenience we inserted the wavenumber k=x\nand solved the phase integral exp\b\n\u0000R\nik\n\u000fdx\t\nwhile assuming zero value at the resonance. The magnitude jAj(x) is\nde\fned asjAj(x) =h\nAr(x)2+Ai(x)2i1=2\n, where the terms on the right hand side denote the real and imaginary\nparts ofA(x). If we split Eq. (82) into its real and imaginary parts, we obtain the two equations\ndAr\ndx=fr(x) +\u000e2\n\u0017[gr(x)Ar\u0000gi(x)Ai]\n\u0000jAj2[lr(x)Ar\u0000li(x)Ai];(84a)Damping of Nonlinear Density Waves in Dense Planetary Rings 21\ndAi\ndx=fi(x) +\u000e2\n\u0017[gi(x)Ar+gr(x)Ai]\n\u0000jAj2[lr(x)Ai+li(x)Ar]:(84b)\nThese coupled equations have to be solved simultaneously under the restriction of a given boundary condition. From\nthe linear theory of satellite forcing (Goldreich and Tremaine (1979); Shu (1984)) we know that a suitable condition\nis given byA(x!\u00001 ) = 0.\nWith equation (82) we are now able to describe the excitation of the nonlinear density waves whose propagation and\ndamping behavior we derived in Section 4. An expression for the (weakly) nonlinear satellite torque can be derived as\nfollows. Recall that the angular momentum luminosity carried by a free wave with amplitude Ais given through [cf.\nEq. (63)]\nL(x) =\u0000mrL\u0002\n4D(\u000frL\nL)2jAj\u00032\n4G: (85)\nLet us consider the radial derivative of Lby using (82)\ndL\ndx=\u0000mrL\u0002\n4D(\u000frL\nL)2\u00032\n4GdjAj2\ndx\n=\u0000mrL\u0002\n4D(\u000frL\nL)2\u00032\n4G\u0012dA\ndx\u0001A\u0003+dA\u0003\ndx\u0001A\u0013\n=\u0000mrL\u0002\n4D(\u000frL\nL)2\u00032\n2G\u0000\nfr(x)Ar+fi(x)Ai+\u000e2\n\u0017gr(x)jAj2\u0000lr(x)jAj4\u0001\n;(86)\nwhere a star denotes complex conjugate. From this equation we can identify the satellite torque density\nT=\u0000mrL\u0002\n4D(\u000frL\nL)2\u00032\n2G(fr(x)Ar+fi(x)Ai): (87)\nThe remaining terms in (86) describe viscous e\u000bects on the angular momentum luminosity of the wave. The torque\ndensity (87) results in an accumulated torque function\nT(x) =Zx\n\u00001T(^x) d^x (88)\nwhich denotes the accumulated satellite torque at radial location x. For all parameter regimes considered in this paper\nwe \fnd that the torque values T(x) for largexare nearly equal to the linear inviscid values (Goldreich and Tremaine\n(1979))\nTLin=\u0000m\u00192\u001b0\nD\n2\nL(\u000frL\nL)4[@x\u001es\u00002m\u001es]2\nrL(89)\nwithin the accuracy of integration, as is expected (Shu et al. (1985a)). Figure 6 shows the scaled accumulated torque\nT=TLinfor am= 2 density wave with forcing parameter \u000es= 0:37.\nIn analogy to the discussion at the end of Section 4 we are interested in solutions of the amplitude equations (84a),\n(84b) which belong to the following parameter regimes\n1. Density waves, damped by constant viscosities ( \f!\u00001).\n2. Linear, (marginally) stable density waves ( \f <\fc), damped by density dependent viscosities.\n3. (Weakly) nonlinear density waves that are (marginally) stable ( \f < \fc). Here the damping is caused both by\nlinear and nonlinear terms.\n4. (Weakly) nonlinear4density waves that are (marginally) unstable (\f >\fc). Here the damping is purely nonlinear\nand follows a power law behavior at large distances.\n4One should note that a wave of this category will eventually become nonlinear, regardless of the value of \u000es. Therefore we do not\ndistinguish linear from nonlinear waves in this case.22 Lehmann et al.\n \n-100 0 100 200\nr-rL [km]0.00.51.01.5T/TLin\nFigure 6 : The scaled accumulated torque following from numerically solving Eqs. (87)-(89) for a m= 2 density wave\nwith the\u001c15-parameters and \u000es= 0:37. Further, we used rL= 96248 km and \u001b0= 600 kg m\u00002.\nCase 1 is considered to illustrate the di\u000berences caused by the density dependence of viscosity on the wave damping and\nbecause such models have been applied to investigate density waves in Saturn's rings [Esposito et al. (1983); Tiscareno\net al. (2007); Colwell et al. (2009a)]. The wave pro\fles corresponding to case 1 are computed by solving the amplitude\nequations (84a), (84b) in the limit \f!\u00001 (cf. Section 2). Therefore their damping is a\u000bected also by the nonlinear\nterms in Eqs. (84a), (84b) for su\u000eciently large amplitude. The model parameters that separate the cases 2, 3 and\n4 are essentially the strength of satellite forcing \u000esand the distance of the ring state to the threshold for viscous\noverstability \u000e\u0017. Values of \u000es\u00181 correspond to strongly nonlinear waves, whereas, if \u000es\u001c1, a stable wave remains\nin the linear regime. Further, positive values \u000e2\n\u0017>0 imply (linear) viscous overstability while in the case \u000e2\n\u0017<0, the\nwave is viscously stable, which formally corresponds to an imaginary value of \u000e\u0017. To illustrate the e\u000bects of di\u000berent\nvalues of\u000esand\u000e\u0017, Figures 7 a, b and c show wave pro\fles for the cases 1, 2, 3 and 4. In these plots the waves damped\nby constant viscosity have substantially shorter damping lengths than those damped by density dependent viscosity.\nHowever, for other choices of model parameters the upper two waves in Figures 7 a, b, c, respectively may look fairly\nsimilar.\n6.COMPARISON WITH THE MODEL OF BGT86\nIn this section we compare the derived weakly nonlinear model (WNL hereafter) with the nonlinear streamline model\nof Borderies et al. (1986) (BGT86 hereafter). To this end we compute density wave pro\fles by applying the method\noutlined in Section IVa in BGT86 where we use the viscosity prescription (8). We perform these calculations to leading\norder of the parameter x[de\fned by (7)], similar to our calculations in Sections 4 and 5. In Appendix D we derive the\npressure tensor components and the viscous coe\u000ecients used in the streamline model.\nFigure 8 shows wave pro\fles derived from both models. From top to bottom panel the forcing strength \u000es[Eq. (75)]\ngradually increases from the linear to the strongly nonlinear regime. The given torques Tare scaled with the linear\ninviscid torque value of the Janus 2:1 resonance [ TS=\u00003:61\u00011011kg m2s\u00002, obtained from (89)], which corresponds\nto a strongly nonlinear wave. The satellite torques serve as input parameters for the BGT model, whereas the wave\nforcing in the WNL model is computed from Eq. (82). For all plots we use rL= 96248 km, \u001b0= 600 kg m\u00002and\nm= 2. For the three lowermost plots we use the \u001c15-parameters, but with a higher viscosity \u00170= 0:0025 m2s\u00001.\nThis value is slightly higher than what is predicted by relation (11) for ice particles at this saturnocentric distance,\nto ensure convergence of the solution procedure for equations (84a), (84b) for the lowermost case with T= 1. For the\ntwo uppermost plots we use the \u001c10-parameters with \u00170= 0:0025 m2s\u00001.\nThroughout the linear and weakly nonlinear regime we observe good agreement between the models. However, with\nincreasing forcing strength, the deviations become larger. This becomes most clear for the case T= 1. Here we observe\nsigni\fcant deviations in the region of maximal wave amplitudes. For the cases T= 0:25 andT= 1 one encounters\nnegative values for the surface mass densities in the WNL pro\fles. This indicates that the weakly nonlinear description\nbreaks down in the regions of maximal wave amplitudes for these cases (but higher order corrections might remedy\nthis shortcoming, see the discussion at the end of Section 4.6). One should note the di\u000berent plot ranges used inDamping of Nonlinear Density Waves in Dense Planetary Rings 23\n \n     \n 0.91.01.1σδs = 0.1 (TS = -3.6 ⋅ 107 kg m2 s-2)\nη = const\n \n     \n 0.91.01.1σδν2   = -0.31\n \n0 100 200 300 400\nr-rL [km]0.91.01.1σδν2   = 0.27\n(a)\n \n      \n 012σδs = 0.22 (TS = -3.6 ⋅ 109 kg m2 s-2)\nη = const\n \n      \n 012σδν2   = -0.31\n \n0 100 200 300 400 500\nr-rL [km]012σδν2   = 0.27\n(b)\n \n      \n 0123σδs = 0.3 (TS = -2.3 ⋅ 1010 kg m2 s-2)\nη = const\n \n      \n 0123σδν2   = -0.31\n \n0 100 200 300 400 500\nr-rL [km]0123σδν2   = 0.27\n(c)\nFigure 7 : Plots ofm= 2 density waves to illustrate the e\u000bects of increasing \u000esand\u000e\u0017. For each panel (a), (b) and\n(c), the pro\fles from top to bottom correspond to the regimes (1, 2, 4), (1, 3, 4) and (1, 3, 4), respectively. For the\n\frst wave in each panel we used \u00170and\rfrom the\u001c10-parameters. For the second and third waves we used the \u001c10-\nand\u001c20-parameters, respectively. Further, rL= 96248 km and \u001b0= 600 kg m\u00002was used.\nthe di\u000berent panels. In contrast, the saturation amplitudes, i.e. the near constant amplitudes at far distances from\nresonance in the cases with viscous overstability, agree very well, with a deviation of about 3% (not shown) for the\nstrongest waves with T= 1. This is remarkable, considering that the approaches behind the two models are entirely\ndi\u000berent.\nApart from the di\u000berences in the amplitude pro\fles, the waves also exhibit di\u000berent wavenumber dispersions. In\nFigure 9 we present Morlet wavelet spectrograms (Torrence and Compo (1998)) of the waves displayed in Fig. 8. In the\nlinear and weakly nonlinear regime, the wave numbers from both models closely follow the linear dispersion relation.24 Lehmann et al.\nIn the nonlinear regime, both models predict longer wavelengths in regions of large wave amplitudes (cf. Figure 1 in\nSection 4.6). To some extent, di\u000berences between the dispersion relations and density pro\fles of the two models arise\nbecause in the BGT model the background surface density changes with radial position, due to the forced conservation\nof angular momentum luminosity. This e\u000bect is not included in the WNL model. Therefore the BGT density wave\npro\fles in Figure 8 are scaled with the corresponding background surface densities \u001b0(r), whereas the WNL model\nwaves are scaled with the constant \u001b0= 600 kg m\u00002. The background densities for the BGT model waves are shown\nin Figure 10.\nFurther, Figure 11 displays the nonlinearity parameter qas a function of radial distance from resonance for the\nwaves. We use (69) to obtain qfor the WNL model waves. For the cases of overstable waves, qsaturates to a \fnite\nvalue. This saturation value is determined by the condition that the radial derivative of the angular momentum\nluminosity d L=dx, carried by the density wave, approaches zero at large distances. In the WNL model, this results in\nthe condition that the two viscous terms in (86) balance each other, which is ful\flled if jAjapproaches the limiting\nfunction (62). This occurs at large distances xwhere the in\ruence of the satellite torque is negligible. In the BGT\nmodel, the wave damping is described through [cf. (27) in BGT86]\ndL\ndx=\u00002\u0019mq\u001b 0(x)T1(q) (90)\nwith the viscous coe\u000ecient T1(q) [Eq. (17) in BGT86 and (D30) in Appendix D]. In the case of viscous overstability\n(\f > \fc) the quantityT1is positive for 0 < q < qcwith some \fnite value qc. This generates a linear instability of\nthe density wave in a similar way as the linear terms /jAj in the amplitude equations (58) and (82) if the condition\n\f >\fcis met. The critical value qcis approached as xbecomes larger. For the \u001c15-parameters one \fnds qc= 0:330 in\nthe BGT model. This behavior can be compared with the fact that the saturation amplitude (62) of the WNL model\nfor which d L=dx!0 corresponds to a qvalue of 4p\n\u000e2\u0017gr=lr, yieldingqc= 0:30 for the\u001c15-parameters. The good\nagreement of these values for qcis re\rected in the agreement of the saturation amplitudes of the overstable density\nwaves in Fig. 8.\nOne should note that the two most nonlinear cases in this comparison are strictly speaking outside the regime of\napplicability of the weakly nonlinear model, the surface density even becoming negative in some regions. The two\nstrongest waves possess values for qgreater than unity within some regions. This shows that in these cases, a second\norder description of the density perturbation is not su\u000ecient to quantitatively describe the wave pro\fle. Nevertheless,\nwe see that for all cases the wave envelope, which is the central quantity of the weakly nonlinear model, is in good\nor at least qualitative agreement with the BGT model, indicating that the amplitude equation remains qualitatively\nvalid.Damping of Nonlinear Density Waves in Dense Planetary Rings 25\n \n      \n 0.951.001.05σ / σ0T =  1.00e-04\nδs =  0.10BGT WNL\n \n      \n 0.71.01.4σ / σ0T =  1.00e-02\nδs =  0.22\n \n      \n 0.51.52.5σ / σ0T =  9.00e-02\nδs =  0.31\n \n      \n 024σ / σ0T =  2.50e-01\nδs =  0.37\n \n0 100 200 300 400 500\nr-rL [km]0246σ / σ0T =   1.0e+00\nδs =  0.47\nFigure 8 : Density wave pro\fles following from the WNL model and the BGT model respectively. Forcing strengths\nof the waves, expressed through \u000esand the scaled torque T, increase gradually from top to bottom. The BGT pro\fles\nhave been divided by the corresponding background surface densities (Fig. 10). In the two lowermost cases the surface\ndensities of the WNL model waves become negative where the amplitudes are largest. This is a consequence of the\nlimitation of the WNL model to second order harmonics of the primary wave solution (Section 4.6). For a quantitative\ndescription of the wave pro\fles of these strongly forced waves more higher harmonics must be included. Nevertheless,\nthe amplitude pro\fle derived by the model remains valid and is una\u000bected by the restriction on second order harmonics.\n7.SUMMARY AND DISCUSSION\nIn this paper we applied the multiple scale approach to derive a weakly nonlinear \ruid model for the excitation and\nviscous damping of spiral density waves in a dense planetary ring. The most important quantity obtained with this\nmodel is the evolution pro\fle of the amplitude of a density wave depending on the distance from resonance location.\nThe model takes into account nonlinearities which are present in the governing \ruid equations and which become\nimportant if the density perturbations are of the same order as the background value. This is the case for many of the\nobserved density waves in Saturn's main rings.\nA linear instability of a density wave arises if the condition for viscous overstability is ful\flled (Schmidt et al.\n(2016)). We \fnd that the damping of such overstable density waves occurs solely due to the nonlinear terms in the\nhydrodynamic balance equations. For large distance from the resonance we derive a power law damping for the scaled\nwave amplitude. As a consequence, the surface mass density perturbation in the model saturates to a \fnite value. In a\ntrue particulate ring one expects that the wave eventually disappears far from resonance. In contrast, an exponential\ndamping relation results from a linearized description with constant viscosity. In general, the resulting density wave26 Lehmann et al.\nFigure 9 : Morlet wavelet spectrograms of the model waves presented in Fig. 8. The dashed line represents the linear\ndispersion relation k=Dx\n2\u0019G\u001b 0for reference. As the satellite torque increases from top to bottom panels, departures\nfrom the linear dispersion relation increase. As already seen in Figure 1, nonlinearity tends to increase the wavelength\nof density waves. Similar to the wave amplitudes, we observe that di\u000berences between the models become larger on a\nquantitative level, as one enters the strongly nonlinear regime.\ndamping lengths are strongly dependent on the distance from the threshold of viscous overstability. We believe that\nthis dependence can in part explain the wide variety of damping lengths observed among waves in Saturn's rings, such\nas the waves at the \frst order resonances with the co-orbitals (Schmidt et al. (2016)).\nOur model predicts, in accordance with existing theories of nonlinear density waves, distinct features of strong waves\nin Saturn's rings. Among these are the sharp peaks and \rat troughs of the radial density pro\fles and the deviations\nfrom the linear dispersion relation in regions of strong wave amplitude. Moreover, our calculations show that longDamping of Nonlinear Density Waves in Dense Planetary Rings 27\n \n      \n 0.81.01.2σ0(r)T=  1.00e-04\n \n      \n 0.81.01.2σ0(r)T=  1.00e-02\n \n      \n 0.81.01.2σ0(r)T=  9.00e-02\n \n      \n 0.81.01.2σ0(r)T=  2.50e-01\n \n0 100 200 300 400 500\nr-rL [km]0.81.01.2σ0(r)T=   1.0e+00\nFigure 10 : Background surface density pro\fles for the waves of the BGT model displayed in Figure 8. These curves\nare scaled with the constant \u001b0= 600 kg m\u00002.\nrange self-gravity contributions, not present in linear WKB-approximation, have the tendency to reduce damping\nlengths.\nThe results from our new approach compare reasonably well with the traditional streamline approach to density\nwave theory. The largest deviations occur for strong forcing and in the highly nonlinear regime. The streamline\napproach is superior at matching the total wave pro\fle, while the newly derived amplitude equation in this paper is a\ncomparably handy tool to gain insight in the evolution of the wave amplitude with distance from resonance, and the\ndi\u000berent regimes of wave formation and the dependence on the parameters of the model.\n \n10 100 1000\nr-rL [km]0.00.51.01.5q(r)BGT\nWNL\nFigure 11 : Nonlinearity parameters for the waves displayed in Figure 8. Eq. (69) was used for the waves of the WNL\nmodel. Curves with higher values of qcorrespond to higher values of \u000esandT. Note that in regions where q>1 the\nWNL model breaks down and relation (69) is no longer a valid approximation for q(see Appendix D).28 Lehmann et al.\nA detailed quantitative reproduction of the observed strongly nonlinear waves in Saturn's rings can be achieved with\nneither one of the models. Both models rely on a isothermal (vertically averaged) \ruid approximation neglecting the\ne\u000bect of the wave on the local velocity dispersion of ring material and variations of the ring thickness with the wave\nphase. Also, the magnitude of gravitational wakes (and their contribution to the viscosity) will be in\ruenced in a\nmore or less complicated manner by the presence of a density wave. Further, we expect that a realistic description\nof self-gravity, which takes into account long range interactions more accurately, leads to further deviations from the\nmodel dispersion relation discussed in this paper.\nring regions simulation.\nOverstability in the inner B ring might explain the remarkable length ( >500 km) of the Janus 2:1 wave train (Colwell\net al. (2009b)). On the other hand, visible wave signatures extend to less than about 200 km for two waves propagating\nin the overstable region of the A ring (Atlas 7:6 and Pan 10:9, Hedman et al. (2014) Figure 5). This appears surprising\nin view of the results from our weakly nonlinear model and from the BGT model. However, in this regard it should\nbe noted that although the models describe the behavior of a density wave if the condition for viscous overstability\nis ful\flled, they do not take into account the presence of overstable oscillations in the wave region. 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Petzschmann, and H. Salo.\nStability analysis of a Keplerian disk of granular grains:\nin\ruence of thermal di\u000busion. Icarus , 145:657{660, 2000.30 Lehmann et al.\nL. J. Spilker, S. Pilorz, A. L. Lane, R. M. Nelson,\nB. Pollard, and C. T. Russell. Saturn A ring surface\nmass densities from spiral density wave dispersion\nbehavior. \"Icarus\" , 171:372{390, 2004.\nG. R. Stewart, D. N. C. Lin, and P. Bodenheimer.\nCollision-induced transport processes in planetary rings.\nIn R. Greenberg and A. Brahic, editors, Planetary Rings ,\npages 447{512, Tucson Arizona, 1984. Univ. of Arizona\nPress.\nF. S. Thomson, E. A. Marouf, G. L. Tyler, R. G. French,\nand N. J. Rappoport. Periodic microstructure in\nSaturn's rings A and B. GRL , 34:24203{+, 2007.M. S. Tiscareno, J. A. Burns, P. D. Nicholson, M. M.\nHedman, and C. C. Porco. Cassini imaging of Saturn's\nrings II: A wavelet technique for analysis of density waves\nand other radial structure in the rings. Icarus , 2007.\nC. Torrence and G. P. Compo. A Practical Guide to\nWavelet Analysis. Bulletin of the American\nMeteorological Society , 79:61{78, 1998.\nJ. Wisdom and S. Tremaine. Local simulations of\nplanetary rings. Astron. J. , 95:925{940, 1988.Damping of Nonlinear Density Waves in Dense Planetary Rings 31\nAPPENDIX\nA.THE NONLINEAR TERMS N2ANDN3\nThe nonlinear Terms N2andN3appearing in Eqs. (29) are given by\nN2(\t1;\t1) =0\nBB@N21\nN22\nN231\nCCA(A1)\nwith\nN21=\u000fk\nD@x(u1\u001e1);\nN22=\u0000\u000fu1@xu1+c2\u000fk2\nD2\u001e1@x\u001e1\n\u0000\u000f2\u000bk\u0017 0\nD\u0000\n[\fc+ 1]@x\u001e1@xu1+\fc\u001e1@2\nxu1\u0001\n;\nN23=\u0000\u000fu1@xv1\n\u0000\u000fk\u00170\n2D2\u0000\n@x\u001e1\u0000\n3\u0002\n\f2\nc\u00001\u0003\nk\n\u001e1+D(\u00003\f1\n + 2 [\fc+ 1]\u000f@xv1)\u0001\n+ 2\fcD\u000f\u001e1@2\nxv1\u0001(A2)\nand\nN3(\t1;\t2) =0\nBB@N31\nN32\nN331\nCCA(A3)\nwith\nN31=\u000f\nD(\u0000D@\u0018u1\u0000i\n@\u0012@\u0018\u001e1\u0000i@t@\u0018\u001e1) +k(@x[u2\u001e1] + 2@x[u1\u001e2]);\nN32=\u0000\u000f(@\u0018\u001e1+@x[u2u1])\n+c2\u000f\nD3\u0000\nD2k@\u0018\u001e1+k3\u001e2\n1@x\u001e1+ 2Dk2\u001e1@x\u001e2+D\u0000\n2k2\u001e2@x\u001e1\u0000iD\u000f@x@\u0018\u001e1\u0001\u0001\n+\u00170\u0012\n\u0000\u000f2\u000bk\nD2\u0000\n2 [\fc+ 1]D@x\u001e2@xu1+@x\u001e1\u0000\u0000\n\f1D+k\u0002\n1\u0000\f2\nc\u0003\n\u001e1\u0001\n@xu1+ [1 +\fc]D@xu2\u0001\u0001\n+\u000f2\u000b\u0012\n2@x@\u0018u1+k\n2D2\u0000\n\u00002\f1D\u001e1+ [\fc\u00001]\fck\u001e2\n1\u00004\fcD\u001e2\u0001\n@2\nxu1\u0000\fck\nD\u001e1@2\nxu2\u0013\u0013\n;\nN33=\u0000\u000f(u2@xv1+u1@xv2)\n+\u00170\u000f\n4D3\u0000\n6 [1 +\fc]D2k\n@\u0018\u001e1+k@x\u001e1\u0000\n3 [\fc\u00002]\u0002\n\f2\nc\u00001\u0003\nk2\n\u001e2\n1\n+ 4Dk\u001e1\u0000\n\u00003\f1\fc\n +\u0002\n\f2\nc\u00001\u0003\n\u000f@xv1\u0001\n+ 2D\u0000\n\u00006\u0002\n\f2\nc\u00001\u0003\nk\n\u001e2\n+D\u0000\n3sgn\u0000\n\u000e2\n\u0017\u0001\n\f2\n\u00002\f1\u000f@xv1\u00002 [\fc+ 1]\u000f@xv2\u0001\u0001\u0001\n\u00002D\u0000\n2k@x\u001e2\u0000\n\u00003\f1D\n + 3\u0002\n\f2\nc\u00001\u0003\nk\n\u001e1+ 2 [1 +\fc]D\u000f@xv1\u0001\n+\u000f\u0000\n3i[\fc+ 1]D\n@x@\u0018\u001e1\u00004D2@x@\u0018v1+k(\u001e1(2\f1D\u0000[\fc\u00001]\fck\u001e1) + 4\fcD\u001e2)@2\nxv1\n+2\fcDk\u001e1@2\nxv2\u0001\u0001\u0001\n:(A4)\nIn the above expressions we de\fned the constant \u000b= 4=3 +\r. Note that the solvability condition (51) leads to a\nsimpli\fcation of above terms.32 Lehmann et al.\nB.SOLUTION OF THE POISSON EQUATION\nTo conduct the multiple scale expansion of the nonlinear \ruid equations (14) one needs to \fnd the relationship\nbetween disk potential \u001eiand disk surface density \u001bifor the multiple scale orders i= 1;2;3. Poisson's equation for a\nthin disk reads (in unscaled form)\n1\nr@\n@r\u0012\nr@\u001e\n@r\u0013\n+1\nr2@2\u001e\n@\u00122+@2\u001e\n@z2= 4\u0019G\u001b\u000e (z): (B5)\nwith the Dirac delta function \u000e(z). If the surface density has the form\n\u001b(r;\u0012;t ) =A(r)\u0001exp(\niZr\nk(^r) d^r)\n\u0001expfi(m\u0012\u0000!t)g; (B6)\nwith rapidly varying phase, such that jkAj\u001d@rA, we can neglect any curvature terms in (B5) [which correspond to\ncorrections on the order of the small parameter ( kr)\u00001] and we are left with\n@2\u001e\n@r2+@2\u001e\n@z2= 4\u0019G\u001b\u000e (z): (B7)\nThe solution of (B7) ful\flling the correct boundary conditions ( \u001e!0 forjzj!1 ) is given by (Shu (1970))\n\u001b(r) =is\n2\u0019G@\u001e\n@r; (B8)\nin the plane z= 0 wheres= sgn (k) is the sign of the wavenumber kin (B6). We can now introduce the expansions\n(23) and (24) in (B8) and collect di\u000berent orders in j\u000e\u0017j. From this directly follows (in dimensionless form)\n\u001b1(x) =is\u000f\nD@\u001e1\n@x; (B9a)\n\u001b2(x) =is\u000f\nD@\u001e2\n@x; (B9b)\n\u001b3(x) =is\u000f\nD@\u001e3\n@x+is\u000f\nD@\u001e1\n@\u0018; (B9c)\nwithD= 3 (m\u00001). The second term in (B9c) accounts for the e\u000bect of the variation of the potential (with the wave\namplitude) on the length scale \u0018.\nFor the solution procedures of the \frst and second order equations in (29) which are presented in Sections 4.4 and\n4.5, respectively, one needs to evaluate the terms d \u001b=d\u001eas these appear in the linear operator (30) and its adjoint\n(31). From expression (35) for the \frst order vector of state directly follows @\u001e1=@x=ik\n\u000f\u001e1, so that with (B9a) we\n\fnd d\u001b=d\u001e=\u0000k\nDfor the \frst order equations. To solve the second order equations in (29) we make the (natural)\nassumption that the second order self gravity potential \u001e2is purely oscillatory and that it consists only of the second\nharmonic of the \frst order potential \u001e1, i.e.\n\u001e2\u0018A(\u0018)2exp\u001a\n2i\u0014Zk\n\u000fdx\u0000!t+m\u0012\u0015\u001b\n:\nWith this assumption one \fnds @\u001e2=@x=2ik\n\u000f\u001e2such hat d\u001b=d\u001e=\u00002k\nDis to be used for the second order equations\n^L\t2=N2(\t1;\t1). One notes that the so derived second order solution [Eqs. (52) and (53)], is consistent with this\nassumption. We do not attempt to \fnd the corresponding expressions for the third order equations since these are not\nrequired to solve (55) from which we obtain the amplitude equation (56).\nSince Poisson's equation is linear, it applies to di\u000berent Fourier-modes exp\b\nj\u0001R\nik\n\u000fdx\t\nseparately with j=\n\u00061;\u00062;\u00063. Hence, all relations that result from solving Poisson's equation apply to isolated modes. In this pa-\nper we use s= 1, since we restrict our analysis to trailing density waves with k>0.Damping of Nonlinear Density Waves in Dense Planetary Rings 33\nC.ANALYTICAL SOLUTION OF THE AMPLITUDE EQUATION\nBefore we attempt to \fnd the solution of the initial value problem\ndjAj\ndx=\u000e2\n\u0017gr(x)jAj\u0000lr(x)jAj3(C10)\nwithjAj(x= 0) =jA0j(derived in Section 4), we can assess its asymptotic behavior by computing the amplitudes\ncorresponding to \fxed points. These are obtained by solving\n0 =\u000e2\n\u0017gr(x)jAj\u0000lr(x)jAj3: (C11)\nAs in Section 4.6 we write gr(x)\u0011^grx2andlr(x)\u0011^lrx4and note that ^ gr>0,^lr>0 for all the parameter\nvalues considered in this paper. If \u000e2\n\u0017<0, corresponding to linear stability, the only \fxed point is jAj= 0 and the\namplitude will asymptotically approach this value for large x. In the case \u000e2\n\u0017>0 we additionally \fnd the \fxed point\njAjsat=q\n\u000e2\u0017^gr=^lrx\u00001and the amplitude will converge to this nonzero value for large x, unless the initial value is\njA0j= 0.\nTo solve (C10), consider the factorization ansatz\njAj=AlAnl;withAl;Anl>0: (C12)\nWe demand Alto satisfy the linear equation\ndAl\ndx=\u000e2\n\u0017^grx2Al: (C13)\nThe corresponding solution is\nAl=Al;0exp(Zx\n0\u000e2\n\u0017^grt2dt)\n\u0011Al;0exp\u001a1\n3\u000e2\n\u0017^grx3\u001b\n(C14)\nwithAl;0\u0011Al(x= 0). With solution (C14), Eq. (C10) can be written as\ndAnl\ndx=\u0000^lrx4A2\nl;0exp\u001a2\n3\u000e2\n\u0017^grx3\u001b\nA3\nnl: (C15)\nThis equation can be integrated with the result\nAnl=\"\n2^lrA2\nl;0Zx\n0t4exp\u001a2\n3\u000e2\n\u0017^grt3\u001b\ndt+1\nA2\nnl;0#\u00001=2\n; (C16)\nwhereAnl;0\u0011Anl(x= 0). Finally, with Eqs. (C14) and (C16), the \fnal solution of (59a) reads\njAj(x) =jA0jexp\b1\n3\u000e2\n\u0017^grx3\t\nq\n2^lrjA0j2Rx\n0t4exp\b2\n3\u000e2\u0017^grt3\t\ndt+ 1(C17)\n=\"\njA0j\u00002exp\u001a\n\u00002\n3\u000e2\n\u0017^grx3\u001b\n+^lr\n^gr\u000e2\u0017\u0012\nx2\u00002Zx\n0texp\u001a2\n3\u000e2\n\u0017^gr\u0000\nt3\u0000x3\u0001\u001b\u0013#\u00001=2\n(C18)\nwhere we de\fned jA0j\u0011Al;0Anl;0and used an integration by parts in the last step. Considering the behavior of\nthe solutions (C18), the \frst aspect to notice is that the \frst term is only relevant for linear waves, for which the\nin\ruence of nonlinearity, expressed through lris negligible. This will be the case as long as the amplitude remains\nmuch smaller than the \fxed point: jAj\u001cjAj satwhich is the saturation amplitude of a nonlinear wave. Thus, for\nsmall initial amplitudes and \u000e2\n\u0017<0 we obtain a linear density wave which damps exponentially due to the \frst term in\n(C18). If\u000e2\n\u0017>0 (which implies viscous overstability) and jA0jis small, the amplitude grows exponentially due to the\n\frst term as long as jAj\u001cjAj sat. As soon asjAjobtains values of the order of jAjsat, the second term proportional\ntolrbecomes signi\fcant and eventually damps the amplitude. For \u000e2\n\u0017>0 the integral function in (C18) is negative\nand has a single minimum. The minimum marks the turnover of the amplitude to a power law as it asymptotically\napproaches the \fxed point jAjsat. Forx!0 the integral behaves as \u0000x2such that nonlinear e\u000bects, represented by\nthe term in round brackets, vanish.34 Lehmann et al.\nD.PRESSURE TENSOR IN THE BGT86 MODEL\nWith Borderies et al. (1986) we assume that the ring dynamics is approximated by particles following streamlines\nof the form\nr=a[1\u0000e(a) cosE] (D1)\nin a cylindrical coordinate system ( r;\u001e) which rotates with angular frequency \n pand origin in the planet's center of\nmass. In this approximation\nE=m('+ \u0001 (a)) (D2)\nis the eccentric anomaly. In the usual notation ais the semi-major axis of the streamline, ethe eccentricity, mis the\nazimuthal mode number, and \u0001( a) is a phase angle. The horizontal compression of the ring material is obtained from\nJ(';a)\u0011@r\n@a\n= 1\u0000\u0014\ne+ade\nda\u0015\ncosE+maed\u0001\ndasinE\n\u00111\u0000qcosE0(D3)\nwith the de\fnitions\nE0=E+\r (D4)\nqcos\r=d(ae)\nda(D5)\nqsin\r=maed\u0001\nda: (D6)\nThe variable \rde\fned in this appendix should not be confused with the de\fnition of \relsewhere in this paper. The\nparameterqis the nonlinearity parameter. For q>1 streamlines start to cross. The streamlines close in this rotating\ncoordinate frame (in this frame the pattern produced by the streamlines looks stationary). In an inertial frame we\nhave the longitude \u0012='+ \nptand the streamlines are described by ellipses with precessing peri-apse angle $\nr=a[1\u0000e(a) cosf\u0012\u0000$g] (D7)\n\u0012=\u00120+ \nt (D8)\n$=$0+ _$t: (D9)\nHere \n is the mean motion and _ $is the precession rate. Comparing the two expressions (D1) and (D7) for the orbits\nyields\nm(\n\u0000\np) = \n\u0000_$ (D10)\nwhich is the condition for a Lindblad resonance and\n$0=\u00120(1\u0000m)\u0000m\u0001: (D11)\nIn order to compute the components of the stress tensor, we need the components of the radial and tangential velocities.\nParticles following streamlines have the velocities (see Eq. (33) and (34) of Borderies et al. (1985))\nur= \naesinE (D12)\nu'=r(\n\u0000\np+ 2\necosE):\nThe hydrodynamic pressure tensor is de\fned as\nP\u000b\f=p\u000e\u000b\f\u00002\u0011D\u000b\f\u0000\u000e\u000b\f\u0010~r\u0001~ u; (D13)Damping of Nonlinear Density Waves in Dense Planetary Rings 35\nwith the trace{less shear tensor\nD\u000b\f=1\n2\u0014\n@x\fu\u000b+@x\u000bu\f\u00002\n3\u000e\u000b\f~r\u0001~ u\u0015\n: (D14)\nIn these expressions pis the isotropic pressure and \u0011and\u0010are the dynamic shear and bulk viscosities, respectively.\nFurther,\u000e\u000b\fis the Kronecker symbol and ~ udenotes the velocity in the ring plane. The components of the pressure\ntensor we need are\nPrr=p+\u00122\n3\u0011\u0000\u0010\u0013\n~r\u0001~ u\u00002\u0011@rur (D15)\nPr'=\u00002\u0011Dr' (D16)\nwith\nDr'=1\n2\u00141\nr@'ur+@ru'\u0000u'\nr\u0015\n: (D17)\nFrom Eqs. (D12) and (D1) we \fnd\n1\nr@'ur=m\necosE=O(\ne;e2) (D18)\n@ru'=\u0014\n2J\u0000(\n + \np)\u0015\n+O(e2;\ne) (D19)\nu'\nr= (\n\u0000\np) +O(\ne;e2); (D20)\nwhere@r\u00111\nJ@awas used. Thus we have\nDr'=\n4J(1\u00004J); (D21)\n~r\u0001~ u\u00111\nr@r(rur) +1\nr@'u' (D22)\n=\nq\nJsinE0+O(\ne;e2): (D23)\nTherefore\nPrr=p\u0000\u00124\n3\u0011+\u0010\u0013\nq\nJsinE0+O(\ne;e2) (D24)\nPr'=\u0000\u0011\n2J(1\u00004J) +O(\ne;e2): (D25)\nWe de\fne\n\u001b(r) =\u001b0(r)\nJ(D26)\nwhere\u001b(r) (in the following \u001b) denotes the perturbed surface density as it has been used elsewhere in this paper.\nThe quantity \u001b0(r) is the radially dependent (background) surface density which is required to maintain the viscous\nangular momentum luminosity in the wave zone r>rLwhich di\u000bers from its value inside the resonance ( r<rL) due\nto the satellite torque which is excited at the resonance and the presence of the density wave [see Eqs. (24), (25) and\n(29) in BGT86]. The value of \u001b0(r) inside the resonance which equals the value of the unperturbed ring is denoted by\n\u001b\u0000\n0. We apply de\fnition (8) for viscosity and the isothermal ideal gas relation for pressure such that\n\u0011=\u001b\u00170\u0012\u001b\n\u001b\u0000\n0\u0013\f\n(D27)\nand\np=\u0014@p\n@\u001b\u0015\n0\u001b\n=p\u001b\u001b(D28)36 Lehmann et al.\nwherep\u001bassumes the values given in Table 1. Inserting (D26)-(D28) in the pressure tensor components (D24) and\n(D25) yields\nPrr=\u001bp\u001b\u0000\u001b\u00170\u0012\u001b\n\u001b\u0000\n0\u0013\f\u00124\n3+\r\u0013\nq\nJsinE0; (D29a)\nPr'=\u0000\u001b\u00170\u0012\u001b\n\u001b\u0000\n0\u0013\f\n2J(1\u00004J): (D29b)\nThe viscous coe\u000ecients needed for the computation of density wave pro\fles, according to the method described in\nSection IVa) in BGT86, are [Equations (17), (18) and (25) in BGT86]5\nT1\u00111\n2\u0019\u001b0(r)Z2\u0019\n0dE0[Prrsin;E0+ 2Pr'cosE0] (D30)\nT2\u00111\n2\u0019\u001b0(r)Z2\u0019\n0dE0[\u0000Prrcos;E0+ 2Pr'sinE0] (D31)\nar'\u00111\n2\u0019\u001b0(r)Z2\u0019\n0dE0Pr': (D32)\nThe behavior for small values of qcan be derived if the above integrals are evaluated with expressions for Prrand\nPr'that are linearized in q. To this extent we use the expressions\n\u0011=\u00170\u001b0\u0014\f+ 1\nJ\u0000\f\u0015\n(D33)\np=p\u001b\u001b0\nJ; (D34)\nfor viscosity and pressure which are valid for small q. Note that in the limit q!0 which occurs for r!rLwe have\n\u001b0(r) =\u001b\u0000\n0. Inserting (D33), (D34) in the pressure tensor components (D24), (D25) and linearizing in qyields\nPrr=p\u001b\u001b0+q\u0014\np\u001b\u001b0cosE0\u0000\u00124\n3+\r\u0013\n\u00170\n\u001b0sinE0\u0015\n(D35)\nPr'=1\n2\u00170\n\u001b0[3 +q(2 + 3\f) cosE0] (D36)\nThe integrals (D30)-(D32) then yield in scaled form (scalings from Table 1)\nT1=3\n2\u0014\n\f\u00001\n3\u0012\n\r\u00002\n3\u0013\u0015\n\u00170q (D37a)\nT2=\u00001\n2p\u001bq (D37b)\nar'=3\n2\u00170 (D37c)\nAs discussed in Section 6 linear instability of a density wave occurs if T1>0 for small q. From (D37a) we \fnd that\nthe criterion for instability equals the hydrodynamic for viscous overstability (10) [Schmit and Tscharnuter (1995)], in\nagreement with our model.\n5Note that the quantity \u0006 0(r) in BGT86 corresponds to our quantity \u001b0(r). The value \u001b\u0000\n0in our paper corresponds to \u0006 \u0000in BGT86."    },    {        "title": "1804.09242v1.Generalisation_of_Gilbert_damping_and_magnetic_inertia_parameter_as_a_series_of_higher_order_relativistic_terms.pdf",        "content": "Generalisation of Gilbert damping and magnetic\ninertia parameter as a series of higher-order\nrelativistic terms\nRitwik Mondalz, Marco Berritta and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-751 20\nUppsala, Sweden\nE-mail: ritwik.mondal@physics.uu.se\nAbstract. The phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion\nremains as the cornerstone of contemporary magnetisation dynamics studies, wherein\nthe Gilbert damping parameter has been attributed to \frst-order relativistic e\u000bects.\nTo include magnetic inertial e\u000bects the LLG equation has previously been extended\nwith a supplemental inertia term and the arising inertial dynamics has been related\nto second-order relativistic e\u000bects. Here we start from the relativistic Dirac equation\nand, performing a Foldy-Wouthuysen transformation, derive a generalised Pauli spin\nHamiltonian that contains relativistic correction terms to any higher order. Using the\nHeisenberg equation of spin motion we derive general relativistic expressions for the\ntensorial Gilbert damping and magnetic inertia parameters, and show that these ten-\nsors can be expressed as series of higher-order relativistic correction terms. We further\nshow that, in the case of a harmonic external driving \feld, these series can be summed\nand we provide closed analytical expressions for the Gilbert and inertial parameters\nthat are functions of the frequency of the driving \feld.\n1. Introduction\nSpin dynamics in magnetic systems has often been described by the phenomenological\nLandau-Lifshitz (LL) equation of motion of the following form [1]\n@M\n@t=\u0000\rM\u0002He\u000b\u0000\u0015M\u0002[M\u0002He\u000b]; (1)\nwhere\ris the gyromagnetic ratio, He\u000bis the e\u000bective magnetic \feld, and \u0015is an\nisotropic damping parameter. The \frst term describes the precession of the local,\nclassical magnetisation vector M(r;t) around the e\u000bective \feld He\u000b. The second term\ndescribes the magnetisation relaxation such that the magnetisation vector relaxes to the\ndirection of the e\u000bective \feld until \fnally it is aligned with the e\u000bective \feld. To include\nzPresent address: Department of Physics, University of Konstanz, D -78457 Konstanz, GermanyarXiv:1804.09242v1  [cond-mat.other]  3 Apr 20182\nlarge damping, the relaxation term in the LL equation was reformulated by Gilbert [2, 3]\nto give the Landau-Lifshitz-Gilbert (LLG) equation,\n@M\n@t=\u0000\rM\u0002He\u000b+\u000bM\u0002@M\n@t; (2)\nwhere\u000bis the Gilbert damping constant. Note that both damping parameters \u000band\u0015\nare here scalars, which corresponds to the assumption of an isotropic medium. Both the\nLL and LLG equations preserve the length of the magnetisation during the dynamics and\nare mathematically equivalent (see, e.g. [4]). Recently, there have also been attempts\nMHeff\nPrecession\nNutationDamping\nFigure 1. Sketch of extended LLG magnetisation dynamics. The green arrow denotes\nthe classical magnetisation vector which precesses around an e\u000bective \feld. The red\nsolid and dotted lines depict the precession and damping. The yellow path signi\fes\nthe nutation, or inertial damping, of the magnetisation vector.\nto investigate the magnetic inertial dynamics which is essentially an extension to the\nLLG equation with an additional term [5{7]. Phenomenologically this additional term of\nmagnetic inertial dynamics, M\u0002I@2M=@t2, can be seen as a torque due to second-order\ntime derivative of the magnetisation [8{11]. The essence of the terms in the extended\nLLG equation is described pictorially in Fig. 1. Note that in the LLG dynamics the\nmagnetisation is described as a classical vector \feld and not as a quantum spin vector.\nIn their original work, Landau and Lifshitz attributed the damping constant \u0015to\nrelativistic origins [1]; later on, it has been more speci\fcally attributed to spin-orbit\ncoupling [12{15]. In the last few decades, several explanations have been proposed\ntowards the origin of damping mechanisms, e.g., the breathing Fermi surface model\n[16, 17], torque-torque correlation model [18], scattering theory formulation [19], e\u000bective\n\feld theories [20] etc. On the other hand, the origin of magnetic inertia is less discussed\nin the literature, although it's application to ultrafast spin dynamics and switching\ncould potentially be rich [9]. To account for the magnetic inertia, the breathing Fermi\nsurface model has been extended [11, 21] and the inertia parameter has been associated\nwith the magnetic susceptibility [22]. However, the microscopic origins of both Gilbert3\ndamping and magnetic inertia are still under debate and pose a fundamental question\nthat requires to be further investigated.\nIn two recent works [23, 24], we have shown that both quantities are of relativistic\norigin. In particular, we derived the Gilbert damping dynamics from the relativistic\nspin-orbit coupling and showed that the damping parameter is not a scalar quantity\nbut rather a tensor that involves two main contributions: electronic and magnetic\nones [23]. The electronic contribution is calculated as an electronic states' expectation\nvalue of the product of di\u000berent components of position and momentum operators;\nhowever, the magnetic contribution is given by the imaginary part of the susceptibility\ntensor. In an another work, we have derived the magnetic inertial dynamics from a\nhigher-order (1 =c4) spin-orbit coupling and showed that the corresponding parameter\nis also a tensor which depends on the real part of the susceptibility [24]. Both these\ninvestigations used a semirelativistic expansion of the Dirac Hamiltonian employing the\nFoldy-Wouthuysen transformation to obtain an extended Pauli Hamiltonian including\nthe relativistic corrections [25, 26]. The thus-obtained semirelativistic Hamiltonian was\nthen used to calculate the magnetisation dynamics, especially for the derivation of the\nLLG equation and magnetic inertial dynamics.\nIn this article we use an extended approach towards a derivation of the\ngeneralisation of those two (Gilbert damping and magnetic inertia) parameters from\nthe relativistic Dirac Hamiltonian, developing a series to fully include the occurring\nhigher-order relativistic terms. To this end we start from the Dirac Hamiltonian in\nthe presence of an external electromagnetic \feld and derive a semirelativistic expansion\nof it. By doing so, we consider the direct \feld-spin coupling terms and show that\nthese terms can be written as a series of higher-order relativistic contributions. Using\nthe latter Hamiltonian, we derive the corresponding spin dynamics. Our results show\nthat the Gilbert damping parameter and inertia parameter can be expressed as a\nconvergent series of higher-order relativistic terms and we derive closed expressions\nfor both quantities. At the lowest order, we \fnd exactly the same tensorial quantities\nthat have been found in earlier works.\n2. Relativistic Hamiltonian Formulation\nTo describe a relativistic particle, we start with a Dirac particle [27] inside a material,\nand, in the presence of an external \feld, for which one can write the Dirac equation\nasi~@ (r;t)\n@t=H (r;t) for a Dirac bi-spinor  . Adopting furthermore the relativistic\ndensity functional theory (DFT) framework we write the corresponding Hamiltonian as\n[23{25]\nH=c\u000b\u0001(p\u0000eA) + (\f\u0000 1)mc2+V 1\n=O+ (\f\u0000 1)mc2+E; (3)\nwhereVis the e\u000bective unpolarised Kohn-Sham potential created by the ion-ion, ion-\nelectron and electron-electron interactions. Generally, to describe magnetic systems, an4\nadditional spin-polarised energy (exchange energy) term is required. However, we have\ntreated e\u000bects of the exchange \feld previously, and since it doesn't contribute to the\ndamping terms we do not consider it explicitly here (for details of the calculations\ninvolving the exchange potential, see Ref. [23, 25]). The e\u000bect of the external\nelectromagnetic \feld has been accounted through the vector potential, A(r;t),cde\fnes\nthe speed of light, mis particle's mass and 1is the 4\u00024 unit matrix. \u000band\fare the\nDirac matrices which have the form\n\u000b= \n0\u001b\n\u001b0!\n; \f = \n10\n0\u00001!\n;\nwhere\u001b= (\u001bx;\u001by;\u001bz) are the Pauli spin matrix vectors and 1is 2\u00022 unit matrix.\nNote that the Dirac matrices form the diagonal and o\u000b-diagonal matrix elements of\nthe Hamiltonian in Eq. (3). For example, the o\u000b-diagonal elements can be denoted as\nO=c\u000b\u0001(p\u0000eA), and the diagonal matrix elements can be written as E=V 1.\nIn the nonrelativistic limit, the Dirac Hamiltonian equals the Pauli Hamiltonian,\nsee e.g. [28]. In this respect, one has to consider that the Dirac bi-spinor can be written\nas\n (r;t) = \n\u001e(r;t)\n\u0011(r;t)!\n;\nwhere the upper \u001eand lower\u0011components have to be considered as \\large\" and \\small\"\ncomponents, respectively. This nonrelativistic limit is only valid for the case when the\nparticle's momentum is much smaller than the rest mass energy, otherwise it gives\nan unsatisfactory result [26]. Therefore, the issue of separating the wave functions of\nparticles from those of antiparticles is not clear for any given momentum. This is mainly\nbecause the o\u000b-diagonal Hamiltonian elements link the particle and antiparticle. The\nFoldy-Wouthuysen (FW) transformation [29] has been a very successful attempt to \fnd\na representation where the o\u000b-diagonal elements have been reduced in every step of the\ntransformation. Thereafter, neglecting the higher-order o\u000b-diagonal elements, one \fnds\nthe correct Hamiltonian that describes the particles e\u000eciently. The FW transformation\nis an unitary transformation obtained by suitably choosing the FW operator [29],\nUFW=\u0000i\n2mc2\fO: (4)\nThe minus sign in front of the operator is because of the property that \fandO\nanticommute with each other. With the FW operator, the FW transformation of the\nwave function adopts the form  0(r;t) =eiUFW (r;t) such that the probability density\nremains the same, j j2=j 0j2. In this way, the time-dependent FW transformed\nHamiltonian can be expressed as [26, 28, 30]\nHFW=eiUFW\u0012\nH\u0000i~@\n@t\u0013\ne\u0000iUFW+i~@\n@t: (5)5\nAccording to the Baker-Campbell-Hausdor\u000b formula, the above transformed Hamilto-\nnian can be written as a series of commutators, and the \fnally transformed Hamiltonian\nreads\nHFW=H+i\u0014\nUFW;H\u0000i~@\n@t\u0015\n+i2\n2!\u0014\nUFW;\u0014\nUFW;H\u0000i~@\n@t\u0015\u0015\n+i3\n3!\u0014\nUFW;\u0014\nUFW;\u0014\nUFW;H\u0000i~@\n@t\u0015\u0015\u0015\n+:::: : (6)\nIn general, for a time-independent FW transformation, one has to work with@UFW\n@t= 0.\nHowever, this is only valid if the odd operator does not contain any time dependency. In\nour case, a time-dependent transformation is needed as the vector potential is notably\ntime-varying. In this regard, we notice that the even operators and the term i~@=@t\ntransform in a similar way. Therefore, we de\fne a term Fsuch thatF=E\u0000i~@=@t.\nThe main theme of the FW transformation is to make the odd terms smaller in every\nstep of the transformation. After a fourth transformation and neglecting the higher\norder terms, the Hamiltonian with only the even terms can be shown to have the form\nas [26, 30{33]\nH000\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6+O6\n16m5c10\u0013\n+E\u00001\n8m2c4[O;[O;F]]\n\u0000\f\n8m3c6[O;F]2+3\n64m4c8\b\nO2;[O;[O;F]]\t\n+5\n128m4c8\u0002\nO2;\u0002\nO2;F\u0003\u0003\n:(7)\nHere, for any two operators AandBthe commutator is de\fned as [ A;B] and the\nanticommutator as fA;Bg. As already pointed out, the original FW transformation\ncan only produce correct and expected higher-order terms up to \frst order i.e., 1 =c4\n[26, 30, 33]. In fact, in their original work Foldy and Wouthuysen derived only the\nterms up to 1 =c4, i.e., only the terms in the \frst line of Eq. (7), however, notably\nwith the exception of the fourth term [29]. The higher-order terms in the original FW\ntransformation are of doubtful value [32, 34, 35]. Therefore, the Hamiltonian in Eq. (7)\nis not trustable and corrections are needed to achieve the expected higher-order terms.\nThe main problem with the original FW transformation is that the unitary operators in\ntwo preceding transformations do not commute with each other. For example, for the\nexponential operators eiUFWandeiU0\nFW, the commutator [ UFW;U0\nFW]6= 0. Moreover, as\nthe unitary operators are odd, this commutator produces even terms that have not been\nconsidered in the original FW transformation [26, 30, 33]. Taking into account those\nterms, the correction of the FW transformation generates the Hamiltonian as [33]\nHcorr:\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6+O6\n16m5c10\u0013\n+E\u00001\n8m2c4[O;[O;F]]\n+\f\n16m3c6fO;[[O;F];F]g+3\n64m4c8\b\nO2;[O;[O;F]]\t\n+1\n128m4c8\u0002\nO2;\u0002\nO2;F\u0003\u0003\n\u00001\n32m4c8[O;[[[O;F];F];F]]: (8)6\nNote the di\u000berence between two Hamiltonians in Eq. (7) and Eq. (8) that are observed\nin the second and consequent lines in both the equations, however, the terms in the\n\frst line are the same. Eq. (8) provides the correct higher-order terms of the FW\ntransformation. In this regard, we mention that an another approach towards the correct\nFW transformation has been employed by Eriksen; this is a single step approach that\nproduces the expected FW transformed higher-order terms [34]. Once the transformed\nHamiltonian has been obtained as a function of odd and even terms, the \fnal form\nis achieved by substituting the correct form of odd terms Oand even termsEin the\nexpression of Eq. (8) and calculating term by term.\nSince we perform here the time-dependent FW transformation, we note that the\ncommutator [O;F] can be evaluated as [ O;F] =i~@O=@t. Therefore, following the\nde\fnition of the odd operator, the time-varying \felds are taken into account through\nthis term. We evaluate each of the terms in Eq. (8) separately and obtain that the\nparticles can be described by the following extended Pauli Hamiltonian [24, 26, 36]\nHcorr:\nFW=(p\u0000eA)2\n2m+V\u0000e~\n2m\u001b\u0001B\u0000(p\u0000eA)4\n8m3c2+(p\u0000eA)6\n16m5c4\n\u0000\u0012e~\n2m\u00132B2\n2mc2+e~\n4m2c2(\n(p\u0000eA)2\n2m;\u001b\u0001B)\n\u0000e~2\n8m2c2r\u0001Etot\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000e~2\n16m3c4\u001a\n(p\u0000eA);@Etot\n@t\u001b\n\u0000ie~2\n16m3c4\u001b\u0001\u0014@Etot\n@t\u0002(p\u0000eA) + (p\u0000eA)\u0002@Etot\n@t\u0015\n+3e~\n64m4c4n\n(p\u0000eA)2\u0000e~\u001b\u0001B;~r\u0001Etot+\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]o\n+e~4\n32m4c6r\u0001@2Etot\n@t2+e~3\n32m4c6\u001b\u0001\u0014@2Etot\n@t2\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002@2Etot\n@t2\u0015\n:\n(9)\nThe \felds in the last Hamiltonian (9) are de\fned as B=r\u0002A, the external magnetic\n\feld,Etot=Eint+Eextare the electric \felds where Eint=\u00001\nerVis the internal \feld\nthat exists even without any perturbation and Eext=\u0000@A\n@tis the external \feld (only\nthe temporal part is retained here because of the Coulomb gauge). It is clear that as the\ninternal \feld is time-independent, it does not contribute to the fourth and sixth lines\nof Eq. (9). However, the external \feld does contribute to the above terms wherever it\nappears in the Hamiltonian.\nThe above-derived Hamiltonian can be split in two parts: (1) a spin-independent\nHamiltonian and (2) a spin-dependent Hamiltonian that involves the Pauli spin matrices.\nThe spin-dependent Hamiltonian, furthermore, has two types of coupling terms. The\ndirect \feld-spin coupling terms are those which directly couples the \felds with the\nmagnetic moments e.g., the third term in the \frst line, the second term in the third\nline of Eq. (9) etc. On the other hand, there are relativistic terms that do not directly\ncouple the spins to the electromagnetic \feld - indirect \feld-spin coupling terms. These7\nterms include e.g., the second term of the second line, the \ffth line of Eq. (9) etc. The\ndirect \feld-spin interaction terms are most important because these govern the directly\nmanipulation of the spins in a system with an electromagnetic \feld. For the external\nelectric \feld, these terms can be written together as a function of electric and magnetic\n\feld. These terms are taken into account and discussed in the next section. The indirect\ncoupling terms are often not taken into consideration and not included in the discussion\n(see Ref. [36, 37] for details). In this context, we reiterate that our current approach of\nderiving relativistic terms does not include the exchange and correlation e\u000bect. A similar\nFW transformed Hamiltonian has previously been derived, however, with a general\nKohn-Sham exchange \feld [23, 25, 26]. As mentioned before, in this article we do not\nintend to include the exchange-correlation e\u000bect, while mostly focussing on the magnetic\nrelaxation and magnetic inertial dynamics.\n2.1. The spin Hamiltonian\nThe aim of this work is to formulate the spin dynamics on the basis of the Hamiltonian\nin Eq. (9). The direct \feld-spin interaction terms can be written together as electric or\nmagnetic contributions. These two contributions can be expressed as a series up to an\norder of 1=m5[36]\nHS\nmagnetic =\u0000e\nmS\u0001\"\nB+1\n2X\nn=1;2;3;4\u00121\n2i!c\u0013n@nB\n@tn#\n+O\u00121\nm6\u0013\n; (10)\nHS\nelectric =\u0000e\nmS\u0001\"\n1\n2mc2X\nn=0;2\u0012i\n2!c\u0013n@nE\n@tn\u0002(p\u0000eA)#\n+O\u00121\nm6\u0013\n; (11)\nwhere the Compton wavelength and pulsation have been expressed by the usual\nde\fnitions \u0015c=h=mc and!c= 2\u0019c=\u0015cwith Plank's constant h. We also have used\nthe spin angular momentum operator as S= (~=2)\u001b. Note that we have dropped\nthe notion of total electric \feld because the the involved \felds ( B,E,A) are external\nonly, the internal \felds are considered as time-independent. The involved terms in the\nabove two spin-dependent Hamiltonians can readily be explained. The \frst term in the\nmagnetic contribution in Eq. (10) explains the Zeeman coupling of spins to the external\nmagnetic \feld. The rest of the terms in both the Hamiltonians in Eqs. (11) and (10)\nrepresent the spin-orbit coupling and its higher-order corrections. We note that these\ntwo spin Hamiltonians are individually not Hermitian, however, it can be shown that\ntogether they form a Hermitian Hamiltonian [38]. As these Hamiltonians describe a\nsemirelativistic Dirac particle, it is possible to derive from them the spin dynamics of\na single Dirac particle [24]. The e\u000bect of the indirect \feld-spin terms is not yet well\nunderstood, but they could become important too in magnetism [36, 37], however, those\nterms are not of our interest here.\nThe electric Hamiltonian can be written in terms of magnetic contributions with\nthe choice of a gauge A=B\u0002r=2. The justi\fcation of the gauge lies in the fact8\nthat the magnetic \feld inside the system being studied is uniform [26]. The transverse\nelectric \feld in the Hamiltonian (10) can be written as\nE=1\n2\u0012\nr\u0002@B\n@t\u0013\n: (12)\nReplacing this expression in the electric spin Hamiltonian in Eq. (11), one can obtain a\ngeneralised expression of the total spin-dependent Hamiltonian as\nHS(t) =\u0000e\nmS\u0001h\nB+1\n21X\nn=1;2;:::\u00121\n2i!c\u0013n@nB\n@tn\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u0012\nr\u0002@n+1B\n@tn+1\u0013\n\u0002(p\u0000eA)i\n: (13)\nIt is important to stress that the above spin-Hamiltonian is a generalisation of the two\nHamiltonians in Eqs. (10) and (11). We have already evaluated the Hamiltonian forms\nforn= 1;2;3;4 and assume that the higher-order terms will have the same form [36].\nThis Hamiltonian consists of the direct \feld-spin interaction terms that are linear and/or\nquadratic in the \felds. In the following we consider only the linear interaction terms,\nthat is we neglect the eAterm in Eq. (13). Here, we mention that the quadratic terms\ncould provide an explanation towards the previously unknown origin of spin-photon\ncoupling or optical spin-orbit torque and angular magneto-electric coupling [38{40].\nThe linear direct \feld-spin Hamiltonian can then be recast as\nHS(t) =\u0000e\nmS\u0001h\nB+1\n21X\nn=1;2;:::\u00121\n2i!c\u0013n@nB\n@tn\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u001a@n+1B\n@tn+1(r\u0001p)\u0000r\u0012@n+1B\n@tn+1\u0001p\u0013\u001bi\n: (14)\nThis is \fnal form of the Hamiltonian and we are interested to describe to evaluate its\ncontribution to the spin dynamics.\n3. Spin dynamics\nOnce we have the explicit form of the spin Hamiltonian in Eq. (14), we can proceed to\nderive the corresponding classical magnetisation dynamics. Following similar procedures\nof previous work [23, 24], and introducing a magnetisation element M(r;t), the\nmagnetisation dynamics can be calculated by the following equation of motion\n@M\n@t=X\njg\u0016B\n\n1\ni~D\u0002\nSj;HS(t)\u0003E\n; (15)\nwhere\u0016Bis the Bohr magneton, gis the Land\u0013 e g-factor that takes a value \u00192 for electron\nspins and \n is a suitably chosen volume element. Having the spin Hamiltonian in Eq.9\n(14), we evaluate the corresponding commutators. As the spin Hamiltonian involves the\nmagnetic \felds, one can classify the magnetisation dynamics into two situations: (a) the\nsystem is driven by a harmonic \feld, (b) the system is driven by a non-harmonic \feld.\nHowever, in the below we continue the derivation of magnetisation dynamics with the\nharmonic driven \felds. The magnetisation dynamics driven by the non-harmonic \felds\nhas been discussed in the context of Gilbert damping and inertial dynamics where it was\nshown that an additional torque contribution (the \feld-derivative torque) is expected\nto play a crucial role [23, 24, 26].\nThe magnetisation dynamics due to the very \frst term of the Hamiltonian in Eq.\n(14) is derived as [24]\n@M(1)\n@t=\u0000\rM\u0002B; (16)\nwith the gyromagnetic ratio \r=gjej=2m. Here the commutators between two spin\noperators have been evaluated using [ Sj;Sk] =i~Sl\u000fjkl, where\u000fjklis the Levi-Civita\ntensor. This dynamics actually produces the precession of magnetisation vector around\nan e\u000bective \feld. To get the usual form of Landau-Lifshitz precessional dynamics, one\nhas to use a linear relationship of magnetisation and magnetic \feld as B=\u00160(M+H).\nWith the latter relation, the precessional dynamics becomes \u0000\r0M\u0002H, where\r0=\r\u00160\nde\fnes the e\u000bective gyromagnetic ratio. We point out that the there are relativistic\ncontributions to the precession dynamics as well, e.g., from the spin-orbit coupling due\nto the time-independent \feld Eint[23]. Moreover, the contributions to the magnetisation\nprecession due to exchange \feld appear here, but are not explicitly considered in this\narticle as they are not in the focus of the current investigations (see Ref. [23] for details).\nThe rest of the terms in the spin Hamiltonian in Eq. (14) is of much importance\nbecause they involve the time-variation of the magnetic induction. As it has been shown\nin an earlier work [23] that for the external \felds and speci\fcally the terms with n= 1\nin the second terms and n= 0 in the third terms of Eq. (14), these terms together\nare Hermitian. These terms contribute to the magnetisation dynamics as the Gilbert\nrelaxation within the LLG equation of motion,\n@M(2)\n@t=M\u0002\u0012\nA\u0001@M\n@t\u0013\n; (17)\nwhere the Gilbert damping parameter Ahas been derived to be a tensor that has mainly\ntwo contributions: electronic and magnetic. The damping parameter Ahas the form\n[23, 24]\nAij=\u0000e\u00160\n8m2c2X\n`;k\u0002\nhripk+pkrii\u0000hr`p`+p`r`i\u000eik\u0003\n\u0002\u0000\n1+\u001f\u00001\u0001\nkj; (18)\nwhere 1is the 3\u00023 unit matrix and \u001fis the magnetic susceptibility tensor that can be\nintroduced only if the system is driven by a \feld which is single harmonic [26]. Note\nthat the electronic contributions to the Gilbert damping parameter are given by the10\nexpectation value hripkiand the magnetic contributions by the susceptibility. We also\nmention that the tensorial Gilbert damping tensor has been shown to contain a scalar,\nisotropic Heisenberg-like contribution, an anisotropic Ising-like tensorial contribution\nand a chiral Dzyaloshinskii-Moriya-like contribution [23].\nIn an another work, we took into account the terms with n= 2 in the second term\nof Eq. (14) and it has been shown that those containing the second-order time variation\nof the magnetic induction result in the magnetic inertial dynamics. Note that these\nterms provide a contribution to the higher-order relativistic e\u000bects. The corresponding\nmagnetisation dynamics can be written as [24]\n@M(3)\n@t=M\u0002\u0012\nC\u0001@M\n@t+D\u0001@2M\n@t2\u0013\n; (19)\nwith a higher-order Gilbert damping tensor Cijand inertia parameter Dijthat have the\nfollowing expressions Cij=\r0~2\n8m2c4@\n@t( 1+\u001f\u00001)ijandDij=\r0~2\n8m2c4( 1+\u001f\u00001)ij. We note\nthat Eq. (19) contains two fundamentally di\u000berent dynamics { the \frst term on the\nright-hand side has the exact form of Gilbert damping dynamics whereas the second\nterm has the form of magnetic inertial dynamics [24].\nThe main aim of this article is to formulate a general magnetisation dynamics\nequation and an extension of the traditional LLG equation to include higher-order\nrelativistic e\u000bects. The calculated magnetisation dynamics due to the second and third\nterms of Eq. (14) can be expressed as\n@M\n@t=e\nmM\u0002h1\n21X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n+1B\n@tn+1\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u001a@n+1B\n@tn+1hr\u0001pi\u0000D\nr\u0012@n+1B\n@tn+1\u0001p\u0013E\u001bi\n: (20)\nNote the di\u000berence in the summation of \frst terms from the Hamiltonian in Eq. (14).\nTo obtain explicit expressions for the Gilbert damping dynamics, we employ a general\nlinear relationship between magnetisation and magnetic induction, B=\u00160(H+M).\nThe time-derivative of the magnetic induction can then be replaced by magnetisation\nand magnetic susceptibility. For the n-th order time-derivative of the magnetic induction\nwe \fnd\n@nB\n@tn=\u00160\u0012@nH\n@tn+@nM\n@tn\u0013\n: (21)\nNote that this equation is valid for the case when the magnetisation is time-dependent.\nSubstituting this expression into the Eq. (20), one can derive the general LLG equation\nand its extensions. Moreover, as we work out the derivation in the case of harmonic\ndriving \felds, the di\u000berential susceptibility can be introduced as \u001f=@M=@H. The\n\frst term ( n-th derivative of the magnetic \feld) can consequently be written by the11\nfollowing Leibniz formula as\n@nH\n@tn=n\u00001X\nk=0(n\u00001)!\nk!(n\u0000k\u00001)!@n\u0000k\u00001(\u001f\u00001)\n@tn\u0000k\u00001\u0001@k\n@tk\u0012@M\n@t\u0013\n; (22)\nwhere the magnetic susceptibility \u001f\u00001is a time-dependent tensorial quantity and\nharmonic. Using this relation, the \frst term and second terms in Eq. (20) assume\nthe form\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=0;1;:::\u00121\n2i!c\u0013n+1nX\nk=0n!\nk!(n\u0000k)!@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\n;\n(23)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u0002\n1X\nn=0;2;:::\u00121\n2i!c\u0013nnX\nk=0n!\nk!(n\u0000k)!h@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\u001b\n\u0001p\u0013Ei\n:(24)\nThese two equations already provide a generalisation of the higher-order magnetisation\ndynamics including the Gilbert damping (i.e., the terms with k= 0) and the inertial\ndynamics (the terms with k= 1) and so on.\n4. Discussion\n4.1. Gilbert damping parameter\nIt is obvious that, as Gilbert damping dynamics involves the \frst-order time derivative of\nthe magnetisation and a torque due to it, kmust take the value k= 0 in the equations\n(23) and (24). Therefore, the Gilbert damping dynamics can be achieved from the\nfollowing equations:\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t; (25)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u00021X\nn=0;2;:::\u00121\n2i!c\u0013nh\u0012@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t\u001b\n\u0001p\u0013Ei\n: (26)\nNote that these equations can be written in the usual form of Gilbert damping as\nM\u0002\u0000\nG\u0001@M\n@t\u0001\n, where the Gilbert damping parameter Gis notably a tensor [2, 23]. The12\ngeneral expression for the tensor can be given by a series of higher-order relativistic\nterms as follows\nGij=e\u00160\n2m1X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n( 1+\u001f\u00001)ij\n@tn\n+e\u00160\n4m2c21X\nn=0;2;:::\u00121\n2i!c\u0013nh@n( 1+\u001f\u00001)ij\n@tn(hrlpli\u0000hrlpii)i\n: (27)\nHere we have used the Einstein summation convention on the index l. Note that there\nare two series: the \frst series runs over even and odd numbers ( n= 0;1;2;3;\u0001\u0001\u0001),\nhowever, the second series runs only over the even numbers ( n= 0;2;4;\u0001\u0001\u0001). Eq. (27)\nrepresents a general relativistic expression for the Gilbert damping tensor, given as a\nseries of higher-order terms. This equation is one of the central results of this article. It\nis important to observe that this expression provides the correct Gilbert tensor at the\nlowest relativistic order, i.e., putting n= 0 the expression for the tensor is found to be\nexactly the same as Eq. (18).\nThe analytic summation of the above series of higher-order relativistic contributions\ncan be carried out when the susceptibility depends on the frequency of the harmonic\ndriving \feld. This is in general true for ferromagnets where a di\u000berential susceptibility\nis introduced because there exists a spontaneous magnetisation in ferromagnets even\nwithout application of a harmonic external \feld. However, if the system is driven by a\nnonharmonic \feld, the introduction of the susceptibility is not valid anymore. In general\nthe magnetic susceptibility is a function of wave vector and frequency in reciprocal space,\ni.e.,\u001f=\u001f(q;!). Therefore, for the single harmonic applied \feld, we use \u001f\u00001/ei!tand\nthen-th order derivative will follow @n=@tn(\u001f\u00001)/(i!)n\u001f\u00001. With these arguments,\none can express the damping parameter of Eq. (27) as (see Appendix A for detailed\ncalculations)\nGij=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n(2!!c+!2)~\ni+!2(hrlpli\u0000hrlpii)\n4!2\nc\u0000!2#\n\u001f\u00001\nij: (28)\nHere, the \frst term in the last expression is exactly the same as the one that has been\nderived in our earlier investigation [23]. As the expression of the expectation value\nhripjiis imaginary, the real Gilbert damping parameter will be given by the imaginary\npart of the susceptibility tensor. This holds consistently for the higher-order terms\nas well. The second term in Eq. (28) stems essentially from an in\fnite series which\ncontain higher-order relativistic contributions to the Gilbert damping parameter. As\n!cscales with c, these higher-order terms will scale with c\u00004or more and thus their\ncontributions will be smaller than the \frst term. Note that the higher-order terms will\ndiverge when != 2!c\u00191021sec\u00001, which means that the theory breaks down at the\nlimit!!2!c. In this limit, the original FW transformation is not de\fned any more\nbecause the particles and antiparticles cannot be separated at this energy limit.13\n4.2. Magnetic inertia parameter\nMagnetic inertial dynamics, in contrast, involves a torque due to the second-order time-\nderivative of the magnetisation. In this case, kmust adopt the value k= 1 in the\nafore-derived two equations (23) and (24). However, if k= 1, the constraint n\u0000k\u00150\ndictates that n\u00151. Therefore, the magnetic inertial dynamics can be described with\nthe following equations:\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=1;2;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)\n@tn\u00001\u0001@2M\n@t2; (29)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u00021X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h\u0012@n\u00001( 1+\u001f\u00001)\n@tn\u00001\u0001@2M\n@t2\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@2M\n@t2\u001b\n\u0001p\u0013Ei\n: (30)\nSimilar to the Gilbert damping dynamics, these dynamical terms can be expressed\nasM\u0002\u0010\nI\u0001@2M\n@t2\u0011\nwhich is the magnetic inertial dynamics [8]. The corresponding\nparameter has the following expression\nIij=e\u00160\n2m1X\nn=1;2;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001(hrlpli\u0000hripli)i\n: (31)\nNote that as ncannot adopt the value n= 0, the starting values of nare di\u000berent in\nthe two terms. Importantly, if n= 1 we recover the expression for the lowest order\nmagnetic inertia parameter Dij, as given in the equation (19) [24].\nUsing similar arguments as in the case of the generalised Gilbert damping\nparameter, when we consider a single harmonic \feld as driving \feld, the inertia\nparameter can be rewritten as follows (see Appendix A for detailed calculations)\nIij=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u0012\u0000!2+ 4!!c\n(2!c\u0000!)2\u0013\n\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001216!!3\nc\n(4!2\nc\u0000!2)2\u0013\n\u001f\u00001\nij: (32)\nThe \frst term here is exactly the same as the one that was obtained in our earlier\ninvestigation [24]. However, there are now two extra terms which depend on the\nfrequency of the driving \feld and that vanish for !!0. Again, in the limit !!2!c,\nthese two terms diverge and hence this expression is not valid anymore. The inertia\nparameter will consistently be given by the real part of the susceptibility.14\n5. Summary\nWe have developed a generalised LLG equation of motion starting from fundamental\nquantum relativistic theory. Our approach leads to higher-order relativistic correction\nterms in the equation of spin dynamics of Landau and Lifshitz. To achieve this, we have\nstarted from the foundational Dirac equation under the presence of an electromagnetic\n\feld (e.g., external driving \felds or THz excitations) and have employed the FW\ntransformation to separate out the particles from the antiparticles in the Dirac equation.\nIn this way, we derive an extended Pauli Hamiltonian which e\u000eciently describes the\ninteractions between the quantum spin-half particles and the applied \feld. The thus-\nderived direct \feld-spin interaction Hamiltonian can be generalised for any higher-order\nrelativistic corrections and has been expressed as a series. To derive the dynamical\nequation, we have used this generalised spin Hamiltonian to calculate the corresponding\nspin dynamics using the Heisenberg equation of motion. The obtained spin dynamical\nequation provides a generalisation of the phenomenological LLG equation of motion\nand moreover, puts the LLG equation on a rigorous foundational footing. The equation\nincludes all the torque terms of higher-order time-derivatives of the magnetisation (apart\nfrom the Gilbert damping and magnetic inertial dynamics). Speci\fcally, however, we\nhave focussed on deriving an analytic expression for the generalised Gilbert damping\nand for the magnetic inertial parameter. Our results show that both these parameters\ncan be expressed as a series of higher-order relativistic contributions and that they\nare tensors. These series can be summed up for the case of a harmonic driving \feld,\nleading to closed analytic expressions. We have further shown that the imaginary part\nof the susceptibility contributes to the Gilbert damping parameter while the real part\ncontributes to the magnetic inertia parameter. Lastly, with respect to the applicability\nlimits of the derived expressions we have pointed out that when the frequency of the\ndriving \feld becomes comparable to the Compton pulsation, our theory will not be valid\nanymore because of the spontaneous particle-antiparticle pair-production.\n6. Acknowledgments\nWe thank P-A. Hervieux for valuable discussions. This work has been supported\nby the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation\n(Contract No. 2015.0060), the European Union's Horizon2020 Research and\nInnovation Programme under grant agreement No. 737709 (FEMTOTERABYTE,\nhttp://www.physics.gu.se/femtoterabyte).15\nAppendix A. Detailed calculations of the parameters for a harmonic \feld\nIn the following we provide the calculational details of the summation towards the results\ngiven in Eqs. (28) and (32).\nAppendix A.1. Gilbert damping parameter\nEq. (27) can be expanded as follows\nGij=e\u00160\n2m1\n2i!c( 1+\u001f\u00001)ij+e\u00160\n4m2c2(hrlpli\u0000hrlpii) ( 1+\u001f\u00001)ij\n+e\u00160\n2m1X\nn=1;2;:::\u00121\n2i!c\u0013n+1\n(i!)n\u001f\u00001\nij+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013n\n(hrlpli\u0000hrlpii) (i!)n\u001f\u00001\nij\n=e\u00160\n2m1\n2i!c( 1+\u001f\u00001)ij+e\u00160\n4m2c2(hrlpli\u0000hrlpii) ( 1+\u001f\u00001)ij\n+e\u00160\n2m1\n2i!c1X\nn=1;2;:::\u0012!\n2!c\u0013n\n\u001f\u00001\nij+e\u00160\n4m2c21X\nn=2;4;:::\u0012!\n2!c\u0013n\n(hrlpli\u0000hrlpii)\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n~\ni1X\nn=1;2;:::\u0012!\n2!c\u0013n\n+ (hrlpli\u0000hrlpii)1X\nn=2;4;:::\u0012!\n2!c\u0013n#\n\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\u0014~\ni!\n2!c\u0000!+ (hrlpli\u0000hrlpii)!2\n4!2\nc\u0000!2\u0015\n\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n(2!!c+!2)~\ni+!2(hrlpli\u0000hrlpii)\n4!2\nc\u0000!2#\n\u001f\u00001\nij: (A.1)\nWe have used the fact that!\n!c<1 and the summation formula\n1 +x+x2+x3+:::=1\n1\u0000x;\u00001<x< 1: (A.2)REFERENCES 16\nAppendix A.2. Magnetic inertia parameter\nEq. (31) can be expanded as follows\nIij=e\u00160\n2m\u00121\n2i!c\u00132\n( 1+\u001f\u00001)ij+e\u00160\n2m1X\nn=2;3;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001(hrlpli\u0000hripli)i\n=e\u00160\n2m\u00121\n2i!c\u00132\n( 1+\u001f\u00001)ij+1X\nn=2;3;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!(i!)n\u00001\u001f\u00001\nij\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!(hrlpli\u0000hripli) (i!)n\u00001\u001f\u00001\nij\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij+e\u00160\n2m\u00121\n2i!c\u001321X\nn=2;3;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!\u001f\u00001\nij\n+e\u00160\n4m2c2\u00121\n2i!c\u00131X\nn=2;4;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!(hrlpli\u0000hripli)\u001f\u00001\nij\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c41X\nn=2;3;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni1X\nn=2;4;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!(hrlpli\u0000hripli)\u001f\u00001\nij\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u001f\u00001\nij\"\n2\u0012!\n2!c\u0013\n+ 3\u0012!\n2!c\u00132\n+ 4\u0012!\n2!c\u00133\n+:::#\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001f\u00001\nij\"\n2\u0012!\n2!c\u0013\n+ 4\u0012!\n2!c\u00133\n+ 6\u0012!\n2!c\u00135\n+:::#\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u0012\u0000!2+ 4!!c\n(2!c\u0000!)2\u0013\n\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001216!!3\nc\n(4!2\nc\u0000!2)2\u0013\n\u001f\u00001\nij: (A.3)\nHere we have used the formula\n1 + 2x+ 3x2+ 4x3+ 5x4+:::=1\n(1\u0000x)2;\u00001<x< 1: (A.4)\nReferences\n[1] Landau L D and Lifshitz E M 1935 Phys. 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Matter 29\n194002 URL http://stacks.iop.org/0953-8984/29/i=19/a=194002\n[40] Paillard C, Mondal R, Berritta M, Dkhil B, Singh S, Oppeneer P M and Bellaiche\nL 2016 Proc. SPIE 9931 99312E{16 URL http://dx.doi.org/10.1117/12.\n2238196"    },    {        "title": "1805.01230v1.Exact_Intrinsic_Localized_Excitation_of_an_Anisotropic_Ferromagnetic_Spin_Chain_in_External_Magnetic_Field_with_Gilbert_Damping__Spin_Current_and_PT_Symmetry.pdf",        "content": "Exact Intrinsic Localized Excitation of an Anisotropic Ferromagnetic Spin Chain in External\nMagnetic Field with Gilbert Damping, Spin Current and PT-Symmetry\nM. Lakshmanan1,a)and Avadh Saxena2,b)\n1)Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University,\nTiruchirappalli - 620 024, India\n2)Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory,\nLos Alamos, NM 87545, USA\nWe obtain the exact one-spin intrinsic localized excitation in an anisotropic Heisenberg\nferromagnetic spin chain in a constant/variable external magnetic field with Gilbert damping\nincluded. We also point out how an appropriate magnitude spin current term in a spin\ntransfer nano-oscillator (STNO) can stabilize the tendency towards damping. Further, we\nshow how this excitation can be sustained in a recently suggested PT-symmetric magnetic\nnanostructure. We also briefly consider more general spin excitations.\na)Electronic mail: lakshman.cnld@gmail.com\nb)Electronic mail: avadh@lanl.gov\n1arXiv:1805.01230v1  [cond-mat.mes-hall]  3 May 2018I. INTRODUCTION\nThe study of dynamics of classical Heisenberg ferromagnetic spin chain with anisotropic inter-\naction is of considerable importance in applied magnetism1,2and from application point of view3,4.\nWhile several continuum versions are known to be completely integrable soliton systems5–7, such\nas the isotropic case, no discrete integrable case is known in the literature, except for a modified\nversion, namely the Ishimori lattice8. On the other hand, the present authors9have shown the exis-\ntence of several classes of exact solutions in terms of Jacobian elliptic functions which exist for the\ncase of the discrete lattice including onsite anisotropy and external magnetic field. Identifying such\ninteresting classes of solutions and their relevance in the context of appropriate physical situations\nconstitute one of the important areas of investigation in spin dynamics9,10.\nFrom another point of view, occurrence of intrinsic localized breathers/oscillations in suitable\nanisotropic ferromagnetic spin chains is of practical relevance11,12and is being explored for the past\nseveral years. Apart from many numerical studies, in recent times the present authors and Subash13\nhave obtained explicit analytical solutions for the Heisenberg anisotropic spin chain with additional\nonsite anisotropy and constant external magnetic field corresponding to excitations of one, two and\nthree spins and also investigated their stability. Additionally, relevant situations were pointed out\nwhere such excitations can be physically identified.\nInrecenttimes,onehasalsoseenthatatnanoscalelevelspintransfernano-oscillator(STNO)14,15,\nwhich essentially consists of a trilayer structure of two nanoscale ferromagnetic films separated by a\nnon-ferromagnetic but conducting layer, can lead to switching of spin angular momentum directions\nand allow for the generation of microwave oscillations16,17. The ferromagnetic film even when it is\nhomogeneous is dominated by anisotropic interactions besides the presence of external magnetic\nfields (both dc and ac) and spin current terms. The equation of motion defining the evolution of\nthe spins is the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation18where the spin current term\nis given by the Slonczewski form. One notices that the LLGS equation is a simple generalization of\nthe Landau-Lifshitz-Gilbert (LLG) equation which describes the nonlinear magnetization dynamics\n2in bulk materials as in the case of ferromagnetic lattices. Then it becomes important to ask what is\nthe influence of spin current term on the spin excitations, particularly intrinsic localized oscillations\n(ILOs) and identify the conditions under which damping effect can be off-set by the spin current\nterm.\nFrom yet another point of view, one may consider the possibility of designing a PT-symmetric\nferromagnetic nanoscale device by considering two nano-film structures interspersed by a nonmag-\nnetic but conducting thinner layer (i.e. a sandwich structure) as suggested by Lee, Kottos and\nShapiro19very recently. These authors have proposed a class of synthetic magnetic nanostructures\nwhich utilize natural dissipation (loss) mechanisms along with suitable chosen gain mechanism so\nas to control the magnetization dynamics. We will also explore how the spin ILOs can be identified\nin these structures.\nIn this paper we deduce an explicit one-spin excitation in an anisotropic ferromagnetic lattice\n(without onsite anisotropy, to start with) in the presence of external magnetic field and explore the\neffect of spin current term to maintain the oscillatory nature of the spin excitation. We then point\nout how this can be generalized to more general spin excitations and in PT-symmetric nanostruc-\ntures.\nThe organization of the paper is as follows. In Sec. 2 we deduce the dynamical equation for an\nanisotropic ferromagnetic spin in the presence of external magnetic field and set up the appropriate\nequation for a one-spin excitation in the presence of Gilbert damping. In Sec. 3, we deduce the\nexplicit one-spin excitation including the damping effect and analyze how the spin excitation gets\naffected by the damping. In Sec. 4, we incorporate the spin current term and point out how an\nappropriate strength of spin current can off-set the effect of damping so as to control the spin\noscillations. In Sec. 5, we point out how the above analysis can be extended to a PT-symmetric\nnanostructure. We briefly indicate how this study can be extended to consider more general spin\nexcitations in Sec. 6. Finally in Sec. 7, we present our conclusions.\n3II. DYNAMICS OF THE ANISOTROPIC SPIN CHAIN AND ONE-SPIN\nEXCITATION\nConsidering the evolution of spins of a one-dimensional anisotropic Heisenberg ferromagnetic\nspin chain modeled by the Hamiltonian12\nH=−N/summationdisplay\n{n}(ASx\nnSx\nn+1+BSy\nnSy\nn+1+CSz\nnSz\nn+1)−D/summationdisplay\nn(Sz\nn)2−/vectorH·/summationdisplay\nn/vectorSn, (1)\nwhere the spin components /vectorSn= (Sx\nn,Sy\nn,Sz\nn)are classical unit vectors satisfying the constant length\ncondition\n(Sx\nn)2+ (Sy\nn)2+ (Sz\nn)2= 1, n = 1,2,...,N. (2)\nHereA,B, andCare the exchange anisotropy parameters, Dis the onsite anisotropy parameter and\nthe external magnetic field /vectorH= (H,0,0)is chosen along the x-axis for convenience. By introducing\nthe appropriate spin-Poisson brackets and deducing the LLG spin evolution equation one can obtain\nthe equation for the spin lattice (1) as\nd/vectorSn\ndt=/vectorSn×/vectorHeff+α/vectorSn×(/vectorSn×/vectorHeff), (3)\nwhere\n/vectorHeff=A(Sx\nn+1+Sx\nn−1)ˆi+B(Sy\nn+1+Sy\nn−1)ˆj+C(Sz\nn+1+Sz\nn−1)ˆk+ 2DSz\nnˆk+/vectorH,(4)\nandαis the Gilbert damping parameter. In component form Eq. (3) with Eq. (4) reads as\ndSx\nn\ndt=CSy\nn(Sz\nn+1+Sz\nn−1)−BSz\nn(Sy\nn+1+Sy\nn−1)−2DSy\nnSz\nn+α/bracketleftbigg\nBSy\nnSz\nn(Sy\nn+1+Sy\nn−1)\n−A(Sx\nn+1+Sx\nn−1)((Sy\nn)2+ (Sz\nn)2) +CSz\nnSx\nn(Sz\nn+1+Sz\nn−1)−2DSx\nn(Sz\nn)2−H((Sx\nn)2+ (Sz\nn)2)/bracketrightbigg\n,(5)\ndSy\nn\ndt=ASz\nn(Sx\nn+1+Sx\nn−1)−CSx\nn(Sz\nn+1+Sz\nn−1) + 2DSx\nnSz\nn+HSz\nn+α/bracketleftbigg\nCSz\nnSy\nn(Sz\nn+1+Sz\nn−1)\n−B((Sx\nn)2+ (Sz\nn)2)(Sy\nn+1+Sy\nn−1) +ASx\nnSy\nn(Sx\nn+1+Sx\nn−1)−2DSy\nn(Sz\nn)2+HSx\nnSy\nn/bracketrightbigg\n, (6)\n4dSz\nn\ndt=BSx\nn(Sy\nn+1+Sy\nn−1)−ASy\nn(Sx\nn+1+Sx\nn−1)−HSy\nn+α/bracketleftbigg\nASx\nnSy\nn(Sx\nn+1+Sx\nn−1)\n−C((Sx\nn)2+ (Sy\nn)2)(Sz\nn+1+Sz\nn−1) +BSy\nnSz\nn(Sy\nn+1+Sy\nn−1) + 2DSz\nn((Sx\nn)2+ (Sy\nn)2) +HSx\nnSz\nn/bracketrightbigg\n.(7)\nNow looking for the one spin excitation for (1) as\n/vectorSn=...,(1,0,0),(1,0,0),(Sx\ni(t),Sy\ni(t),Sz\ni(t)),(1,0,0),(1,0,0),..., (8)\nwhere we have used nto denote a general spin in the lattice and used ito specify the localized spin\nexcitation, and redesignating (Sx\ni(t),Sy\ni(t),Sz\ni(t))as(Sx\n0(t),Sy\n0(t),Sz\n0(t)), the equation of motion\n(LLG equation) for the excited spin can be given as\ndSx\n0\ndt=−2DSy\n0Sz\n0−α/bracketleftbigg\n(2A+H)((Sy\n0)2+ (Sz\n0)2) + 2DSx\n0(Sz\n0)2/bracketrightbigg\n, (9)\ndSy\n0\ndt= (2A+H)Sz\n0+ 2DSx\n0Sz\n0+α/bracketleftbigg\n(2A+H)Sx\n0Sy\n0−2DSy\n0(Sz\n0)2/bracketrightbigg\n, (10)\ndSz\n0\ndt=−(2A+H)Sy\n0+α/bracketleftbigg\n(2A+H)Sx\n0Sz\n0+ 2D((Sx\n0)2+ (Sy\n0)2)Sz\n0/bracketrightbigg\n. (11)\nNote that from Eqs. (9) - (11), one can check that\nSx\n0dSx\n0\ndt+Sy\n0dSy\n0\ndt+Sz\n0dSz\n0\ndt= 0, (12)\nso that/vectorS2= (Sx\no)2+ (Sy\n0)2+ (Sz\n0)2=Constant = 1is conserved.\nNext, further confining to the case where the onsite anisotropy vanishes, D= 0, we have the\nLLG equation for the one-spin excitation,\ndSx\n0\ndt=−α(2A+H)(1−(Sx\n0)2), (13)\ndSy\n0\ndt= (2A+H)Sz\n0+α(2A+H)Sx\n0Sy\n0, (14)\ndSz\n0\ndt=−(2A+H)Sy\n0+α(2A+H)Sx\n0Sz\n0, (15)\nwith the constraint /vectorS2= (Sx\no)2+ (Sy\n0)2+ (Sz\n0)2= 1. The system (13) - (15) can be exactly solved\nas shown below.\n5III. EXPLICIT ONE-SPIN EXCITATION\nNow the above system of nonlinear differential equations can be straightforwardly solved. Inte-\ngrating (14) we obtain\nSx\n0(t) =c2e−2α(2A+H)t−1\nc2e−2α(2A+H)t+ 1, (16)\nwherecis an arbitrary constant. We also note that when α= 0, that is no damping, Sx\n0(t) =\n(c2−1)/(c2+ 1) =const =√\n1−a2as noted in ref. [13], Eq. (11). Also we note that Sx\n0(0) =\n(c2−1)/(c2+ 1)andSx\n0(∞) =−1, indicating a switching from a given initial value to the other\nground state, Sx\n0=−1.\nTo findSy\n0andSz\n0, we proceed as follows. Considering Eq. (14) and differentiating once with\nrespect toton both sides to obtain (d2Sy\n0/dt2), after making use of the forms of (dSx\n0/dt)and\n(dSz\n0/dt)from (13) and (15), respectively, we have\nd2Sy\n0\ndt2=−(2A+H)2(1 +α2)Sy\n0+ 2α(2A+H)2Sx\n0Sz\n0+ 2α2(2A+H)2(Sx\n0)2Sy\n0,(17)\nso that\nSz\n0(t) =1\n2α(2A+H)2Sx\n0/bracketleftBiggd2Sy\n0\ndt2+ (2A+H)2(1 +α2)Sy\n0−2α2(2A+H)2(Sx\n0)2Sy\n0/bracketrightBigg\n.(18)\nAlso from (14) we can write\nSz\n0(t) =1\n2A+H/bracketleftBiggdSy\n0\ndt−αSx\n0Sy\n0/bracketrightBigg\n. (19)\nEquating the right hand sides of (18) and (19), we obtain\nd2Sy\n0\ndt2=−2α(2A+H)Sx\n0dSy\n0\ndt+ (2A+H)2(1 +α2)Sy\n0= 0. (20)\nAfterastandardtransformationandtwointegrations(asindicatedinAppendixA),wecanexplicitly\nwrite the solution for Sy\n0as\nSy\n0=cexp(−α(2A+H)t)\nc2exp(−2α(2A+H)t) + 1ˆacos(Ωt+δ),Ω = 2A+H, (21)\n6where ˆais an arbitrary constant. Also from (14) we have\nSz\n0(t) =1\n(2A+H)/bracketleftBiggdSy\n0\ndt−αSx\n0Sy\n0/bracketrightBigg\n=−ˆasin(Ωt+δ)ce−α(2A+H)t\nc2e(−2α(2A+H)t+ 1. (22)\nNowinordertofixtheconstant ˆawedemandthatthespinlengthconstraint (Sx\n0)2+(Sy\n0)2+(Sz\n0)2= 1\nbe valid. This leads to\nˆa2= 4orˆa= 2, (23)\nso that we have now the complete solution of the excited spin as\nSx\n0(t) =c2e−2α(2A+H)t−1\nc2e−2α(2A+H)t+ 1, (24)\nSy\n0(t) =2ce−α(2A+H)t\nc2e−2α(2A+H)t+ 1cos[(2A+H)t+δ], (25)\nSz\n0(t) =−2ce−2α(2A+H)t\nc2e−2α(2A+H)t+ 1sin[(2A+H)t+δ]. (26)\nNote that the arbitrary constant corresponding to the undamped case ( α= 0) is\nˆa=c2−1\nc2+ 1. (27)\nIt is obvious from the above that for α= 0,Sx\n0=constant =c2−1\nc2+1, whileSy\n0andSz\n0are periodic\nfunctions of t. In this case Eqs. (14)-(15) are linear in Sy\n0andSz\n0so that the perturbation around\nthe origin in the ( Sy\n0−Sz\n0) plane admits pure imaginary eigenvalues. When α/negationslash= 0, they get damped\nas shown in Fig. 1 corresponding to the explicit forms (24)-(26). Note that in the above we have\nassumed the external magnetic field to be a constant in time. However, even in the case where the\nfield is a variable function of time, say\nH(t) =h0+h1cosωt, (28)\nwhereh0,h1andωare constants, we observe from the equations of motion of the spin components\n(13) - (15), that Hoccurs always as a linear combination 2A+H(t) = (2A+h0+h1cosωt).\nTherefore by redefining the time (2A+H)tas\nτ= (2A+h0)t−h1ωsinωt, (29)\n7all the previous analysis goes through. The final spin excitations are of the same form as (24) - (26)\nbut with the transformed time variable given by Eq. (29).\ntSx\n0(t)(i)\n2000 1500 1000 500 01\n0.5\n0\n-0.5\n-1\ntSy\n0(t)(ii)\n2000 1500 1000 500 01\n0.5\n0\n-0.5\n-1\ntSz\n0(t)(iii)\n2000 1500 1000 500 01\n0.5\n0\n-0.5\n-1\nFIG. 1. Damped spin excitation: One-spin excitation (Eqs. (24)-(26)) showing the three spin components\nfor the damped cases ( α= 0.005).\nIV. EFFECT OF SLONCZEWSKI SPIN CURRENT\nNext we consider the influence of spin current term in a trilayer structured STNO (see Fig. 2),\nwhere we consider the excitation of a single spin of magnetization in the outer uniformly magnetized\nlayer under anisotropic interaction and external magnetic field in the presence of spin current. The\ncorresponding spin excitation is given by the Landau-Lifshitz-Gilbert-Slonczewski equation for the\nspin as\nFIG. 2. A schematic representation of STNO.\nd/vectorSn\ndt=/vectorSn×/vectorHeff+α/vectorSn×(/vectorSn×/vectorHeff) +j/vectorSn×(/vectorSn×/vectorSp), (30)\n8where/vectorHeffis the effective field given by Eq. (4) and jis the magnitude of the spin current and\nthe polarization vector /vectorSpis\n/vectorSp= (1,0,0), (31)\ncorresponding to the flow of electrons in the x-direction. Consequently,\n/vectorSn×(/vectorSn×/vectorSp) =/vectorSn(/vectorSn·/vectorSp)−/vectorSp=−((Sy\nn)2+ (Sz\nn)2)ˆi+Sx\nnSy\nnˆj+Sx\nnSz\nnˆk, (32)\nwhere (ˆi,ˆj,ˆk)form the unit orthonormal trihedral. As a result, the equations for the one-spin\nexcitations get modified from (13) - (15) as\ndSx\n0\ndt=−[α(2A+H)−j](1−(Sz\n0)2), (33)\ndSy\n0\ndt= (2A+H)Sz\n0+ [α(2A+H)−j]Sx\n0Sy\n0, (34)\ndSz\n0\ndt=−(2A+H)Sy\n0+ [α(2A+H)−j]Sx\n0Sz\n0. (35)\nNow choosing the spin current as\nj=α(2A+H), (36)\none can check that\ndSx\n0\ndt= 0, (37)\ndSy\n0\ndt= (2A+H)Sz\n0, (38)\ndSz\n0\ndt=−(2A+H)Sz\n0. (39)\nConsequently, the spin vector evolves as\n/vectorS0=/parenleftBig√\n1−ˆa2,ˆacos(Ωt+δ),−ˆasin(Ωt+δ/parenrightBig\n, (40)\nwhere ˆa=constant and Ω = (2A+H), and the effect of damping is exactly offset by the spin\ncurrent term. Thus the spin current acts effectively as an “external magnetic field plus anisotropy\"\nand the system can generate microwave oscillations. When j <α (2A+H), damping will overtake\nasymptotically and the spin will switch its direction.\n9V.PT-SYMMETRIC MAGNETIC DEVICE\nRecentlyaclassofsyntheticmagneticnanostructuresthatmakesuseofthenatureofloss/dissipation\nmechanism together with appropriate amplification (gain) process has been suggested by Lee, Kot-\ntos and Shapiro19to control magnetization dynamics. The suggested arrangement consists of two\ncoupled nano-ferromagnetic films, n= 1,2(when separated by a spacer) in the presence of an\nexternal magnetic field along the x-axis, for example out-of-plane geometry (so that the z-axis is\nperpendicular to the films) as shown in Fig. 3.\nFIG.3. APT-symmetrictrilayerstructurecomprisingtwomagneticthinfilmsandaspacerlayersuggested\nby Lee, Kottos and Shapiro19.\nConsidering the effective instantaneous local fields as /vectorH1effand/vectorH2efffor the two layers 1 and 2,\nrespectively, associated with the homogeneous magnetization vectors /vectorS1= (/vectorM1/|/vectorM1|)and/vectorS2=\n(/vectorM2/|/vectorM2|), we have the associated dynamical equations\nd/vectorS1\ndt=/vectorS1×/vectorH1eff+k/vectorS1×/vectorS2+α/vectorS1×d/vectorS1\ndt, (41)\nd/vectorS2\ndt=/vectorS2×/vectorH2eff+k/vectorS2×/vectorS1−α/vectorS2×d/vectorS2\ndt, (42)\nwherekistheferromagneticcouplingand αisthedamping/gaincoefficient. Notethatthecombined\nsystems (41)-(42) are invariant under the simultaneous changes of the variables /vectorS1,2→ −/vectorS2,1,\n/vectorH1,2eff→−/vectorH2,1effandt→−t, which may be treated as equivalent to combined PT-symmetry\n10operation19. Now we choose the two layers such that\n/vectorS2=/vectorS1×/vectorSp,/vectorS1=−/vectorS2×/vectorSp, (43)\nwhere/vectorSp= (1,0,0)is a fixed polarization vector. Equation (43) implies that /vectorSpis perpendicular\nto the plane of /vectorS1and/vectorS2. Then, similar to the analysis in Sec. 4, we can choose the ferromagnetic\ncouplingksuch that for simple anisotropy as in Eqs. (13) - (15) and external magnetic field H, we\ncan choose\nk=α(2A+H), (44)\nso that the gain/loss terms are exactly cancelled by the ferromagnetic coupling, leaving out\nd/vectorS1\ndt=/vectorS1×/vectorH1eff, (45)\nd/vectorS2\ndt=/vectorS2×/vectorH2eff, (46)\nleading to spin oscillations and thereby to an effective control of magnetization oscillations.\nVI. MORE GENERAL SPIN EXCITATIONS\nOne can consider more general localized spin excitations like two, three, etc. spin excitations.\nFor example, in the case of localized two-spin excitations,\n/vectorSn=...,(1,0,0),(1,0,0),(Sx\ni,Sy\ni,Sz\ni),(Sx\ni+1,Sy\ni+1,Sz\ni+1),(1,0,0),(1,0,0),...\n=...,(1,0,0),(1,0,0),(Sx\n0,Sy\n0,Sz\n0),(Sx\n1,Sy\n1,Sz\n1),(1,0,0),(1,0,0),..., (47)\n11we obtain the dynamical equations from (5) - (7) as\ndSx\n0\ndt=CSy\n0Sz\n1−BSz\n0Sy\n1−2DSy\n0Sz\n0\n+α[BSx\n0Sy\n0Sy\n1−(A(Sx\n1+ 1) +H)((Sy\n0)2+ (Sz\n0)2) +CSx\n0Sz\n0Sz\n1−2DSx\n0(Sz\n0)2], (48)\ndSy\n0\ndt=ASz\n0(Sx\n1+ 1)−CSx\n0Sz\n1+ 2DSx\n0Sz\n0+HSz\n0\n+α[(A(Sx\n1+ 1) +H)Sx\n0Sy\n0−BSy\n1((Sx\n0)2+ (Sz\n0)2) +CSy\n0Sz\n0Sz\n1−2DSy\n0(Sz\n0)2], (49)\ndSz\n0\ndt=BSx\n0Sy\n1−ASy\n0(Sx\n1+ 1)−HSy\n0\n+α[(A(Sx\n1+ 1) +H)Sx\n0Sz\n0+BSy\n0Sz\n0Sy\n1−CSz\n1((Sx\n0)2+ (Sy\n0)2) + 2DSz\n0((Sx\n0)2+ (Sy\n0)2)],(50)\ndSx\n1\ndt=CSz\n0Sy\n1−BSy\n0Sz\n1−2DSy\n1Sz\n1\n+α[BSy\n0Sx\n1Sy\n1−(A(Sx\n0+ 1) +H)((Sy\n1)2+ (Sz\n1)2) +CSz\n0Sx\n1Sz\n1−2DSx\n1(Sz\n1)2], (51)\ndSy\n1\ndt=ASz\n1(Sx\n0+ 1)−CSx\n1Sz\n0+ 2DSx\n1Sz\n1+HSz\n1\n+α[(A(Sx\n0+ 1) +H)Sx\n1Sy\n1−BSy\n0((Sx\n1)2+ (Sz\n1)2) +CSz\n0Sy\n1Sz\n1−2DSy\n1(Sz\n1)2], (52)\ndSz\n1\ndt=BSx\n1Sy\n0−ASy\n1(Sx\n0+ 1)−HSy\n1\n+α[(A(Sx\n0+ 1) +H)Sx\n1Sz\n1+BSy\n0Sy\n1Sz\n1−CSz\n0((Sx\n1)2+ (Sy\n1)2) + 2DSz\n1((Sx\n1)2+ (Sy\n1)2)].(53)\nNote that the terms proportional to αare generalizations for the present two-spin excitation case\ncompared to those given in Eqs. (9) - (11). As such the system (48) - (53) does not seem to be\nanalytically solvable. In Fig. 4, we numerically integrate the system for both the undamped case\n(α= 0) and the damped case ( α/negationslash= 0) for nonzero Dand present the solutions in the undamped\nand damped cases to show the existence of more general internal localized excitations. The analysis\ncan be extended to even more general situations, which will be presented elsewhere.\nVII. CONCLUSION\nBy looking at the simplest internal localized excitations in an anisotropic Heisenberg ferromag-\nnetic spin chain in external magnetic field with additional Gilbert damping, we deduced the explicit\nsolutions which characteristically show the effect of damping. Then applying a spin current in an\n12tSx\n0(t)(i)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n0(t)(ii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n0(t)(iii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSx\n1(t)(iv)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n1(t)(v)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n1(t)(vi)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1Fig. 4 (a): Undamped two-spin excitations\ntSx\n0(t)(i)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n0(t)(ii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n0(t)(iii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSx\n1(t)(iv)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n1(t)(v)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n1(t)(vi)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\nFig. 4 (b): Damped two-spin excitations\nFIG. 4. Solution of Eqs. (48)-(53) for two-spin excitations S0andS1for the (a) undamped ( α= 0) and\n(b) damped cases ( α= 0.005), with the anisotropy parameters A= 0.1,B= 0.23,C= 1.0andD= 0.3\nand the magnetic field H= 113Oe.\nSTNO of appropriate magnitude, we pointed out how the tendency toward damping can be offset\nexactly and thereby sustaining the magnetic oscillations. Our prediction about a PT-symmetric\n13STNO could be tested in magnetic multilayer structures with carefully balanced gain and loss. We\nhave also pointed out how such controlled oscillations can be effected in a recently suggested nano-\nmagnetic trilayer device. It will be insightful to observe these oscillations in appropriate magnetic\nsystems experimentally. Finally, in a related context we note that nonreciprocal optical modes can\nexist at an interface between two PT-symmetric magnetic domains near a frequency corresponding\nto almost zero effective permeability20.\nVIII. ACKNOWLEDGMENTS\nThe authors wish to thank Dr. D. Aravinthan for his help in the numerical analysis. The\nresearch work of ML was supported by a NASI Senior Scientist Platinum Jubilee Fellowship\n(NAS 69/5/2016-17) and a DST-SERB Distinguished Fellowship (No.: SERB/F/6717/2017-18).\nML was also supported by a Council of Scientific and Industrial Research, India research project\n(No.: 03/1331/15/EMR-II) and a National Board for Higher Mathematics research project (No.:\n2/48(5)/2015/NBHM(R.P.)/R&D II/14127). ML also wishes to thank the Center for Nonlinear\nStudies, Los Alamos National Laboratory, USA for its warm hospitality during his visit in the\nsummer of 2017. This work was supported in part by the U.S. Department of Energy.\nAPPENDIX A\nHere we briefly point out how to solve Eq. (20). Introducing the transformation\nSy\n0(t) =eα(2A+H)/integraltext\nSx\n0dt·ˆSy\n0(t) (A. 1)\ninto Eq. (20), we obtain\nd2ˆSy\n0(t)\ndt2+ (2A+H)2ˆSy\n0(t) = 0. (A. 2)\nConsequently, we have\nˆSy\n0(t) = ˆacos(Ωt+δ),Ω = 2A+H, (A. 3)\n14where ˆaandδare arbitrary constants. Then, the prefactor on the right hand side of (21) can be\ndeduced as follows. Since\nI=/integraldisplay\nSx\n0dt=/integraldisplayc2exp(−2α(2A+H)t)−1\nc2exp(−2α(2A+H)t) + 1dt=−1\n2α(2A+H)log(c2exp(−2α(2A+H)t) + 1)2\nc2exp(−2α(2A+H)t,\n(A. 4)\nthe prefactor becomes\nexp/bracketleftbigg\nα(2A+H)/integraldisplay\nSx\n0dt/bracketrightbigg\n=cexp(−α(2A+H)t)\nc2exp(−2α(2A+H)t) + 1. (A. 5)\nCorrespondingly\nSy\n0=cexp(−α(2A+H)t)\nc2exp(−2α(2A+H)t) + 1ˆacos(Ωt+δ),Ω = 2A+H, (A. 6)\nwhich is Eq. (21).\nREFERENCES\n1B. Hillerbrands and K. Ounadjela, Spin Dynamics in Confined Magnetic Structures , Vols. I & II\n(Springer, Berlin) 2002.\n2M. Lakshmanan, Philos. Trans. R. Soc. A 369(2011) 1280.\n3B. Georges, V. Cros and A. Fert, Phys. Rev. B 73(2006) 0604R.\n4Z. Yang, S. Zhang and Y. C. Li, Phys. Rev. Lett. 99(2007) 134101.\n5M. Lakshmanan, Phys. Lett. A 61(1977) 53.\n6K. Nakamura and T. Sasada, J. Phys. C 15(1982) L915.\n7E. K. Sklyanin, LOMI preprint E-3-79, Leningrad (1979).\n8Y. Ishimori, Prog. Theor. Phys. 72(1984) 33.\n9M. Lakshmanan and A. Saxena, Physica D 237(2008) 885.\n10H. Zabel and M. Farle (Eds.), Magnetic Nanostructures: Spin Dynamics and Spin Transport\n(Springer, Berlin) 2013.\n11A. Sievers and S. Takeno, Phys. Rev. Lett. 61(1988) 970.\n1512Y. Zolotaryuk, S. Flach and V. Fleurov, Phys. Rev. B 63(2003) 214422.\n13M. Lakshmanan, B. Subash and A. Saxena, Phys. Lett. A 378(2014) 1119.\n14G. Bertotti, I. Mayergoyz and C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems\n(Elsevier, Amsterdam) 2009.\n15B. Georges, J. Grollier, V. Cros and A. Fert, Appl. Phys. Lett. 92(2008) 232504.\n16B. Subash, V. K. Chandrasekar and M. Lakshmanan, Europhys. Lett. 102(2013) 17010; 109\n(2015) 17009.\n17J. Turtle, K. Beauvais, R. Shaffer, A. Palacios, V. In, T. Emery and P. Langhini, J. Appl. Phys.\n113(2013) 114901.\n18J. C. Slonczewski, J. Magn. & Magn. Mater. 159(1996) L261.\n19J. M. Lee, T. Kottos and B. Shapiro, Phys. Rev. B 91(2015) 094416.\n20J. Wang, H. Y. Dong, C. W. Ling, C. T. Chan and K. H. Fung, Phys. Rev. B 91(2015) 235410.\n16"    },    {        "title": "1805.01776v2.Superparamagnetic_Relaxation_Driven_by_Colored_Noise.pdf",        "content": "arXiv:1805.01776v2  [cond-mat.stat-mech]  7 May 2018Superparamagnetic Relaxation Driven by Colored Noise\nJ. G. McHugh,1R. W. Chantrell,1I. Klik,2and C. R. Chang2\n1Department of Physics, The University of York, York, YO10 5D D, UK\n2Department of Physics, National Taiwan University, Taipei , Taiwan\nA theoretical investigation of magnetic relaxation proces ses in single domain particles driven\nby colored noise is presented. Two approaches are considere d; the Landau-Lifshitz-Miyazaki-Seki\nequation, which is a Langevin dynamics model based on the int roduction of an Ornstein-Uhlenbeck\ncorrelated noise into the Landau-Lifshitz-Gilbert equati on and a Generalized Master Equation ap-\nproach whereby the ordinary Master Equation is modified thro ugh the introduction of an explicit\nmemory kernel. It is found that colored noise is likely to bec ome important for high anisotropy\nmaterials where the characteristic system time, in this cas e the inverse Larmor precession frequency,\nbecomes comparable to the correlation time. When the escape time is much longer than the corre-\nlation time, the relaxation profile of the spin has a similar e xponential form to the ordinary LLG\nequation, while for low barrier heights and intermediate da mping, for which the correlation time is\na sizable fraction of the escape time, an unusual bi-exponen tial decay is predicted as a characteristic\nof colored noise. At very high damping and correlation times , the time profile of the spins exhibits\na more complicated, noisy trajectory.\nI. INTRODUCTION\nThermally-activated magnetization reversal over an\nanisotropic energy barrier is the driving force for switch-\ning in magnetic materials. Theoretical understanding\nwasfirst developed by N´ eel1basedon the transition state\ntheory (TST) leading to an Arrhenius-like relaxation\ntime proportional to exp( EB/kBT) whereEBis the en-\nergy barrier, kBthe Boltzmann constant and Tthe tem-\nperature. Brown2provided further insight through the\nconstruction ofthe Langevinequation for the problem by\nthe introduction of white-noise fields into the Landau-\nLifshitz equation with Gilbert damping, leading to the\nstochastic Landau-Lifshitz-Gilbert (LLG) equation, An\nexpression for the relaxation time of thermally-driven\nescape over the energy barrier is then found through\nthe lowest eigenvalue of the correspondingFokker-Planck\nequation(FPE)governingthetime-evolutionoftheprob-\nability density function of the magnetization orientation.\nThe routeto the Arrhenius-likerelaxationtime expres-\nsion is one of two directions leading from the Langevin\nequation. The second, Langevin Dynamics (LD) ap-\nproach is the direct numerical solution of the Langevin\nequation3–6. There is a natural separation of timescales,\nwith LD used for high frequency applications such as\nmagnetic recording and the Arrhenius-like relaxation\ntimeusedforslowdynamicbehaviorarisingfromthermal\nactivationoverenergybarriers. The twoapproacheshave\nbeen compared by Kalmykov et. al.,7who calculated es-\ncape times for both cases giving excellent agreement for\nthe variation of escape time with damping constant and\ndemonstrating the importance of starting the LD calcu-\nlations from the correct thermal equilibrium distribution\nwithin the energy minimum.The LLG equation for a single spin takes the well-\nknown form\ndS\ndt=−γ\n1+α2/parenleftbig\nS×H+αS×(S×H)/parenrightbig\n,(1)\nwhereαis the phenomenological damping constant, γ=\n1.7611T−1s−1andSisaunitvectorinthedirectionofthe\nspin,S=µ/µs. The local magnetic field, H, is derived\nfrom the first derivative of the spin Hamiltonian Hwith\nrespect to the spin degree of freedom,\nH=−1\nµs∂H\n∂S. (2)\nThermal fluctuations are necessary to incorporate the\ndeviations of a particular spin from the average tra-\njectory. This is done via the formal inclusion of ran-\ndom fields in the LLG equation. In order to realize the\nFluctuation-Dissipation theorem for this system, these\nthermal fields must also be proportionalto the same phe-\nnomenological damping constant, αthat occurs in the\ndamping. The moments of the thermal field are then\ngiven by\n∝angbracketleftHth,i(t)∝angbracketright= 0 (3)\n∝angbracketleftHth,i(t)Hth,j(t′)∝angbracketright=2αkBT\nγµsδ(t−t′)δij(4)\nwherei,jlabel the spin components.\nIn all numerical simulations, we interpret the stochas-\ntic equation in the Stratonovich sense and employ the\nHeun method An implicit assumption of this approach is\nthe presence of white noise, which exists in the zero cor-\nrelation time limit for some physical noise process with a2\nwell-defined correlation time. Such a colored noise may\nbe implemented for a magnetic system through the use\nof the Landau-Lifshitz-Miyazaki-Seki pair of Langevin\nequations, which take the form\ndS\ndt=γS×/parenleftbig\nH+η/parenrightbig\n, (5)\ndη\ndt=−1\nτc(η−χS)+R, (6)\nwhereτcis the correlation time and χis a spin-bath cou-\npling which is related to the phenomenological damping\nparameter as α=γχτcin the limit of small correlation\ntimes. The autocorrelation of the white noise field, R, is\ngiven by\n∝angbracketleftRi(t)Rj(t′)∝angbracketright=2χkBT\nτcµsδijδ(t−t′).(7)\nThis pair of Langevin equations leads to a frequency-\ndependent damping of the spin together with an expo-\nnentially correlated noise term in the spin-only space,\n∝angbracketleftˆηi(t)ˆηj(t′)∝angbracketright=χkBT\nµse−(t−t′)\nτcδij=χkBT\nµsK(t−t′)δij\n(8)\nwhereK(t) = exp−(t−t′)\nτcis the exponential memory ker-\nnel. For completeness, additional background on the\nLLMS Langevin equation and colored noise is included\nin Appendix A.\nAn alternative approach to the Langevin equation is\nthe discrete orientation approximation, whereby, in the\nlimit oflarge barriers, the detailed dynamics are replaced\nby phenomenological rate equations describing transi-\ntions between the minima of the magnetic potential. We\nmay augment this description by the introduction of a\nmemory kernel into the rates, thus replacing the master\nequation description with a generalized master equation\nwhich explicitly incorporates the retardation effect into\nthe rate equations.\nHere we investigate the introduction of colored noise\ninto the calculation of escape rates. This leads to sig-\nnificant effects for materials with large magnetocrys-\ntalline anisotropy energies, including the prediction of\nbi-exponential behavior at intermediate damping, when\nthe characteristic time of the relaxation process becomes\ncomparable to the heat bath correlationtime. The paper\nis organized as follows. We first outline thermally acti-\nvated escape times for single nanoparticles, followed by\nanintroductionofcolorednoiseintotheLangevinformal-\nism via the LLMS equations. We then derive the relax-\nation profile from the non-Markovian generalized exten-\nsion of the rate equation, followed by a systematic inves-\ntigationoftheeffects ofthebarrierheightandcorrelation\ntimes on the relaxation profile from LLMS simulations.A. Thermally-Assisted Magnetization Reversal\nWe will investigate here the effect that colored noise\nhas on the dynamics of the thermal escape problem for a\nmagnetic nanoparticle. The spin Hamiltonian of the sys-\ntem contains both an applied field and anisotropy term,\ntaking the form\nH=−KVS2\nz−µs/vectorH·S, (9)\nwhereKis the anisotropy constant and Vis the particle\nvolume. For the escape problem we have a spin energy\npotential of the form\nV(θ,φ) =σβ−1/parenleftbig\nsin2θ−2h(cosψcosθ(10)\n+sinψsinθcosφ)/parenrightbig\n,\nwhereθ,φare respectively the polar and azimuthal\ncomponents of the spin in spherical coordinates, σ=\nKV/k BTis the reduced barrier height parameter, h=\nH/2σis the reduced field, β= (kBT)−1andψis\nthe angle between the easy-axis and the applied field.\nThis potential has a bistable character under the condi-\ntion than the critical applied field value, h < h c(ψ) =\n((cos2/3ψ+sin2/3ψ)−3/213, in which case there are local\nand global minima in the north and south polar regions,\nwith an equatorial saddle point between them. We are\nthen interested in the calculation of the characteristic es-\ncape time of a spin initialized in one such minimum.\nFor the special case of aligned field and easy axis, for\nwhichψ= 0 the potential is\nV(θ) =σβ−1/parenleftbig\nsin2θ−2hcosθ/parenrightbig\n. (11)\nIn this case the escape time takes the Arrhenius form,\nwhere the barrier energy, EB, is proportional to the\nanisotropy energy, leading to an escape time\nτ∝f−1\n0eKV/k BT(12)\nwheref0is the attempt frequency, the frequency of Lar-\nmor gyromagnetic precession at the bottom of the well.\nWeinvestigatetheescapetimeinthecoloredandwhite\nnoise cases through repeated numerical integration of the\nLangevin equations for a spin initialized in a potential\nminimum. An important consideration for such simula-\ntions is the choice of initial and switching condition for\nthe spin. We will initialize the spins with the Boltz-\nmann distribution at the bottom of the well in order to\navoidinconsistenciesat lowdamping, while the switching\ncondition is chosen such that Sz<−0.5, with the spin\ninitialized in the positive z-direction, so that the spin is\nsufficiently deep in the well such that it has escaped.\nII. LLMS ESCAPE TIMES & COLORED NOISE\nA. System time τsvs.τccharacteristic bath time.\nFor the uniaxial escape problem the external field in\nthe LLMS will consist of an external applied part and an3\nanisotropy contribution\nH=Ha+H0 (13)\nthe magnitude of the anisotropic contribution depends\non the orientation of the spin and is given by Ha=\n2ku\nµs/vectorSz·/vectorz=Hk/vectorSz·/vectorzwhere/vectorzis the direction of easy\nmagnetization and kuis the anisotropy energy. To gain\nintuition into the relevant timescales for the relaxation\nproblem, we will assume the uniaxial case in the follow-\ning, where the external field is applied along the same\ndirection as the easy axis, such that both fields only have\ncomponents in the z-direction.\nWe note that the anisotropic field contribution varies\nwith the projection of the spin on to the easy-axis as\nHa=Hk(S·z)/vectorz= (Hkcosθ)/vectorz, (14)\nThelargest field magnitude and consequently the fastest\ntimescale of the problem is set by the value for which\nthe anisotropic field contribution is at its largest, which\nis when the spin and the easy-axis precisely coalign. For\nany other orientation, the field will be smaller and the\ntimescale of oscillation hence slower. We may then take\nthe spin-only Langevin equation,\ndS\ndt=γS(t)×/parenleftbig\n(Hkcos(θ))/vectorz+¯η−χ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′/parenrightbig\n,\n(15)\nand proceed to scale this equation by the maximum\nanisotropy field value. Defining the system time for the\nspin asτs= (γHk)−1then\ndS\ndt=1\nτsS(t)×/parenleftbig\ncos(θ)/vectorz+H−1\nk¯η\n−H−1\nkχ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′/parenrightbig\n.(16)\nWe may scale the time variable in the Langevin equa-\ntionsothatthesystemtimeisremovedbytaking ζ=τst.\nThen we have\ndS\ndζ=S(ζ)×cos(θ)/vectorz+S(ζ)×/parenleftbigg\nH−1\nk¯η(ζ)\n+H−1\nkχ/integraldisplayζ′\n−∞dζ′e−(ζ−ζ′)τs\nτcdS(ζ′)\ndζ′/parenrightbigg\n(17)\nThe autocorrelation of the noise is similarly transformed\nto become\n∝angbracketleft¯η(ζ)¯η(ζ′)∝angbracketright=τs\nτc¯De−ζ−ζ′)τs\nτc=¯D\nτe−(ζ−ζ′)/τ(18)\nwhereτs/τc=τand¯D=D/τs=χτkBT/µs. We\ncan then write the coupling as ¯ χ=χ/Hk, and ab-\nsorb theHkfactor into the diffusion constant for the\nthermal field. Since the thermal fields are given by\n¯η(ζ) =√\n2D\nτ/integraltextζ\n−∞K(ζ−ζ′)Γ(ζ′), the diffusion constant\nbecomes\n¯D=χτkBT\nµsH2\nk=¯χτkBT\n2ku=¯χτ\n2σ(19)11.21.41.61.82\n0.01 0.1 1τ/τLLG\nγ Hk τcσ = 2\nσ = 5\nσ = 7.5\nFIG. 1. Escape time, normalized to the uncorrelated LLG\nescape time vs correlation time, from LLMS simulations for a\nCo nanoparticle with α= 0.05 and different reduced barrier\nheights, σ,\nwhereσ=ku/kBT. The final expression for the\nLangevin equation is then\ndS(ζ)\ndt=S(ζ)×/parenleftbig\ncos(θ)/vectorz+¯η−¯χ/integraldisplayζ\n−∞dζ′K(ζ−ζ′)dS(ζ′)\ndζ′/parenrightbig\n.\n(20)\nIn the case that τ≪1 andτc≪τs, the memory\nkernels appearing in the noise and damping terms are\nreduced to delta functions and the white noise behav-\nior is restored. Additionally the bath coupling and the\nstrength of the thermal fluctuations are reduced by the\nanisotropy field, so that in the event of a very large\nanisotropy the precessional dynamics of the spin dom-\ninate the thermal and damping parts. We then conclude\nthat the condition τc/greaterorsimilar(γHk)−1dictates whether the\neffect of correlations are relevant in the system dynamics\nin the high barrier limit.\nThis prediction is borne out in numerical simulations\nof the LLMS equation. Figure 1 depicts the escape time\ncalculated using the LLMS model for a Co nanoparti-\ncle of volume V= 8×10−27m3, with anisotropy en-\nergyKV= 1.12×1021J, and a magnetic moment µs=\n1.12×10−20J/T.wherethe correlationtime is normalized\nby the inverse of the Larmor precession frequency, and\nthe escape time in the LLMS is normalized by the escape\ntime calculated from the Markovian LLG equation. The\nescape rate departs from the LLG escape rate only once\nthe correlation time is some significant fraction of the\nLarmor time, and for increasing barrier height the corre-\nlation time must be a larger fraction of the gyromagnetic\nprecession before the escape rate departs from the LLG\nprediction.\nFigure 2 shows a comparison of the escape time for the\nCo nanoparticle and a SmCo 5nanoparticle of the same4\n110\n1.0E-15 1.0E-14 1.0E-13 1.0E-12 1.0E-11 1.0E-10τ/τLLG\nτcSmCo\nCo\nFIG. 2. Comparison of simulation results for systems with pa -\nrameters chosen tobe similar toSmCo 5and Co nanoparticles,\nrespectively, for large reduced barriers σ= 13.5, and a fixed\nα= 0.05. The higher anisotropy SmCo 5exhibits departure\nfrom LLG behavior at smaller correlation times.\nvolume. The SmCo 5material parameters are taken to be\nµs= 6.4×10−18J/T, and anisotropy KV= 2.16×10−16,\na much higher anisotropy energy density than Co. This\nhigher anisotropy gives the nanoparticle a faster system\ntime, which causes the LLMS to depart from the LLG\nfor smaller bath correlation times, τc, on the order of\n50−100fsfor the SmCo 5particle, while it is approxi-\nmately 1psfor the Co nanoparticle. The fact that the\nsystem time is inversely proportional to the magnitude\nof the anisotropy field is exhibited in the simulations by\nthe difference between LLMS and LLG escape rates at\nsmaller values of the bath correlation time for the mate-\nrial with higher magnetic anisotropy.\nB. Arrhenius Behavior\nCrucially, it is found that the Arrhenius behavior of\nthe escape rate is recovered from LLMS simulations in\nthe limit of large barrier height. In figure 3 we show the\ntemperature- dependence of the escape time vs reduced\nbarrier height.\nIn the high damping case, we see that the escape rates\nbegin to convergeas the temperature tends towardszero.\nAs the escape time between the wells becomes much\nlonger than the bath correlation time, the detailed dy-\nnamics of the spin within the well becomes less relevant.\nAt low damping, the LLMS and LLG appear not to\nconverge even at the larger barrier heights considered\nhere. We attribute this difference to the difference in\ndamping regimes and the physically distinct mechanisms\ninvolved in the escape process between the two regimes.\nEscape at high damping is mediated by thermal fluctu-0.010.1110100100010000100000\n0246810121416τkr γ Hk\nσLLMS, h=0.2, α = 0.01\nLLG, h=0.2, α = 0.01 Ψ = 1/4\n0.010.1110100100010000\n0246810121416τkr γ Hk\nσLLMS, h=0.3, α = 1, Ψ = 1/4\nLLG, h=0.3, α = 0.01, Ψ = 1/4\n0.010.1110100100010000\n0246810121416τkr γ Hk\nσLLMS, h=0.2, α = 1\nLLG, h=0.2, α = 1 Ψ = 1/4\n0.010.11101001000\n0246810121416τkr γ Hk\nσLLMS, h=0.3, α = 1, Ψ = 1/4\nLLG, h=0.3, α = 1, Ψ = 1/4\nFIG. 3. Escape time, τγHkvs reduced barrier height, σ,\nfrom LLMS and LLG simulations, for different values of the\napplied field h=µsH/σand damping, α, with a fixed angle\nof Ψ = π/4 between the applied field and the easy axis of\nmagnetization. 1:Low damping, h= 0.2,2:Low damping,\nh= 0.3.3:High damping, h= 0.2,4:High damping,\nh= 0.3.\nations, which liberate the bound spin. In the limit of\nvanishing temperature the infrequency of thermal oscil-\nlations of sufficient energy dominate the escape behavior\nand the escape rates converge.\nIn contrast, the energy-controlled diffusion regime is\ncharacterized by the almost-free precessional motion of\nthe spin in the well. In the highly correlated case, the5\nsimple damping is replaced with a frequency-dependent\ndamping, an effect which increases the overall effective\ndamping. In the limit T→0, this inhibits the escape\nratebetweenthewellsbydecreasingtherateatwhichthe\nspin is able to attain a trajectory with sufficient escape\nenergy.\nIII. RATE EQUATIONS FOR\nTHERMALLY-ACTIVATED MAGNETIZATION\nREVERSAL\nA. Master Equation\nThe master equation is a phenomenological set of first-\norder differential rate equations for a multi-level system,\nwhich takes the form\ndni\ndt= Γij(t)nj(t), (21)\nwhereniis a probability vector representing the proba-\nbility that the system is in one of a discrete set of states,\nandi,jlabel those discrete states, while the matrix of\ncoefficients Γ i,jdictates the transition rate from state i\nto the state jof the system.\nThe dynamics of the thermally-assisted escape prob-\nlem in a magnetic system may be approximated by such\na master equation under the condition that the energy\nbarrier is large compared to the thermal energy, σ >1,\nbut not too large such that it would inhibit inter-well\ntransitions. This approximation to the Langevin dynam-\nics is called the discrete orientation approximation . The\nspin orientations are assumed to be restricted only to the\n2 minima of the potential energy dictated by the spin\nHamiltonian. The time evolution of the occupation of\neach state follows from Eq. 21, where i,j= 1,2. The\ntransition matrix elements follow from the applied field,\nanisotropy and temperature. In particular, we will as-\nsume a fixed applied field, such that the transition rates\nare constant in time and the matrix takes the form\nΓij=/parenleftbigg\n−κ12κ21\nκ12−κ21/parenrightbigg\n, (22)\nIn the uniaxial case these rates are given by κ1→2=\nκ12=f0exp(−σ(1+h)2) andκ2→1=κ21=\nf0exp(−σ(1−h)2), whereσandhare the reduced bar-\nrier height and applied field, respectively. The time evo-\nlution of the population of the state n1is then explicitly\ngiven by\ndn1\ndt=−κ12n1+κ21n2= (κ12+κ21)n1+κ21.(23)\nThe time-evolution of the magnetization follows from\nthe individual rates for the two wells, where the magne-\ntization is given by m(t) =n1(t)−n2(t) and is subject\nto the normalization condition n1(t) +n2(t) = 1. The\ndifferential equation for the magnetization is then\ndm\ndt=−Γ1m(t)−Γ2, (24)where Γ 1=κ12+κ21and Γ 2=κ12−κ21. This is the\nsame form as the rate for the individual wells, Eq. 23.\nFor an initial magnetization m0=n1(t= 0)−n2(t=\n0), the magnetization as a function of time is a simple\nexponential,\nm(t) =e−Γ1t(Γ1m0+Γ2)\nΓ1−Γ2\nΓ1,(25)\nwhich tends to the value\n−Γ1\nΓ2=κ21−κ12\nκ12+κ21. (26)\nIn the long-time limit, the steady state magnetization\ncorresponding to the difference in the transition rates\nbetween the wells, if κ2→1> κ1→2, the transition rate\ninto well 1 is greater than the rate out, and we have a\npositive magnetization, as expected.\nB. Generalized Master Equation\nThe non-Markovian extension of the master equation\nformalismiswhat iscalledageneralizedmasterequation.\nUnder this model, the set of i×jrates represented in the\ntransition matrix in Eq. 21 are promoted to a set of i×j\nmemory kernels for the transitions between the wells i,j,\nreplacing the set of first-order differential equations with\na set of integro-differential equations for the population\nof each well,\ndni\ndt=/integraldisplay∞\n0Mij(t−τ)n(τ)dτ. (27)\nWe will consider the simplified case\nMij(t) =e−t/Θ\nΘAij=K(t)Γij, (28)\nwhere Γ ijare the same constant transition rates consid-\nered in the Markovian master equations, now modified\nby a simple exponential kernel over the recent popula-\ntion of the well. The integro-differential expression for\nthe magnetization then becomes\ndm\ndt=−Γ1/integraldisplay∞\n0K(t−τ)m(τ)dτ−Γ2/integraldisplay∞\n0K(t−τ)dτ.\n(29)\nWhere we note that for the exponential kernel, K(t) =\ne−t/Θ\nΘ, the uncorrelated form of the master equation\nis recovered in the limit of vanishing correlation time,\nlimΘ→0K(t) =δ(t).\nThe Laplace transform of this equation is\nωm(ω)−m0=−Γ1K(ω)m(ω)−Γ2\nωK(ω),(30)\nwhereK(ω) =L(K(t)) is the Laplace transform of the\nmemory kernel,\nK(ω) =Θ−1\nω+Θ−1=1\n1+Θω, (31)6\nwe then have\nm(ω) =−Γ2\nωK(ω)+m0\nω+Γ1K(ω). (32)\nAfter inserting the expression for the Laplace transform\nof the kernel we find\nm(ω) =−Γ2\nω+m0(1+Θω)\nΘω2+ω+Γ1. (33)\nFinally we solve for the time-dependence of the mag-\nnetization by taking the inverse Laplace transform,\nm(t) =L−1[(1+Θω)\nΘω2+ω+Γ1] =φ(t)(Γ1m0+Γ2)\nΓ1−Γ2\nΓ1,\n(34)\nwe note that this bears a strong resemblance to the\nMarkovian expression, Eq. 25, with the exponential be-\ning replaced by the function φ(t), which is\nφ(t) =1\n2β/parenleftBig\n(β−1)e−t(1+β)/2Θ+(β+1)e−t(1−β)/2Θ/parenrightBig\n,\n(35)\nwhereβ=√1−4Γ1Θ. In the limit t→ ∞, the value of\nthe magnetization again tends to−Γ2\nΓ1. To see that this\nagrees with the uncorrelated solution for small correla-\ntion times, we may expand βin Θ for small Θ, hence\nβ= 1−2Γ1Θ, inserting into the magnetization it be-\ncomes\nm(t) =β−1\n2βe−t/2ΘeΓ1t+(β+1)\n2βe−Γ1t.(36)\nAs Θ→0,β→1, and only the second term in the\nexpression for the magnetization remains, m(t) =e−Γ1t,\nso the small correlation time limit of the spin evolution\nagrees with the non Markovian master equation.\n00.10.20.30.40.50.60.70.80.91\n00.511.522.53m(t)\nΓ1tR < 0.01\nR=0.1\nR = 0.24\nFIG. 4. m(t) vst, forR= 0,0.1,0.2, under the initial condi-\ntionm= 1, with transition rates κ12= 1,κ21= 0Finally, we note that the solution for the magnetiza-\ntion breaks down into two regimes. First, we note that\nthe expression for βdepends only on the product of the\ncorrelation time, Θ, and the rate Γ 1, and not on their\nspecific individual values. We may then discuss the be-\nhavior of the model in terms of only the ratio parameter\nR= Γ1Θ = Θ/Γ−1\n1, which gives the ratio of the well\ncorrelation time to the escape time. Rewriting the Eq.35\nfor the spin vs time,\nm(t) =(Γ1m0+Γ2)\nΓ1/parenleftBig\n(e−t/2Θ([eβt/2Θ(37)\n−e−βt/2Θ]/2β+[e−βt/2Θ+eβt/2Θ]/2)/parenrightBig\n−Γ2\nΓ1,\nwhich may be simplified in terms of hyperbolic trigono-\nmetric functions,\nm(t) =e−t/2Θ/parenleftBigsinh(βt/2Θ)\nβ+cosh(βt/2Θ)/parenrightBig\n.(38)\nFor smaller R <1\n4, we have a real value of β=√1−4R, and the time-dependence of the spin corre-\nsponds to Eq. 38. In Figure 4, we plot the time-evolution\nfor values of R <1\n4. Once the correlation time is some\nsizable fraction of the escape time, the behavior begins\nto depart from the simple exponential behaviorpredicted\nin the Markovian system. At early times the magnetiza-\ntion decays more slowly than the exponential decay and\nat later times it decays more quickly, while the timescale\nover which the decay occurs (Γ 1) remains the same. The\neffect of the increasing correlation time between the pop-\nulations of the wells is then to shift the process to differ-\nent, lower frequencies.\nIn the case that R >1\n4, we have an imaginary argu-\nment to sinh and cosh, we then have an expression for\nm(t)\nm(t) =e−t/2Θ(sin(bt/2Θ)\nb+cos(bt/2Θ)) (39)\nwhereb=√\n4R−1. We note that the solutions take\nthe form of damped oscillations which tends toward the\nequilibrium value of the magnetization. However, these\nsolutions are unphysical as the occupation in individual\nwells maybecome lessthan 0 for these values. This is not\nsurprising, as for longer correlation times the generalized\nmaster equation will overestimate the population in each\nwell and generate a time evolution which will continue to\nreduce the population of a well, even when that well is\npresentlyempty. Itisalsounclearwhatitwouldmeanfor\nthe correlation time of the well population to exceed or\nbe on the order of the overall escape time, as this would\nimplythatthetimescaleoverwhichthespinpopulationis\ncorrelatedexceeds the overallescape time for the system,\nwhich is itself determined by changes in the individual\nwell populations.7\nIV. COMPARISON\nWe may now directly compare the magnetic relaxation\nprofiles calculated from explicit numerical integration of\nEqs. 5, 6 at various barrier heights, damping and cor-\nrelation times, to the biexponential decay predicted by\nthe generalized master equation. In all of the present\nsimulations we again use simulation parameters compa-\nrable to the Co nanoparticle of volume V= 8×10−27m3,\nanisotropy energy density K= 4.2×105J/m3giving an\nanisotropy energy KV= 1.12×1021J, and a magnetic\nmomentµs= 1.12×10−20J/T, while no external applied\nfield is assumed, Hext= 0.\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)σ = 2, α = 0.01\ne-t/τ\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)σ = 6, α = 0.5\ne-t/τ\nFIG. 5. Spin relaxation profiles from LLG simulations for\nTOP:σ= 2,α= 0.01, giving an exponential decay with\ncharacteristic escape time τ= 5×10−9sandBOTTOM :\nσ= 6,α= 0.5,τ= 4.5×10−9s\n.\nThe spins are initialized in the equilibrium Boltz-\nmann distribution in one of the minima of the po-\ntential energy, according to the distribution P(θ)∝\nsin(θ)exp(−ku/kBTsin2(θ)). To ensure that the noise\nis equilibrated with the spin at the correct temperature,\nthe noise is initially set to ηi,j,k= 0, and is then evolved\nin the presence of the equilibrium distribution in the well\nuntiltheycomeintothermalequilibrium. Theinitialcon-\ndition of the noise is important, as, for example, a choice\nofη(t= 0) = 0, will result in a field which quickly alignswith the spins in the potential minimum and give an un-\nphysical increase in the well population from equilibrium\nat short times.\nThe time-evolution of the magnetization, M(t) =\n∝angbracketleftSz(i)∝angbracketrightis then plotted, normalized by the initial rema-\nnent magnetization inside of the well, Mr=M(0).\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)τc = 1, σ = 2, α = 0.01\ne-t/τ\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)τc = 1, σ = 6, α = 0.5\ne-t/τ\nFIG. 6. Exponential behavior from LLMS simulations, for\nTOP:τc= 1,σ= 2.α= 0.01 we have an exponential decay\nwith escape time τ= 5.5×10−9, andBOTTOM :τc= 1,\nσ= 6.α= 0.5τ= 5.3×0−9s For low damping and large\nbarrier heights, the correlation time is much smaller than t he\nescape time.\nIn Figure 5, we depict the numerical calculation of the\nrelaxation profile from the LLG. This gives rise to an ex-\nponential behavior with a single relaxation time, which\nis directly comparable to the exponential decay of the\nmaster equation. In general, the relaxation profile from\nthe LLG may be non-exponential, with both the inte-\ngral relaxation time and the decay profile depending on\nthe higher-order eigenvalues of the Fokker-Planck opera-\ntor and the equilibrium correlation functions of the spin,\nτi\nint=/summationtext\nkτi\nkλk. However, the relaxation is dominated\nby the first eigenvalue in the high-barrier limit and for\nsmall applied fields , for σ>1, with good agreement be-\ntween the LLG and exponential decay for σas low as 2,\nas is shown in Figure 5.\nFigure 6 shows the relaxation from LLMS simulations\nof the Co nanoparticle, where the correlation time is cho-8\n0.10.20.30.40.50.60.70.80.91\n0e+005e-111e-101e-102e-103e-10M(t)/Mr\nt (s)τc = 1, σ = 2, α = 0.5\ne-t/τ\nFIG. 7. Biexponential behavior from LLMS simulations for\nτ= 1,σ= 2,α= 0.5 andτ= 1.48x10−10\nsen to be of the order of the inverse Larmor precession\ntime such that τc≈(γHk)−1. In both the cases of low\ndamping and higher barriers, we see that the ordinary\nexponential behavior of the LLG is retained. In this case\nthe escape time is much larger than the correlation time\nofthenoise,andtherelaxationaldynamicsareunaffected\nby the intra-well dynamics of the spin which occur on a\nmuch faster timescale than the relaxation, τc/τ≈0.01\nfor both simulations.\nIn the intermediate-to-high damping and high damp-\ning regimes, the behavior of the magnetization becomes\nmuchmoreinterestinganddepartsfromthe LLG. In par-\nticular,forarelativelysmallbarrierof σ= 2,α= 0.5and\nacorrelationtimeagainoftheorderoftheinverseLarmor\nfrequency. In this case the ratio of the escape to the cor-\nrelationtime is τc/τ= 9.4×10−12s/1.48×10−10s≈0.06.\nThe influence of the spin correlation is now visible in the\nrelaxation profile of the escape, as shown in Figure 7,\nwhich is similar to the biexponential deviation predicted\nby the generalized master equation, with the relaxation\nproceeding more slowly at earlier times and speeding up\nat later times.\nFinally, for very long correlationtimes and high damp-\ning, the correlation time remains a sizable fraction of\nthe escape time. However the biexponential behavior\nis no longer evident as shown in figure 8. The decay\nremains approximately exponential with a highly noisy\npath, a possible indication that the precise decay profile\nis extremely dependent on the initial conditions for such\nstrong coupling between the spin and bath.\nV. CONCLUSIONS\nWe have investigated thermal relaxation in magnetic\nnanoparticles introducing colored noise. Two models00.20.40.60.81\n0e+002e-104e-106e-108e-101e-09M(t)/Mr\nt (s)τc = 5, σ = 2, α = 0.5\ne-t/τ\n00.20.40.60.81\n0e+001e-102e-103e-104e-105e-10M(t)/Mr\nt (s)τc = 5, σ = 2, α = 5\ne-t/τ\nFIG. 8. LLMS simulations at high damping and long corre-\nlation times. The behavior continues to depart from a purely\nexponential decay, but now exhibits a noisy, more compli-\ncated time-dependence. TOP:τc= 5,σ= 2 ,α= 0.5 and\nτ= 4.5×10−10,BOTTOM :τc= 5,σ= 2 ,α= 5 and\nτ= 1.8×10−10\n.\nare considered. The first is an approach based on the\nnumerical solution of the Landau-Lifshitz-Miyazaki-Seki\n(LLMS) model, which replaces the white noise approx-\nimation associated with the use of LLB-equation based\nmodels. Due to computational requirements the LLMS\napproach is useful for relatively short timescales, conse-\nquently a second approach is derived based on a general-\nizedmasterequationapproachinvolvingtheintroduction\nof a memory kernel. We find that the importance of col-\nored noise is determined by the ratio of the correlation\ntimeτcto the characteristic system time τs= (γHk)−1,\nwhich is essentially the Larmor precession time. Con-\nsequently correlated noise should become important for\nmaterials with large magnetic anisotropy such as SmCo 5\nwhere the characteristic time approaches femtoseconds.\nBoth models, the LLMS-based approach and the mas-\nter equation, although derived for different timescales,\nexhibit an unusual bi-exponential decay of the magneti-\nzation, which represents an interesting signature of the\npresence of colored noise.9\nAppendix A: Colored Noise\nIn this appendix we present some relevant background\nmaterial on the LLMS equation and colored noise.\n1. Landau-Lifshitz-Miyazaki-Seki\nThe LLMS equations constitute an implementation of\na colored noise in a system with a thermalization con-\ndition represented through the Fluctuation-Dissipation\ntheorem. We reproduce here the original derivation by\nMiyazaki and Seki10, of the spin-only expression of the\nLLMS, which allows us to compare the LLMS thermal\nfluctuations directly to the Ornstein-Uhlenbeck. The\ntime evolution of the LLMS noise term is similar to the\nOU, with an additional term which couples explicitly to\nthe spin,\ndη\ndt=−1\nτc/parenleftBig\nη(t)−χS(t)/parenrightBig\n+R. (A1)\nTakingD=χkBT\nµs, then the autocorrelation of the field\nRis∝angbracketleftR(t)R(t′)∝angbracketright= 2D\nτcδ(t−t′), and proceeding to solve\nas a first-order linear differential equation in the same\nmanner as the OU noise, we have\nη(t) =χ\nτc/integraldisplayt\n−∞dt′K(t−t′)S(t′) (A2)\n+/radicalbigg\n2D\nτc/integraldisplayt\n−∞dt′K(t−t′)Γ(t).\nAfter integrating the first term by parts, we have\nη(t) =/radicalbigg\n2D\nτc/integraldisplayt\n−∞dt′K(t−t′)Γ(t) (A3)\n−χ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′,\nand by inserting this into the precessional equation for\nthe spin, we get the spin-only form for the LLMS equa-\ntion,\ndS\ndt=γS(t)×/parenleftBig\nH+¯η−χ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′/parenrightBig\n,(A4)\nwhere we now label the thermal fluctuations by ¯η(t),\n¯η(t) =/radicalbigg\n2D\nτc/integraldisplayt\n−∞dt′K(t−t′)Γ(t′).(A5)\nThe autocorrelation of this thermal field is\n∝angbracketleft¯η(t)¯η(t′)∝angbracketright=DK(t−t′) (A6)\n=χkBT\nµsK(t−t′) =β−1\nµsχK(t−t′),\nRecognizing χK(t−t′) as the damping term, we see that\nthis is a representation of the Fluctuation-Dissipationtheorem for the colored noise, where the additional fac-\ntor ofµsarises from the spin normalization. Taking the\nzero correlation time limit,\nlim\nτc→0∝angbracketleft¯η(t)¯η(t′)∝angbracketright= 2Dτcδ(t−t′).(A7)\n00.20.40.60.81\n00.511.522.533.5P(θ)\nθLLMS, σ = 1\nAnalytical\n00.20.40.60.81\n00.511.522.533.5P(θ)\nθLLMS, σ = 10\nAnalytical\nFIG. 9. P(θ) vsθ, from numerical simulations of the LLMS\nequation for TOP:σ= 1 and BOTTOM :σ= 10, with\nτcγHk= 2.\n.\nWe note that the LLMS thus derived from the physi-\ncal consideration of the spin-field interaction is not im-\nmediately comparable with the typical expression for the\nOrnstein-Uhlenbeck colored noise, owing to the fact that\nthe 1/τcterm has been implicitly absorbed in the white\nnoise term. If we rescale the driving noise such that\nQ(t) =τcR(t), we then have a pair of Langevin equa-\ntions\ndS\ndt=γ(S×(H+η)), (A8)\nwhile the noise evolves as,\ndη\ndt=−1\nτc/parenleftBig\nη(t)−χS(t)+Q/parenrightBig\n. (A9)\nThe autocorrelation of the white noise is\n∝angbracketleftQ(t)Q(t′)∝angbracketright=2χτckBT\nµsδ(t−t′) = 2Dδ(t−t′),(A10)10\nwithD=χτckBT\nµs, while the limit of the autocorrelation\nof the thermal term in the spin-only expression is now,\nlim\nτc→0∝angbracketleft¯Q(t)¯Q(t′)∝angbracketright=D\nτcδ(t−t′),(A11)\nwhich is directly comparable to the Ornstein-Uhlenbeck\nform of the colored noise. The expression of the LLMS\nin terms of the bath variable Qhas the additional benefit\nthat/bracketleftbig\nQ/bracketrightbig\n=Tand so we can interpret Qas the thermal\nmagnetic field contribution to the evolution of the bath\nfield.\nFinally, we may see that the limit of the LLMS equa-\ntion for vanishing correlation time is the LLG equation.\nFor small correlation times we can then take the Taylor\nexpansion about the time tint′, so that the damping\nterm becomes,\n/integraldisplayt\n−∞K(t−t′)dS(t′)\ndt′dt′=/bracketleftBig/integraldisplayt\n−∞K(t′)dt′/bracketrightBigdS(t)\ndt+...\n(A12)\nHence the spin and memory kernel decouple in the small\ncorrelation time limit, and the Langevin equation be-\ncomes\ndS\ndt=γS(t)×/parenleftBig\nH+¯η−/bracketleftBig\nχ/integraldisplayt\n−∞dt′K(t−t′)/bracketrightBigdS(t)\ndt/parenrightBig\n,\n(A13)\nAfter performing the integration over t′, the damping is\nχ/integraldisplayt\n−∞dt′e−(t−t′)/τc=χτc. (A14)\nand by direct comparison of the damping terms in this\nexpression and in Gilbert’s equation we have the rela-\ntionship of the phenomenological damping to the LLMS\nparameters α=χγτc. We note also that this expression\ncan be seen if we identify the driving white noise in the\nbath field of the LLMS with the thermal magnetic fieldsof the LLG.\n∝angbracketleftQ(t)Q(t′)∝angbracketright=2χτckBT\nµsδ(t−t′)\n=2αkBT\nγµsδ(t−t′)\n=∝angbracketleftHth(t)Hth(t′)∝angbracketright(A15)\nunder the assumption that α=γχτc.\n2. Thermalization\nAs a quantitative evaluation of the LLMS model and\nour implementation thereof, we compare the equilibrium\nbehavior to the appropriate analytical Boltzmann distri-\nbution, which the Markovian LLG equation also satis-\nfies. We simulate a single spin under the influence of\nanisotropy only. The Boltzmann distribution for such a\nsystem is\nP(θ)∝sinθexp(−kusin2θ\nkBT) (A16)\nwhereθis the angle between the spin and the easy-\naxis and the factor of sin θarises from normalizing the\nprobability distribution on the sphere. WE initialize the\nspin along the easy-axis direction, then allow the spin to\nevolve for 108steps after equilibration and evaluate the\nprobability distribution by recording the number of steps\nthe spin spends at each angle to the easy-axis.\nIn Figure 9, we compare the numerical results to the\nanalytical expression for both the LLMS model and the\nstandard LLG augmented by Ornstein-Uhlenbeck fields\nof the type generated by the Langevin equation in Eq. 4.\nThe simulations using the LLMS model agree with the\nanticipated Boltzmann distribution at equilibrium, while\nthe LLG with Ornstein-Uhlenbeck fails to reproduce the\ncorrect distribution. This is because, as we have argued,\nthis does not comprise a correct implementation of the\nFluctuation-Dissipation theorem, with deviations corre-\nsponding to the missing high-frequency components of\nthe damping.\n1L. N´ eel, Ann. G´ eophys. C.N.R.S. 5, 99 (1949).\n2W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963).\n3A. Lyberatos, D.V. Berkov and R.W. Chantrell, J Phys:\nCondens. Matter 5, 8911 (1993)\n4A. Lyberatos and R. W. Chantrell, J. Appl. Phys. 73, 6501\n(1993).\n5J. L. Garc´ ıa-Palacios and F. J. L´ azaro, Phys. Rev. B 58,\n14937 (1998).\n6D. V. Berkov, IEEE Trans. Magn. 38, 2489 (2002).7Y. P. Kalmykov, W. T. Coffey, U. Atxitia, O. Chubykalo-\nFesenko, P. M. D´ ejardin, and R. W. Chantrell, Phys. Rev.\nB82, 024412 (2010).\n8R. Street and J. C. Woolley, Proc. Phys. Soc. A 62, 562\n(1949).\n9U Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U\nNowak and A. Rebei, Phys. Rev. Lett. 102, 057203 (2009).\n10K. Miyazaki and K. Seki, J. Appl. Phys. 112, 121301\n(2012).11\n11U. Atxitia and O. Chubykalo-Fesenkoo, Phys. Rev. B 84,\n144414 .(2011)\n12P. H¨ anggi, P. Jung, Adv. Chem. Phys. 89, 239 (1995).\n13U. Nowak, Annual Reviews of Computational Physics IX,\npg. 105-151 , World Scientific (2001).\n14U. Nowak, R. W. Chantrell, and E. C. Kennedy, Phys.\nRev. Lett. 84, 163 (2000).15W. T. Coffey and Y. P. Kalmykov, J. Appl. Phys. 112,\n121301 (2012).\n16R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis and R. W. Chantrell, J. Phys.: Condens.\nMatter26, 103202 (2014).\n17I. M. Sokolov, Phys. Rev. E 66, 041101 (2002).\n18I. M. Sokolov, Phys. Rev. E 63, 056111 (2001)."    },    {        "title": "1805.01815v2.Effective_damping_enhancement_in_noncollinear_spin_structures.pdf",        "content": "Effective damping enhancement in noncollinear spin structures\nLevente Rózsa,1,∗Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: August 29, 2021)\nDamping mechanisms in magnetic systems determine the lifetime, diffusion and transport prop-\nerties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological\nLandau–Lifshitz–Gilbert equation, here the effective damping parameter in noncollinear magnetic\nsystems is determined describing the linewidth in resonance experiments or the decay parameter\nin time-resolved measurements. It is shown how the effective damping can be calculated from the\nelliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded\nthat the effective damping is larger than the Gilbert damping, and it may significantly differ be-\ntween excitation modes. Numerical results for the effective damping are presented for the localized\nmagnons in isolated skyrmions, with parameters based on the Pd/Fe/Ir(111) model-type system.\nSpinwaves(SW)ormagnonsaselementaryexcitations\nof magnetically ordered materials have attracted signifi-\ncant research attention lately. The field of magnonics[1]\nconcerns the creation, propagation and dissipation of\nSWs in nanostructured magnetic materials, where the\ndispersion relations can be adjusted by the system ge-\nometry. A possible alternative for engineering the prop-\nerties of magnons is offered by noncollinear (NC) spin\nstructures[2] instead of collinear ferro- (FM) or antifer-\nromagnets (AFM). SWs are envisaged to act as informa-\ntion carriers, where one can take advantage of their low\nwavelengths compared to electromagnetic waves possess-\ning similar frequencies[3]. Increasing the lifetime and the\nstability of magnons, primarily determined by the relax-\nation processes, is of crucial importance in such applica-\ntions.\nThe Landau–Lifshitz–Gilbert (LLG) equation[4, 5] is\ncommonly applied for the quasiclassical description of\nSWs, where relaxation is encapsulated in the dimen-\nsionless Gilbert damping (GD) parameter α. The life-\ntime of excitations can be identified with the resonance\nlinewidth in frequency-domain measurements such as fer-\nromagnetic resonance (FMR)[6], Brillouin light scatter-\ning (BLS)[7] or broadband microwave response[8], and\nwith the decay speed of the oscillations in time-resolved\n(TR) experiments including magneto-optical Kerr effect\nmicroscopy (TR-MOKE)[9] and scanning transmission x-\nray microscopy (TR-STXM)[10]. Since the linewidth is\nknowntobeproportionaltothefrequencyofthemagnon,\nmeasuring the ratio of these quantities is a widely ap-\nplied method for determining the GD in FMs[3, 6]. An\nadvantage of AFMs in magnonics applications[11, 12] is\ntheir significantly enhanced SW frequencies due to the\nexchange interactions, typically in the THz regime, com-\npared to FMs with GHz frequency excitations. However,\nit is known that the linewidth in AFM resonance is typ-\nically very wide because it scales with a larger effective\ndamping parameter αeffthan the GD α[13].\nThe tuning of the GD can be achieved in magnonic\ncrystals by combining materials with different values of\nα. It was demonstrated in Refs. [14–16] that this leadsto a strongly frequency- and band-dependent αeff, based\non the relative weights of the magnon wave functions in\nthe different materials.\nMagnetic vortices are two-dimensional NC spin config-\nurations in easy-plane FMs with an out-of-plane magne-\ntized core, constrained by nanostructuring them in dot-\norpillar-shapedmagneticsamples. Theexcitationmodes\nofvortices, particularlytheirtranslationalandgyrotropic\nmodes, havebeeninvestigatedusingcollective-coordinate\nmodels[17] based on the Thiele equation[18], linearized\nSW dynamics[19, 20], numerical simulations[21] and ex-\nperimental techniques[22–24]. It was demonstrated theo-\nretically in Ref. [21] that the rotational motion of a rigid\nvortex excited by spin-polarized current displays a larger\nαeffthan the GD; a similar result was obtained based on\ncalculating the energy dissipation[25]. However, due to\nthe unbounded size of vortices, the frequencies as well\nas the relaxation rates sensitively depend on the sample\npreparation, particularly because they are governed by\nthe magnetostatic dipolar interaction.\nIn magnetic skyrmions[26], the magnetic moment di-\nrections wrap the whole unit sphere. In contrast to vor-\ntices, isolated skyrmions need not be confined for stabi-\nlization, and are generally less susceptible to demagneti-\nzation effects[3, 27]. The SW excitations of the skyrmion\nlattice phase have been investigated theoretically[28–30]\nand subsequently measured in bulk systems[3, 8, 31]. It\nwas calculated recently[32] that the magnon resonances\nmeasured via electron scattering in the skyrmion lattice\nphase should broaden due to the NC structure. Calcula-\ntions predicted the presence of different localized modes\nconcentrated on the skyrmion for isolated skyrmions\non a collinear background magnetization[33–35] and for\nskyrmions in confined geometries[20, 36, 37]. From the\nexperimental side, the motion of magnetic bubbles in a\nnanodisk was investigated in Ref. [38], and it was pro-\nposed recently that the gyration frequencies measured in\nIr/Fe/Co/Pt multilayer films is characteristic of a dilute\narray of isolated skyrmions rather than a well-ordered\nskyrmion lattice[6]. However, the lifetime of magnons in\nskyrmionic systems based on the LLG equation is appar-arXiv:1805.01815v2  [cond-mat.mtrl-sci]  15 Oct 20182\nently less explored.\nIt is known that NC spin structures may influence the\nGD via emergent electromagnetic fields[29, 39, 40] or via\nthe modified electronic structure[41, 42]. Besides deter-\nmining the SW relaxation process, the GD also plays\na crucial role in the motion of domain walls[43–45] and\nskyrmions[46–48] driven by electric or thermal gradients,\nboth in the Thiele equation where the skyrmions are\nassumed to be rigid and when internal deformations of\nthe structure are considered. Finally, damping and de-\nformations are also closely connected to the switching\nmechanisms of superparamagnetic particles[49, 50] and\nvortices[51], as well as the lifetime of skyrmions[52–54].\nTheαeffin FMs depends on the sample geometry due\nto the shape anisotropy[13, 55, 56]. It was demonstrated\nin Ref. [56] that αeffis determined by a factor describing\nthe ellipticity of the magnon polarization caused by the\nshape anisotropy. Elliptic precession and GD were also\ninvestigated by considering the excitations of magnetic\nadatomsonanonmagneticsubstrate[57]. Thecalculation\nof the eigenmodes in NC systems, e.g. in Refs. [6, 20, 35],\nalso enables the evaluation of the ellipticity of magnons,\nbut this property apparently has not been connected to\nthe damping so far.\nAlthough different theoretical methods for calculating\nαeffhave been applied to various systems, a general de-\nscriptionapplicabletoallNCstructuresseemstobelack-\ning. Here it is demonstrated within a phenomenological\ndescription of the linearized LLG equation how magnons\nin NC spin structures relax with a higher effective damp-\ning parameter αeffthan the GD. A connection between\nαeffand the ellipticity of magnon polarization forced by\nthe NC spin arrangement is established. The method\nis illustrated by calculating the excitation frequencies\nof isolated skyrmions, considering experimentally deter-\nmined material parameters for the Pd/Fe/Ir(111) model\nsystem[58]. It is demonstrated that the different local-\nized modes display different effective damping parame-\nters, with the breathing mode possessing the highest one.\nThe LLG equation reads\n∂tS=−γ/primeS×Beff−αγ/primeS×/parenleftBig\nS×Beff/parenrightBig\n,(1)\nwithS=S(r)the unit-length vector field describing\nthe spin directions in the system, αthe GD and γ/prime=\n1\n1+α2ge\n2mthe modified gyromagnetic ratio (with gbeing\ntheg-factor of the electrons, ethe elementary charge and\nmthe electron mass). Equation (1) describes the time\nevolution of the spins governed by the effective magnetic\nfieldBeff=−1\nMδH\nδS, withHthe Hamiltonian or free\nenergy of the system in the continuum description and\nMthe saturation magnetization.\nThe spins will follow a damped precession relaxing\nto a local minimum S0ofH, given by the condition\nS0×Beff=0. Note that generally the Hamiltonian rep-\nresents a rugged landscape with several local energy min-\nima, corresponding to e.g. FM, spin spiral and skyrmionlattice phases, or single objects such as vortices or iso-\nlated skyrmions. The excitations can be determined by\nswitching to a local coordinate system[20, 34, 47] with\nthe spins along the zdirection in the local minimum,\n˜S0= (0,0,1), and expanding the Hamiltonian in the\nvariablesβ±=˜Sx±i˜Sy, introduced analogously to spin\nraising and lowering or bosonic creation and annihila-\ntion operators in the quantum mechanical description of\nmagnons[59–61]. The lowest-order approximation is the\nlinearized form of the LLG Eq. (1),\n∂tβ+=γ/prime\nM(i−α)/bracketleftbig\n(D0+Dnr)β++Daβ−/bracketrightbig\n,(2)\n∂tβ−=γ/prime\nM(−i−α)/bracketleftbig\nD†\naβ++ (D0−Dnr)β−/bracketrightbig\n.(3)\nFor details of the derivation see the Supplemental\nMaterial[62]. The term Dnrin Eqs. (2)-(3) is respon-\nsible for the nonreciprocity of the SW spectrum[2]. It\naccounts for the energy difference between magnons\npropagating in opposite directions in in-plane oriented\nultrathin FM films[63, 64] with Dzyaloshinsky–Moriya\ninteraction[65, 66] and the splitting between clockwise\nand counterclockwise modes of a single skyrmion[20].\nHere we will focus on the effects of the anomalous\nterm[34]Da, which couples Eqs. (2)-(3) together. Equa-\ntions (2)-(3) may be rewritten as eigenvalue equations by\nassuming the time dependence\nβ±(r,t) =e−iωktβ±\nk(r). (4)\nForα= 0, the spins will precess around their equilib-\nriumdirection ˜S0. Iftheequationsareuncoupled, the ˜Sx\nand ˜Syvariables describe circular polarization, similarly\nto the Larmor precession of a single spin in an exter-\nnal magnetic field. However, the spins are forced on an\nelliptic path due to the presence of the anomalous terms.\nThe effective damping parameter of mode kis defined\nas\nαk,eff=/vextendsingle/vextendsingle/vextendsingle/vextendsingleImωk\nReωk/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (5)\nwhich is the inverse of the figure of merit introduced in\nRef. [15]. Equation (5) expresses the fact that Im ωk,\nthe linewidth in resonance experiments or decay coeffi-\ncient in time-resolved measurements, is proportional to\nthe excitation frequency Re ωk.\nInterestingly, there is a simple analytic expression con-\nnectingαk,effto the elliptic polarization of the modes at\nα= 0. Forα/lessmuch1, the effective damping may be ex-\npressed as\nαk,eff\nα≈/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\ndr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\ndr=/integraltext\na2\nk(r) +b2\nk(r)dr/integraltext\n2ak(r)bk(r)dr,\n(6)3\n0.0 0.2 0.4 0.6 0.8 1.00246810\nFIG. 1. Effective damping parameter αk,effas a function of\ninverseaspectratio bk/akofthepolarizationellipse, assuming\nconstantakandbkfunctions in Eq. (6). Insets illustrate the\nprecession for different values of bk/ak.\nwhere the (0)superscript denotes that the eigenvectors\nβ±\nk(r)defined in Eq. (4) were calculated for α= 0, while\nak(r)andbk(r)denote the semimajor and semiminor\naxes of the ellipse the spin variables ˜Sx(r)and ˜Sy(r)\nare precessing on in mode k. Details of the derivation\nare given in the Supplemental Material[62]. Note that\nan analogous expression for the uniform precession mode\nin FMs was derived in Ref. [56]. The main conclusion\nfrom Eq. (6) is that αk,effwill depend on the considered\nSW mode and it is always at least as high as the GD\nα. Although Eq. (6) was obtained in the limit of low\nα, numerical calculations indicate that the αk,eff/αratio\ntends to increase for increasing values of α; see the Sup-\nplementalMaterial[62]foranexample. Theenhancement\nof the damping from Eq. (6) is shown in Fig. 1, with the\nspace-dependent ak(r)andbk(r)replaced by constants\nfor simplicity. It can be seen that for more distorted po-\nlarization ellipses the spins get closer to the equilibrium\ndirectionafterthesamenumberofprecessions, indicating\na faster relaxation.\nSince the appearance of the anomalous terms Dain\nEqs. (2)-(3) forces the spins to precess on an elliptic\npath, it expresses that the system is not axially sym-\nmetric around the local spin directions in the equilib-\nrium state denoted by S0. Such a symmetry breaking\nnaturally occurs in any NC spin structure, implying a\nmode-dependent enhancement of the effective damping\nparameter in NC systems even within the phenomeno-\nlogical description of the LLG equation. Note that the\nNC structure also influences the electronic properties of\nthe system, which can lead to a modification of the GD\nitself, see e.g. Ref. [42].\nIn order to illustrate the enhanced and mode-\ndependent αk,eff, we calculate the magnons in isolated\nchiralskyrmionsinatwo-dimensionalultrathinfilm. Thedensity of the Hamiltonian Hreads[67]\nh=/summationdisplay\nα=x,y,z/bracketleftBig\nA(∇Sα)2/bracketrightBig\n+K(Sz)2−MBSz\n+D(Sz∂xSx−Sx∂xSz+Sz∂ySy−Sy∂ySz),(7)\nwithAthe exchange stiffness, Dthe Dzyaloshinsky–\nMoriya interaction, Kthe anisotropy coefficient, and B\nthe external field.\nIn the following we will assume D>0andB≥0\nwithout the loss of generality, see the Supplemental\nMaterial[62] for discussion. Using cylindrical coordi-\nnates (r,ϕ)in real space and spherical coordinates S=\n(sin Θ cos Φ ,sin Θ sin Φ,cos Θ)in spin space, the equi-\nlibrium profile of the isolated skyrmion will correspond\nto the cylindrically symmetric configuration Θ0(r,ϕ) =\nΘ0(r)andΦ0(r,ϕ) =ϕ, the former satisfying\nA/parenleftbigg\n∂2\nrΘ0+1\nr∂rΘ0−1\nr2sin Θ 0cos Θ 0/parenrightbigg\n+D1\nrsin2Θ0\n+Ksin Θ 0cos Θ 0−1\n2MBsin Θ 0= 0 (8)\nwith the boundary conditions Θ0(0) =π,Θ0(∞) = 0.\nThe operators in Eqs. (2)-(3) take the form (cf.\nRefs. [34, 35, 47] and the Supplemental Material[62])\nD0=−2A/braceleftBigg\n∇2+1\n2/bracketleftbigg\n(∂rΘ0)2−1\nr2/parenleftbig\n3 cos2Θ0−1/parenrightbig\n(∂ϕΦ0)2/bracketrightbigg/bracerightBigg\n−D/parenleftbigg\n∂rΘ0+1\nr3 sin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n−K/parenleftbig\n3 cos2Θ0−1/parenrightbig\n+MBcos Θ 0, (9)\nDnr=/parenleftbigg\n4A1\nr2cos Θ 0∂ϕΦ0−2D1\nrsin Θ 0/parenrightbigg\n(−i∂ϕ), (10)\nDa=A/bracketleftbigg\n(∂rΘ0)2−1\nr2sin2Θ0(∂ϕΦ0)2/bracketrightbigg\n+D/parenleftbigg\n∂rΘ0−1\nrsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n+Ksin2Θ0.(11)\nEquation (11) demonstrates that the anomalous terms\nDaresponsible for the enhancement of the effective\ndamping can be attributed primarily to the NC arrange-\nment (∂rΘ0and∂ϕΦ0≡1) and secondarily to the\nspins becoming canted with respect to the global out-\nof-plane symmetry axis ( Θ0∈ {0,π}) of the system.\nTheDnrtermintroducesanonreciprocitybetweenmodes\nwith positive and negative values of the azimuthal quan-\ntum number (−i∂ϕ)→m, preferring clockwise rotat-\ning modes ( m < 0) over counterclockwise rotating ones\n(m > 0) following the sign convention of Refs. [20, 34].\nBecauseD0andDnrdepend onmbutDadoes not, it is\nexpected that the distortion of the SW polarization el-\nlipse and consequently the effective damping will be more\nenhanced for smaller values of |m|.\nThe different modes as a function of external field\nare shown in Fig. 2(a), for the material parameters de-\nscribing the Pd/Fe/Ir(111) system. The FMR mode at4\n0.7 0.8 0.9 1.0 1.1 1.20255075100125150175\n(a)\n0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.53.0\n(b)\nFIG. 2. Localized magnons in the isolated skyrmion, with the\ninteraction parameters corresponding to the Pd/Fe/Ir(111)\nsystem[58]:A = 2.0pJ/m,D =−3.9mJ/m2,K =\n−2.5MJ/m3,M= 1.1MA/m. (a) Magnon frequencies f=\nω/2πforα= 0. Illustrations display the shapes of the excita-\ntion modes visualized on the triangular lattice of Fe magnetic\nmoments, with red and blue colors corresponding to positive\nand negative out-of-plane spin components, respectively. (b)\nEffective damping coefficients αm,eff, calculated from Eq. (6).\nωFMR =γ\nM(MB−2K), describing a collective in-phase\nprecession of the magnetization of the whole sample, sep-\narates the continuum and discrete parts of the spectrum,\nwith the localized excitations of the isolated skyrmion\nlocated below the FMR frequency[34, 35]. We found a\nsingle localized mode for each m∈{0,1,−2,−3,−4,−5}\nvalue, so in the following we will denote the excita-\ntion modes with the azimuthal quantum number. The\nm=−1mode corresponds to the translation of the\nskyrmion on the field-polarized background, which is a\nzero-frequency Goldstone mode of the system and not\nshown in the figure. The m=−2mode tends to zero\naroundB= 0.65T, indicating that isolated skyrmions\nbecome susceptible to elliptic deformations and subse-\nquently cannot be stabilized at lower field values[68].\nThe values of αm,effcalculated from Eq. (6) for the\ndifferent modes are summarized in Fig. 2(b). It is impor-\ntant to note that for a skyrmion stabilized at a selected\n0 20 40 60 80 100-0.04-0.020.000.020.040.06\n-0.03 0.00 0.03-0.030.000.03FIG. 3. Precession of a single spin in the skyrmion in the\nPd/Fe/Ir(111) system in the m= 0andm=−3modes at\nB= 0.75T, from numerical simulations performed at α=\n0.1. Inset shows the elliptic precession paths. From fitting\nthe oscillations with Eq. (4), we obtained |Reωm=0|/2π=\n39.22GHz,|Imωm=0|= 0.0608ps−1,αm=0,eff= 0.25and\n|Reωm=−3|/2π= 40.31GHz,|Imωm=−3|= 0.0276ps−1,\nαm=−3,eff= 0.11.\nfield value, the modes display widely different αm,effval-\nues, with the breathing mode m= 0being typically\ndamped twice as strongly as the FMR mode. The ef-\nfective damping tends to increase for lower field values,\nand decrease for increasing values of |m|, the latter prop-\nerty expected from the m-dependence of Eqs. (9)-(11)\nas discussed above. It is worth noting that the αm,eff\nparameters are not directly related to the skyrmion size.\nWealsoperformedthecalculationsfortheparametersde-\nscribing Ir|Co|Pt multilayers[69], and for the significantly\nlargerskyrmionsinthatsystemweobtainedconsiderably\nsmaller excitation frequencies, but quantitatively similar\neffective damping parameters; details are given in the\nSupplemental Material[62].\nThe different effective damping parameters could pos-\nsibly be determined experimentally by comparing the\nlinewidths of the different excitation modes at a selected\nfield value, or investigating the magnon decay over time.\nAn example for the latter case is shown in Fig. 3, dis-\nplaying the precession of a single spin in the skyrmion,\nobtained from the numerical solution of the LLG Eq. (1)\nwithα= 0.1. AtB= 0.75T, the frequencies of the\nm= 0breathing and m=−3triangular modes are close\nto each other (cf. Fig. 2), but the former decays much\nfaster. Because in the breathing mode the spin is follow-\ning a significantly more distorted elliptic path (inset of\nFig. 3) than in the triangular mode, the different effective\ndamping is also indicated by Eq. (6).\nIn summary, it was demonstrated within the phe-\nnomenological description of the LLG equation that the\neffective damping parameter αeffdepends on the consid-\nered magnon mode. The αeffassumes larger values if5\nthe polarization ellipse is strongly distorted as expressed\nby Eq. (6). Since NC magnetic structures provide an\nanisotropic environment for the spins, leading to a dis-\ntortion of the precession path, they provide a natural\nchoice for realizing different αeffvalues within a single\nsystem. The results of the theory were demonstrated for\nisolated skyrmions with material parameters describing\nthe Pd/Fe/Ir(111) system. The results presented here\nmay stimulate further experimental or theoretical work\non the effective damping in skyrmions, vortices, domain\nwalls or spin spirals.\nThe authors would like to thank U. Atxitia and G.\nMeier for fruitful discussions. Financial support by the\nAlexander von Humboldt Foundation, by the Deutsche\nForschungsgemeinschaft via SFB 668, by the European\nUnion via the Horizon 2020 research and innovation pro-\ngram under Grant Agreement No. 665095 (MAGicSky),\nand by the National Research, Development and Inno-\nvation Office of Hungary under Project No. K115575 is\ngratefully acknowledged.\n∗rozsa.levente@physnet.uni-hamburg.de\n[1] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010).\n[2] M.Garst, J.Waizner, andD.Grundler, J.Phys.D:Appl.\nPhys. 50, 293002 (2017).\n[3] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I.\nStasinopoulos, H. Berger, C. Pfleiderer, and D. Grundler,\nNat. Mater. 14, 478 (2015).\n[4] L. 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Mater.\n138, 255 (1994).\n[68] A. Bogdanov and A. Hubert, Phys. Stat. Sol. B 186, 527\n(1994).\n[69] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sam-\npaio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K.\nGarcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M.\nGeorge, M. Weigand, J. Raabe, V. Cros, and A. Fert,\nNat. Nanotechnol. 11, 444 (2016).\n[70] F. Romá, L. F. Cugliandolo, and G. S. Lozano, Phys.\nRev. E 90, 023203 (2014).\n[71] A. O. Leonov, T. L. Monchesky, N. Romming, A. Ku-betzka, A. N. Bogdanov, and R. Wiesendanger, New J.\nPhys. 18, 065003 (2016).\n[72] J. Hagemeister, A. Siemens, L. Rózsa, E. Y. Vedme-\ndenko, and R. Wiesendanger, Phys. Rev. B 97, 174436\n(2018).Supplemental Material to\nEffective damping enhancement in noncollinear spin structures\nLevente Rózsa,1,∗Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: August 29, 2021)\nIn the Supplemental Material the derivation of the linearized equations of motion and the effective\ndamping parameter are discussed. Details of the numerical determination of the magnon modes in\nthe continuum model and in atomistic spin dynamics simulations are also given.\nS.I. LINEARIZED\nLANDAU–LIFSHITZ–GILBERT EQUATION\nHere we will derive the linearized form of the Landau–\nLifshitz–Gilbert equation given in Eqs. (2)-(3) of the\nmaintextanddiscussthepropertiesofthesolutions. The\ncalculation is similar to the undamped case, discussed in\ndetail in e.g. Refs. [1–3]. Given a spin configuration sat-\nisfying the equilibrium condition\nS0×Beff=0, (S.1)\nthe local coordinate system with ˜S0= (0,0,1)may be\nintroduced, andtheHamiltonianbeexpandedinthevari-\nables ˜Sxand˜Sy. Thelineartermmustdisappearbecause\nthe expansion is carried out around an equilibrium state.\nThe lowest-order nontrivial term is quadratic in the vari-\nables and will be designated as the spin wave Hamilto-\nnian,\nHSW=/integraldisplay\nhSWdr, (S.2)\nhSW=1\n2/bracketleftbig˜Sx˜Sy/bracketrightbig/bracketleftbiggA1A2\nA†\n2A3/bracketrightbigg/bracketleftbigg˜Sx\n˜Sy/bracketrightbigg\n=1\n2/parenleftBig\n˜S⊥/parenrightBigT\nHSW˜S⊥. (S.3)\nThe operator HSWis self-adjoint for arbitrary equi-\nlibrium states. Here we will only consider cases where\nthe equilibrium state is a local energy minimum, mean-\ning thatHSW≥0; the magnon spectrum will only be\nwell-defined in this case. Since hSWis obtained as an\nexpansion of a real-valued energy density around the\nequilibrium state, and the spin variables are also real-\nvalued, fromtheconjugateofEq.(S.3)onegets A1=A∗\n1,\nA2=A∗\n2, andA3=A∗\n3.\nThe form of the Landau–Lifshitz–Gilbert Eq. (1) in\nthe main text may be rewritten in the local coordinates\nby simply replacing Sby˜S0everywhere, including the\ndefinitionoftheeffectivefield Beff. TheharmonicHamil-\ntonianHSWin Eq. (S.2) leads to the linearized equation\nof motion\n∂t˜S⊥=γ/prime\nM(−iσy−α)HSW˜S⊥,(S.4)\n∗rozsa.levente@physnet.uni-hamburg.dewithσy=/bracketleftbigg\n0−i\ni0/bracketrightbigg\nthe Pauli matrix.\nBy replacing ˜S⊥(r,t)→˜S⊥\nk(r)e−iωktas usual, for\nα= 0the eigenvalue equation\nωk˜S⊥\nk=γ\nMσyHSW˜S⊥\nk (S.5)\nis obtained. If HSWhas a strictly positive spectrum,\nthenH−1\n2\nSWexists, and σyHSWhas the same eigenvalues\nasH1\n2\nSWσyH1\n2\nSW. Since the latter is a self-adjoint ma-\ntrix with respect to the standard scalar product on the\nHilbert space, it has a real spectrum, consequently all ωk\neigenvalues are real. Note that the zero modes of HSW,\nwhich commonly occur in the form of Goldstone modes\ndue to the ground state breaking a continuous symme-\ntry of the Hamiltonian, have to be treated separately.\nFinally, we mention that if the spin wave expansion is\nperformed around an equilibrium state which is not a\nlocal energy minimum, the ωkeigenvalues may become\nimaginary, meaning that the linearized Landau–Lifshitz–\nGilbert equation will describe a divergence from the un-\nstable equilibrium state instead of a precession around\nit.\nEquations (2)-(3) in the main text may be obtained\nby introducing the variables β±=˜Sx±i˜Syas described\nthere. The connection between HSWand the operators\nD0,Dnr, andDais given by\nD0=1\n2(A1+A3), (S.6)\nDnr=1\n2i/parenleftBig\nA†\n2−A2/parenrightBig\n, (S.7)\nDa=1\n2/bracketleftBig\nA1−A3+i/parenleftBig\nA†\n2+A2/parenrightBig/bracketrightBig\n.(S.8)\nAn important symmetry property of Eqs. (2)-(3) in\nthe main text is that if (β+,β−) =/parenleftbig\nβ+\nke−iωkt,β−\nke−iωkt/parenrightbig\nis an eigenmode of the equations, then (β+,β−) =/parenleftBig/parenleftbig\nβ−\nk/parenrightbig∗eiω∗\nkt,/parenleftbig\nβ+\nk/parenrightbig∗eiω∗\nkt/parenrightBig\nis another solution. Following\nRefs. [1, 3], this can be attributed to the particle-hole\nsymmetry of the Hamiltonian, which also holds in the\npresence of the damping term. From these two solutions\nmentioned above, the real-valued time evolution of the\nvariables ˜Sx,˜Symay be expressed as\n˜Sx\nk=eImωktcos (ϕ+,k−Reωkt)/vextendsingle/vextendsingleβ+\nk+β−\nk/vextendsingle/vextendsingle,(S.9)\n˜Sy\nk=eImωktsin (ϕ−,k−Reωkt)/vextendsingle/vextendsingleβ+\nk−β−\nk/vextendsingle/vextendsingle,(S.10)arXiv:1805.01815v2  [cond-mat.mtrl-sci]  15 Oct 20182\nwithϕ±,k= arg/parenleftbig\nβ+\nk±β−\nk/parenrightbig\n. As mentioned above, the\nImωkterms are zero in the absence of damping close to\na local energy minimum, and Im ωk<0is implied by\nthe fact that the Landau–Lifshitz–Gilbert equation de-\nscribes energy dissipation, which in the linearized case\ncorresponds to relaxation towards the local energy min-\nimum. In the absence of damping, the spins will precess\non an ellipse defined by the equation\n/parenleftBig\n˜Sx\nk/parenrightBig2\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk+β−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ncos2(ϕ+,k−ϕ−,k)\n+2˜Sx\nk˜Sy\nksin (ϕ+,k−ϕ−,k)/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk−β−(0)\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk+β−(0)\nk/vextendsingle/vextendsingle/vextendsinglecos2(ϕ+,k−ϕ−,k)\n+/parenleftBig\n˜Sy\nk/parenrightBig2\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk−β−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ncos2(ϕ+,k−ϕ−,k)= 1,(S.11)\nwhere the superscript (0)indicatesα= 0. The semima-\njor and semiminor axes of the ellipse akandbkmay be\nexpressed from Eq. (S.11) as\nakbk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (S.12)\na2\nk+b2\nk= 2/parenleftbigg/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\n.(S.13)\nNote thatβ+\nkandβ−\nk, consequently the parameters of\nthe precessional ellipse akandbk, are functions of the\nspatial position r.\nS.II. CALCULATION OF THE EFFECTIVE\nDAMPING PARAMETER FROM\nPERTURBATION THEORY\nHere we derive the expression for the effective damping\nparameter αeffgiven in Eq. (6) of the main text. By\nintroducingβk=/parenleftbig\nβ+\nk,−β−\nk/parenrightbig\n,\nD=/bracketleftbiggD0+Dnr−Da\n−D†\naD0−Dnr/bracketrightbigg\n,(S.14)\nand using the Pauli matrix σz=/bracketleftbigg\n1 0\n0−1/bracketrightbigg\n, Eqs. (2)-(3)\nin the main text may be rewritten as\n−ωkσzβk=γ/prime\nM(D+iασzD)βk(S.15)\nin the frequency domain. Following standard perturba-\ntion theory, we expand the eigenvalues ωkand the eigen-\nvectorsβkin the parameter α/lessmuch1. For the zeroth-order\nterms one gets\n−ω(0)\nkσzβ(0)\nk=γ\nMDβ(0)\nk, (S.16)\n0.0 0.1 0.2 0.3 0.4 0.50.00.51.01.52.02.5FIG. S1. Effective damping coefficients αm,effof the isolated\nskyrmion in the Pd/Fe/Ir(111) system at B= 1T, calcu-\nlated from the numerical solution of the linearized Landau–\nLifshitz–Gilbert equation (S.15), as a function of the Gilbert\ndamping parameter α.\nwith realω(0)\nkeigenvalues as discussed in Sec. S.I. The\nfirst-order terms read\n−ω(0)\nk/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(1)\nk/angbracketrightBig\n−ω(1)\nk/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(0)\nk/angbracketrightBig\n=γ\nM/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleD/vextendsingle/vextendsingle/vextendsingleβ(1)\nk/angbracketrightBig\n+iαγ\nM/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσzD/vextendsingle/vextendsingle/vextendsingleβ(0)\nk/angbracketrightBig\n,\n(S.17)\nafter taking the scalar product with β(0)\nk. The first terms\non both sides cancel by letting Dact to the left, then\nusing Eq. (S.16) and the fact that the ω(0)\nkare real. By\napplying Eq. (S.16) to the remaining term on the right-\nhand side one obtains\nω(1)\nk=−iαω(0)\nk/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndr,(S.18)\nby writing in the definition of the scalar product. By\nusing the definition αk,eff=|Imωk/Reωk|≈/vextendsingle/vextendsingle/vextendsingleω(1)\nk/ω(0)\nk/vextendsingle/vextendsingle/vextendsingle\nand substituting Eqs. (S.12)-(S.13) into Eq. (S.18), one\narrives at Eq. (6) in the main text as long as/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndoes not change sign under the integral.\nIt is worthwhile to investigate for which values of α\ndoes first-order perturbation theory give a good estimate\nforαk,effcalculated from the exact solution of the lin-\nearized equations of motion, Eq. (S.15). In the materials\nwhere the excitations of isolated skyrmions or skyrmion\nlattices were investigated, significantly different values of\nαhave been found. For example, intrinsic Gilbert damp-\ning parameters of α= 0.02-0.04were determined experi-\nmentallyforbulkchiralmagnetsMnSiandCu 2OSeO 3[4],\nα= 0.28was deduced for FeGe[5], and a total damp-\ning ofαtot= 0.105was obtained for Ir/Fe/Co/Pt mag-\nnetic multilayers[6], where the latter value also includes3\nvarious effects beyond the Landau–Lifshitz–Gilbert de-\nscription. Figure S1 displays the dependence of αm,eff\nonαfor the eigenmodes of the isolated skyrmion in the\nPd/Fe/Ir(111) system, shown in Fig. 2 of the main text.\nMost of the modes show a linear correspondence between\nthe two quantities with different slopes in the displayed\nparameter range, in agreement with Eq. (6) in the main\ntext. For the breathing mode m= 0the convex shape\nof the curve indicates that the effective damping param-\neter becomes relatively even larger than the perturbative\nexpression Eq. (6) as αis increased.\nS.III. EIGENMODES OF THE ISOLATED\nSKYRMION\nHere we discuss the derivation of the skyrmion profile\nEq. (8) and the operators in Eqs. (9)-(11) of the main\ntext. The energy density Eq. (7) in polar coordinates\nreads\nh=A/bracketleftbigg\n(∂rΘ)2+ sin2Θ (∂rΦ)2+1\nr2(∂ϕΘ)2\n+1\nr2sin2Θ (∂ϕΦ)2/bracketrightbigg\n+D/bracketleftbigg\ncos (ϕ−Φ)∂rΘ\n−1\nrsin (ϕ−Φ)∂ϕΘ + sin Θ cos Θ sin ( ϕ−Φ)∂rΦ\n+1\nrsin Θ cos Θ cos ( ϕ−Φ)∂ϕΦ/bracketrightbigg\n+Kcos2Θ−MBcos Θ.\n(S.19)\nThe Landau–Lifshitz–Gilbert Eq. (1) may be rewritten\nas\nsin Θ∂tΘ =γ/primeBΦ+αγ/primesin ΘBΘ,(S.20)\nsin Θ∂tΦ =−γ/primeBΘ+αγ/prime1\nsin ΘBΦ,(S.21)\nwith\nBχ=−1\nMδH\nδχ\n=−1\nM/bracketleftbigg\n−1\nr∂r/parenleftbigg\nr∂h\n∂(∂rχ)/parenrightbigg\n−∂ϕ∂h\n∂(∂ϕχ)+∂h\n∂χ/bracketrightbigg\n,\n(S.22)\nwhereχstands for ΘorΦ. Note that in this form it is\ncommon to redefine BΦto include the 1/sin Θfactor in\nEq. (S.21)[7]. The first variations of Hfrom Eq. (S.19)may be expressed as\nδH\nδΘ=−2A/braceleftbigg\n∇2Θ−sin Θ cos Θ/bracketleftbigg\n(∂rΦ)2+1\nr2(∂ϕΦ)2/bracketrightbigg/bracerightbigg\n−2Ksin Θ cos Θ +MBsin Θ\n−2Dsin2Θ/bracketleftbigg\nsin (ϕ−Φ)∂rΦ + cos (ϕ−Φ)1\nr∂ϕΦ/bracketrightbigg\n,\n(S.23)\nδH\nδΦ=−2A/braceleftbigg\nsin2Θ∇2Φ + sin 2Θ/bracketleftbigg\n∂rΘ∂rΦ +1\nr2∂ϕΘ∂ϕΦ/bracketrightbigg/bracerightbigg\n+ 2Dsin2Θ/bracketleftbigg\nsin (ϕ−Φ)∂rΘ + cos (ϕ−Φ)1\nr∂ϕΘ/bracketrightbigg\n,\n(S.24)\nTheequilibriumconditionEq.(8)inthemaintextmay\nbe obtained by setting ∂tΘ =∂tΦ = 0in Eqs. (S.20)-\n(S.21) and assuming cylindrical symmetry, Θ0(r,ϕ) =\nΘ0(r)and Φ0(r,ϕ) =ϕ. In the main text D>0\nandB≥0were assumed. Choosing D<0switches\nthe helicity of the structure to Φ0=ϕ+π, in which\ncaseDshould be replaced by |D|in Eq. (8). For the\nbackground magnetization pointing in the opposite di-\nrectionB≤0, one obtains the time-reversed solutions\nwith Θ0→π−Θ0,Φ0→Φ0+π,B→−B. Time rever-\nsal also reverses clockwise and counterclockwise rotating\neigenmodes; however, the above transformations do not\ninfluence the magnitudes of the excitation frequencies.\nFinally, we note that the frequencies remain unchanged\neven if the form of the Dzyaloshinsky–Moriya interaction\nin Eq. (S.19), describing Néel-type skyrmions common in\nultrathin films and multilayers, is replaced by an expres-\nsion that prefers Bloch-type skyrmions occurring in bulk\nhelimagnets – see Ref. [3] for details.\nFordeterminingthelinearizedequationsofmotion,one\ncan proceed by switching to the local coordinate system\nas discussed in Sec. S.I and Refs. [1, 3]. Alternatively,\nthey can also directly be derived from Eqs. (S.20)-(S.21)\nby introducing Θ = Θ 0+˜Sx,Φ = Φ 0+1\nsin Θ 0˜Syand\nexpanding around the skyrmion profile from Eq. (8) up\nto first order in ˜Sx,˜Sy– see also Ref. [2]. The operators\nin Eq. (S.3) read\nA1=−2A/parenleftbigg\n∇2−1\nr2cos 2Θ 0(∂ϕΦ0)2/parenrightbigg\n−2D1\nrsin 2Θ 0∂ϕΦ0−2Kcos 2Θ 0+MBcos Θ 0,\n(S.25)\nA2=4A1\nr2cos Θ 0∂ϕΦ0∂ϕ−2D1\nrsin Θ 0∂ϕ,(S.26)\nA3=−2A/braceleftbigg\n∇2+/bracketleftbigg\n(∂rΘ0)2−1\nr2cos2Θ0(∂ϕΦ0)2/bracketrightbigg/bracerightbigg\n−2D/parenleftbigg\n∂rΘ0+1\nrsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n−2Kcos2Θ0+MBcos Θ 0, (S.27)4\nwhich leads directly to Eqs. (9)-(11) in the main text via\nEqs. (S.6)-(S.8).\nThe excitation frequencies of the ferromagnetic state\nmay be determined by setting Θ0≡0in Eqs. (9)-(11) in\nthe main text. In this case, the eigenvalues and eigenvec-\ntors can be calculated analytically[1],\nωk,m=γ/prime\nM(1−iα)/bracketleftbig\n2Ak2−2K+MB/bracketrightbig\n,(S.28)\n/parenleftBig\nβ+\nk,m(r),β−\nk,m(r)/parenrightBig\n= (0,Jm−1(kr)),(S.29)\nwithJm−1theBesselfunction ofthefirstkind, appearing\ndue to the solutions being regular at the origin. Equa-\ntion (S.28) demonstrates that the lowest-frequency exci-\ntation of the background is the ferromagnetic resonance\nfrequencyωFMR =γ\nM(MB−2K)atα= 0. Since the\nanomalous term Dadisappears in the out-of-plane mag-\nnetized ferromagnetic state, all spin waves will be circu-\nlarly polarized, see Eq. (S.29), and the effective damping\nparameterwillalwayscoincidewiththeGilbertdamping.\nRegarding the excitations of the isolated skyrmion, for\nα= 0the linearized equations of motion in Eq. (S.15)\nare real-valued; consequently, β±\nk,m(r)can be chosen to\nbe real-valued. In this case Eqs. (S.9)-(S.10) take the\nform\n˜Sx\nk,m= cos (mϕ−ωk,mt)/parenleftBig\nβ+\nk,m(r) +β−\nk,m(r)/parenrightBig\n,(S.30)\n˜Sy\nk,m= sin (mϕ−ωk,mt)/parenleftBig\nβ+\nk,m(r)−β−\nk,m(r)/parenrightBig\n.(S.31)\nThis means that modes with ωk,m>0form> 0will\nrotate counterclockwise, that is, the contours with con-\nstant ˜Sx\nk,mand ˜Sy\nk,mwill move towards higher values of\nϕastis increased, while the modes with ωk,m>0for\nm < 0will rotate clockwise. Modes with m= 0corre-\nspond to breathing excitations. This sign convention for\nmwas used when designating the localized modes of the\nisolated skyrmion in the main text, and the kindex was\ndropped since only a single mode could be observed be-\nlow the ferromagnetic resonance frequency for each value\nofm.\nS.IV. NUMERICAL SOLUTION OF THE\nEIGENVALUE EQUATIONS\nThe linearized Landau–Lifshitz–Gilbert equation for\nthe isolated skyrmion, Eqs. (2)-(3) with the operators\nEqs.(9)-(11)inthemaintext, weresolvednumericallyby\na finite-difference method. First the equilibrium profile\nwas determined from Eq. (8) using the shooting method\nfor an initial approximation, then obtaining the solution\non a finer grid via finite differences. For the calculationswe used dimensionless parameters (cf. Ref. [8]),\nAdl= 1, (S.32)\nDdl= 1, (S.33)\nKdl=KA\nD2, (S.34)\n(MB)dl=MBA\nD2, (S.35)\nrdl=|D|\nAr, (S.36)\nωdl=MA\nγD2ω. (S.37)\nThe equations were solved in a finite interval for\nrdl∈[0,R], with the boundary conditions Θ0(0) =\nπ,Θ0(R) = 0. For the results presented in Fig. 2 in the\nmain text the value of R= 30was used. It was confirmed\nbymodifying Rthattheskyrmionshapeandthefrequen-\ncies of the localized modes were not significantly affected\nby the boundary conditions. However, the frequencies of\nthe modes above the ferromagnetic resonance frequency\nωFMR =γ\nM(MB−2K)did change as a function of\nR, since these modes are extended over the ferromag-\nnetic background – see Eqs. (S.28)-(S.29). Furthermore,\nin the infinitely extended system the equations of mo-\ntion include a Goldstone mode with/parenleftbig\nβ+\nm=−1,β−\nm=−1/parenrightbig\n=/parenleftbig\n−1\nrsin Θ 0−∂rΘ0,1\nrsin Θ 0−∂rΘ0/parenrightbig\n, corresponding to\nthe translation of the skyrmion on the collinear\nbackground[1]. This mode obtains a finite frequency in\nthe numerical calculations due to the finite value of R\nand describes a slow clockwise gyration of the skyrmion.\nHowever, this frequency is not shown in Fig. 3 of the\nmain text because it is only created by boundary effects.\nIn order to investigate the dependence of the effective\ndamping on the dimensionless parameters, we also per-\nformed the calculations for the parameters describing the\nIr|Co|Pt multilayer system[9]. The results are summa-\nrized in Fig. S2. The Ir|Co|Pt system has a larger di-\nmensionless anisotropy value ( −KIr|Co|Pt\ndl = 0.40) than\nthe Pd/Fe/Ir(111) system ( −KPd/Fe/Ir(111)\ndl = 0.33). Al-\nthough the same localized modes are found in both cases,\nthe frequencies belonging to the m= 0,1,−3,−4,−5\nmodes in Fig. S2 are relatively smaller than in Fig. 2\ncompared to the ferromagnetic resonance frequency at\nthe elliptic instability field where ωm=−2= 0. This\nagrees with the two limiting cases discussed in the lit-\nerature: it was shown in Ref. [1] that for Kdl= 0the\nm= 1,−4,−5modes are still above the ferromagnetic\nresonance frequency at the elliptic instability field, while\nin Ref. [2] it was investigated that all modes become soft\nwithfrequenciesgoingtozeroat (MB)dl= 0inthepoint\n−Kdl=π2\n16≈0.62,belowwhichaspinspiralgroundstate\nis formed in the system. Figure S2(b) demonstrates that\nthe effective damping parameters αm,effare higher at the\nellipticinstabilityfieldinIr|Co|PtthaninPd/Fe/Ir(111),\nshowing an opposite trend compared to the frequencies.\nRegarding the physical units, the stronger exchange\nstiffness combined with the weaker Dzyaloshinsky–5\n0.03 0.04 0.05 0.06 0.07 0.080246810\n(a)\n0.03 0.04 0.05 0.06 0.07 0.081.01.52.02.53.03.5\n(b)\nFIG. S2. Localized magnons in the isolated skyrmion, with\nthe interaction parameters corresponding to the Ir|Co|Pt\nmultilayer system from Ref. [9]: A= 10.0pJ/m,D=\n1.9mJ/m2,K=−0.143MJ/m3,M = 0.96MA/m. The\nanisotropy reflects an effective value including the dipolar in-\nteractions as a demagnetizing term, −K =−K 0−1\n2µ0M2\nwithK0=−0.717MJ/m3. (a) Magnon frequencies f=ω/2π\nforα= 0. Illustrations display the shapes of the excitation\nmodes visualized as the contour plot of the out-of-plane spin\ncomponentsona 1×1nm2grid,withredandbluecolorscorre-\nsponding to positive and negative Szvalues, respectively. (b)\nEffective damping coefficients αm,eff, calculated from Eq. (6)\nin the main text.Moriya interaction and anisotropy in the multilayer sys-\ntem leads to larger skyrmions stabilized at lower field val-\nues and displaying lower excitation frequencies. We note\nthat demagnetization effects were only considered here\nas a shape anisotropy term included in K; it is expected\nthat this should be a relatively good approximation for\nthe Pd/Fe/Ir(111) system with only a monolayer of mag-\nnetic material, but it was suggested recently[6] that the\ndipolar interaction can significantly influence the excita-\ntion frequencies of isolated skyrmions in magnetic multi-\nlayers.\nS.V. SPIN DYNAMICS SIMULATIONS\nFor the spin dynamics simulations displayed in Fig. 3\nin the main text we used an atomistic model Hamiltonian\non a single-layer triangular lattice,\nH=−1\n2/summationdisplay\n/angbracketlefti,j/angbracketrightJSiSj−1\n2/summationdisplay\n/angbracketlefti,j/angbracketrightDij(Si×Sj)−/summationdisplay\niK(Sz\ni)2\n−/summationdisplay\niµBSz\ni, (S.38)\nwith the parameters J= 5.72meV for the Heisenberg\nexchange,D=|Dij|= 1.52meV for the Dzyaloshinsky–\nMoriya interaction, K= 0.4meV for the anisotropy,\nµ= 3µBfor the magnetic moment, and a= 0.271nm\nfor the lattice constant. For the transformation be-\ntween the lattice and continuum parameters in the\nPd/Fe/Ir(111) system see, e.g., Ref. [10]. The simula-\ntionswereperformedbynumericallysolvingtheLandau–\nLifshitz–Gilbert equation on an 128×128lattice with\nperiodic boundary conditions, which was considerably\nlarger than the equilibrium skyrmion size to minimize\nboundary effects. The initial configuration was deter-\nmined by calculating the eigenvectors in the continuum\nmodel and discretizing it on the lattice, as shown in the\ninsets of Fig. 2 in the main text. It was found that such\na configuration was very close to the corresponding exci-\ntation mode of the lattice Hamiltonian Eq. (S.38), simi-\nlarly to the agreement between the continuum and lattice\nequilibrium skyrmion profiles[10].\n[1] C. Schütte and M. Garst, Phys. Rev. B 90, 094423\n(2014).\n[2] V. P. Kravchuk, D. D. Sheka, U. K. Rössler, J. van den\nBrink, andYu.Gaididei, Phys.Rev.B 97, 064403(2018).\n[3] S.-Z. Lin, Phys. Rev. B 96, 014407 (2017).\n[4] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I.\nStasinopoulos, H. Berger, C. Pfleiderer, and D. Grundler,\nNat. Mater. 14, 478 (2015).\n[5] M. Beg, M. Albert, M.-A. Bisotti, D. Cortés-Ortuño, W.\nWang, R. Carey, M. Vousden, O. Hovorka, C. Ciccarelli,\nC. S. Spencer, C. H. Marrows, and H. Fangohr, Phys.\nRev. B95, 014433 (2017).\n[6] B. Satywali, F. Ma, S. He, M. Raju, V. P. Kravchuk,\nM. Garst, A. Soumyanarayanan, and C. Panagopoulos,arXiv:1802.03979 (2018).\n[7] F. Romá, L. F. Cugliandolo, and G. S. Lozano, Phys.\nRev. E90, 023203 (2014).\n[8] A. O. Leonov, T. L. Monchesky, N. Romming, A. Ku-\nbetzka, A. N. Bogdanov, and R. Wiesendanger, New J.\nPhys.18, 065003 (2016).\n[9] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sam-\npaio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K.\nGarcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M.\nGeorge, M. Weigand, J. Raabe, V. Cros, and A. Fert,\nNat. Nanotechnol. 11, 444 (2016).\n[10] J. Hagemeister, A. Siemens, L. Rózsa, E. Y. Vedme-\ndenko, and R. Wiesendanger, Phys. Rev. B 97, 174436\n(2018)."    },    {        "title": "1805.11468v1.Gilbert_damping_in_non_collinear_magnetic_system.pdf",        "content": "arXiv:1805.11468v1  [cond-mat.mtrl-sci]  29 May 2018APS/123-QED\nGilbert damping in non-collinear magnetic systems\nS. Mankovsky, S. Wimmer, H. Ebert\nDepartment of Chemistry/Phys. Chemistry, LMU Munich,\nButenandtstrasse 11, D-81377 Munich, Germany\n(Dated: May 30, 2018)\nThe modification of the magnetization dissipation or Gilber t damping caused by an inhomoge-\nneous magnetic structure and expressed in terms of a wave vec tor dependent tensor α(/vector q) is in-\nvestigated by means of linear response theory. A correspond ing expression for α(/vector q) in terms of\nthe electronic Green function has been developed giving in p articular the leading contributions to\nthe Gilbert damping linear and quadratic in q. Numerical results for realistic systems are pre-\nsented that have been obtained by implementing the scheme wi thin the framework of the fully\nrelativistic KKR (Korringa-Kohn-Rostoker) band structur e method. Using the multilayered system\n(Cu/Fe 1−xCox/Pt)nas an example for systems without inversion symmetry we demo nstrate the\noccurrence of non-vanishing linear contributions. For the alloy system bcc Fe 1−xCoxhaving inver-\nsion symmetry, on the other hand, only the quadratic contrib ution is non-zero. As it is shown, this\nquadratic contribution does not vanish even if the spin-orb it coupling is suppressed, i.e. it is a direct\nconsequence of the non-collinear spin configuration.\nPACS numbers: 71.15.-m,71.55.Ak, 75.30.Ds\nI. INTRODUCTION\nThe magnetization dissipation in magnetic materi-\nals is conventionally characterized by means of the\nGilbert damping (GD) tensor αthat enters the Landau-\nLifshitz-Gilbert (LLG) equation [1]. This positive-\ndefinite second-rank tensor depends in general on the\nmagnetization direction. It is well established that in\nthe case of spatially uniformly magnetized ferromagnetic\n(FM) metals two regimes of slow magnetization dynam-\nics can be distinguished, which are governed by differ-\nent mechanisms of dissipation [2–4]: a conductivity-like\nbehaviour occuring in the limiting case of ordered com-\npounds that may be connected to the Fermi breathing\nmechanism and a resistivity-likebehaviourshown by ma-\nterials with appreciable structural, chemical or tempera-\nture induced disorder and connected to a spin-flip scat-\nteringmechanism. Animportantissueisthatbothmech-\nanisms are determined by the spin-orbit coupling in the\nsystem (see e.g. [2, 4, 5]). During the last years, it was\ndemonstrated by variousauthors that first-principles cal-\nculationsforthe GD parameterforcollinearferromagntic\nmaterials allow to cover both regimes without use of any\nphenomenological parameters. In fact, in spite of the dif-\nferences concerning the formulation for the damping pa-\nrameter and the corresponding implementaion [6–8], the\nnumerical results are in generalin rathergood agreement\nwith each other as well as with experiment.\nIn the case of a pronounced non-collinear magnetic\ntexture, e.g. in the case of domain walls or topologi-\ncally nontrivial magnetic configurations like skyrmions,\nthe description of the magnetization dissipation assum-\ning a spatial-invariant tensor αis incomplete, and a non-\nlocal character of GD tensor in such systems has to be\ntaken into account [9–11]. This implies that the dissipa-\ntive torque on the magnetization should be representedby the expression of the following general form [12]:\nτGD= ˆm(/vector r,t)×/integraldisplay\nd3r′α(/vector r−/vector r′)∂\n∂tˆm(/vector r′,t).(1)\nIn the case of a magnetic texture varying slowly in space,\nhowever, an expansion of the damping parameter in\nterms of the magnetization density and its gradients [11]\nis nevertheless appropriate:\nαij=αij+αkl\nijmkml+αklp\nijmk∂\n∂rlmp(2)\n+αklpq\nij∂\n∂rkml∂\n∂rpmq+... ,\nwhere the first term αijstands for the conventional\nisotropic GD and the second term αkl\nijmkmlis associated\nwith the magneto-crystalline anisotropy (MCA). The\nthird so-called chiral term αklp\nijmk∂\n∂rlmpis non-vanishing\nin non-centrosymmetric systems. The important role of\nthis contribution to the damping was demonstrated ex-\nperimentally when investigating the field-driven domain\nwall(DW)motioninasymmetricPt/Co/Pttrilayers[13].\nAs an alternative to the expansion in Eq. (2) one can\ndiscuss the Fourier transform α(/vector q) of the damping pa-\nrametercharacterizinginhomogeneousmagneticsystems,\nwhich enter the spin dynamics equation\n∂\n∂t/vector m(/vector q) =−γ/vector m(/vector q)×/vectorH−/vector m(/vector q)×α(/vector q)∂\n∂t/vector m(/vector q).(3)\nIn this formulation the term linear in qis the first chiral\nterm appearing in the expansion of α(/vector q) in powers of q.\nFurthermore, it is important to note that it is directly\nconnected to the αklp\nijmk∂\n∂rlmpterm in Eq. (2).\nBy applying a gauge field theory, the origin of the\nnon-collinear corrections to the GD can be ascribed to\nthe emergent electromagnetic field created in the time-\ndependent magnetic texture [14, 15]. Such an emergent2\nelectromagneticfieldgivesrisetoaspincurrentwhosedi-\nvergence characterizes the change of the angular momen-\ntum in the system. This allows to discuss the impact of\nnon-collinearity on the GD via a spin-pumping formula-\ntion[9,14,16]. Somedetailsofthephysicsbehind thisef-\nfect depend on the specific propertiesofthe materialcon-\nsidered. Accordingly, different models for magnetisation\ndissipation were discussed in the literature [9, 12, 14, 17–\n19]. Non-centrosymmetric two-dimensional systems for\nwhich the Rashba-like spin-orbit coupling plays an im-\nportant role havereceived special interest in this context.\nThey have been discussed in particular by Akosa et al.\n[19], in order to explain the origin of chiral GD in the\npresence of a chiral magnetic structure.\nThe fourth term on the r.h.s. of Eq. (2) corresponds\nto a quadratic term of an expansion of α(/vector q) with re-\nspect to q. It was investigated for bulk systems with\nnon-magnetic [20] and magnetic [9] impurity atoms, for\nwhich the authors have shown on the basis of model con-\nsideration that it can give a significant correction to the\nhomogeneous GD in the case of weak metallic ferromag-\nnets. In striking contrast to the uniform part of the GD\nthis contribution does not require a non-vanishing spin-\norbit interaction.\nTo our knowledge, only very few ab-initio investiga-\ntions on the Gilbert damping in non-collinear magnetic\nsystems along the lines sketched above have been re-\nported so far in the literature. Yuan et al. [21] calcu-\nlated the in-plane and out-of-plane damping parameters\nin terms of the scattering matrix for permalloy in the\npresence of N´ eel and Bloch domain walls. Freimuth et\nal. [22], discuss the properties of a q-dependent Gilbert\ndamping α(/vector q) calculated for the one-dimensional Rashba\nmodelinthepresenceofthe N´ eel-typenon-collinearmag-\nnetic exchange field, demonstrating different GD for left-\nhanded and right-handed DWs. Here we extend the for-\nmalism developed before to deal with the GD in ferro-\nmagnets [6], to get access to non-collinear system. The\nformalism based on linear response theory allows to ex-\npand the GD parameters with respect to a modulation\nof the magnetization expressed in terms of a wave vector\n/vector q. Correspondingnumerical results will be presented and\ndiscussed.\nII. GILBERT DAMPING FOR\nNON-COLLINEAR MAGNETIZATION\nIn the following we focus on the intrinsic contribution\nto the Gilbert damping, excluding spin current induced\nmagnetizationdissipationwhich occursin the presenceof\nan external electric field. For the considerations on the\nmagnetization dissipation an adiabatic variation of the\nmagnetization in the time and space domain is assumed.\nMoreover, it is assumed that the magnitude of the local\nmagnetic moments is unchanged during a change of the\nmagnetization, i.e. the exchange field should be strong\nenough to separate transverse and longitudinal parts ofthe magnetic susceptibility. With these restrictions, the\nnon-local Gilbert damping can be determined in terms of\nthe spin susceptibility tensor\nχαβ(/vector q,ω) =i1\nV∞/integraldisplay\n0dt∝angbracketleftˆSα(/vector q,t)ˆSα(−/vector q,0)∝angbracketright0ei(ω−δ)t,(4)\nwhereˆSα(/vector q,t) is the /vector q- andt-dependent spin operator\nand reduced units havebeen used ( /planckover2pi1= 1). With this, the\nFourier transformationofthe real-spaceGilbert damping\ncan be represented by the expression [23, 24]\nααβ(/vector q) =γ\nM0Vlim\nω→0∂ℑ[χ−1]αβ(/vector q,ω)\n∂ω.(5)\nHereγ=gµBis the gyromagneticratio, M0=µtotµB/V\nis the equilibrium magnetization and Vis the volume of\nthe system. In order to avoid the calculation of the dy-\nnamical magnetic susceptibility tensor χ(/vector q,ω), which is\nthe Fourier transformed of the real space susceptibility\nχ(/vector r−/vector r′,ω), it is convenient to represent χ(/vector q,ω) in Eq.\n(5), in terms of a correlation function of time deriva-\ntives ofˆS. As˙ˆScorresponds to the torque /vectorT, that may\ninclude non-dissipative and dissipative parts, one may\nconsider instead the torque-torque correlation function\nπ(/vector q,ω) [24–27].\nAssuming the magnetization direction parallelto ˆ zone\nobtains the expression for the Gilbert damping α(/vector q)\nα(/vector q) =γ\nM0Vlim\nω→0∂ℑ[ǫ·π(/vector q,ω)·ǫ]\n∂ω. (6)\nwhereǫ=/bracketleftbigg\n0 1\n−1 0/bracketrightbigg\nis the transverse Levi-Civita tensor.\nThisimpliesthefollowingrelationshipofthe αtensorele-\nments with the elements of the torque-torque correlation\ntensorπ:αxx∼ −πyyandαyy∼ −πxx[24].\nUsing Kubo’s linear response theory in the Matsubara\nrepresentation and taking into account the translational\nsymmetry of a solid the torque-torque correlation func-\ntionπαβ(/vector q,ω) can be expressed by (see, e.g. [28]):\nπαβ(/vector q,iωn) =1\nβ/summationdisplay\npm∝angbracketleftTαG(/vectork+/vector q,iωn+ipm)\nTβG(/vectork,ipm)∝angbracketrightc,(7)\nwhereG(/vectork,ip) is the Matsubara Green function and ∝angbracketleft...∝angbracketrightc\nindicates a configurational average required in the pres-\nence of any disorder (chemical, structural or magnetic)\nin the system. Using a Lehman representation for the\nGreen function [28]\nG(/vectork,ipm) =/integraldisplay+∞\n−∞dE\nπℑG+(/vectork,E)\nipm−E(8)\nwithG+(/vectork,E) the retarded Green function and using the\nrelation\n1\nβ/summationdisplay\npm1\nipm+iωn−E11\nipm−E2=f(E2)−f(E1)\niωn+E2−E13\nfor the sum over the Matsubara poles in Eq. (7), the torque-torq ue correlation function is obtained as:\nπαβ(/vector q,iωn) =1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE1\nπ+∞/integraldisplay\n−∞dE2\nπTr/angbracketleftbigg\nTαℑG(/vectork,E1)TβℑG(/vectork,E2)f(E2)−f(E1)\niωn+E2−E1/angbracketrightbigg\nc. (9)\nPerfoming finally the analytical continuation iωn→ω+iδone arrives at the expression\nΓαβ(/vector q,ω) =−π\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE1\nπ+∞/integraldisplay\n−∞dE2\nπTr/angbracketleftbigg\nTαℑG(/vectork+/vector q,E1)TβℑG(/vectork,E2)/angbracketrightbigg\nc(f(E2)−f(E1))δ(ω+E2−E1)\n=−π\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπTr/angbracketleftbigg\nTαℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg\nc(f(E)−f(E+ω)) (10)\nfor the imaginary part of the correlation function with Γ αβ(/vector q,ω) =−πℑπαβ(/vector q,ω). Accordingly one gets for the\ndiagonal elements of Gilbert damping tensor the expression\nααα(/vector q) =γ\nM0Vlim\nω→0∂[ǫ·Γ(/vector q,ω)·ǫ]\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαα\n=γπ\nM0Vlim\nω→0∂\n∂ω1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπ2(f(E+ω)−f(E))Tr/angbracketleftbigg\nTβℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg\nc\n=γ\nM0V1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπδ(E−EF)Tr/angbracketleftbigg\nTβℑG(/vectork+/vector q,E)TβℑG(/vectork,E)/angbracketrightbigg\nc\n=1\n4[ααα(/vector q,G+,G+)+ααα(/vector q,G−,G−)−ααα(/vector q,G+,G−)−ααα(/vector q,G−,G+)], (11)\nwhere the index βof the torque operator Tβis related to the index αaccording to Eq. 6, and the auxiliary functions\nααα(/vector q,G±,G±) =γ\nM0Vπ1\nΩBZ/integraldisplay\nd3kTr/angbracketleftbigg\nTβG±(/vectork+/vector q,EF)TβG±(/vectork,EF)/angbracketrightbigg\nc(12)\nexpressed in terms of the retarded and advanced Green function s,G+andG−, respectively.\nTo account properly for the impact of spin-orbit coupling when dealin g with Eqs. (11) and (12) a description of\nthe electronic structure based on the fully relativistic Dirac formalis m is used. Working within the framework of local\nspin density formalism (LSDA) this implies for the Hamiltonian the form [2 9]:\nˆHD=c/vectorα·/vector p+βmc2+V(/vector r)+β/vectorσ·ˆ/vector mBxc(/vector r). (13)\nHereαiandβare the standard Dirac matrices, /vectorσdenotes the vector of relativistic Pauli matrices, /vector pis the relativistic\nmomentum operator [30] and the functions V(/vector r) and/vectorBxc=/vectorσ·ˆ/vector mBxc(/vector r) are the spin-averaged and spin-dependent\nparts, respectively, of the LSDA potential [31] with ˆ/vector mgiving the orientation of the magnetisation.\nWith the Dirac Hamiltonian given by Eq. (13), the torque operator ma y be written as /vectorT=β[/vector σ׈/vector m]Bxc(/vector r).\nFurthermore, the Green functions entering Eqs. (11) and (12) a re determined using the spin-polarized relativistic\nversion of multiple scattering theory [29, 32] with the real space re presentation of the retarded Green function given\nby:\nG+(/vector r,/vector r′,E) =/summationdisplay\nΛΛ′Zn\nΛ(/vector r,E)τnm\nΛΛ′(E)Zm×\nΛ′(/vector r′,E)\n−δnm/summationdisplay\nΛ/bracketleftbig\nZn\nΛ(/vector r,E)Jn×\nΛ′(/vector r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(/vector r,E)Zn×\nΛ′(/vector r′,E)Θ(rn−r′\nn)/bracketrightbig\n. (14)4\nHere/vector r,/vector r′refertoatomiccellscenteredatsites nandm, respectively,where Zn\nΛ(/vector r,E) =ZΛ(/vector rn,E) =ZΛ(/vector r−/vectorRn,E) isa\nfunction centered at the corresponding lattice vector /vectorRn. The four-component wave functions Zn\nΛ(/vector r,E) (Jn\nΛ(/vector r,E)) are\nregular (irregular) solutions to the single-site Dirac equation labeled by the combined quantum numbers Λ = ( κ,µ),\nwithκandµbeing the spin-orbit and magnetic quantum numbers [30]. Finally, τnm\nΛΛ′(E) is the so-called scattering\npath operator that transfers an electronic wave coming in at site minto a wave going out from site nwith all possible\nintermediate scattering events accounted for.\nUsing matrix notation with respect to Λ, this leads to the following exp ression for the auxilary damping parameters\nin Eq. (12):\nααα(/vector q,G±,G±) =γ\nM0Vπ1\nΩBZ/integraldisplay\nd3kTr/angbracketleftbigg\nTβτ(/vectork+/vector q,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc. (15)\nIn the case of a uniform magnetization, i.e. for q= 0 one obviously gets an expression for the Gilbert damping tensor\nas it was worked out before [7]. Assuming small wave vectors, the te rmτ(/vectork+/vector q,E±\nF) can be expanded w.r.t. /vector qleading\nto the series\nτ(/vectork+/vector q,EF) =τ(/vectork,E)+/summationdisplay\nµ∂τ(/vectork,E)\n∂kµqα+1\n2/summationdisplay\nµν∂τ(/vectork,E)\n∂kµ∂kνqµqν+... (16)\nthat results in a corresponding expansion for the Gilbert damping:\nα(/vector q) =α+/summationdisplay\nµαµqµ+1\n2/summationdisplay\nµναµνqµqν+... (17)\nwith the following expansion coefficients:\nα0±±\nαα=g\nπµtot1\nΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβτ(/vectork,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc(18)\nαµ±±\nαα=g\nπµtot1\nΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβτ(/vectork,E±\nF)/angbracketrightbigg\nc(19)\nαµν±±\nαα=g\nπµtot1\n2ΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβ∂2τ(/vectork,E±\nF)\n∂kµ∂kνTβτ(/vectork,E±\nF)/angbracketrightbigg\nc, (20)\nand with the g-factor 2(1+ µorb/µspin) in terms of the spin and orbital moments, µspinandµorb, respectively, and the\ntotal magnetic moment µtot=µspin+µorb. The numerically cumbersome term in Eq. (20), that involves the sec ond\norder derivative of the matrix of /vectork-dependent scattering path operator τ(/vectork,E), can be reformulated by means of an\nintegration by parts:\n1\nΩBZ/integraldisplay\nd3kTβτ(/vectork,EF)Tβ∂2τ(/vectork,EF)\n∂kµ∂kν=/bracketleftBigg/integraldisplay /integraldisplay\ndkβdkγTi\nβτ(/vectork,E)Tj\nβ∂τ(/vectork,E)\n∂kβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleKα\n2\n−Kα\n2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=0\n−/integraldisplay /integraldisplay /integraldisplay\ndkαdkβdkγTβ∂τ(/vectork,EF)\n∂kµTβ∂τ(/vectork,EF)\n∂kν/bracketrightBigg\n=−1\nΩBZ/integraldisplay\nd3kTβ∂τ(/vectork,EF)\n∂kµTβ∂τ(/vectork,EF)\n∂kν\nleading to the much more convenient expression:\nαµν±±\nαα=−g\n2πµtot/integraldisplay\nd3kTr/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβ∂τ(/vectork,E±\nF)\n∂kν/angbracketrightbigg\nc. (21)\nIII. RESULTS AND DISCUSSIONS\nThe scheme presented above to deal with the Gilbert\ndamping in non-collinear systems has been implementedwithin the SPR-KKR program package [33]. To exam-5\nine the importance of the chiral correction to the Gilbert\ndamping a first application of Eq. (19) has been made\nfor the multilayer system (Cu/Fe 1−xCox/Pt)nseen as a\nnon-centrosymmetricmodelsystem. Thecalculatedzero-\norder (uniform) GD parameter αxxand the correspond-\ning first-order (chiral) αx\nxxcorrection term for /vector q∝bardblˆxare\nplotted in Fig. 1 top and bottom, respectively, as a func-\ntion of the Fe concentration x. Both terms, αxxandαx\nxx,\n0 0.2 0.4 0.6 0.8 100.20.4αxx\n0 0.2 0.4 0.6 0.8 1xCo0123αxxx (a.u.)\nFIG. 1: The Gilbert damping parameters αxx(top) and\nαx\nxx(bottom) calculated for the model multilayer system\n(Cu/Fe 1−xCox/Pt)nusing Eqs. (18) and (19), respectively.\nincrease approaching the pure limits w.r.t. the Fe 1−xCox\nalloy subsystem. In the case of the uniform parame-\nterαxx, this increase is associated with the dominating\nbreathing Fermi-surface damping mechanism. This im-\nplies that the modification of the Fermi surface (FS) in-\nduced by the spin-orbit coupling (SOC) follows the mag-\nnetization direction that slowly varies with time. An ad-\nditional contribution to the GD, having a similar origin,\noccurs for the non-centrosymmertic systems with heli-\nmagnetic structure. In this case, the features of the elec-\ntronicstructure governedby the lackofinversionsymme-\ntry result in a FS modification dependent on the helicity\nof the magnetic structure. This implies a chiral contri-\nbution to the GD which can be associated with the term\nproportional to the gradient of the magnetization. Ob-\nviously, this additional modification of the FS and the\nassociated mechanism for the GD does not show up for\na uniform ferromagnet. As αis caused by the SOC one\ncan expect that it vanishes for vanishing SOC. This was\nindeed demonstrated before [5]. The same holds also for\nαxthat is cased by SOC as well.\nAnother system considered is the ferromagnetic alloy\nsystem bcc Fe 1−xCox. As this system has inversion sym-\nmetry the first-order term αµshould vanish. This expec-\ntation could also be confirmed by calculations that ac-count for the SOC. The next non-vanishing term of the\nexpansion of the GD is the term ∝q2. The correspond-\ning second-order term αxx\nxxis plotted in Fig. 2 (bottom)\ntogether with the zero-order term αxx(top). The bot-\n0 0.1 0.2 0.3 0.4 0.500.511.52αxx× 103\n0 0.1 0.2 0.3 0.4 0.5xCo012αxxxx ((a.u.)2)Fe1-xCox\nFIG. 2: The Gilbert damping terms αxx(top) and αxx\nxx(bot-\ntom) calculated for bcc Fe 1−xCox.\ntom panel shows in addition results for αxx\nxxthat have\nbeen obtained by calculations with the SOC suppressed.\nAs one notes the results for the full SOC and for SOC\nsuppressed are very close to each other. The small dif-\nference between the curves for that reason have to be as-\ncribed to the hybridization of the spin-up and spin-down\nsubsystems due to SOC. As discussed in the literature\n[9, 17, 20] a non-collinear magnetic texture has a corre-\nsponding consequence but a much stronger impact here.\nIn contrastto the GDin uniform FM systemswhereSOC\nisrequiredto breakthe totalspin conservationin the sys-\ntem,αxx\nxxis associated with the spin-pumping effect that\ncan be ascribed to an emergent electric field created in\nthe non-uniform magnetic system. In this case magnetic\ndissipation occurs due to the misalignment of the elec-\ntron spin following the dynamic magnetic profile and the\nmagnetization orientation at each atomic site, leading to\nthe dephasing of electron spins [16]\nIV. SUMMARY\nTo summarize, expressions for corrections to the GD\nofhomogeneoussystems werederived which areexpected\nto contribute in the case of non-collinear magnetic sys-\ntems. The expression for the GD parameter α(/vector q) seen\nas a function of the wave vector /vector qis expanded in powers\nofq. In the limit of weakly varying magnetic textures,\nthis leads to the standard uniform term, α, and the first-\nand second-order corrections, αµandαµν, respectively.\nModel calculations confirmed that a non-vanishing value6\nforαµcan be expected for systems without inversion\nsymmetry. In addition, SOC has been identified as the\nmajor source for this term. The second-order term, on\nthe other hand, may also show up for systems with inver-\nsion symmetry. In this case it was demonstrated by nu-\nmerical work, that SOC plays only a minor role for αµν,\nwhile the non-collinearity of the magnetization plays the\ncentral role.V. ACKNOWLEDGEMENT\nFinancial support by the DFG via SFB 1277 (Emer-\ngente relativistische Effekte in der Kondensierten Ma-\nterie) is gratefully acknowledged.\n[1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).\n[2] V. 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Ebert et al., The Munich\nSPR-KKR package , version 7.7,\nhttp://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR\n(2017), URL http://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR ."    },    {        "title": "1806.00151v1.Dirac_Surface_State_Modulated_Spin_Dynamics_in_a_Ferrimagnetic_Insulator_at_Room_Temperature.pdf",        "content": "  \nDirac -Surface -State Modulated Spin Dynamics in a  Ferrimagnetic Insulator  at Room \nTemperature  \n \nChi Tang1†, Qi Song2, 8†, Cui -Zu Chang3, 4, Yadong Xu1, Yuichi Ohnuma5, Mamoru \nMatsuo5, 6, Yawen Liu1, Wei Yuan2, 8, Yunyan Yao2, 8, Jagadeesh S. Moodera3, 7, \nSadamichi Maekawa5, Wei Han2, 8 and Jing Shi1* \n1Department of Physics & Astronomy, University of California, Riverside, Riverside, CA 92521, \nUSA  \n2International Center for Quantum Materials, School  of Physics , Peking University, Beijing \n100871, China  \n3Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, MA \n02139, USA  \n4Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA  \n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319 -1195, Ibaraki, \nJapan  \n6Advanced Institute for Materials Research, Tohoku University, Sendai 980 -8577, Miyagi, Japan \n7Department of Physics, Massachusetts Institute of Technology, C ambridge, MA 02139, USA  \n8Collaborative Innovation Center of Quantum Matter, Beijing 100871, China  \n†These are co -first authors.  \n*Correspondence to:  jing.shi@ucr.edu .   \n \n \n \nAbstract: This work demonstrates d ramatic ally modified spin dynamics of  magnetic \ninsulator (MI) by the spin -momentu m locked Dirac surface states of  the adjacent \ntopological insulator (TI) which can be harnessed for spintronic applica tions. As the \nBi-concentration x is systematically tuned in  5 nm thick (Bi xSb1-x)2Te3 TI film, the \nweight of the surface relative to bulk states peaks at x = 0.32 when the chemical \npotential approaches the Dirac point. At this concentration, the Gilbert damping \nconstant of the  precessing magnetization in  10 nm thick Y 3Fe5O12 MI film in the \nMI/TI heterostructures  is enhanced by an order of magnitude, the largest amo ng all \nconcentrations. In addition, the MI acquires additional strong magnetic anisotropy \nthat favors the in -plane orientation with similar Bi -concentration dependence. These \nextraordinary effects  of the Dirac surface states distinguish TI from other mater ials \nsuch as heavy metals in modulat ing spin dynamics of the neighboring magnetic layer.   INTRODUCTION  \n \nTopological insulators (TI) are a new state  of quantum matter with unique spin and charge properties \nowing to the non -trivial band topology an d strong spin -orbit  coupling  (1). These properties can lead to a \nvariety of exotic phenomena including topological magneto -electric effects  (2), quantum anomalous Hall \neffect  (3), image  magnetic monopoles  (4), etc. A remarkable feature that profoundly affect s spin and \ncharge transport in TI is that electrons in Dirac surface states have their  spin locked orthogonally  to the ir \nmomenta in two dimensions  (5, 6). Various spin and charge transport effects of such spin -momentum \nlocking have been observed in spin valves with TI  (7), spin Seebeck effect  (8), inverse Edelstein effect  (9-\n13) and spin -orbit torque switching  (14, 15). Clearly, strong coupling between ele ctron spin and \ntranslational degrees of freedom can be exploited as an efficient way of manipulating spins and vice versa, \nwhich are essential for spintronics.  \nDevices revealing the aforementioned effects are often constructed in heterostructures containin g TI and a \nferromagnetic layer which serves as either a pure spin current source or a spin detector. Any spin -\ndependent effect on transport properties is conveniently measured through the metallic surface states. In \nthe same heterostructures, the reverse e ffect, i.e. the effect of the spin -momentum locked Dirac surface \nstates on the spin dynamics of the magnetic layer, has not yet been systematically studied. It is known that \na thin heavy metal such as Pt or W in contact with a magnetic material can cause b roadening of the \nferromagnetic resonance (FMR) linewidth, which is mainly attributed to the excess flow of spin current \ndue to the spin pumping effect  (16, 17); however, the linewidth broadening is generally insignificant (~10 \nOe) (18, 19). In this regime, the effect is quantitatively described by the momentum sum of the imaginary \npart of the dynamical transverse spin susceptibility  (16) or alternatively the spin -mixing conductance at the \ninterface  (17).  \nIn this work, we investigate the room temperature spin dynamics of yttrium iron garnet ( Y3Fe5O12 or YIG), \na ferrimagnetic insulator, in heterostructures containing (Bi xSb1-x)2Te3, a TI material . By systematically \ntuning the chemical potential of the TI  with a fixed thickness, i.e. five quintuple layers (QL) or ~ 5  nm, via \nvarying Bi concentration x, we control the weight of the surface states  relative to the bulk states  (20). An \nalternative way of controlling the surface state weight is to use electrostatic gating which requires \nextensive materials work. In TI layers dominated by surface states,  the direction of spins pumped into the \nDirac  surface states by FMR has to be locked in the plane. On the other hand, the spins in TI and those of \nYIG are exchange coupled to each other at the interface . As a result,  the direction of the  precessing spins \nin YIG is forced to be aligned in the plane (Fig. 1(a)). Consequently, spin pumping into Dirac surface \nstates  is expected to give rise to  strong damping of the precessing magnetization . Indeed, we observe \nunprecedentedly large FMR linewidth broadening of YIG (up to 111 Oe at 9.6 GHz) when the chemical \npotential of the TI is tuned close to the Dirac point which corresponds to a 15 times larger Gilbert damping \nconstant. This dramatic enhancement in damping is accompanied by an anomalously large increase (~ \n100%) in the easy-plane anisotropy of the YIG layer.    \nRESULTS  \n \nWe choose YIG for the magnetic constituent in our heterostructures for several reasons  to be described in \nthe Materials and Methods section . In this study, we have fabricated seven heterostructure samples with \nvarious Bi concentration s, x, ranging from 0 to 1 .  x = 0.32 sam ple is most insulating as indicated by the \nlargest resistance at 300 K (Fig. 1(b)) and the largest negative slope in its temperature dependence (Fig. \n1(c)), suggesting that the chemical potential is located close to the Dirac point. With x deviating from 0 .32, \nthe chemical potential is tuned away from the Dirac point  (20). In Bi 2Te3 (x = 1) and Sb 2Te3 (x = 0), the \nchemical potential is located in the bulk conduction and valence bands, respectively .  \nWe measure FMR spectra with both cavity and broad -brand co -planar waveguide FMR setups. Prior to TI \nlayer deposition,  cavity  FMR  measurements are performed on all 10 nm thick YIG films at room \ntemperature using a Bruker 9. 6 GHz X -band EMX EPR spectrometer. Fig. 2( b) (red curve) shows a FMR \nderivative absorption spectrum of a  representative YIG film  with an in -plane magnetic field, which can be \nwell fitted with a single Lorentzian derivative. T he peak -to-peak  linewidth ∆𝐻𝑝𝑝 of 7.0 Oe and  resonance \nfield 𝐻𝑟𝑒𝑠 of 2434 Oe  are obtained from the fitting . Such a narrow linewidth indicates  high YIG f ilm \nquality . The mean values of both ∆𝐻𝑝𝑝 and 𝐻𝑟𝑒𝑠 for all seven  samples are 10.2 ± 3.1 Oe and 2394.5 ± 40.2 \nOe, respectively. When FMR is performed as a function of the polar angles 𝜃𝐻 (defined in Fig. 2(a )), the \n𝜃𝐻-dependence of both quantitie s shows very small variatio ns for all seven YIG samples (fig. S2 ), \nindicating a tight control over the YIG film quality.  \nTo study the effect of the  TI Dirac surface state s on YIG spin dynamics, we compare the YIG FMR spectra \ntaken before and after the TI growth. Fig. 2( b) shows a direct comparison between the two representative \nFMR spectra taken with in -plane fields before and after the growth of a 5 QL Sb 2Te3 (x = 0) film. Two \ndistinct differences stand out. First, 𝐻𝑟𝑒𝑠 is shifted to a lower field  by 207 Oe , i.e. from 2434  Oe to 2227 \nOe, indicating a large effect on magnetic anisotropy upon adding the 5 QL Sb 2Te3. Second, the F MR \nlinewidth ∆𝐻𝑝𝑝 is broadened  by seven times. To more accurately determine the effective anisotropy field \nchange, we me asure  FMR of  both YIG/Sb 2Te3 and a YIG reference sample as a function of  polar angle  𝜃𝐻 \n(21, 22). Fig. 2(c) shows the spectra at selected polar angles between the  in-plane ( 𝜃𝐻 = 90˚) and out -of-\nplane ( 𝜃𝐻 = 0˚) magnetic field orientations. The YIG reference sample has  relatively narrow FMR  \nlinewidth  for all polar angles.  In the meantime, 𝐻𝑟𝑒𝑠 decreases monotonically from ~5560 Oe for 𝐻 out-of-\nplane to ~ 2434  Oe for 𝐻 in-plane,  consistent with the beh avior of nanometer thick YIG films with easy-\nplane magnetic anisotropy  (23). With 5 QL Sb 2Te3 on top, however, dramatic differences can be readily \nidentified  as shown in the top panel of Fig. 2(c): significant broadening of ∆𝐻𝑝𝑝 and large shift in 𝐻𝑟𝑒𝑠. \n𝐻𝑟𝑒𝑠 shift occurs at all angles; hence, the overall 𝐻𝑟𝑒𝑠 range is greatly expanded, i.e. between 2227 Oe and \n6000 Oe. Such marked  effects are not  seen in heterostructures containing thin heavy metal layers such as \nPt (18).   The differences  in both ∆𝐻𝑝𝑝 and 𝐻𝑟𝑒𝑠 caused by the 5 QL Sb 2Te3 suggest  their origin in  TI’s band \nstructure. In order to study the effects of the TI Dirac surface states, we compare the FMR results in all \nseven samples  in which the surface -to-bulk ratio systematically varies . We first study the effect of varying \nx on the easy-plane magnetic anisotropy. We plot 𝐻𝑟𝑒𝑠 as a function of θH in Fig. 3(a)  for all seven \nYIG/(Bi xSb1-x)2Te3 samples plus a YIG reference film. T he angular dependence data can be fitted \nreasonably well with the shape anisotropy plus  a uniaxial anisotropy term, both in the form of cos2𝜃. As is \nroutinely done in literature (21), we solve three transcendental equations  numerically and seek  the least -\nsquare fitting results . Note that if the uniaxial anisotropy  term is negative, it simply represents  easy-plane \nanisotropy. We find that  it is indeed negative, i.e.  there is additional  easy-plane anisotropy  in all seven \nsamples .  By fitting the p olar angle dependence, we obtain the two best -fit parameters, 4𝜋𝑀 𝑒𝑓𝑓 and the g-\nfactor for each sample. The effective anisotropy field is defined as  4𝜋𝑀 𝑒𝑓𝑓= 4𝜋𝑀 𝑠+ 𝐻𝑎𝑛, where 4𝜋𝑀 𝑠 \nand 𝐻𝑎𝑛 denote the demagnetizing field and an effective easy-plane (positive) magnetic anisotropy field, \nrespectively  (24). The best -fit parameters are s ummarized  in table. S1 . The extracted 4𝜋𝑀 𝑒𝑓𝑓 is plotted in \nFig. 3(b). Since the demagnetizing field  4𝜋𝑀 𝑠 is about 1780 Oe, 4𝜋𝑀 𝑒𝑓𝑓 is clearly larger than 4𝜋𝑀 𝑠 in all \nsamples. Furthermore, 4𝜋𝑀 𝑒𝑓𝑓 depends on x. For the two most metallic TI films, i.e. x = 0 and x = 1, \n4𝜋𝑀 𝑒𝑓𝑓 is increased by 420 and 460 Oe, which accounts for 23% and 26% of its demagnetizing field \n4𝜋𝑀 𝑠, respectively. In comparison, the corresponding increase is only 5%  in YIG/Pt (fig. S3(a)). More \ninterestingly, as the chemical potential is tuned into the band gap, i.e. surface states becoming dominant, \n4𝜋𝑀 𝑒𝑓𝑓 increases further and peaks in the most insulating sample ( x = 0.32), reaching 3800 Oe. This \nincrease repr esents nearly a 100% enhancement over the YIG demagnetizing field 4𝜋𝑀 𝑠.  \nA common origin of enhanced magnetic anisotropy in thin films is related to the interface strain. In \n(Bi xSb1-x)2Te3, the interaction between the neighboring Te -Bi/Sb -Te-Bi/Sb -Te qu intuple layers is of the \nvan der Waals type. Between TIG and TI , there is no  epitaxial relation due to widely different lattice \nstructures; therefore, the strain and strain -induced anisotropy are expected to be small at the YIG -TI \ninterface for all samples . Another possibility of the enhanced 4𝜋𝑀 𝑒𝑓𝑓 is an increased demagnetizing field. \nIf part of TI becomes ferromagnetic, it can in principle cause an increase in 4𝜋𝑀 𝑠. We exclude this \npossibility with the following arguments. First, s uch a proximity induced moment, if exists, can only come  \nfrom a few atomic layers at the interface and is clearly too small to account for the  observed  100% \nincrease. Our magnetization measurements do not support this possibility either (f ig. S5) . The same  \nmagnetometry results do not show any clear  Bi-concentration dependence  within experimental uncertainty.  \nMore importantly, the proximity effect in YIG/TI heterostructures occurs at much lower temperatures (< \n150 K) as reported in previous studies  (8, 25). Additionally , the Tc of the induced ferromagnetism in TI \nwas found to be uncorrected with TI’s chemical potential position  (25). In our data, the 4𝜋𝑀 𝑒𝑓𝑓 \nenhancement follows the same trend as the re sistivity as shown in Fig. 1(c) . From these analyses, w e \nconclude that the enhanced effective  anisotropy originates from the Dirac surface states.   The cavity -FMR meas urements have already indicated anomalously broadened FMR linewidth at a \nparticular microwave frequency. In order to extract the Gilbert damping constant α, we perform broad -\nband FMR measurements using a coplanar waveguide setup for all YIG/TI samples up t o 12 GHz. \nRepresentative transmission data 𝑆21 in Fig. 4(a) show the FMR absorption of  YIG/Sb 2Te3 with several \nfrequencies. Both FMR resonance field shift and linewidth broadening display the same trend in Fig. 4(b). \nThe half width at half maximum, ∆𝐻= √3∆𝐻𝑝𝑝/2 , is extracted by fitting a Lorentzian function to each \n𝑆21 spectrum up to 12 GHz, and then plotted in Fig. 4(c) for all  samples . A linear relation between ∆𝐻 and \nfrequency is observed, and α can be calculated by (24) \n \n∆𝐻=2𝜋\n𝛾𝛼𝑓+ ∆𝐻0 ,      (1) \n \nwhere 𝛾 and ∆𝐻0 are the gyromagnetic ratio and  the inhomogeneity linewidth broadening, respectively. \nFig. 4(d) shows α vs. x for all samples. I nterestingly, 𝛼 peaks at x = 0.32 as well, i.e. in the most insulating \nsample, similar to the resistivity and the effective  anisotropy field. Compared to the two most metallic \nsamples with 𝛼 ×for x = 0 (or Sb 2Te3) and 𝛼 ×for x = 1 (or Bi 2Te3), 𝛼 reaches the \nmaximum value of 2.2 ×  for x = 0.32, which is an order of magnitude larger than that of the bare YIG \nfilms (average value of ×). In comparison, 𝛼 in YIG/Pt is only twice as large as that in the YIG \nreference, as shown in f ig. S4(b). Additionally, 𝛼 shows the same trend as that of the resistivity and \neffective  anisotropy. These fact s suggest a common origin, i.e. the special band structures of the TI surface \nstates rather than the spin -orbit coupling of the constituent elements. The latter effect would imply a \nmonotonically increasing trend as more Bi atoms are incorporated.  \n \nDISCUSS ION  \n \nWe now show that these three effects are actually connected and given by the spin -momentum locking \nproperties of  the TI Dirac surface state s. The spins ( 𝜎⃗) of TI and the spins ( 𝑆⃗) of YIG are coupled by the \nexchange interaction expressed as  (26, 27), 𝐻=𝐽𝑠𝑑𝜎⃗ ∙ 𝑆⃗, with 𝐽𝑠𝑑 being the exchange constant at the \ninterface.  We note that the spins in TI lie  in the plane due to the spin -momentum locking.  Therefore, the \nspins in YIG are pulled towards  the plane as well.  This pulling effect induces th e easy -plane anisotropy \ngiven by  \n \n                            𝐸𝑎𝑛= −1\n2 (𝜒∕∕−𝜒⊥)𝑀2                                                (2) \n     \nwhere 𝜒∕∕ and 𝜒⊥ are the in -plane  and out -of-plane components of the spin susceptibility in TI, \nrespectively, and M is the magnetization of  YIG.  In particular, when the chemical potential  is close to  the  Dirac point , 𝜒⊥ is strongly suppressed due to the gap (28) and, thus, 𝜒∕∕≫ 𝜒⊥. TI’s spin susceptibility \ngives rise to the easy-plane magnetic anisotropy field by  𝐻𝑎𝑛= −𝜕𝐸𝑎𝑛\n𝜕𝑀. \nThe same interfacial exchange interaction also affects  spin pumping  (16); the spin precession of YIG i n \nFMR results in the motion of spins in TI.  The induced spin current ( 𝐼𝑠) in TI is expressed as  \n  \n     𝐼𝑠 ∝ 𝐽𝑠𝑑2𝐼𝑚𝜒+−(𝜔)     (3) \n \nwhere 𝜒+−(𝜔) is the transverse component in TI to the mag netization direction of YIG, and  ω is the FMR \nfrequenc y.  We note that since the spin Seebeck effect is due to the spin pumping by heat  (26, 27), it is also \ngiven by 𝜒+−(𝜔) but integrated over ω in the range of the thermal distribution of spin fluctuations.   The \nsusceptibility is calculated by taking into account the di rect transition near the Fermi level  because of the \nspin-momentum locking in TI (see SI  for details).  Since the resonance frequency  (a few GHz) is much \nsmaller than the energy gap of TI (~ 0.3  eV) (29), the direct transition is more effective when the Fermi \nenergy is near the Dirac point.  As a result, the spin pumping (the Gilbert damping) is significantly \nenhanced near the Dirac point. This model also explain s enhanced spin Seebeck effect reported previously  \n(8). \nIn summary , we have observe d dramatic modifications of YIG spin dynamics by spin -momentum locked \nsurfaces of a thin TI layer in high -quality YIG/TI heterostructures  with different Bi/Sb ratios . The spin -\nmomentum locking in TI provides not only a sensitive detection of the magnetic s tate in magnetic \nmaterials serving as a spin current source, but also an active way of manipulating ultrafast magnetization \ndynamics and magnetic anisotropy with the unique properties of the topological Dirac surface states, \nwhich offers exciting opportuni ties for potential spintronic applications.    \n \nMATERIALS  AND METHODS  \n \nChoice of YIG : We choose 10 nm thick YIG films as the MI layers in all MI/TI heterostructures. First, \nYIG in general has a very small Gilbert damping constant 𝛼~ 3 × 10-5 in crystals  and ~ 10-3 in 10 nm \nthick YIG films); and the FMR linewidth is relatively narrow (~ 10 Oe  at ~ 10 GHz for thin films). \nTherefore, small linewidth changes can be easily detected. Second, YIG films are prepared  first with high \ntemperatures (~ 800 ºC) with pulsed laser deposition and rapid thermal annealing  and the TI layers are \ngrown at much lower temperatures (~230 ºC) after  with molecular beam epitaxy . This growth sequence \nand the large temperature difference prevent serious intermixing across the interface. In our \nheterostructures, YIG is atomically flat, which ensures the flat YIG -TI interface. Third, similar to our \nprevious spin Seebeck effect study  (8), here we conduct FMR measurements at room temperature which is \nwell above that of the induced ferromagnetism in the TI surface layer. Consequently, the dynamic behavior \nof YIG is not affected by the induced ferromagnetism in TI  (18).   Heterostructure growth:  Thin YIG films are grown on epi -ready lattice -matched single crystal GGG \n(111) substrates via pulsed laser deposition. The detailed growth recipe of YIG films is described in a \nprevious p aper (30). To fabricate high -quality YIG/(Bi xSb1-x)2Te3 heterostructures with various Bi  \nconcentrations (x = 0, 0.22, 0.27, 0.32, 0.46, 0.67 and 1 in this study) , YIG (111) films are then transferred \nto an ultra-high vacuum molecular beam epitaxy (MBE) system with the base pressure better than 5×10-10 \nTorr for TI growth. High -purity Bi (99.999%), Sb (99.9999%) and Te (99.9999%) are evaporated from \nKnudsen effusion cells. During the growth, the YIG substrate  is kept at 230 °C and the growth rate  of TI  is \n~0.2 QL/min. The heterostructure film is covered with a 5 nm Te protection layer before taken out of the \nMBE chamber for the FMR measurements.    \nFMR measurements:  The polar angle dependent FMR  measurements fo r all samples are performed using \na Bruker 9.6 GHz X -band EMX EPR spectrometer. Samples can be rotated with respect to the static field \ndirection from the in -plane to out -of-plane geometry with a protractor  reading the angle precisely.  \nThe Gilbert damping  constant measurements are conducted by a broad -band FMR using coplanar \nwaveguide setup. The forward amplitude of complex transmission coefficients ( S21) is recorded by the \nvector network analyzer (VNA, Agilent E5071C) connected to a straight -line coplanar  waveguide  (31). \nSample is attached to the waveguide and the measurement is performed with the frequency sweeping from \n1 GHz to 12 GHz at a fixed magnetic field which can be varied up to 4000 Oe.  \n \n \nSUPPLEMENTARY  MATERIALS  \n \nfig. S1. Crystal structure and surface morphology.  \nfig. S2. FMR resonance field and linewidth for all YIG films.  \nfig. S3. Effect of Pt on YIG resonance characteristics.  \nfig. S4. Comparison of FMR between YIG/TI and YIG/Pt.  \nfig. S5. Comparison of total effective anisotropy field and demagnetizing field.  \nfig. S6. High -resolution transmission electron microscope image of a representative YIG/TI sample.  \nfig. S7 . TI surface state dispersion.  \nfig. S8 . Direct transition of TI conduction electrons driven by spi n pumping.  \ntable S1. Two parameters (i.e. 4𝜋𝑀 𝑒𝑓𝑓 and γ/2π) obtained from fitting.  \n  REFERENCES AND NOTES  \n \n1. X. L. Qi, S. C. Zhang, Topological Insulators and Superconductors. Rev. Mod. Phys.  83, \n1057 (2011).  \n2. X. L. Qi, T. L. Hughes, S. C. Zhang, Topological Field Theory of Time -Reversal Invariant \nInsulators. Phys. Rev. B  78, 195424 (2008).  \n3. C. Z. Chang  et al. , Experimental Observation of the Quantum Anomalous Hall Effect in a \nMagnetic Topological Insulator. Science  340, 167 (2013).  \n4. X. L. Qi, R. D. Li, J. D. Zang, S. C. Zhang, Inducing a Magnetic Monopole with \nTopological Surface States. Science  323, 1184 (2009).  \n5. D. 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Kally for the sample preparation and useful \ndiscussions with Z. Shi, J. Li, V. Ortiz and M. Aldosary .  \nFunding: This work was  supported as part of the  SHINES, an Energy Frontier Research Center funded by \nthe U.S. Department of Energy, Office of Science, Basic Energy Sci ences under Award No. SC0012670  \n(CT, YDX, YWL  and JS) . QS, WY, YYY and WH  acknowledge the support of the National Basic \nResearch Program s of China (973 Grants 2014CB920902 and 2015CB921104) and the National Natural \nScience Foundation of China (NSFC Grant 11574006).  CZC and JSM acknowledge the support from NSF \nGrants No. DMR -1207469, ONR Grant No. N00014 -16-1-2657, and the STC Center for In tegrated \nQuantum Materials under NSF Grant No. DMR -1231319.  \nAuthor contributions:  JS conceived and supervised the experiments. CT grew the YIG magnet ic insulator \nthin films  and performed cavity FMR measurements and data analysis with the help of YWL . QS \nperformed the broadband FMR measurements and data analysis with the help of WY and YYY and the \nsupervision of WH. YDX performed TEM sample preparation. CZC grew the topological insulator thin  films on YIG to form heterostructures  with JSM’s supervision . YO, MM and SM  did theoretical \ncalculations . All authors  participated in the preparation of the final manuscript.  \nCompeting interests:  The authors declare that they have no competing interests. Data and materials \navailability: All data needed to evaluate t he conclusions in the paper are present in the paper and/or the \nSupplementary Materials. Additional data related to this paper may be requested from the authors.  \n  FIGURES AND TABLES  \n \n \n \n \n \nFig. 1.  FMR measurement principle  and TI properties .  (a) Schematic drawing of \nmagnetization dynamics in YIG interfaced with TI in which the spin of the surface state electron is \nlocked to momentum. (b) Room temperature sheet resistance of (Bi xSb1-x)2Te3 with different Bi \nconcentrations. (c) Temperature depe ndence of the sheet resistance of seven YIG (10 nm)/(Bi xSb1-\nx)2Te3 (5 QL) heterostructures.  \n  \n \n \nFig. 2.  YIG FMR spectra with and without TI .  (a) Definition of polar angles, 𝜃𝐻  and 𝜃𝑀 in \nFMR measurements.  (b) FMR derivative absorption spectra of YIG/Sb 2Te3 and YIG reference \nsample at a frequency of 9.6 GHz with magnetic field applied in -plane ( 𝜃𝐻 = 90˚). The solid lines \nare the best fits to extract the resonance field 𝐻𝑟𝑒𝑠 and peak -to-peak linewidth ∆𝐻𝑝𝑝. (c) FMR \nderivative absorption spe ctra of YIG/Sb 2Te3 and YIG reference sample with the polar angle θH \nranging from 0˚(out -of-plane) to 90˚ (in -plane) at 300 K. The extra peak -like feature on the high \nfield side of the resonance at 0˚ and 10˚ is also observed some other samples, which could  be \ncaused by minor inhomogeneity change in YIG due to the presence of the TI layer.  \n  \n \n \n \nFig. 3. Extracted 4𝜋𝑀 𝑒𝑓𝑓 from FMR polar angle dependence fitting.  (a) Polar angle 𝜃𝐻 \ndependence of FMR resonance field 𝐻𝑟𝑒𝑠 for all seven YIG (10 nm)/(Bi xSb1-x)2Te3 (5 QL) samples \nand YIG reference sample.  Solid curves are the best fits.  (b) Bi -concentration dependence of \nextracted effective anisotropy field 4𝜋𝑀 𝑒𝑓𝑓 obtained from fitting in (a) for all seven YIG (10 \nnm)/(Bi xSb1-x)2Te3 (5 QL) samples. Th e black dash ed line is the 4𝜋𝑀 𝑒𝑓𝑓 value for the YIG \nreference sample.  \n \n  \n Fig. 4.  Extracted Gilbert damping from FMR linewidth fitting . (a) FMR transmission spectra \n𝑆21 for YIG/Sb 2Te3 for different chosen frequencies: 4, 6, 8, 10 and 12 GHz at 300 K after \nbackground subtraction. (b) Normalized FMR spectra 𝑆21at a fixed frequency of 10 GHz with an \napplied in -plane static field for YIG/(Bi xSb1-x)2Te3 samples with different Bi concentrat ions and \nYIG reference sample. (c) Frequency dependence of FMR linewidth for all seven YIG/TI samples \nand YIG reference sample. The resonance peak height is reduced in samples with increased \ndamping constant , which causes poor Lorent zian fitting and conseq uently large apparent noise in \nextracted linewidth. (d) Bi concentration ( x)-dependence of the Gilbert damping constant α \nextracted from the slope of the straight lines in (c).  \n \n "    },    {        "title": "1806.00658v1.Ultra_low_damping_insulating_magnetic_thin_films_get_perpendicular.pdf",        "content": "1 \n Ultra -low damping  insulating magnetic  thin films get perpendicular  \n \nLucile Soumah1, Nathan Beaulieu2, Lilia Qassym3, Cécile Carrétero1, Eric Jacquet1, Richard Lebourgeois3, \nJamal Ben Youssef2, Paolo Bortolotti1, Vincent Cros1, Abdelmadjid Anane1* \n \n1 Unité Mixte de Physique CNRS , Thales, Univ.  Paris -Sud, Université Paris Saclay, 91767 Palaiseau, France  \n2LABSTICC, UMR 6285 CNRS, Université de Bretagne Occidentale, 29238 Brest, France  \n3 Thales Research and Technology, Thales 91767 Palaiseau , France  \n* Email : madjid.anane@u -psud.fr  \n \nA magnetic material combining low losses and  large Perpendicular Magnetic Anisotropy  (PMA) is still a \nmissing brick in the magnonic  and spintronic field s. We report  here  on the growth of ultrathin Bismuth \ndoped Y 3Fe5O12 (BiYIG ) films on Gd 3Ga 5O12 (GGG) and substituted GGG (sGGG) (111) oriented \nsubstrates. A fine tuning of the PMA is obtained  using both epitaxial strain and growth induced \nanisotrop ies. Both spontaneously in -plane and out -of-plane magnetized thin films can be elaborated . \nFerromagnetic Resonance  (FMR)  measurement s demonstrate the  high dynamic quality of these BiYIG \nultrathin  films , PMA films with Gilbert damping values as low as 3 10-4 and FMR linewidth  of 0.3 mT at \n8 GHz are achieved  even for films that do not exceed 30 nm in thickness . Moreover, w e measure \nInverse Spin Hall Effect (ISHE) on Pt/BiYIG stack s showing  that the magnetic insulator ’s surface  is \ntransparent to  spin current  making it appealing  for spintronic applications . \n \n  2 \n Introduction.  \nSpintronic s exploit s the electron’s spin in ferromagnetic transition metal s for data storage and data \nprocessing. Interestingly, as spintronics  codes information in the angular momentum degre es of \nfreedom , charge transport and therefore the use of conducting materials is not a requirement, opening \nthus electronics to  insulators . In magnetic insulators  (MI),  pure spin currents  are described using \nexcitation  states of the ferromagnetic background named magnons (or spin waves). Excitation, \npropagation  and  detection  of magnons are   at  the  confluent of the emerging concepts of magnonics  \n1,2, caloritronics3 and  spin -orbitronics4. Magnons, and their classical counterpart , the  spin waves (SWs) \ncan carry information over distances as large as millimeters in high quality thick YIG films, with \nfrequencies extending from the GHz to the THz regime5–7. The main figure of merit for magnonic \nmaterials is the Gilbert damping  1,5,8 which  has to be as small as possible. This makes  the number of \nrelevant materials for SW propagation quite limited and none of them has yet been found to possess a \nlarge enough perpendicular magnetic anisotropy (PMA ) to induce  spontaneous out -of-plane \nmagnetization . We report  here  on the  Pulsed  Laser  Deposition  (PLD)  growth  of  ultra -low loss   MI  \nnanometers -thick films  with large PMA : Bi substituted Yttrium Iron Garnet ( BixY3-xFe5O12 or BiYIG ) where  \ntunability of the PMA  is achieved through epitaxial strain and Bi doping level.  The peak -to-peak FMR  \nlinewidth  (that characterize the losses)  can be as low as  𝜇0𝛥𝐻pp=0.3 mT at 8 GHz for 30 nm thick \nfilms. This material thus opens new perspectives for both spintronic s and magnonic s fields as the SW \ndispersion relation can now be easily tuned through magnetic anisotropy without the need of  a large \nbias magnetic field. Moreover, energy efficient data storage devices based on magn etic textures existing \nin PMA materials like magnetic bubble s, chiral domain walls  and magnetic skyrmions  would benefit from \nsuch a low loss material for efficient operation9. \nThe study of micron -thick YIG films grown by liquid phase epitaxy (LPE) was among the hottest topics in \nmagnetism few decades ago . At this time,  it has been already noticed that unlike rare earths (Thulium, \nTerbium, Dysprosium …) substitutions, Bi substitution does not overwhelmingly increase the magnetic \nlosses10,11 even though it induces high uniaxial magnetic anisotropy12–14 . Very recently, ultra -thin MI \nfilms showing PMA have been the subject of an increasing interes t 15,16: Tm 3Fe5O12 or BaFe 12O19 \n(respectively a garnet and an hexaferrite) have been used to demonstrate spin -orbit -torque \nmagnetization reversal using a Pt over -layer as a source of spin current 4,17,18. However, their large \nmagnetic losses prohibit their use as a spin -wave medium  (reported va lue o f 𝜇0𝛥𝐻pp of TIG is  16.7  mT at \n9.5 GHz)19. Hence, whether  it is possible to fabricate ultra -low loss thin films  with a large PMA  that can \nbe used for both magnonics and spintronics applications remains to be demonstrated . Not only l ow 3 \n losses are important for long range spin wave propagation but they are also necessary for spin transfer \ntorque oscillators (STNOs) as the threshold current scales with the Gilbert damping20.  \nIn the quest for the  optimal  material platform , we explore here the growth of Bi doped YIG ultra -thin \nfilms using PLD  with different substitution;  BixY3-xIG (x= 0.7, 1 and 1.5)  and having a thickness ranging \nbetween 8 and 50 nm. We demonstrate fine tuning of the magnetic anisotropy using epitaxial strain  and  \nmeasure  ultra low Gilbert damping values ( 𝛼=3∗10−4) on ultrathin films with PMA . \nResults  \nStructural and magnetic characterization s \nThe two substrates that are used are Gallium Gadolinium Garnet  (GGG)  which is best lattice matched to \npristine YIG and substituted GGG  (sGGG) which is  traditionally used to accommodate  substituted YIG \nfilms for photonics  applications . The  difference between  Bi and Y ionic radii ( rBi = 113 pm and rY = 102 \npm)21 leads to a linear increase of  the  BixY3-xIG bulk lattice  parameter  with Bi content  (Fig. 1 -(a) and Fig. \n1-(b)). In Fig. 1,  we present the  (2−) X-ray diffraction patterns (Fig.1 -(c) and 1 -(d)) and reciprocal \nspace maps (RSM) (Fig.1 -(e) and 1 -(f)) of BiYIG on sGGG(111) and GGG(111) substrates  respectively . The \npresence of ( 222) family peaks in the diffraction spectra shown in Fig. 1 -(b) and 1 -(c) is a signature of  the \nfilms’ epitaxial quality  and the presence of Laue fringes attest s the coherent crystal structure existing \nover the whole thickness. As expected, all films on GGG are under compressive strain, whereas films \ngrown on sGGG exhibit a transition from a tensile (for x= 0.7 and 1) towards a compressive ( x= 1.5) \nstrain . Reciprocal Space Mapping  of these BiYIG samples  shown in Fig.1 -(e) and 1 -(f) evidences the \npseudomorphic nature of the growth for all films , which confirms the good epitaxy.  \nThe static magnetic properties of the films have been characterized using SQUID ma gnetometry, Faraday \nrotation measurements and Kerr microscopy. As the Bi doping has the effect of enhancing the magneto -\noptical response 22–24, we measure on average a large  Faraday rotation coefficients reaching up to 𝜃F =\n−3 °.𝑚−1 @ 632 nm  for  x= 1 Bi doping level  and 15 nm film thickness . Chern et al .25 performed PLD \ngrowth of BixY3-xIG on GGG  and reported an increase of 𝜃F\n𝑥= −1.9 °.𝜇𝑚−1 per Bi substitution  x @ 632 \nnm. The Faraday rotation coefficients we find  are slightly larger  and m ay be due to the much lower \nthickness of our films as 𝜃F is also dependent on  the film  thickness26. The saturation magnetization ( Ms) \nremains constant for all Bi content  (see Table 1)  within the  10%  experimental errors . We observe a clear  \ncorrelation between the strain and the shape of the in -plane and out -of-plane hysteresis loop s reflecting \nchanges in the magnetic anisotropy. Wh ile films under compressive strain exh ibit in -plane anisotropy, \nthose under  tensile strain show a large out -of-plane anisotrop y that can eventually lead to an out -of-\nplane easy axis  for x= 0.7 and x= 1 grown on sGGG.  The transition can be either induced by ch anging the 4 \n substrate  (Fig.2 -(a)) or the Bi content ( Fig. 2-(b)) since both act on the misfit strain. We ascribe the \nanisotropy change  in our films to  a combination of magneto -elastic anisotropy and growth induced \nanisotropy, this later term being the domin ant one (see Supplementary Note 1).  \nIn Fig. 2 -(c), we  show the magnetic domains structures at remanance  observed using polar Kerr \nmicroscopy  for Bi1Y2IG films  after demagnetization : µm-wide maze -like magnetic domains demonstrate s \nunambiguously that the magnetic easy axis is perpendicular to the film surface . We observe  a decrease \nof the domain width (Dwidth) when the film thickness ( tfilm) increases as expected from magnetostatic  \nenergy considerations. In fact, a s Dwidth is severa l order s of magnitude larger than tfilm, a domain wall \nenergy of σDW  0.7 and 0.65  mJ.m-2 (for x= 0.7 and 1 Bi doping) can inferred using the Kaplan and \nGerhing model27  (the fitting procedure is detailed  in the Supplementary Note 2).  \n \nDynamical characterization and spin transparency . \nThe most striking feature of these large PMA films is their extremely low magne tic losses that we \ncharacterize  using Ferromagnetic Resonance (FMR) measurements. First of all, we quantify  by in-plane  \nFMR the anisotropy field HKU deduced from the effective magnetization ( Meff): HKU = M S – Meff (the \nprocedure to derive Meff from in plan e FMR is presented in Supplementary Note 3 ). HKU values for BiYIG \nfilms with different doping levels grown on various substrates  are summarized  in Table 1.  As expected \nfrom out-of-plane  hysteresis curves , we observe different signs for HKU. For spontaneously out -of-plane \nmagnetized samples , HKU is positive and large enough to fully compensate the demagnetizing field while \nit is negative for in -plane magnetized films. From these results , one can expect that fine tuning of the Bi \ncontent allow s fine tuning of the effective magnetization and consequently of the FMR resonan ce \nconditions. We measure magnetic losses on a 30nm thick Bi1Y2IG//sGGG film  under tensile strain with \nPMA (Fig. 3 -(a)). We use the FMR absorption line shape  by extracting  the peak -to-peak linewidt h (𝛥𝐻pp) \nat different out-of-plane  angle  for a 30nm thick perpendicularly magnetized Bi 1Y2IG//sGGG film at 8 GHz \n(Fig.  3-(b)). This  yields an optimal value of  𝜇0𝛥𝐻pp  as low as 0.3 mT  (Fig. 3-(c)) for 27° out-of-plane  polar \nangle. We stress here that state of the art PLD grown YIG//GGG films exhibit similar values for 𝛥𝐻pp at \nsuch resonant conditions28. This angular dependence of 𝛥𝐻pp that shows pronounced variations at \nspecific angle is characteristic of a two magnons scattering relaxation process with few \ninhomogenei ties29. The value of this angle is sample dependent  as it is related to the distribution of the \nmagnetic inhomogeneities .  The dominance in our films of those two i ntrinsic relaxation processes \n(Gilbert damping and two magnons  scattering) confirms  the high films quality . We also derived the \ndamping value of th is film (Fig.  3-(d)) by selecting the lowest linewidth (corresponding to a specific out of 5 \n plane angle) at each frequency, the spread of the out of plane angle is ±3.5 ° around 30.5 °. The obtained \nGilbert damping value is α = 3.10-4 and the peak -to-peak extrinsic linew idth 𝜇0𝛥𝐻0 =0.23 mT a re \ncomparable to the one obtained for  the best PLD grown YIG//GGG nanometer thick films28 (α =2.10-4). \nFor x= 0.7  Bi doping, the smallest observed FMR linewidth is 0.5 mT  at 8 G Hz. \nThe low magnetic losses of BiYIG films could open  new perspectives for magnetization dynamics control \nusing spin-orbit torques20,30,31. For such phenomenon  interface transparency to spin curr ent is then the \ncritical parameter which is defined using the effective spin -mixing conductance ( 𝐺↑↓). We use spin \npumping experiments to estimate the increase of the Gilbert damping due to Pt deposition on Bi1Y2IG \nfilms. The spin mixing conductance can thereafter be calculated using 𝐺↑↓=4𝜋𝑀s𝑡film\n𝑔eff𝜇B(𝛥𝛼) where 𝑀s and \n𝑡film are the BiYIG magnetization saturation and thickness, 𝑔eff  is the effective Landé factor ( 𝑔eff=2), \n𝜇B is the Bohr magneton and 𝛥𝛼 is the increase in the Gilbert damping constant induced by the Pt top \nlayer. We obtain  𝐺↑↓=3.9 1018m−2  which is comparable to what is obtained on PLD grown  YIG//GGG \nsystems  28,32,33. Consequently , the doping in Bi should  not alter the spin orbit -torque efficiency  and spin \ntorque devices made out of  BiYIG will be as energy efficient as their  YIG counterpart . To further confirm \nthat spin current cross es the Pt/BiYIG interface , we measure Inverse Spin Hall Effect (ISHE) in Pt for a Pt/  \nBi1.5Y1.5IG(20nm)/ /sGGG in -plane magnetized film (to fulfill the ISHE geometry requirements  the \nmagnetization needs to be in -plane and perp endicular to the measured voltage ). We measure a \ncharacteristic voltage peak due to ISHE  that reverses its sign when the static in-plane  magnetic field is \nreversed (Fig. 4). We emphasize here that the amplitude of the s ignal is similar to that of Pt/ YIG//GGG in \nthe same experimental conditions.  \nConclusion  \nIn summary, this new material platform will be highly beneficial for magnon -spintronics and  related \nresearch fields like  caloritronics. In many aspects , ultra -thin BiYIG films offer new leverages for  fine \ntuning of the magnetic properties with no  drawbacks compared to the reference materials of th ese \nfields: YIG.  BiYIG with its higher Faraday rotation coefficient (almost two orders of magnitude more than \nthat of  YIG) will increase the sensitivity of light based detection technics  that can be used  (Brillouin light \nspectroscopy (BLS) or time resolved Kerr microscopy34). Innovative scheme s for on -chip magnon -light \ncoupler could be now  developed bridging the field of magnonics to th e one  of photonics. From a \npractical point of view , the design of future active devices will be much more flexible as it is possible to \neasily engineer the spin waves dispersion relation through magnetic anis otropy tuning without the need \nof large bias magnetic fields. For instance, working in the forward volume waves configuration comes 6 \n now cost free, whereas in sta ndard in -plane magnetized media  one has to overcome the demagnetizing  \nfield. As the development  of PMA tunnel junctions  was key in developing today  scalable MRAM \ntechnology , likewise, we believe that P MA in nanometer -thick low loss  insulator s paves the path to new \napproaches where the magnonic medium material could also be used to store information locally \ncombining therefore the memory and computational functions, a most desirable feature for the brain -\ninspired neuromorphic paradigm . \n  7 \n Methods  \nPulsed Laser Deposition (PLD) growth  \nThe PLD growth of BiYIG films is realized using stoichiometric BiYIG  target. The laser used is a frequency \ntripled Nd:YAG laser ( λ =355nm), of a 2.5Hz repetition rate and a fluency E varying from 0.95 to 1.43 \nJ.cm-2 depending upon the Bi doping in the target. The distance between target and substrate is fixed at \n44mm. Pri or to the deposition the substrate is annealed at 700°C under 0.4 mbar of O 2. For the growth, \nthe pressure is set at 0.25 mbar O 2 pressure. The optimum growth temperature varies with the Bi \ncontent from 400 to 550°C. At the end of the growth, the sample is  cooled down under 300 mbar of O 2. \n \nStructural characterization  \nAn Empyrean diffractometer with Kα 1 monochromator  is used  for measurement in Bragg -Brentano  \nreflection mode  to derive the (111) interatomic plan distance. Reciprocal Space Mapping is performed on \nthe same diffractometer and we used the diffraction along the (642) plane direction which allow to gain \ninformation on the in-plane  epitaxy relation along [20 -2] direc tion.  \n \nMagnetic characterization  \nA quantum design SQUID magnetometer was used to measure the films’ magnetic moment ( Ms) by \nperforming hysteresis curves along the easy magnetic direction at room temperature. The linear \ncontribution of the paramagnetic (sGG G or GGG) substrate is linearly subtracted.  \nKerr microscope (Evico Magnetics) is used in the polar mode to measure out-of-plane  hysteresis curves \nat room temperature.  The same microscope is also used to image the magnetic domains structure after \na demagn etization procedure. The spatial resolution of the system is 300 nm.  \nA broadband FMR setup with a motorized rotation stage was used. Frequencies from 1 to 20GHz have \nbeen explored. The FMR is measured as the derivative of microwave power absorption via a low \nfrequency modulation of the DC magnetic field. Resonance spectra were recorded with the applied static \nmagnetic field oriented in different geometries (in plane or tilted of an angle 𝜃 out of the strip line \nplane).  For o ut of plane magnetized samples the Gilbert damping parameter has been obtained by \nstudying the  angular  linewidth dependence.  The procedure assumes that close to the minimum \nlinewidth (Fig 3a) most of the linewidth angular dependence is dominated by the inhomogeneous \nbroadening, thus opt imizing the angle for each frequency within few degrees allows to estimate  better  8 \n the intrinsic contribution. To do so we varied the out of plane angle of the static field from 2 7° to 34 ° for \neach frequency  and w e select the lowest value of   𝛥𝐻pp.  \nFor Inverse spin Hall effect measurements, the same FMR setup was used, however here the modulation \nis no longer applied to the magnetic field but to the RF power at a frequency of 5kHz. A Stanford  \nResearch SR860 lock -in was used a signal demodulator.  \nData  availability : \nThe data that support the findings of this study are available within the article or from the corresponding \nauthor upon reasonable request . \nAcknowledgements:  \n We acknowledge J. Sampaio for preliminary Faraday rotation measurements and N. Rey ren and A. \nBarthélémy for fruitful discussions. This research was supported by the ANR Grant ISOLYIG (ref 15 -CE08 -\n0030 -01). LS is partially supported by G.I.E III -V Lab. France.   \n \nAuthor Contributions : \nLS performed the growth, all the measurements, the da ta analysis and wrote the manuscript  with AA . NB \nand JBY conducted the quantitative Faraday Rotation measurements and participated in the FMR data \nanalysis. LQ and fabricated the PLD targets . RL supervised the target fabrication and participated in the \ndesign of the study . EJ participated in the optimization of the film growth conditions. CC supervised the \nstructural characterization experiments. AA conceived the study and w as in charge of overall direction . \nPB and VC contributed to the design and implem entation of the research . 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Lett.  117,  1–5 (2016).  \n 11 \n Figures Captions :  \n \nFigure 1 -Structural properties  of ultra -thin BiYIG films . \n(a) and (b) : Evolution of the target  cubic lattice parameter of BixY3-xIG, the dashed line represents the \nsubstrate (sGGG  and GGG respectively) lattice parameter and allow s to infer the expected tensile or \ncompressive strain arising for each substrate/target combination.  \n(c) and ( d): 2𝜃−𝜔 X-Ray diffraction scan along the (111) out-of-plane  direction for BixY3-xIG films gr own \non sGGG (111) and GGG (111) respectively. From the film and substrate diffraction peak position , we can \nconclude about the nature of the strain. Compressive strain is observed for 1.5 doped films grown on \nsGGG substrate and for all films grown on GGG w hereas tensile strain occurs for films with x=  0.7 and x= \n1 Bi content grown on sGGG.  \n(e) and (f) : RSM along the evidence the (642) oblique plan showing pseudomorphic growth in films: both \nsubstrate and film the diffraction peak are aligned along the qx\\\\[20-2] direction. The relative position of \nthe diffraction peak of the film (up or down) along qx is related to the out-of-plane  misfit between the \nsubstrate and the film (tensile or compressive).  \nFigure 2 -Static magnetic  properties . \n(a) Out-of plane Kerr hysteresis loop performed in the polar mode for Bi0.7Y2.3IG films grown on the two \nsubstrates: GGG and sGGG  \n(b) Same measurement for BixY3-xIG grown on sGGG with  the three different Bi doping ( x= 0.7, 1 and 1.5) . \nBi0.7Y2.3IG//GGG is in -plane magnetized whereas perpendicular magnetic anisotropy (PMA) occurs for x= \n0.7 and x= 1 films grown on sGGG: square shaped loops with low saturation field ( µ0Hsat about 2.5 mT) are \nobserved. Those two films are experiencing tensile strain . Whereas the inset shows that the Bi1.5Y1.5IG \nfilm saturates at a much higher field wi th a curve characteristic of in -plane easy magnetization direction. \nNote that for Bi1.5Y1.5IG//sGGG µ0Hsat ≈290mT >µ0Ms≈162mT which points toward a negative uniaxial \nanisotropy term ( µ0HKU) of 128mT which is coherent with the values obtained from in plane FMR \nmeasurement .  \n(c) Magnetic domains structure imaged on Bi 1Y2IG//sGGG films of three different thicknesses at \nreman ant state after demagnetization . The scale bar, displayed in blue , equal s 20 µm . Periods of the \nmagnetic domains structure ( Dwidth) are derived using 2D Fast Fourier Transform . We obtained Dwidth =3.1,  \n1.6 and 0.4  µm for tBi1Y2IG= 32, 47 and 52  nm respectively. We note a decrease of Dwidth with increasing \ntBi1Y2IG  that is coherent with the Kaplan and Gehring model valide in the case  Dwidth>>tBiYIG.  12 \n Figure 3-Dynamical properties  of BiYIG films with PMA.  \n(a) Sketch of the epitaxial configuration for Bi1Y2IG films , films are grown under tensi le strain giving rise \nto tetragonal distortion of the unit cell.  \n(b) Out-of-plane  angular depend ence of the peak -to-peak FMR linewidth ( 𝛥𝐻pp) at 8 GHz on a 30 nm \nthick Bi1Y2IG//sGGG with PMA  (the continuous  line is a guide for the eye) . The geometry of the \nmeasurement is shown in top right of the graph. The wide disparity of the value for the peak to peak \nlinewidth 𝛥𝐻pp is attributed to the two magnons scattering process  and inhomogeneties in the sample . \n(c) FMR absorption linewidth o f 0.3 mT for the same film at measured at  𝜃=27°. (d) Frequency \ndependence of the FMR linewidth . The calculated  Gilbert damping parameter and the extrinsic linewidth \nare displayed  on the graph .  \nFigure 4- Inverse Spin Hall Effect  of BiYIG  films with in plane magnetic anisotropy.  \nInverse Spin Hall Effect (ISHE) voltage vs magnetic  field measured on the Pt / Bi1.5Y1.5IG//sGGG  sample in \nthe FMR resonant condition at 6 GHz proving  the interface transparency to spin current. The rf excitation \nfield is about 10-3 mT which corresponds to a linear regime of excitation. Bi1.5Y1.5IG//sGGG present s an in-\nplane  easy magnetization axis due to a growth under compressive strain.  \n  \n  13 \n Table 1 - Summary of the magnetic properties of BixY3-xIG films on GGG and sGGG \nsubstrates.  \nThe saturation magnetization is roughly unchanged. The effective magnetization Meff obtained through \nbroad -Band FMR measurements allow to deduce the out -of-plane anisotropy fields HKU (HKU =Ms-Meff) \nconfirming the dramatic changes of the out-of-plane magnetic anisotropy variations observed in the \nhysteresis curves . \n   \nBi doping  Substrate  µ0MS(mT)  µ0Meff(mT)  µ0HKU(mT)  \n0 GGG  157 200 -43 \n0.7 sGGG  180 -151 331 \n0.7 GGG  172 214 -42 \n1 sGGG  172 -29 201 \n1 GGG  160 189 -29 \n1.5 sGGG  162 278 -116 14 \n Figure 1  \n (f) \n0.5 1.0 1.51.2351.2401.2451.2501.255  Cubic Lattice parameter in nmBi content \n  sGGG\n45 50 55103106109\n Intensity (cps)\n2 angle(°)0.711.5\n0.711.5\n1.70 1.724.254.304.35qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+01 1E+05\n1.74 1.754.234.274.30qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n1.73 1.764.214.254.29qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n0.7 1 1.5\nsGGGBi0.7Y2.3IG\nsGGGBi1Y2IG\nBi1.5Y1.5IGsGGG\n45 50 55102105108\n Intensity (cps)\n2 angle (°)\n0.5 1.0 1.51.2351.2401.2451.2501.255  Cubic Lattice parameter in nmBi content \n  \nGGG0.711.5\n0.7511.5\n1.75 1.774.204.304.40qz//[444] (rlu)*10\nqx//[2-20] (rlu)*101E+01 1E+05\n1.750 1.7754.284.324.35qz//[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n0.7\nGGG\nBi0.7Y2.3IGGGG\nBi1Y2IG1sGGG\nGGG(a)\n(b)(c)\n(d)(e)\n(f)15 \n Figure 2  \n \n \n \n16 \n Figure 3  \n \n \n  \n17 \n Figure 4  \n-130 -120 -110 110 120 130-4000400800V (nV)\nµ0H (mT)2 \n Supplementary Notes 1- Derivation of the magneto -elastic anisotropy  \n \nThe out -of-plan e anisotropy constant KU is ascribed to be a result  of, at least , two contributions : a \nmagneto -elastic anisotropy  term  induced by strain (𝐾MO) and a term that is due to preferential \noccupation of Bi atoms of non equivalent dodecahedral sites of the  cubic u nit cell. This last term is \nknown as the growth induced anisotropy term 𝐾GROWTH . From X -ray characterizations and f rom the \nknown properties of the thick BiYIG LPE grown films, it is possible to calculate the expected values of  \n𝐾MO in each doping/substra te combination. We thereafter deduce 𝐾GROWTH  from the relation  \n𝐾U=𝐾MO+ 𝐾GROWTH . KMO is directly proportional to the misfit between the film and the substrate:  \n𝐾MO=3\n2∙𝐸\n1−𝜇∙𝑎film −𝑎substrate\n𝑎film∙𝜆111 \nWhere E, µ and λ111 are respectively the Young modulus, the Poisson coefficient, and the \nmagnetostrictive constant along the (111) direction.  Those constant are well established for the bulk1: E= \n2.055.1011 J.m-3, µ= 0.29 . The magnetostriction  coefficient  λ111 for the thin film case is slightly higher \nthan that of the bulk and depends  upon the Bi rate x: 𝜆111(𝑥)=−2.819 ∙10−6(1+0.75𝑥) 2. The two \nlattice par ameter entering in the equation: afilm and asubstrate  correspond to the lattice parameter of the \nrelaxed film structure and of the subs trate. Under an elastic deformation afilm can be derived with the \nPoisson coefficient:  \n𝑎film =𝑎substrate −[1−𝜇111\n1+𝜇111]𝛥𝑎⊥  with  ∆𝑎⊥=4√3𝑑444film−𝑎substrate  \nAll values for the different target/substrate combinations are displayed in the Table S-1. We note here \nthat a negative (positive) misfit corresponding to a tensile (compressive) strain will favor an out-of-plane  \n(in-plane ) magnetic anisotropy which is coherent with what is observed in our samples. To estimate the \ncontribution to the magnetic  energy of the magneto elastic anisotropy term we compare it to the \ndemagnetizing field 𝜇0𝑀s that favors in-plane  magnetic anisotropy in thin films. Interestingly the \nmagneto elastic field ( 𝜇0𝐻MO) arising from 𝐾MO (𝜇0𝐻MO=2𝐾MO\n𝑀s) never exceed 30% o f the \ndemagnetizing fields and therefore cannot alon e be responsible of the observed PMA.  \n \nStudies on µm-thick BiYIG films  grown  by LPE showed that PMA in BiYIG arises due to the growth \ninduced anisotropy term 𝐾GROWTH  , this term is positive 3 for the case of Bi substitution . We have \ninferred 𝜇0𝐻GROWTH  values for all films using 𝐾U constants  measured by FMR. The results are \nsummarized in Supplementary Figure 1. One can clearly see that  𝐾GROWTH  is strongly substrate \ndependent and therefore does not depend sol ely on the Bi content. We conclude that strain play s a role \nin Bi3+ ion ordering within the unit cell . \n \n 3 \n  \n \nSupplementary Figure 1- Summary of the inferred values of the effective magnetic \nanisotropy out -of-plan fields.  \n \nHorizontal dash lines represent the magnitude of the demagnetization field µ0Ms. When µ0HKU is larger \nthan µ0Ms (dot line) films have a PMA, they are in -plane magnetized otherwise .   \n0200400µ0HKuvs Bi content \n HKu = Hstrain+ Hgrowth\nµ0Ms \n1.50.7µ0HKu (mT) µ0 Hstrain\n µ0 Hgrowth 1\nµ0Ms\n-1000100200\n0.7 µ0HKu (mT) µ0Hstrain\n µ0Hgrowth\n14 \n Supplementary Table 1 - Summary of the films’ calculated magneto -elastic \nanisotropy constant ( KMO) and the corresponding anisotropy field HMO \n \n \n  Bi content  substrate  afilm(Å) Δ a┴/afilm KMO (J.m-3) µ0 HMO(mT)  µ0 Hdemag (mT)  \n0.7 sGGG  12.45  0.6 5818  81 179 \n0.7 GGG  12.41  -0.4 -4 223  -61 172 \n1.0 sGGG  12.47  0.3 3 958  57 172 \n1.0 GGG  12.42  -0.6 -6 500  -102 160 \n1.5 sGGG  12.53  -0.6 -8 041  -124 157 5 \n Supplementary Notes 2- Derivation of the domain wall energy  \n \nTo derive the characteristic domain wall energy σDW for the maze shape like magnetic domains , we use \nthe Kaplan et al. model4. This model applies in our case as the ratio of the film thickness ( tfilm) to the  \nmagnetic domain width ( Dwidth) is small (𝑡film\n𝐷width~ 0.01). The domain wall width and the film thickness are \nthen expected to be linked by:  \n𝐷width =𝑡filme−π1.33𝑒𝜋𝐷0\n2𝑡film  where 𝐷0=2𝜎w\n𝜇0𝑀s2 is the dipolar length.  \nHence we expect a linear dependence of ln(𝐷width\n𝑡film) vs  1\n𝑡film : \nln(𝐷width\n𝑡film)=π𝐷0\n2∙ 1\n𝑡film+Cst.  (1) \nThe magnetic domain width of Bi xY3-xIG//sGGG ( x = 0.7 and 1) for several thicknesses are extracted from \n2D Fourier Transform of the Kerr microscopy images at remanence. In Supplementary Fig ure 2, we plot \nln(𝐷width\n𝑡film) vs  1\n𝑡film which follow s the expected linear dependence  of Equation (1) . We infer from the zero \nintercept an estimat ion of the dipolar length D0 of BiYIG films doped at 0.7 and 1 in Bi : D0 x=0.7= 16.5  µm \nand D0 x=1= 18.9µm. The corresponding domain wall energy are respectively 0.7 mJ .m-2 and 0.65 mJ .m-2. \nEven if the small  difference in domain wall energy between the two Bi content  may not be significant \nregarding the statistical fitting errors, it correlates to the decrease of the out-of-plane  anisotropy ( KU) \nwith increasing the Bi content.  6 \n Supplementary Figure 2- Evolution of the domain width vs film thickness  \n \nln(𝐷width\n𝑡film) vs  1\n𝑡film for Bi xY3-xIG//sGGG films doped at 0.7 (a) and 1 (b) in Bi. Dots correspond to the \nexperimental values. The dashed line is the linear fit that allows to extract the D0 parameter.  \n  \n25 50 752468\n  ln(Dwidth/tfilm)\ntfilm(µm-1)50 1002468\n  ln(Dwidth/tfilm)\n/tfilm(µm-1)Bi0.7Y2.3IG//sGGG\nD0=16.5µm\nBi1Y2IG//sGGG\nD0=18.9µm\n(a) \n(b) 7 \n  \nSupplementary Notes 3 - Damping and effective magnetic field derivation  \n \nFrom In Plane frequency dependent of FMR we can derive the effective magnetization  (𝑀eff) using the \nKittel law:  \n𝑓res=𝜇0𝛾√𝐻res(𝐻res+𝑀-eff) \nWhere  γ is the gyromagnetic ratio of the BiYIG  (assumed to be same as the one of the YIG) : γ=28 GHz.T-1. \n𝐻res  and 𝑓res are respectively the FMR resonant field and frequency . The uniaxial magnetic anisotropy \ncan thereafter  be derived using the saturation magnetizatio n form squid magnetometry using :   \nMeff=Ms-HKU. The Gilbert damping ( α) and the inhomogeneous linewidth ( ΔH 0) which are the two \nparamaters defining the magnetic relaxation are obtained from the evolution of the peak to peak \nlinewidth ( ΔHpp) vs the resonant frequency ( fres): \n𝛥𝐻pp=𝛥𝐻0+2\n√3𝛼𝑓res\n𝜇0𝛾  (2) \nThe first term is frequency independent and often attributed magnetic inhomogeneity’s (anisotropy, \nmagnetization).  \n  8 \n Supplementary Figure 3 - µ0ΔHpp vs fres on 18 nm thick Bi15.Y1.5IG//sGGG  \n \nThe l inewidth  frequency dependence from 5 to 19 GHz for in plane magnetized Bi 1.5Y1.5IG//sGGG  sample \nallow to ex tract the damping and the inhomogeneous linewidth parameter  using the Equation (2) . \n \n  \n0 5 10 150.00.51.01.5 \n Hpp(mT)\nfres (GHz)H0=0.5 mT\n=1.9*10-39 \n Supplementary References:  \n \n1. Hansen, P., Klages, C. ‐P. & Witter, K. Magnetic and magneto ‐optic properties of praseodymium ‐ \nand bismuth ‐substituted yttrium iron garnet films. J. Appl. Phys.  60, 721–727 (1986).  \n2. Ben Youssef, J., , Legall, H. & . U. P. et M. C. Characterisation and physical study of bismuth \nsubstituted thin garnet films grown by liquid phase epitaxy (LPE). (1989).  \n3. Fratello, V. J., Slusky, S. E. G., Brandle, C. D. & Norelli, M. P. Growth -induced anisotropy in \nbismuth: Rare -earth iron garnets. J. Appl. Phys.  60, 2488 –2497 (1986).  \n4. Kaplan, B. & Gehring, G. A. The domain structure in ultrathin magnetic films. J. Magn. Magn. \nMater.  128,  111–116 (1993).  \n "    },    {        "title": "1806.03172v1.Brownian_motion_of_magnetic_domain_walls_and_skyrmions__and_their_diffusion_constants.pdf",        "content": "Brownian motion of magnetic domain walls and skyrmions,\nand their di\u000busion constants\nJacques Miltat,\u0003Stanislas Rohart, and Andr\u0013 e Thiaville\nLaboratoire de Physique des Solides, Universit\u0013 e Paris-Sud,\nUniversit\u0013 e Paris-Saclay, CNRS, UMR 8502, F-91405 Orsay Cedex, France\n(Dated: October 8, 2018)\nExtended numerical simulations enable to ascertain the di\u000busive behavior at \fnite temperatures\nof chiral walls and skyrmions in ultra-thin model Co layers exhibiting symmetric - Heisenberg - as\nwell as antisymmetric - Dzyaloshinskii-Moriya - exchange interactions. The Brownian motion of\nwalls and skyrmions is shown to obey markedly di\u000berent di\u000busion laws as a function of the damping\nparameter. Topology related skyrmion di\u000busion suppression with vanishing damping parameter,\nalbeit already documented, is shown to be restricted to ultra-small skyrmion sizes or, equivalently,\nto ultra-low damping coe\u000ecients, possibly hampering observation.\nI. INTRODUCTION\nThe prospect of ultra-small stable information bits in\nmagnetic layers in presence of the Dzyaloshinskii-Moriya\n(DM) interaction [1] combined to the expectation of their\nminute current propagation [2], notably under spin-orbit\ntorques [3], builds up a new paradigm in information\ntechnology. In stacks associating a metal with strong\nspin-orbit interactions e.g. Pt and a ferromagnetic metal\nsuch as Co, that may host isolated skyrmions, large do-\nmain wall velocities have also been forecast [4] and ob-\nserved [5]. The DM interaction induces chiral magnetiza-\ntion textures, walls or skyrmions, that prove little prone\nto transformations of their internal structure, hence their\nextended stability and mobility.\nIn order, however, to achieve low propagation cur-\nrents, steps will need to be taken towards a reduction\nof wall- or skyrmion-pinning. Recent experimental stud-\nies indicate that skyrmions fail to propagate for cur-\nrents below a threshold roughly equal to 2 1011Am\u00002for\n[Pt/Co/Ta]nand [Pt/CoFeB/MgO]nmultilayers [6], or\n2:5 1011Am\u00002for [Pt/(Ni/Co/Ni)/Au/(Ni/Co/Ni)/Pt]\nsymmetrical bilayers [7]. Only in one seldom instance did\nthe threshold current fall down to about 2 :5 1010Am\u00002\nfor a [Ta/CoFeB/TaO] stack, still probably, however, one\norder of magnitude higher than currents referred to in\nsimulation work applying to perfect samples [8].\nIn a wall within a Co stripe 50 nm wide, 3 nm thick, the\nnumber of spins remains large, typically 216for a 5 nm\nwide wall. A skyrmion within a Co monolayer (ML) over\nPt or Ir, on the other hand, contains a mere 250 spins,\nsay 28. Assuming that a sizeable reduction of pinning\nmight somehow be achieved, then a tiny structure such\nas a skyrmion is anticipated to become sensitive, if not\nextremely sensitive, to thermal \ructuations.\nIn this work, we show, on the basis of extended nu-\nmerical simulations, that both chiral walls and skyrmions\nwithin ferromagnets obey a di\u000busion law in their Brow-\nnian motion at \fnite temperature [9, 10]. The di\u000busion\n\u0003jacques.miltat@u-psud.fr\nx z q \nwS tS L \na) b) q FIG. 1. a) Wall within a narrow stripe: wSis the stripe width,\ntSits thickness. The stripe element length Lis solely de\fned\nfor computational purposes. qis the wall displacement; b)\nsnapshot of the magnetization distribution: color coding after\nmx. The wall region mx\u00191 appears red. Thermal \ructua-\ntions are visible within domains: T= 25 K,wS= 100 nm,\ntS= 0:6 nm,\u000b= 0:5.\nlaw is shown to be valid over a broad range of damp-\ning parameter values. The thermal di\u000busion of domain\nwalls seems to have attracted very little attention, ex-\ncept for walls in 1D, double potential, structurally un-\nstable, lattices [11], a source of direct inspiration for\nthe title of this contribution. Chiral magnetic domain\nwalls are found below to behave classically with a mobil-\nity inversely proportional to the damping parameter. As\nshown earlier [12, 13], such is not the case for skyrmions,\na behavior shared by magnetic vortices [14]. Vortices and\nskyrmions in ferromagnetic materials are both charac-\nterized by a de\fnite topological signature. In contradis-\ntinction, skyrmions in antiferromagnetic compounds are\ncharacterized by opposite sign spin textures on each sub-\nlattice, with, as a result, a classical, wall-like, dependence\nof their di\u000busion constant [15]. Lastly, ferrimagnets do\ndisplay reduced skyrmion Hall angles [16], most likely\nconducive to modi\fed di\u000busion properties.arXiv:1806.03172v1  [cond-mat.mes-hall]  8 Jun 20182\nII. DOMAIN WALL DIFFUSION\nWe examine here, within the micromagnetic frame-\nwork, the Langevin dynamics of an isolated domain wall\nwithin a ferromagnetic stripe with thickness tS, width\nwSand \fnite length L(see Fig. 1). The wall is located\nat mid-position along the stripe at time t= 0. Thermal\nnoise is introduced via a stochastic \feld ~HRduncorrelated\nin space, time and component-wise, with zero mean and\nvariance\u0011proportional to the Gilbert damping parame-\nter\u000band temperature T[17] :\nh~HRdi=~0\nhHi\nRd(~ r;t)Hj\nRd(~r0;t0)i=\u0011\u000eij\u000e(~ r\u0000~r0)\u000e(t\u0000t0)\n\u0011=2kBT\n\r0\u00160MS\u000b(1)\nwhere,kBis Boltzmann constant, \u00160and\r0are the vac-\nuum permability and gyromagnetic ratio, respectively,\nMSthe saturation magnetization. Written as such, the\nfunctions\u000e(~ r\u0000~r0) and\u000e(t\u0000t0) have the dimension of\nreciprocal volume and time, respectively. Applied to nu-\nmerical simulations, the variance of the stochastic \feld\nbecomes\u0011=2kBT\n\r0\u00160MSVdt\u000b, whereVis the computation\ncell volume and dtthe integration time step.\nA. Simulation results\nThe full set of numerical simulations has been per-\nformed by means of an in-house code ported to graph-\nical processing units (GPU's). Double precision has\nbeen used throughout and the GPU-speci\fc version of\nthe \"Mersenne twister\" [18] served as a source of long-\nsequence pseudo-random numbers generator.\nMaterial parameters have been chosen such as to mimic\na 3-ML Co layer (thickness tS= 0:6 nm) on top of Pt\nwith an exchange constant equal to A= 10\u000011J/m, a\nMs= 1:09 106A/m saturation magnetization, a Ku=\n1:25 106J/m3uniaxial anisotropy constant allowing for\na perpendicular easy magnetization axis within domains,\nand a moderate-to-high DM interaction (DMI) constant\nDDM= 2 mJ/m2. In order to temper the neglect of short\nwavelength excitations [19], the cell size has been kept\ndown toLx=Ly= 1 nm, whilst Lz=tS= 0:6 nm.\nThe stripe length has been kept \fxed at L= 1\u0016m,\na value compatible with wall excursions within the ex-\nplored temperature range. The latter has, for reasons\nto be made clear later, been restricted to \u00191=3 of the\npresumed Curie temperature for this model Co layer. Fi-\nnally, the integration time constant, also the \ructuating\n\feld refresh time constant, has been set to dt= 25 fs.\nAs shown by the snapshot displayed in Fig. 1b, the wall\nmay acquire some (moderate) curvature and/or slanting\nduring its Brownian motion. Because wall di\u000busion is\ntreated here as a 1D problem, the wall position qis de-\n0510152025\n487488489Time [ns]Wall Position [nm]ΔtqFIG. 2. Excerpt of a wall trace displaying wall position \ructu-\nations vstime:T= 77 K,\u000b= 0:5,wS= 100 nm,tS= 0:6 nm.\nqis the wall displacement during time interval \u0001 t.\n\fned as the average position owing to :\nq=L\nNxNyPNx\ni=1PNy\nj=1mz(i;j)\n[hmziL\u0000hmziR](2)\nwhere,iandjare the computation cell indices, Nxand\nNythe number of cells along the length and the width\nof the stripe, respectively, hmziLis the \ructuations aver-\naged value of the zmagnetization component far left of\nthe domain wall,hmziRthe average value of mzfar right.\nRegardless of sign, hmziRandhmziLare expected to be\nequal in the absence of any Hz\feld.\nFig. 2 displays the position as a function of time of a\nwall within a wS= 100 nm wide stripe immersed in a\nT= 77 K temperature bath. A 2 ns physical time win-\ndow has been extracted from a simulation set to run for\n1:5\u0016s. The \fgure shows short term wall position \ruc-\ntuations superimposed onto longer time di\u000busion. Ac-\ncording to Einstein's theory of Brownian motion [9], the\nprobability P(x;t) of \fnding a particle at position xat\ntimetobeys the classical di\u000busion equation @tP(x;t) =\nD@2\nx2P(x;t) with, as a solution, a normal (gaussian) dis-\ntributionP(x;t) = 1=p\n4\u0019Dtexp(\u0000x2=4Dt), whereDis\nthe di\u000busion constant.\nSo does the raw probability of \fnding a (sti\u000b) wall in\na narrow stripe at position qafter a time interval \u0001 t, as\nshown in Fig. 3 (see Fig. 2 for variable de\fnition). It\nought to be mentioned that the average wall displace-\nmenthq(\u0001t)iis always equal to 0, with an excellent ac-\ncuracy, provided the overall computation time is large\nenough. The \ft to a normal distribution proves rather\nsatisfactory, with, however, as seen in Fig. 3, a slightly\nincreasing skewness in the distributions as a function of\nincreasing \u0001 t. Skewness, however, 1) remains moderate3\n05 1031 1041.5 1042 104\n-30-20-100102030Δt = 0.2 nsN\nq - <q>  [nm]05 1031 1041.5 1042 104\n-30-20-100102030Δt = 0.5 ns\nq - <q>  [nm]N\n05 1031 1041.5 1042 104\n-30-20-100102030Δt = 1.0 ns\nq - <q>  [nm]N\n05 1031 1041.5 1042 104\n-30-20-100102030q - <q>  [nm]Δt = 2.0 nsN\nFIG. 3. Wall within stripe: event statistics with time interval\n\u0001tas a parameter; \u000b= 0:5,wS= 100 nm, tS= 0:6 nm,\nT= 25K. The continuous blue lines are \fts to a gaussian\ndistribution, the variance of which increases with \u0001 t.\nup to \u0001tvalues typically equal to 5 \u000010 ns, 2) is seen to\nreverse sign with time interval (compare Fig. 3b and c),\nexcluding intrinsic biasing. The distributions standard\ndeviation is clearly seen to increase with increasing \u0001 t.\nAlternatively, one may represent the variance hq2i\n(hqi= 0) as a function of the time interval \u0001 t: if di\u000bu-\nsion applies, then a linear dependence is expected, with\na 2Dslope for a one-dimensional di\u000busion. Fig.4a shows,\nfor various temperatures, that a linear law is indeed ob-\nserved. Lastly, as shown in Fig.4b, the di\u000busion constant\nincreases linearly with increasing temperature. The er-\nror bars measuring the departure from strict linearity in\nFig.4a remain limited in extent. For the stripe width\nand damping parameter considered here ( wS= 100 nm,\n\u000b= 0:5), the ratio of di\u000busion constant to temperature\nis found to amount to D=T= 0:187 nm2ns\u00001K\u00001.\nB. Wall di\u000busion constant (analytical)\nThiele's equation [20] states that a magnetic texture\nmoves at constant velocity ~ vprovided the equilibrium of\n3 forces be satis\fed:\n~G\u0002~ v+\u000bD~ v=~F (3)\nwhere,~Fis the applied force, ~FG=~G\u0002~ vis the gyrotropic\nforce,~Gthe gyrovector, ~FD=\u000bD~ vthe dissipation force,\nDthe dissipation dyadic.\nFor the DMI hardened N\u0013 eel wall considered here : ~G=\n050100150200250300\n012345Δt [ns]< q2 > [nm2]\n25° K50° K77° K120° K150° K\na)0102030\n050100150T [K]D  [nm2 ns-1]\nb)FIG. 4. a) Variance hq2i(nm2) of the wall displacement vs\ntime interval \u0001 twith temperature Tas a parameter. Thick\nlines represent a linear \ft to data; b) Di\u000busion constant D\nas a function of temperature (square full symbols). Dis pro-\nportional to the slope of the hq2ivs\u0001tcurves in Fig.4a (see\ntext for details). The error bars are deduced from the slopes\nof straight lines through the origin that encompass all data\npoints in Fig.4a for a given temperature and the \ft time\nbracket, 1\u00005 ns. For the sake of legibility, the error bars\nhave been moved-up by 2 :5 units. Continuous line: linear\n\ft through the origin. The dashed line is the analytical ex-\npectation in the \"low\" noise limit. \u000b= 0:5,wS= 100 nm,\ntS= 0:6 nm.\n~0. For a 1D wall, the Thiele equation simply reads :\n\u000bDxxvx=Fx (4)\nwhere,Dxx=\u00160MS\n\r0R\nV(@~ m\n@x)2d3r.\nThe calculation proceeds in two steps, \frst evaluate\nthe force, hence, according to Eqn.4, the velocity auto-\ncorrelation functions, then integrate vstime in order to\nderivehq2i. The force, per de\fnition, is equal to minus\nthe partial derivative of the energy Ew.r.t. the displace-\nmentq, namelyFx=\u0000@E\n@q=\u0000\u00160MSR\nV@~ m\n@x\u0001~H d3r.\nFormally,\nhFx(t)Fx(t0)i= (\u00160MS)2\u0002 (5)*Z\nV@~ m(~ r;t)\n@x\u0001~H(~ r;t)d3rZ\nV@~ m(~r0;t0)\n@x\u0001~H(~r0;t0)d3r0+\nAs noticed earlier [14], since the random \feld noise is\n\"multiplicative\" [17], moving the magnetization vector\nout of the average brackets is, strictly speaking, not al-\nlowed, unless considering the magnetization vector to\nonly marginally di\u000ber from its orientation and modulus\nin the absence of \ructuations (the so-called \"low\" noise\nlimit [14]):\nhFx(t)Fx(t0)i= (\u00160MS)2\u0002 (6)\nZ\nVX\ni;j\"\n@mi(~ r;t)\n@x@mj(~r0;t0)\n@xD\nHi(~ r;t)Hj(~r0;t0)E#\nd3r d3r0\nIf due account is being taken of the fully uncorrelated4\ncharacter of the thermal \feld (Eqn.1), the force auto-\ncorrelation function becomes:\nhFx(t)Fx(t0)i= 2\u000bkBTDxx\u000e(t\u0000t0) (7)\nThe velocity auto-correlation function follows from\nEqn.4. Lastly, time integration ( q(t) =Rt\n0vx(t0)dt0)\nyields :\nhq2(t)i= 2Dt;D=kBT\n\u000bDxx(8)\nIn order to relate the di\u000busion constant to a more directly\nrecognizable wall mobility, Dxxmay be expanded as :\nDxx=\u00160MS\n\r02wStS\n\u0001T(9)\nwhere, \u0001Thas been called the Thiele wall width (implic-\nitly de\fned in [21]). Dmay thus be expressed as :\nD=kBT\n2\u00160MS1\nwStS\r0\u0001T\n\u000b(10)\nthus, proportional to the wall mobility \r0\u0001T=\u000b.\nA directly comparable result may be obtained after\nconstructing a full Langevin equation from the ( q;\u001e)\nequations of domain wall motion (Slonczewski's equa-\ntions [22]), where \u001eis the azimuthal magnetization angle\nin the wall mid-plane. In this context, the wall mobility\nis\u0016W=\r0\u0001=\u000b, where \u0001 is the usual wall width, inci-\ndentally equal to the Thiele wall width in the case of a\npure Bloch wall. The Langevin equation [10] here reads:\nmD\n2wStSd2\nq2\u000b\ndt2+1\n22\u00160Ms\n\u0016WwStSd\nq2\u000b\ndt=kBT(11)\nwhere,mDis D oring's wall mass density (kg =m2):\nmD=\u0000\n1 +\u000b2\u0001\u0012\r0\n2\u00160Ms\u0013\u000021\n\u0019jDDMj(12)\nan expression valid in the limit jDDMj\u001dKE\u000b=\nKu\u00001\n2\u00160M2\ns. Note that the DMI constant DDMex-\nplicitly enters the expression of the wall mass, as a con-\nsequence of the wall structure sti\u000bening by DMI. In the\nstationary regime, hq2iis proportional to time tand the\nwall di\u000busion constant exactly matches Eqn.10, after sub-\nstitution of \u0001 Tby \u0001. Finally, the characteristic time for\nthe establishment of stationary motion is:\nt0=mD1\n2\u00160Ms\r0\u0001\n\u000b(13)\nFor the parameters of our model 3-ML Co layer on top\nof Pt, D oring's mass density is equal to \u00183 10\u00008kg=m2\nfor\u000b= 0:5, and the characteristic time amounts to\nt0'25 ps. Still for \u000b= 0:5,wS= 100 nm and\ntS= 0:6 nm,D=Tamounts to 0 :153 nm2ns\u00001K\u00001for\n\u0001T= 4:13 nm, i.e. the value computed from a properly\nconverged wall pro\fle at T= 0. The relative di\u000berence\n0255075\n050100150T [K]a)wS = 25 nmwS = 50 nmwS = 100 nmD  [nm2 ns-1]\n00.250.50.751\n00.010.020.030.040.051/wS [nm-1]D /T  [nm2 ns-1 K-1]\nb)FIG. 5. a) Di\u000busion constant Das a function of temperature\nwith the stripe width wSas a parameter (full symbols); b)\nD=Tas a function of the inverse of the stripe width. \u000b= 0:5,\ntS= 0:6 nm, throughout. Solid blue lines: linear \ft through\nthe origin, dashed line: analytical expectation.\nbetween simulation and theoretical values is found to be\nof the order of\u001920%.\nOwing to Eqn.10, Dis expected to prove inversely pro-\nportional to both the stripe width wSand the Gilbert\ndamping parameter \u000b, a behavior con\frmed by simula-\ntions. Fig.5a displays the computed values of the dif-\nfusion coe\u000ecient as a function of temperature with the\nstripe width as a parameter, whilst Fig.5b states the lin-\near behavior ofDvswS\u00001. The slope proves, however,\nsome 13:5% higher than anticipated from Eqn.10. Lastly,\nthe 1=\u000bdependence is veri\fed in Fig.6 showing the com-\nputed variation of Dvstemperature with \u000bas a param-\neter for a narrow stripe ( wS= 25 nm) as well as the\ncorresponding \u000bdependence ofD=T. The dotted line\nrepresents Eqn.10 without any adjusting parameter. The\nrelative di\u000berence between simulation data and theoret-\nical expectation is beyond, say \u000b= 0:25, seen to grow\nwith increasing \u000bbut also appears to be smaller for a\nnarrow stripe as compared to wider tracks.\nAltogether, simulation results only moderately depart\nfrom theoretical predictions. The Brownian motion of\na DMI-sti\u000bened wall in a track clearly proves di\u000busive.\nThe di\u000busion constant is classically proportional to the\nwall mobility and inversely proportional to the damping\nparameter. Unsurprisingly, the smaller the track width,\nthe larger the di\u000busion constant. In order to provide an\norder of magnitude, the di\u000busion induced displacement\nexpectation,p\n2D\u0001t, for a wall sitting in a 100 nm-wide,\npinning-free, track for 25 ns at T= 300 K proves essen-\ntially equal to\u0006the stripe width.\nIII. SKYRMION DIFFUSION\nOutstanding observations, by means of Spin Polarized\nScanning Tunneling Microscopy, have revealed the exis-\ntence of isolated, nanometer size, skyrmions in ultra-thin5\n0255075100125150175200\n020406080100α = 0.125α = 0.25α = 0.5α = 0.075α = 0.05\nα = 0.8a)T [K]D  [nm2 ns-1]\n0246810\n-20-1001020\n00.20.40.60.81D /T  [nm2 ns-1 K-1](%)\nαb)wS = 25 nmwS = 50 nmwS = 100 nm\nFIG. 6. a) Di\u000busion constant Das a function of temperature\nwith the damping constant \u000bas a parameter ( wS= 25 nm,\ntS= 0:6 nm). Solid blue lines: linear \ft through the ori-\ngin; b)D=T(large semi-open symbols) as a function of \u000bfor\nwS= 25 nm and tS= 0:6 nm; dotted blue curve: analyt-\nical expectation. Full symbols: relative di\u000berence between\ncomputational and analytical results (%).\nFIG. 7. a) Snapshot of a skyrmion immersed in a 12.5 K\ntemperature bath ( \u000b= 0:5), together with the underlying\nlattice. Red cells: sz\u0019+1, blue cells: sz\u0019\u00001. The white\ncross indicates the barycenter of lattice site positions satisfy-\ningsz\u00150:5.\n\flms such as a PdFe bilayer on an Ir(1111) single crystal\nsubstrate [23] [24]. We analyse below the thermal motion\nof skyrmions in a model system made of a Co ML on top\nof Pt(111). We deal with skyrmions with a diameter of\nabout 2:5 nm containing at T= 0 about 250 spins.\nA. Simulation results\nIn order to monitor the Brownian motion of an iso-\nlated skyrmion, rather than micromagnetics, it is pre-\nferred to simulate the thermal agitation of classical\nspins,~ s(jsj= 1), on a triangular lattice. Lat-\ntice e\u000bects and frequency cuto\u000bs in thermal excitations\nare thus avoided. Such simulations have already been\nused e.g. for the determination of the barrier to col-\nlapse of an isolated skyrmion [25, 26]. The parame-\nFIG. 8. Example of skyrmion trajectory. Distances in atomic\nunits (1 at:u:= 2:51\u0017A). The trajectory started at the origin\nof coordinates at time t= 0 and stopped at the cross location\nat physical time t\u0019100 ns.T= 25 K,\u000b= 1.\nters are: lattice constant a= 2:51\u0017A, magnetic mo-\nment\u0016At= 2:1\u0016B/atom, Heisenberg exchange nearest\nneighbor constant J= 29 meV/bond, Dzyaloshinskii-\nMoriya exchange d=\u00001:5 meV/bond, magnetocrys-\ntalline anisotropy 0 :4 meV/atom. The stochastic \feld\nis still de\fned by Eqn.1 after substitution of the prod-\nuctMSVby the magnetic moment per atom. The code\nfeatures full magnetostatic (dipole-dipole) interactions.\nFast Fourier Transforms implementation ensues from the\ndecomposition of the triangular lattice into two rectangu-\nlar sublattices, at the expense of a multiplication of the\nnumber of dipole-dipole interaction coe\u000ecients. Lastly,\nthe base time step, also the stochastic \feld refresh time,\nhas been given a low value in view of the small atomic\nvolume, namely dt= 2:5 fs for\u000b\u00150:1,dt= 1 fs below.\nTime steps that small may be deemed little compatible\nwith the white thermal noise hypothesis [17]. They are in\nfact dictated by the requirement for numerical stability,\nprimarily w.r.t. exchange interactions.\nFig.7 is a snapshot of an isolated skyrmion in the\nmodel Co ML with a temperature raised to 12 :5 K. The\nskyrmion is at the center of a 200 at. u.- i.e. \u001950 nm-size\nsquare computation window, that contains 46400 spins\nand is allowed to move with the di\u000busing skyrmion. Do-\ning so alleviates the computation load without restricting\nthe path followed by the skyrmion. Free boundary con-\nditions (BC's) apply. The window, however, proves su\u000e-\nciently large to render the con\fning potential created by\nBC's ine\u000bective. The skyrmion position as a function of\ntime is de\fned simply as the (iso)barycenter of the con-\ntiguous lattice site positions x(k),y(k), wheresz\u00150:5:\nqSk\nx=1\nKKX\nk=1x(k) ;qSk\ny=1\nKKX\nk=1y(k) (14)6\n01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050NΔt = 0.2 ns\nqx,y - <qx,y>  [at. u.]01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050Δt = 0.5 nsN\nqx,y - <qx,y>  [at. u.]\n01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050NΔt = 1.0 ns\nqx,y - <qx,y>  [at. u.]01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050N\nqx,y - <qx,y>  [at. u.]Δt = 2.0 ns\nFIG. 9. Skyrmion: event statistics with time interval \u0001 t\nas a parameter for the displacement components qx(black\nfull symbols) and qy(red open symbols), labeled qx;yin the\n\fgures. In each panel, the curves have been o\u000bset vertically\nfor legibility. Solid lines: \ft to a gaussian distribution. \u000b=\n0:25,T= 25 K\nwhere,kis the lattice site index, Kthe number of lattice\nsites satisfying the above condition. Such a de\fnition\nproves robust vsthermal disorder such as displayed in\nFig. 7. Similarly to the case of wall di\u000busion, we analyze\n\frst the distributions of the displacement components\nqx;qy. The event statistics for each value of the time\ninterval is clearly gaussian (see Fig.9). However, the noise\nin the distributions appears larger when compared to the\nwall case. It also increases faster with \u0001 t. On the other\nhand, the raw probabilities for hq2\nxiandhq2\nyibarely di\u000ber\nas anticipated from a random process. The behavior of\nhq2i(q2=q2\nx+q2\ny)vs\u0001tis displayed in Fig.10a.\nThe range of accessible temperatures is governed by\nthe thermal stability of the tiny skyrmion within a Co\nML: with a lifetime of '1\u0016s at 77 K [25{28], tem-\nperatures have been con\fned to a \u001450 K range. When\ncompared to the wall case (Fig.4a), the linear dependence\nofhq2iwith respect to \u0001 tappears less satisfactory, al-\nthough, over all cases examined, the curves do not display\na single curvature, but rather meander gently around a\nstraight line. The slope is de\fned as the slope of the\nlinear regression either for time intervals between 0 :25\nand 2:5 ns (thick line segments in Fig.10a) or for the full\nrange 0 to 5 ns (dashed lines). Then, the ratio of the\ndi\u000busion constant to temperature, D=T, for an isolated\nskyrmion within the model Co ML considered here is\nequal to 0:250 and 0:249 nm2ns\u00001K\u00001, respectively, for\n\u000b= 0:5 (see Fig.10b). The di\u000berence proves marginal.\nLastly, error bars appear even narrower than in the wall\n01000200030004000\n01234550° K25° K12.5° K4.2° K< q2 > [at.u.2]\nΔt [ns]a)01020\n0255075T [K]b)D  [nm2 ns-1]FIG. 10. a) Variance (at :u:2) of the skyrmion displacement\nhq2ivstime interval \u0001 twith temperature Tas a parameter.\nThick and dashed lines represent a linear \ft to data with\ndi\u000berent time coverage, namely [0 :25\u00002:5 ns] and [0\u00005 ns];\nb) Di\u000busion constant Das a function of temperature for a\n[0:25\u00002:5 ns]- (open symbols) and [0 \u00005 ns]- (full symbols)\nlinear \ft. Solid blue line: linear \ft through the origin. Dashed\nline: analytical expectation in the \"low\" noise limit. In order\nto ensure legibility, the error bars as de\fned in the caption\nof Fig.4 and pertaining to the [0 :25\u00002:5 ns] \ft time bracket\nhave been moved-up by one unit. \u000b= 0:5.\ncase.\nB. Skyrmion di\u000busion constant (analytical)\nThe gyrovector ~Gin Thiele's equation (Eqn.3) has in\nthe case of a skyrmion or a vortex, and in many other\ninstances such as lines within walls, a single non-zero\ncomponent, here Gz. Thiele's equation, in components\nform, reads:\n\u0000Gzvy+\u000b[Dxxvx+Dxyvy] =Fx\n+Gzvx+\u000b[Dyxvx+Dyyvy] =Fy(15)\nBecause of the revolution symmetry of a skyrmion at rest,\nDxyorDyxmay safely be neglected and Dyy=Dxx.\nAccordingly, the velocities may be expressed as:\nvx=\u000bDFx+GFy\nG2+ (\u000bD)2;vy=\u000bDFy\u0000GFx\nG2+ (\u000bD)2(16)\nwhere,G=Gz,D=Dxx=Dyy.\nSimilarly to the stochastic \feld, the force components\nare necessarily uncorrelated. The velocity autocorrela-\ntion functions may now be obtained following the same\nlines as in the wall case, yielding, in the low noise ap-\nproximation:\nhvx(t)vx(t0)i=hvy(t)vy(t0)i= 2kBT\u000bD\nG2+ (\u000bD)2\u000e(t\u0000t0)\n(17)7\n00.050.10.150.20.250.30.35\n-20-1001020304050\n0246810α(%)D /T  [nm2 ns-1 K-1]\nFIG. 11. Computed values of D=Tvs\u000b(large open symbols);\nblack line: guide to the eye; blue (resp. red) solid curves: an-\nalytical values with [ \r0SAt=\u00160\u0016At]D= 4\u0019(resp. 14:5). The\nblue curve thus corresponds to the Belavin-Polyakov pro\fle\nlimit. The relative di\u000berence between simulation and theory\nis indicated by small full symbols (% : right scale).\nThe average values of the displacements squared, hq2\nxi\nandhq2\nyifollow from time integration:\n\nq2\nx(t)\u000b\n=\nq2\ny(t)\u000b\n= 2kBT\u000bD\nG2+ (\u000bD)2t (18)\nAs shown previously [12, 13], the di\u000busion constant for a\nskyrmion thus reads:\nD=kBT\u000bD\nG2+ (\u000bD)2(19)\nThe following relations do apply:\n\nq2\nx(t)\u000b\n=\nq2\ny(t)\u000b\n= 2Dt\n\nq2(t)\u000b\n=\nq2\nx(t) +q2\ny(t)\u000b\n= 4Dt(20)\nRelation (19) implies a peculiar damping constant de-\npendence with, assuming for the time being DandGto\nhave comparable values, a gradual drop to zero of the\ndi\u000busion constant with decreasing \u000b(\u000b\u00141), termed\n\"di\u000busion suppression by G\" by C. Sch utte et al. [12].\nDi\u000busion suppression is actually not a complete surprise\nsince, for electrons in a magnetic \feld, a similar e\u000bect is\nleading to the classical magnetoresistance. A similar de-\npendenceD(\u000b) is also expected for a vortex. Boundary\nconditions, however, add complexity to vortex di\u000busion.\nWhat nevertheless remains, is a linear dependence of D\nvs\u000b[14], namely, di\u000busion suppression.\nThe classical expressions for GzandDxxvalid for a\nmagnetization continuum need to be adapted when deal-\ning with discrete spins. We obtain:\nGz=\u00160\u0016At\n\r0X\nk[~ s(k)\u0001[@x~ s(k)\u0002@y~ s(k)]]\nDxx=\u00160\u0016At\n\r0X\nkh\n[@x~ s(k)]2i (21)where,\u0016Atis the moment per atom.\nThe dimensionless product\r0SAt\n\u00160\u0016AtGz(Eqn.21), where\nSAtis the surface per atom, amounts to 4 \u0019, irrespec-\ntive of the skyrmion size in a perfect material at T= 0.\nStated otherwise, the skyrmion number is 1 [29]. In\nthe Belavin-Polyakov pro\fle limit [30], the dimention-\nless product\r0SAt\n\u00160\u0016AtDxx(Eqn.21) also amounts to 4 \u0019. In\nthis limit,Dis proportional to \u000b=(1+\u000b2).Dxxincreases\nwith skyrmion radius beyond the Belavin-Polyakov pro-\n\fle limit (see supplementary material in [7]). For a\nskyrmion at rest in the model Co ML considered here,\nD=Dxx\u001914:5\u00160\u0016At=(\r0SAt). For that value of\nDxx, and for the parameters used in the simulations,\nD=T, the ratio of the theoretical skyrmion di\u000busion con-\nstant to temperature, is equal 0 :234 nm2ns\u00001K\u00001, for\n\u000b= 0:5 (SAt=a2p\n3=2), to be compared to the 0 :250\nvalue extracted from simulations. More generally, Fig.11\ncompares numerical D=Tvalues calculated for a broad\nspectrum of \u000bvalues with theoretical expectations for\nD= 14:5\u00160\u0016At=(\r0SAt) and in the Belavin-Poliakov\nlimit. The average di\u000berence between analytical and sim-\nulation results is, in the \u000b= (0;1) interval, seen to be of\nthe order of'15%.\nIV. DISCUSSION\nIn the present study of thermal di\u000busion characteris-\ntics, satisfactory agreement between simulations and the-\nory has been attained for DMI sti\u000bened magnetic tex-\ntures, be it walls in narrow tracks or skyrmions. The\n\u000bdependence of the di\u000busion constants has been thor-\noughly investigated, with, as a result, a con\frmation of\nBrownian motion suppression in the presence of a non-\nzero gyrovector or, equivalently, a topological signature.\nThe theory starts with the Thiele relation applying to\na texture moving under rigid translation at constant ve-\nlocity. Furthermore, the chosen values of the components\nof the dissipation dyadic, are those valid for textures at\nrest, atT= 0. The\u000bdependence of the di\u000busion con-\nstants clearly survives these approximations. And, yet, a\nwall within a narrow stripe or a skyrmion in an ultra-thin\nmagnetic layer are deformable textures, as obvious from\nFigs.1,7. Simulations, on the other hand, rely on the\npioneering analysis of Brownian motion, here meaning\nmagnetization/spin orientation \ructuations [17], within\na particle small enough to prove uniformly magnetized\nand then extend the analysis to ultra-small computation\ncell volumes down to the single spin. Both approaches\nrely on the hypothesis of a white -uncorrelated- noise at\n\fnite temperature.\nThe discussion of results is organized in two parts. In\nthe \frst, results are analyzed in terms of a sole action of\nstructure plasticity on the diagonal elements of the dis-\nsipation dyadic. In the second, we envisage, without fur-\nther justi\fcation, how the present results are amended if,\nin the di\u000busion constants of walls and skyrmions (Eqns.8\nand 19), the gyrotropic and dissipation terms are re-8\n01 10-102 10-10\n20406080100f (GHz)S\n 12.5 K  25 K  T = 50 K \na)0123\n010203040506070T(K)< rEq > [nm]\nb)\nFIG. 12. a) Power spectrum Sof the time series rEq(t) for\nthree temperatures. The hatched area corresponds to the fre-\nquency range where a signature of the fundamental skyrmion\nbreathing mode is anticipated to be observed ( \u001939:3 GHz,\nin the present case); b) Equivalent skyrmion radius hrEqias\na function of temperature. Error bars correspond to \u00061\u001b\nof the gaussian distribution, itself a function of temperature.\n\u000b= 0:5, throughout.\nplaced by their time average as deduced from simulations.\nA. Size e\u000bects\nThe integral de\fnition of wall position adopted in this\nwork (Eqn.2) allows for a 1D treatment of wall di\u000busion,\nthus ignoring any di\u000busion characteristics potentially as-\nsociated with wall swelling, tilting, curving or meander-\ning. Additional information is, however, available in the\ncase of skyrmions. We concentrate here on the number,\nn, of spins within the skyrmion satisfying the condition\nsz\u00150:5, and its \ructuations as a function of time. The\nsurface of the skyrmion is nSAtand its equivalent radius,\nrEq, is de\fned by r2\nEq=nSAt=\u0019. The skyrmion radius\nrEqis found to \ructuate with time around its average\nvalue, according to a gaussian distribution that depends\non temperature, but becomes independent of the autocor-\nrelation time interval beyond \u001925 ps. The power spec-\ntrum of the time series rEq(t), shown in Fig.12a, excludes\nthe existence of a signi\fcant power surge around the\nfundamental breathing mode frequency of the skyrmion\n(\u001939:3 GHz for the present model Co ML) [31]. The\nskyrmion radius as de\fned from the discrete ndistribu-\ntion is thus subject to white noise. The average radius\nhrEqi, on the other hand, varies signi\fcantly with tem-\nperature, increasing from \u00191:6 nm to 2:4 nm when the\ntemperature is increased from 4 :2 K to 50 K (Fig.12b)\nand the diagonal element of the dissipation dyadic is ex-\npected to increase with increasing skyrmion radius [3, 7].\nOwing to relations (19,21), the maximum of D(\u000b) is\nfound for\u000b=Gz=Dxx=G=D . For\u000b < G=D , resp.\n\u000b > G=D ,Dincreases, resp. decreases, with D, hence\nthe relative positions of the blue and black continuous\ncurves in Fig.11. At maximum, Dis independent of D\nand amounts to kBT\r0SAt\n\u00160\u0016At1\n2G=kBT\r0SAt\n\u00160\u0016At1\n8\u0019. It ensues\nz\t\r \nf\t\r α\t\n\r\nR/Δ\t\n\ra) b) 00.20.40.60.81\n01020304050R /!\"#D / #\" < 0#D / #\" > 0FIG. 13. Di\u000busion suppression: a) general shape of function\nf(\u000b;R= \u0001) with 0 < \u000b < 1, 1< R= \u0001<50; b) crest line\nseparating the region of di\u000busion suppression ( @D=@\u000b > 0)\nfrom region @D=@\u000b< 0.\nthat the discrepancy between numerical and analytical\nDvalues around \u000b= 1 may not be relaxed by a sole\nvariation of D. On the other hand, allowing Dto increase\nwith skyrmion radius, itself a function of temperature,\nleads to an increase (decrease) of the di\u000busion coe\u000ecient\nfor\u000b<G=D (\u000b>G=D ).\nLikely more important is the reduction, as a function\nof skyrmion size, of the \u000bwindow where di\u000busion sup-\npression is expected. If including the ( R=\u0001 + \u0001=R) de-\npendence of Dxx(see supplementary material in [7]; \u0001 is\nthe wall width and Rthe skyrmion radius), the skyrmion\ndi\u000busion constant may be expressed as:\nD=kBT\r0SAt\n\u00160\u0016At1\n8\u0019f\u0012\n\u000b;R\n\u0001\u0013\n\u0011=R\n\u0001;\u0018=1\n2\u00121 +\u00112\n\u0011\u0013\n;f(\u000b;\u0011) =2\u000b\u0018\n1 + (\u000b\u0018)2(22)\nThe general shape of function f(\u000b;R= \u0001) is shown in\nFig.13a. The maximum of f(\u000b;R= \u0001) is equal to 1 for\nall values of \u000bandR=\u0001. The crest line R\u000b= \u0001 is\nseen to divide the parameter space into two regions (see\nFig.13b), a region close to the axes where @D=@\u000b > 0,\ni.e. the region of di\u000busion suppression, from the much\nwider region where @D=@\u000b < 0, that is, the region of\nwall-like behavior for skyrmion di\u000busion. Clearly, the \u000b\nwindow for di\u000busion suppression decreases dramatically\nwith increasing skyrmion size R=\u0001. A \frst observation\nof skyrmion Brownian motion at a video recording time\nscale (25 ms) may be found in the Supplementary Ma-\nterial of Ref.[32]. Skyrmions are here unusually large\nand most likely escape the di\u000busion suppression window\n(\u000b<0:02 forR=\u0001 = 50). Combining skyrmion thermal\nstability with general observability and damping parame-\nter tailoring may, as a matter of fact, well prove extremely\nchallenging for the observation of topology related di\u000bu-\nsion suppression.9\n0.750.80.850.90.951\n101214161820\n020406080100120140160T (K)< DVF >< mz / mz(T=0) >< sz / sz(T=0) >1 ML3 ML : 0.6 nm\nFIG. 14. Average reduced zmagnetization or spin component\nas a function of temperature (left scale) and time averaged\nvalue of the sole vector function, hDVFi, within the diagonal\nelement of the dissipation tensor in the skyrmion case (right\nscale). These results prove independent of the damping pa-\nrameter provided the time step in the integration of the LLG\nequation be suitably chosen.\nB. Time averaging\nOne certainly expects from the simulation model a fair\nprediction of the average magnetization hMziorhSzivs\ntemperature T, at least for temperatures substantially\nlower than the Curie temperature TC. Fig.14 shows the\nvariation ofhMzi=Mz(T= 0) orhSzi=Sz(T= 0) with\ntemperature for the two model magnetic layers of this\nwork. Although simulation results do not compare unfa-\nvorably with published experimental data [33{35], where,\ntypically, the Curie temperature amounts to \u0019150Kfor\n1 ML, and proves larger than 300 Kfor thicknesses above\n2 ML, a more detailed analysis, potentially including dis-\norder, ought to be performed.\nhGzi=\u00160\u0016Athszi\n\r0hX\nk[~ s(k)\u0001[@x~ s(k)\u0002@y~ s(k)]]i\n=\u00160\u0016Athszi\n\r0SAthGVF\nzi\nhDxxi=\u00160\u0016Athszi\n\r0hX\nk[@x~ s(k)]2i\n=\u00160\u0016Athszi\n\r0SAthDVF\nxxi(23)\nLet us now, without further justi\fcation, substitute in\nthe expression of the skyrmion di\u000busion coe\u000ecient time\naveraged values of GandD, owing to relations (23).\nKeeping in mind the geometrical meaning of GVF\nz, the\ndimensionless vector function in G,hGziis anticipated\nto be a sole function of hszi. Inversely, DVF\nxx, the (di-\nmensionless) vector function in hDxxi, a de\fnite posi-\ntive quantity, steadily increases with thermal disorder.\nIt is even found to be proportional to temperature (notshown). Its time averaged value for the sole skyrmion\nmay only be obtained by subtraction of values computed\nin the presence and absence of the skyrmion.\nFor the skyrmion in our model Co monolayer, hDVF\nxxi\nis found to increase moderately with temperature (see\nFig.14), a result also anticipated from an increase with\ntemperature of the skyrmion radius. Besides, both hGzi\nandhDxxiare expected to decrease with temperature\ndue to their proportionality to hszi.hDxxiis thus sub-\nject to two competing e\u000bects of temperature T. Present\nevidence, however, points at a dominating in\ruence of\nhsz(T)i.\nV. SUMMARY AND OUTLOOK\nSummarizing, it has been shown that the Brownian\nmotion of chiral walls and skyrmions in DMI materials\nobeys di\u000busion equations with markedly di\u000berent damp-\ning parameter ( \u000b) dependence. Although not a new re-\nsult, skyrmions Brownian motion suppression with de-\ncreasing\u000b(\u000b<G=D ) is substantiated by a wide explo-\nration of the damping parameter space. The observation\nof this astonishing topological property might, however,\nbe hampered by the restriction to ultra-small skyrmion\nsizes or ultra-low \u000bvalues. The discrepancy (up to 20%)\nbetween simulation results and theoretical expectations\ncould be reduced by the introduction of time averaged\nvalues for the gyrotropic and dissipation contributions\nto the analytical di\u000busion coe\u000ecients in the \"low\" noise\nlimit, at the expense of a tiny upwards curvature in the\nD(T) curves. A strong theoretical justi\fcation for doing\nso remains, however, lacking at this stage.\nIn this work, the sample has been assumed to be per-\nfect, i.e. devoid of spatial variations of the magnetic\nproperties, even though the lifting of such a restriction is\nanticipated to prove mandatory for a proper description\nof experiments. Di\u000busion in the presence of disorder has\nbeen theoretically studied for a number of disorder and\nrandom walk types [36, 37]. Generally, disorder changes\nthe linear growth with time of the position variance into\na power law, a behavior called superdi\u000busion if the ex-\nponent is larger than 1 and subdi\u000busion if smaller. For\ninstance, if the skyrmion motion in a disordered system\nmay be mapped onto a 2D random walk with an onsite\nresidence time \u001c, probability/\u001c\u0000(1+\u0016)(\u0016<1), then the\ndi\u000busion exponent will be \u0016, meaning subdi\u000busion. Be-\nsides, choosing a physically realistic disorder model for a\nCo monolayer might well prove equally arduous [38]. Al-\ntogether, skyrmion di\u000busion in the presence of disorder\nhas been left out for future work.\nACKNOWLEDGMENTS\nSupport by the Agence Nationale de la Recherche\n(France) under Contracts No. ANR-14-CE26-0012 (Ul-\ntrasky), No. ANR-17-CE24-0025 (TopSky) is gratefully10\nacknowledged.\n[1] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Ku-\nbetzka, R. Wiesendanger, G. Bihlmayer, and S. Bl ugel,\nNat. Phys. 7, 713 (2011)\n[2] F. Jonietz, S. M uhlbauer, C. P\reiderer, A. Neubauer,\nW. M unzer, A. Bauer, T. Adams, R. Georgii, P. B oni,\nR. A. Duine, K. Everschor, M. Garst, and A. Rosch, Sci-\nence330, 1648 (2010)\n[3] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert,\nNat. Nanotech. 8, 839 (2013)\n[4] A. Thiaville, S. Rohart, \u0013E. Ju\u0013 e, V. Cros, and A. Fert,\nEPL100, 57002 (2012)\n[5]\u0013E. Ju\u0013 e, A. Thiaville, S. Pizzini, J. Miltat, J. 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B 74, 195411 (2006)"    },    {        "title": "1806.04782v3.Dynamical_and_current_induced_Dzyaloshinskii_Moriya_interaction__Role_for_damping__gyromagnetism__and_current_induced_torques_in_noncollinear_magnets.pdf",        "content": "arXiv:1806.04782v3  [cond-mat.other]  9 Dec 2020Dynamical and current-induced Dzyaloshinskii-Moriya int eraction: Role for damping,\ngyromagnetism, and current-induced torques in noncolline ar magnets\nFrank Freimuth1,2,∗Stefan Bl¨ ugel1, and Yuriy Mokrousov1,2\n1Peter Gr¨ unberg Institut and Institute for Advanced Simula tion,\nForschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y and\n2Institute of Physics, Johannes Gutenberg University Mainz , 55099 Mainz, Germany\nBoth applied electric currents and magnetization dynamics modify the Dzyaloshinskii-Moriya in-\nteraction (DMI), which we call current-induced DMI (CIDMI) and dynamical DMI (DDMI), respec-\ntively. We report a theory of CIDMI and DDMI. The inverse of CI DMI consists in charge pumping\nbyatime-dependentgradient ofmagnetization ∂2M(r,t)/∂r∂t, while theinverseofDDMIdescribes\nthe torque generated by ∂2M(r,t)/∂r∂t. In noncollinear magnets CIDMI and DDMI depend on\nthe local magnetization direction. The resulting spatial g radients correspond to torques that need\nto be included into the theories of Gilbert damping, gyromag netism, and current-induced torques\n(CITs) in order to satisfy the Onsager reciprocity relation s. CIDMI is related to the modification of\norbital magnetism induced by magnetization dynamics, whic h we call dynamical orbital magnetism\n(DOM), and spatial gradients of DOM contribute to charge pum ping. We present applications of\nthis formalism to the CITs and to the torque-torque correlat ion in textured Rashba ferromagnets.\nI. INTRODUCTION\nSince the Dzyaloshinskii-Moriya interaction (DMI)\ncontrols the magnetic texture of domain walls and\nskyrmions, methods to tune this chiral interaction by\nexternal means have exciting prospects. Application of\ngatevoltage[1–3]orlaserpulses[4]arepromisingwaysto\nmodify DMI. Additionally, theory predicts that in mag-\nnetic trilayer structures the DMI in the top magnetic\nlayer can be controlled by the magnetization direction\nin the bottom magnetic layer [5]. Moreover, methods to\ngeneratespin currentsmaybe usedto induceDMI, which\nis predicted by the relations between the two [6, 7]. Re-\ncent experiments show that also electric currents modify\nDMI in metallic magnets, which leads to large changes in\nthe domain-wallvelocity [8, 9]. However,a rigoroustheo-\nretical formalism for the investigation of current-induced\nDMI (CIDMI) in metallic magnets has been lacking so\nfar, and the development of such a formalism is one goal\nof this paper.\nRecently, a Berry phase theory of DMI [6, 10, 11]\nhas been developed, which formally resembles the mod-\nern theory of orbital magnetization [12–14]. Orbital\nmagnetism is modified by the application of an electric\nfield, which is known as the orbital magnetoelectric re-\nsponse [15]. In the case of insulators it is straightfor-\nward to derive the expressions for the magnetoelectric\nresponse directly. However, in metals it is much easier\nto derive expressions instead for the inverse of the mag-\nnetoelectric response, i.e., for the generation of electric\ncurrents by time-dependent magnetic fields [16]. The in-\nverse current-induced DMI (ICIDMI) consists in charge\npumping by time-dependent gradients of magnetization.\nDue to the analogies between orbital magnetism and the\nBerryphasetheoryofDMIonemayexpectthatinmetals\nit is convenient to obtain expressions for ICIDMI, which\ncanthen be used todescribe the CIDMI byexploitingthereciprocity between CIDMI and ICIDMI. We will show\nin this paper that this is indeed the case.\nIn noncentrosymmetric ferromagnets spin-orbit inter-\naction (SOI) generates torques on the magnetization –\nthe so-called spin-orbit torques (SOTs) – when an elec-\ntric current is applied [17]. The Berry phase theory of\nDMI [6, 10, 11] establishes a relation to SOTs. The for-\nmal analogies between orbital magnetism and DMI have\nbeen shown to be a very useful guiding principle in the\ndevelopment of the theory of SOTs driven by heat cur-\nrents [18]. In particular, it is fruitful to consider the DMI\ncoefficients as a spiralization, which is formally analo-\ngous to magnetization. In the theory of thermoelectric\neffects in magnetic systems the curl of magnetization de-\nscribes a bound current, which cannot be measured in\ntransport experiments and needs to be subtracted from\nthe Kubo linear response in order to obtain the mea-\nsurable current [19–21]. Similarly, in the theory of the\nthermal spin-orbit torque spatial gradients of the DMI\nspiralization, which result from the temperature gradi-\nent together with the temperature dependence of DMI,\nneed to be subtracted in order to obtain the measurable\ntorqueandto satisfyaMott-likerelation[10, 18]. In non-\ncollinear magnets the question arises whether gradients\nof the spiralization that are due to the magnetic texture\ncorrespond to torques like those from thermal gradients.\nWe will show that indeed the spatial gradients of CIDMI\nneed to be included into the theory of current-induced\ntorques (CITs) in noncollinear magnets in order to sat-\nisfy the Onsager reciprocity relations [22].\nWhen the system is driven out of equilibrium by mag-\nnetization dynamics rather than electric current one may\nexpect DMI to be modified as well. The inverse effect of\nthis dynamical DMI (DDMI) consists in the generation\nof torques by time-dependent magnetization gradients.\nIn noncollinear magnets the DDMI spiralization varies\nin space. We will show that the resulting gradient cor-2\nresponds to a torque that needs to be considered in the\ntheory of Gilbert damping and gyromagnetism in non-\ncollinear magnets.\nThis paper is structured as follows. In section IIA\nwe give an overview of CIT in noncollinear magnets and\nintroduce the notation. In section IIB we describe the\nformalism used to calculate the response of electric cur-\nrent to time-dependent magnetization gradients. In sec-\ntion IIC we show that current-induced DMI (CIDMI)\nand electric current driven by time-dependent magneti-\nzation gradients are reciprocal effects. This allows us\nto obtain an expression for CIDMI based on the formal-\nism of section IIB. In section IID we discuss that time-\ndependent magnetization gradients generate additionally\ntorques on the magnetization and show that the inverse\neffect consists in the modification of DMI by magnetiza-\ntion dynamics, which we calldynamical DMI (DDMI). In\nsection IIE we demonstrate that magnetization dynam-\nics induces orbital magnetism, which we call dynamical\norbitalmagnetism (DOM) and showthat DOM is related\nto CIDMI. In section IIF we explain how the spatial gra-\ndients of CIDMI and DOM contribute to the direct and\nto the inverse CIT, respectively. In section IIG we dis-\ncuss how the spatial gradients of DDMI contribute to the\ntorque-torque correlation. In section IIH we complete\nthe formalism used to calculate the CIT in noncollinear\nmagnets by adding the chiral contribution of the torque-\nvelocity correlation. In section III we finalize the theory\nof the inverse CIT by adding the chiral contribution of\nthe velocity-torque correlation. In section IIJ we fin-\nish the computational formalism of gyromagnetism and\ndamping by adding the chiral contribution of the torque-\ntorque correlation and the response of the torque to the\ntime-dependent magnetization gradients. In section III\nwe discuss the symmetry properties of the response to\ntime-dependent magnetizationgradients. In section IVA\nwe present the results for the chiral contributions to the\ndirectand the inverseCITin the Rashbamodel andshow\nthat both the perturbation by the time-dependent mag-\nnetization gradient and the spatial gradients of CIDMI\nand DOM need to be included to ensure that they are\nreciprocal. In section IVB we present the results for the\nchiralcontributiontothe torque-torquecorrelationinthe\nRashba model and show that both the perturbation by\nthe time-dependent magnetization gradient and the spa-\ntialgradientsofDDMI need tobe included toensurethat\nit satisfies the Onsager symmetry relations. This paper\nends with a summary in section V.II. FORMALISM\nA. Direct and inverse current-induced torques in\nnoncollinear magnets\nEven in collinearmagnets the application of an electric\nfieldEgenerates a torque TCIT1on the magnetization\nwhen inversion symmetry is broken [17, 23]:\nTCIT1\ni=/summationdisplay\njtij(ˆM)Ej, (1)\nwheretij(ˆM) is the torkance tensor, which depends on\nthe magnetization direction ˆM. This torque is called\nspin-orbit torque (SOT), but we denote it here CIT1,\nbecause it is one contribution to the current-induced\ntorques (CITs) in noncollinear magnets. Inversely, mag-\nnetization dynamics pumps a charge current JICIT1ac-\ncording to [24]\nJICIT1\ni=/summationdisplay\njtji(−ˆM)ˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n,(2)\nwhereˆejis a unit vector that points into the j-th spa-\ntial direction. Generally, JICIT1can be explained by\nthe inverse spin-orbit torque [24] or the magnonic charge\npumping [25]. We denote it here by ICIT1, because it\nis one contribution to the inverse CIT in noncollinear\nmagnets. In the special case of magnetic bilayers one im-\nportantmechanism responsiblefor JICIT1arisesfrom the\ncombination of spin pumping and the inverse spin Hall\neffect [26, 27].\nIn noncollinear magnets there is a second contribution\nto the CIT, which is proportional to the spatial deriva-\ntives of magnetization [28]:\nTCIT2\ni=/summationdisplay\njklχCIT2\nijklEjˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n.(3)\nThe description of noncollinearity by the derivatives\n∂ˆM/∂rlisonlyapplicablewhenthe magnetizationdirec-\ntion changes slowly in space like in magnetic skyrmions\nwith large radius and in wide magnetic domain walls. In\norder to treat noncollinear magnets such as Mn 3Sn [29],\nwhere the magnetization direction varies strongly on the\nscale of one unit cell, Eq. (3) needs to be modified, which\nis beyond the scope of the present paper. The adia-\nbatic and the non-adiabatic [30] spin transfer torques\nare two important contributions to χCIT2\nijkl, but the in-\nterplay between broken inversion symmetry, SOI, and\nnoncollinearity can lead to a large number of additional\nmechanisms [22, 31]. Similarly, the current pumped\nby magnetization dynamics contains a contribution that\nis proportional to the spatial derivatives of magnetiza-3\ntion [22, 32, 33]:\nJICIT2\ni=/summationdisplay\njklχICIT2\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n.\n(4)\nTCIT2\niandJICIT2\nican be considered as chiral contribu-\ntionsto the CIT and to the ICIT, respectively, because\nthey distinguish between left- and right-handed spin spi-\nrals. Due to the reciprocity between direct and inverse\nCIT [22, 24] the coefficients χICIT2\nijklandχCIT2\njiklare related\naccording to\nχICIT2\nijkl(ˆM) =χCIT2\njikl(−ˆM). (5)\nB. Response of electric current to time-dependent\nmagnetization gradients\nIn order to compute JICIT2based on the Kubo lin-\near response formalism it is necessary to split it into\ntwo contributions, JICIT2aandJICIT2b. While JICIT2a\nis obtained as linear response to the perturbation by\natime-dependent magnetization gradient in a collinear\nferromagnet, JICIT2bis obtained as linear response to\nthe perturbation by magnetization dynamics in a non-\ncollinear ferromagnet. Therefore, as will become clear\nbelow,JICIT2acan be expressed by a correlation func-\ntion of two operators, because it describes the response\nof the current to a time-dependent magnetization gradi-\nent: A time-dependent magnetization gradient is a single\nperturbation, which is described by a single perturbing\noperator. In contrast, JICIT2binvolves the correlation\nof three operators, because it describes the response of\nthe current to magnetization dynamics in the presence of\nperturbation by noncollinearity. These are twoperturba-\ntions: One perturbation by the magnetization dynamics,\nandasecondperturbationtodescribethenoncollinearity.\nIn the Kubo formalism the expressions for the response\nonethe onehand toatime-dependent magnetizationgra-\ndient, which is described by a single perturbing operator,\nand the response on the other hand to a time-dependent\nmagnetization in the presence of a magnetization gradi-\nent, which is described by two perturbing operators, are\ndifferent. Therefore, we split JICIT2into these two con-\ntributions, which we call JICIT2aandJICIT2b. In the\nremainder of this section we discuss the calculation of\nthe contribution JICIT2a. The contribution JICIT2bis\ndiscussed in section III below.\nJICIT2ais determined by the second derivative of mag-\nnetization with respect to time and space variables and\ncan be written as\nJICIT2a\ni=/summationdisplay\njkχICIT2a\nijk∂2ˆMj\n∂rk∂t. (6)\nAnonzerosecondderivative∂2ˆMj\n∂rk∂tis what we referto asa\ntime-dependent magnetization gradient . Wewillshowbe-low that in special cases∂2ˆMj\n∂rk∂tcan be expressed in terms\nof the products∂ˆMl\n∂rk∂ˆMl\n∂t, which will allow us to rewrite\nJICIT2a\niin the form of Eq. (4) in the cases relevant for\nthe chiral ICIT. However, as will become clear below,\nEq. (6) is the most general expression for the response\nto time-dependent magnetization gradients, and it can-\nnot generally be rewritten in the form of Eq. (4): This\nis only possible when it describes a contribution to the\nchiral ICIT.\nJICIT2aoccurs in two different situations, which need\nto be distinguished. In one case the magnetization gra-\ndient varies in time like sin( ωt) everywhere in space. An\nexample is\nˆM(r,t) =\nηsin(q·r)sin(ωt)\n0\n1\n, (7)\nwhereηis the amplitude and the derivatives at t= 0 and\nr= 0 are\n∂ˆM(r,t)\n∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=∂ˆM(r,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0= 0 (8)\nand\n∂2ˆM(r,t)\n∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\nηqiω\n0\n0\n. (9)\nIn the other case the magnetic texture varies like a\npropagating wave, i.e., proportional to sin( q·r−ωt). An\nexample is given by\nˆM(r,t) =\nηsin(q·r−ωt)\n0\n1−η2\n2sin2(q·r−ωt)\n,(10)\nwhere the derivatives at t= 0 and r= 0 are\n∂ˆM(r,t)\n∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\nηqi\n0\n0\n, (11)\n∂ˆM(r,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\n−ηω\n0\n0\n (12)\nand\n∂2ˆM(r,t)\n∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\n0\n0\nη2qiω\n. (13)\nIn the latter example, Eq. (10), the second derivative,\nEq. (13), is along the magnetization ˆM(r= 0,t= 0),\nwhile in the former example, Eq. (7), the second deriva-\ntive, Eq. (9), is perpendicular to the magnetization when\nr= 0 andt= 0.4\nWe assume that the Hamiltonian is given by\nH(r,t) =−/planckover2pi12\n2me∆+V(r)+µBˆM(r,t)·σΩxc(r)+\n+1\n2ec2µBσ·[∇V(r)×v],\n(14)\nwhere the first term describes the kinetic energy, the sec-\nond term is a scalar potential, Ωxc(r) in the third term is\nthe exchange field, and the last term describes the spin-\norbit interaction. Around t= 0 and r= 0 we can de-\ncompose the Hamiltonian as H(r,t) =H0+δH(r,t),\nwhereH0is obtained from H(r,t) by replacing ˆM(r,t)\nbyˆM(r= 0,t= 0) and\nδH(r,t) =∂H0\n∂ˆMxηsin(q·r)sin(ωt)\n=µBΩxc(r)σxηsin(q·r)sin(ωt)(15)\nin the case of the first example, Eq. (7). In the case of\nthe second example, Eq. (10),\nδH(r,t)≃∂H\n∂ˆMxηsin(q·r−ωt)\n+∂H\n∂ˆMzη2sin(q·r)sin(ωt),(16)\nwhereforsmall randtonlythesecondterm ontheright-\nhand side contributes to∂2H(r,t)\n∂rk∂t. We consider here only\nthe time-dependence of the exchange field direction and\nignore the time-dependence of the exchange field mag-\nnitude Ωxc(r) that is induced by the time-dependence\nof the exchange field direction. While the variation of\nthe exchange field magnitude drives currents and torques\nas well, as shown in Ref. [34], the variation of the ex-\nchange field magnitude is a small response and therefore\nthese secondary responses are suppressed in magnitude\nwhen compared to the direct primary responses of the\ncurrent and torque to the variation in the exchange field\ndirection. We will use the perturbations Eq. (15) and\nEq. (16) in order to compute the response of current and\ntorque within the Kubo response formalism. An alterna-\ntive approach for the calculation of the response to time-\ndependent fields is variational linear-response, which has\nbeen applied to the spin susceptibility by Savrasov [35].\nThe perturbation by the time-dependent gradient can\nbe written as\nδH=∂H\n∂ˆM·∂2ˆM\n∂ri∂tsin(qiri)\nqisin(ωt)\nω,(17)\nwhich turns into Eq. (15) when Eq. (9) is inserted. When\nEq. (13) is inserted it turns into the second term in\nEq. (16).\nIn Appendix A we derive the linear response to pertur-\nbations of the type of Eq. (17) and show that the corre-\nsponding coefficient χICIT2a\nijkin Eq. (6) can be expressedas\nχICIT2a\nijk=ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nviRvkRROjR+viRRvkROjR+\n−viRROjRvkR−viRvkROjAA\n+viROjAvkAA+viROjAAvkA\n−viRvkRROjA−viRRvkROjA\n+viRROjAvkA+viAvkAOjAA\n−viAOjAvkAA−viAOjAAvkA/bracketrightBig\n,(18)\nwhereR=GR\nk(E) andA=GA\nk(E) are shorthands for the\nretarded and advanced Green’s functions, respectively,\nandOj=∂H/∂ˆMj.e >0 is the positive elementary\ncharge.\nIn the case of the perturbation of the type Eq. (7) the\nsecond derivative∂2ˆM\n∂ri∂tis perpendicular to M. In this\ncase it is convenient to rewrite Eq. (6) as\nJICIT2a\ni=/summationdisplay\njkχICIDMI\nijkˆej·/bracketleftBigg\nˆM×∂2ˆM\n∂rk∂t/bracketrightBigg\n,(19)\nwhere the coefficients χICIDMI\nijkare given by\nχICIDMI\nijk=ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nviRvkRRTjR+viRRvkRTjR+\n−viRRTjRvkR−viRvkRTjAA\n+viRTjAvkAA+viRTjAAvkA\n−viRvkRRTjA−viRRvkRTjA\n+viRRTjAvkA+viAvkATjAA\n−viATjAvkAA−viATjAAvkA/bracketrightBig\n,(20)\nand\nT=ˆM×∂H\n∂ˆM(21)\nis the torque operator. In Sec. IIC we will explain that\nχICIDMI\nijkdescribes the inverse of current-induced DMI\n(ICIDMI).\nIn the case of the perturbation of the type of Eq. (10)\nthe second derivative∂2ˆMj\n∂rk∂tmay be rewritten as product\nof the first derivatives∂ˆMl\n∂tand∂ˆMl\n∂rk. This may be seen5\nas follows:\n∂H\n∂ˆM·∂2ˆM\n∂ri∂t=∂2H\n∂t∂ri=\n=∂\n∂t/bracketleftBigg/parenleftbigg\nˆM×∂H\n∂ˆM/parenrightbigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg\n∂ˆM\n∂t×∂H\n∂ˆM/parenrightBigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n׈M/parenrightBigg\n×∂H\n∂ˆM/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg\n=\n=−/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n=\n=−∂ˆM\n∂t·∂ˆM\n∂ri/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n.\n(22)\nThis expression is indeed satisfied by Eq. (11), Eq. (12)\nand Eq. (13):\n∂ˆM\n∂ri·∂ˆM\n∂t=−∂2ˆM\n∂ri∂t·ˆM (23)\natr= 0,t= 0. Consequently, Eq. (6) can be rewritten\nas\nJICIT2a\ni=/summationdisplay\njkχICIT2a\nijk∂2ˆMj\n∂rk∂t=\n=−/summationdisplay\njklχICIT2a\nijk∂ˆMl\n∂rk∂ˆMl\n∂t[1−δjl]\n=/summationdisplay\njklχICIT2a\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(24)\nwhere\nχICIT2a\nijkl=−/summationdisplay\nmχICIT2a\niml[1−δjm]δjk.(25)\nThus, Eq. (24) and Eq. (25) can be used to express\nJICIT2a\niin the form of Eq. (4).\nC. Direct and inverse CIDMI\nEq. (20) describes the response of the electric current\nto time-dependent magnetization gradients of the type\nEq. (15). The reciprocal process consists in the current-\ninduced modification of DMI. This can be shown by ex-\npressing the DMI coefficients as [10]\nDij=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Dijψkn(r)\n=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Ti(r)rjψkn(r),\n(26)where we defined the DMI-operator Dij=Tirj. Using\nthe Kubo formalism the current-induced modification of\nDMI may be written as\nDCIDMI\nij=/summationdisplay\nkχCIDMI\nkijEk (27)\nwith\nχCIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(28)\nwhere\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) =−i∞/integraldisplay\n0dteiωt∝an}bracketle{t[Dij(t),vk(0)]−∝an}bracketri}ht(29)\nis the Fourier transform of a retarded function and Vis\nthe volume of the unit cell.\nSince the position operator rin the DMI operator\nDij=Tirjis not compatible with Bloch periodic bound-\nary conditions, we do not use Eq. (28) for numerical\ncalculations of CIDMI. However, it is convenient to use\nEq. (28) in order to demonstrate the reciprocity between\ndirect and inverse CIDMI.\nInverseCIDMI (ICIDMI) describes the electric current\nthat responds to the perturbation by a time-dependent\nmagnetization gradient according to\nJICIDMI\nk=/summationdisplay\nijχICIDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n.(30)\nThe perturbation by a time-dependent magnetization\ngradient may be written as\nδH=−/summationdisplay\njm·∂2ˆM\n∂t∂rjrjΩxc(r)sin(ωt)\nω=\n=/summationdisplay\njT·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nrjsin(ωt)\nω\n=/summationdisplay\nijDijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nsin(ωt)\nω.(31)\nConsequently, the coefficient χICIDMI\nkijis given by\nχICIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n.(32)\nUsing\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,ˆM) =−∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,−ˆM) (33)\nwe find that CIDMI and ICIDMI are related through the\nequations\nχCIDMI\nkij(ˆM) =−χICIDMI\nkij(−ˆM). (34)\nIn order to calculate CIDMI we use Eq. (20) for ICIDMI\nand then use Eq. (34) to obtain CIDMI.6\nThe perturbation Eq. (16) describes a different kind\nof time-dependent magnetization gradient, for which the\nreciprocaleffect consists in the modification of the expec-\ntation value ∝an}bracketle{tσ·ˆMrj∝an}bracketri}ht. However, while the modification\nof∝an}bracketle{tTirj∝an}bracketri}htby an applied current can be measured [8, 9]\nfrom the change of the DMI constant Dij, the quantity\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}hthas not been considered so far in ferromagnets.\nIn noncollinear magnets the quantity ∝an}bracketle{tσrj∝an}bracketri}htcan be used\ntodefinespintoroidization[36]. Therefore,whiletheper-\nturbation of the type of Eq. (15) is related to CIDMI and\nICIDMI, which are both accessible experimentally [8, 9],\nin the case of the perturbation of the type of Eq. (16)\nwe expect that only the effect of driving current by the\ntime-dependent magnetization gradient is easily accessi-\nble experimentally, while its inverse effect is difficult to\nmeasure.\nD. Direct and inverse dynamical DMI\nNot only applied electric currents modify DMI, but\nalso magnetization dynamics, which we call dynamical\nDMI (DDMI). DDMI can be expressed as\nDDDMI\nij=/summationdisplay\nkχDDMI\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n.(35)\nIn Sec. IIG we will show that the spatial gradient of\nDDMI contributes to damping and gyromagnetism in\nnoncollinear magnets. The perturbation used to describe\nmagnetization dynamics is given by [24]\nδH=sin(ωt)\nω/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n·T.(36)\nConsequently, the coefficients χDDMI\nkijmay be written as\nχDDMI\nkij=−1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n.(37)\nSince the position operator in Dijis not compatible\nwith Bloch periodic boundary conditions, we do not use\nEq. (37) for numerical calculations of DDMI, but instead\nwe obtain it from its inverse effect, which consists in the\ngeneration of torques on the magnetization due to time-\ndependent magnetization gradients. These torques can\nbe written as\nTIDDMI\nk=/summationdisplay\nijχIDDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n,(38)\nwhere the coefficients χIDDMI\nkijare\nχIDDMI\nkij=1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n,(39)becausethe perturbationby the time-dependent gradient\ncan be expressed in terms of Dijaccording to Eq. (31)\nand because the torque on the magnetizationis described\nby−T[23]. Consequently,DDMIandIDDMIarerelated\nby\nχDDMI\nkij(ˆM) =−χIDDMI\nkij(−ˆM). (40)\nFor numerical calculations of IDDMI we use\nχIDDMI\nijk=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRTjR+TiRRvkRTjR+\n−TiRRTjRvkR−TiRvkRTjAA\n+TiRTjAvkAA+TiRTjAAvkA\n−TiRvkRRTjA−TiRRvkRTjA\n+TiRRTjAvkA+TiAvkATjAA\n−TiATjAvkAA−TiATjAAvkA/bracketrightBig\n,(41)\nwhichisderivedinAppendix A. InordertoobtainDDMI\nwecalculateIDDMIfromEq.(41)andusethereciprocity\nrelation Eq. (40).\nEq.(38)is validfortime-dependent magnetizationgra-\ndients that lead to perturbations of the type of Eq. (15).\nPerturbations of the second type, Eq. (16), will induce\ntorques on the magnetization as well. However, the in-\nverse effect is difficult to measure in that case, because it\ncorresponds to the modification of the expectation value\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}htby magnetization dynamics. Therefore, while\nin the case of Eq. (15) both direct and inverse response\nare expected to be measurable and correspond to ID-\nDMI and DDMI, respectively, we expect that in the case\nof Eq. (16) only the direct effect, i.e., the response of the\ntorque to the perturbation, is easy to observe.\nE. Dynamical orbital magnetism (DOM)\nMagnetization dynamics does not only induce DMI,\nbut also orbital magnetism, which we call dynamical or-\nbital magnetism (DOM). It can be written as\nMDOM\nij=/summationdisplay\nkχDOM\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n,(42)\nwhere we introduced the notation\nMDOM\nij=e\nV∝an}bracketle{tvirj∝an}bracketri}htDOM, (43)\nwhich defines a generalized orbital magnetization, such\nthat\nMDOM\ni=1\n2/summationdisplay\njkǫijkMDOM\njk (44)7\ncorresponds to the usual definition of orbital magnetiza-\ntion. The coefficients χDOM\nkijare given by\nχDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvirj;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(45)\nbecause the perturbation by magnetization dynamics is\ndescribed by Eq. (36). We will discuss in Sec. IIF that\nthe spatial gradient of DOM contributes to the inverse\nCIT. Additionally, we will show below that DOM and\nCIDMI are related to each other.\nIn order to obtain an expression for DOM it is conve-\nnient to consider the inverse effect, i.e., the generation of\natorquebythe applicationofa time-dependent magnetic\nfieldB(t) that actsonly onthe orbitaldegreesoffreedom\nof the electrons and not on their spins. This torque can\nbe written as\nTIDOM\nk=1\n2/summationdisplay\nijlχIDOM\nkijǫijl∂Bl\n∂t, (46)\nwhere\nχIDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;virj∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(47)\nbecause the perturbation by the time-dependent mag-\nnetic field is given by\nδH=−e\n2/summationdisplay\nijkǫijkvirj∂Bk\n∂tsin(ωt)\nω.(48)\nTherefore, thecoefficientsofDOMandIDOM arerelated\nby\nχDOM\nkij(ˆM) =−χIDOM\nkij(−ˆM). (49)\nIn Appendix A we show that the coefficient χIDOM\nijkcan\nbe expressed as\nχIDOM\nijk=−ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRvjR+TiRRvkRvjR+\n−TiRRvjRvkR−TiRvkRvjAA\n+TiRvjAvkAA+TiRvjAAvkA\n−TiRvkRRvjA−TiRRvkRvjA\n+TiRRvjAvkA+TiAvkAvjAA\n−TiAvjAvkAA−TiAvjAAvkA/bracketrightBig\n.(50)\nEq. (50) and Eq. (20) differ only in the positions of\nthe two velocity operators and the torque operator be-\ntween the Green functions. As a consequence, IDOM\nare ICIDMI are related. In Table I and Table II we list\nthe relations between IDOM and ICIDMI for the Rashba\nmodel Eq. (83). We will explain in Sec. III that IDOM\nandICIDMI arezeroin the Rashbamodel when themag-\nnetization is along the zdirection. Therefore, we discussin Table I the case where the magnetization lies in the xz\nplane, and in Table II we discuss the case where the mag-\nnetization lies in the yzplane. According to Table I and\nTable II the relation between IDOM and ICIDMI is of\nthe formχIDOM\nijk=±χICIDMI\njik. This is expected, because\nthe indexiinχIDOM\nijkis connected to the torque operator,\nwhile the index jinχICIDMI\nijkis connected to the torque\noperator.\nTABLEI:Relations betweentheinverseofthemagnetization -\ndynamics induced orbital magnetism (IDOM) and inverse\ncurrent-inducedDMI (ICIDMI)in the 2d Rashbamodel when\nˆMlies in the zxplane. The components of χIDOM\nijk(Eq. (50))\nandχICIDMI\nijk(Eq. (20)) are denoted by the three indices ( ijk).\nICIDMI IDOM\n(211) (121)\n(121) (211)\n-(221) (221)\n(112) (112)\n-(212) (122)\n-(122) (212)\n(222) (222)\n(231) (321)\n(132) (312)\n-(232) (322)\nTABLE II: Relations between IDOM and ICIDMI in the 2d\nRashba model when ˆMlies in the yzplane.\nICIDMI IDOM\n(111) (111)\n-(211) (121)\n-(121) (211)\n(221) (221)\n-(112) (112)\n(212) (122)\n(122) (212)\n-(131) (311)\n(231) (321)\n(132) (312)\nF. Contributions from CIDMI and DOM to direct\nand inverse CIT\nIn electronic transport theory the continuity equation\ndetermines the current only up to a curl field [37]. The\ncurl of magnetization corresponds to a bound current\nthat cannot be measured in electron transport experi-\nments such that\nJ=JKubo−∇×M (51)\nhastobeusedtoextractthetransportcurrent Jfromthe\ncurrentJKuboobtained from the Kubo linear response.8\nThe subtraction of ∇×Mhas been shown to be impor-\ntant when calculating the thermoelectric response [37]\nand the anomalous Nernst effect [20]. Similarly, in the\ntheory of the thermal spin-orbit torque [10, 18] the gra-\ndients of the DMI spiralization have to be subtracted in\norder to obtain the measurable torque:\nTi=TKubo\ni−/summationdisplay\nj∂\n∂rjDij, (52)\nwhere the spatial derivative of the spiralization arises\nfrom its temperature dependence and the temperature\ngradient.\nSince CIDMI and DOM depend on the magnetization\ndirection, they vary spatially in noncollinear magnets.\nSimilar to Eq. (52) the spatial derivatives of the current-\ninduced spiralization need to be included into the theory\nof CIT. Additionally, the gradients of DOM correspond\ntocurrentsthatneedtobeconsideredinthetheoryofthe\ninverse CIT, similar to Eq. (51). In section IV we explic-\nitly show that Onsager reciprocity is violated if spatial\ngradients of DOM and CIDMI are not subtracted from\nthe Kubo response expressions. By trial-and-error we\nfind that the following subtractions are necessary to ob-\ntain response currents and torques that satisfy this fun-\ndamental symmetry:\nJICIT\ni=JKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂MDOM\nij\n∂ˆM(53)\nand\nTCIT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DCIDMI\nij\n∂ˆM,(54)\nwhereJICIT\niis the current driven by magnetization dy-\nnamics, and TCIT\niis the current-induced torque.\nInterestingly, we find that also the diagonal elements\nMDOM\niiare nonzero. This shows that the generalized def-\ninition Eq. (43) is necessary, because the diagonal ele-\nmentsMDOM\niido not contribute in the usual definition\nofMiaccording to Eq. (44). These differences in the\nsymmetry properties between equilibrium and nonequi-\nlibrium orbital magnetism can be traced back to sym-\nmetry breaking by the perturbations. Also in the case\nof the spiralization tensor Dijthe nonequilibrium cor-\nrectionδDijhas different symmetry properties than the\nequilibrium part (see Sec. III).\nThe contribution of DOM to χICIT2\nijklcan be written as\nχICIT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDOM\njil\n∂ˆM/bracketrightBigg\n(55)\nand the contribution of CIDMI to χCIT2\nijklis given by\nχCIT2b\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χCIDMI\njil\n∂ˆM/bracketrightBigg\n.(56)G. Contributions from DDMI to gyromagnetism\nand damping\nThe response to magnetization dynamics that is de-\nscribed by the torque-torque correlation function con-\nsists of torques that are related to damping and gyro-\nmagnetism [24]. The chiral contribution to these torques\ncan be written as\nTTT2\ni=/summationdisplay\njklχTT2\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(57)\nwhere the coefficients χTT2\nijklsatisfy the Onsager relations\nχTT2\nijkl(ˆM) =χTT2\njikl(−ˆM). (58)\nSinceDDMIdependsonthemagnetizationdirection,it\nvaries spatially in noncollinear magnets and the resulting\ngradients of DDMI contribute to the damping and to the\ngyromagnetic ratio:\nTTT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DDDMI\nij\n∂ˆM.(59)\nThe resulting contribution of the spatial derivatives of\nDDMI to the coefficient χTT2\nijklis\nχTT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDDMI\njil(ˆM)\n∂ˆM/bracketrightBigg\n.(60)\nH. Current-induced torque (CIT) in noncollinear\nmagnets\nThe chiral contribution to CIT consists of the spatial\ngradient of CIDMI, χCIT2b\nijklin Eq. (56), and the Kubo\nlinear response of the torque to the applied electric field\nin a noncollinear magnet, χCIT2a\nijkl:\nχCIT2\nijkl=χCIT2a\nijkl+χCIT2b\nijkl. (61)\nIn orderto determine χCIT2a\nijkl, we assume that the magne-\ntization direction ˆM(r) oscillates spatially as described\nby\nˆM(r) =\nηsin(q·r)\n0\n1\n1/radicalBig\n1+η2sin2(q·r),(62)\nwherewewilltakethelimit q→0attheendofthecalcu-\nlation. Since the spatial derivative of the magnetization\ndirection is\n∂ˆM(r)\n∂ri=\nηqicos(q·r)\n0\n0\n+O(η3),(63)9\nthe chiralcontributiontothe CIToscillatesspatiallypro-\nportional to cos( q·r). In order to extract this spatially\noscillating contribution we multiply with cos( q·r) and\nintegrate over the unit cell. The resulting expression for\nχCIT2a\nijklis\nχCIT2a\nijkl=−2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(64)\nwhereVis the volume of the unit cell, and\nthe retarded torque-velocity correlation function\n∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) needs to be evaluated in the\npresence of the perturbation\nδH=Tkηsin(q·r) (65)\ndue to the noncollinearity (the index kin Eq. (65) needs\nto match the index kinχCIT2a\nijkl).\nIn Appendix B we show that χCIT2a\nijklcan be written as\nχCIT2a\nijkl=−2e\n/planckover2pi1Im/bracketleftBig\nW(surf)\nijkl+W(sea)\nijkl/bracketrightBig\n,(66)\nwhere\nW(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)vjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)vjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)vjGA\nk(E)TkGA\nk(E)vlGA\nk(E)\n+/planckover2pi1\nmeδjlTiGR\nk(E)GA\nk(E)TkGA\nk(E)/bracketrightBigg(67)\nis a Fermi surface term ( f′(E) =df(E)/dE) and\nW(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)/bracketleftBigg\n−Tr[TiRvlRRvjRTkR]−Tr[TiRvlRTkRRvjR]\n−Tr[TiRRvlRvjRTkR]−Tr[TiRRvjRvlRTkR]\n+Tr[TiRRvjRTkRvlR]+Tr[TiRRTkRvjRvlR]\n+Tr[TiRRTkRvlRvjR]−Tr[TiRRvlRTkRvjR]\n−Tr[TiRvlRRTkRvjR]+Tr[TiRTkRRvjRvlR]\n+Tr[TiRTkRRvlRvjR]+Tr[TiRTkRvlRRvjR]\n−/planckover2pi1\nmeδjlTr[TiRRRTkR]−/planckover2pi1\nmeδjlTr[TiAAATkA]\n−/planckover2pi1\nmeδjlTr[TiAATkAA]/bracketrightBigg(68)\nis a Fermi sea term.I. Inverse CIT in noncollinear magnets\nThe chiral contribution JICIT2(see Eq. (4)) to the\ncharge pumping is described by the coefficients\nχICIT2\nijkl=χICIT2a\nijkl+χICIT2b\nijkl+χICIT2c\nijkl,(69)\nwhereχICIT2a\nijkldescribes the response to the time-\ndependentmagnetizationgradient(seeEq.(18),Eq.(25),\nand Eq. (24)) and χICIT2c\nijklresults from the spatial gra-\ndient of DOM (see Eq. (55)). χICIT2b\nijkldescribes the re-\nsponseto the perturbation bymagnetizationdynamics in\na noncollinear magnet. In order to derive an expression\nforχICIT2b\nijklwe assume that the magnetization oscillates\nspatially as described by Eq. (62). Since the correspond-\ning response oscillates spatially proportional to cos( q·r),\nwe multiply by cos( q·r) and integrate over the unit cell\nin order to extract χICIT2b\nijklfrom the retarded velocity-\ntorque correlation function ∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω), which\nis evaluated in the presence of the perturbation Eq. (65).\nWe obtain\nχICIT2b\nijkl=2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(70)\nwhich can be written as (see Appendix B)\nχICIT2b\nijkl=2e\n/planckover2pi1Im/bracketleftBig\nV(surf)\nijkl+V(sea)\nijkl/bracketrightBig\n,(71)\nwhere\nV(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBig\nviGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+viGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−viGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBig(72)\nis the Fermi surface term and\nV(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\n−Tr[viRvlRRTjRTkR]−Tr[viRvlRTkRRTjR]\n−Tr[viRRvlRTjRTkR]−Tr[viRRTjRvlRTkR]\n+Tr[viRRTjRTkRvlR]+Tr[viRRTkRTjRvlR]\n+Tr[viRRTkRvlRTjR]−Tr[viRRvlRTkRTjR]\n−Tr[viRvlRRTkRTjR]+Tr[viRTkRRTjRvlR]\n+Tr[viRTkRRvlRTjR]+Tr[viRTkRvlRRTjR]/bracketrightBig(73)\nis the Fermi sea term.\nIn Eq. (70) we use the Kubo formula to describe the\nresponse to magnetization dynamics combined with per-\nturbation theory to include the effect of noncollinearity.10\nThereby, the time-dependent perturbation and the per-\nturbation by the magnetization gradient are separated\nand perturbations of the form of Eq. (15) or Eq. (16)\nare not automatically included. For example the flat cy-\ncloidal spin spiral\nˆM(x,t) =\nsin(qx−ωt)\n0\ncos(qx−ωt)\n (74)\nmoving inxdirection with speed ω/qand the helical spin\nspiral\nˆM(y,t) =\nsin(qy−ωt)\n0\ncos(qy−ωt)\n (75)\nmovinginydirectionwith speed ω/qbehavelikeEq.(10)\nwhentandraresmall. Thus, these movingdomainwalls\ncorrespond to the perturbation of the type of Eq. (10)\nand the resulting contribution JICIT2afrom the time-\ndependent magnetization gradient is not described by\nEq. (70) and needs to be added, which we do by adding\nχICIT2a\nijklin Eq. (69).\nJ. Damping and gyromagnetism in noncollinear\nmagnets\nThe chiral contribution Eq. (57) to the torque-torque\ncorrelation function is expressed in terms of the coeffi-\ncient\nχTT\nijkl=χTT2a\nijkl+χTT2b\nijkl+χTT2c\nijkl, (76)\nwhereχTT2c\nijklresults from the spatial gradient of DDMI\n(see Eq. (60)), χTT2a\nijkldescribes the response to a time-\ndependent magnetization gradient in a collinear magnet,\nandχTT2b\nijkldescribes the response to magnetization dy-\nnamics in a noncollinear magnet.\nIn order to derive an expression for χTT2b\nijklwe as-\nsume that the magnetization oscillates spatially accord-\ning to Eq. (62). We multiply the retarded torque-torque\ncorrelation function ∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) with cos(qlrl)\nand integrate over the unit cell in order to extract the\npart of the response that varies spatially proportional to\ncos(qlrl). We obtain:\nχTT2b\nijkl=2\nVηlim\nql→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n.\n(77)\nIn Appendix B we discuss how to evaluate Eq. (77) in\nfirst order perturbation theory with respect to the per-\nturbation Eq.(65) and showthat χTT2b\nijklcan be expressedas\nχTT2b\nijkl=2\n/planckover2pi1Im/bracketleftBig\nX(surf)\nijkl+X(sea)\nijkl/bracketrightBig\n,(78)\nwhere\nX(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBigg(79)\nis a Fermi surface term and\nX(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(TiRvlRRTjRTkR)−(TiRvlRTkRRTjR)\n−(TiRRvlRTjRTkR)−(TiRRTjRvlRTkR)\n+(TiRRTjRTkRvlR)+(TiRRTkRTjRvlR)\n+(TiRRTkRvlRTjR)−(TiRRvlRTkRTjR)\n−(TiRvlRRTkRTjR)+(TiRTkRRTjRvlR)\n+(TiRTkRRvlRTjR)+(TiRTkRvlRRTjR)/bracketrightBigg\n(80)\nis a Fermi sea term.\nThe contribution χTT2a\nijklfrom the time-dependent gra-\ndients is given by\nχTT2a\nijkl=−/summationdisplay\nmχTT2a\niml[1−δjm]δjk,(81)\nwhere\nχTT2a\niml=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvlRROmR+TiRRvlROmR+\n−TiRROmRvlR−TiRvlROmAA\n+TiROmAvlAA+TiROmAAvlA\n−TiRvlRROmA−TiRRvlROmA\n+TiRROmAvlA+TiAvlAOmAA\n−TiAOmAvlAA−TiAOmAAvlA/bracketrightBig\n,(82)\nwithOm=∂H/∂ˆMm(see Appendix A).\nIII. SYMMETRY PROPERTIES\nIn this section we discuss the symmetry properties of\nCIDMI, DDMI and DOM in the case of the magnetic\nRashba model\nHk(r) =/planckover2pi12\n2mek2+α(k׈ez)·σ+∆V\n2σ·ˆM(r).(83)11\nAdditionally, we discuss the symmetry properties of the\ncurrents and torques induced by time-dependent magne-\ntization gradients of the form of Eq. (10).\nWe consider mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation\naround the zaxis. When ∆ V= 0 these operations leave\nEq. (83) invariant, but when ∆ V∝ne}ationslash= 0 they modify the\nmagnetization direction ˆMin Eq. (83), as shown in Ta-\nble III. At the same time, these operations affect the\ntorqueTandthecurrent Jdrivenbythe time-dependent\nmagnetization gradients (see Table III). In Table IV and\nTable V we show how ˆM×∂ˆM/∂rkis affected by the\nsymmetry operations.\nAflat cycloidalspin spiralwith spinsrotatingin the xz\nplane is mapped by a c2 rotation around the zaxis onto\nthe same spin spiral. Similarly, a flat helical spin spiral\nwith spins rotating in the yzplane is mapped by a c2 ro-\ntationaroundthe zaxisontothesamespinspiral. There-\nfore, when ˆMpoints inzdirection, a c2 rotation around\nthezaxis does not change ˆM×∂ˆM/∂ri, but it flips the\nin-plane current Jand the in-plane components of the\ntorque,TxandTy. Consequently, ˆM×∂2ˆM/∂ri∂tdoes\nnot induce currents or torques, i.e., ICIDMI, CIDMI, ID-\nDMI and DDMI are zero, when ˆMpoints inzdirection.\nHowever, they become nonzero when the magnetization\nhas an in-plane component (see Fig. 1).\nSimilarly, IDOM vanishes when the magnetization\npoints inzdirection: In that case Eq. (83) is invariant\nunder the c2 rotation. A time-dependent magnetic field\nalongzdirection is invariant under the c2 rotation as\nwell. However, TxandTychange sign under the c2 rota-\ntion. Consequently, symmetryforbidsIDOM inthiscase.\nHowever, when the magnetization has an in-plane com-\nponent, IDOM and DOM become nonzero (see Fig. 2).\nThat time-dependent magnetization gradients of the\ntype of Eq. (7) do not induce in-plane currents and\ntorqueswhen ˆMpoints inzdirectioncan alsobe seendi-\nrectly from Eq. (7): The c2 rotation transforms q→ −q\nandMx→ −Mx. Since sin( q·r) is odd in r, Eq. (7) is in-\nvariantunder c2rotation, whilethe in-planecurrentsand\ntorques induced by time-dependent magnetization gradi-\nents change sign under c2 rotation. In contrast, Eq. (10)\nis not invariant under c2 rotation, because sin( q·r−ωt)\nis not odd in rfort>0. Consequently, time-dependent\nmagnetization gradients of the type of Eq. (10) induce\ncurrents and torques also when ˆMpoints locally into\nthezdirection. These currents and torques, which are\ndescribed by Eq. (24) and Eq. (82), respectively, need to\nbe added to the chiral ICIT and the chiral torque-torque\ncorrelation. While CIDMI, DDMI, and DOM are zero\nwhen the magnetization points in zdirection, their gra-\ndients are not (see Fig. 1 and Fig. 2). Therefore, the gra-\ndients of CIDMI, DOM, and DDMI contribute to CIT, to\nICIT and to the torque-torque correlation, respectively,\neven when ˆMpoints locally into the zdirection.TABLE III: Effect of mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation around\nthezaxis. The magnetization Mand the torque Ttransform\nlike axial vectors, while the current Jtransforms like a polar\nvector.\nMxMyMzJxJyTxTyTz\nMxz−MxMy−MzJx−Jy−TxTy−Tz\nMyzMx−My−Mz−JxJyTx−Ty−Tz\nc2-Mx-MyMz-Jx−Jy−Tx−TyTz\nTABLE IV: Effect of symmetry operations on the magneti-\nzation gradients. Magnetization gradients are described b y\nthree indices ( ijk). The first index denotes the magnetiza-\ntion direction at r= 0. The third index denotes the di-\nrection along which the magnetization changes. The second\nindex denotes the direction of ∂ˆM/∂rkδrk. The direction of\nˆM×∂ˆM/∂rkis specified by the number below the indices\n(ijk).\n(1,2,1) (1,3,1) (2,1,1) (2,3,1) (3,1,1) (3,2,1)\n3-2 -3 1 2-1\nMxz(-1,2,1)(-1,-3,1) (2,-1,1) (2,-3,1) (-3,-1,1) (-3,2,1)\n-3 -2 3 -1 2 1\nMyz(1,2,1) (1,3,1)(-2,-1,1) (-2,3,1) (-3,-1,1) (-3,2,1)\n3-2 -3 -1 2 1\nc2(-1,2,1)(-1,-3,1) (-2,1,1) (-2,-3,1) (3,1,1) (3,2,1)\n-3 -2 3 1 2-1\n.\nTABLE V: Continuation of Table IV\n(1,2,2) (1,3,2) (2,1,2) (2,3,2) (3,1,2) (3,2,2)\n3 -2 -3 1 2-1\nMxz(-1,-2,2) (-1,3,2) (2,1,2) (2,3,2)(-3,1,2)(-3,-2,2)\n3 2-3 1-2 -1\nMyz(1,-2,2) (1,-3,2) (-2,1,2)(-2,-3,2) (-3,1,2)(-3,-2,2)\n-3 2 3 1-2 -1\nc2(-1,2,2) (-1,-3,2) (-2,1,2)(-2,-3,2) (3,1,2) (3,2,2)\n-3 -2 3 1 2-1\nA. Symmetry properties of ICIDMI and IDDMI\nInthefollowingwediscusshowTableIII,TableIV,and\nTable V can be used to analyze the symmetry of ICIDMI\nandIDDMI.AccordingtoEq.(19)thecoefficient χICIDMI\nijk\ndescribes the response of the current JICIT2a\nito the time-\ndependent magnetization gradient ˆej·[ˆM×∂2ˆM\n∂rk∂t]. Since\nˆM×∂2ˆM\n∂rk∂t=∂\n∂t[ˆM×∂ˆM\n∂rk] fortime-dependent magnetiza-\ntion gradients of the type Eq. (7) the symmetry proper-\nties ofχICIDMI\nijkfollow from the transformation behaviour\nofˆM×∂ˆM\n∂rkandJunder symmetry operations.\nWe consider the case with magnetization in xdirec-\ntion. The component χICIDMI\n132describes the current in x\ndirection induced by the time-dependence of a cycloidal\nmagnetizationgradientin ydirection(withspinsrotating12\nFIG. 1: ICIDMI in a noncollinear magnet. (a) Arrows illus-\ntrate the magnetization direction. (b) Arrows illustrate t he\ncurrentJyinduced by a time-dependent magnetization gra-\ndient, which is described by χICIDMI\n221. When ˆMpoints in z\ndirection, χICIDMI\n221andJyare zero. The sign of χICIDMI\n221and\nofJychanges with the sign of Mx.\nFIG. 2: DOM in a noncollinear magnet. (a) Arrows illustrate\nthe magnetization direction. (b) Arrows illustrate the orb ital\nmagnetization induced by magnetization dynamics (DOM).\nWhenˆMpoints in zdirection, DOM is zero. The sign of\nDOM changes with the sign of Mx.\nin thexyplane).Myzflips both ˆM×∂ˆM\n∂yandJx, but\nit preserves ˆM.Mzxpreserves ˆM×∂ˆM\n∂yandJx, but it\nflipsˆM. A c2 rotation around the zaxis flips ˆM×∂ˆM\n∂y,\nˆMandJx. Consequently, χICIDMI\n132(ˆM) is allowed by\nsymmetry and it is even in ˆM. The component χICIDMI\n122\ndescribes the current in xdirection induced by the time-\ndependence of a helical magnetization gradient in ydi-\nrection (with spins rotating in the xzplane).Myzflips\nˆM×∂ˆM\n∂yandJx, but it preserves ˆM.MzxflipsˆM×∂ˆM\n∂y\nandˆM, but it preserves Jx. A c2 rotation around the z\naxis flipsJxandˆM, but it preserves ˆM×∂ˆM\n∂y. Conse-\nquently,χICIDMI\n122is allowed by symmetry and it is odd in\nˆM. The component χICIDMI\n221describes the current in y\ndirection induced by the time-dependence of a cycloidal\nmagnetization gradient in xdirection (with spins rotat-\ning in thexzplane).Mzxpreserves ˆM×∂ˆM\n∂x, but it flipsJyandˆM.Myzpreserves ˆM,Jy, andˆM×∂ˆM\n∂x. The\nc2 rotation around the zaxis preserves ˆM×∂ˆM\n∂x, but\nit flipsˆMandJy. Consequently, χICIDMI\n221is allowed by\nsymmetry and it is odd in ˆM. The component χICIDMI\n231\ndescribes the current in ydirection induced by the time-\ndependence of a cycloidal magnetization gradient in xdi-\nrection (with spins rotating in the xyplane).Mzxflips\nˆM×∂ˆM\n∂x,ˆM, andJy.Myzpreserves ˆM×∂ˆM\n∂x,ˆMand\nJy. The c2 rotation around the zaxis flips ˆM×∂ˆM\n∂x,Jy,\nandˆM. Consequently, χICIDMI\n231is allowed by symmetry\nand it is even in ˆM.\nThese properties are summarized in Table VI. Due to\nthe relations between CIDMI and DOM (see Table I and\nTable II), they can be used for DOM as well. When the\nmagnetization lies at a general angle in the xzplane or in\ntheyzplaneseveraladditionalcomponentsofCIDMIand\nDOMarenonzero(seeTableIandTableII,respectively).\nTABLE VI: Allowed components of χICIDMI\nijkwhenˆMpoints\ninxdirection. + components are even in ˆM, while - compo-\nnents are odd in ˆM.\n132 122 221 231\n+ - - +\nSimilarly, one can analyze the symmetry of DDMI. Ta-\nble VII lists the components of DDMI, χDDMI\nijk, which are\nallowed by symmetry when ˆMpoints inxdirection.\nTABLEVII:Allowedcomponentsof χDDMI\nijkwhenˆMpointsin\nxdirection. +componentsareevenin ˆM, while -components\nare odd in ˆM.\n222 232 322 332\n- + + -\nB. Response to time-dependent magnetization\ngradients of the second type (Eq. (10))\nAccording to Eq. (13) the time-dependent magneti-\nzation gradient is along the magnetization. Therefore,\nin contrast to the discussion in section IIIA we can-\nnot use ˆM×∂2ˆM\n∂rk∂tin the symmetry analysis. Eq. (24)\nand Eq. (25) show that χICIT2a\nijjldescribes the response of\nJICIT2a\nitoˆej·/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\nˆej·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nwhileχICIT2a\nijkl=\n0 forj∝ne}ationslash=k. According to Eq. (23) the symmetry prop-\nerties of/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\n·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nagree to the symmetry\nproperties of ˆM·∂2ˆM\n∂rl∂t. Therefore, in order to under-\nstand the symmetry properties of χICIT2a\nijjlwe consider\nthe transformation of JandˆM·∂2ˆM\n∂rl∂tunder symmetry\noperations.\nWe consider the case where ˆMpoints inzdirection.\nχICIT2a\n1jj1describes the current driven in xdirection, when13\nthe magnetization varies in xdirection. MxzflipsˆM,\nbut preserves JxandˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,Jx,\nandˆM·∂2ˆM/(∂x∂t). c2 rotation flips ˆM·∂2ˆM/(∂x∂t)\nandJx, but preserves ˆM. Consequently, χICIT2a\n1jj1is al-\nlowed by symmetry and it is even in ˆM.\nχICIT2a\n2jj1describes the current flowing in ydirection,\nwhen magnetization varies in xdirection. MxzflipsˆM\nandJy, but preserves ˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,\nandˆM·∂2ˆM/(∂x∂t), but preserves Jy. c2 rotation\nflipsˆM·∂2ˆM/(∂x∂t) andJy, but preserves ˆM. Conse-\nquently,χICIT2a\n2jj1is allowed by symmetry and it is odd in\nˆM.\nSimilarly, one can show that χICIT2a\n1jj2is odd in ˆMand\nthatχICIT2a\n2jj2is even in ˆM.\nAnalogously, one can investigate the symmetry prop-\nerties ofχTT2a\nijjl. We find that χTT2a\n1jj1andχTT2a\n2jj2are odd\ninˆM, whileχTT2a\n2jj1andχTT2a\n1jj2are even in ˆM.\nIV. RESULTS\nIn the following sections we discuss the results for the\ndirect and inverse chiral CIT and for the chiral torque-\ntorque correlation in the two-dimensional (2d) Rashba\nmodel Eq. (83), and in the one-dimensional (1d) Rashba\nmodel [38]\nHkx(x) =/planckover2pi12\n2mek2\nx−αkxσy+∆V\n2σ·ˆM(x).(84)\nAdditionally, we discuss the contributions of the time-\ndependent magnetization gradients, and of DDMI, DOM\nand CIDMI to these effects.\nWhile vertex corrections to the chiral CIT and to\nthe chiral torque-torque correlation are important in the\nRashba model [38], the purpose of this work is to show\nthe importance ofthe contributionsfrom time-dependent\nmagnetization gradients, DDMI, DOM and CIDMI. We\ntherefore consider only the intrinsic contributions here,\ni.e., we set\nGR\nk(E) =/planckover2pi1[E −Hk+iΓ]−1, (85)\nwhere Γ is a constant broadening, and we leave the study\nof vertex corrections for future work.\nThe results shown in the following sections are ob-\ntained for the model parameters ∆ V= 1eV,α=2eV˚A,\nand Γ = 0 .1Ry = 1.361eV, when the magnetization\npoints inzdirection, i.e., ˆM=ˆez. The unit of χCIT2\nijkl\nis charge times length in the 1d case and charge in the\n2d case. Therefore, in the 1d case we discuss the chiral\ntorkance in units of ea0, wherea0is Bohr’s radius. In the\n2d case we discuss the chiral torkance in units of e. The\nunit ofχTT2\nijklis angular momentum in the 1d case and\nangular momentum per length in the 2d case. Therefore,\nwe discussχTT2\nijklin units of /planckover2pi1in the 1d case, and in units\nof/planckover2pi1/a0in the 2d case.-2 -1 0 1 2\nFermi energy [eV]-0.02-0.0100.010.020.030.040.05χijklCIT2 [ea0]2121\n1121\n2121 (gauge-field)\n1121 (gauge-field)\nFIG. 3: Chiral CIT in the 1d Rashba model for cycloidal gra-\ndients vs. Fermi energy. General perturbation theory (soli d\nlines) agrees to the gauge-field approach (dashed lines).\nA. Direct and inverse chiral CIT\nIn Fig. 3 we show the chiral CIT as a function of the\nFermi energyfor cycloidalmagnetization gradients in the\n1d Rashba model. The components χCIT2\n2121andχCIT2\n1121are\nlabelled by 2121 and 1121, respectively. The component\n2121ofCITdescribesthe non-adiabatictorque, while the\ncomponent 1121 describes the adiabatic STT (modified\nby SOI). In the one-dimensional Rashba model, the con-\ntributionsχCIT2b\n2121andχCIT2b\n1121(Eq. (56)) from the CIDMI\nare zero when ˆM=ˆez(not shown in the figure). For cy-\ncloidal spin spirals, it is possible to solve the 1d Rashba\nmodel by a gauge-field approach [38], which allows us to\ntest the perturbation theory, Eq. (66). For comparison\nwe show in Fig. 3 the results obtained from the gauge-\nfield approach, which agree to the perturbation theory,\nEq. (66). This demonstrates the validity of Eq. (66).\nIn Fig. 4 we show the chiral ICIT in the 1d Rashba\nmodel. The components χICIT2\n1221andχICIT2\n1121are labelled\nby 1221and 1121, respectively. The contribution χICIT2a\n1221\nfrom the time-dependent gradient is of the same order of\nmagnitude as the total χICIT2\n1221. Comparison of Fig. 3 and\nFig. 4 shows that CIT and ICIT satisfy the reciprocity\nrelationsEq. (5), that χCIT2\n1121is odd in ˆM, and thatχCIT2\n2121\nis even in ˆM, i.e.,χCIT2\n2121=χICIT2\n1221andχCIT2\n1121=−χICIT2\n1121.\nThe contribution χICIT2a\n1221from the time-dependent gradi-\nents is crucial to satisfy the reciprocity relations between\nχCIT2\n2121andχICIT2\n1221.\nIn Fig. 5 and Fig. 6 we show the CIT and the ICIT, re-\nspectively, for helical gradients in the 1d Rashba model.\nThe components χCIT2\n2111andχCIT2\n1111are labelled 2111 and\n1111, respectively, in Fig. 5, while χICIT2\n1211andχICIT2\n1111\nare labelled 1211 and 1111, respectively, in Fig. 6. The\ncontributions χCIT2b\n2111andχCIT2b\n1111from CIDMI are of the14\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.04χijklICIT2 [ea0]1221 \n1121\nχ1221ICIT2a\nFIG. 4: Chiral ICIT in the 1d Rashba model for cycloidal\ngradients vs. Fermi energy. Dashed line: Contribution from\nthe time-dependent gradient.\nsame order of magnitude as the total χCIT2\n2111andχCIT2\n1111.\nSimilarly, the contributions χICIT2c\n1211andχICIT2c\n1111from\nDOM are of the same order of magnitude as the to-\ntalχICIT2\n1211andχICIT2\n1111. Additionally, the contribution\nχICIT2a\n1111from the time-dependent gradient is substantial.\nComparisonofFig.5andFig.6showsthatCITandICIT\nsatisfy the reciprocity relation Eq. (5), that χCIT2\n2111is odd\ninˆM, and thatχCIT2\n1111is even in ˆM, i.e.,χCIT2\n1111=χICIT2\n1111\nandχCIT2\n2111=−χICIT2\n1211. These reciprocity relations be-\ntween CIT and ICIT are only satisfied when CIDMI,\nDOM, and the response to time-dependent magnetiza-\ntion gradients are included. Additionally, the compar-\nison between Fig. 5 and Fig. 6 shows that the contri-\nbutions of CIDMI to CIT ( χCIT2b\n1111andχCIT2b\n2111) are re-\nlated to the contributions of DOM to ICIT ( χICIT2c\n1111and\nχICIT2c\n1211). These relations between DOM and ICIT are\nexpected from Table I.\nIn Fig. 7 and Fig. 8 we show the CIT and the ICIT,\nrespectively, for cycloidal gradients in the 2d Rashba\nmodel. In this case there are contributions from CIDMI\nand DOM in contrast to the 1d case with cycloidal gra-\ndients (Fig. 3). Comparison between Fig. 7 and Fig. 8\nshows that χCIT2\n1121andχCIT2\n2221are odd in ˆM, thatχCIT2\n1221\nandχCIT2\n2121are even in ˆM, and that CIT and ICIT sat-\nisfy the reciprocity relation Eq. (5) when the gradients\nof CIDMI and DOM are included, i.e., χCIT2\n1121=−χICIT2\n1121,\nχCIT2\n2221=−χICIT2\n2221,χCIT2\n1221=χICIT2\n2121, andχCIT2\n2121=χICIT2\n1221.\nχCIT2\n1121describesthe adiabatic STT with SOI, while χCIT2\n2121\ndescribes the non-adiabatic STT. Experimentally, it has\nbeen found that CITs occur also when the electric field\nis applied parallel to domain-walls (i.e., perpendicular to\ntheq-vector of spin spirals) [39]. In our calculations, the\ncomponents χCIT2\n2221andχCIT2\n1221describe such a case, where\nthe applied electric field points in ydirection, while the-2 -1 0 1 2\nFermi energy [eV]-0.04-0.0200.020.040.06χijklCIT2 [ea0]1111\n2111\nχ1111CIT2b\nχ2111CIT2b\nFIG. 5: Chiral CIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.040.06χijklICIT2 [ea0]1111\n1211\nχ1111ICIT2a\nχ1111ICIT2c\nχ1211ICIT2c\nFIG. 6: Chiral ICIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted line: Contribution from the time-\ndependent magnetization gradient.\nmagnetization direction varies with the xcoordinate.\nIn Fig. 9 and Fig. 10 we show the chiral CIT and\nICIT, respectively, for helical gradients in the 2d Rashba\nmodel. The component χCIT2\n2111describes the adiabatic\nSTT with SOI and the component χCIT2\n1111describes the\nnon-adiabatic STT. The components χCIT2\n2211andχCIT2\n1211\ndescribe the case when the applied electric field points\ninydirection, i.e., perpendicular to the direction along\nwhich the magnetization direction varies. Comparison\nbetween Fig. 9 and Fig. 10 shows that χCIT2\n1111andχCIT2\n2211\nare even in ˆM, thatχCIT2\n1211andχCIT2\n2111are odd in ˆMand\nthat CIT andICIT satisfythe reciprocityrelationEq.(5)\nwhenthegradientsofCIDMIandDOMareincluded, i.e.,15\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]1121\n2221\n1221\n2121\nχ2221CIT2b\nχ1221CIT2b\nχ2121CIT2b\nFIG. 7: Chiral CIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklICIT2 [e]1121\n1221\n2121\n2221\nχ2221ICIT2a\nχ1221ICIT2a\nχ2121ICIT2c\nχ1221ICIT2c\nχ2221ICIT2c\nFIG. 8: Chiral ICIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradients.\nχCIT2\n1111=χICIT2\n1111,χCIT2\n2211=χICIT2\n2211,χCIT2\n1211=−χICIT2\n2111, and\nχCIT2\n2111=−χICIT2\n1211.\nB. Chiral torque-torque correlation\nIn Fig. 11 we show the chiral contribution to the\ntorque-torque correlation in the 1d Rashba model for\ncycloidal gradients. We compare the perturbation the-\nory Eq. (78) plus Eq. (82) to the gauge-field approach\nfrom Ref. [38]. This comparison shows that perturba-\ntion theory provides the correct answer only when the\ncontribution χTT2a\nijkl(Eq. (82)) from the time-dependent-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]2211\n1111\n1211\n2111\nχ2111CIT2b\nχ1211CIT2b\nχ2211CIT2b\nχ1111CIT2b\nFIG. 9: Chiral CIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.004-0.00200.0020.0040.006χijklICIT2 [e]1111\n1211\n2111\n2211\nχ1111ICIT2a\nχ2221ICIT2a\nχ1111ICIT2c\nχ2111ICIT2c\nχ1211ICIT2c\nχ2211ICIT2c\nFIG. 10: Chiral ICIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradient.\ngradients is taken into account. The contributions χTT2a\n1221\nandχTT2a\n2221fromthe time-dependent gradientsarecompa-\nrable in magnitude to the total values. In the 1d Rashba\nmodel the DDMI-contribution in Eq. (60) is zero for cy-\ncloidal gradients (not shown in the figure). The compo-\nnentsχTT2\n2121andχTT2\n1221describe the chiral gyromagnetism\nwhile the components χTT2\n1121andχTT2\n2221describe the chi-\nral damping [38, 40, 41]. The components χTT2\n2121and\nχTT2\n1221are odd in ˆMand they satisfy the Onsagerrelation\nEq. (58), i.e., χTT2\n2121=−χTT2\n1221.\nIn Fig. 12 we show the chiral contributions to the\ntorque-torque correlation in the 1d Rashba model for\nhelical gradients. In contrast to the cycloidal gradients16\n-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]2121\n1221\n2221\n1121\nχ1221TT2a\nχ2221TT2a\n2121 (gf)\n1221 (gf)\n1121 (gf)\n2221 (gf)\nFIG. 11: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 1d Rashba model vs. Fermi\nenergy. Perturbation theory (solid lines) agrees to the gau ge-\nfield (gf) approach (dotted lines). Dashed lines: Contribut ion\nfrom the time-dependent gradient.\n(Fig. 11) there are contributions from the spatial gra-\ndients of DDMI (Eq. (60)) in this case. The Onsager\nrelation Eq. (58) for the components χTT2\n2111andχTT2\n1211is\nsatisfied only when these contributions from DDMI are\ntaken into account, which are of the same order of mag-\nnitude as the total values. The components χTT2\n2111and\nχTT2\n1211are even in ˆMand describe chiral damping, while\nthe components χTT2\n1111andχTT2\n2211are odd in ˆMand de-\nscribe chiral gyromagnetism. As a consequence of the\nOnsager relation Eq. (58) we obtain χTT2\n1111=χTT2\n2211= 0\nfor the total components: Eq. (58) shows that diagonal\ncomponents of the torque-torque correlation function are\nzero unless they are even in ˆM. However, χTT2a\n1111,χTT2c\n1111,\nandχTT2b\n1111=−χTT2a\n1111−χTT2c\n1111are individually nonzero.\nInterestingly, the off-diagonal components of the torque-\ntorquecorrelationdescribechiraldampingforhelicalgra-\ndients, while for cycloidal gradients the off-diagonal ele-\nments describe chiral gyromagnetism and the diagonal\nelements describe chiral damping.\nIn Fig. 13 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for cy-\ncloidal gradients. In contrast to the 1d Rashba model\nwith cycloidal gradients (Fig. 11) the contributions from\nDDMIχTT2c\nijkl(Eq.(60))arenonzerointhiscase. Without\nthesecontributionsfromDDMI theOnsagerrelation(58)\nχTT2\n2121=−χTT2\n1221is violated. The DDMI contribution is\nof the same order of magnitude as the total values. The\ncomponents χTT2\n2121andχTT2\n1221are odd in ˆMand describe\nchiral gyromagnetism, while the components χTT2\n1121and\nχTT2\n2221are even in ˆMand describe chiral damping.\nIn Fig. 14 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for he-\nlical gradients. The components χTT2\n1211andχTT2\n2111are even-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]1111\n2111\n1211\n2211\nχ1111TT2c\nχ2111TT2c\nχ1211TT2c\nχ2211TT2c\nχ1111TT2a\nχ2111TT2a\nFIG. 12: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 1d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_/a0]1121\n2121\n1221\n2221\nχ1221TT2a\nχ2221TT2a\nχ2121TT2c\nχ1221TT2c\nFIG. 13: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\ninˆMand describe chiral damping, while the compo-\nnentsχTT2\n1111andχTT2\n2211are odd in ˆMand describe chiral\ngyromagnetism. The Onsager relation Eq. (58) requires\nχTT2\n1111=χTT2\n2211= 0 andχTT2\n2111=χTT2\n1211. Without the\ncontributions from DDMI these Onsager relations are vi-\nolated.17\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_ /a0]1111\n2111\n1211\n2211\nχ1111TT2a\nχ2111TT2a\nχ1111TT2c\nχ1211TT2c\nχ2211TT2c\nFIG. 14: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\nV. SUMMARY\nFinding ways to tune the Dzyaloshinskii-Moriya inter-\naction (DMI) by external means, such as an applied elec-\ntriccurrent,holdsmuchpromiseforapplicationsinwhich\nDMI determines the magnetic texture of domain walls or\nskyrmions. In order to derive an expression for current-\ninduced Dzyaloshinskii-Moriya interaction (CIDMI) we\nfirst identify its inverse effect: When magnetic textures\nvary as a function of time, electric currents are driven by\nvarious mechanisms, which can be distinguished accord-\ningtotheirdifferentdependenceonthetime-derivativeof\nmagnetization, ∂ˆM(r,t)/∂t, and on the spatial deriva-\ntive∂ˆM(r,t)/∂r: One group of effects is proportional\nto∂ˆM(r,t)/∂t, a second group of effects is propor-\ntional to the product ∂ˆM(r,t)/∂t ∂ˆM(r,t)/∂r, and\na third group is proportional to the second derivative\n∂2ˆM(r,t)/∂r∂t. We show that the response of the elec-\ntric current to the time-dependent magnetization gradi-\nent∂2ˆM(r,t)/∂r∂tcontais the inverse of CIDMI. We\nestablish the reciprocity relation between inverse and di-\nrectCIDMI and therebyobtainan expressionforCIDMI.\nWe find that CIDMI is related to the modification of\norbital magnetism induced by magnetization dynamics,\nwhich we call dynamical orbital magnetism (DOM). We\nshow that torques are generated by time-dependent gra-\ndients of magnetization as well. The inverse effect con-\nsists in the modification of DMI by magnetization dy-\nnamics, which we call dynamical DMI (DDMI).\nAdditionally, we develop a formalism to calculate the\nchiral contributions to the direct and inverse current-\ninduced torques (CITs) and to the torque-torque correla-tion in noncollinear magnets. We show that the response\nto time-dependent magnetization gradients contributes\nsubstantially to these effects and that the Onsager reci-\nprocityrelationsareviolated when it is not takeninto ac-\ncount. InnoncollinearmagnetsCIDMI,DDMIandDOM\ndepend on the local magnetization direction. We show\nthat the resulting spatial gradients of CIDMI, DDMI\nand DOM have to be subtracted from the CIT, from\nthe torque-torque correlation, and from the inverse CIT,\nrespectively.\nWe apply our formalism to study CITs and the torque-\ntorque correlation in textured Rashba ferromagnets. We\nfind that the contribution of CIDMI to the chiral CIT is\noftheorderofmagnitudeofthe totaleffect. Similarly, we\nfind that the contribution of DDMI to the chiral torque-\ntorque correlation is of the order of magnitude of the\ntotal effect.\nAcknowledgments\nWeacknowledgefinancialsupportfromLeibnizCollab-\norative Excellence project OptiSPIN −Optical Control\nofNanoscaleSpin Textures. Weacknowledgefundingun-\nder SPP 2137 “Skyrmionics” of the DFG. We gratefully\nacknowledge financial support from the European Re-\nsearch Council (ERC) under the European Union’s Hori-\nzon 2020 research and innovation program (Grant No.\n856538, project ”3D MAGiC”). The work was also sup-\nported by the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) −TRR 173 −268565370\n(project A11). We gratefully acknowledge the J¨ ulich\nSupercomputing Centre and RWTH Aachen University\nfor providing computational resources under project No.\njiff40.\nAppendix A: Response to time-dependent gradients\nIn this appendix we derive Eq. (18), Eq. (20), Eq. (41),\nand Eq. (82), which describe the response to time-\ndependent magnetization gradients, and Eq. (50), which\ndescribesthe responsetotime-dependentmagneticfields.\nWe consider perturbations of the form\nδH(r,t) =Bb1\nqωsin(q·r)sin(ωt).(A1)\nWhenweset B=∂H\n∂ˆMkandb=∂2ˆMk\n∂ri∂t, Eq.(A1)turnsinto\nEq. (17), while when we set B=−eviandb=1\n2ǫijk∂Bk\n∂t\nwe obtain Eq. (48). We need to derive an expression for\nthe response δA(r,t) of an observable Ato this pertur-\nbation, which varies in time like cos( ωt) and in space like\ncos(q·r), because∂2ˆM(r,t)\n∂ri∂t∝cos(q·r)cos(ωt). There-\nfore, weusethe Kubolinearresponseformalismtoobtain18\nthe coefficient χin\nδA(r,t) =χcos(q·r)cos(ωt), (A2)\nwhich is given by\nχ=i\n/planckover2pi1qωV/bracketleftBig\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n−∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(−/planckover2pi1ω)/bracketrightBig\n,(A3)\nwhere∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) is the retarded\nfunction at frequency ωandVis the volume of the unit\ncell.\nThe operator Bsin(q·r) can be written as\nBsin(q·r) =1\n2i/summationdisplay\nknm/bracketleftBig\nB(1)\nknmc†\nk+nck−m−B(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A4)\nwherek+=k+q/2,k−=k−q/2,c†\nk+nis the cre-\nation operator of an electron in state |uk+n∝an}bracketri}ht,ck−mis the\nannihilation operator of an electron in state |uk−m∝an}bracketri}ht,\nB(1)\nknm=1\n2∝an}bracketle{tuk+n|[Bk++Bk−]|uk−m∝an}bracketri}ht(A5)\nand\nB(2)\nknm=1\n2∝an}bracketle{tuk−n|[Bk++Bk−]|uk+m∝an}bracketri}ht.(A6)\nSimilarly,\nAcos(q·r) =1\n2/summationdisplay\nknm/bracketleftBig\nA(1)\nknmc†\nk+nck−m+A(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A7)\nwhere\nA(1)\nknm=1\n2∝an}bracketle{tuk+n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk−m∝an}bracketri}ht(A8)\nand\nA(2)\nknm=1\n2∝an}bracketle{tuk−n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk+m∝an}bracketri}ht.(A9)\nIt is convenient to obtain the retarded response func-\ntion in Eq. (A3) from the correspondingMatsubarafunc-\ntion in imaginary time τ\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(τ) =\n=1\n4i/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/bracketleftBig\nA(1)\nknmB(2)\nkn′m′Z(1)\nknmn′m′(τ)\n−A(2)\nknmB(1)\nkn′m′Z(2)\nknmn′m′(τ)/bracketrightBig\n,\n(A10)\nwhered= 1,2 or 3 is the dimension,\nZ(1)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(0)ck+m′(0)∝an}bracketri}ht\n=−GM\nm′n(k+,−τ)GM\nmn′(k−,τ),\n(A11)Z(2)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(0)ck−m′(0)∝an}bracketri}ht\n=−GM\nm′n(k−,−τ)GM\nmn′(k+,τ),\n(A12)\nand\nGM\nmn′(k+,τ) =−∝an}bracketle{tTτck+m(τ)c†\nk+n′(0)∝an}bracketri}ht(A13)\nis the single-particle Matsubara function. The Fourier\ntransform of Eq. (A10) is given by\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=i\n4/planckover2pi1β/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\np/bracketleftBig\nA(1)\nknmB(2)\nkn′m′GM\nm′n(k+,iEp)GM\nmn′(k−,iEp+iEN)\n−A(2)\nknmB(1)\nkn′m′GM\nm′n(k−,iEp)GM\nmn′(k+,iEp+iEN)/bracketrightBig\n,\n(A14)\nwhereEN= 2πN/βandEp= (2p+ 1)π/βare bosonic\nandfermionicMatsubaraenergypoints, respectively, and\nβ= 1/(kBT) is the inverse temperature.\nIn order to carry out the Matsubara summation over\nEpwe make use of\n1\nβ/summationdisplay\npGM\nmn′(iEp+iEN)GM\nm′n(iEp) =\n=i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′+iδ)\n+i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iδ)GM\nm′n(E′−iEN)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′−iδ)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′−iδ)GM\nm′n(E′−iEN),(A15)\nwhereδis a positive infinitesimal. The retarded function\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(ω) is obtained from the Mat-\nsubara function ∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) by the\nanalytic continuation iEN→/planckover2pi1ωto real frequencies. The\nright-hand side of Eq. (A15) has the following analytic\ncontinuation to real frequencies:\ni\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GR\nm′n(E′)\n+i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω)\n−i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GA\nm′n(E′)\n−i\n2π/integraldisplay\ndE′f(E′)GA\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω).(A16)\nTherefore, we obtain\nχ=−i\n8π/planckover2pi12qω/integraldisplayddk\n(2π)d[Zk(q,ω)−Zk(−q,ω)\n−Zk(q,−ω)+Zk(−q,−ω)],(A17)19\nwhere\nZk(q,ω) =\n=/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGR\nk+(E′)/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGA\nk+(E′)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n.(A18)\nWe consider the limit lim q→0limω→0χ. In this limit\nEq. (A17) may be rewritten as\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)d∂2Zk(q,ω)\n∂q∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=ω=0.(A19)\nThe frequency derivative of Zk(q,ω) is given by\n1\n/planckover2pi1∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGR\nk+(E′)/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGR\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGA\nk+(E′)/bracketrightBigg\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGA\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n.\n(A20)\nUsing∂GR(E)/∂E=−GR(E)GR(E)//planckover2pi1we obtain\n∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGR\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−BkGA\nk+GA\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGA\nk+/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−BkGA\nk+GA\nk+/bracketrightBig\n.\n(A21)\nMaking use of\nlim\nq→0∂GR\nk+\n∂q=1\n2GR\nkv·q\nqGR\nk (A22)we finally obtain\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)dlim\nq→0lim\nω→0∂2Z(q,ω)\n∂q∂ω=\n=−i\n4π/planckover2pi12q\nq·/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nAkRvRRBkR+AkRRvRBkR\n−AkRRBkRvR−AkRvRBkAA\n+AkRBkAvAA+AkRBkAAvA\n−AkRvRRBkA−AkRRvRBkA\n+AkRRBkAvA\n+AkAvABkAA−AkABkAvAA\n−AkABkAAvA/bracketrightBig\n,(A23)\nwhere we use the abbreviations R=GR\nk(E) andA=\nGA\nk(E). When we substitute B=∂H\n∂ˆMj,A=−evi, and\nq=qkˆek, we obtain Eq. (18). When we substitute B=\nTj,A=−evi, andq=qkˆek, we obtain Eq. (20). When\nwe substitute A=−Ti,B=Tj, andq=qkˆek, we obtain\nEq. (41). When we substitute B=−evj,A=−Ti,\nandq=qkˆek, we obtain Eq. (50). When we substitute\nB=∂H\n∂ˆMj,A=−Ti, andq=qkˆek, we obtain Eq. (82).\nAppendix B: Perturbation theory for the chiral\ncontributions to CIT and to the torque-torque\ncorrelation\nIn this appendix we derive expressionsfor the retarded\nfunction\n∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) (B1)\nwithin first-orderperturbation theory with respect to the\nperturbation\nδH=Bηsin(q·r), (B2)\nwhich may arise e.g. from the spatial oscillation of the\nmagnetization direction. As usual, it is convenient to ob-\ntain the retarded response function from the correspond-\ning Matsubara function\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ) =−∝an}bracketle{tTτcos(q·r)A(τ)C(0)∝an}bracketri}ht.\n(B3)\nThe starting point for the perturbative expansion is\nthe equation\n−∝an}bracketle{tTτcos(q·r)A(τ1)C(0)∝an}bracketri}ht=\n=−Tr/bracketleftbig\ne−βHTτcos(q·r)A(τ1)C(0)/bracketrightbig\nTr[e−βH]=\n=−Tr/braceleftbig\ne−βH0Tτ[Ucos(q·r)A(τ1)C(0)]/bracerightbig\nTr[e−βH0U],(B4)20\nwhereH0is the unperturbed Hamiltonian and we con-\nsider the first order in the perturbation δH:\nU(1)=−1\n/planckover2pi1/integraldisplay/planckover2pi1β\n0dτ1Tτ{eτ1H0//planckover2pi1δHe−τ1H0//planckover2pi1}.(B5)\nThe essentialdifference between Eq. (A3) and Eq. (B4) is\nthat in Eq. (A3) the operator Benters together with the\nfactor sin( q·r)sin(ωt) (see Eq. (A1)), while in Eq. (B4)\nonly the factor sin( q·r) is connected to Bin Eq. (B2),\nwhile the factor sin( ωt) is coupled to the additional op-\neratorC.\nWe use Eq. (A4) and Eq. (A7) in order to express\nAcos(q·r) andBsin(q·r) in terms of annihilation and\ncreation operators. In terms of the correlators\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk−n′′ck−m′′∝an}bracketri}ht\n(B6)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B7)\nand\nZ(5)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B8)\nand\nZ(6)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk−n′′ck−m′′∝an}bracketri}ht\n(B9)\nEq. (B4) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay/planckover2pi1β\n0dτ/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3)\nknmn′m′n′′m′′(τ,τ1)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6)\nknmn′m′n′′m′′(τ,τ1)/bracketrightBigg(B10)\nwithin first-order perturbation theory, where we de-\nfinedCk−n′′m′′=∝an}bracketle{tuk−n′′|C|uk−m′′∝an}bracketri}htandCk+n′′m′′=\n∝an}bracketle{tuk+n′′|C|uk+m′′∝an}bracketri}ht.\nNote that Z(5)can be obtained from Z(3)by replac-\ningk−byk+andk+byk−. Similarly, Z(6)can be\nobtained from Z(4)by replacing k−byk+andk+by\nk−. Therefore, we write down only the equations forZ(3)andZ(4)in the following. Using Wick’s theorem\nwe find\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nm′n(k−,τ1−τ)GM\nmn′(k+,τ−τ1)GM\nm′′n′′(k−,0)\n+GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1)\n(B11)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nmn′(k+,τ−τ1)GM\nm′n(k−,τ1−τ)GM\nm′′n′′(k+,0)\n+GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1).\n(B12)\nThe Fourier transform\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1)(B13)\nof Eq. (B10) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3a)\nknmn′m′n′′m′′(iEN)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6a)\nknmn′m′n′′m′′(iEN)/bracketrightBigg(B14)\nin terms of the integrals\nZ(3a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′(iEp)GM\nk−m′′n(iEp)GM\nk−m′n′′(iEp+iEN)\n(B15)\nand\nZ(4a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′′(iEp)GM\nk−m′n(iEp)GM\nk+m′′n′(iEp−iEN),\n(B16)\nwhereEN= 2πN/βis a bosonic Matsubara energy point\nand we used\nGM(τ) =1\n/planckover2pi1β∞/summationdisplay\np=−∞e−iEpτ//planckover2pi1GM(iEp),(B17)21\nwhereEp= (2p+1)π/βis a fermionic Matsubara point.\nAgain Z(5a)is obtained from Z(3a)by replacing k−by\nk+andk+byk−andZ(6a)is obtained from Z(4a)in\nthe same way.\nSummation overMatsubarapoints Epin Eq.(B15) and\nin Eq. (B16) and analytic continuation iEN→/planckover2pi1ωyields\n2πi/planckover2pi1Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′(E)GR\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E)GA\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GR\nk−m′n′′(E)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GA\nk−m′n′′(E)\n(B18)\nand\n2πi/planckover2pi1Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E)GR\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′′(E)GA\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GR\nk+m′′n′(E)\n+/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GA\nk+m′′n′(E).\n(B19)\nIn the next step we take the limit ω→0 (see Eq. (64),\nEq. (70), and Eq. (77)):\n−1\nVlim\nω→0Im∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ω=\n=η\n4/planckover2pi1Im/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n,(B20)where we defined\nY(3)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck−n′′m′′×\n×∂Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(4)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck+n′′m′′×\n×∂Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(5)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck+n′′m′′×\n×∂Z(5a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(6)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck−n′′m′′×\n×∂Z(6a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\n(B21)\nwhich can be expressed as Y(3)=Y(3a)+Y(3b)and\nY(4)=Y(4a)+Y(4b), where\n2π/planckover2pi1Y(3a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)\n+AkGR\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)/bracketrightBig\n(B22)\nand\n2π/planckover2pi1Y(3b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GR\nk−(E)BkGR\nk+(E)\n+AkGA\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n.(B23)22\nSimilarly,\n2π/planckover2pi1Y(4a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n−AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)\n−AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)/bracketrightBig\n(B24)\nand\n2π/planckover2pi1Y(4b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)BkGA\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GR\nk+(E)\n+AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GR\nk+(E)/bracketrightBigg\n.(B25)\nWe call Y(3a)andY(4a)Fermi surface terms and Y(3b)\nandY(4b)Fermi sea terms. Again Y(5)is obtained from\nY(3)by replacing k−byk+andk+byk−andY(6)is\nobtained from Y(4)in the same way.\nFinally, we take the limit q→0:\nΛ =−2\n/planckover2pi1VηIm lim\nq→0lim\nω→0∂\n∂ω∂\n∂qi∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n=1\n2/planckover2pi1lim\nq→0∂\n∂qiIm/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n=1\n2/planckover2pi1Im/bracketleftBig\nX(3)+X(4)−X(5)−X(6)/bracketrightBig\n,\n(B26)\nwhere we defined\nX(j)=∂\n∂qi/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=0Y(j)(B27)\nforj= 3,4,5,6. Since Y(4)andY(6)are related by\nthe interchange of k−andk+it follows that X(6)=\n−X(4). Similarly, since Y(3)andY(5)arerelated by the\ninterchange of k−andk+it follows that X(5)=−X(3).\nConsequently, we need\nΛ =1\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)+X(4a)+X(4b)/bracketrightBig\n,(B28)\nwhere X(3a)andX(4a)are the Fermi surface terms and\nX(3b)andX(4b)are the Fermi sea terms. The Fermisurface terms are given by\nX(3a)=−1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nAkGR\nk(E)vkGR\nk(E)CkGA\nk(E)BkGA\nk(E)\n+AkGR\nk(E)CkGA\nk(E)vkGA\nk(E)BkGA\nk(E)\n−AkGR\nk(E)CkGA\nk(E)BkGA\nk(E)vkGA\nk(E)\n+AkGR\nk(E)∂Ck\n∂kGA\nk(E)BkGA\nk(E)/bracketrightBigg(B29)\nand\nX(4a)=−/bracketleftBig\nX(3a)/bracketrightBig∗\n. (B30)\nThe Fermi sea terms are given by\nX(3b)=−1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(ARvRRCRBR)+(AACAABAvA)\n−(ARRvRCRBR)−(ARRCRvRBR)\n+(ARRCRBRvR)−(AAvACABAA)\n−(AACAvABAA)+(AACABAvAA)\n+(AACABAAvA)−(AAvACAABA)\n−(AACAvAABA)−(AACAAvABA)\n−(ARR∂C\n∂kRBR)−(AA∂C\n∂kAABA)\n−(AA∂C\n∂kABAA)/bracketrightBigg(B31)\nand\nX(4b)=−/bracketleftBig\nX(3b)/bracketrightBig∗\n. (B32)\nIn Eq. (B31) we use the abbreviations R=GR\nk(E),A=\nGA\nk(E),A=Ak,B=Bk,C=Ck. It is important\nto note that Ck−andCk+depend on qthrough k−=\nk−q/2 andk+=k+q/2 . Theqderivative therefore\ngenerates the additional terms with ∂Ck/∂kin Eq. (B29)\nand Eq. (B31). In contrast, AkandBkdo not depend\nlinearly on q.\nEq. (B28) simplifies due to the relations Eq. (B30) and\nEq. (B32) as follows:\nΛ =2\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)/bracketrightBig\n. (B33)\nIn order to obtain the expression for the chiral con-\ntribution to the torque-torque correlation we choose the\noperators as follows:\nB→ Tk\nA→ −Ti\nC→ Tj\n∂C\n∂k= 0\nv→vl.(B34)23\nThis leads to Eq. (78), Eq. (79) and Eq. (80) of the main\ntext.\nIn order to obtain the expression for the chiral contri-\nbution to the CIT, we set\nB→ Tk\nA→ −Ti\nC→ −evj\n∂C\n∂k→ −e/planckover2pi1\nmδjl\nv→vl.(B35)\nThis leads to Eq. (66), Eq. (67) and Eq. (68).\nIn order to obtain the expression for the chiral contri-\nbution to the ICIT, we set\nB→ Tk\nA→ −evi\nC→ Tj\n∂C\n∂k→0\nv→vl.(B36)\nThis leads to Eq. (71), Eq. (72) and Eq. (73).\n∗Corresp. author: f.freimuth@fz-juelich.de\n[1] K. Nawaoka, S. 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B 93, 214429 (2016)."    },    {        "title": "1806.04881v1.Low_magnetic_damping_of_ferrimagnetic_GdFeCo_alloys.pdf",        "content": "1 \n Low magnetic damping of ferrimagnetic GdFeCo alloys \nDuck-Ho Kim1†*, Takaya Okuno1†, Se Kwon Kim2, Se-Hyeok Oh3, Tomoe Nishimura1, \nYuushou Hirata1, Yasuhiro Futakawa4, Hiroki Yoshikawa4, Arata Tsukamoto4, Yaroslav \nTserkovnyak2, Yoichi Shiota1, Takahiro Moriyama1, Kab-Jin Kim5, Kyung-Jin Lee3,6,7, and \nTeruo Ono1,8* \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 6 11-0011, Japan \n2Department of Physics and Astronomy, University of California, Los Angeles, California \n90095, USA \n3Department of Nano-Semiconductor and Engineering, Korea Univers ity, Seoul 02841, \nRepublic of Korea \n4College of Science and Technology, Nihon University, Funabashi,  Chiba 274-8501, Japan \n5Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea \n6Department of Materials Science & Engineering, Korea University , Seoul 02841, Republic \nof Korea \n7KU-KIST Graduate School of Converging Science and Technology, K orea University, Seoul \n02841, Republic of Korea \n8Center for Spintronics Research Network (CSRN), Graduate School  of Engineering Science, \nOsaka University, Osaka 560-8531, Japan \n \n† These authors contributed equally to this work. \n* E-mail: kim.duckho.23z@st.kyoto-u.ac.jp , ono@scl.kyoto-u.ac.jp  2 \n We investigate the Gilbert damping parameter  for rare earth (RE)–\ntransition metal (TM) ferrimagnets over a wide temperature rang e. Extracted from the \nfield-driven magnetic domain-wall mobility,  was as low as 7.2 × 10-3 and was almost \nconstant across the angular momentum compensation temperature 𝑻𝐀, starkly \ncontrasting previous predictions that  should diverge at 𝑻𝐀 due to vanishing total \nangular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to \nthe total angular momentum but is dominated by electron scatter ing at the Fermi level \nwhere the TM has a dominant damping role.  \n  3 \n Magnetic damping, commonly described by the Gilbert damping par ameter, \nrepresents the magnetization relaxation phenomenon, describing how quickly magnetization \nspins reach equilibrium [1–3]. Understanding the fundamental or igin of the damping as well \nas searching for low damping materials has been a central theme  of magnetism research. \nSeveral theoretical models for magnetic damping have been propo sed [4–11] and compared \nwith experiments [12–20]. Ultra-low damping was predicted in fe rromagnetic alloys using a \nlinear response damping model [11] and was demonstrated experim entally for CoFe alloys \n[20]. However, the majority of these studies have focused only on ferromagnetic systems. \nAntiferromagnets, which have alt ernating orientations of their neighboring magnetic \nmoments, have recently received considerable attention because of their potential importance \nfor spintronic applications [21– 30]. Antiferromagnetic spin sys tems can have much faster \nspin dynamics than their ferromagnetic counterparts, which is a dvantageous in spintronic \napplications [21, 25, 31–39]. However, the manipulation and con trol of antiferromagnets is \nchallenging because the net magnetic moment is effectively zero . Recently, antiferromagnetic \nspin dynamics have been successfully demonstrated using the mag netic domain-wall (DW) \ndynamics in ferrimagnets with finite magnetization in the vicin ity of the angular momentum \ncompensation temperature, at which the net angular momentum van ishes [38]. This field-\ndriven antiferromagnetic spin dyn amics is possible because the time evolution of the \nmagnetization is governed by the commutation relation of the an gular momentum rather than \nthe commutation relation of the magnetic moment.  \nMotivated by the aforementioned result, in this letter, we inve stigate the magnetic \ndamping of ferrimagnets across th e angular momentum compensatio n temperature, which \nwill allow us to understand magnetic damping in antiferromagnet ically coupled system. We 4 \n selected rare earth (RE)–transition metal (TM) ferrimagnets for  the material platforms \nbecause they have an angular momentum compensation temperature 𝑇 w h e r e  \nantiferromagnetic spin dynamics are achieved [38, 40, 41]. The magnetic-field-driven DW \nmotion was explored over a wide range of temperatures including  𝑇, and the Gilbert \ndamping parameter was extracted from the measured DW mobility a t each temperature by \nemploying the collective coordina te model initially developed f or ferrimagnetic spin \ndynamics [38]. Contrary to the previous prediction that the Gil bert damping parameter would \ndiverge at 𝑇 due to the vanishing of the total angular momentum [42, 43], w e found that the \nGilbert damping parameter remained nearly constant over a wide range of temperatures \nacross 𝑇 with the estimated value as low as 7.2 × 10-3, which was similar to the reported \nvalues of TM-only ferromagnets [20]. These results suggested th at Gilbert damping was \nmainly governed by electron scattering at the Fermi level, and hence, the 4f electron of the \nR E  e l e m e n t ,  w h i c h  l i e s  f a r  b e l o w  t h e  F e r m i  l e v e l ,  d i d  n o t  p l a y  an important role in the \nmagnetic damping of RE–TM ferrimagnets. \nFor this study, we prepared perpendicularly magnetized ferrimag netic GdFeCo films \nin which the Gd and FeCo moments were coupled antiferromagnetic ally. Specifically, the \nfilms were 5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/100-nm SiN on an intrinsic Si substrate. The \nGdFeCo films were then patterned  into 5-µm-wide and 500-µm-long  microwires with a Hall \ncross structure using electron beam lithography and Ar ion mill ing. For current injection, \n100-nm Au/5-nm Ti electrodes were stacked on the wire. A Hall b ar was designed to detect \nthe DW velocity via the anomalous Hall effect (AHE). \nWe measured the magnetic DW motion using a real-time DW detecti on technique [38, \n40, 41, 44, 45] [see Fig. 1(a) for a schematic]. We first appli ed a magnetic field of –200 mT 5 \n to saturate the magnetization al ong the –z direction. Subsequen tly, a constant perpendicular \nmagnetic field 𝜇𝐻, which was lower than the coercive field, was applied along +z  direction. \nNext, a d.c. current was applied along the wire to measure the anomalous Hall voltage. Then, \na current pulse (12 V , 100 ns) was injected through the writing  line to nucleate the DW in the \nwire. The created DW was moved along the wire and passed throug h the Hall bar because of \nthe presence of 𝜇𝐻. The DW arrival time was detected by monitoring the change in the Hall \nvoltage using a real-time oscillo scope. The DW velocity could t hen be calculated from the \narrival time and the travel dis tance between the writing line a nd Hall bar (500 µm). \nFigure 1(b) shows the averaged DW velocity 〈𝑣〉 as a function of the perpendicular \nmagnetic field 𝜇𝐻 for several temperatures 𝑇∗. Here, we used the d.c. current density of \n|𝐽|ൌ1.3×1010 A / m2 to measure the AHE change due to DW motion. Note that 𝑇∗ i s  a n  \nelevated temperature that considers Joule heating by d.c. curre nt [46]. To eliminate the \nundesired current-induced spin-transfer-torque effect, we avera ged the DW velocity for 𝐽 \nand –𝐽, i.e., 〈𝑣〉ൌሾ𝑣ሺ𝐽ሻ𝑣ሺെ𝐽ሻሿ/2. Figure 1(b) shows that 〈𝑣〉 increases linearly with \n𝜇𝐻 for all 𝑇∗. Such linear behavior can be described by 〈𝑣〉ൌ𝜇ሾ𝜇𝐻െ𝜇 𝐻ሿ, where 𝜇 \nis the DW mobility and 𝜇𝐻 is the correction field, which generally arises from \nimperfections in the sample or complexities of the internal DW structure [47, 48]. We note \nthat 𝜇𝐻 can also depend on the temperature dependence of the magnetic properties of \nferrimagnets [45]. Figure 1(c) shows 𝜇 as a function of 𝑇∗ at several current densities \n(|𝐽|ൌ1.3, 1.7, and 2.0 ×1010 A / m2). A sharp peak clearly occurs for 𝜇 a t  𝑇∗ൌ241.5 K \nirrespective of |𝐽|. The drastic increase of 𝜇 is evidence of antiferromagnetic spin dynamics \nat 𝑇, as demonstrated in our pre vious report [38, 40, 41]. \nThe obtained DW mobility was theoretically analyzed as follows.  The DW velocity 6 \n of ferrimagnets in the precessional regime is given by [38, 39]  \n          𝑉 ൌ 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ\nሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶ𝜇𝐻,                                                                                ሺ1ሻ  \nwhere 𝑉 is the DW velocity, 𝜆 is the DW width, 𝜇𝐻 is the perpendicular magnetic field, \n𝛼 is the Gilbert damping parameter, 𝑀 and 𝑠 are the magnetization and the spin angular \nmomentum of one sublattice, respectively. The spin angular mome ntum densities are given \nby 𝑠ൌ𝑀 /𝛾 [49], where 𝛾ൌ𝑔 𝜇/ℏ is the gyromagnetic ratio of lattice 𝑖, 𝑔 i s  t h e  \nLandé g factor of lattice 𝑖, 𝜇 is the Bohr magneton, and ℏ is the reduced Plank’s constant. \nThe Gilbert damping is in principle different for two sublattic e s ,  b u t  f o r  s i m p l i c i t y ,  w e  \nassume that it is the same, which can be considered as the aver age value of the damping \nparameters for the two sublattices weighted by the spin angular  momentum density. We note \nthat this assumption does not alter our main conclusion: low da mping and its insensitivity to \nthe temperature. Equation (1) gives the DW mobility 𝜇 a s  𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ/\nሼሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶሽ, which can be rearranged as \n          𝜇 ሺ𝑠ଵ𝑠 ଶሻଶ𝛼ଶെ𝜆ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ𝛼𝜇 ሺ𝑠ଵെ𝑠 ଶሻଶൌ 0                                           ሺ2ሻ  \nUsing Eq. (2) to find the solution of 𝛼, we find \n          𝛼 േൌ𝜆ሺ𝑀ଵെ𝑀 ଶሻേඥሾ𝜆ଶሺ𝑀ଵെ𝑀 ଶሻଶെ4 𝜇ଶሺ𝑠ଵെ𝑠 ଶሻଶሿ\n2𝜇ሺ𝑠ଵ𝑠 ଶሻ.                                              ሺ3ሻ  \nEquation (3) allows us to estimate 𝛼 for the given 𝜇. We note that for each value of 𝜇, 𝛼 \nca n h av e t w o v a lu e s,  𝛼ା and 𝛼ି because of the quadratic nature of Eq. (2). Only one of \nthese two solutions is physically sound, which can be obtained using the following energy \ndissipation analysis.  7 \n  The energy dissipation (per unit cross section) through the DW  dynamics is given by \n𝑃ൌ2 𝛼 ሺ 𝑠 ଵ𝑠 ଶሻ𝑉ଶ/𝜆  2𝛼ሺ𝑠 ଵ𝑠 ଶሻ 𝜆Ωଶ [38, 39], where Ω is the angular velocity of the \nDW. The first and the second terms represent the energy dissipa tion through the translational \nand angular motion of the DW, respectively. In the precessional  regime, the angular velocity \nis proportional to the translational velocity: Ωൌ ሺ𝑠ଵെ𝑠 ଶሻ𝑉/𝛼ሺ𝑠 ଵ𝑠 ଶሻ𝜆. Replacing Ω b y  \nthe previous expression yields 𝑃ൌ𝜂 𝑉ଶ w h e r e  𝜂ൌ2 ሺ 𝑀 ଵെ𝑀 ଶሻ/𝜇 is the viscous \ncoefficient for the DW motion:  \n          𝜂 ൌ2\n𝜆ቊ𝛼ሺ𝑠ଵ𝑠 ଶሻ ሺ𝑠ଵെ𝑠 ଶሻଶ\n𝛼ሺ𝑠ଵ𝑠 ଶሻቋ .                                                                                         ሺ4ሻ  \nThe first and the second terms in parenthesis capture the contr ibutions to the energy \ndissipation from the translational and angular dynamics of the DW, respectively. The two \nsolutions for the Gilbert damping parameter,  𝛼ା and 𝛼ି, can yield the same viscous \ncoefficient 𝜂. The case of the equal solutions,  𝛼ାൌ𝛼 ି, corresponds to the situation when \nthe two contributions are identical:  𝛼േൌሺ 𝑠 ଵെ𝑠 ଶሻ/ሺ𝑠ଵ𝑠 ଶሻ. For the larger solution 𝛼ൌ\n𝛼ା, the energy dissipation is dominated by the first term, i.e., through the translational DW \nmotion, which should be the case in the vicinity of 𝑇 where the net spin density ሺ𝑠ଵെ𝑠 ଶሻ \nis small and thus the angular velocity is negligible. For examp le, at exact 𝑇, the larger \nsolution 𝛼ା is the only possible solution because the smaller solution is zero, 𝛼ିൌ0, and \nthus unphysical. For the smaller solution 𝛼ൌ𝛼 ି, the dissipation is dominated by the second \nterm, i.e., through the precessional motion, which should descr ibe cases away from 𝑇. \nTherefore, in the subsequent analysis, we chose the larger solu tion  𝛼ା in the vicinity of 𝑇 \nand the smaller solution 𝛼ି far away from 𝑇 and connected the solution continuously in \nbetween. 8 \n The other material parameters such as 𝑀ଵ, 𝑀ଶ, 𝑠ଵ, and 𝑠ଶ a r e  e s t i m a t e d  b y  \nmeasuring the net magnetic moment of GdFeCo film, |𝑀୬ୣ୲|, for various temperatures. \nBecause 𝑀୬ୣ୲ includes contributions from both  the Gd and FeCo sub-moments, the sub-\nmagnetic moments, 𝑀ଵ a n d  𝑀ଶ, could be decoupled based on the power law criticality [see \ndetails in refs. 38, 40]. The spin angular momentums, 𝑠ଵ and 𝑠ଶ, were calculated using the \nknown Landé g factor of FeCo and Gd (the Landé g factor of FeCo  is 2.2 and that of Gd is \n2.0) [50–52]. \nFigures 2(a)–(c) show the temperature-dependent DW mobility 𝜇, sub-magnetic \nmoment 𝑀, and sub-angular momentum  𝑠, respectively. Here, we used the relative \ntemperature defined as ∆𝑇 ൌ 𝑇∗െ𝑇  to investigate the Gilbert damping near 𝑇. The \nGilbert damping parameter 𝛼 was obtained based on Eq. (3) and the information in Fig. \n2(a)–(c). Figure 2(d) shows the resulting values of 𝛼േ as a function of ∆𝑇. For ∆𝑇ଵ൏\n∆𝑇 ൏ ∆𝑇 ଶ, 𝛼ା is nearly constant, while 𝛼ି varies significantly. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 \n∆𝑇ଶ, on the other hand,  𝛼ି is almost constant, while 𝛼ା varies significantly. At ∆𝑇 ൌ ∆𝑇 ଵ \nand ∆𝑇 ൌ ∆𝑇 ଶ, the two solutions are equal, corresponding to the aforementio ned case when \nthe energy dissipation through the translational and angular mo tion of the DW are identical. \nThe proper damping solution can be selected by following the gu ideline obtained \nfrom the above analysis. For ∆𝑇ଵ൏∆ 𝑇൏∆ 𝑇 ଶ, which includes 𝑇, the energy dissipation \nshould be dominated by the translational motion, and thus 𝛼ା is a physical solution. Note \nalso that 𝛼ି becomes zero at 𝑇, which results in infinite DW mobility in contradiction with \nthe experimental observation. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇  ∆𝑇 ଶ, where the energy dissipation is \ndominated by the angular motion of the DW, 𝛼ି is the physical solution. 9 \n Figure 3 shows the resultant Gil bert damping parameter in all t ested temperature \nranges. The Gilbert damping parameter was almost constant acros s 𝑇 with 𝛼ൌ7.2 × 10-3 \n(see the dotted line in Fig. 3). This result is in stark contra st to the previous prediction. In ref. \n[42], Stanciu et al. investigated the temperature dependence of the effective Gilb ert damping \nparameter based on a ferromagnet-based model and found that the  damping diverged at 𝑇. \nBecause they analyzed the magnetic resonance in ferrimagnetic m aterials based on a \nferromagnet-based model, which cannot describe the antiferromag netic dynamics at 𝑇 a t  \nwhich the angular momentum vanis hes, it exhibits unphysical res ults. However, our \ntheoretical analysis for field-driven ferromagnetic DW motion b ased on the collective \ncoordinate approach can properly describe both the antiferromag netic dynamics in the \nvicinity of 𝑇 and the ferromagnetic dynamics away from 𝑇 [38]. Therefore, the \nunphysical divergence of the Gilbert damping parameter at 𝑇 is absent in our analysis. \nOur results, namely the insensitivity of damping to the compens ation condition and \nits low value, have important implications not only for fundame ntal physics but also for \ntechnological applications. From the viewpoint of fundamental p hysics, nearly constant \ndamping across 𝑇 indicates that the damping is almost independent of the total angular \nmomentum and is mostly determined by electron spin scattering n ear the Fermi level. \nSpecifically, our results suggest that the 4f electrons of RE e lements, which lie in a band far \nbelow the Fermi level, do not play an important role in the mag netic damping of RE-TM \nferrimagnets, whereas the 3d and 4s bands of TM elements have a  governing role in magnetic \ndamping. This result is consistent with the recently reported t heoretical and experimental \nresults in FeCo alloys [20]. From the viewpoint of practical ap plication, we note that the \nestimated damping of 𝛼ൌ7.2 × 10-3 is the upper limit, as the damping estimated from DW 10 \n dynamics is usually overestimated due to disorders [53]. The ob tained value of the Gilbert \ndamping parameter is consistent with our preliminary ferromagne t i c  r e s o n a n c e  ( F M R )  \nmeasurements. The experimental results from FMR measurements an d the corresponding \ntheoretical analysis will be publ ished elsewhere. This low valu e of the Gilbert damping \nparameter suggests that ferrimagne ts can serve as versatile pla t f o r m s  f o r  l o w - d i s s i p a t i o n  \nhigh-speed magnetic devices such as spin-transfer-torque magnet ic random-access memory \nand terahertz magnetic oscillators. \nIn conclusion, we investigated the field-driven magnetic DW mot ion in ferrimagnetic \nG d F e C o  a l l o y s  o v e r  a  w i d e  r a n g e  o f  t e m p e r a t u r e s  a c r o s s  𝑇 and extracted the Gilbert \ndamping parameter from the DW mobility. The estimated Gilbert d amping parameter was as \nlow as 7.2 × 10-3 and almost constant over the temperature range including 𝑇, which is in \nstark contrast to the previous prediction in that the Gilbert d amping parameter would diverge \nat 𝑇 due to the vanishing total angular momentum. 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Shinjo, Science 284, 468 \n(1999). \n[49] In this Letter , the parameters such as the spin angular mo mentum density 𝑠 r e p r e s e n t  \nthe magnitudes of the quantities.  Their directions are separate ly handled through the signs in \nthe equations of motion. \n[50] C. Kittel, Phys. Rev. 76, 743 (1949). \n[51] G. G. Scott, Rev. Mod. Phys. 34, 102 (1962). \n[52] B. I. Min and Y.-R. Jang, J. Phys. Condens. Matter 3, 5131 (1991). \n[53] H. Min, R. D. McMichael, M. J. Donahue, J. Miltat, and M. D. Stiles, Phys. Rev. Lett. \n104, 217201 (2010). \n   16 \n Figure Captions \nFigure 1(a) Schematic illustration of the GdFeCo microwire devi ce. (b) The averaged DW \nvelocity 〈𝑣〉 as a function of the perpendicular magnetic field 𝜇𝐻 for several temperatures \n𝑇∗ (202, 222, 242, 262, and 282 K). The dots indicate the best li n e a r  f i t s .  ( c )  T h e  D W  \nmobility 𝜇 as a function of 𝑇∗ at several current densities ( |𝐽|ൌ1.3, 1.7, and 2.0 ×1010 \nA/m2). \nFigure 2 The temperature-dependent (a) DW mobility 𝜇, (b) sub-magnetic moment 𝑀, and \n(c) sub-angular momentum  𝑠. Here, we use the relative temperature defined as ∆𝑇 ൌ 𝑇∗െ\n𝑇. (d) The Gilbert damping parameter 𝛼േ as a function of ∆𝑇. Here, we use 𝜆ൌ15 nm for \nproper solutions of Eq. (3). \nFigure 3 The resultant Gil bert damping parameter 𝛼 in all tested temperature ranges. \n 17 \n Acknowledgements  \nThis work was supported by the JSPS KAKENHI (Grant Numbers 15H0 5702, 26103002, and \n26103004), Collaborative Research Program of the Institute for Chemical Research, Kyoto \nUniversity, and R & D project for ICT Key Technology of MEXT fr om the Japan Society for \nthe Promotion of Science (JSPS). This work was partly supported  by The Cooperative \nResearch Project Program of the Research Institute of Electrica l Communication, Tohoku \nUniversity. D.H.K. was supported as an Overseas Researcher unde r the Postdoctoral \nFellowship of JSPS (Grant Number P16314). S.H.O. and K.J.L. wer e supported by the \nNational Research Foundation of Korea (NRF-2015M3D1A1070465, 20 17R1A2B2006119) \nand the KIST Institutional Program (Project No. 2V05750). S.K.K . was supported by the \nArmy Research Office under Contract No. W911NF-14-1-0016. K.J.K . was supported by the \nNational Research Foundation of Korea (NRF) grant funded by the  Korea Government \n(MSIP) (No. 2017R1C1B2009686).  \nCompeting financial interests \nThe authors declare no competing financial interests. 200 225 250 275 3000.00.51.01.52.0\n 1.3\n1.7\n2.0\n [104 m/sT]\nT* [K]J [1010 A/m2]0 50 100 1500.00.51.01.5\n  202\n 222\n 242\n 262\n 282<v> [km/s]\n0H [mT]T* [K]\nFigure 1b\nca\nWriting line\n\tܫ\nܸ\nߤܪ\ny xz-60 -40 -20 0 20 40 60 801.52.02.53.0 s1\n s2s [10-6 Js/m3]\nT [K]-60 -40 -20 0 20 40 60 800.00.51.01.52.0\n [104 m/sT]\nT [K]\n-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \n +\n -\nT [K]T1T2-60 -40 -20 0 20 40 60 800.30.40.50.6  M1\n M2M [MA/m]\nT [K]a\nb\nc\nd\nFigure 2Figure 3-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \nT [K]"    },    {        "title": "1807.04977v1.Gilbert_damping_of_high_anisotropy_Co_Pt_multilayers.pdf",        "content": "Gilbert damping of high anisotropy Co/Pt multilayers\nThibaut Devolder\u0003\nCentre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,\nUniversité Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France\nS. Couet, J. Swerts, and G. S. Kar\nimec, Kapeldreef 75, 3001 Heverlee, Belgium\n(Dated: June 16, 2021)\nUsing broadband ferromagnetic resonance, we measure the damping parameter of [Co(5 Å)/Pt(3 Å)] \u00026mul-\ntilayers whose growth was optimized to maximize the perpendicular anisotropy. Structural characterizations in-\ndicate abrupt interfaces essentially free of intermixing despite the miscible character of Co and Pt. Gilbert damp-\ning parameters as low as 0.021 can be obtained despite a magneto-crystalline anisotropy as large as 106J/m3.\nThe inhomogeneous broadening accounts for part of the ferromagnetic resonance linewidth, indicating some\nstructural disorder leading to a equivalent 20 mT of inhomogenity of the effective field. The unexpectedly rel-\natively low damping factor indicates that the presence of the Pt heavy metal within the multilayer may not be\ndetrimental to the damping provided that intermixing is avoided at the Co/Pt interfaces.\nI. INTRODUCTION\nThanks to their large perpendicular magnetic anisotropy,\ntheir confortable magneto-optical signals and their easy\ngrowth by physical vapor deposition1, the [Co/Pt] multilay-\ners are one of the most popular system in spintronics. Early\nin spintronics history this model system was used to study\nthe physics of domain wall propagation2, for the develop-\nment of advanced patterning techniques3and for the assess-\nment of micromagnetic theories4. More recently they have\nbeen extensively used as high quality fixed layers in per-\npendicularly magnetized tunnel junctions, in particular in the\nmost advanced prototypes of spin-transfer-torque magnetic\nrandom access memories memories5. Despite the widespread\nuse of Co/Pt multilayers, their high frequency properties,\nand in particular their Gilbert damping parameter remains\nlargely debated with experimental values that can differ by\norders of magnitude from60.02 to 20 times larger7and the-\noretical calculations from circa 0.035 in Co 50Pt50alloys8\nto slightly smaller or substantially larger values in multilay-\ners made of chemically pure layers9. Direct measurements\nby conventional ferromagnetic resonance (FMR) are scarce\nas the high anisotropy of the material pushes the FMR fre-\nquencies far above610 GHz and results in a correlatively\nlow permeability that challenges the sensitivity of commer-\ncial FMR instruments10. As a result most of the measure-\nments of the damping of Co/Pt systems were made by all-\noptical techniques11,12in small intervals of applied fields. Un-\nfortunately this technique requires the static magnetization to\nbe tilted away from the out-of-plane axis and this tilt ren-\nders difficult the estimation of the contribution of the ma-\nterial disorder to the observed FMR linewidth using the es-\ntablished protocols13; this is problematic since in Co-Pt sys-\ntems there contributions of inhomogeneity line broadening\nand two-magnon scattering by the structural disorder (rough-\nness, interdiffusion, granularity,...) are often large14,15.\nIt is noticeable that past reports on the damping of Co/Pt\nsystems concluded that it ought to be remeasured in sam-\nples with atomically flat interfaces11. Besides, this measure-\nment should be done in out-of-plane applied field since thiseases the separation of the Gilbert damping contribution to\nthe linewidth from the contribution of structural disorder16.\nIn this paper, we measure the damping parameter of [Co(5\nÅ)/Pt(3 Å)]\u00026multilayers whose growth was optimized to\nmaximize perpendicular anisotropy anisotropy. The sputter-\ndeposition is performed at an extremely low17Argon pressure\nin remote plasma conditions which enables very abrupt inter-\nfaces that are essentially free of intermixing. We show that in\ncontrast to common thinking, the Gilbert damping parameter\nof Co/Pt multilayers can be low; its effective value is 0.021\nbut it still likely16includes contributions from spin-pumping\nthat our protocol can unfortunately not suppress.\nII. EXPERIMENTAL\nOur objective is to report the high frequency properties of\nCo/Pt multilayers that were optimized for high anisotropy.\nThe multilayer is grown by sputter-deposition on a Ru (50 Å)\nbuffer and capped with a Ru(70 Å)/Ta(70 Å)/Ru(100 Å)/Ta(10\nÅ, cap) sequence (bottom to top order). The Ru buffer was\nchosen because it does not mix with Co-based multilayers\neven under tough annealing conditions18. The stacks were\ndeposited by physical vapor deposition in a Canon-Anelva\nEC7800 300 mm system on oxidized silicon substrates at\nroom temperature. The Argon plasma pressure is kept at 0.02\nPa, i.e. substantially lower than the usual conditions of 0.1-0.5\nPa used in typical deposition machines17. As this multilayer\nis meant to be the reference layer of bottom-pinned magnetic\ntunnel junctions, in some samples (fig. 1) the non-magnetic\ncap is replaced the following sequence: Ta cap / Fe 60Co20B20\n/ MgO / Fe 60Co20B20/ Ta / [Co(5 Å)/Pt(3 Å)] \u00024/ Ru sim-\nilar to as in ref. 19 and 20 to form a bottom-pinned mag-\nnetic tunnel junction with properties designed for spin-torque\napplications21. All samples were annealed at 300\u000eC for 30\nminutes in an out-of-plane field of 1 T.arXiv:1807.04977v1  [cond-mat.mtrl-sci]  13 Jul 20182\nRu[Co5Å/Pt3Å]×6 RuMgOFeCoBFeCoBTaRuTa[Co5Å/Pt3Å]×4Co5Å(a)(b)(c)\nFIG. 1. (Color online). Structure and anisotropy of a Co-Pt multi-\nlayer. (a) Transmission Electron Micrograph of a magnetic tunnel\njunction that embodies our Co/Pt as hard multilayer at the bottom of\nthe reference synthetic antiferromagnet, similar to that of ref. 21. (b)\nEasy axis and (c) hard axis hysteresis loops of the hard multilayer\nwhen covered with Ru(70 Å)/Ta(70 Å)/Ru(100 Å)/Ta(10 Å, cap)\nIII. STRUCTURE\nX-ray reflectivity scans (not shown) indicate Bragg reflex-\nions at 2\u0012= 11 , 22.2 and 33.6 deg., consistent with the mul-\ntilayer periodicity of 8 Å. Consistently, the Pt to Co inter-\nmixing is sufficiently low that well formed 3Å Pt spacers can\nbe seen the Transmission Electron Micrograph after anneal-\ning [Fig. 1(a)]. Almost no roughness is observed throughout\nthe Co/Pt multilayer. We emphasize that this quality of inter-\nfaces is almost equivalent to that obtained in Molecular Beam\nEpitaxy conditions22. Indeed Co and Pt are strongly miscible\nsuch that hyperthermal (high energy) deposition techniques\nlike sputter deposition do not easily yield this low degree of\nintermixing, except when the deposition is conducted under\nsufficiently low plasma pressure in remote plasma conditions,\ni.e. when the substrate-to-target distance is large to avoid di-\nrect plasma exposure to the film being deposited.\nIV . ANISOTROPY\nThe magnetic material properties were measured by vibrat-\ning sample magnetometry (VSM) and Vector Network Ana-\nlyzer ferromagnetic resonance23in both easy (z) and hard axis\n(x) configurations. For VNA-FMR the sample is mechanically\npressed on the surface of a 50 microns wide coplanar waveg-\nuide terminated by an open circuit; data analysis is conducted\nfollowing the methods described in ref. 24. The VSM signal\nindicated a magnetization Ms= 8:5\u0002105kA/m if assuming\na magnetic thickness of 48 Å, i.e. assuming that the [Co(5\nÅ)/Pt(3 Å)]\u00026multilayer can be described as a single mate-\nrial. The loops indicate a perpendicular anisotropy with full\nremanence. The reversal starts at 46.8 mT and completes be-\n/s45/s50 /s45/s49 /s48 /s49 /s50/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48\n/s72/s101/s102/s102\n/s107/s49/s43/s72\n/s107/s50\n/s105/s110/s45/s112/s108/s97/s110/s101/s32\n/s102/s105/s101/s108/s100/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s48/s72/s32/s40/s84/s41/s72/s101/s102/s102\n/s107/s49/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32\n/s102/s105/s101/s108/s100FIG. 2. (Color online). FMR frequencies versus in-plane (cross sym-\nbols) or out-of-plane (square symbols) applied field. The bold lines\nare fits using Eq. 1 and 2, yielding \u00160(Hk1\u0000Ms) = 1:320\u00060:005T\nand\u00160Hk2= 0:120\u00060:015T.\nfore 48 mT with a tail-free square hysteresis loop. Careful\nattempts to demagnetize the sample using an acperpendicular\nfield failed to produce a multidomain state at remanence. This\nindicates that the lowest nucleation field in the whole sample\nis larger that the domain wall propagation field everywhere in\nthe film. This low propagation field indicates qualitatively that\nthe effective anisotropy field is very uniform. The hard axis\nloop indicates an in-plane saturation field of \u00191:3\u00060:1T in\nline with the expectations for such composition3. The round-\ning of the hard axis loop near saturation and its slight hys-\nteretic remanence [Fig. 1(c)] impedes a more precise deduc-\ntion of the anisotropy fields from the sole hard axis loop.\nWe shall instead use the ferromagnetic resonance data\nbecause magnetization eigenfrequencies constitute absolute\nmeasurements of the effective fields acting on the mag-\nnetization. Fig. 2 gathers the measured FMR frequen-\ncies measured for in-plane and out-of-plane applied fields\nfrom -2.5 to 2.5 T. To analyze the microwave susceptibil-\nity data, we assume an energy density that reads E=\n1\n2\u00160Hk1MSsin2\u0012+1\n4\u00160Hk2MSsin4\u0012with\u0012the (suppos-\nedly uniform) angle between the magnetization and the sam-\nple normal. Our convention is that the first and second order\nmagneto-crystalline anisotropy fields Hk1= 2K1=(\u00160MS)\nandHk2= 4K2=(\u00160MS)are positive when they favor per-\npendicular magnetization, i.e. \u0012= 0.\nIn that framework, the ferromagnetic resonance frequencies\nin out-of-plane and in-plane applied fields saturating the mag-\nnetization as:\n!perp=\r0(Hz+Hk1\u0000Ms) (1)\nand\n!in-plane =\r0p\nHx(Hx\u0000Hk1\u0000Hk2+Ms); (2)\nwhere\r0=j\rj\u00160is the gyromagnetic ratio. (For in-plane\nfieldsHxlower thanHx;sat=Hk1\u0000Hk2\u0000Msthe magne-\ntization is tilted. A straightforward energy minimization was3\nused to yield magnetization tilt \u0012that was subsequently in-\njected to a Smit and Beljers equation to yield the FMR fre-\nquency). The best fit to the experimental data is obtained\nfor\u00160(Hk1\u0000Ms) = 1:320\u00060:005 T (corresponding to\nK1= 106J/m3) and\u00160Hk2= 0:120\u00060:015T. Note that\nthe second order anisotropy is small but non negligible such\nthat the effective anisotropy fields deduced from easy and axis\naxis measurements would differ by circa 10% if Hk2was dis-\nregarded.\nV . GILBERT DAMPING\nA. Models\nWe now turn to the analysis of the FMR linewidth (Fig.\n3). As common in FMR, the linewidth comprises an intrin-\nsic Gilbert damping part and an extrinsic additional contribu-\ntion linked to the lateral non uniformity of the local effective\nfieldsHk1\u0000Ms. This can be gathered in a characteristic\nfield \u0001H0measuring the disorder relevant for FMR. In out-\nof-plane field FMR experiments, the proportionality between\neffective fields and resonance frequencies (Eq. 1) allows to\nwrite simply \u0001H0=1\n\r0\u0001!j!!0, and for the perpendicular\nmagnetization we follow the usual convention16and write:\n1\n\r0\u0001!perp= 2\u000b(Hz+Hk1\u0000Ms) + \u0001H0 (3)\nor equivalently \u0001!perp= 2\u000b!perp+\r0\u0001H0.\nFor in-plane magnetization, the intrinsic linewidth above\nthe in-plane saturation field is\n1\n\r0\u0001!Gilbert\nin-plane =\u000b(2Hx\u0000Hk1\u0000Hk2+Ms) (4)\nThe resonance frequency (Eq. 2) is non linear with the ef-\nfective fields such that the non uniformity \u0001H0of the local\neffective fields translates in a linewidth broadening through\nthe term\n1\n\r0\u0001!disorder\nin-plane =d!in-plane\nd(Ms\u0000Hk1)\u0001H0 (5)\nwhere the derivative term ispHx\n2pHx\u0000Hk1\u0000Hk2+Ms. In case of\nfinite disorder, this factor diverges at the spatially-averaged\nin-plane saturation field Hx;sat.\nB. Results\nFor each applied field, the real and imaginary parts of the\ntransverse permeability \u0016(f)were fitted with the one expected\nfor the uniform precession mode25with three free param-\neters: the FMR frequency !FMR=(2\u0019), the FMR linewidth\n\u0001!=(2\u0019))and a scaling (sensitivity) factor common to both\nreal and imaginary parts of \u0016(f)as illustrated in Fig. 3b.When plotting the symmetric lorentzian-shaped imagi-\nnary part of the transverse permeability versus the asymet-\nric lorentzian-shaped real part of the permeability for fre-\nquencies ranging from dcto infinity, a circle of diameter\nMs=[2\u000b(Hz+Hk1\u0000Ms)]should be obtained for a spatially\nuniform sample18. The finite disorder \u0001H0distorts the exper-\nimental imaginary part of the permeability towards a larger\nand more gaussian shape. It can also damp and smoothen the\npositive and negative peaks of the real part of the permeabil-\nity; when the applied field is such that the inhomogeneous\nbroadening is larger than the intrinsic Gilbert linewidth, this\nresults in a visible ellipticity of the polar plot of \u0016(f). In our\nexperimental polar plot of \u0016(f)(Fig. 3a) the deviations from\nperfect circularity are hardly visible which indicates that the\ninhomogeneous broadening is not the dominant contribution\nto the sample FMR linewidth in out-of-plane field conditions.\nTo confirm this point we have plotted in Fig. 3c the de-\npendence of FMR linewidth with FMR frequency for out-of-\nplane applied fields. A linear fit yields \u000b= 0:021\u00060:002and\n\u0001H0\u001940mT. A substantial part of the measured linewidth\nthus still comes from the contribution of the lateral inhomo-\ngeneity of the effective anisotropy field within the film. As a\nresult, low field measurements of the FMR linewidth would\nbe insufficient to disentangle the Gilbert contribution and the\nstructural disorder contributions to the total FMR linewidth.\nThe in-plane applied field FMR linewidth can in principle\nbe used to confirm this estimate of the damping factor. Un-\nfortunately we experience a weak signal to noise ratio in in-\nplane field FMR experiments such that only a crude estimation\nof the linewidth was possible.Within the error bar, it is inde-\npendent from the applied field from 1.7 to 2.5 T (not shown)\nwhich indicates that the disorder still substantially contributes\nto the linewidth even at our maximum achievable field. At\n2.5 T the linewidth was1\n2\u0019\u0001!in-plane\u00193:0\u00060:3GHz. This\nis consistent width the expectations of that would predict 2.2\nGHz of intrinsic contribution (Eq. 4) and 0.4 GHz of intrinsic\ncontribution (Eq. 5).\nVI. DISCUSSION\nWe conclude that the damping of Co/Pt multilayers can be\nof the order of 0.02 even for multilayers with anisotropies\namong the strongest reported (see ref. 26 for a survey of\nthe anisotropy of Co/Pt multilayers). Note that \u000b\u00190:021is\nstill a higher bound, as we are unable to measure and subtract\nthe spin-pumping contribution. Measuring the spin-pumping\ncontribution would require to vary the cap and buffer layer\nthicknesses without affecting the multilayer structure which\nis difficult to achieve. Still, we can conclude that the damp-\ning of Co/Pt multilayers lies in the same range as other high\nanisotropy multilayers like Co/Ni (ref. 18 and 27) and Co/Pd\n(ref. 16) systems.\nThis conclusion is in stark contrast with the common\nthinking7that Co/Pt systems alway s have a large damping.\nThis widespread opinion is based on the standard models\nof magneto-crystalline anisotropy28and damping29that pre-\ndicts that they both scale with the square of the spin-orbit4\n/s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s52/s52 /s52/s54 /s52/s56 /s53/s48 /s53/s50 /s53/s52 /s53/s54/s45/s49/s48/s48/s49/s48/s45/s49/s48 /s48 /s49/s48\n/s45/s49/s48/s48/s72/s97/s108/s102/s32/s70/s77/s82/s32/s108/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s71/s72/s122/s41\n/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s48/s46/s54/s32/s71/s72/s122/s32/s43/s32/s48/s46/s48/s50/s49 /s102/s80/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121/s32/s40/s114/s101/s97/s108/s32/s112/s97/s114/s116/s41/s80/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121/s32\n/s40/s105/s109/s97/s103/s105/s110/s97/s114/s121/s32/s112/s97/s114/s116/s41\n/s73/s109/s97/s103/s105/s110/s97/s114/s121/s32/s48/s46/s48/s52\n/s32/s50/s102/s32 /s102\n/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32/s80/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121/s82/s101/s97/s108\n/s40/s99/s41/s40/s98/s41/s77/s97/s99/s114/s111/s115/s112/s105/s110/s32/s102/s105/s116/s32/s32\n/s40/s97/s41\nFIG. 3. (Color online). Gilbert damping of the Co/Pt multilayer. (a)\nImaginary part versus real part of the permeability for a field of 0.45\nT applied perpendicularly to the plane. The bold lines are theoretical\nmacrospin permeability curves with linewidth parameters (i.e. effec-\ntive damping) of 0.04. (b) Same data but versus frequency. (c): FMR\nhalf linewidth versus FMR frequency. The bold line is a guide to the\neye with a slope \u000b= 0:021and zero frequency intercept of 0.6 GHz.\ncoupling\u0018, which is particularly large in the Pt atoms. We\nemphasize that this expectation of large damping is not sys-\ntematically verified: in studies that make a thorough anal-\nysis of the effects of structural disorder, no correlation wasfound between anisotropy and damping in comparable mate-\nrial systems11,16. Rather, a large correlation was found be-\ntweenHk1and\u0001H0, indicating that when the anisotropy is\nstrong, any local inhomogeneity thereof has a large impact\non the FMR linewidth. Owing to the difficulty of achiev-\ning well-defined Co/Pt interfaces, we believe that past con-\nclusions on the large damping of Co/Pt systems were based\non systems likely to present some intermixing at the inter-\nface; indeed the presence of impurities with large spin-orbit\ncoupling considerably degrades (increases) the damping of a\nmagnetic material30and synchonously degrades (decreases)\nthe magneto-crystalline anisotropy31.\nVII. CONCLUSION\nIn summary, we have studied high anisotropy [Co(5 Å)/Pt(3\nÅ)]\u00026multilayers grown by low pressure remote plasma\nsputter deposition. The deposition conditions were tuned\nto achieve abrupt interfaces with little intermixing. Broad-\nband ferromagnetic resonance was used to measure the first\nand second order uniaxial anisotropy fields. With the mag-\nnetization measured by vibrating sample magnetometry, this\nyields an anisotropy energy of 1MJ/m3. The inhomogeneous\nbroadening accounts for part of the ferromagnetic resonance\nlinewidth, indicating some structural disorder leading to a\nequivalent 40 mT (or equivalently 600 MHz) of inhomogenity\nof the effective field in out-of-plane applied fields. This FMR-\nrelevant inhomogeneity is comparable to the coercivity of 47\nmT. Despite the large anisotropy a Gilbert damping parameter\nas low as 0.021\u00060.002 is obtained. This unexpectedly rela-\ntively low damping factor indicates that the presence of the Pt\nheavy metal within the multilayer can in some condition not\nbe detrimental to the damping. 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Kambersky, Physical Review B 76(2007), 10.1103/Phys-\nRevB.76.134416.\n30J. O. Rantscher, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F.\nEgelhoff, B. B. Maranville, D. Pulugurtha, A. P. Chen, and L. M.\nConnors, Journal of Applied Physics 101, 033911 (2007).\n31T. Devolder, Physical Review B 62, 5794 (2000)."    },    {        "title": "1807.07897v2.Another_view_on_Gilbert_damping_in_two_dimensional_ferromagnets.pdf",        "content": "Another view on Gilbert damping in two-dimensional\nferromagnets\nAnastasiia A. Pervishko1, Mikhail I. Baglai1,2, Olle Eriksson2,3, and Dmitry Yudin1\n1ITMO University, Saint Petersburg 197101, Russia\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75 121 Uppsala, Sweden\n3School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden\nABSTRACT\nA keen interest towards technological implications of spin-orbit driven magnetization dynamics requests a proper theoretical\ndescription, especially in the context of a microscopic framework, to be developed. Indeed, magnetization dynamics is so far\napproached within Landau-Lifshitz-Gilbert equation which characterizes torques on magnetization on purely phenomenological\ngrounds. Particularly, spin-orbit coupling does not respect spin conservation, leading thus to angular momentum transfer to\nlattice and damping as a result. This mechanism is accounted by the Gilbert damping torque which describes relaxation of the\nmagnetization to equilibrium. In this study we work out a microscopic Kubo-St ˇreda formula for the components of the Gilbert\ndamping tensor and apply the elaborated formalism to a two-dimensional Rashba ferromagnet in the weak disorder limit. We\nshow that an exact analytical expression corresponding to the Gilbert damping parameter manifests linear dependence on the\nscattering rate and retains the constant value up to room temperature when no vibrational degrees of freedom are present\nin the system. We argue that the methodology developed in this paper can be safely applied to bilayers made of non- and\nferromagnetic metals, e.g., CoPt.\nIntroduction\nIn spite of being a mature field of research, studying magnetism and spin-dependent phenomena in solids still remains one of\nthe most exciting area in modern condensed matter physics. In fact, enormous progress in technological development over the\nlast few decades is mainly held by the achievements in spintronics and related fields1–11. However the theoretical description of\nmagnetization dynamics is at best accomplished on the level of Landau-Lifshitz-Gilbert (LLG) equation that characterizes\ntorques on the magnetization. In essence, this equation describes the precession of the magnetization, mmm(rrr;t), about the effective\nmagnetic field, HHHeff(rrr;t), created by the localized moments in magnetic materials, and its relaxation to equilibrium. The latter,\nknown as the Gilbert damping torque12, was originally captured in the form ammm\u0002¶tmmm, where the parameter adetermines the\nrelaxation strength, and it was recently shown to originate from a systematic non-relativistic expansion of the Dirac equation13.\nThus, a proper microscopic determination of the damping parameter a(or, the damping tensor in a broad sense) is pivotal to\ncorrectly simulate dynamics of magnetic structures for the use in magnetic storage devices14.\nFrom an experimental viewpoint, the Gilbert damping parameter can be extracted from ferromagnetic resonance linewidth\nmeasurements15–17or established via time-resolved magneto-optical Kerr effect18, 19. In addition, it was clearly demonstrated\nthat in bilayer systems made of a nonmagnetic metal (NM) and a ferromagnet material (FM) the Gilbert damping is drastically\nenhanced as compared to bulk FMs20–24. A strong magnetocrystalline anisotropy, present in CoNi, CoPd, or CoPt, hints\nunambiguously for spin-orbit origin of the intrinsic damping. A first theoretical attempt to explain the Gilbert damping\nenhancement was made in terms of sdexchange model in Ref.25. Within this simple model, magnetic moments associated with\nFM layer transfer angular momentum via interface and finally dissipate. Linear response theory has been further developed\nwithin free electrons model26, 27, while the approach based on scattering matrix analysis has been presented in Refs.28, 29. In\nthe latter scenario spin pumping from FM to NM results in either backscattering of magnetic moments to the FM layer or\ntheir further relaxation in the NM. Furthermore, the alternative method to the evaluation of the damping torque, especially\nin regard of first-principles calculations, employs torque-correlation technique within the breathing Fermi surface model30.\nWhile a direct estimation of spin-relaxation torque from microscopic theory31, or from spin-wave spectrum, obtained on the\nbasis of transverse magnetic field susceptibility32, 33, are also possible. It is worth mentioning that the results of first-principles\ncalculations within torque-correlation model34–38and linear response formalism39, 40reveal good agreement with experimental\ndata for itinerant FMs such as Fe, Co, or Ni and binary alloys.\nLast but not least, an intensified interest towards microscopic foundations of the Gilbert parameter ais mainly attributed\nto the role the damping torque is known to play in magnetization reversal41. In particular, according to the breathing Fermi\nsurface model the damping stems from variations of single-particle energies and consequently a change of the Fermi surfacearXiv:1807.07897v2  [cond-mat.mes-hall]  21 Nov 2018z\nyx\nFM\nNMFigure 1. Schematic representation of the model system: the electrons at the interface of a bilayer, composed of a\nferromagnetic (FM) and a nonmagnetic metal (NM) material, are well described by the Hamiltonian (1). We assume the\nmagnetization of FM layer depicted by the red arrow is aligned along the zaxis.\nshape depending on spin orientation. Without granting any deep insight into the microscopic picture, this model suggests that\nthe damping rate depends linearly on the electron-hole pairs lifetime which are created near the Fermi surface by magnetization\nprecession. In this paper we propose an alternative derivation of the Gilbert damping tensor within a mean-field approach\naccording to which we consider itinerant subsystem in the presence of nonequilibrium classical field mmm(rrr;t). Subject to the\nfunction mmm(rrr;t)is sufficiently smooth and slow on the scales determined by conduction electrons mean free path and scattering\nrate, the induced nonlocal spin polarization can be approached within a linear response, thus providing the damping parameter\ndue to the itinerant subsystem. In the following, we provide the derivation of a Kubo-St ˇreda formula for the components of\nthe Gilbert damping tensor and illustrate our approach for a two-dimensional Rashba ferromagnet, that can be modeled by\nthe interface between NM and FM layers. We argue that our theory can be further applied to identify properly the tensorial\nstructure of the Gilbert damping for more complicated model systems and real materials.\nMicroscopic framework\nConsider a heterostructure made of NM with strong spin-orbit interaction covered by FM layer as shown in Fig. 1, e.g., CoPt.\nIn general FMs belong to the class of strongly correlated systems with partially filled dorforbitals which are responsible\nfor the formation of localized magnetic moments. The latter can be described in terms of a vector field mmm(rrr;t)referred to as\nmagnetization, that in comparison to electronic time and length scales slowly varies and interacts with an itinerant subsystem.\nAt the interface (see Fig. 1) the conduction electrons of NM interact with the localized magnetic moments of FM via a certain\ntype of exchange coupling, sdexchange interaction, so that the Hamiltonian can be written as\nh=p2\n2m+a(sss\u0002ppp)z+sss\u0001MMM(rrr;t)+U(rrr); (1)\nwhere first two terms correspond to the Hamiltonian of conduction electrons, on condition that the two-dimensional momentum\nppp= (px;py) =p(cosj;sinj)specifies electronic states, mis the free electron mass, astands for spin-orbit coupling strength,\nwhile sss= (sx;sy;sz)is the vector of Pauli matrices. The third term in (1) is responsible for sdexchange interaction with the\nexchange field MMM(rrr;t) =Dmmm(rrr;t)aligned in the direction of magnetization and Ddenoting sdexchange coupling strength. We\nhave also included the Gaussian disorder, the last term in Eq. (1), which represents a series of point-like defects, or scatterers,\nhU(rrr)U(rrr0)i= (mt)\u00001d(rrr\u0000rrr0)with the scattering rate t(we set ¯h=1throughout the calculations and recover it for the final\nresults).\nSubject to the norm of the vector jmmm(rrr;t)j=1remains fixed, the magnetization, in broad terms, evolves according to (see,\ne.g., Ref.42),\n¶tmmm=fff\u0002mmm=gHHHeff\u0002mmm+csss\u0002mmm; (2)\nwhere fffcorresponds to so-called spin torques. The first term in fffdescribes precession around the effective magnetic field\nHHHeffcreated by the localized moments of FM, whereas the second term in (2) is determined by nonequilibrium spin density of\nconduction electrons of NM at the interface, sss(rrr;t). It is worth mentioning that in Eq. (2) the parameter gis the gyromagnetic\nratio, while c= (gmB=¯h)2m0=dis related to the electron g\u0000factor ( g=2), the thickness of a nonmagnetic layer d, with mBand\nm0standing for Bohr magneton and vacuum permeability respectively. Knowing the lesser Green’s function, G<(rrrt;rrrt), one\ncan easily evaluate nonequilibrium spin density of conduction electrons induced by slow variation of magnetization orientation,\nsm(rrr;t) =\u0000i\n2Tr\u0002\nsmG<(rrrt;rrrt)\u0003\n=Qmn¶tmn+:::; (3)\n2/8where summation over repeated indexes is assumed ( m;n=x;y;z). The lesser Green’s function of conduction electrons\ncan be represented as G<=\u0000\nGK\u0000GR+GA\u0001\n=2, where GK,GR,GAare Keldysh, retarded, and advanced Green’s functions\nrespectively.\nKubo-St ˇreda formula\nWe further proceed with evaluating Qmnin Eq. (3) that describes the contribution to the Gilbert damping due to conduction\nelectrons. In the Hamiltonian (1) we assume slow dynamics of the magnetization, such that approximation MMM(rrr;t)\u0019\nMMM+(t\u0000t0)¶tMMMwith MMM=MMM(rrr;t0)is supposed to be hold with high accuracy,\nH=p2\n2m+a(sss\u0002ppp)z+sss\u0001MMM+U(rrr)+(t\u0000t0)sss\u0001¶tMMM; (4)\nwhere first four terms in the right hand side of Eq. (4) can be grouped into the Hamiltonian of a bare system, H0, which\ncoincides with that of Eq. (1), provided by the static magnetization configuration MMM. In addition, the expression (4) includes the\ntime-dependent term V(t)explicitly, as the last term. In the following analysis we deal with this in a perturbative manner. In\nparticular, the first order correction to the Green’s function of a bare system induced by V(t)is,\ndG(t1;t2) =Z\nCKdtZd2p\n(2p)2gppp(t1;t)V(t)gppp(t;t2); (5)\nwhere the integral in time domain is taken along a Keldysh contour, while gppp(t1;t2) =gppp(t1\u0000t2)[the latter accounts for the\nfact that in equilibrium correlation functions are determined by the relative time t1\u0000t2] stands for the Green’s function of the\nbare system with the Hamiltonian H0in momentum representation. In particular, for the lesser Green’s function at coinciding\ntime arguments t1=t2\u0011t0, which is needed to evaluate (3), one can write down,\ndG<(t0;t0) =i\n2¥Z\n\u0000¥de\n2pZd2p\n(2p)2n\ngR\npppsm¶g<\nppp\n¶e\u0000¶gR\nppp\n¶esmg<\nppp+g<\npppsm¶gA\nppp\n¶e\u0000¶g<\nppp\n¶esmgA\npppo\n¶tMm; (6)\nwhere m=x;y;z, while gR,gA, and g<are the bare retarded, advanced, and lesser Green’s functions respectively. To derive the\nexpression (6) we made use of Fourier transformation gppp=Rd(t1\u0000t2)gppp(t1\u0000t2)eie(t1\u0000t2)and integration by parts.\nTo finally close up the derivation we employ the fluctuation-dissipation theorem according to which g<(e) = [gA(e)\u0000\ngR(e)]f(e), where f(e) = [eb(e\u0000m)+1]\u00001stands for the Fermi-Dirac distribution with the Fermi energy m. Thus, nonequi-\nlibrium spin density of conduction electrons (3) within linear response theory is determined by Qmn=Q(1)\nmn+Q(2)\nmn, where\nQ(1)\nmn=1\n4Trh\nsm¥Z\n\u0000¥de\n2pZd2p\n(2p)2n¶gR\nppp\n¶esngR\nppp\u0000gR\npppsn¶gR\nppp\n¶e+gA\npppsn¶gA\nppp\n¶e\u0000¶gA\nppp\n¶esngA\npppo\nf(e)i\n; (7)\nwhich involves the integration over the whole Fermi sea, and\nQ(2)\nmn=1\n4Trh\nsm¥Z\n\u0000¥de\n2p\u0012\n\u0000¶f(e)\n¶e\u0013Zd2p\n(2p)2n\ngR\npppsngR\nppp+gA\npppsngA\nppp\u00002gR\npppsngA\npppoi\n; (8)\nwhich selects the integration in the vicinity of the Fermi level. Generally, the form of Qmnbelongs to the class of Kubo-St ˇreda\nformula, and, in essence, represents the response to the external stimulus in the form of ¶tMn. We can immediately establish a\nquantitative agreement between the result given by Eq. (8) and the previous studies within a Kubo formalism40, 43–46which allow\na direct estimation within the framework of disordered alloys. Formally, the expression (7) corresponds to the so-called St ˇreda\ncontribution. Such a term was originally identified in Ref.47when studying quantum-mechanical conductivity. Notably, in\nEq. (7) each term represents the product of either retarded or advanced Green’s functions. In this case the poles of the integrand\nfunction are positioned on the same side of imaginary plane, making disorder correction smaller in the weak disorder limit (see,\ne.g., Ref.48). Meanwhile, having no classical analog this contribution appears to be important enough when the spectrum of the\nsystem is gapped and the Fermi energy is placed exactly in the gap47. It is worth mentioning that the contribution due to Eq. (7)\nhas never been discussed in this context before. In the meantime, Kubo-St ˇreda expression for the components of the Gilbert\ndamping tensor has been addressed from the perspective of first-principles calculations49and current-induced torques50.\n3/8Results and discussion\nLet us apply the formalism developed in the previous section to a prototypical model: we work out the Gilbert damping tensor\nfor a Rashba ferromagnet with the magnetization mmm=zzzaligned along the zaxis. In the limit of weak disorder the Green’s\nfunction of a bare system can be expressed as\ngR\nppp(e) =\u0000\ne\u0000H0\u0000SR\u0001\u00001=e\u0000eppp+id+a(sss\u0002ppp)z+(D+ih)sz\n(e\u0000eppp+id)2\u0000a2p2\u0000(D+ih)2; (9)\nwhere eppp=p2=(2m)is the electron kinetic energy. We put the self-energy SRdue to scattering off scalar impurities into Eq. (9),\nwhich is determined from SR=\u0000i(d\u0000hsz)(see, e.g., Ref.51). In particular, for jej>jDjwe can establish that d=1=(2t)\nandh=0 in the weak disorder regime to the leading order.\nWithout loss of generality, in the following we restrict the discussion to the regime m>jDj, which is typically satisfied with\nhigh accuracy in experiments. As previously discussed, the contribution owing to the Fermi sea, Eq. (7), can in some cases be\nignored, while doing the momentum integral in Eq. (8) results in,\n1\nmtZd2p\n(2p)2gR\nppp(e)sssgA\nppp(e) =D2\nD2+2ersss+Dd\nD2+2er(sss\u0002zzz)+D2\u0000er\nD2+2er(sss\u0002zzz)\u0002zzz; (10)\nwhere r=ma2. Thus, thanks to the factor of delta function d(e\u0000m) =\u0000¶f(e)=¶e, to estimate Q(2)\nmnat zero temperature one\nshould put e=min Eq. (10). As a result, we obtain,\nQ(2)\nmn=\u00001\n4pm\nD2+2mr0\n@2tmr D 0\n\u0000D 2tmr 0\n0 0 2 tD21\nA: (11)\nMeanwhile, to properly account the correlation functions which appear when averaging over disorder configuration one has\nto evaluate the so-called vertex corrections, which from a physical viewpoint makes a distinction between disorder averaged\nproduct of two Green’s function, hgRsngAidis, and the product of two disorder averaged Green’s functions, hgRidissnhgAidis, in\nEq. (8). Thus, we further proceed with identifying the vertex part by collecting the terms linear in dexclusively,\nGGGs=Asss+B(sss\u0002zzz)+C(sss\u0002zzz)\u0002zzz; (12)\nprovided A=1+D2=(2er),B= (D2+2er)Dd=(D2+er)2, and C=D2=(2er)\u0000er=(D2+er). To complete our derivation\nwe should replace snin Eq. (8) by Gs\nnand with the aid of Eq. (10) we finally derive at e=m,\nQ(2)\nmn=0\n@Qxx Qxy 0\n\u0000QxyQxx\n0 0\u0000mtD2=(4pmr)1\nA: (13)\nWe defined Qxx=\u0000mtmr=[2p(D2+mr)]andQxy=\u0000mD(D2+2mr)=[4p(D2+mr)2], which unambiguously reveals that\naccount of vertex correction substantially modifies the results of the calculations. With the help of Eqs. (3), (11), and (13) we\ncan write down LLG equation. Slight deviation from collinear configurations are determined by xandycomponents ( mxand\nmyrespectively, so that jmxj;jmyj\u001c1). The expressions (11) and (13) immediately suggest that the Gilbert damping at the\ninterface is a scalar, aG,\n¶tmmm=˜gHHHeff\u0002mmm+aGmmm\u0002¶tmmm; (14)\nwhere the renormalized gyromagnetic ratio and the damping parameter are,\n˜g=g\n1+cDQxy;aG=\u0000cDQxx\n1+cDQxy\u0019\u0000cDQxx: (15)\nIn the latter case we make use of the fact that mc\u001c1for the NM thickness d\u0018100mm — 100 nm. In Eq. (14) we have\nredefined the gyromagnetic ratio g, but we might have renormalized the magnetization instead. From physical perspective,\nthis implies the fraction of conduction electrons which become associated with the localized moment owing to sdexchange\ninteraction. With no vertex correction included one obtains\naG=mc\n2p¯htmrD\nD2+2mr; (16)\n4/8t=1ns\nt=10ns\nD=0.2meV\nD=0.3meV\nD=1meV\n501001502002503000.0000.0010.0020.0030.004\nT,KaGFigure 2. Gilbert damping, obtained from numerical integration of Eq. (8), shows almost no temperature dependence\nassociated with thermal redistribution of conduction electrons. Dashed lines are plotted for D=1meV for t=1andt=10ns,\nwhereas solid lines stand for D=0:2, 0:3, and 1 meV for t=100 ns.\nwhile taking account of vertex correction gives rise to a different result,\naG=mc\n2p¯htmrD\nD2+mr: (17)\nTo provide a quantitative estimate of how large the St ˇreda contribution in the weak disorder limit is, on condition that m>jDj,\nwe work out Q(1)\nmn. Using ¶gR=A(e)=¶e=\u0000[gR=A(e)]2and the fact that trace is invariant under cyclic permuattaions we conclude\nthat only off-diagonal components m6=ncontribute. While the direct evaluation results in Q(1)\nxy=3mD=[2(D2+2mr)]in the\nclean limit. It has been demonstrated that including scattering rates dandhdoes not qualitatively change the results, leading to\nsome smearing only52.\nInterestingly, within the range of applicability of theory developed in this paper, the results of both Eqs. (16) and (17)\ndepend linearly on scattering rate, being thus in qualitative agreement with the breathing Fermi surface model. Meanwhile, the\nlatter does not yield any connection to the microscopic parameters (see, e.g., Ref.53for more details). To provide with some\nquantitative estimations in our simulations we utilize the following set of parameters. Typically, experimental studies based on\nhyperfine field measurements equipped with DFT calculations54reveal the sdStoner interaction to be of the order of 0.2 eV ,\nwhile the induced magnetization of s-derived states equals 0.002–0.05 (measured in the units of Bohr magneton, mB). Thus,\nthe parameter of sdexchange splitting, appropriate for our model, is D\u00180.2–1 meV . In addition, according to first-principles\nsimulations we choose the Fermi energy m\u00183 eV . The results of numerical integration of (8) are presented in Fig. 2 for several\nchoices of sdexchange and scattering rates, t. The calculations reveal almost no temperature dependence in the region up to\nroom temperature for any choice of parameters, which is associated with the fact that the dominant contribution comes from the\nintegration in a tiny region of the Fermi energy. Fig. 2 also reveal a non-negligible dependence on the damping parameter with\nrespect to both Dandt, which illustrates that a tailored search for materials with specific damping parameter needs to address\nboth the sdexchange interaction as well as the scattering rate. From the theoretical perspective, the results shown in Fig. 2\ncorrespond to the case of non-interacting electrons with no electron-phonon coupling included. Thus, the thermal effects are\naccounted only via temperature-induced broadening which does not show up for m>jDj.\nConclusions\nIn this paper we proposed an alternative derivation of the Gilbert damping tensor within a generalized Kubo-St ˇreda formula.\nWe established the contribution stemming from Eq. (7) which was missing in the previous analysis within the linear response\ntheory. In spite of being of the order of (mt)\u00001and, thus, negligible in the weak disorder limit developed in the paper, it should\nbe properly worked out when dealing with more complicated systems, e.g., gapped materials such as iron garnets (certain half\nmetallic Heusler compounds). For a model system, represented by a Rashba ferromagnet, we directly evaluated the Gilbert\ndamping parameter and explored its behaviour associated with the temperature-dependent Fermi-Dirac distribution. In essence,\nthe obtained results extend the previous studies within linear response theory and can be further utilized in first-principles\ncalculations. We believe our results will be of interest in the rapidly growing fields of spintronics and magnonics.\n5/8References\n1.Žuti´c, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323 (2004).\n2.Bader, S. D. & Parkin, S. S. P. 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Rev.\nLett.117, 046601 (2016).\n52.Nunner, T. S., et al. Anomalous Hall effect in a two-dimensional electron gas. Phys. Rev. B 76, 235312 (2007).\n53.Eriksson, O., Bergman, A., Bergqvist, L. & Hellsvik, J. Atomistic spin dynamics: Foundations and applications (Oxford\nUniversity Press, Oxford, 2017).\n54.Brooks, M. S. S. & Johansson, B. Exchange integral matrices and cohesive energies of transition metal atoms. J. Phys. F:\nMet. Phys. 13, L197 (1983).\n7/8Acknowledgements\nA.A.P. acknowledges the support from the Russian Science Foundation Project No. 18-72-00058. O.E. acknowledges support\nfrom eSSENCE, the Swedish Research Council (VR), the foundation for strategic research (SSF) and the Knut and Alice\nWallenberg foundation (KAW). D.Y . acknowledges the support from the Russian Science Foundation Project No. 17-12-01359.\nAuthor contributions statement\nD.Y . conceived the idea of the paper and contributed to the theory. A.A.P. wrote the main manuscript text, performed numerical\nanalysis and prepared figures 1-2. M.I.B. and O.E. contributed to the theory. All authors reviewed the manuscript.\nAdditional information\nCompeting interests The authors declare no competing interests.\n8/8"    },    {        "title": "1807.11808v3.Comparative_study_of_methodologies_to_compute_the_intrinsic_Gilbert_damping__interrelations__validity_and_physical_consequences.pdf",        "content": "Comparative study of methodologies to compute the intrinsic Gilbert damping:\ninterrelations, validity and physical consequences\nFilipe S. M. Guimar~ aes,\u0003J. R. Suckert, Jonathan Chico, Juba Bouaziz, Manuel dos Santos Dias, and Samir Lounis\nPeter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, 52425 J ulich, Germany\n(Dated: December 20, 2018)\nRelaxation e\u000bects are of primary importance in the description of magnetic excitations, leading\nto a myriad of methods addressing the phenomenological damping parameters. In this work, we\nconsider several well-established forms of calculating the intrinsic Gilbert damping within a uni\fed\ntheoretical framework, mapping out their connections and the approximations required to derive\neach formula. This scheme enables a direct comparison of the di\u000berent methods on the same footing\nand a consistent evaluation of their range of validity. Most methods lead to very similar results for\nthe bulk ferromagnets Fe, Co and Ni, due to the low spin-orbit interaction strength and the absence\nof the spin pumping mechanism. The e\u000bects of inhomogeneities, temperature and other sources of\n\fnite electronic lifetime are often accounted for by an empirical broadening of the electronic energy\nlevels. We show that the contribution to the damping introduced by this broadening is additive, and\nso can be extracted by comparing the results of the calculations performed with and without spin-\norbit interaction. Starting from simulated ferromagnetic resonance spectra based on the underlying\nelectronic structure, we unambiguously demonstrate that the damping parameter obtained within\nthe constant broadening approximation diverges for three-dimensional bulk magnets in the clean\nlimit, while it remains \fnite for monolayers. Our work puts into perspective the several methods\navailable to describe and compute the Gilbert damping, building a solid foundation for future\ninvestigations of magnetic relaxation e\u000bects in any kind of material.\nI. INTRODUCTION\nDynamical processes lie at the core of magnetic manip-\nulation. From the torques acting on the magnetic mo-\nments to how fast they relax back to their equilibrium\norientations, a material-speci\fc time-dependent theory\nis essential to describe and predict their behavior. In\nmost cases, the description of the time evolution of the\nmagnetization is done via micromagnetics1or atomistic\nspin dynamics (ASD)2,3approaches, in which the mag-\nnetization is considered either as a classical continuous\nvector \feld or as individual 3D vectors on a discrete\nlattice, respectively. They have been successfully used\nto describe a plethora of magnetic phenomena, ranging\nfrom spin waves in low dimensional magnets4, domain\nwalls5and skyrmion6dynamics to thermal stability of\nmagnetic textures7. These approaches model the mag-\nnetization dynamics via a phenomenological equation of\nmotion that contains both precessional and relaxation\nterms.\nA \frst attempt to address these processes was per-\nformed by Landau and Lifshitz (LL), by considering\na Larmor-like precessional torque and adding to it a\n(weaker) damping term of relativistic origin8. Since its\nphenomenological inception in 1935, the precise nature\nof the relaxation processes has been a source of intense\ndebate. In particular, the original LL formulation was\nfound to not properly describe situations in which the\ndamping was large. This problem was addressed by\nGilbert, who introduced a Rayleigh-like dissipation term\ninto the magnetic Lagrangian, thus obtaining the now-ubiquitous Landau-Lifshitz-Gilbert (LLG) equation9,\ndM\ndt=\u0000\rM\u0002B+\u000b\nMM\u0002dM\ndt\n=\u0000e\rM\u0002B\u0000\u000be\r\nMM\u0002(M\u0002B):(1)\nwhere\r >0 is the gyromagnetic factor, Mis the (spin)\nmagnetic moment, Bis the time-dependent e\u000bective\nmagnetic \feld acting on M, and\u000bis the scalar damping\nparameter named after Gilbert. The upper form of the\nLLG equation is due to Gilbert, and the lower one shows\nthat it is equivalent to a LL equation with a renormalized\ngyromagnetic factor, e\r=\r=(1 +\u000b2). The \frst term in\nthe right-hand side of Eq. (1) describes the precession of\nthe magnetic moments around the e\u000bective \feld, while\nthe second term is the Gilbert damping one, that de-\nscribes the relaxation of the magnetic moments towards\nB. This equation corrects the previously mentioned issue\nfor large values of \u000b, for which the original LL equation\nis expected to fail10,11.\nThe ferromagnetic resonance (FMR) technique is one\nof the most common procedures to probe magnetiza-\ntion dynamics12, in which the damping parameter is re-\nlated to the linewidth of the obtained spectra13. Al-\nthough many measurements have been carried out in bulk\nmaterials12,14{18, their description at low temperatures is\nstill controversial19{22. This can be attributed to the dif-\nferent intrinsic and extrinsic mechanisms that can con-\ntribute to the relaxation processes23{36. When varying\nthe temperature, two distinct regimes could be identi-\n\fed in the measured relaxation parameters37. For high\ntemperatures, a proportionality between the linewidth\nand the temperature was observed in most of the exper-arXiv:1807.11808v3  [cond-mat.mes-hall]  19 Dec 20182\niments. It was called resistivity-like, due to the simi-\nlarity with the temperature dependence of this quantity.\nA conductivity-like regime (linewidth inversely propor-\ntional to the temperature) was identi\fed at low temper-\natures for certain materials such as Ni15,17, but not for\nFe18,38. It was also seen that di\u000berent concentrations\nof impurities a\u000bected this low-temperature regime, even\nsuppressing it altogether16.\nFrom the theoretical point-of-view, the calculation of\nthe Gilbert parameter is a challenging problem due to\nthe many di\u000berent mechanisms that might be at play for\na given material39,40. Perhaps this is why most of the\ntheoretical approaches have focused on contributions to\nthe damping from electronic origin. The ultimate goal\nthen becomes the development of a predictive theory of\nthe Gilbert damping parameter, based on the knowledge\nof a realistic electronic structure of the target magnetic\nmaterial. The ongoing e\u000borts to complete this quest\nhave resulted in the development of a myriad of tech-\nniques21,22,37,41{43. Comparisons between a few of these\napproaches are available44,45, including experimental val-\nidation of some methods24,46, but a complete picture is\nstill lacking.\nWe clarify this subject by addressing most of the well-\nestablished methods to calculate the Gilbert damping\nfrom \frst principles. First, we connect the many dif-\nferent formulas, highlighting the approximations made\nin each step of their derivations, determining what con-\ntributions to the damping they contain, and establish-\ning their range of validity. These are schematically illus-\ntrated in Fig. 1. Second, we select a few approaches and\nevaluate the Gilbert damping within a uni\fed and con-\nsistent framework, making use of a multi-orbital tight-\nbinding theory based on \frst-principles electronic struc-\nture calculations. FMR simulations and the mapping of\nthe slope of the inverse susceptibility are used to bench-\nmark the torque correlation methods based on the ex-\nchange and spin-orbit torques. We apply these di\u000berent\ntechniques to bulk and monolayers of transition metals\n(Fe, Ni and Co), for which the spin pumping mecha-\nnism is not present and only the spin-orbit interaction\n(SOI) contributes to the relaxation. Disorder and tem-\nperature e\u000bects are included by an empirical broadening\nof the electronic energy levels37,43,47,48. Third, we engage\na longstanding question regarding the behavior of the\ndamping in the low-temperature and low-disorder limits:\nshould the intrinsic contribution to the Gilbert damping\ndiverge for clean systems? Our results using the con-\nstant broadening model demonstrate that the divergence\nis present in the clean limit of 3D systems but not of\nthe 2D ones49, which we attest by eliminating the pos-\nsibility of them being caused by numerical convergence\nissues or di\u000berent anisotropy \felds. Our results also in-\ndicate that the limit !!0 is not responsible for the\ndivergence of the intrinsic damping, as it is commonly\nattributed19,37,43,50. Finally, we propose a new way to\nobtain the spin-orbit contribution that excludes the \fc-\ntitious temperature/disorder contribution caused by thearti\fcial broadening51,52: they can be discounted by sub-\ntracting the values of damping calculated without SOI.\nFor bulk systems, this yields the total damping, while in\nlayered materials this method should also discount part\nof the spin pumping contribution. In Ref. 20, where tem-\nperature and disorder are included via a CPA analogy, a\nsimilar arti\fcial increase of \u000bfor high temperatures was\nremoved by including vertex corrections.\nThis work is organized as follows. We start, in Sec. II,\nwith a brief overview of the di\u000berent methods proposed\nin the literature. In Sec. III, we explain the theory used\nto calculate the response functions. We then turn to\nthe distinct theoretical forms of calculating the damping:\nIn Sec. IV, we analyze the di\u000berent approaches related\nto the spin-spin responses, while in Sec. V, the torque\nmethods are explored. We then discuss the obtained re-\nsults and conclude in Sec. VII. The Hamiltonian used in\nthe microscopic theory is given in Appendix A, while the\nanisotropy \felds for the 3D and 2D systems together with\nthe transverse dynamical magnetic susceptibility given\nby the LLG equation are given in Appendix B.\nII. OVERVIEW OF METHODS ADDRESSING\nINTRINSIC GILBERT DAMPING\nWe now focus on the di\u000berent methods to describe\nthe microscopic contributions to the Gilbert parameter,\nwhich encompasses e\u000bects that transfers energy and an-\ngular momentum out of the magnetic system. Within\nthese mechanisms, the relativistic SOI comes to the fore.\nThis is often referred to as the intrinsic contribution to\nthe damping, and was \frst identi\fed by Landau and Lif-\nshitz8. The origin of this damping mechanism lies in\nthe non-hermiticity of the relativistic corrections to the\nspin Hamiltonian when the magnetization precesses26,27.\nThe elementary magnetic excitations, called magnons,\ncan also be damped via Stoner excitations (electron-hole\npairs with opposite spins)33,34,53. Alternatively, the con-\nduction electrons can carry spin angular momentum even\nin absence of the SOI. This leads to damping via the spin-\npumping mechanism32,54{56.\nEarly models proposed to describe these processes al-\nready argued that the interaction between the magnetic\nmoments and the conduction electrons is a key ingre-\ndient57. This led to the so-called breathing Fermi sur-\nface model, where the shape of the Fermi surface de-\npends on the orientation of the magnetization through\nthe SOI41. This approach, however, could only capture\nthe conductivity-like regime, which diverges at low tem-\nperatures. The decay of magnons into Stoner excitations\nwas also considered early on39, describing well the exper-\nimental behavior of Ni but also missing the increase at\nlarger temperatures of other materials.\nAn important progress was made by Kambersky us-\ning the spin-orbit torque correlation function to calculate\nthe damping parameter37. This approach captures both\nconductivity- and resistivity-like behaviors, which were3\n↵\u0000↵noSOI\n<latexit sha1_base64=\"QiCQ2bOsL/EV3Sx85lkyZMPneI4=\">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</latexit><latexit sha1_base64=\"QiCQ2bOsL/EV3Sx85lkyZMPneI4=\">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</latexit><latexit sha1_base64=\"QiCQ2bOsL/EV3Sx85lkyZMPneI4=\">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</latexit><latexit sha1_base64=\"QiCQ2bOsL/EV3Sx85lkyZMPneI4=\">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</latexit>\nInversion\nInversion for  low frequencies + sum ruleDyson equation of susceptibilityEquation of motion of susceptibility\nSpectral representation  at T=0KSpectral representation at T=0KLow SOI\nLow spin pumpingNo spin pumpingComputational costsFull SOI Spin pumpingFMR linewidthSlope of inverse susceptibility\nSlope of inverse mean-field susceptibility\nExchange torque correlation at Fermi surfaceSlope of mean-field  SO-torque susceptibility with SOIProduct of spectral functions of opposite spinsSlope of mean-field susceptibilitySpin correlation  at Fermi surfaceSO-torque correlation at Fermi surface with SOISpin response methodsTorque response methods\nEquation of motion of susceptibility + perturbation theory\nDyson equation for Green function + Orbital quenchingSlope of SO-torque susceptibility  without SOI\nSpectral representation at T=0KSlope of mean-field  SO-torque susceptibility without SOISO-torque correlation at Fermi surface  without SOILarge broadening\nFigure 1. Diagram exhibiting the di\u000berent methods investigated in this work, their connections and range of validity. Two\ngroups are identi\fed: one related to the spin susceptibility (spin response methods), including the ferromagnetic resonance\nand the slope of the inverse susceptibility that involves a direct mapping of this quantity to the LLG equation; and the other\nassociated with torque responses, for which approximations need to be taken. The steps indicated by solid lines represent\nexact connections, while dashed arrows involve some kind of approximation. The arrow on the left points from the methods\nthat require less computational power (lower part) to the more demanding ones (upper part). Boxes are hyper-linked with the\nrespective equations and sections.\nshown to originate from the intra- and interband transi-\ntions, respectively58. Recently, this so-called torque cor-\nrelation method was re-obtained using a di\u000berent per-\nturbative approach19, spurring discussions about the va-\nlidity of the obtained results, specially the divergence\ncaused by the intraband transitions22. A similar method\nalso based on torque correlation functions was developed\nusing a scattering theory approach42involving the ex-\nchange torque operator instead of the spin-orbit torque\none. Results obtained in this way also present diverg-\ning behavior in the clean limit of 3D structures20. A\nsimilar scattering framework was used to explain the\nenhancement of the Gilbert damping due to the spin\npumping in thin \flms32. Yet another method relating\nthe Gilbert damping to the spin-spin response was pro-\nposed and related to the existing spin-orbit torque cor-\nrelation method43. It also presented diverging intraband\ncontributions when the parameter used to broaden the\ndelta functions (which mimics the e\u000bect of disorder or\ntemperature) was taken to zero59. The vertex correc-\ntions proposed in Ref. 59 did not remove this diver-\ngence. More recently, Costa and Muniz21showed that\nthe damping parameters of layered structures remain \f-nite in the zero broadening limit, when extracted directly\nfrom the linewidth of the dynamical magnetic suscepti-\nbility (within the random phase approximation).\nSeveral of these methods have been implemented\nfor material-speci\fc calculations20,47,49,58,60{62, and some\napproaches were compared and related43{45,63. In this\nwork, we start our analysis with the uniform frequency-\ndependent spin-spin susceptibility, which is measured ex-\nperimentally in FMR setups, to derive the other expres-\nsions for the damping parameter based on the spin- and\ntorque-correlation methods.\nIII. MICROSCOPIC THEORY\nWe begin by setting the grounds of the theory we use\nto evaluate the di\u000berent formulas of the Gilbert damping\non equal footing. The electronic structure of the system\nis described by the mean-\feld Hamiltonian\n^H=^H0+^Hxc+^HSOI+^Hext: (2)\nThe paramagnetic band structure is described by ^H0\nwithin a multi-orbital tight-binding parametrization. An4\ne\u000bective local electron-electron interaction within the\nmean-\feld approximation is included in ^Hxc, which is re-\nsponsible for ferromagnetism. We also account for spin-\norbit interaction through ^HSOI, and the interaction with\nexternal static magnetic \felds via ^Hext. The explicit\nforms of all the terms are given in Appendix A.\nIn this work, we investigate the di\u000berent methods to\ncompute the intrinsic Gilbert damping utilizing the pro-\ntotypical bulk magnets Fe (bcc), Co (fcc) and Ni (fcc),\nand also square lattices corresponding to the (001) planes\nof those materials, with the same nearest-neighbor dis-\ntances as in its bulk forms.\nFor simplicity, we consider the spin-orbit interaction\nand the local e\u000bective Coulomb interaction only on the\ndorbitals, with U= 1 eV64{66for all systems, and the\nspin-orbit strengths \u0015Fe\nSOI= 54 meV67,\u0015Co\nSOI= 70 meV68,\nand\u0015Ni\nSOI= 133 meV68. The magnetic ground state is\nfound by self-consistently enforcing charge neutrality for\nthe bulk materials69. For the monolayer cases, the total\nnumber of electrons in the atomic plane is decreased to\nn= 7:3 (Fe),n= 8:1 (Co) and n= 9:0 (Ni), as the re-\nmaining charge spills into the vacuum (which we are not\nexplicitly taking into account within the model). The\nground-state properties (spin moment M, orbital mo-\nmentM`and magnetic anisotropy energy K) obtained\nwithin this framework are listed in Table I. The easy axis\nfor all the bulk systems and the monolayers were found\nto be along the (001) direction. We emphasize that our\ngoal is not to achieve the most realistic description of the\nelectronic structure of these materials, but rather to de-\n\fne a concrete set of cases that allow us to compare the\ndi\u000berent methods to compute the Gilbert damping.\nThe magnetic excitations are described using linear re-\nsponse theory, where the transverse magnetic response\n\u000eM(t) due to an oscillatory magnetic \feld \u000eB(t) is given\nby70\n\u000eM\u000b(t) =Z\ndt0\u001f\u000b\f(t\u0000t0)\u000eB\f(t0); (3)\nwhere the convention to sum over repeated indices of\nthe components \f=fx;y;zgis used. This approach\ncaptures the orbitally-averaged part of the response. The\nbulk monolayer\nbcc Fe fcc Co fcc Ni Fe Co Ni\nM(\u0016B) 2.32 1.48 0.43 2.90 1.90 0.96\nM`(\u0016B) 0.072 0.079 0.055 0.28 0.22 0.20\nK(meV) 0.19 0.26 0.084 1.7 1.8 1.9\nTable I. Ground state properties of the investigated systems.\nMandM`denotes the spin and orbital magnetic moments,\nrespectively. Values obtained for \u0011= 1:36 meV. The mag-\nnetic anisotropy constant Kis obtained from the anisotropy\n\felds given by Eq. (B3).magnetic susceptibility is given by\n\u001f\u000b\f(t\u0000t0) =\u00004\n\n^S\u000b(t);^S\f(t0)\u000b\u000b\n= 4i\n\u0002^S\u000b(t);^S\f(t0)\u0003\u000b; (4)\nin atomic units. ^S\u000b(t) is the\u000b-component of the spin op-\nerator. In the \frst line of the equation above, we reprise\nthe double-bracket notation of Zubarev for the spin-spin\nretarded Green function71. This notation is convenient\nfor the derivations of Sec. V.\nFor the crystal symmetries of the systems we are in-\nterested in, it is convenient to work in the circular ba-\nsis^S\u0006=^Sx\u0006i^Sy, which diagonalizes the susceptibil-\nity matrix with components \u001f\u0000+(t) and\u001f+\u0000(t). The\nfrequency- and wave vector-dependent transverse suscep-\ntibility\u001f\u0000+(q;!) is obtained within the random phase\napproximation (RPA), which captures the collective spin\nwave modes21,72, as well as the possible decay into\nparticle-hole excitations (Stoner modes) described by the\nsingle-particle response function \u001f\u0000+\n0(!). Considering\nmatrices that take into account the orbital dependency,\nthe two susceptibilities are related by\n[\u001f\u0000+]\u00001= [\u001f\u0000+\n0]\u00001\u00001\n4U: (5)\nHere,U\u0016\u0017=U\u000e\u0016\u0017is a matrix with the e\u000bective lo-\ncal Coulomb interaction strength within the dorbitals.\nIt plays a similar role to the exchange-correlation ker-\nnel in the adiabatic local-density approximation of time-\ndependent DFT calculations73. We de\fne the transverse\nmagnetic response of the system by summing the suscep-\ntibility matrix over all the dorbitals.\nThe uniform single particle transverse susceptibility\n\u001f\u0000+\n0(!) =\u001f\u0000+\n0(q= 0;!), obtained within the mean-\n\feld approximation, is expressed in terms of the single-\nparticle Green functions as\n\u001f\u0000+\n0;\u0016\u0017(!) =1\n\u0019NX\nkZ\u000fF\nd\"\b\nG\"\"\n\u0016\u0017(k;\"+!) ImG##\n\u0017\u0016(k;\")\n+ ImG\"\"\n\u0016\u0017(k;\")\u0002\nG##\n\u0016\u0017(k;\"\u0000!)\u0003\u0003o\n:\n(6)\nThe sum is over the wave vectors in the \frst Brillouin\nzone, with Ntheir number. The indices \u0016;\u0017represent\norbitals and \u000fFis the Fermi level.\nIn the spirit of many preceding works37,43,47,48, the\ne\u000bect of temperature and disorder is modeled by in-\ntroducing a constant band broadening \u0011on the en-\nergy levels, such that G(!)!G(!+ i\u0011). The imag-\ninary part of the Green function is then de\fned as\nImG\u0016\u0017(!) =1\n2ifG\u0016\u0017(!+ i\u0011)\u0000G\u0016\u0017(!\u0000i\u0011)g. This ap-\nproach attempts to capture all the intrinsic e\u000bects origi-\nnated from the electronic structure of the system.\nThe imaginary part of the susceptibility is related\nto the energy dissipation of the system74, encoding\nthe relaxation mechanism of the magnetization towards\nequilibrium. The damping parameter is then obtained5\nby mapping the transverse magnetic susceptibility ob-\ntained from the quantum mechanical calculation de-\nscribed above to the phenomenological form provided by\nthe LLG, Eq. (1). On the following sections, we present\ndi\u000berent mapping procedures involving several approx-\nimations and explore their range of validity when the\nbroadening \u0011is taken to zero (clean limit).\nIV. SPIN RESPONSE METHODS\nA. Ferromagnetic resonance\nMagnetic excitations can be investigated by applying\ntime-dependent perturbations. This is done in FMR ex-\nperiments where the magnetic sample is subjected to a\nstatic magnetic \feld and an oscillatory radio-frequency\none. By varying either the strength of the static compo-\nnent or the frequency of the oscillatory \feld, the system\ncan be driven through magnetic resonance. This setup\nyields the uniform mode of the transverse magnetic sus-\nceptibility. As the Gilbert parameter describes the relax-\nation mechanisms of the magnetization, it is related to\nthe linewidth of the resonance peak21,75.\nWe simulate this kind of experiments by calculating\nthe transverse magnetic response relying on the linear\nresponse theory discussed in Sec. III, and mapping the\nimaginary part of the susceptibility into the result ob-\ntained from the LLG equation (see Appendix B),\nIm\u001f\u0000+(!) =\u00002\u000b\r!M\n[!\u0000\r(Bext+Ban)]2+ (\u000b!)2:(7)\nWhen \fxing the frequency and varying Bext;z, this func-\ntion presents a resonance at Bres= (!\u0000\rBan;z)=\r\nwith linewidth given by the full width at half maxi-\nmum \u0001B= 2\u000b!=\r . On the other hand, when the\n\feld is kept \fxed and the frequency is varied, the res-\nonance is located at !res=\r(Bext;z+Ban;z)=p\n1 +\u000b2\nwith full width at half maximum approximately given by\n\u0001!\u00192\u000b\rjBext;z+Ban;zj, in the limit \u000b\u001c175.\nThe Gilbert parameter can then be obtained either by\n\ftting Eq. (7) or through the ratio between the linewidth\nand the resonance position. In this sense, a divergence of\nthe damping when \u0011!0 seems counter-intuitive, since\nthis would imply that either the resonance position ( Bres\nor!res) goes to zero or that the corresponding linewidth\nincreases drastically. In the presence of SOI, the SU(2)\nrotational symmetry is broken and the anisotropy \feld\nBan;zshifts the resonance position to a \fnite value | it\ncosts a \fnite amount of energy to set the magnetization\ninto precession76. Therefore, the divergence of the damp-\ning parameter can only happen if the linewidth increases\nand goes to in\fnity.\nTo verify this claim, we simulate FMR experiments\nin fcc Co bulk by calculating the imaginary part of the\ntransverse magnetic susceptibility as a function of the\nfrequency!, in the presence of the spin-orbit interac-\ntion. In Fig. 2a, we present the obtained spectra fordi\u000berent values of the broadening \u0011. When a relatively\nlarge value of the broadening is used, \u0011= 13:6 meV (solid\ncurve), the spectra displays a broad resonance peak,\nwhich can be characterized by a value \u000b= 1:3\u000210\u00002,\nobtained by \ftting the linear response data with Eq. (7).\nWhen the broadening of the energy levels is decreased\nto\u0011= 4:1 meV (dashed curve), the peak shifts and be-\ncomes sharper ( \u000b= 3:8\u000210\u00003), as one intuitively ex-\npects when disorder and/or temperature decreases. No-\ntice that most of the change in \u000bis due to the change\nin the peak width, while the resonance shift is relatively\nsmall. This can be viewed as a consequence of the smaller\nenergy overlap between the bands, which decrease possi-\nble interband transitions58. Surprisingly, by further de-\ncreasing the broadening to \u0011= 0:41 meV (dotted curve),\nthe peak becomes broader when compared to the pre-\nvious case, with \u000b= 5:6\u000210\u00003. This counter-intuitive\nresult represents a shorter lifetime of the magnetic excita-\ntion when the electronic lifetime (mean time between two\nsuccessive scattering events) \u001c=\u0011\u00001becomes longer.\nObtaining the damping from the FMR curves is com-\nputationally demanding, though. The response function\nmust be calculated for many frequencies (or magnetic\n\felds) to resolve the peak. For the case of low broaden-\nings that require many k-points in the Brillouin zone for\na converged result, this task becomes prohibitive. In the\nnext section, we provide alternative methods to obtain\nthe Gilbert parameter based on the static limit of the\nsusceptibility, and compare their outcomes with the ones\nobtained using the resonance approach.\nB. Inverse Susceptibility Method\nWe proceed now to investigate a di\u000berent mapping\nof the microscopic transverse susceptibility to the LLG\nequation and possible approximations to simplify the cal-\nculation of the Gilbert damping. From Eq. (B4), one can\nsee that\u000bde\fnes the slope of the imaginary part of the\ninverse susceptibility43, i.e.,\n\u000b= 2\rMlim\n!!0Im[\u001f\u0000+(!)]\u00001\n!: (8)\nWe will refer to this as the inverse susceptibility method\n(ISM). The mapping to the LLG model of the slope at\nsmall frequencies has a great advantage over the FMR\none since it only requires a single frequency-point calcula-\ntion, instead of a full sweep over frequencies or magnetic\n\felds for the \ftting procedure.\nIn Fig. 2b, we display the damping parameter for bcc\nFe, fcc Co and fcc Ni bulk systems calculated as a func-\ntion of the electronic energy broadening. We also include\nthe results obtained from the FMR approach (solid sym-\nbols), which compare well with the ISM given in Eq. (8).\nNote that although Eq. (8) has an explicit linear depen-\ndence on the spin moment M, the susceptibility implic-\nitly depends on its value. The obtained curves are in-\nversely related to M: highest for Ni ( M\u00180:45\u0016B), low-6\n0.5 0.55 0.6 0.6502468·105\nFrequencyω(meV)−Imχ−+(ω) (states/eV)η1= 13.6 meV→α= 1.3·10−2\nη2= 4.1 meV→α= 3.8·10−3\nη3= 0.41 meV→α= 5.6·10−3\n1 10 10010−210−1100101102\nBroadening η(meV)Gilbert damping αXC-TCM ISM\nFe\nFe 5·λSOI\nFe 10·λSOI\n1 10 10010−310−210−1100101\nBroadening η(meV)Gilbert damping αISM Fe\nISM Co\nISM Co no SOI\nFMR Co\nISM Ni\n1 10 10010−310−210−1100101\nBroadening η(meV)Gilbert damping αFe monolayer\nCo monolayer\nNi monolayer10 100 1,000Temperature (K)\n10 100 1,000Temperature (K)\n10 100 1,000Temperature (K)(a)\n(b)(c)\n(d)\nFigure 2. Characteristics of the Gilbert damping in 3D and 2D systems in presence and absence of SOI. (a) Ferromagnetic\nresonance spectra for fcc Co, in presence of spin-orbit interaction and no external \feld, calculated for three di\u000berent decreasing\nbroadenings \u00111= 13:6 meV (solid), \u00112= 4:1 meV (dashed) and \u00113= 0:41 meV (dotted). The values of the Gilbert damping\ngiven in the legend box, obtained by \ftting to Eq. (7), decrease from the \frst case to the second, but increases again when \u0011is\nfurther decreased. (b) Gilbert damping in presence of spin-orbit interaction for bcc Fe (blue triangles), fcc Co (red circles, solid\nline) and fcc Ni (green squares) as a function of the broadening, obtained from the slope of the inverse susceptibility, Eq. (8).\nAll values were computed with 108k-points in the full Brillouin zone. Solid red circles are the values obtained from the FMR\nspectra in (a), while the open red circles connected by dashed lines represent the damping parameter for fcc Co when SOI is\nnot included in the calculations. (c) Damping parameter for bcc Fe for di\u000berent SOI strenghts: \u0015SOI= 54:4 meV, 5 \u0002\u0015SOI,\nand 10 \u0002\u0015SOI. (d) Gilbert damping of Fe, Co and Ni monolayers in the presence of SOI. No increase in the Gilbert damping\nis seen when the broadening \u0011is decreased.\nest for Fe (M\u00182:3\u0016B) and Co in-between ( M\u00181:5\u0016B).\nThis trend is con\frmed by setting the SOI strength \u0015SOI\nto the same values for all the elements (not shown). The\nposition of the minimum value of \u000bis connected with\n\u0015SOI, which determines when the intraband or interband\ntransitions become more important58. To substantiate\nthis claim, we employed the technique of arti\fcially scal-\ning the\u0015SOI, as previously done in connection to the\nmagnetic anisotropy energy77. The results are shown in\nFig. 2c, where the SOI strength \u0015SOIof Fe bulk is mag-\nni\fed by factors of 5 and 10. Indeed, the minimum can\nclearly be seen to shift to larger values of \u0011.An important aspect to be considered is the conver-\ngence of Eq. (6) | failing to achieve numerical precision\nmay give rise to spurious results49,78. This can be partly\nsolved using sophisticated schemes to perform those cal-\nculations79,80. When the broadening is lowered, the con-\nvergence of the wave vector summation is a\u000bected by\nthe increasingly dominant role of the poles of the Green\nfunctions in the vicinity of the Fermi energy. For that\nreason, to capture the intricacies of the electronic states\n| in particular, the important contributions from the\nsmall gaps opened by the weak SOI |, we calculated\nthe slope of the response function using a very \fne in-7\ntegration mesh on the Brillouin zone reaching up to 109\nk-points. The results in Fig. 2c also demonstrate that the\ndivergence is not an issue of numerical convergence, since\nthis behavior is shifted to larger values of broadenings,\nfor which the convergence is more easily achieved.\nNevertheless, such diverging e\u000bect only occurs in the\npresence of spin-orbit interaction. In Fig. 2b we also dis-\nplay the values of \u000bfor Co fcc obtained using the ISM\nwhen the SOI is not included in the calculations (cir-\ncles connected by dashed lines). In this case, \u000bnoSOI lin-\nearly goes to zero when the broadening is decreased21.\nThe non-vanishing damping when SOI is not present\ncan be interpreted as originating from the \fnite elec-\ntronic lifetimes introduced by the constant broadening\nparameter. As it stands, \u0011represents the coupling to a\n\fctitious reservoir51,52providing dissipation mechanisms\nthat physically should originate from disorder or temper-\nature, for example.\nObtaining the damping from the FMR spectra when\nSOI is not present requires an applied magnetic \feld,\nsuch that the resonance frequency becomes \fnite and\navoiding an in\fnite response at zero frequency (repre-\nsenting no cost of energy due to the rotational symmetry,\ni.e., the Goldstone mode). Nevertheless, the results pre-\nsented in Fig. 2d were obtained using the ISM without\nany applied \feld. Calculations with an applied magnetic\n\feld shifting the peak to the original anisotropy energy\nwere indistinguishable from those values (with variations\nsmaller than 3%). This is accordance to the phenomeno-\nlogical expectations expressed through Eq. (B4), where\nthe slope is independent of the magnetic \feld.\nOne can put our results for bulk ferromagnets into\nperspective by comparing with low dimensional systems.\nWe investigated this case within our linear response ap-\nproach, using monolayers of Fe, Co and Ni. The calcu-\nlations follow the same procedure, except that the sum\noverkvectors in Eq. (6) is restricted to the 2D Brillouin\nzone. The results are presented as triangles (Fe), cir-\ncles (Co) and squares (Ni) connected by dotted lines in\nFig. 2d, and once again exhibit a monotonous decay with\nthe decrease of \u0011. We note that previous calculations of\nthe damping parameter in thin \flms also did not \fnd it\nto increase rapidly for decreasing broadening21,49.\nBesides the dimensionality, another main di\u000berence\nfrom the bulk to the layered case is the larger anisotropy\n\felds of the latter (see Table I). Nevertheless, this can-\nnot explain the non-diverging behavior in the monolay-\ners. We have already shown that by arti\fcially increas-\ning the SOI strength of the bulk | and, consequently, its\nanisotropy \feld |, the conductivity-like behavior of the\ndamping occurs at even larger broadenings (see Fig. 2c).\nOn the other hand, to rule out a possible divergence hap-\npening at lower broadenings ( \u0011<0:1 meV, not reachable\nin our calculations), we have also scaled up \u0015SOIof the\nmonolayers by one order of magnitude. This resulted in\nlarger dampings, nonetheless, the same decreasing be-\nhaviour with \u0011!0 was observed (not shown). There-\nfore, the divergence can only be attributed to the three-dimensionality of the ferromagnet.\nC. Approximate static limit methods\nWe now look back to Fig. 1 and proceed to perform\napproximations on Eq. (8) in order to simplify the calcu-\nlations of the damping parameter. Here we follow Ref. 43.\nFirst, we use Eq. (5) that relates the RPA susceptibility\nmatrix to the mean-\feld response matrix \u001f0, such that\nIm\u001f\u00001\u0019Im\u001f\u00001\n0. Although Uis a real matrix, the sum\nover orbitals ( \u001f=P\n\u0016\u0017\u001f\u0016\u0017) ends up mixing the real\nand imaginary parts of the matrix elements. Only when\nRe\u001f\u00001\n0=U=4 the relation above becomes an equality.\nThis means that, within our model with Uacting only on\nthedorbitals,\u001fmust also be de\fned by summing over\nthose orbitals only. Under the previous assumption, we\nobtain\n\u000b\u00192\rMlim\n!!0Im[\u001f\u0000+\n0(!)]\u00001\n!: (9)\nThis relation is only valid when \u001f\u0000+\n0is decoupled from\nthe other types of susceptibilities (transverse and longi-\ntudinal), as in the systems we investigate in this work.\nThe damping parameter can therefore be obtained from\nthe single-particle transverse susceptibility \u001f0.\nFor frequencies !in the meV range (where the col-\nlective spin excitations are located), \u001f\u0000+\n0has a simple\n!-dependence81:\n\u001f\u0000+\n0(!)\u0019Re\u001f0(0) + i!Im\u001f0\n0(0): (10)\nwhere\u001f0\n0(0) =d\u001f\u0000+\n0\nd!\f\f\f\n!=0. These results are valid also in\nthe presence of spin-orbit coupling. Using Eq. (10), the\nGilbert damping can be written as\n\u000b\u0019\u00002\rM\u0002\nRe\u001f\u0000+\n0(0)\u0003\u00002lim\n!!0Im\u001f\u0000+\n0(!)\n!:(11)\nAlthough the expansion of the susceptibility for low fre-\nquencies was used, no extra approximation is employed,\nsince Eq. (9) is calculated in the limit !!0. Re\u001f\u0000+\n0(0)\ncan be obtained using the sum rule that relates the\nstatic susceptibility with the magnetic moments76. For\n3dtransition metals, the external and the spin-orbit\n\felds are three orders of magnitude smaller than U, and\nso the static susceptibility of the bulk systems reads\nRe\u001f\u0000+\n0(0)\u00194=U. Thus,\n\u000b\u0019\u0000\rMU2\n8lim\n!!0Im\u001f\u0000+\n0(!)\n!: (12)\nFinally, from Eq. (6) it is possible to show that Eq. (12)8\nsimpli\fes as\n\u000b\u0019\rMU2\n2\u0019NX\nk;\u0016\u0017TrfImG\u0017\u0016(k;\u000fF)^S\u0000ImG\u0016\u0017(k;\u000fF)^S+g\n=\r\n2M\u0019NX\nk;\u0016\u0017TrfImG\u0017\u0016(k;\u000fF)^T\u0000\nxcImG\u0016\u0017(k;\u000fF)^T+\nxcg\n=\rMU2\u0019\n8NX\nk;\u0016\u0017n#\n\u0017\u0016(k;\u000fF)n\"\n\u0016\u0017(k;\u000fF):\n(13)\nwheren\u001b\n\u0016\u0017(k;\u000fF) =\u00001\n\u0019ImG\u001b\u001b\n\u0016\u0017(k;\u000fF) is the matrix el-\nement of the spectral function of spin \u001bcalculated at\nkand\u000fF. The second equation is written in terms\nof the \\exchange-correlation torque operator\", T\u0006\nxc=\n\u0000i\u0002^S\u0006;^Hxc\u0003\n=\u0007iUM^S\u0006. This result is equivalent\nto the one obtained in Ref. 42, which we reference as\ntheexchange torque correlation method (XC-TCM) |\nalthough, in reality, it relates \u000bwith the spin-spin re-\nsponse. The last step in Eq. (13) connects the damping\nwith the product of spectral functions of opposite spins\nat the Fermi level, as shown theoretically in Ref. 81 and\ncon\frmed experimentally in Ref. 46.\nIn Fig. 2c, we compare the results obtained with this\napproximated method with the ISM described before, for\nthe di\u000berent values of SOI scalings. For the bulk tran-\nsition metals we investigate, the approximation is very\ngood, since the SOI is relatively small. In fact, even\nwhen the SOI is scaled one order of magnitude higher,\nthe results of the XC-TCM are still very good.\nThe formulas in Eq. (13) show that we have arrived\nat the bottom of the triangle in Fig. 1. These forms\ndo not involve an integral over energy, which simpli\fes\nsubstantially the calculation of \u000b. For that reason, they\nare suitable for \frst-principles approaches (e.g., Refs. 20\nand 62). This concludes our investigations of the spin\nresponse methods. In the next section, we take a di\u000berent\npath to calculate the Gilbert damping.\nV. TORQUE RESPONSE METHODS\nDespite the simplicity of the methods based on the spin\nsusceptibility discussed in the previous section, seminal\nwork was based on a di\u000berent type of response function.\nThe main idea, \frst proposed by Kambersky37, is to di-\nrectly relate \u000bto the spin-orbit interaction. Here, our\naim is twofold. First, we connect the spin susceptibility\nwith the spin-orbit torque response via the equation of\nmotion, clarifying the damping mechanisms captured by\nthis formalism. Second, we compare the results obtained\nwith both types of methods.\nWe start with the equation of motion for the spin-spin\nsusceptibility. Its time-Fourier transform can be written\nas19\n!\n\n^S\u0000;^S+\u000b\u000b\n!=M+\n\n\u0002^S\u0000;^H\u0003\n;^S+\u000b\u000b\n!; (14)whereM=\u00002\n^Sz\u000b\n. From the Hamiltonian given in\nEq. (2), the commutator [ ^S\u0000;^H\u0003\nhas four contributions:\nkinetic (spin currents, from ^H0), exchange torque (from\n^Hxc), external torque (from ^Hext) and spin-orbit torque\n(from ^HSOI). In presence of SOI, the total spin magnetic\nmoment is not a conserved quantity and spin angular\nmomentum can be transferred to the orbital degrees of\nfreedom. For bulk systems subjected to static external\n\felds and in the present approximation for the electron-\nelectron interaction, the only two non-vanishing torques\nare due to the external \feld and the spin-orbit interac-\ntion. It also follows from these assumptions that the\nmechanisms that contribute to the relaxation arises then\nfrom the spin-orbit torques ^T\u0006\nSOI=\u0000i\u0002^S\u0006;^HSOI\u0003\nand\nfrom the broadening of the energy levels \u0011.\nIt can be shown19that the inverse of the susceptibility\n\u001f\u0000+(!) =\n\n^S\u0000;^S+\u000b\u000b\n!is given by\n\u0002\n\u001f\u0000+(!)\u0003\u00001=\u0002\n\u001f\u0000+\nnoSOI (!)\u0003\u00001\u0002\n1 +\u001f\u0000+\nnoSOI (!) \u0000(!)\u0003\u00001\n\u0019\u0002\n\u001f\u0000+\nnoSOI (!)\u0003\u00001\u0000\u0000(!):\n(15)\nHere,\u001f\u0000+\nnoSOI (!) is the susceptibility calculated excluding\nthe SOI contribution to the Hamiltonian. The connection\nbetween the two susceptibilities in Eq. (15) is provided\nby the quantity\nM2\u0000(!) = i\n\u0002^T\u0000\nSOI;^S+\u0003\u000b\n+\n\n^T\u0000\nSOI;^T+\nSOI\u000b\u000b\n!:(16)\nUsing Eq. (8), and noticing that the \frst term on the\nright-hand side of the equation above does not contribute\nto the imaginary part, we \fnd\n\u000b=\u000bnoSOI\u00002\r\nMlim\n!!0Im\n\n^T\u0000\nSOI;^T+\nSOI\u000b\u000b\n!\n!: (17)\n\u000bnoSOI is the contribution obtained by inputting\n\u001f\u0000+\nnoSOI (!) into Eq. (8), which is \fnite due to the broad-\nening\u0011.\nKambersky37\frst obtained this same result following a\ndi\u000berent approach. In our framework, this would involve\nstarting from Eq. (5) and exploiting the consequences of\nthe fact that the collective spin excitations ( !\u0018meV)\nhave low frequencies when compared to the exchange en-\nergy (U\u0018eV). On the other hand, Hankiewicz et al.19\ndescribed the same expansion for low SOI, and justi\fed\nits use for !.\rBext. Finally, Edwards22shows that\nthis formula is equivalent to a perturbation theory cor-\nrect to\u00152\nSOI(compared to \rBext\u0000!). For that rea-\nson, he suggests that the states used in the calculation\nof\n\n^T\u0000\nSOI;^T+\nSOI\u000b\u000b\n!should not include SOI, since the op-\nerator ^T\u0000\nSOI/\u0015SOI. Due to the orbital quenching in the\nstates without SOI, this leads to the absence of intra-\nband contributions and, consequently, of the divergent\nbehavior for \u0011!082.\nIn this approach, temperature and disorder e\u000bects are\nincluded in \u000bnoSOI (shown in Fig. 2d for Co), while the9\nspin-orbit intrinsic broadening is calculated by the sec-\nond term in Eq. (17), which can also be obtained as\n\u000b\u0000\u000bnoSOI . An extra advantage of calculating the damp-\ning as the aforementioned di\u000berence is that one explic-\nitly subtracts the contributions introduced by \u0011, provid-\ning similar results to those obtained with vertex correc-\ntions20. Considering the torque-torque response within\nthe mean-\feld approximation (an exact result in the per-\nturbative approach22), we obtain, similarly to Eq. (13),\n\u000b\u0000\u000bnoSOI =\n2\r\nM\u0019NX\nkTrfImG(k;\u000fF)^T\u0000\nSOIImG(k;\u000fF)^T+\nSOIg:\n(18)\nIn this formula, the involved quantities are matrices in\nspin and orbital indices and the trace runs over both.\nThis is known as Kambersy's formula, commonly used in\nthe literature43,44,47,49,58, which we refer to as spin-orbit\ntorque correlation method (SO-TCM). As in Eq. (13), it\nrelates the damping to Fermi level quantities only. When\nthe SOI is not included in the calculation of the Green\nfunctionsG(k;\u000fF) and enters only through the torque\noperators, we name it perturbative SO-TCM22. These\nmethods are placed at the bottom right of Fig. 1, with\nthe main approximations required indicated by the long\ndashed arrows.\nWe now proceed to compare these approaches with the\nISM explained in Sec. IV B. Fig. 3 presents the calcula-\ntions of the SOI contribution to the damping parameter\nof bulk Fe (a), Co (b) and Ni (c) using the SO-TCM ob-\ntained in Eq. (18), when no external \feld is applied. Both\napproaches, including SOI (red curve with squares) in\nthe Green functions or not (green curve with triangles),\nare shown. For a meaningful comparison, we compute\n\u000b\u0000\u000bnoSOI within the ISM.\nWe \frst note that the perturbative approach suggested\nby Edwards22describes reasonably well the large broad-\nening range (i.e., mostly given by the interband transi-\ntions), but deviates from the other approaches for low \u0011.\nThis is an expected behaviour since it does not include\nthe intraband transitions that display the \u0011\u00001behav-\nior within the constant broadening model. In the clean\nlimit, the Gilbert damping computed from the pertur-\nbative SO-TCM approaches zero for all elements, in a\nvery monotonic way for Co and Ni, but not for Fe. This\nmethod is thus found to be in agreement with the other\nones only when \u0015SOI\u001c\u0011. The SO-TCM formula in-\ncluding the SOI in the states (i.e., Kambersky's formula)\nmatches very well \u000bobtained within ISM in the whole\nrange of broadenings.\nFinally, after demonstrating that the SO-TCM pro-\nvides very similar results to the ISM, we can use it to\nresolve the wave-vector-dependent contributions to the\nGilbert parameter by planes in the reciprocal space, as\n\u000b(kmax\nz) =kmax\nzX\njkzj\u000b(kz); (19)\n1 10 10010−310−2Gilbert damping α(a)Fe\nISM ( α−αnoSOI )\nSO-TCM\nperturbative SO-TCM10 100 1,000Temperature [K]\n1 10 10010−410−310−2Gilbert damping α(b)Co\n10 100 1,000\n1 10 10010−410−310−210−1100\nBroadening η[meV]Gilbert damping α(c)Ni\n10 100 1,000Figure 3. Comparison between \u000b\u0000\u000bnoSOI for (a) Fe bcc, (b)\nCo fcc and (c) Ni fcc, obtained using the inverse susceptibility\nmethod (ISM) with the spin-orbit-torque correlation method\n(SO-TCM) with and without SOI in the states (perturbative\nSO-TCM). All the points were computed with 108k-points in\nthe full Brillouin zone.\nwhere\u000b(kz) is given by the right-hand side of Eq. (18)\nsummed over kx;ky. The result, displayed in Fig. 4, uses\n100 million k-points for all curves and shows the expected\ndivergence in presence of SOI and a decrease with \u0011when\nthis interaction is absent. In every case, most of the con-\ntribution arises from the \frst half ( kmax\nz<0:4). Note\nthat when the broadening of the energy levels is low, the\nintegrated alpha without SOI (Fig. 4b) displays step-\nlike contributions, while when SOI is present, they are\nsmoother. This is a consequence of the damping being10\n01234Gilbert damping α(·10−2)(a) With SOI\nη(meV):\n0.14\n0.41\n0.68\n1.1\n1.4\n4.1\n6.8\n10.9\n13.6\n40.8\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\nkmax\nzGilbert damping α(·10−3)(b) Without SOIDecreasingη\nDecreasingη\nFigure 4. Integrated Gilbert damping for fcc Co as a function\nofkzplotted against the maximum value, kmax\nz(see Eq. (19)),\nwith SOI (a) and without SOI (b). The curves were obtained\nusing the SO-TCM given in Eq. (18). Colors represent di\u000ber-\nent values of the broadening \u0011(in units of meV). The value of\n\u000bforkmax\nz= 0 (i.e., a single value of kzin the sum) represents\na two-dimensional system, whilst for kmax\nz= 1 the sum covers\nthe whole 3D Brillouin zone. In the latter case, the damping\ndecreases when \u0011is decreased without SOI, while it increases\ndrastically when SOI is present. For 2D systems, the\ncaused by interband transitions in the former and intra-\nband in the latter.\nThe convergence of the previous results for the smallest\n\u0011including SOI were tested with respect to the total\nnumber of k-points in the Brillouin zone in Fig. 5. By\ngoing from 10 million to 10 billion k-points, the results\nvary\u001820%. However, compared with the result shown\nin Fig. 4a, the damping gets even larger, corroborating\nonce more the divergent results.\nVI. DISCUSSIONS\nIn this section, we make a few \fnal remarks on the pre-\nviously obtained results and we go beyond bulk systems\nto comment on the approximations taken and additional\nphysical mechanisms that may come into play in other\n0 0.2 0.4 0.6 0.8 1024\nkmax\nzGilbert damping α(·10−2)\nk-points:\n107\n108\n109\n1010Figure 5. Integrated Gilbert damping for fcc Co as a func-\ntion ofkzplotted against the maximum value, kmax\nz, for\n\u0011= 0:14 meV and di\u000berent amount of k-points (up to 10\nbillion) in the Brillouin zone.\nmaterials. We also provide a new analytical explanation\nfor the divergence of the damping parameter within the\nconstant broadening model.\nOur \frst comment regards the application of static\nmagnetic \felds B. As described in Refs. 19 and 22, the\napproximations done in Eq. (15) to derive an expression\nfor\u000binvolves comparisons between the excitation en-\nergy andB. However, all the results we have presented\nhere were obtained in absence of static \felds. We also\nperformed calculations including external magnetic \felds\nup toB\u00187 T, and the computed damping parameter is\nweakly in\ruenced by their presence. We conclude that\nthe validity of the SO-TCM formula given in Eq. (18)\ndoes not hinge on having a magnetic \feld, supporting\nthe arguments already given in Ref. 19.\nA further remark concerns the approximations made\nto obtain the mean-\feld result in Eq. (12). We assumed\nthat SOI is weak when using the magnetic sum rule.\nThis approximation may break down when this is not\nthe case. The spin pumping also a\u000bects the magnetic\nsum rule, which may worsen the agreement with the ISM\nresults. Although this contribution is not present in the\ninvestigated (bulk-like) systems, it plays an important\nrole in magnetic multilayers. This e\u000bect enhances the\ndamping factor32,54,55. Furthermore, the SO-TCM ex-\nplicitly excludes spin pumping, as this is described by\n^I\u0000\nS=\u0000i\u0002^S\u0000;^H0\u0003\n, dropped from the equation of mo-\ntion. These validity conditions are indicated in Fig. 1 by\nthe large blue rectangle (low SOI), red triangle (low spin\npumping) and green rectangle (no spin pumping).\nAnother mechanism that opens new spin relaxation\nchannels is the coupling between transverse and longi-\ntudinal excitations induced by the SOI. This was one of\nthe reasons raised in Ref. 21 to explain the divergence of\nthe damping parameter. However, this is absent not only11\nwhen the system has full spin rotational symmetry83, but\nalso when rotational symmetry is broken by the SOI in\n2D and 3D systems for the symmetries and materials we\ninvestigated. Even though the damping is \fnite in the\n\frst two cases (as shown in Fig. 2d), the divergence is\nstill present in the latter (Fig. 2b).\nWe can also recognize that the mathematical expres-\nsion for\u000bin terms of the mean-\feld susceptibility given\nin Eq. (12) is similar to the conductivity one (i.e., the\nslope of a response function)84| which leads to the same\nissues when approaching the clean limit ( \u0011!0). How-\never, the physical meaning is the exact opposite: While\nthe divergence of the conductivity represents an in\fnite\nacceleration of an ideal clean system, in\fnite damping\ndenotes a magnetic moment that is instantly relaxed in\nwhichever direction it points (as d M=dt!0 for\u000b!1 )\n| i.e., no dynamics10,11. This means that a clean 3D\nspin system is in\fnitely viscous. Within the constant\nbroadening model, the divergence of the Gilbert damp-\ning can also be seen analytically by comparing Eq. (12)\nwith the calculations of the torkance done in Ref. 48.\nBy replacing the torque operator and the current density\nby the spin lowering and raising operators, respectively,\nthe even contribution (in the magnetization) to the re-\nsponse function vanishes and only the odd one remains.\nIn this approximation, it is also seen that only the Fermi\nsurface quantities are left, while the Fermi sea does not\ncontribute85. In the limit of low broadenings, this con-\ntribution is shown to diverge as \u0011\u00001. This divergence\narises from intraband transitions which are still present\nin the clean limit, and originate from the \fnite electronic\nlifetimes introduced by the constant broadening approx-\nimation.\nThe static limit ( !!0) is another reason that many\nauthors considered to be behind the divergent damping\nbehavior19,37,43,50. This limit is taken in Eq. (8) in or-\nder to eliminate the contribution of terms nonlinear in\nfrequency from the inverse susceptibility (e.g., inertia ef-\nfects68,86). They can be present in the full microscopic\ncalculation of the susceptibility but are not included in\nthe phenomenological model discussed in Appendix B.\nAdding the quadratic term in frequency leads to an in-\nverse susceptibility given by\nIm[\u001f\u0000+(!)]\u00001=\u0000!\n2\rM(\u000b\u0000!I)\nwhereIis the o\u000b-diagonal element of the moment of iner-\ntia tensor86. The \ft to the expression linear in frequency\nthen yields an e\u000bective \u000be\u000b(!). In the vicinity of the res-\nonance frequency, \u000be\u000b(!res) =\u000b\u0000!resI, which is clearly\nreduced in comparison to the one obtained in the static\nlimit,\u000be\u000b(0) =\u000b. According to Ref. 68, I\u0018\u000b=\u0011, which\nexplains the discrepancy between the FMR and the ISM\nseen in Fig. 2b as \u0011!0. We can then conclude that the\nstatic limit is not the culprit behind the divergence of \u000b\nin the clean limit.VII. CONCLUSIONS\nIn this work, we presented a study of di\u000berent meth-\nods to calculate the intrinsic Gilbert damping \u000b, o\u000bering\na panorama of how the approaches are related and their\nrange of validity (see Fig. 1). They can be grouped into\nthree main categories: the methods that directly employ\nthe results of full microscopic calculations of the dynam-\nical magnetic susceptibility \u001f(!) (FMR and ISM); the\nexchange-torque method (XC-TCM), which is also based\non\u001f(!) but making use of the mean-\feld approximation;\nand the spin-orbit torque-correlation method (SO-TCM),\nobtained from the (spin-orbit) torque-torque response via\nan equation of motion for \u001f(!). While the FMR, ISM\nand XC-TCM include all the contributions to the mag-\nnetic relaxation, the SO-TCM provides only the intrinsic\ncontribution due to the angular momentum transfer to\nthe orbital degrees of freedom (not including, for exam-\nple, the spin pumping mechanism). The XC- and SO-\nTCM, given by Eqs. (13) and (18), are predominant in\nthe literature due to their simplicity in obtaining \u000bin\nterms of Fermi level quantities. It is important to note,\nhowever, that they rely on approximations that may not\nalways be full\flled21.\nIn order to implement and compare the di\u000berent meth-\nods, we constructed a uni\fed underlying framework\nbased on a multi-orbital tight-binding Hamiltonian using\nas case studies the prototypical bulk 3D systems: bcc Fe,\nfcc Co and fcc Ni. For this set of materials, the di\u000berent\nmethods lead to similar results for \u000b, showing that the\ncorresponding approximations are well-founded. Even\nwhen the SOI strength is scaled up by one order of mag-\nnitude, this excellent agreement remains, as we explic-\nitly veri\fed for bcc Fe. We found one method that falls\nout-of-line with the others in the clean limit, namely the\nperturbative form of the SO-TCM formula22,82. In this\ncase, although the equation is identical to the well-known\nKambersky formula, Eq. (18), the electronic states used\nto evaluate it do not include SOI. By comparison with\nthe other methods, we conclude that the results obtained\nby the perturbative SO-TCM are only valid in the large\nbroadening regime (compared to the SOI strength). Cen-\ntral to our analysis was a careful study of the convergence\nof our results with respect to the number of k-points,\nreaching up to 1010k-points in the full Brillouin zone.\nThe behavior of \u000bis intimately connected with the con-\nstant broadening approximation for the electronic life-\ntimes. For high temperatures, the Gilbert damping in-\ncreases with increasing temperature ( \u000b\u0018\u0011), while for\nlow temperatures it diverges for 3D ferromagnets ( \u000b\u0018\n1=\u0011), but not for 2D (ferromagnetic monolayers). Our\ncalculations revealed that the high temperature values\nof\u000barise mostly from the broadening of the electronic\nstates. In Ref. 20, the strongly increasing behaviour of\n\u000bfor high temperatures was found to be spurious, and\ncured employing a more realistic treatment of disorder\nand temperature, and the so-called vertex corrections.\nWe found that the contribution of the intrinsic SOI to \u000b12\nis additive to the one arising from the broadening, and\ncan be easily extracted by performing a calculation of\n\u000bwithout SOI and subtracting this result from the SOI\none,\u000b\u0000\u000bnoSOI . Combined with the ISM, this provides\na relatively simple and accurate way to obtain the in-\ntrinsic damping, which discounts contributions from the\nadditional broadening \u0011. This establishes an alternative\nway of accessing the high temperature regime of \u000b.\nThe low-temperature divergence of \u000bwhen approach-\ning the clean limit for 3D ferromagnets has also been the\nsubject of much discussion. The \frst di\u000eculty is in es-\ntablishing numerically whether this quantity actually di-\nverges or not. Our results consistently show an increase\nof\u000bwith decreasing \u0011, down to the smallest achievable\nvalue of\u0011= 0:14 meV (Fig. 5), with no hints of a plateau\nbeing reached, but only when accounting for SOI. This\ndivergence arises from the intraband contributions to \u000b,\nas discussed in Ref. 58. Refs. 22 and 82 used pertur-\nbation theory arguments to claim that such intraband\ncontributions should be excluded. However, as we dis-\ncussed in Sec. IV B, adapting the formalism of Ref. 48 to\nthe calculation of \u000bshows that these intraband terms are\nenabled by the constant broadening approximation, and\nso should be included in the calculations. Contrary to\nthe high temperature regime, works that employ a more\nrealistic treatment of disorder and temperature still \fnd\nthe diverging behavior of \u000b20,52.\nIn real experiments, any kind of material disturbance\nsuch as disorder or temperature e\u000bects leads to a \fnite\nvalue of the damping. Besides that, a non-uniform com-\nponent of the oscillatory magnetic \feld (either from the\napparatus itself or due to its limited penetration into the\nsample) induces excitations with \fnite wave vectors and\n\fnite linewidths39,87. A di\u000berent way to determine the\ndamping parameter is using the time-resolved Magneto-\nOptic Kerr E\u000bect (TR-MOKE)40,88. It has the advan-\ntage that, as it accesses a smaller length scale ( \u00181µm)\nthan FMR experiments (which probe the whole magnetic\nvolume), the measured magnetic properties are more ho-\nmogeneous and thus the e\u000bect of linewidth broadening\nmay be weaker. The magnetic excitations in nanomag-\nnets can also be probed by recent re\fnements of FMR\nexperimental setups89,90.\nAlthough the methods we described here are gen-\neral, we did not explicitly addressed non-local sources\nof damping such as the spin-pumping32. As a future\nproject, we plan to ascertain whether our conclusions\nhave to be modi\fed for systems where this mechanism\nis present. Systems that combine strong magnetic el-\nements with heavy ones possessing strong SOI are ex-\npected to have anisotropic properties, as well-known for\nthe magnetic interactions91. It is then natural to explore\nwhen the Gilbert damping can also display signi\fcant\nanisotropy, becoming a tensor instead of a scalar quan-\ntity47,78. Indeed, this has been observed experimentally\nin magnetic thin \flms92,93. As the SOI, magnetic non-\ncollinearity can also lead to other forms of damping in do-\nmain walls and skyrmions50,94{98. From the microscopicpoint of view, the potential coupling between transverse\nand longitudinal degrees of freedom allowed by the non-\ncollinear alignment should also be considered. Lastly,\nhigher order terms in frequency, such as the moment of\ninertia68,86,99{101, might also become important in the\ndynamical magnetic susceptibility for large frequencies\nor for antiferromagnets, for instance.\nThe description of magnetization dynamics of real ma-\nterials helps to design new spintronic devices able to con-\ntrol the \row of information. Our work sheds light on fun-\ndamental questions about the main relaxation descrip-\ntions used in the literature and sets ground for future\ntheoretical predictions.\nAppendix A: Ground-state Hamiltonian\nIn this Appendix, we give the explicit forms of the\nterms in the Hamiltonian written in Eq. 2. As the inves-\ntigated systems only have one atom in the unit cell, the\nsite indices are omitted.\nThe electronic hoppings in the lattice are described by\n^H0=1\nNX\nk\u001bX\n\u0016\u0017t\u0016\u0017(k)cy\n\u0016\u001b(k)c\u0017\u001b(k); (A1)\nwithcy\n\u0016\u001b(k) andc\u0017\u001b(k) being the creation and annihila-\ntion operators of electrons with spin \u001band wave vector\nkin the orbitals \u0016and\u0017, respectively. The tight-binding\nparameters t\u0016\u0017(k) were obtained by \ftting paramagnetic\nband structures from \frst-principles calculations up to\nsecond nearest neighbors102, within the two-center ap-\nproximation103.\nThe electron-electron interaction is characterized by\na local Hubbard-like104interaction within the Lowde-\nWindsor approximation105, resulting in the mean-\feld\nexchange-correlation term\n^Hxc=\u0000X\n\u00162d\n\u001bU\n2(\nM\u000b\u001b\u000b\n\u001b\u001b0+X\n\u00172d\u000en\u0017(2\u000e\u001b\u001b0\u000e\u0016\u0017\u0000\u000e\u001b\u001b0))\ncy\n\u0016\u001b(k)c\u0016\u001b0(k):\n(A2)\nHere,Uis the local e\u000bective Coulomb interaction, M\u000b\nand\u001b\u000bare the\u000b-component of the magnetic moment\nvector (summed over the dorbitals) and of the Pauli\nmatrix, respectively. \u000en\u0016is the change in the occupation\nof orbital\u0016compared to the DFT calculations included\nin Eq. A1. M\u000band\u000en\u0016are determined self-consistently.\nThe atomic SOI is described by\n^HSOI=\u0015X\n\u0016\u0017\n\u001b\u001b0^L\u000b\n\u0016\u0017^S\u000b\n\u001b\u001b0cy\n\u0016\u001b(k)c\u0017\u001b0(k);(A3)\nwhereL\u000bandS\u000bare the\u000bcomponents of the orbital and\nspin vector operators, respectively. The strength of the\nSOI,\u0015, is also obtained from \frst-principles calculations.13\nThe interaction with a static magnetic \feld Bextis\ndescribed by\n^Hext=B\u000b\nextX\n\u0016\u0017\n\u001b\u001b0(^L\u000b\n\u0016\u0017\u000e\u001b\u001b0+\u001b\u001b\u001b0\u000e\u0016\u0017)cy\n\u0016\u001b(k)c\u0016\u001b0(k);\n(A4)\nwhere\u0016Bis absorbed to B\u000b\nextand we used gL= 1 and\ngS= 2 as the Land\u0013 e factors for the orbital and spin\nangular momentum.\nAppendix B: Phenomenology of FMR\nThe semi-classical description of the magnetization is\nobtained using the Landau-Lifshitz-Gilbert (LLG) equa-\ntion (1)9. The e\u000bective \feld acting on the magnetic mo-\nment is obtained from the energy functional of the system\nasBe\u000b(t) =\u0000@E=@M. For the symmetries we investi-\ngate, the model energy106for the 3D cubic cases77can\nbe written as\nE3D(M) =K4\nM4(M2\nxM2\ny+M2\nyM2\nz+M2\nxM2\nz)\u0000M\u0001Bext;\n(B1)\nwhile for 2D systems,\nE2D(M) =\u0000K2\nM2M2\nz\u0000M\u0001Bext: (B2)\nPositive values of K4andK2yield easy magnetization\ndirection along the (001) direction.\nWe consider magnetic moments pointing along the easy\naxis, which de\fnes the ^ zdirection. Static magnetic \felds\nare applied along the same orientation. The magnetic\nmoment is set into small angle precession, M=M^ z+\u000eMx(t)^ x+\u000eMy(t)^ y, by an oscillatory \feld in the trans-\nverse plane, i.e., Bext(t) =Bext^ z+\u000eBext(t). In this form,\nthe e\u000bective \feld (linear in the transverse components of\nthe magnetization) is given by Be\u000b(t) =Ban(t)+Bext(t),\nwith\nB3D\nan(t) =\u00002K4\nM2(\u000eMx^ x+\u000eMy^ y) , and B2D\nan=2K2\nM^ z\n(B3)\nbeing the anisotropy \felds for 3D and 2D systems, respec-\ntively. In the following expressions, K4andK2appear\nin the same way, so they are denoted by K.\nThe Fourier transform of the linearized equation of mo-\ntion can be written using the circular components \u000eM\u0006=\n\u000eMx\u0006i\u000eMy. Within this convention, \u000eM\u0000=\u000eB\u0000=\n\u001f\u0000+=2 and\n\u001f\u0000+(!) =\u00002\rM\n[!\u0000\r(Bext+Ban)]\u0000i\u000b!; (B4)\nwhereBan= 2K=M .\nACKNOWLEDGMENTS\nWe are very grateful to R. B. Muniz, A. T. Costa\nand D. M. Edwards for fruitful discussions. 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Troncoso,1Wolfgang Belzig,2and Arne Brataas1,y\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Department of Physics, University of Konstanz, D-78457 Konstanz, Germany\nAbstract\nWe present a systematic phenomenological description of Gilbert damping in two-sublattice mag-\nnets. Our theory covers the full range of materials from ferro- via ferri- to antiferromagnets. Fol-\nlowing a Rayleigh dissipation functional approach within a Lagrangian classical \feld formulation,\nthe theory captures intra- as well as cross-sublattice terms in the Gilbert damping, parameterized\nby a 2\u00022 matrix. When spin-pumping into an adjacent conductor causes dissipation, we obtain\nthe corresponding Gilbert damping matrix in terms of the interfacial spin-mixing conductances.\nOur model reproduces the experimentally observed enhancement of the ferromagnetic resonance\nlinewidth in a ferrimagnet close to its compensation temperature without requiring an increased\nGilbert parameter. It also predicts new contributions to damping in an antiferromagnet and sug-\ngests the resonance linewidths as a direct probe of the sublattice asymmetry, which may stem from\nboundary or bulk.\n1arXiv:1808.04385v2  [cond-mat.mtrl-sci]  6 Nov 2018I. INTRODUCTION\nThe fundamental connection1between magnetic moment and spin angular momentum\nunderlies the important role for magnets in nearly all spin-based concepts. An applied mag-\nnetic \feld provides the means to manipulate the state of a ferromagnet (FM), and thus the\nassociated spin. Conversely, a spin-polarized current absorbed by the FM a\u000bects its mag-\nnetization2{5. Exploiting a related phenomenon, switching the state of an antiferromagnet\n(AFM) has also been achieved6. Emboldened by this newly gained control, there has been\nan upsurge of interest in AFMs7{10, which o\u000ber several advantages over FMs. These include\nthe absence of stray \felds and a larger anisotropy-induced gap in the magnon spectrum. The\ntwo-sublattice nature of the AFMs further lends itself to phenomena distinct from FMs11.\nConcurrently, ferrimagnets (FiMs) have been manifesting their niche in a wide range of\nphenomena such as ultrafast switching12{14and low-dissipation spin transport15{22. A class of\nFiMs exhibits the so-called compensation temperature23{28, at which the net magnetization\nvanishes, similar to the case of AFMs. Despite a vanishing magnetization in the compensated\nstate, most properties remain distinct from that of AFMs29. Thus, these materials can be\ntuned to mimic FMs and AFMs via the temperature. In conjunction with the possibility of a\nseparate angular-momentum compensation, when the magnetization does not vanish but the\ntotal spin does, FiMs provide a remarkably rich platform for physics and applications. An\nincreased complexity in the theoretical description29,30hence accompanies these structurally\ncomplicated materials, and may be held responsible for comparatively fewer theoretical\nstudies. Nevertheless, a two-sublattice model with distinct parameters for each sublattice\nqualitatively captures all the phenomena mentioned above.\nDissipation strongly in\ruences the response of a magnet to a stimulus and is thus cen-\ntral to the study of magnetic phenomena such as switching, domain wall motion and spin\ntransport. Nevertheless, magnetic damping has conventionally been investigated via the\nferromagnetic resonance (FMR) linewidth. It is accounted for phenomenologically in the\nLandau-Lifshitz description of the magnetization dynamics via the so-called Gilbert damp-\ning term31, which produces a good agreement with experiments for a wide range of systems.\nThe Gilbert damping represents the viscous contribution and may be `derived' within a\nLagrangian formulation of classical \feld theory by including the Rayleigh dissipation func-\ntional31. While the magnetic damping for FMs has been studied in great detail29,31{35,\n2from phenomenological descriptions to microscopic models, a systematic development of an\nanalogous description for ferri- and antiferromagnets has been lacking in literature. Further-\nmore, recent theoretical results on spin pumping in two-sublattice magnets36and damping\nin AFMs37suggest an important role for the previously disregarded29cross-sublattice terms\nin Gilbert damping, and thus set the stage for the present study. Yuan and co-workers have\nrecently presented a step in this direction focussing on spin torques in AFMs38.\nHere, we formulate the magnetization dynamics equations in a general two-sublattice\nmagnet following the classical Lagrangian approach that has previously been employed for\nFMs31. The Gilbert damping is included phenomenologically via a Rayleigh dissipation\nfunctional appropriately generalized to the two-sublattice system, which motivates intra-\nas well as cross-sublattice terms. The Gilbert damping parameter thus becomes a 2 \u00022\nmatrix, in contrast with its scalar form for a single-sublattice FM. Solving the system of\nequations for spatially homogeneous modes in a collinear ground state, we obtain the decay\nrates of the two eigenmodes \fnding direct pathways towards probing the dissipation mech-\nanism and asymmetries in the system. Consistent with recent experiments28,39, we \fnd an\nenhancement in the decay rates39close to the magnetization compensation in a FiM with\nan unaltered damping matrix28. The general description is found to be consistent with the\nspin pumping mediated damping in the magnet34{36, and allows for relating the Gilbert\ndamping matrix with the interfacial spin-mixing conductances. Focusing on AFMs, we ex-\npress the magnetization dynamics in terms of the Neel variable thus clarifying the origin\nof the di\u000berent damping terms in the corresponding dynamical equations38,40. Apart from\nthe usually considered terms, we \fnd additional contributions for the case when sublattice-\nsymmetry is broken in the AFM36,41{45. Thus, FMR linewidth measurements o\u000ber a direct,\nparameter-free means of probing the sublattice asymmetry in AFMs, complementary to the\nspin pumping shot noise36.\nThis paper is organized as follows. We derive the Landau-Lifshitz-Gilbert (LLG) equa-\ntions for the two-sublattice model in Sec. II. The ensuing equations are solved for the\nresonance frequencies and decay rates of the uniform modes in a collinear magnet in Sec.\nIII. Section IV presents the application of the phenomenology to describe a compensated\nferrimagnet and spin pumping mediated Gilbert damping. The case of AFMs is discussed in\nSec. V. We comment on the validity and possible generalizations of the theory in Sec. VI.\nThe paper is concluded with a summary in Sec. VII. The discussion of a generalized Rayleigh\n3dissipation functional and properties of the damping matrix is deferred to the appendix.\nII. MAGNETIZATION DYNAMICS AND GILBERT DAMPING\nWe consider a two-sublattice magnet described by classical magnetization \felds MMMA\u0011\nMMMA(rrr;t) andMMMB\u0011MMMB(rrr;t) corresponding to the sublattices AandB. The system is\ncharacterized by a magnetic free energy F[MMMA;MMMB] with the magnetization \felds assumed\nto be of constant magnitudes MA0andMB0. Here, the notation F[ ] is employed to emphasize\nthat the free energy is a functional over the magnetization \felds, i.e. an integration of the\nfree energy density over space.\nThe undamped magnetization dynamics is described by equating the time derivative of\nthe spin angular momentum associated with the magnetization to the torque experienced\nby it. The resulting Landau-Lifshitz equations for the two \felds may be written as:\nd\ndt\u0012MMMA;B\n\u0000j\rA;Bj\u0013\n=\u0000_MMMA;B\nj\rA;Bj=MMMA;B\u0002\u00160HHHA;B; (1)\nwhere\rA;B(<0) are the gyromagnetic ratios for the two sublattices, and HHHA;Bare the\ne\u000bective magnetic \felds experienced by the respective magnetizations. This expression of\nangular momentum \row may be derived systematically within the Lagrangian classical \feld\ntheory31. The same formalism also allows to account for a restricted form of damping via\nthe so-called dissipation functional R[_MMMA;_MMMB] in the generalized equations of motion:\nd\ndt\u000eL[\u0001]\n\u000e_MMMA;B\u0000\u000eL[\u0001]\n\u000eMMMA;B=\u0000\u000eR[_MMMA;_MMMB]\n\u000e_MMMA;B; (2)\nwhereL[\u0001]\u0011 L [MMMA;MMMB;_MMMA;_MMMB] is the Lagrangian of the magnetic system. Here,\n\u000eL[\u0001]=\u000eMMMArepresents the functional derivative of the Lagrangian with respect to the var-\nious components of MMMA, and so on. The left hand side of Eq. (2) above represents the\nconservative dynamics of the magnet and reproduces Eq. (1) with31\n\u00160HHHA;B=\u0000\u000eF[MMMA;MMMB]\n\u000eMMMA;B; (3)\nwhile the right hand side accounts for the damping.\nThe Gilbert damping is captured by a viscous Rayleigh dissipation functional parame-\nterized by a symmetric matrix \u0011ijwithfi;jg=fA;Bg:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (4)\n4whereVis the volume of the magnet. The above form of the functional assumes the damping\nto be spatially homogeneous, isotropic, and independent of the equilibrium con\fguration.\nA more general form with a lower symmetry is discussed in appendix A. Including the\ndissipation functional via Eq. (2) leads to the following replacements in the equations of\nmotion (1):\n\u00160HHHA!\u00160HHHA\u0000\u0011AA_MMMA\u0000\u0011AB_MMMB; (5)\n\u00160HHHB!\u00160HHHB\u0000\u0011BB_MMMB\u0000\u0011AB_MMMA: (6)\nHence, the LLG equations for the two-sublattice magnet become:\n_MMMA=\u0000j\rAj(MMMA\u0002\u00160HHHA) +j\rAj\u0011AA\u0010\nMMMA\u0002_MMMA\u0011\n+j\rAj\u0011AB\u0010\nMMMA\u0002_MMMB\u0011\n; (7)\n_MMMB=\u0000j\rBj(MMMB\u0002\u00160HHHB) +j\rBj\u0011AB\u0010\nMMMB\u0002_MMMA\u0011\n+j\rBj\u0011BB\u0010\nMMMB\u0002_MMMB\u0011\n:(8)\nThese can further be expressed in terms of the unit vectors ^mmmA;B=MMMA;B=MA0;B0:\n_^mmmA=\u0000j\rAj(^mmmA\u0002\u00160HHHA) +\u000bAA\u0010\nmmmA\u0002_^mmmA\u0011\n+\u000bAB\u0010\n^mmmA\u0002_^mmmB\u0011\n; (9)\n_^mmmB=\u0000j\rBj(^mmmB\u0002\u00160HHHB) +\u000bBA\u0010\n^mmmB\u0002_^mmmA\u0011\n+\u000bBB\u0010\n^mmmB\u0002_^mmmB\u0011\n; (10)\nthereby introducing the Gilbert damping matrix ~ \u000bfor a two-sublattice system:\n~\u000b=0\n@\u000bAA\u000bAB\n\u000bBA\u000bBB1\nA=0\n@j\rAj\u0011AAMA0j\rAj\u0011ABMB0\nj\rBj\u0011ABMA0j\rBj\u0011BBMB01\nA; (11)\n\u000bAB\n\u000bBA=j\rAjMB0\nj\rBjMA0: (12)\nAs elaborated in appendix B, the positivity of the dissipation functional implies that the\neigenvalues and the determinant of ~ \u000bmust be non-negative, which is equivalent to the\nfollowing conditions:\n\u0011AA;\u0011BB\u00150; \u0011 AA\u0011BB\u0015\u00112\nAB=)\u000bAA;\u000bBB\u00150; \u000b AA\u000bBB\u0015\u000bAB\u000bBA: (13)\nThus, Eqs. (9) and (10) constitute the main result of this section, and introduce the damping\nmatrix [Eq. (11)] along with the constraints imposed on it [Eq. (12) and (13)] by the\nunderlying formalism.\n5III. UNIFORM MODES IN COLLINEAR GROUND STATE\nIn this section, we employ the phenomenology introduced above to evaluate the resonance\nfrequencies and the decay rates of the spatially homogeneous modes that can be probed in a\ntypical FMR experiment. We thus work in the macrospin approximation, i.e. magnetizations\nare assumed to be spatially invariant. Considering an antiferromagnetic coupling J(>0)\nbetween the two sublattices and parameterizing uniaxial easy-axis anisotropies via KA;B(>\n0), the free energy assumes the form:\nF[MMMA;MMMB] =Z\nVd3r\u0002\n\u0000\u00160H0(MAz+MBz)\u0000KAM2\nAz\u0000KBM2\nBz+JMMMA\u0001MMMB\u0003\n;(14)\nwhereH0^zzzis the applied magnetic \feld. The magnet is assumed to be in a collinear ground\nstate:MMMA=MA0^zzzandMMMB=\u0000MB0^zzzwithMA0>M B0. Employing Eq. (3) to evaluate the\ne\u000bective \felds, the magnetization dynamics is expressed via the LLG equations (9) and (10).\nConsidering MMMA=MAx^xxx+MAy^yyy+MA0^zzz,MMMB=MBx^xxx+MBy^yyy\u0000MB0^zzzwithjMAx;Ayj\u001c\nMA0,jMBx;Byj \u001cMB0, we linearize the resulting dynamical equations. Converting to\nFourier space via MAx=MAxexp (i!t) etc. and switching to circular basis via MA\u0006(B\u0006)=\nMAx(Bx)\u0006iMAy(By), we obtain two sets of coupled equations expressed succinctly as:\n0\n@\u0006!\u0000\nA\u0000i!\u000b AA\u0000\u0010\nj\rAjJMA0+i!\u000b ABMA0\nMB0\u0011\n\u0010\nj\rBjJMB0+i!\u000b BAMB0\nMA0\u0011\n\u0006!+ \n B+i!\u000b BB1\nA0\n@MA\u0006\nMB\u00061\nA=0\n@0\n01\nA; (15)\nwhere we de\fne \n A\u0011j\rAj(JMB0+ 2KAMA0+\u00160H0) and \n B\u0011j\rBj(JMA0+ 2KBMB0\u0000\n\u00160H0). Substituting !=!r\u0006+i!i\u0006into the ensuing secular equation, we obtain the\nresonance frequencies !r\u0006to the zeroth order and the corresponding decay rates !i\u0006to the\n\frst order in the damping matrix elements:\n!r\u0006=\u0006(\nA\u0000\nB) +p\n(\nA+ \n B)2\u00004J2j\rAjj\rBjMA0MB0\n2; (16)\n!i\u0006\n!r\u0006=\u0006!r\u0006(\u000bAA\u0000\u000bBB) +\u000bAA\nB+\u000bBB\nA\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000: (17)\nIn the expression above, Eq. (16) and Eq. (17), we have chosen the positive solutions of\nthe secular equations for the resonance frequencies. The negative solutions are equal in\nmagnitude to the positive ones and physically represent the same two modes. The positive-\npolarized mode in our notation corresponds to the typical ferromagnetic resonance mode,\nwhile the negative-polarized solution is sometimes termed `antiferromagnetic resonance'25.\n6020406080100120\n0 0.2 0.4 0.6 0.8 10.040.050.060.070.080.090.1FIG. 1. Resonance frequencies and normalized decay rates vs. the applied \feld for a quasi-\nferromagnet ( MA0= 5MB0).j\rAj=j\rBj= 1;1:5;0:5 correspond to solid, dashed and dash-dotted\nlines respectively. The curves in blue and red respectively depict the + and \u0000modes. The damping\nparameters employed are \u000bAA= 0:06,\u000bBB= 0:04 and\u000bAB= 0.\nIn order to avoid confusion with the ferromagnetic or antiferromagnetic nature of the un-\nderlying material, we call the two resonances as positive- and negative-polarized. The decay\nrates can further be expressed in the following form:\n!i\u0006\n!r\u0006=\u0016\u000b(\nA+ \n B)\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000\u0006\u0001\u0016\u000b; (18)\nwith \u0016\u000b\u0011(\u000bAA+\u000bBB)=2 and \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2. Eq. (18) constitutes the main result\nof this section and demonstrates that (i) asymmetric damping in the two sublattices is\nmanifested directly in the normalized decay rates of the two modes (Figs. 1 and 2), and\n(ii) o\u000b-diagonal components of the damping matrix may reduce the decay rates (Fig. 2).\nFurthermore, it is consistent with and reproduces the mode-dependence of the decay rates\nobserved in the numerical studies of some metallic AFMs37.\nTo gain further insight into the results presented in Eqs. (16) and (18), we plot the\n7resonance frequencies and the normalized decay rates vs. the applied magnetic \feld for a\ntypical quasi-ferromagnet, such as yttrium iron garnet, in Fig. 1. The parameters employed\nin the plot arej\rBj= 1:8\u00021011,MB0= 105,KA=KB= 10\u00007, andJ= 10\u00005in SI units,\nand have been chosen to represent the typical order of magnitude without pertaining to a\nspeci\fc material. The plus-polarized mode is lower in energy and is raised with an increasing\napplied magnetic \feld. The reverse is true for the minus-polarized mode whose relatively\nlarge frequency makes it inaccessible to typical ferromagnetic resonance experiments. As\nanticipated from Eq. (18), the normalized decay rates for the two modes di\u000ber when \u000bAA6=\n\u000bBB. Furthermore, the normalized decay rates are independent of the applied \feld for\nsymmetric gyromagnetic ratios for the two sublattices. Alternately, a measurement of the\nnormalized decay rate for the plus-polarized mode is able to probe the sublattice asymmetry\nin the gyromagnetic ratios. Thus it provides essential information about the sublattices\nwithout requiring the measurement of the large frequency minus-polarized mode.\nIV. SPECIFIC APPLICATIONS\nWe now examine two cases of interest: (i) the mode decay rate in a ferrimagnet close to\nits compensation temperature, and (ii) the Gilbert damping matrix due to spin pumping\ninto an adjacent conductor.\nA. Compensated ferrimagnets\nFMR experiments carried out on gadolinium iron garnet23,39\fnd an enhancement in the\nlinewidth, and hence the mode decay rate, as the temperature approaches the compensation\ncondition, i.e. when the two e\u000bective46sublattices have equal saturation magnetizations.\nThese experiments have conventionally been interpreted in terms of an e\u000bective single-\nsublattice model thereby ascribing the enhancement in the decay rate to an increase in the\nscalar Gilbert damping constant allowed within the single-sublattice model24. In contrast,\nexperiments probing the Gilbert parameter in a di\u000berent FiM via domain wall velocity\n\fnd it to be essentially unchanged around compensation28. Here, we analyze FMR in a\ncompensated FiM using the two-sublattice phenomenology developed above and thus address\nthis apparent inconsistency.\n8020406080100120\n1 2 3 4 500.050.10.150.20.250.3FIG. 2. Resonance frequencies and normalized decay rates vs. relative saturation magnetizations\nof the sublattices. The curves which are not labeled as + or \u0000represent the common normalized\ndecay rates for both modes. The parameters employed are the same as for Fig. 1 with \rA=\rB.\nThe compensation behavior of a FiM may be captured within our model by allowing\nMA0to vary while keeping MB0\fxed. The mode frequencies and normalized decay rates\nare examined with respect to the saturation magnetization variation in Fig. 2. We \fnd an\nenhancement in the normalized decay rate, consistent with the FMR experiments23,39, for a\n\fxed Gilbert damping matrix. The single-sublattice interpretation ascribes this change to a\nmodi\fcation of the e\u000bective Gilbert damping parameter24, which is equal to the normalized\ndecay rate within that model. In contrast, the latter is given by Eq. (18) within the\ntwo-sublattice model and evolves with the magnetization without requiring a modi\fcation\nin the Gilbert damping matrix. Speci\fcally, the enhancement in decay rate observed at\nthe compensation point is analogous to the so-called exchange enhancement of damping in\nAFMs47. Close to compensation, the FiM mimics an AFM to some extent.\nWe note that while the spherical samples employed in Ref. 23 are captured well by our\nsimple free energy expression [Eq. (14)], the interfacial and shape anisotropies of the thin\n9\flm sample employed in Ref. 39 may result in additional contributions to decay rates. The\nsimilarity of the observed linewidth trends for the two kinds of samples suggests that these\nadditional anisotropy e\u000bects may not underlie the observed damping enhancement. Quan-\ntitatively accounting for these thin \flm e\u000bects requires a numerical analysis, as discussed\nin Sec VI below, and is beyond the scope of the present work. Furthermore, domain forma-\ntion may result in additional damping contributions not captured within our single-domain\nmodel.\nB. Spin pumping mediated Gilbert damping\nSpin pumping34from a FM into an adjacent conductor has been studied in great detail35\nand has emerged as a key method for injecting pure spin currents into conductors48. The\nangular momentum thus lost into the conductor results in a contribution to the magnetic\ndamping on top of the intrinsic dissipation in the bulk of the magnet. A variant of spin\npumping has also been found to be the dominant cause of dissipation in metallic magnets37.\nThus, we evaluate the Gilbert damping matrix arising due to spin pumping from a two-\nsublattice magnet36into an adjacent conductor acting as an ideal spin sink.\nWithin the macrospin approximation, the total spin contained by the magnet is given by:\nSSS=\u0000MA0V^mmmA\nj\rAj\u0000MB0V^mmmB\nj\rBj: (19)\nThe spin pumping current emitted by the two-sublattice magnet has the following general\nform36:\nIIIs=~\neX\ni;j=fA;BgGij\u0010\n^mmmi\u0002_^mmmj\u0011\n; (20)\nwithGAB=GBA, where the spin-mixing conductances Gijmay be evaluated within di\u000berent\nmicroscopic models36,49{51. Equating the spin pumping current to \u0000_SSSand employing Eqs.\n(9) and (10), the spin pumping contribution to the Gilbert damping matrix becomes:\n\u000b0\nij=~Gijj\rij\neMi0V; (21)\nwhich in turn implies\n\u00110\nij=~Gij\neMi0Mj0V; (22)\n10for the corresponding dissipation functional. The resulting Gilbert damping matrix is found\nto be consistent with its general form and constraints formulated in Sec. II. Thus, employing\nthe phenomenology developed above, we are able to directly relate the magnetic damping in\na two-sublattice magnet to the spin-mixing conductance of its interface with a conductor.\nV. ANTIFERROMAGNETS\nDue to their special place with high symmetry in the two-sublattice model as well as the\nrecent upsurge of interest7{10,52{54, we devote the present section to a focused discussion on\nAFMs in the context of the general results obtained above. It is often convenient to describe\nthe AFM in terms of a di\u000berent set of variables:\nmmm=^mmmA+^mmmB\n2; nnn=^mmmA\u0000^mmmB\n2: (23)\nIn contrast with ^mmmAand ^mmmB,mmmandnnnare not unit vectors in general. The dynamical\nequations for mmmandnnnmay be formulated by developing the entire \feld theory, starting with\nthe free energy functional, in terms of mmmandnnn. Such a formulation, including damping,\nhas been accomplished by Hals and coworkers40. Here, we circumvent such a repetition and\ndirectly express the corresponding dynamical equations by employing Eqs. (9) and (10) into\nEq. (23):\n_mmm=\u0000(mmm\u0002\rm\u00160HHHm)\u0000(nnn\u0002\rn\u00160HHHn) +X\np;q=fm;ng\u000bm\npq(ppp\u0002_qqq); (24)\n_nnn=\u0000(mmm\u0002\rn\u00160HHHn)\u0000(nnn\u0002\rm\u00160HHHm) +X\np;q=fm;ng\u000bn\npq(ppp\u0002_qqq); (25)\nwith\n\rm\u00160HHHm\u0011j\rAj\u00160HHHA+j\rBj\u00160HHHB\n2; (26)\n\rn\u00160HHHn\u0011j\rAj\u00160HHHA\u0000j\rBj\u00160HHHB\n2; (27)\n\u000bm\nmm=\u000bn\nnm=\u000bAA+\u000bBB+\u000bAB+\u000bBA\n2; (28)\n\u000bm\nmn=\u000bn\nnn=\u000bAA\u0000\u000bBB\u0000\u000bAB+\u000bBA\n2; (29)\n\u000bm\nnn=\u000bn\nmn=\u000bAA+\u000bBB\u0000\u000bAB\u0000\u000bBA\n2; (30)\n\u000bm\nnm=\u000bn\nmm=\u000bAA\u0000\u000bBB+\u000bAB\u0000\u000bBA\n2: (31)\n11A general physical signi\fcance, analogous to \rA;B, may not be associated with \rm;nwhich\nmerely serve the purpose of notation here. The equations obtained above manifest new\ndamping terms in addition to the ones that are typically considered in the description\nof AFMs. Accounting for the sublattice symmetry of the antiferromagnetic bulk while\nallowing for the damping to be asymmetric, we may assume \rA=\rBandMA0=MB0, with\n\u0016\u000b\u0011(\u000bAA+\u000bBB)=2, \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2, and\u000bAB=\u000bBA\u0011\u000bod. Thus, the damping\nparameters simplify to\n\u000bm\nmm=\u000bn\nnm=\u0016\u000b+\u000bod; (32)\n\u000bm\nmn=\u000bn\nnn=\u0001\u0016\u000b; (33)\n\u000bm\nnn=\u000bn\nmn=\u0016\u000b\u0000\u000bod; (34)\n\u000bm\nnm=\u000bn\nmm=\u0001\u0016\u000b; (35)\nthereby eliminating the \\new\" terms in the damping when \u000bAA=\u000bBB. However, the sublat-\ntice symmetry may not be applicable to AFMs, such as FeMn, with non-identical sublattices.\nFurthermore, the sublattice symmetry of the AFM may be broken at the interface41{43via,\nfor example, spin mixing conductances36,45,55resulting in \u000bAA6=\u000bBB.\nThe resonance frequencies and normalized decay rates [Eqs. (16) and (18)] take a simpler\nform for AFMs. Substituting KA=KB\u0011K,\rA=\rB\u0011\r, andMA0=MB0\u0011M0:\n!r\u0006=\u0006j\rj\u00160H0+ 2j\rjM0p\n(J+K)K; (36)\n!i\u0006\n!r\u0006=J(\u0016\u000b\u0000\u000bod) + 2K\u0016\u000b\n2p\n(J+K)K\u0006\u0001\u0016\u000b\u0019(\u0016\u000b\u0000\u000bod)\n2r\nJ\nK+ \u0016\u000br\nK\nJ\u0006\u0001\u0016\u000b; (37)\nwhere we have employed J\u001dKin the \fnal simpli\fcation. The term /p\nK=J has typically\nbeen disregarded on the grounds K\u001cJ. However, recent numerical studies of damping in\nseveral AFMs37\fnd \u0016\u000b\u001d\u0016\u000b\u0000\u000bod>0 thus suggesting that this term should be comparable\nto the one proportional top\nJ=K and hence may not be disregarded. The expression above\nalso suggests measurement of the normalized decay rates as a means of detecting the sublat-\ntice asymmetry in damping. For AFMs symmetrical in the bulk, such an asymmetry may\narise due to the corresponding asymmetry in the interfacial spin-mixing conductance36,45,55.\nThus, decay rate measurements o\u000ber a method to detect and quantify such interfacial e\u000bects\ncomplementary to the spin pumping shot noise measurements suggested earlier36.\n12VI. DISCUSSION\nWe have presented a phenomenological description of Gilbert damping in two-sublattice\nmagnets and demonstrated how it can be exploited to describe and characterize the system\ne\u000bectively. We now comment on the limitations and possible generalizations of the formal-\nism presented herein. To begin with, the two-sublattice model is the simplest description of\nferri- and antiferromagnets. It has been successful in capturing a wide range of phenomenon.\nHowever, recent measurements of magnetization dynamics in nickel oxide could only be ex-\nplained using an eight-sublattice model56. The temperature dependence of the spin Seebeck\ne\u000bect in yttrium iron garnet also required accounting for more than two magnon bands57.\nA generalization of our formalism to a N-sublattice model is straightforward and can be\nachieved via a Rayleigh dissipation functional with N2terms, counting \u0011ijand\u0011jias sepa-\nrate terms. The ensuing Gilbert damping matrix will be N \u0002N while obeying the positive\ndeterminant constraint analogous to Eq. (13).\nIn our description of the collinear magnet [Eq. (14)], we have disregarded contributions\nto the free energy which break the uniaxial symmetry of the system about the z-axis. Such\nterms arise due to spin-nonconserving interactions58, such as dipolar \felds and magnetocrys-\ntalline anisotropies, and lead to a mixing between the plus- and minus-polarized modes30.\nIncluding these contributions converts the two uncoupled 2 \u00022 matrix equations [(15)] into\na single 4\u00024 matrix equation rendering the solution analytically intractable. A detailed\nanalysis of these contributions30shows that their e\u000bect is most prominent when the two\nmodes are quasi-degenerate, and may be disregarded in a \frst approximation.\nIn evaluating the resonance frequencies and the decay rates [Eqs. (16) and (18)], we\nhave assumed the elements of the damping matrix to be small. A precise statement of the\nassumption employed is !i\u001c!r, which simply translates to \u000b\u001c1 for a single-sublattice\nferromagnet. In contrast, the constraint imposed on the damping matrix within the two-\nsublattice model by the assumption of small normalized decay rate is more stringent [Eq.\n(18)]. For example, this assumption for an AFM with \u000bAB= \u0001\u0016\u000b= 0 requires \u0016 \u000b\u001c\np\nK=J\u001c1. This stringent constraint may not be satis\fed in most AFMs37, thereby\nbringing the simple Lorentzian shape description of the FMR into question. It can also be\nseen from Fig. 2 that the assumption of a small normalized decay rate is not very good for\nthe chosen parameters.\n13VII. SUMMARY\nWe have developed a systematic phenomenological description of the Gilbert damping\nin a two-sublattice magnet via inclusion of a Rayleigh dissipation functional within the La-\ngrangian formulation of the magnetization dynamics. Employing general expressions based\non symmetry, we \fnd cross-sublattice Gilbert damping terms in the LLG equations in con-\nsistence with other recent \fndings36{38. Exploiting the phenomenology, we explain the en-\nhancement of damping23,39in a compensated ferrimagnet without requiring an increase in\nthe damping parameters28. We also demonstrate approaches to probe the various forms\nof sublattice asymmetries. Our work provides a uni\fed description of ferro- via ferri- to\nantiferromagnets and allows for understanding a broad range of materials and experiments\nthat have emerged into focus in the recent years.\nACKNOWLEDGMENTS\nA. K. thanks Hannes Maier-Flaig and Kathrin Ganzhorn for valuable discussions. We\nacknowledge \fnancial support from the Alexander von Humboldt Foundation, the Research\nCouncil of Norway through its Centers of Excellence funding scheme, project 262633, \\QuS-\npin\", and the DFG through SFB 767 and SPP 1538.\nAppendix A: Generalized Rayleigh dissipation functional\nAs compared to the considerations in Sec. II, a more general approach to parameterizing\nthe dissipation functional is given by:\nR[_MMMA;_MMMB] =1\n2Z\nVZ\nVd3r0d3rX\np;q=fA;BgX\ni;j=fx;y;zg_Mpi(rrr)\u0011ij\npq(rrr;rrr0)_Mqj(rrr0): (A1)\nThis form allows to capture the damping in an environment with a reduced symmetry.\nHowever, the larger number of parameters also makes it di\u000ecult to extract them reliably\nvia typical experiments. The above general form reduces to the case considered in Sec. II\nwhen\u0011ij\npq(rrr;rrr0) =\u0011pq\u000eij\u000e(rrr\u0000rrr0) and\u0011pq=\u0011qp. Furthermore, the coe\u000ecients \u0011ij\npqmay depend\nuponMMMA(rrr) andMMMB(rrr) as has been found in recent numerical studies of Gilbert damping\nin AFMs37.\n14Appendix B: Damping matrix\nThe Rayleigh dissipation functional considered in the main text is given by:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (B1)\nwhich may be brought into the following concise form with the notation~_MMM\u0011[_MMMA_MMMB]|:\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_MMM|~\u0011~_MMM; (B2)\nwhere ~\u0011is the appropriate matrix given by:\n~\u0011=0\n@\u0011AA\u0011AB\n\u0011AB\u0011BB1\nA: (B3)\nConsidering an orthogonal transformation~_MMM=~Q~_M, the dissipation functional can be\nbrought to a diagonal form\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_M|~Q|~\u0011~Q~_M; (B4)\nwhere ~Q|~\u0011~Qis assumed to be diagonal. 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Here we use theoretical calculations\nand numerical simulations to determine the mean switching time of antiferromagnetic nanoparticles\nin the superparamagnetic limit. It is demonstrated that the thermal stability is drastically reduced\ncompared to ferromagnetic particles in the limit of low Gilbert damping, attributed to the exchange\nenhancement of the attempt frequencies. It is discussed how the system parameters have to be\nengineered in order to optimize the switching rates in antiferromagnetic nanoparticles.\nI. INTRODUCTION\nIn the field of spintronics, the storage, transfer and\nprocessing of information is based on the spin magnetic\nmoment of electrons. Conventional spintronic devices are\nmainly based on ferromagnetic (FM) systems. However,\nrecent advances in understanding and controlling anti-\nferromagnetic (AFM) materials have led to an increasing\ninterest in AFM spintronics [1–7]. Possible advantages\nof spintronic devices based on AFM materials include\ntheir lack of stray fields, which normally destroys single-\ndomain states and leads to an interaction between bit\npatterns; the low susceptibility to external fields; and the\nrich choice of new materials, including a variety of AFM\ninsulators. Moreover, AFM spin dynamics are found to\nbe faster than those of FMs [4, 8–10].\nFor many applications, the size of magnetic structures\nwill have to be scaled down to the nanometer regime,\nwhere, eventually, thermal excitations will reduce the\nstability of the magnetic state. In single-domain FM\nnanoparticles this is known as the superparamagnetic\nlimit[11],wherethewholestructurecanbedescribedasa\nsinglemacroscopicmagneticdipole. Besidestheirtechno-\nlogicalrelevance, superparamagneticparticleshavefound\ntheir uses in biomedical applications [12] as well as in\nrock magnetism [13]. Analogously, a single-domain AFM\nnanoparticlemaybedescribedbyamacroscopicNéelvec-\ntor, being the difference of the two sublattice magneti-\nzations. The spontaneous switching of the Néel vector\nunder thermal fluctuations constitutes the superparam-\nagnetic limit in AFMs. In this context, it was shown re-\ncently [14] that thermally activated superparamagnetic\nreversal enhances the current-induced switching rates in\nAFM Hall cross devices. Furthermore, AFM nanoparti-\ncles play an important role in biological molecules such\nas the natural [15] and synthetic forms [16] of the iron-\nstorage protein ferritin, and in the field of geochemistry\n\u0003rozsa.levente@physnet.uni-hamburg.de\nyunai.atxitia@fu-berlin.de[17].\nThe thermal stability of FM nanoparticles has been\nstudied extensively in the past [18–23]. An analytical\nformula for the thermal switching rate in the superpara-\nmagnetic limit was first given by Brown [24] based on\nthe stochastic Landau–Lifshitz–Gilbert equation [25, 26].\nThe mean switching time in AFM nanoparticles has been\ninvestigated in significantly less works so far [27], and the\nanalytical studies [28, 29] have been restricted to the case\nofuncompensatedAFMswithafinitemagnetization. For\ncurrenttechnologicalapplicationsofcompensatedAFMs,\na simple but accurate formula explaining the role of the\ninteraction parameters in the reversal process seems to\nbe lacking.\nHere we theoretically investigate the switching rate in\ncompensated AFM nanoparticles. By deriving an ana-\nlytical expression, it is demonstrated that the coupling\nbetween the Néel vector and the magnetization leads to\nsignificantly faster dynamical processes than in FMs. In\nthe limit of low Gilbert damping, this causes strong os-\ncillations in the Néel vector direction during the reversal\nprocess and an exchange enhancement of the switching\nrate compared to Brown’s formula applicable to FMs.\nThe accuracy of the analytical formula is confirmed by\nspin dynamics simulations. By analyzing the effect of\ndifferent material parameters on the switching rate, the\nadvantages and disadvantages of AFMs over FMs are dis-\ncussed for various applications. Our findings contribute\nto the understanding of thermal effects in AFM nanos-\ntructures, their stability as well as switchability, where\nthe latter is often affected by heating effects due to ap-\nplied currents or laser excitation.\nThe paper is organized as follows. The analytical for-\nmulae for the switching times in uniaxial FM and AFM\nnanoparticles are discussed in Sec. II. The spin dynam-\nics simulations are introduced in Sec. III. The necessary\nconditions for the application of the macrospin model\nto the results of atomistic simulations are detailed in\nSec. IV. The switching times between FMs and AFMs\nare compared in Sec. V, and the results are summarized\nin Sec. VI.arXiv:1808.07665v3  [cond-mat.mes-hall]  27 Aug 20192\nII. ANALYTICAL MODEL\nA. Axially symmetric FM nanoparticle\nFor the analytical investigations we will focus on the\nsimplest example of a magnetic nanoparticle, which\nswitches by coherent rotation between two stable mag-\nnetic states separated by an energy barrier \u0001E, as\nsketched in Fig. 1(a). We will rely on the so-called single-\ndomain approximation, where the total magnetization of\nthe nanoparticle is described by a single magnetic mo-\nment or macrospin in the FM case. This remains valid\nif the particle size stays below the exchange length, cor-\nresponding to the characteristic size of domain walls in\nthe system. The dynamics can be calculated within the\nframework of the macroscopic Landau–Lifshitz–Gilbert\nequation [25, 26]\n_m=\u0000m\u0002\u0012\n\rhm\u0000\u000b_m\nm0\u0013\n; (1)\nwhere\u000bis the Gilbert damping constant, \ris the gyro-\nmagnetic ratio, mthe magnetization of the nanoparti-\ncle,m0the magnitude of the magnetization, and hm=\n\u0000\u000emFthe effective field, where Fis the magnetic free\nenergy of the system. For simplicity, in this work we\nrestrain the discussion to a uniaxial particle, with the\nfree-energy density f=\u0000Ham2\nz=(2m0), whereHa=\n2DzN=(Vm0)is the anisotropy field, Dzis the anisotropy\nenergy of a single spin, Nis the number of spins in the\nnanoparticle and Vis its volume. Equation (1) describes\nthe rotational motion of the macrospin, with its length\nfixed atjmj=m0. In this case, the free energy has two\nminima,mz=m0= 1andmz=m0=\u00001, with the energy\nbarrier between them being \u0001E=Ham0V=2 =DzN.\nThermalactivationallowsthenanoparticletojumpbe-\ntween the free energy minima with a characteristic time\nscale. In the limit of low temperatures, kBT\u001c\u0001E, the\nswitching time for coherent rotation over the barrier was\nderived by Brown [24],\n\u001cfm=1 +\u000b2\n\u000b!\u00001\nar\n\u0019kBT\nDzNeDzN\nkBT: (2)\nThis expression is of the form of the exponential Néel–\nArrhenius law \u001c=\u001c0e\u0001E=k BT, with the energy barrier\n\u0001E=DzNdetermined above. The prefactor \u001c0is called\nthe inverse attempt frequency. Its first factor is related\nto the damping dependence of the switching time, clearly\nwith a minimum at \u000b= 1. The second factor is the\nprecessional time scale of the system, with !a=\rHa=\n\r2DzN=(Vm0). The weak temperature dependence of\nthe prefactor is attributed to the Goldstone mode of the\nsystem at the top of the energy barrier in the axially\nsymmetric free-energy expression.\n(a)\n(b)FIG. 1. Comparison of reversal mechanisms in AFM (left)\nand FM (right) nanoparticles. While the energy barriers \u0001E\nforcoherentrotationareidentical(a), theattemptfrequencies\nstrongly differ caused by the different dynamical properties of\nthe eigenmodes (b). Springs in the AFM case represent that\nenergy may be transferred between anisotropy and exchange\ncontributions, while the latter is not present in the macrospin\ndescription of FMs.\nB. Nonaxially symmetric FM nanoparticles\nEquation (2) is valid for all values of the damping pa-\nrameter\u000b. As pointed out in, e.g., Ref. [30], its simple\nformcanbeattributedtothefactthattheFokker–Planck\nequation derived from the stochastic Landau–Lifshitz–\nGilbert equation simplifies to an ordinary differential\nequation for the polar angle variable cos#=mz=m0.\nIf the rotational symmetry of the system is broken, for\nexample, by a tilted external magnetic field [31], then the\nfreeenergyFmustdescribethecouplingbetweenpolar #\nand azimuthal 'variables, or longitudinal and transver-\nsal degrees of freedom. This transforms the Fokker–\nPlanck equation into a partial differential equation which\nis significantly more difficult to handle.\nFormassiveparticles, escaperatesfromanenergymin-\nimum were first systematically derived by Kramers [32],\nwho differentiated between intermediate-to-high damp-\ning (IHD) and very-low-damping (VLD) limits. For IHD,\nit can be assumed that the system is in thermal equilib-\nrium both close to the energy minimum (min) and in the\nvicinity of the saddle point (sp) which has to be crossed\nduring the escape. The IHD limit of nonaxially symmet-\nric FM nanoparticles was derived by Brown [33], which\nwas later revealed to be [34–39] a special case of Langer’s\n[40] expression for multiple degrees of freedom. Within\nthis description, the Hamiltonian or the free energy is\napproximated by a harmonic expansion around the mini-\nmum and close to the saddle point, while the equations of\nmotion are linearized near the saddle point. The energy\nscale of thermal fluctuations is required to be much lower3\nthan the energy barrier protecting the metastable state,\nleading to an Arrhenius-like formula. Applications to\nmagnetic systems can be found in, e.g., Refs. [34–37, 41].\nThe generalization to an arbitrary number of Goldstone\nmodesaspresentedinEq.(3)belowisbasedonharmonic\ntransition-state theory [42], which differs from Langer’s\ntheory in applying a dynamical prefactor independent of\nthe damping.\nThe switching time \u001cIHDmay be expressed by the for-\nmula\n\u001cIHD=2\u0019\n\u0015+;spVmin\nVsp(2\u0019kBT)Psp\u0000Pmin\n2vuutQ0\njj\"j;spj\nQ0\nj\"j;mineEsp\u0000Emin\nkBT;\n(3)\nwhereEis the energy of the given configuration and \"j\ndenotes the eigenvalues of the harmonic Hamiltonian in\nthe equilibrium state. Ideally, all eigenvalues in the min-\nimum are positive, and there is a single negative eigen-\nvalue (hence the absolute value) in the first-order saddle\npoint, along which direction the transition takes place.\nHowever, the system may possess zero-energy Goldstone\nmodeswhicharetobehandledseparately. Thesemustbe\nleft out of the eigenvalue products, hence the prime nota-\ntion. Each of these will contribute ap2\u0019kBTfactor in-\nstead, with Pdenoting the number of Goldstone modes.\nVis the phase space volume belonging to the Goldstone\nmodes. Finally, \u0015+;spis the single positive eigenvalue of\nthe linearized equations of motion in the saddle point.\nThis determines how fast the system crosses the transi-\ntion state. The derivation of Eq. (2) based on Eq. (3) is\ngiven in Appendix A.\nHowever, in the VLD limit the approximations of\nLanger’s theory break down, since the weak coupling be-\ntween the system and the heat bath encapsulated in the\ndamping parameter is no longer sufficient for ensuring\nthermal equilibrium at higher energy values. In order to\nachieve agreement with the fluctuation–dissipation theo-\nrem, one has to calculate the energy dissipated during a\nsingle precession along the energy contour including the\nsaddle point, and ensure that this is low compared to the\nthermal energy kBTin the VLD case. Such a calcula-\ntion for FM nanoparticles was carried out by Klik and\nGunther [43]. Finally, the missing connection between\nthe VLD and IHD limits, the solution of the so-called\nKramers turnover problem, was derived by Mel’nikov\nand Meshkov [44] for massive particles, and adapted to\nnonaxially symmetric FM nanoparticles by Coffey et al.\n[35, 41]. This can be summarized in the formula\n\u001c=A\u00001\u0012\u000bS\nkBT\u0013\n\u001cIHD; (4)\nwhereA\u0012\u000bS\nkBT\u0013\nis the depopulation factor,\nA(x) =e1\n2\u0019R1\n\u00001ln\u0012\n1\u0000e\u0000x(1\n4+y2)\u0013\n1\n1\n4+y2dy;(5)and\u001cIHDis the switching time in the IHD limit given by\nEq. (3). The validity of the general formula for FMs was\nlater thoroughly confirmed by the numerical solution of\nthe Fokker–Planck equation, spin dynamics simulations\nand experiments; see, e.g., Refs. [29, 30, 39].\nC. Axially symmetric AFM nanoparticle\nFor AFMs, to the best of our knowledge, analytical\nformulae similar to Eq. (2) remain unknown. Only a few\nrecent works have addressed the problem [28, 29]. How-\never, they assumed AFM nanoparticles with uncompen-\nsated magnetic moments, attributed to finite-size effects\nand lattice defects in naturally occurring nanoparticles\n[45, 46]. In this limit, the AFM was effectively described\nas a FM with a very small magnetic moment. In spin-\ntronics applications, it is possible to prepare completely\ncompensated AFM structures, for example by atom ma-\nnipulation as demonstrated in Ref. [27]. The dynamics in\nAFMs are described by coupled equations of motion for\nthe Néel vector and the magnetization [47–50], expected\nto lead to a qualitatively different behavior. Dissipa-\ntive dynamics in two-sublattice AFMs may be derived by\nconsidering two coupled Landau–Lifshitz–Gilbert equa-\ntions (1) for the sublattice magnetizations m1andm2.\nThese are transformed to the dynamical variables of the\nmagnetization m= (m1+m2)=2and the Néel vector\nn= (m1\u0000m2)=2. At low temperature, it is reason-\nable to assume that the Néel vector conserves its length\njnj=m0and only undergoes rotational motion. The\nmagnetization remains perpendicular to the Néel vector,\nn\u0001m= 0, since in a compensated AFM a finite magneti-\nzation may only be formed by canting the two sublattice\nmagnetizations perpendicularly to their original antipar-\nallel orientation. This leads to the equations of motion\n[48]\n_n=\u0000n\u0002\u0012\n\rhm\u0000\u000b_m\nm0\u0013\n; (6)\n_m=\u0000m\u0002\u0012\n\rhm\u0000\u000b_m\nm0\u0013\n\u0000n\u0002\u0012\n\rhn\u0000\u000b_n\nm0\u0013\n;(7)\nwhere hm;n=\u0000\u000em;nFare the effective fields acting on\nthe magnetization and the Néel vector, respectively. The\nfree-energydensityofanaxiallysymmetricsingle-domain\nAFM particle reads f=Hem2=(2m0)\u0000Han2\nz=(2m0),\nwithHe=qJN=(Vm0)being the exchange field de-\nscribing the coupling between the sublattices, where q\nis the number of nearest neighbors and Jthe exchange\nconstant in the corresponding atomistic model. In the\nfollowing, we will assume that qJ \u001dDz, which is true\nfor practically all magnetic materials.\nAlthough the AFM nanoparticle is still axially sym-\nmetric, it fundamentally differs from its FM counter-\npart described in Sec. IIA. As illustrated in Fig. 1(b),\nin AFMs the anisotropy energy assigned to the zcom-\nponent of the Néel vector nzmay transform into the ex-4\nchange energy between the sublattices, leading to a fi-\nnite magnetization m, even in the conservative case. In\ncomparison, the FM particle may only perform a preces-\nsion around the easy axis with a constant polar angle\n#. Consequently, one has to rely on the theory for cou-\npled degrees of freedom, such as in the case of nonaxially\nsymmetric FM systems in Sec. IIB, when deriving the\nswitching time in AFMs. Applying Eq. (4) to this prob-\nlem leads to the expression\n\u001cafm=A\u00001\u0012\u000bS\nkBT\u00131 +\u000b2\n\u000b!\u00001\nafmr\n\u0019kBT\nDzNeDzN\nkBT;(8)\nwith the derivation given in Appendix B.\nIn comparison with Eq. (2), one can observe that\nthe energy barrier \u0001E=DzNbetween the minima at\nnz=m0= 1andnz=m0=\u00001remainsthesameinEq.(8),\nas long as all individual spins rotate coherently during\nswitching. Similarly, the temperature-dependent square-\nroot term attributed to the axial symmetry is preserved.\nOn the other hand, the frequency !ais replaced by !afm,\n!afm=\n\rN\nVm02\n4\u0012\nDz\u00001\n2qJ\u0013\n+s\u00121\n2qJ+Dz\u00132\n+2DzqJ\n\u000b23\n5:\n(9)\nThe transition between the IHD and the VLD limits is\ngoverned by the ratio of the thermal energy kBTand the\nenergy loss per cycle on the contour including the saddle\npoint,\n\u000bS=\u000bN\u001216D2\nz\n3p2DzqJ+ 4p\n2DzqJ\u0013\n;(10)\nfor the derivation see Appendix C.\nIn order to highlight the differences and similarities\nbetween the FM and AFM cases, appropriate asymp-\ntotic expressions are derived. On the one hand, in the\nlimit of high damping \u000b\u001d1, one has!afm\u0019!aand\nA(\u000bS=(kBT))\u00191due to the strong energy dissipation\n\u000bS\u001dkBT, leading to\n\u001cafm;\u000b\u001d1\u0019\u000b!\u00001\nar\n\u0019kBT\nDzNeDzN\nkBT; (11)\nwhich coincides with the asymptotic behavior for FMs,\nEq. (2).\nOn the other hand, significant deviations may be ob-\nserved between the two types of systems in the limit of\nlow damping. For \u000b\u001c1, the characteristic frequency\nof AFMs is \u000b!afm\u0019p2DzqJ=p\nqJ=(2Dz)!a, indi-\ncating that the dynamics are exchange-enhanced com-\npared to FMs. Furthermore, the depopulation factor\nmay be approximated as A(\u000bS=(kBT))\u0019\u000bS=(kBT)\u0019\n\u000b4Np2DzqJ=(kBT)for slow energy dissipation and\nqJ\u001dDz. The VLD limit of Eq. (8) reads\n\u001cafm;\u000b\u001c1\u00191\n\u000bkBT\n4qJN!\u00001\nar\n\u0019kBT\nDzNeDzN\nkBT:(12)In Eq. (12), the switching time is inversely propor-\ntional to the damping parameter, as expected from the\nfluctuation–dissipation theorem [32]. Furthermore, it is\nreduced by a factor of kBT=(4qJN)compared to the ap-\npropriate limit of Eq. (2). The typical value of intrinsic\ndamping in magnetic materials is \u000b= 0:001\u00000:01, e.g.,\n\u000b= 0:0025was determined for Mn 2Au in Ref. [51]. This\nmeans that the switching time in AFMs could be up to\nseveral orders of magnitude shorter than in FMs, which\neffectively means much less thermal stability.\nThe high- and low-damping limits of the AFM switch-\ningtime, definedbyEqs.(11)and(12), maybeconnected\nby the simplified formula\n\u001cafm;asymptotic =kBT\n4qJN+\u000b2\n\u000b!\u00001\nar\n\u0019kBT\nDzNeDzN\nkBT;(13)\nwhich has an analogous form to Eq. (2). This clearly\nexpresses the difference in the behavior between FMs\nand AFMs: while for the former the minimal switch-\ning time is found for \u000bfm,min = 1, for the latter this\nvalue now depends on the system parameters, \u000bafm,min =p\nkBT=(4qJN), being decreased due to the exchange in-\nteraction. Since for high \u000bvalues the switching times in\nFMsandAFMscoincide, whiletheminimumisshiftedto\nlower\u000bvalues in AFMs, this implies that AFM nanopar-\nticles are significantly less resistant against thermal fluc-\ntuations at low damping compared to their FM counter-\nparts. However, note that in the immediate vicinity of\n\u000bafm,min, Eq. (8) is expected to give a more accurate de-\nscription than Eq. (13), since the former includes a more\nprecise interpolation between the VLD and IHD limits\nexactly in this turnover regime.\nIII. SPIN DYNAMICS SIMULATIONS\nTo test the validity of Eqs. (2) and (8), we performed\natomistic spin dynamics simulations. For the description\nof the magnetic system, we introduce the classical atom-\nistic spin Hamiltonian\nH=\u00071\n2X\nhi;jiJSiSj\u0000X\niDzS2\ni;z:(14)\nHere the Sivariables denote unit vectors on a simple cu-\nbic lattice and Jis the Heisenberg exchange interaction\nbetween atoms at nearest-neighbor sites iandj. For the\n\u0000sign in Eq. (14) the ground state is FM, while for the\n+sign it is AFM. Dz>0is the single-ion magnetocrys-\ntalline anisotropy, implying that the ground state of the\nsystem lies along the zdirection.\nThe time evolution of the unit vectors Siis described\nby the Landau–Lifshitz–Gilbert equation,\n(1 +\u000b2)\u0016s_Si=\u0000\rSi\u0002[Hi+\u000b(Si\u0002Hi)];(15)\nwhere\u0016sdenotes the magnetic moment of a single spin\nand\u000bis the Gilbert damping as in the macrospin model.5\nBy including a Langevin thermostat, the equilibrium\nand nonequilibrium thermodynamic properties can be\nobtained in the classical approximation. The effective\nlocal magnetic field at lattice site iis\nHi=\u0000@H\n@Si+\u0018i(t); (16)\nwhereHis given by Eq. (14) in the present case and \u0018i\nis a field-like stochastic process. Here we consider the\nwhite-noise limit [52], with the first and second moments\nh\u0018i(t)i=0;h\u0018i;a(0)\u0018j;b(t)i=2\u000bkBT\u0016s\n\r\u000eij\u000eab\u000e(t);\n(17)\nwhereaandbdenote the Cartesian components.\nIV. CORRESPONDENCE BETWEEN THEORY\nAND SIMULATIONS\nA. Temperature-dependent effective parameters\nFor a direct comparison of Eqs. (2) and (8) with the\nresults of the spin dynamics simulations, it has to be en-\nsuredthat theassumptionswhichtheanalyticalformulae\narebasedonaresatisfiedbytheatomisticmodel. Aslong\nas the linear size of the system is shorter than a charac-\nteristic length scale on the order of the exchange length,\nLe\u0018p\nJ=Dz, it is expected that coherent rotation is\nthe primary mechanism of magnetization reversal in the\nnanoparticles. Above this threshold, the nucleation of\na pair of domain walls becomes energetically favorable\ncompared to the energy barrier which has to be overcome\nby coherent rotation [53–57].\nEven for small particles, one has to take into account\nthat in the atomistic model the thermal fluctuations de-\ncreasem0, the equilibrium length of the magnetization in\nFMsandoftheNéelvectorinAFMs[58]. Inearlierpubli-\ncations for FM systems [59], it was found that the dimen-\nsionless magnetization may be well approximated by the\nphenomenological relation m0V=(N\u0016s) = (1\u0000T=Tc)1=3\nfor 3d Heisenberg models. Furthermore, one has to ac-\ncount for finite-size effects. Small systems such as the\nnanoparticles considered here have a reduced magneti-\nzation compared to the bulk at a given temperature,\nas a result of lower coordination numbers at the sur-\nfaces. For 3d Heisenberg spin models, finite-size-scaling\ntheory provides a value for the apparent Curie temper-\nature as a function of the size L(linear characteristic\nsize of the nanoparticle), Tc(L)=T1\nc= 1\u0000(d0=L)1=\u0017,\nwhere the parameter d0corresponds to the characteristic\nexchange length, and \u0017to the critical exponent. A re-\ncentwork[60]inFMFePtnanoparticlesusingsimilarpa-\nrameters to our simulations has estimated d0= 0:4nm,\nto be compared to a lattice constant of a= 0:38nm,\nand\u0017= 0:856. The critical temperature of the cu-\nbic Heisenberg model in the thermodynamic limit wasfound to be kBT1\nc= 1:443J[61]. In this work we per-\nformed simulations for a cubic nanoparticle composed of\nN= 43= 64spins; therefore, the lateral size is 4 spins,\nmeaningd0=L= 0:4=(0:38\u00024) = 0:238in the finite-\nsize-scaling expression, leading to Tc(L) = 1:173J. We\nfound that the phenomenological relation using this crit-\nical temperature was in agreement with spin dynamics\nsimulations of the dimensionless order parameter at the\ntemperature ranges where coherent reversal is dominant.\nIn the analytical expressions the effect of the re-\nduced order parameter may be considered by assuming\ntemperature-dependent magnetic parameters in Eqs. (2)\nand (8) [58],\nDz=Dz\u0012m0V\nN\u0016s\u00133\n; (18)\nJ=J\u0012m0V\nN\u0016s\u00132\n: (19)\nThe cubic dependence of the anisotropy on the dimen-\nsionless magnetization expressed in Eq. (18) is the result\nof the Callen–Callen theory [62, 63]. The quadratic de-\npendence of the exchange in Eq. (19) may be derived\nfrom the random phase approximation [64].\nFurthermore, the reduced coordination number qat\nthe surface also directly affects Eq. (8). Here we substi-\ntuted the mean value of the number of nearest neighbors:\nfor a nanoparticle composed of N= 64spins in simple\ncubic arrangement, q= 6for the spins inside (23= 8),\nq= 5for the spins at the faces (6\u00022\u00022 = 24),q= 4\nfor the spin at the edges (12\u00022 = 24), andq= 3for the\nspins at the corners (8), thusqavg= 4:5.\nB. Oscillations in the order parameter in the VLD\nlimit of AFM nanoparticles\nA further requirement for an accurate comparison be-\ntween simulations and analytical expressions is that the\nidentified switching events in the simulations have to cor-\nrespond to the reversals described by the theory [39]. For\nuniaxial nanoparticles with easy axis along the zdirec-\ntion, the following criteria may be identified. First, the z\ncomponent of the order parameter mornhas to change\nsign, and thereafter cross a threshold value governed by\nthe equilibrium value m0at the given temperature. Dur-\ning the process, the energy of the particle increases while\ncrossing the energy barrier, before decreasing again when\ncoming to rest in the other energy minimum; see Sup-\nplemental Videos 1 and 2 [65] for an illustration of this\nprocess.\nFor FMs, the sign change of mzis always accompa-\nnied by an increase in the anisotropy energy. On the\nother hand, in AFM nanoparticles the energy can be\ntransformed between the anisotropy contribution of the\nNéel vector and the exchange contribution of the magne-\ntization, meaning that nzmay switch sign even if the\ntotal energy of the system remains constant. In the6\n(a)\n(b)\nFIG. 2. Illustration of the switching events in the AFM\nnanoparticle for (a) low ( \u000b= 0:0005) and (b) intermediate\n(\u000b= 0:1) damping. The other simulation parameters are\nT= 0:6J=k B,Dz= 0:1J, for a cubic nanoparticle consisting\nofN= 43= 64spins. The threshold values for the switching\nare chosen to be\u00060:75hj~nzji, where ~nzis thezcomponent of\nthe dimensionless order parameter and hj~nzjiis the thermal\naverageofitsabsolutevalue. ~navg\nzwasobtainedbyperforming\na moving average on the ~nzdata using a window of width\n\u0001t= 8:8\u0016s=(\rJ).\nlow-damping limit such an oscillatory motion can indeed\nbe observed, where the zcomponent of the Néel vector\nswitches sign and crosses the threshold value many times\nbefore coming to rest in one of the minima, see Fig. 2(a)\nand Supplemental Video 3 [65]. This is analogous to a\nmechanical particle in a double-well potential, where the\nenergy is transformed between the kinetic and potential\nparts during the motion. During these oscillations in the\nNéel vector, the energy of the system is slowly varied\ndue to the weak coupling to the heat bath, meaning that\nthe oscillations take place on a roughly constant energy\nsurface and hence they only represent a single switch-\ning event. For an estimate of the oscillation periods see\nAppendix D.\nTo determine the actual switching events in the low-\ndamping limit in the simulations, we therefore used a\ntime average of the data, where the time window was\nlarger than the period of the fast oscillations of the Néel\n0.0001 0.001 0.01 0.1 1 1010100100010000100000FIG. 3. Damping dependence of the switching time for\nFM and AFM nanoparticles. The system parameters are\nT= 0:6J=k B,Dz= 0:1J,N= 64. Symbols correspond\nto simulations using atomistic spin dynamics methods and\nlines to the analytical formulae Eqs. (2), (8), and (13).\nvector while crossing the energy barrier. As shown in\nFig. 2(a), in the averaged data the zcomponent of the\ndimensionless order parameter only crosses the threshold\nvalue once after its sign change during a single rever-\nsal. In contrast, for intermediate-to-high values of \u000bthe\nenergy fluctuates strongly on the time scale of a single ro-\ntation, and the oscillatory switching is absent as shown\nin Fig. 2(b). In this case, the same number of switching\nevents are registered both with and without the averag-\ning procedure.\nV. COMPARISON OF SWITCHING TIMES\nIn order to validate the damping dependence of the\nswitching time in both FMs and AFMs, we performed\ncomputer simulations by varying the damping value \u000bat\nafixedtemperature T= 0:6J=kB, showninFig.3. Inor-\nder to enable an accurate comparison, the same absolute\nvalue of the exchange interaction Jand the anisotropy\nDz= 0:1Jwas considered during the simulations, per-\nformed for a cubic nanoparticle consisting of N= 64\nspins. As can be seen in the figure, Eq. (2) gives good\nagreement with the simulation results for the FM case,\nwhile Eq. (8) is accurate for the AFM case over the whole\nparameter range. While the switching times are similar\nforhighdamping, theminimalswitchingtimeisfoundfor\nsignificantly lower \u000bvalues in the AFM case, leading to\na reduced thermal stability in the limit of low damping.\nNote that the asymptotic expression Eq. (13) for AFMs,\nwhich has an analogous form to Eq. (2) for FMs, un-\nderestimates the switching time in the turnover regime.\nIn particular, for the present simulation parameters the\nVLD limit, characterized by the relation \u001cafm;\u000b\u001c1/\u000b\u000017\n0.0001 0.001 0.01 0.1 1 10100100010000100000\nFIG. 4. Damping dependence of the switching time\nfor the antiferromagnetic nanoparticle, using the parame-\ntersT= 0:6J=k B,Dz= 0:1J,N= 64. Circles and\nsquares correspond to the same simulation data as in Fig. 3,\nwith and without performing the time averaging. Lines show\nEq. (8), expected to hold for all \u000bvalues, and Eq. (B17) with-\nout the depopulation factor, which is only applicable in the\nintermediate-to-high-damping limit.\nin Eq. (12), is not reached yet for \u000b\u00190:001, and the\nAFM switching time shows a weaker dependence on the\ndamping in this turnover regime.\nFigure 4 illustrates the effect of time averaging of the\nsimulation data on the obtained switching times. Moving\naverages were performed on a time interval of \u0001t= 8:8\n\u0016s=(\rJ). Without performing the time averaging, the\nmean time between sign changes of the zcomponent of\nthe order parameter converges to a constant value at low\ndamping, similarly to the intermediate-to-high-damping\nformula, given by Eq. (B17) in Appendix B. However,\nthis behavior is in contradiction with the fluctuation–\ndissipation theorem. The range in \u000bwhere the time-\naveraging starts to play a significant role in the sim-\nulation data coincides with the interval where the de-\npopulation factor in Eq. (8) becomes important in the\ntheoretical description. This emphasizes the necessity of\ncorrectly determining the switching time in the very-low-\ndamping-limit both in the analytical model as well as in\nthe numerical simulations.\nA further important difference between FMs and\nAFMs, as can be deduced from Eqs. (2) and (8), is\nthat the switching time in AFMs depends on the mi-\ncroscopic exchange interaction J, while this parame-\nter is absent in the single-domain description of FMs.\nThe analytical expressions Eqs. (2) and (8) for differ-\nent values of Jare compared in Fig. 5 as a function\nof temperature, using the parameters Dz= 0:1J0and\n\u000b= 0:0005. For the AFM case simulation results are\nalso presented, confirming the assumed Néel–Arrhenius\nlaw in this parameter range. For the FM case with the\n1 1.5 2 2.510210410610810101012FIG. 5. Dependence of the switching time on the exchange\ninteraction Jfor FM and AFM nanoparticles. The system\nparameters are \u000b= 0:0005,Dz= 0:1J0,N= 64. Sym-\nbols correspond to simulations using atomistic spin dynamics\nmethodsfortheAFMcaseandlinestotheanalyticalformulae\nEqs. (2) and (8).\nsignificantly longer switching times only the analytical\nformula Eq. (2) is shown, which had been confirmed in\nearlier publications [30] and in Fig. 3 here for a differ-\nent damping regime. Note that while Eq. (2) does not\nexplicitly depend on J, the predicted analytical curves\nare still different for J=J0andJ= 10J0, since the\nequilibrium magnetization m0is higher in the latter case\n(cf. Eqs. (18) and (19)). As indicated in the figure, at\nthe lowest temperature where the simulations were per-\nformed (J0=kBT= 2), the ratio \u001cfm=\u001cafmis about 10\ntimes larger for 10times higher exchange interaction, in\nagreement with the VLD damping limits of Eqs. (2) and\n(12).\nVI. CONCLUSION\nIn summary, we investigated the superparamagnetic\nlimit of AFM nanoparticles analytically as well as by\nmeans of computer simulations. The derived analyti-\ncal expression, Eq. (8), for the mean switching time in-\ndicates a drastically reduced thermal stability of AFM\nnanostructures as compared to their FM counterparts\nbecause of the exchange enhancement of the attempt fre-\nquency. The latter is caused by the coupling between the\nanisotropy term connected to the Néel vector and the\nexchange term connected to the magnetization in the\nfree-energy density of single-domain AFMs, which also\ncauses a strong oscillation of the Néel vector direction at\nlow damping values during the switching process.\nThe significantly faster dynamics in AFMs is one of\ntheir main proposed advantages over FMs in spintron-\nics applications [4]. However, this enhanced speed also8\nleads to an increased susceptibility to thermal fluctua-\ntions as demonstrated here; for realistic materials with a\nlow damping value, the switching times of AFMs can be\nexpected to be four to five orders of magnitude shorter\nthan those of FMs, a finding that is in agreement with\na work on antiferromagnetic grains in exchange bias sys-\ntems [66]. Furthermore, the procedures capable of in-\ncreasing the switching times in FMs may be less efficient\nin AFMs. The energy barrier in the Arrhenius expres-\nsions Eqs. (2) and (8), which is the leading term in the\ntemperature dependence, may be increased by choosing\na higher anisotropy value Dz, a larger system size N,\na lower temperature T, or at larger order parameters\nm0achieved by coupling the microscopic spins stronger\nto each other by a higher exchange coupling J. Ac-\ncording to the VLD limit Eq. (12), all of these meth-\nods except increasing the anisotropy lead to a decrease\nin the inverse attempt frequency, meaning that they de-\ncrease the\u001cafm=\u001cfmratio assuming the same system pa-\nrameters. Furthermore, for FMs damping values in the\nrange\u000b= 0:001\u00000:01, typical for materials suggested\nfor spintronic devices [51], will surely fall into the VLD\nregime, where lower \u000bvalues lead to an enhanced switch-\ning time. On the other hand, for AFMs similar values\nmay belong to the turnover region where the lifetime\nis minimal and the dependence on \u000bis weak, around\n\u000bafm,min\u0019p\nkBT=(4qJN). These problems may be\ncircumvented by selecting materials with a high damp-\ning value, where the difference between FM and AFM\nswitching times disappears.\nHowever, fast reversal of the nanoparticles may also be\ndesired in specific applications. Since thermal activation\nfacilitates the current-induced switching in spintronic de-\nvices[14], ahigherattemptfrequencynecessitatesalower\ncurrent density for achieving the same switching rate. In\nmagnetic hyperthermia [12], the reversal of nanoparticles\nis used to provide targeted warming of tissues, which can\nbecome more efficient at higher frequencies. For such\npurposes, AFM nanoparticles may provide advantages\nover their FM counterparts.\nACKNOWLEDGMENTS\nFinancial support for this work at the University of\nKonstanz was provided by the Deutsche Forschungs-\ngemeinschaft via SFB 1214. At the FU Berlin sup-\nport by the Deutsche Forschungsgemeinschaft through\nSFB/TRR 227 \"Ultrafast Spin Dynamics\", Project A08\nis gratefully acknowledged. L.R. would like to acknowl-\nedge the Alexander von Humboldt Foundation and the\nNational Research, Development and Innovation Office\nof Hungary via Project No. K115575 for support.Appendix A: FM switching time in the IHD limit\nHere the switching time of axially symmetric FM\nnanoparticles, given by Eq. (2), is derived based on the\ngeneral expression Eq. (3). The free-energy density is\ngiven byf=\u0000Hem2\nz=(2m0)whereHe= 2DzN=(Vm0),\nand the normalization jmj=m0is assumed. The expres-\nsion has a minimum at mz=m0= 1, and the expansion\nis performed in the small variables mx=m0;my=m0\u001c1.\nThis yields\nFmin=\u0000DzN; (A1)\n\"1;min=\"2;min= 2DzN: (A2)\nThe saddle point is at mx=m0= 1with the expansion\nvariablesmy=m0;mz=m0\u001c1, which results in\nFsp= 0; (A3)\n\"1;sp=\u00002DzN; (A4)\n\"2;sp= 0: (A5)\nNote that\"1;spis negative, corresponding to the un-\nstable mode in the saddle point. The other eigenvalue\n\"2;spdescribes a Goldstone mode, representing the fact\nthat the saddle point can be arbitrarily chosen along the\ncirclem2\nx+m2\ny=m2\n0. The corresponding phase space\nvolume is\nVsp= 2\u0019; (A6)\nthe circumference of the circle.\nThe linearized Landau–Lifshitz–Gilbert equation in\nthe saddle point reads\n@tmy=1\n1 +\u000b2\rN\nVm02Dzmz=1\n1 +\u000b2!amz;(A7)\n@tmz=1\n1 +\u000b2\rN\nVm0\u000b2Dzmz=\u000b\n1 +\u000b2!amz;(A8)\nwith the eigenvalues\n\u00151;sp=\u000b\n1 +\u000b2!a; (A9)\n\u00152;sp= 0; (A10)\nwhere\u0015+;sp=\u00151;spis the single positive eigenvalue.\nSubstituting Eqs. (A1)-(A6) and Eq. (A9) into Eq. (3)\ngives precisely Eq. (2), which in this special case is valid\nfor all values of the damping.\nAppendix B: AFM switching time in the IHD limit\nHere Eq. (8) without the depopulation factor will be\nderived based on Eq. (3). We will use the free-energy\ndensityf=Hem2=(2m0)\u0000Han2\nz=(2m0), withHe=\nqJN=(Vm0)being the exchange field, where qis the\nnumber of nearest neighbors and Jthe exchange con-\nstant in the corresponding atomistic spin model, rescaled\nbyaccountingforthethermallyreducedorderparameter.9\nThe minimum of the free energy Fis at n=m0=\n(0;0;1);m=m0= (0;0;0)with\nFmin=\u0000DzN; (B1)\n\"1;min=\"2;min= 2DzN; (B2)\n\"3;min=\"4;min=qJN: (B3)\nForDz\u001cqJthe saddle point is n=m0=\n(1;0;0);m=m0= (0;0;0), where the expansion yields\nFsp= 0; (B4)\n\"1;sp=\u00002DzN; (B5)\n\"2;sp= 0; (B6)\n\"3;sp=\"4;sp=qJN; (B7)\nHere\"1;spis the unstable mode and \"2;spis the Gold-\nstone mode with\nVsp= 2\u0019: (B8)\nThe linearized equations of motion in the saddle point\nread\n@tmy=\rN\nVm02Dznz\u0000\u000b@tnz; (B9)\n@tmz=\u000b@tny; (B10)\n@tny=\u0000\rN\nVm0qJmz\u0000\u000b@tmz;(B11)\n@tnz=\rN\nVm0qJmy+\u000b@tmy; (B12)\nleading to the eigenvalues\n\u00151;sp=0; (B13)\n\u00152;sp=\u00001\n1 +\u000b2\rN\nVm0\u000bqJ; (B14)\n\u00153;sp=1\n1 +\u000b2\rN\nVm0\"\n\u000b\u0012\nDz\u00001\n2qJ\u0013\n+s\n\u000b2\u00121\n2qJ+Dz\u00132\n+ 2DzqJ#\n;(B15)\n\u00154;sp=\u00001\n1 +\u000b2\rN\nVm0\"\n\u000b\u00121\n2qJ\u0000Dz\u0013\n+s\n\u000b2\u00121\n2qJ+Dz\u00132\n+ 2DzqJ#\n;(B16)\nwhere the positive eigenvalue is \u0015+;sp=\u00153;sp.\nSubstitutingEqs.(B1)-(B8)andEq.(B15)intoEq.(3)\nproduces\n\u001cIHD\nafm=1 +\u000b2\n\u000bVm0\n\rN\u0014\u0012\nDz\u00001\n2qJ\u0013\n+s\u00121\n2qJ+Dz\u00132\n+2DzqJ\n\u000b23\n5\u00001r\n\u0019kBT\nDzNeDzN\nkBT;\n(B17)the intermediate-to-high-damping limit of Eqs. (8) and\n(9). Note that since the eigenvalues \"3;min;\"4;mincancel\nwith\"3;sp;\"4;sp, the difference between the ferromagnetic\nand antiferromagnetic cases only comes from the dynam-\nical prefactor \u0015+;sp, which is exchange-enhanced at low\nand intermediate damping for the latter.\nAppendix C: Energy dissipation per cycle when\npassing through the saddle point\nHere the depopulation factor in Eq. (4) will be cal-\nculated for the AFM particle. The variable Sin the\nargument of Ain Eq. (8) denotes the action of the un-\ndampedmotioncrossingthroughthesaddlepoint. Equa-\ntion(4)expressesthatif \u000bS, theenergydissipatedduring\na single cycle of motion over the saddle point [41, 43], is\nsmall compared to the thermal energy kBT, then it takes\nlonger for the particle to cross the energy barrier since it\ncan no longer be assumed that the equilibrium Maxwell–\nBoltzmann distribution is formed in the region close to\nthe saddle point.\nIn order to calculate this energy dissipation, Eqs. (6)\nand (7) are linearized in \u000bat low damping, yielding\n_n=\u0000\rn\u0002\u0012\nhm+\u000bm\nm0\u0002hm+\u000bn\nm0\u0002hn\u0013\n;(C1)\n_m=\u0000\rm\u0002\u0012\nhm+\u000bm\nm0\u0002hm+\u000bn\nm0\u0002hn\u0013\n\u0000\rn\u0002\u0012\nhn+\u000bn\nm0\u0002hm\u0013\n: (C2)\nThe free energy dissipation per cycle may be written\nas\n\u0000\u0001F=\u000bS=\u0000ZT\n0_Fdt=ZT\n0Z\nhm_m+hn_ndrdt\n=\u000b\rm 0VZT\n0\u0012m\nm0\u0002hm+n\nm0\u0002hn\u00132\n+\u0012n\nm0\u0002hm\u00132\ndt: (C3)\nIntroducing the renormalized variables ^m =\nm=m0,^n=n=m0, and substituting hm =\n\u0000qJN=(Vm0)^m;hn= 2DzN=(Vm0)^nzezfor the\nconsidered system one obtains\n\u000bS=\u000b\rN2\nVm0ZT\n04D2\nz\u0000\n1\u0000^n2\nz\u0001\n^n2\nz+ (qJ)2^m2dt;(C4)\nwith the integral to be evaluated along the trajectory of\nthe undamped motion crossing the saddle point.10\nFor\u000b= 0, Eqs. (C1) and (C2) may be written as\n@t^mx=\u0000\rN\nVm02Dz^ny^nz; (C5)\n@t^my=\rN\nVm02Dz^nx^nz; (C6)\n@t^mz= 0; (C7)\n@t^nx=\rN\nVm0qJ(^ny^mz\u0000^nz^my);(C8)\n@t^ny=\rN\nVm0qJ(^nz^mx\u0000^nx^mz);(C9)\n@t^nz=\rN\nVm0qJ(^nx^my\u0000^ny^mx);(C10)\nfor the axially symmetric AFM nanoparticle. Since the\nconstraintj^nj= 1issatisfiedbythedynamicalequations,\nthe normalized Néel vector may be rewritten in spherical\ncoordinates, (^nx;^ny;^nz) = (sin#cos';sin#sin';cos#).\nFor the variable ^mone has ^n\u0001^m= 0, and itszcompo-\nnent is a constant of motion as expressed by Eq. (C7).\nWithout the damping, the free energy of the system is\nalso conserved during the time evolution,\nF=qJ\n2N^m2\u0000DzNcos2#: (C11)\nUsing the conserved quantities Fand ^mz, Eqs. (C5)-\n(C10) may be expressed as\n@t#=\u0007s\n!2\nF\u0000!2\n0sin2#\u0000!2\n^mz\nsin2#;(C12)\n@t'=\u00001\nsin2#!^mz; (C13)\nwith\n!0=\rN\nVm0p\n2DzqJ; (C14)\n!F=\rN\nVm0s\n2\u0012F\nN+Dz\u0013\nqJ;(C15)\n!^mz=\rN\nVm0qJ^mz: (C16)\nFor the trajectory including the saddle point one has\nF= 0, see Eq. (B4), and ^mz= 0. Equation (C12) may\nbe used to change the parametrization from the time tto\nthe polar angle #, which simplifies Eq. (C4) to\n\u000bS=\u000bNZ2\u0019\n04D2\nzp2DzqJ\u0010\njcos#j\u0000jcos#j3\u0011\n+p\n2DzqJjcos#jd#: (C17)\nEvaluating the integral Eq. (C17) yields Eq. (10).\nAppendix D: Oscillations in the Néel vector based\non the theoretical model\nAsshowninFig.2, significantoscillationsinthe zcom-\nponent of the order parameter were observed in the spin\n00.511.522.533.5\n0 0 .5 1 1 .5 2|F(ω)|\nfrequency ω(γJ/µ s)FIG. 6. Fourier spectrum of the oscillations of the zcom-\nponent of the order parameter from the spin dynamics sim-\nulations. The same simulation parameters were used as for\nFig. 2(a),\u000b= 0:0005,T= 0:6J=k B,Dz= 0:1J,N= 64.\nThe characteristic oscillation frequency from Eq. (C14) is\n!0= 0:66\rJ=\u0016 s.\ndynamics simulations of antiferromagnetic nanoparticles\nat very low damping values. This can be explained by\nthe fact that forF>0where the switching occurs, even\nin the conservative system ^nwill perform full rotations\nduring which its zcomponent changes sign, as described\nbyEq.(C12). For ^mz= 0, theperiodoftheseoscillations\nmay be evaluated in a closed form,\nTF=Z2\u0019\n01q\n!2\nF\u0000!2\n0sin2#d#=4\n!FK\u0012!F\n!0\u0013\n;(D1)\nwithKthe complete elliptic integral of the first kind.\nIt can be seen from Eq. (D1) that the oscillation fre-\nquency will change as the free energy varies due to the\ncoupling to the heat bath. If the thermal fluctuations\nare weak as required for the application of Arrhenius-\nlike expressions such as Eqs. (2) and (8), the free energy\ndoesnotbecomesignificantlyhigherthanitssaddle-point\nvalue during the switching, and in this case the oscilla-\ntion frequencies will be comparable to !0. For example,\n0:01\u0014F=(DzN)\u00140:2yields 0:39\u00142\u0019=(TF!0)\u00140:65.\nThe adiabatic variation of the energy leads to a wide dis-\ntribution of frequency values if the oscillations are inves-\ntigated in Fourier space, as displayed in Fig. 6. Using the\ntemperature-dependent effective parameters described in\nSec. IVA, for the model coefficients in Fig. 6 one obtains\n!0= 0:66\rJ=\u0016 s, roughly corresponding to the peak in\nthe frequency distribution. 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Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n(Dated: September 5, 2018)\nWe present angle dependent measurements of the damping prop erties of epitaxial Fe layers with\nMgO, Al and Pt capping layers. Based on the preferential dist ribution of lattice defects following the\ncrystal symmetry, we make use of a model of the defect density to separate the contribution of two-\nmagnon scattering to the damping from the isotropic contrib ution originating in the spin pumping\neffect, the viscous Gilbert damping and the magnetic proximi ty effect. The separation of the two-\nmagnon contribution, which depends strongly on the defect d ensity, allows for the measurement of\na value of the effective spin mixing conductance which is clos er to the value exclusively due to spin\npumping. The influence of the defect density for bilayers sys tems due to the different capping layers\nand to the unavoidable spread in defect density from sample t o sample is thus removed. This shows\nthe potential of studying spin pumping phenomena in fully or dered systems in which this separation\nis possible, contrary to polycrystalline or amorphous meta llic thin films.\nINTRODUCTION\nIn bilayers systems formed by a ferromagnetic (FM)\nlayer in contact with a metallic non-magnetic (NM) one,\na pure spin current can be generated and injected in the\nlatterwhen the ferromagneticresonanceisexcited. Typi-\ncally, a microwavemagnetic field is used for this purpose.\nThe whole processis commonly referredto as spin pump-\ning [1, 2]. If the non-magnetic layer is formed by a heavy\nmetal with large spin-orbit coupling (Pt, Ta or similar),\nthe spin current can be detected by using the inversespin\nHall effect (ISHE) for conversion into a charge current.\nSince the spin current leaving the magnetic layer car-\nries away angular momentum from the magnetization\nprecession,it representsanadditionallosschannelforthe\nmagnetic system and consequently causes an increase in\nthe measured Gilbert damping parameter α[1]:\n∆αsp=γ/planckover2pi1\n4πMsdFMg↑↓(1)\nwhereg↑↓is the real part of the spin mixing conductance\nwhich is controlling the magnitude of the generated spin\ncurrent and γis the gyromagnetic ratio.\nThis expression is only valid for sufficiently thick NM\nlayers where no reflection of the spin current takes place\nat the film surface or interface with other materials, i.e.\nno spin current is flowing back into the magnetic layer.\nIn principle, it allows the estimation of g↑↓by measur-\ning the increase in damping compared to the intrinsic\nvalue. However, to perform this measurement is not\nstraightforward. If the estimation of g↑↓for a FM/Pt\nsystem is needed, ideally one should measure the effec-\ntive Gilbert damping parameter α0for a single stand-\ning magnetic layer acting as a reference sample with no\nlosses due to spin pumping and repeat the same after de-\npositing a thick Pt layer. However, most of the commonferromagnetic materials, with exception of the magnetic\ninsulators like YIG, will change its properties due to ox-\nidation processes. Therefore, a capping layer is required\nand one has to find an appropriate one, in the sense that\nits introduction must not modify the damping properties\nof the magnetic layer. Examples in the literature show\nthat this is far to be a trivial task [3–5]. In addition to\nthis, the emergence of a finite magnetic polarization in\nPt in contact with a ferromagnetic layers has an impact\non damping which further hinders the estimation of g↑↓\n[5–12].\nFor the reference layers, the most convenient candi-\ndates as capping material are oxides like MgO, for which\nit has been proven that they are able to block the flow\nof spin current and therefore to deactivate spin pumping\n[13–15], or metals with weak spin-orbit interaction like\nAl or Ru. But even for these cases, it has been shown\nthat an increase of damping not related to spin pumping\nis possible. Ruiz et al.show for instance that a MgO\ncapping layer increases strongly the damping in permal-\nloy while this is not the case for Al capping layer [5].\nThe reason has nothing to do with the metallic char-\nacter of the capping layer since the increase for Ru is\neven larger than with MgO. The same work [5] already\nprovides a hint for a possible reason since the increase\nof damping roughly scales with the value of the inter-\nface perpendicular anisotropy constant K⊥\nS. Theoretical\nworks [16] show that the counterplay between the de-\nmagnetizing field responsible for the in-plane orientation\nof the magnetization and the perpendicular anisotropy\nfield can induce inhomogeneous magnetization states for\ncertain field strengths combinations which are responsi-\nble for an increased damping. In this sense, this effect\nhas been also adduced to explain the damping thickness\ndependence in Co 2FeAl/MgO systems [3].\nHere we present angle dependent measurements of the\ndamping properties of epitaxial Fe layers with MgO, Al2\nFIG. 1. (Color online) Dependence of the FMR linewidth on the frequency for different orientations φHof the external magnetic\nfield with respect to the [100] crystallographic axis of Fe fo r (a) Fe/Al and (b) Fe/Pt systems. The lines correspond to a li near\nfit to extract the effective damping parameter αeff. Forφ= 30◦a strong non-linearity due to magnetic dragging is observed .\nFor visibility reasons, each data set is shifted vertically by 1.25 mT with respect the previous one.\nand Pt capping layers. Fully epitaxial systems consti-\ntute a perfect ordered model with almost ideal and well\ndefined interfaces. Here, we will show that the angle de-\npendence of damping allows for a measurement of the\nstrength of the two-magnon scattering and of its contri-\nbution to the effective damping parameter. With the\nseparation of this contribution we access the increase\nof damping caused only by spin pumping and magnetic\nproximity effect and to an estimation of g↑↓without the\ncontamination of defects effects.\nEXPERIMENTAL DETAILS\nThe samples were deposited by e-beam evaporation\non MgO(100) substrates in a molecular beam epitaxy\n(MBE) chamber with a base pressure P b= 5×\n10−10mbar. A set of Fe/Pt bilayers with fixed Fe thick-\nness (12 nm) and varying Pt thickness were prepared.\nAdditional reference samples, where Pt is substituted by\nMgO or Al, have also been prepared. The Fe and Pt\nfilms were grown with a deposition rate of 0.05 ˚A/s. The\nsamples were deposited with a substrate temperature of\n300◦C and subsequently annealed at the same tempera-\nture.\nThe characterization by X-ray diffractometry (XRD)\n(presented elsewhere [17]) shows that the Fe/Pt bilayers\nare fully epitaxial with the Fe unit cell rotated by 45◦\nwith respect to the MgO substrate unit cell and with\nPt rotated again 45◦with respect to Fe. In the case of\nFe/Al, epitaxial growth of the upper layer could not be\nachieved.\nThe dynamic properties and material parameters were\nstudied by measuring the ferromagnetic resonance using\na strip-line vector network analyzer (VNA-FMR). Forthis, the samples were placed facing the strip-line and\nthe˜S12transmission parameter was recorded.\nRESULTS AND DISCUSSION\nFigures 1 shows the dependence of the measured FMR\nline width ∆ Hon the frequency for the reference layer\nwith Al capping (a) and a Fe/Pt system (b). The data\nis shown for different orientation of the external static\nmagnetic field varying from φH= 0◦([100], easy axis) to\nφH= 45◦([110], hard axis). For visibility reasons, each\ndata set is shifted vertically by 1.25 mT with respect to\nthe previous one.\nAs commented before, the choice of capping layer can\nhave a large influence on the linewidth and effective\ndampingofthe magneticlayer,evenforlightmetals. The\nmagnetic proximity effect (MPE) in the case of Pt also\ncontributes to an increase on damping, [5, 9–12] which\nadditionally challenges the measurement of the contri-\nbution from the spin pumping. Taking into account all\nthese considerations, the effective increase on damping\nwhen comparing a reference system and a system with a\nheavy metal can be separated as follows:\nαeff=α0+αmpe+αsp+αi. (2)\nHereα0is the intrinsic damping parameter which can\nbe defined as characteristic of the material under in-\nvestigation (growth conditions however may influence it\nstrongly) and it is the sum of the losses by two-magnon\nscattering and by energy transfer to the phonon system.\nαmpeis the contribution due to the dynamic coupling be-\ntween the ordered spins in Pt due to the MPE and the3\nFIG. 2. (Color online) (a) Dependence of the FMR resonance\nfieldHFMRon the in-plane direction of the static magnetic\nfield for two values of the resonant frequency. (b) Dependenc e\nof the in-plane angle of the magnetization vector φMon the\nexternal field direction φH. Both angles are measured relative\ntothe[100] axis. Thedottedline represents thecase of perf ect\ncollinearity between magnetization and external field.\nmagnetization in the magnetic layer. αspis the result\nof the losses by the spin current generated in the fer-\nromagnetic layer by the precession of the magnetization\nand that flows into the Pt layer (spin pumping). The last\ntermαisummarizes the increase of damping due to other\ninterfacial effects such as interface PMA as commented\nabove, spin memory loss [18] or isotropic scattering at\ninterface defects [19].\nSeveral efforts have been made in order to separate\nsome of the contributions to αeff. In a recent work with\nCoFeB/Pt [9] we were able to separate αmpedue to the\ndependence on the Pt thickness. As already reported by\nCaminale et al.[11], a linear Pt thickness dependence of\nthe spin-current absorption in spin-sink layers exhibiting\nMPE and of αmpeis expected [12]. A detailed vector\nnetwork analyzer FMR study has also been recently re-\nported to separate the different contributions in NiFe/Pt\nsystems [20].\nTheterm α0isaresultoftwocontributions[22]. Oneis\nthe pure Gilbert damping, which is of viscous nature and\ngenerates a dissipation of energy and angular momentum\ntothe lattice. The secondoneisthe transfertospin-wavemodes with k/negationslash= 0 from the FMR mode via two-magnon\nscattering. For a pure Gilbert-like viscous damping the\nlinewidth dependence on the frequency is purely linear:\nµ0∆H=µ0∆H0+4παf\nγ. (3)\nHere, ∆H0is the inhomogeneous broadening and is re-\nlated to film quality.\nThe lines in Figs. 1 (a) and (b) are a fit to this ex-\npression. It has to be mentioned that although a viscous\ndamping generates a linear dependence, on the contrary\nit is not possible to assume that the observation of a lin-\nearbehavior provesthat only viscous damping is present.\nThe reason for that is that two-magnon scattering can\nmimic also a linear dependence [21–23]. For both sam-\nples, and for the MgO capped sample not shown here,\nforφ= 30◦a strongly non linear behavior with a large\nincrease in linewidth values for smaller frequencies is ob-\nserved. For this reason, the hollow points in Fig. 1 have\nbeen excluded from the fit. The non-linearity at low fre-\nquencies cannot be explained by viscous damping and it\nis caused by magnetic dragging. The magnetic dragging\neffect describesthe increaseofthe linewidth of precessing\nmagneticlayerswithlargemagneticcrystalineanisotropy\ndue to the non-collinearity of the magnetization and the\nexternal magnetic field. In Fig. 2 (a), the dependence\nof the resonance field HFMRon the in-plane direction of\nthe external magnetic field is shown for two fixed fre-\nquency values. As a result of the four-fold anisotropy ex-\npectedfromthecubiclatticeofFeandassumingaperfect\ncollinearity between magnetization vector and external\nfield,HFMRcan be modeled as: [10, 24]\nµ0HFMR=µ0˜HFMR+2K1\nMscos(4φ),(4)\nwhereK1is the cubic anisotropyconstant, φthe in-plane\nazimuthal angle and ˜HFMRis the averaged resonance\nfield value. The fraction2K1\nMsis directly the anisotropy\nfield H B. In Fig. 2(a) a deviation from this model is ob-\nserved for angles between the hard and easy axis and it\nis due to magnetic dragging, i.e., the magnetization is\nnot aligned to the external field due to the effect of the\nanisotropy field. The fact that the deviation from the\nmodel in Eq.4 is smallerfor largerfrequencies(i.e. larger\napplied field) alsosupportsthis interpretation. The same\nbehavior observed for φ= 30◦has been also been re-\nported for ultrathin Fe films [25] or for insulating LSMO\nfilms [28] and attributed to magnetic dragging. The de-\ngree of non-collinearity can be estimated by solving the\nequilibrium condition for the angle defining the orienta-\ntion of the magnetization φMfor each value of φH:\nHsin(φM−φH)+HB\n4sin(4φM) = 0,(5)4\nwhere the value for the cubic anisotropy field was taken\nfrom [10]. Fig. 2(b) shows the obtained value of φMfor\nthe data shown in Fig. 2(a). The angle between mag-\nnetization and magnetic field can be as large as 10◦for\n13 GHz and it is decreased to a maximum around 4.5◦\nfor 18 GHz. The magnetic dragging effect is largest for\nφHbetween the easy and hard axis and vanishes along\nthe main crystallographic axes.\nFigure 3 shows the value of the effective damping pa-\nrameter αeffas obtained from the fits in Fig. 1 for the\nthree capping layers. In all of them, an eight-fold sym-\nmetry on the in-plane angle φHis observed with maxima\nalong the easy and hard axis of the Fe layers and min-\nima in between. For the Fe/Al and Fe/MgO samples,\nwhere spin pumping has no influence, αeff=α0+αi\nwhile for the Fe/Pt sample, where both losses through\nspin pumping and due to the MPE are active, we obtain\nthe situation shown in Eq. 2. It is remarkable that the\ndifferent origins of the damping do not change the overall\nsymmetry of the angular dependence. It has though an\nimpact on the absolute values, which are larger for the\nFe/Pt sample.\nIn the literature concerning epitaxial layers, it is possi-\nble to find different symmetries for the dependence of the\nFMRlinewidthorthedampingparameteronthein-plane\nfield direction. For the Heusler alloy Co 2FeAl both four-\nand eight-fold symmetries for the linewidth have been\nreported. The situation differs depending on the thick-\nnessofthe film [23] andalsobetween different groups[30]\npointing out to a role of the growth conditions. For Fe 3Si\nfilms and Fe/V multilayer systems a four-fold symmetry\nis reported [22, 26] and for ultrathin Fe layers, where the\nrole of the interface is strong, a two-fold symmetry of\nαeffhas been measured [25]. Eight-fold symmetry has\nbeen also observed in epitaxial FeSi systems [26, 29]. In\na different work on Fe layers, a decrease on the obtained\nαvalue along the intermediate orientation between the\ntwomainaxisrelativetothe onemeasuredalongtheeasy\nand hard axis was reported [27], pointing to an angular\ndependence very similar to ours. Concerning insulating\nsystems, two- and four-fold symmetries have been ob-\nserved in LSMO films [28].\nTwo-magnon scattering can only occur if scattering\ncenters in form of defects are present. If, as expected,\nthese are present as point lattice defects or dislocation\nlinesalongthemaincrystallographicdirections, itisclear\nthat the scattering intensity should reflect the symmetry\nof the lattice. This fact would for certain explain a four-\nor eight-fold anisotropy in damping observed in some on\nthe reports mentioned above and the maxima in αefffor\nour samples for φ= 0◦,45◦,90◦,135◦.\nFollowingZakeri et al. andAria et al., the contribution\nto damping due to two-magnonscattering can be written\nas [21, 26]:α2M=/summationdisplay\n/angbracketleftxi/angbracketrightΓ/angbracketleftxi/angbracketrightf(φH−φ/angbracketleftxi/angbracketright), (6)\nwhere Γ /angbracketleftxi/angbracketrightrepresents the strength of the two-magnon\nscattering contribution along the in-plane crystallo-\ngraphic direction /angbracketleftxi/angbracketright. The function f(φH−φ/angbracketleftxi/angbracketright) al-\nlows for an angle dependent two-magnon contribution to\ndamping with respect to the orientation of the external\nfieldHrelative to the crystallographic directions /angbracketleftxi/angbracketright.\nThe physical interpretation of the function f(φH−φ/angbracketleftxi/angbracketright)\nlays in the Fourier transform of the defects in the film\n[26, 34]. By using the ansatz f(φH−φ/angbracketleftxi/angbracketright) = cos2(4φH−\nφ/angbracketleftxi/angbracketright) we can fit the damping dependence using a simpli-\nfied version:\nαeff=αiso+α2M=αiso+Γ2Mcos2(4φH−φ[100]) (7)\nwhereαisoincludes now all the isotropic contributions to\ndamping, i.e. αmpe,αsp, pure Gilbert damping and po-\ntentially isotropic interface contributions from the term\nαi, mainly spin memory loss and interface PMA related\neffects.\nThe red lines in Fig. 3 show the fit to this model. The\nobtained parameters are summarized in Table. I. A very\nlow value below 1 ×10−3is obtained for αisofor the Fe/Al\nsample. Since αsp,MPE= 0 is expected and due to the\nlow value we consider that the obtained αisomust be\nvery close to the value corresponding only to pure vis-\ncous Gilbert damping corresponding to high quality Fe.\nHowever, strictly speaking, the obtained value is only\nan upper limit since still other effects might contribute.\nConcerning 3d metals with no half-metallic character, a\nvery low damping value of 0.7 ×10−3has been reported\nby Leeet al.for CoFe [35]. This value is comparable\nto theαisomeasured here for Fe/Al. The fact that the\nCoFe samples in which the low value was obtained are\nalso fully epitaxial with an exceptionally high crystalline\nquality explains the similarity in values. The low defect\ndensity in CoFe almost suppresses two-magnon scatter-\ning in the CoFe samplesand thereforeis comparablewith\nourαisowhere that contribution is already separated.\nFor the Fe/MgO sample the value for αisoincreases by\na factor larger than 2 although also here αsp,MPE= 0.\nαiso Γ2M\n(10−3) (10−3)\nFe/Al 0.8 ±0.3 3.6 ±0.4\nFe/Pt 3.4 ±0.3 2.4 ±0.4\nFe/MgO 1.9 ±0.1 1.3 ±0.1\nTABLE I. Isotropic contribution αisoand two-magnon scat-\ntering contribution Γ 2Mto the total effective damping param-\neterαeff.5\nFIG. 3. (Color online) Angular dependence of the effective da mping parameter αeffin the in-plane direction of the static\nmagnetic field φHfor (a) Fe/Al, (b) Fe/Pt and (c) Fe/MgO. The red lines are a fit t o Eq. 7.\nThe main differences between Fe/Al and Fe/MgO are\nthat the MgO is single crystalline while Al is polycrys-\ntalline and the contrast between the metallic character\nof Al with the insulating oxide. The lattice mismatch\nbetween MgO and Fe is around 4% and introduces there-\nfore a certain degree of stress in the Fe layer which is\nnot present when the capping is polycrystalline Al and\nwhichcanhaveanimpactondamping. Atthesametime,\nsince the Gilbert damping is sensitive to the density of\nstates and this one is modified at the interface by the\nkind of bonds between the Fe atom and the atoms from\nthe cappinglayer, the simple materialdifferencemay also\nexplain the difference. In this sense it is remarkable that\nthe low damping value by Lee et al.commented before is\nonly observed for CoFe with a MgO capping layer and a\nlargervalue is measuredwhen MgAl 2O4is used [35]. Our\ndata confirms the important role of the capping layer on\ndamping observed in other works [5].\nA further increase in the value of αisois observed for\nthe Fe/Pt sample where additional losses through spin\npumping and MPE are present. Unfortunately the data\npresented in this paper does not allow to disentangle\nthese two contributions. For this reason, when using\nEq. 1 for the calculation of spin mixing conductance, it\nmakes sense to refer to an effective value g↑↓\neffwhich is at\nthe same time an upper limit for the corresponding value\nfor spin pumping alone. Using the Fe/Al sample as a\nreference we obtain a value for the spin mixing conduc-\ntance of 3 .7±0.9×1019m−2. This value is lower than\nthe one presented in our previous report [10] and shows\nthat the value of g↑↓\neffcan be easily overestimated if the\neffect of two-magnon scattering on damping is not sepa-\nrated, with the consequent overestimation of the injected\nspin current and underestimation of the spin Hall angle\nfrom the ISHE voltage [17]. The advantage of using epi-\ntaxial magnetic layers is that they allow the separation\nof the contribution of the two-magnon scattering due to\nthe strongangulardependenceand welldefinedcrystallo-\ngraphicdirections. Thesameisnotpossibleincommonly\nused material as CoFeB or NiFe where the amorphous or\npolycrystalline nature of the layers blends the scatteringdependence on the in-plane angle.\nThe parameter Γ 2Mprovides further insight into the\norigin of total damping in the samples. This parameter\nis larger for the Fe/Al sample in comparison to the fully\nepitaxial bilayers being almost three times larger than\nfor Fe/MgO. As a result, the total damping in the Fe/Al\nsample is dominated by the two-magnon scattering due\nalsoto the low αisowhile the sameis not true in the other\ntwo systems. It has to be taken into account that, since\nas scattering centers for magnon scattering the defects at\nthe interfaces play a role, they can be dominant in thin\nfilms. From TEM images (presented for instance in [10]),\nwe can prove the existence of a highly ordered interface\nin the fully epitaxial samples. Of course, the same is not\ntrue for the case with polycrystalline Al capping. We\nbelieve that the dominant role of the interface here is\npossible, also due to the overall low defect density in the\nbulk of the Fe layer.\nFor completeness we want to discuss two additional\neffects potentially affecting the linewidth and damping.\nDue to the spread of internal and anisotropy field due to\nmosaicity in the film, there is a contribution to the line\nbroadening which has the following form [26, 33]:\n∆Hmosaic=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂HFMR\n∂φH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆φH, (8)\nwhere ∆φHis the average spread of the direction of the\neasy axes in the film plane. From Fig. 1(c) it is clear\nthat this contribution should increase the linewidth in\ntheregion φ= 15−30◦andequivalentonesbutthis isnot\nobserved pointing to a weak impact of mosaicity. In any\ncase, the mosaicity term is frequency independent and\nwill be only visible in the inhomogeneous linebroadening\n∆H0and will not affect the determination of αeff.\nThe discussion followingthe introduction ofEqs.6 and\n7 was focused on crystalline lattice defects as the origin\nof two-magnon scattering. However any kind on inhomo-\ngeneity in the magnetic state of the sample may play the\nsame role. The presence of magnetic dragging, visible for\ninstance for φ= 30◦in Fig. 1 can create a slight inhomo-\ngeneity in the magnetization state for field orientations6\nclose to the hard axis direction and an increase of damp-\ning around the hard axis orientation. In any case, this\ncontribution follows also the symmetry of the lattice and\nit is accounted in the Γ 2Mparameter.\nAlthough certain theoretical works point to an\nanisotropic Gilbert damping in fully epitaxial systems\ndue to its dependence on the density of states at the\nFermi energy [31, 32], experimentally this has been only\nseen in ultrathin Fe films [22] due to the modification of\nthe electronic structure induced by the interfacial spin-\norbit coupling. The anisotropy in αeffpresented here can\nbe fully explained by two-magnon scattering, and there-\nfore an isotropic Gilbert damping can be assumed.\nCONCLUSIONS\nMaking use of the well defined dependence of the two-\nmagnon scattering mechanism on the in-plane field di-\nrection, we have been able to separate this contribution\nto damping from the isotropic contributions originating\nfrom the viscous Gilbert damping mechanism, from spin\npumping and from the magnetic proximity effect in Pt.\nThe method can be implemented thanks to the pref-\nerential ordering of crystalline defects with respect to\nthe crystallographic directions in epitaxial systems and\ntherefore cannot be extended to amorphous or polycrys-\ntalline magnetic films. This shows the potential of the\nstudy of spin pumping related phenomena in ordered\nsystems. 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In this work an analytical formula for the speed of the domain wall that\nexplains this behavior is derived by means of an asymptotic study of the Landau Lifshitz Gilbert\nequation for thin nanotubes. We show that the dynamics can be reduced to a one dimensional\nhyperbolic reaction di\u000busion equation, namely, the damped double Sine Gordon equation, which\nshows the transition to the magnonic regime as the domain wall speed approaches the speed of\nspin waves. This equation has been previously found to describe domain wall propagation in weak\nferromagnets with the mobility proportional to the Dzyaloshinskii-Moriya interaction constant, for\nPermalloy nanotubes the mobility is proportional to the nanotube radius.\nPACS numbers: 75.78.-n, 75.78.Fg\n1arXiv:1809.06278v3  [cond-mat.mes-hall]  17 Sep 2019I. INTRODUCTION\nMagnetic domain wall propagation is a subject of much current interest due to its possible\napplications in magnetic memory devices. Understanding and controlling the motion of\ndomain walls is essential for applications. In the micromagnetic approach, the magnetization\nis governed by the Landau Lifshitz Gilbert (LLG) equation [1, 2]\n@~ m\n@t=\u0000\r0~ m\u0002~He\u000b(~ m) +\u000b~ m\u0002@~ m\n@t(1)\nwhere~ mis the unit magnetization vector, that is, the magnetization ~M=Ms~ m, whereMsis\nthe constant saturation magnetization, a property of the material. The constant \r0=j\rj\u00160,\nwhere\ris the gyromagnetic ratio of the electron and \u00160is the magnetic permeability of\nvacuum. The parameter \u000b>0 is the dimensionless phenomenological Gilbert damping con-\nstant. The e\u000bective magnetic \feld ~He\u000bincludes the physical interactions and the external\napplied \feld ~Ha. The di\u000berent physical phenomena that must be included in the e\u000bective\n\feld and the geometry of the ferromagnetic material together with the intrinsic nonlinear-\nity of the problem imply that exact analytical solutions are generally nonexistent so that\nnumerical and approximate analytic methods have been developed to understand experi-\nmental results and predict new phenomena. The exact solution of Walker [3, 4] developed\nfor an in\fnite medium with an easy axis, a local approximation for the demagnetizing \feld,\nincluding exchange interaction and under the action of an external magnetic \feld along the\neasy axis, shows that when the applied \feld is small, the speed of the domain wall increases\nlinearly with the \feld. When the applied \feld reaches a critical value, the Walker \feld Hw;\nthe magnetization enters into a precessing motion. This behavior, which is encountered\neven when additional physical e\u000bects and di\u000berent geometries are studied, puts a limit to\nthe maximum speed that a domain wall can achieve.\nFor applications it is desirable to have stable domain walls and to reach high propagation\nvelocities. For such purpose di\u000berent physical e\u000bects and geometries have been considered.\nNumerical simulations for thin Permalloy nanotubes under the action of an external \feld\nalong the nanotube axis showed unexpected behavior [5, 6]. For small \felds the speed in-\ncreases linearly with the \feld, reaching a plateau at relatively low applied \feld and very\nhigh velocity. No instability nor Walker breakdown of the domain wall was observed for\nthis material in the parameter regime studied. This unexpected behavior occurs for a spe-\n2ci\fc chirality of the domain wall, namely right handed domain walls, for which the radial\ncomponent of the magnetization remains small throughout the motion [6].\nThe main result of this manuscript is the derivation of an analytical expression for the\nspeed of the domain wall which explains the linear increase at small \felds, the reaching of a\nplateau and the high values of the velocity. For Permalloy, a material of negligible uniaxial\nanisotropy, we \fnd that the speed is given by\nv=\r0RHap\n\u000b2+\u00160R2H2\na=(2A); (2)\nwhereAis the exchange constant [7] and R the thin nanotube radius. For small applied\n\feld we recover the linear regime [8],\nvL=\r0\n\u000bRHa; (3)\nwhereas for large applied \feld the speed tends to the constant value\nv1=\r0s\n2A\n\u00160= 1006 ms\u00001for Permalloy (4)\nwhich we identify with the minimal phase speed of spin waves. Following the notation used\nfor weak ferromagnets, notice that Eqn. (2) can be written as v=\u0016Ha=p\n1 +\u00162H2\na=v2\n1with\nmobility\u0016= (dv=dHa)jHa=0=\r0R=\u000b. The rise of the speed with the \feld is very fast; for a\nPermalloy nanotube of radius R= 55 nanometers, for an external \feld \feld B=\u00160Ha= 2\nmT, Eqn. (2) yields v= 893 m s\u00001.\nThe suppression of the Walker breakdown together with a slowdown of a domain wall\nas it approaches the phase speed of spin waves has been encountered previously in di\u000berent\nproblems. In antiferromagnets with Dzyaloshinskii-Moriya interaction (DMI) the mobility\nwas found to be proportional to the DMI constant [9{11]. See the recent review [12] for ad-\nditional references. Similar behavior was found in rough nanowires [13, 14] and in nanowires\nwith a strong hard axis perpendicular to the wire when an external \feld is applied along the\nwire [15{17]. A theoretical explanation for the e\u000bect of spin waves on Bloch walls was given\nin [18] where it was shown that the transition to the magnonic regime may occur before or\nafter the Walker breakdown depending on the parameters of the problem. See also [19, 20].\nIn all these works bulk matter or thin \flms were the subject of study. For Permalloy nan-\notubes the numerical simulations of [5, 6] show that the sudden change of slope in the rate\n3of increase of the speed of domain walls is accompanied by Cherenkov spin wave emission\nonce the DW speed exceeds the phase speed of the spin waves.\nIn the present work we study theoretically the DW propagation in Permalloy nanotubes\nand \fnd that curvature acts as an additional anisotropy and plays an equivalent role to the\nDMI in weak ferromagnets. The parallel between curvature of a nanotube and DMI has\nbeen observed in [21{23] among others. In [23] it is shown that the analytical expression of\nthe dispersion relation for spin waves in a nanotube has the same mathematical form as the\ndispersion relation for spin waves in thin \flms with DMI. Here we \fnd this mathematical\nanalogy in the mobility of the domain wall. See [24] for a recent comprehensive review on\nthe dynamics of magnetic nanotubes.\nAlthough simulations have been carried out for Permalloy, in the derivation below we will\nallow a material with non negligible uniaxial anisotropy for greater generality.\nII. STATEMENT OF THE PROBLEM\nConsider a thin nanotube with an easy direction along the nanotube axis which we\nchoose as the zaxis. The dynamic evolution of the magnetization is governed by the LLG\nequation (1). A right handed orthogonal cylindrical coordinate system ( \u001a;';z ) is introduced\nas shown in Fig.II in terms of which the unit magnetization vector is written as ~ m=\nm\u001a(\u001a;';z )^\u001a+m'(\u001a;';z ) ^'+mz(\u001a;';z )^z.\nFIG. 1: Cylindrical coordinate system in the nanotube.\nFor su\u000eciently thin tubes the demagnetizing \feld can be approximated by a local ex-\npression with the saturation magnetization acting as an e\u000bective radial hard axis anisotropy\n[8, 25, 26]. In this approximation and including exchange energy, uniaxial anisotropy energy,\n4demagnetization energy and Zeeman energy, the micromagnetic energy can be written as\n[7, 8]\nE=Z\n\nd3x(Ajr~ mj2+Ku(1\u0000m2\nz) +\u00160M2\ns\n2m2\n\u001a\u0000Hamz); (5)\nwhere \n is the material volume of the nanotube, Ais the exchange constant, Kuthe uniaxial\nanisotropy and an external \feld ~Ha=Ha^zhas been applied along the axis. The e\u000bective\n\feld is given by\n~He\u000b=\u00001\n\u00160Ms\u000eE\n\u000e~ m:\nIn a very thin nanotube variations of the magnetization with radius may be neglected so\nthat the unit magnetization depends only on the polar coordinate 'and the axial position\nz. With~ m=~ m(';z), the e\u000bective magnetic \feld can be written as [8]\n~He=2A\n\u00160Ms\u00141\nR2@2~ m\n@'2+@2~ m\n@z2\u0015\n+2Ku\n\u00160Msmz^z\u0000Msm\u001a^\u001a+Ha^z: (6)\nIntroducing Msas unit of magnetic \feld, and introducing the dimensionless space and time\nvariables\u0018=z=R and\u001c=\r0Mstwe rewrite equations (1) and (6) in dimensionless form as\nd~ m\nd\u001c=\u0000~ m\u0002~he\u000b+\u000b~ m\u0002d~ m\nd\u001c(7)\nwith\n~he\u000b=A0\u0014@2~ m\n@'2+@2~ m\n@\u00182\u0015\n+kumz^z\u0000m\u001a^\u001a+ha^z (8)\nwherehais the dimensionless applied \feld. The dimensionless numbers that have appeared\nareku= 2Ku=(\u00160M2\ns) andA0, the square of the ratio between the exchange length lex=\np\n2A=\u0016 0M2\nsand the radius, that is, A0= 2A=(\u00160M2\nsR2). Equations (7) and (8) describe\nthe dynamics of the problem.\nNumerical simulations [5, 6] have been performed for Permalloy for which the exchange\nconstantA= 1:3\u000210\u000011J m\u00001,Ms= 8\u0002105A m\u00001,Ku\u00190 and the external applied\n\feld does not exceed 10\u00002Ms. The nanotube used in simulations has inner radius R, and\nwidthwwithw << R: Here we neglect the variations with radius and consider the range\nR= 55\u0000100\u000210\u00009m. The vacuum permeability \u00160= 4\u0019\u000210\u00007N A\u00002so that\u00160Ms\u00191T.\nWe take the value \r0= 2:21\u0002105s\u00001T\u00001. For Permalloy the exchange length is lex= 5:68\nnm and for a radius of 80 nm lex=R= 0:071. The uniaxial anisotropy vanishes, ku= 0, and\nthe dimensionless applied \feld is in the range 0 <ha<10\u00002. The damping parameter \u000bin\nnumerical simulations ranges from 0 :01 to 0:03.\n5In the numerical simulations of Permalloy nanotubes [5, 6] it is observed that a right\nhanded vortex-type domain wall is stable against the Walker breakdown when driven with\na magnetic \feld. The handedness of the domain wall in [5, 6] is de\fned in relation to the\napplied \feld, a propagating domain wall is called right handed if ( ~ m\u0002~Ha)\u0001^\u001a > 0. Left\nhanded domain walls become unstable and convert into the other, stable chirality. In this\nwork we are interested in the speed of the stable DW, which will be selected through the\nscaling in the asymptotic solution.\nIII. ASYMPTOTIC SOLUTION\nIn this section we perform an asymptotic analysis of the LLG equation to \fnd a reduced\nmodel for the evolution of the domain wall as the applied \feld increases. The reduced model\nwill be valid for a restricted parameter range which is chosen based on the numerical results\ndescribed above for Permalloy. We are interested in right handed vortex walls for which\nthe radial component of the magnetization is small, m\u001a\u001c1 [5, 6]. Introducing a small\ndimensionless parameter \u000fwe write this condition as\nm\u001a=\u000f~m\u001a: (9)\nThe normalization condition ~ m2= 1 becomes\nm2\n'+m2\nz= 1\u0000\u000f2~m2\n\u001a: (10)\nWe will model a situation in which the ratio lex=Rand the Gilbert constant are of the same\norder in\u000fas the radial component of the magnetization. We assume that the applied \feld\nand uniaxial anisotropy are of an order smaller. Let then\nA0=\u000f2~A; k u=\u000f2~ku; ha=\u000f2~ha; \u000b =\u000f~\u000b: (11)\nIt is found that a consistent asymptotic approach can be obtained if a new time scale s=\u000f\u001c\nis introduced. With these scalings, the components of the e\u000bective magnetic \feld can be\nwritten as\n6(~he\u000b)\u001a=\u0000\u000f~m\u001a\u00002\u000f2~A@m'\n@'+\u000f3~A\u0000\nr2\ns~m\u001a\u0000~m\u001a\u0001\n=\u000fH0\n\u001a+\u000f2H1\n\u001a+\u000f3H2\n\u001a;(12a)\n(~he\u000b)'=\u000f2~A(r2\nsm'\u0000m') + 2\u000f3~A@~m\u001a\n@'\n=\u000f2H1\n'+\u000f3H2\n';(12b)\n(~he\u000b)z=\u000f2\u0010\n~ha+~Ar2\nsmz+~kumz\u0011\n=\u000f2H1\nz; (12c)\nwherer2\ns=@\u0018\u0018+@''and where we grouped terms according to the power of \u000fso that\nH0\n\u001a=\u0000~m\u001a,H1\n'=~A(r2\nsm'\u0000m') andH1\nz=~ha+~Ar2\nsmz+~kumz:In obtaining these\nexpressions for the e\u000bective \feld the property @^\u001a=@' = ^'; @^'=@' =\u0000^\u001ais used.\nIntroducing the scaling for \u000bandm\u001ain the LLG equation, we obtain at leading order in\n\u000f;\n_~m\u001a=\u0000(m'H1\nz\u0000mzH1\n') + ~\u000b(m'_mz\u0000mz_m'); (13a)\n_m'=\u0000mzH0\n\u001a; (13b)\n_mz=m'H0\n\u001a; (13c)\nwhere a dot represents a derivative with respect to the scaled time variable sand the\nsubindices represent the components of each vector.\nThe normalization condition (10) implies that, at leading order, we may write\nm'= sin\u0012(\u0018;';s ); m z= cos\u0012(\u0018;';s ): (14)\nIt follows then that equations (13b) and (13c) are equivalent and imply\n_\u0012=\u0000H0\n\u001a= ~m\u001a: (15)\nReplacing the value of ~ m\u001afrom (15) in (13a) together with the expressions for the e\u000bective\n\feldH1\n';H1\nz, the evolution equation for \u0012is found to be\n\u0012+ ~\u000b_\u0012=~A(\u0012\u0018\u0018+\u0012'')\u0000sin\u0012\u0010\n~ha+ (~A+~ku) cos\u0012\u0011\n; (16)\n7where the subscripts in \u0012denote derivatives with respect to \u0018and'respectively. Notice\nthat one may go back to the original unscaled variables and the small parameter \u000fcancels\nout.\nIn what follows we study cylindrically symmetric domain walls, for which \u0012'= 0 and\nidentify the evolution equation with the damped double Sine Gordon equation,\n@2\u0012\n@\u001c2+\u000b@\u0012\n@\u001c=A0\u0012\u0018\u0018\u0000sin\u0012(ha+ (A0+ku) cos\u0012); (17)\na particular case of hyperbolic reaction di\u000busion equation, for which the existence and\nstability of traveling waves have been studied rigorously in [27, 28].\nThis equation has been derived in the analysis of domain wall propagation in weak ferro-\nmagnets, [9{11, 29] and in systems with a strong easy plane [16, 30]. In [11] the dependence\nof mobility on the Dzyaloshinskii constant is derived with great detail. A common feature\nin these problems is the sudden decrease in the rate of increase of the speed with the applied\n\feld.\nThis equation has the same traveling wave solutions as the reaction di\u000busion equation\n\u000b_\u0012=A0\u0012\u0018\u0018\u0000sin\u0012(ha+ (A0+ku) cos\u0012) but with velocity c=cr=p\n1 +c2\nr=A0wherecris\nthe speed of fronts of the reaction di\u000busion equation [27]. We give the explicit expression\nfor the head to head (HH) domain wall, the tail to tail solution is similar. The HH solution\nis found to be the usual domain wall pro\fle,\n\u0012(\u0018;t) = 2 arctan\u0014\nexp\u0012\u0018\u0000c\u001c\n\u0001\u0013\u0015\n(18)\nwith the speed cand domain wall width \u0001 given by\nc=r\nA0\nA0+kuhap\n\u000b2+ (A0+ku)\u00001h2\na; \u0001 =\u000bc\nha: (19)\nThe leading order magnetization ~ m=m'^'+mz^zis given by\n~ m= sech\u0012\u0018\u0000c\u001c\n\u0001\u0013\n^'\u0000tanh\u0012\u0018\u0000c\u001c\n\u0001\u0013\n^z: (20)\nThe external \feld is applied along the zaxis so the magnetization is a right handed ( m'\u00150)\nhead to head domain wall as de\fned in [6]. The small radial component of the magnetization\nis calculated from (15).\nFor small applied \feld we recover the linear regime, that is, the speed increases linearly\nwith the \feld, and the domain wall width tends to a constant value, that is,\nlim\nha!0c=r\nA0\nA0+kuha\n\u000b;lim\nha!0\u0001 =r\nA0\nA0+ku: (21)\n8In this limit the dynamics is primarily governed by the reaction di\u000busion equation \u000b\u0012\u001c=\nA0\u0012\u0018\u0018\u0000sin\u0012(ha+ (A0+ku) cos\u0012) as already found in [8].\nIn terms of the physical parameters the dimensional domain wall width for small \feld\n\u000e= \u0001Rand speedvLcan be written as\n\u000e=s\nA\nA\nR2+Ku; vL=\r0Ha\n\u000b\u000e\nFor Permalloy, Ku= 0 and\u000e=Rin agreement with the results for a static domain\nwall in a thin nanotube [31]. The speed vLcoincides with the low \feld Walker solution\nvW=\r0Hap\nA=K=\u000b , withKan e\u000bective anisotropy A=R2. In the limit of large radius the\ndomain wall width for an in plane magnetized thin \flm,p\nA=Kuis recovered.\nFor large applied \feld the speed tends to a constant value and the domain wall width\ndecreases as the \feld increases,\nlim\nha!1c=p\nA0lim\nha!1\u0001 =\u000bpA0\nha: (22)\nThis limiting value for the speed corresponds to the minimal value of the phase velocity\nfor spin waves, vpmin. In e\u000bect, consider a material like Permalloy with vanishing uniaxial\nanisotropy for simplicity. The dispersion relation for the DSG equation (17), for vanishing\ndamping and vanishing applied \feld, is given by !DSG=pA0p\n1 +k2, so that the phase\nspeed is a decreasing function of kwhich tends asymptotically topA0askgrows. The full\ndispersion relation for spin waves in a thin nanotube, in the absence of damping and applied\n\feld, with vanishing uniaxial anisotropy, is given by [32]\n!=pA0\n2p\n(1 +k2) +A0(1 +k2)2 (23)\nin the units used in this work. We see that for small A0the full dispersion relation coincides,\nup to a constant, with !DSG.\nThe evolution equation (17) shows the transition from the low \feld regime where the\nspeed of the domain wall increases linearly with the \feld to the regime where the domain wall\nspeed approaches vpminand is slowed down by emitting spin waves. In order to capture the\nlarge applied \feld regime where the DW speed exceeds vpminand Cherenkov emission occurs\na di\u000berent scaling is needed. The DW width shrinks with increasing \feld, \u0001 \u0019\u000bpA0=ha,\nwhich indicates that at larger \felds a new scaling for the longitudinal coordinate \u0018is required.\n9v[m/s]\n<latexit sha1_base64=\"j9SroW/XuGmsdepu2GtGB3geGE0=\">AAACDHicbZC7SgNBFIbPejfe4qWzGWIEQYi7NlqKNpYKxgSyS5idnDWDsxdmzgbjklfwGWy1thNb38HSN3GSWGj0h4GP/z+HOfxhpqQh1/1wpqZnZufmFxZLS8srq2vl9Y1rk+ZaYF2kKtXNkBtUMsE6SVLYzDTyOFTYCG/Phnmjh9rINLmifoZBzG8SGUnByVrt8la1ynrMJ7yjgrXiAxMMqtV2ecetuSOxv+B9w85Jxd9/BICLdvnT76QijzEhobgxLc/NKCi4JikUDkp+bjDj4pbfYMtiwmM0QTG6fsB2rdNhUartS4iN3J8bBY+N6cehnYw5dc1kNjT/y1o5RcdBIZMsJ0zE+KMoV4xSNqyCdaRGQapvgQst7a1MdLnmgmxhJb+DkU+FT10kPijZTrzJBv7C9WHNc2vepS3nFMZagG2owB54cAQncA4XUAcB9/AIT/DsPDgvzqvzNh6dcr53NuGXnPcvE2+bZw==</latexit><latexit sha1_base64=\"Y2klHEeoJsELu4qSFIuhBBCz4zM=\">AAACDHicbZC7SgNBFIZnvbve4qWzGZIVBCHu2mgZtLFUMCpklzA7OZsMzl6YORuMS17Bygew1dpObH2HlL6Jk0uh0R8GPv7/HObwh5kUGl13YM3Mzs0vLC4t2yura+sbpc2ta53mikOdpzJVtyHTIEUCdRQo4TZTwOJQwk14dzbMb7qgtEiTK+xlEMSsnYhIcIbGapZ2HId2qY9wjwVtxIc66DtOs1Rxq+5I9C94E6jUyv7B06DWu2iWvvxWyvMYEuSSad3w3AyDgikUXELf9nMNGeN3rA0NgwmLQQfF6Po+3TNOi0apMi9BOnJ/bhQs1roXh2YyZtjR09nQ/C9r5BidBIVIshwh4eOPolxSTOmwCtoSCjjKngHGlTC3Ut5hinE0hdl+CyIfCx87gKxvm0686Qb+wvVR1XOr3qUp55SMtUR2SZnsE48ckxo5JxekTjh5IM/khbxaj9ab9W59jEdnrMnONvkl6/MbM7Oc7Q==</latexit><latexit sha1_base64=\"Y2klHEeoJsELu4qSFIuhBBCz4zM=\">AAACDHicbZC7SgNBFIZnvbve4qWzGZIVBCHu2mgZtLFUMCpklzA7OZsMzl6YORuMS17Bygew1dpObH2HlL6Jk0uh0R8GPv7/HObwh5kUGl13YM3Mzs0vLC4t2yura+sbpc2ta53mikOdpzJVtyHTIEUCdRQo4TZTwOJQwk14dzbMb7qgtEiTK+xlEMSsnYhIcIbGapZ2HId2qY9wjwVtxIc66DtOs1Rxq+5I9C94E6jUyv7B06DWu2iWvvxWyvMYEuSSad3w3AyDgikUXELf9nMNGeN3rA0NgwmLQQfF6Po+3TNOi0apMi9BOnJ/bhQs1roXh2YyZtjR09nQ/C9r5BidBIVIshwh4eOPolxSTOmwCtoSCjjKngHGlTC3Ut5hinE0hdl+CyIfCx87gKxvm0686Qb+wvVR1XOr3qUp55SMtUR2SZnsE48ckxo5JxekTjh5IM/khbxaj9ab9W59jEdnrMnONvkl6/MbM7Oc7Q==</latexit><latexit sha1_base64=\"/+j9p+rP1jjQly2r7+bfVNMThgg=\">AAACDHicbZC5TgMxFEU9YQthC0tHY5EgUYUZGigjaCiDRBYpM4o8zpvEimeR/SYijPILfAMt1HSIln+g5E9wlgISrmTp6N735KfrJ1JotO0vK7eyura+kd8sbG3v7O4V9w8aOk4VhzqPZaxaPtMgRQR1FCihlShgoS+h6Q9uJnlzCEqLOLrHUQJeyHqRCARnaKxO8ahcpkPqIjxgRtvhufbG5XKnWLIr9lR0GZw5lMhctU7x2+3GPA0hQi6Z1m3HTtDLmELBJYwLbqohYXzAetA2GLEQtJdNrx/TU+N0aRAr8yKkU/f3RsZCrUehbyZDhn29mE3M/7J2isGVl4koSREiPvsoSCXFmE6qoF2hgKMcGWBcCXMr5X2mGEdTWMHtQuBi5mIfkI0LphNnsYFlaFxUHLvi3Nml6vW8nTw5JifkjDjkklTJLamROuHkkTyTF/JqPVlv1rv1MRvNWfOdQ/JH1ucP6dqZ3g==</latexit>µ0Ha[mT]\n<latexit sha1_base64=\"x36GB0uNt1OORYSkO/833Q6joLA=\">AAACE3icbZA7SwQxFIXv+HZ9rVraBF1BEJYZGy1FG0sFV4WdYchk77jBZGZI7ojLsIU/wt7OVms7sfUHWPpPzO5a+DoQ+DjnXpKcpFDSku+/e2PjE5NT0zOztbn5hcWl+vLKmc1LI7AlcpWbi4RbVDLDFklSeFEY5DpReJ5cHQ7y82s0VubZKfUKjDS/zGQqBSdnxfW1RoOFuox9dhRzFhLeUMXa+jTqNxpxfcNv+kOxvxB8wcb+erh9DwDHcf0j7OSi1JiRUNzaduAXFFXckBQK+7WwtFhwccUvse0w4xptVA0/0WebzumwNDfuZMSG7veNimtrezpxk5pT1/7OBuZ/WbukdC+qZFaUhJkYXZSWilHOBo2wjjQoSPUccGGkeysTXW64INdbLexgGlIVUheJ92uuk+B3A3/hbKcZ+M3gxJVzACPNwBqswxYEsAv7cATH0AIBt/AAj/Dk3XnP3ov3Ohod8752VuGHvLdPusud3g==</latexit><latexit sha1_base64=\"49/6L556qISZOKAPMkLNKSKyNQ4=\">AAACE3icbZDNSgMxFIUz/lv/qi7dBFtBEMqMG10W3XSp0KrQGYZMeqcNJjNDckcsQxe+gz6DW127E7c+QJe+iWnrQlsPBD7OuZckJ8qkMOi6Q2dufmFxaXlltbS2vrG5Vd7euTJprjm0eCpTfRMxA1Ik0EKBEm4yDUxFEq6j2/NRfn0H2og0aWI/g0CxbiJiwRlaKyzvVavUV3no0kbIqI9wjwVtq2YwqFbDcsWtuWPRWfB+oFLf948eh/X+RVj+8jspzxUkyCUzpu25GQYF0yi4hEHJzw1kjN+yLrQtJkyBCYrxJwb0wDodGqfangTp2P29UTBlTF9FdlIx7JnpbGT+l7VzjE+DQiRZjpDwyUVxLimmdNQI7QgNHGXfAuNa2LdS3mOacbS9lfwOxD4WPvYA2aBkO/GmG5iFq+Oa59a8S1vOGZloheyRfXJIPHJC6qRBLkiLcPJAnskLeXWenDfn3fmYjM45Pzu75I+cz2/bD59k</latexit><latexit sha1_base64=\"49/6L556qISZOKAPMkLNKSKyNQ4=\">AAACE3icbZDNSgMxFIUz/lv/qi7dBFtBEMqMG10W3XSp0KrQGYZMeqcNJjNDckcsQxe+gz6DW127E7c+QJe+iWnrQlsPBD7OuZckJ8qkMOi6Q2dufmFxaXlltbS2vrG5Vd7euTJprjm0eCpTfRMxA1Ik0EKBEm4yDUxFEq6j2/NRfn0H2og0aWI/g0CxbiJiwRlaKyzvVavUV3no0kbIqI9wjwVtq2YwqFbDcsWtuWPRWfB+oFLf948eh/X+RVj+8jspzxUkyCUzpu25GQYF0yi4hEHJzw1kjN+yLrQtJkyBCYrxJwb0wDodGqfangTp2P29UTBlTF9FdlIx7JnpbGT+l7VzjE+DQiRZjpDwyUVxLimmdNQI7QgNHGXfAuNa2LdS3mOacbS9lfwOxD4WPvYA2aBkO/GmG5iFq+Oa59a8S1vOGZloheyRfXJIPHJC6qRBLkiLcPJAnskLeXWenDfn3fmYjM45Pzu75I+cz2/bD59k</latexit><latexit sha1_base64=\"xBgp5p2BvHq86gbNtgDnUBYYwik=\">AAACE3icbZC9TsMwFIWd8lfKX4Cxi0WLxFQlLDBWsDCC1BakJooc94ZatZPIvkFUUQcegmdghZkNsfIAjLwJbunA35EsfTrnXtk+cS6FQc97dyoLi0vLK9XV2tr6xuaWu73TM1mhOXR5JjN9FTMDUqTQRYESrnINTMUSLuPR6TS/vAFtRJZ2cJxDqNh1KhLBGVorcuvNJg1UEXn0LGI0QLjFkvZVJ5w0m5Hb8FreTPQv+HNokLnOI/cjGGS8UJAil8yYvu/lGJZMo+ASJrWgMJAzPmLX0LeYMgUmLGefmNB96wxokml7UqQz9/tGyZQxYxXbScVwaH5nU/O/rF9gchyWIs0LhJR/XZQUkmJGp43QgdDAUY4tMK6FfSvlQ6YZR9tbLRhAEmAZ4BCQTWq2E/93A3+hd9jyvZZ/4TXaJ/N2qqRO9sgB8ckRaZMzck66hJM78kAeyZNz7zw7L87r12jFme/skh9y3j4BkUWcVQ==</latexit>02468020040060080010001200FIG. 2: Speed of the domain wall versus applied \feld in millitesla for a thin Permalloy nanotube of\nradius R= 55 nanometers. At low \feld the speed increases linearly with the \feld, after a sudden\nchange in slope the speed tends to a constant value v1at large \felds. The dashed lines show the\nlimiting speeds vLandv1.\nThe damped DSG equation captures the emission of spin waves that occurs below but close\ntovpmin.\nA di\u000berent transition occurs at hKPP\na= 2A0when the speed of the reaction di\u000busion\nequationcrchanges from a pushed to a pulled or KPP front [33] and crbecomes proportional\nto the square root of the applied \feld.\nIn what follows consider Permalloy for which ku= 0. Going back to dimensional quanti-\nties, the speed of the domain wall for Permalloy is given by Eq. (2) with the limiting values\nat low and high \felds Eq. (3) and Eq. (4). In Fig. 2 the graph of the speed as a function\nof the applied \feld shows the gradual change from the linear to the magnonic regime. We\nhave used the values given above for Permalloy.\nAn approximate estimate of the \feld H\u0003\naat which this transition occurs is obtained by\nthe intersection vL(H\u0003\na) =v1which yields\nH\u0003\na=\u000b\nRs\n2A\n\u00160:\nFor Permalloy we obtain v1= 1006 m s\u00001, andB\u0003\na=\u00160H\u0003\na= 0:001 T. The transition to the\nKPP regime occurs at a much higher \feld, BKPP\na=\u00160HKPP\na= 0:021 T and is not associated\nto the transition from the linear to the magnonic regime. In this simple model the order of\nmagnitude of the speed and the value of the \feld at which the transition from the linear to\nthe magnonic regime occurs agrees with the order of magnitude of the numerical simulations\nof the LLG equation.\n10IV. SUMMARY\nWe studied the dynamics of a vortex domain wall in a thin nanotube by means of an\nasymptotic study of the Landau-Lifshitz Gilbert equation in a parameter regime based on\nexisting numerical simulations [5, 6]. The numerical simulations on Permalloy nanotubes in\na certain range of radii showed that when an external \feld is applied along the axis, domain\nwalls of one type of chirality, for which the radial magnetization remains small during the\nmotion, are stable and can reach high speeds. Initially the speed increases linearly with\nthe applied \feld, and at higher \felds the rate of increase is slowed down by the emission\nof spin waves. No Walker breakdown was observed in the parameter range considered in\nthe numerical studies. Domain walls of the opposite chirality are unstable and as the \feld\nincreases they convert into DW of stable chirality.\nThe purpose of this work was to understand analytically the behavior of the speed of\ndomain walls of stable chirality as a function of the applied \feld. Through an asymptotic\nanalysis the LLG was reduced to the damped double sine-Gordon equation from which an\nexplicit analytic formula for the speed as a function of the applied \feld was obtained together\nwith the leading order DW pro\fle. This model captures the initial regime of linear growth\nof the speed followed by a slowdown in the rate of increase through the emission of spin\nwaves before reaching the minimal phase speed of the spin waves, which is an upper bound\non the speed of the DW in this model. The order of magnitude of the speed and the value\nof the applied \feld where the transition from the linear to the magnonic regime occurs is in\nagreement with the numerical results of [5, 6]. In order to reach higher \felds and capture\nthe Cherenkov spin wave emission process a di\u000berent asymptotic regime is necessary.\nFor Permalloy, which has vanishing uniaxial anisotropy, the ratio of the exchange constant\nwith the square of the radius of the nanotube A=R2plays the role of an e\u000bective uniaxial\nanisotropy which leads to a mobility proportional to the nanotube radius. In constrast, for\nweak ferromagnets the mobility is proportional to de Dzyaloshinkii-Moriya constant. That\nthe e\u000bect of curvature acts as an equivalent e\u000bective anisotropy was already shown in [8, 22],\nand an analogy between the e\u000bect of DMI and curvature was found in the dispersion relation\nof spin waves in a nanotube [23]. The results in this manuscript show a similar e\u000bect when\nstudying the transition from the linear to the magnonic DW regime in nanotubes.\nAn analytical approach to the regime of higher \feld, where Cherenkov emission occurs,\n11will be the subject of future study.\nV. ACKNOWLEDGMENTS\nThis work was partially supported by Fondecyt (Chile) project 116{0856.\n[1] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet., 8, 153 (1935).\n[2] T. L. Gilbert, Ph.D. Thesis, Illinois Institute of Technology (1956), partially reprinted in IEEE\nTrans. Mag., 40, 3443 (2004).\n[3] L.R. Walker, Bell Telephone Laboratories Memorandum, 1956 (unpublished). An account of\nthis work is given in J.F. Dillon, Jr., Magnetism, Vol. III, edited by G.T. Rado and H. Suhl\n(Academic, New York, 1963).\n[4] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974).\n[5] M. Yan, A. C. K\u0013 akay, F. Garc\u0013 \u0010a-Sanchez and R. Hertel, Appl. Phys. Lett. 99, 122505 (2011).\n[6] R Hertel, J. 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J. Mikeska, J. Phys. C 13, 2913 (1980).\n[30] H. How, R. C. O'Handley and F. R. Morgenthaler, Phys. Rev. B 40, 4808 (1989).\n[31] P. Landeros and A. S. Nu~ nez, J. Appl. Phys. 108033917 (2010).\n[32] A. L. Gonz\u0013 alez, P. Landeros and A. S. N\u0013 u~ nez, J. Mag. Mag. Mater. 322, 530 (2010).\n[33] M. C. Depassier, EPL 108, 37008 (2014).\n13"    },    {        "title": "1809.08453v1.Optimizing_a_Generalized_Gini_Index_in_Stable_Marriage_Problems__NP_Hardness__Approximation_and_a_Polynomial_Time_Special_Case.pdf",        "content": "arXiv:1809.08453v1  [cs.CC]  22 Sep 2018manuscript No.\n(will be inserted by the editor)\nOptimizing a Generalized Gini Index in Stable Marriage\nProblems: NP-Hardness, Approximation and a Polynomial\nTime Special Case\nHugo Gilbert ·Olivier Spanjaard\nthe date of receipt and acceptance should be inserted later\nAbstract This paper deals with fairness in stable marriage problems. The idea studied here\nis to achieve fairness thanks to a Generalized Gini Index (GG I), a well-known criterion in\ninequality measurement, that includes both the egalitaria n and utilitarian criteria as special\ncases. We show that determining a stable marriage optimizin g a GGI criterion of agents’\ndisutilities is an NP-hard problem. We then provide a polyno mial time 2-approximation\nalgorithm in the general case, as well as an exact algorithm w hich is polynomial time in the\ncase of a constant number of non-zero weights parametrizing the GGI criterion.\nKeywords Stable marriage problem ·Fairness ·Generalized Gini index ·Complexity\n1 Introduction\nSince the seminal work of Gale and Shapley [1962] on stable ma rriages, matching problems\nunder preferences have been extensively studied both by eco nomists and computer scien-\ntists. These problems involve two sets of agents (also calle d individuals in the sequel) that\nshould be matched with each other while taking agents’ prefe rences into account. The results\nobtained in the field have a tremendous number of application s, among which the National\nResident Matching Program in the US (for allocating junior d octors to hospitals), the teacher\nallocation in France (for allocating newly tenured teacher s to schools) or the allocation of\nlawyers in Germany (for assigning graduating lawyers to leg al internship positions). For an\noverview of the applications of matching models under prefe rences, the interested reader\ncan refer to a recent book chapter on this topic [Bir´ o, 2017] .\nThe stable marriage problem involves nmen and nwomen, each of whom ranks the\nmembers of the opposite sex in order of preference. The goal i s to find a stable matching, i.e.,\na matching between men and women such that there is no man and w oman that prefer each\nother to their current match. Gale and Shapley [1962] provid ed an algorithm that computes\na stable marriage. However, it is well-known that this algor ithm favours one group (men or\nwomen, according to the way the algorithm is applied) over th e other.\nH. Gilbert·O. Spanjaard\nSorbonne Universit´ e, CNRS, Laboratoire d’Informatique d e Paris 6, LIP6, F-75005 Paris, France\nE-mail:{hugo.gilbert,olivier.spanjaard }@lip6.fr2 Hugo Gilbert, Olivier Spanjaard\nWe are interested here in fairstable marriage algorithms, i.e., in procedures favouring\nstable marriages that fairly share dissatisfactions –also called disutilities– among individuals\n(irrespective of their sex), the dissatisfaction being defi ned for each woman (resp. man) as\na function of the rank, in order of preferences, of the man (re sp. woman) to whom she is\npaired with. Given the vector of individuals’ dissatisfact ions induced by a matching, there\nare several ways of formalizing the notion of “fairness”. We mean here by fair stable mar-\nriage that the vector of individuals’ dissatisfactions sho uld be well-balanced. For example,\nconsider the following instance of the stable marriage prob lem.\nExample 1 The instance consists of 10 men {m1,...,m 10}and women {w1,...,w 10}\nwith the following preferences, where i≻m\nkj(resp.i≻w\nkj) means that mk(resp.wk)\npreferswitowj(resp.mitomj):\nm1: 1≻m\n12≻m\n13≻m\n14≻m\n15≻m\n16≻m\n17≻m\n18≻m\n19≻m\n110\nm2: 2≻m\n21≻m\n23≻m\n24≻m\n25≻m\n26≻m\n27≻m\n28≻m\n29≻m\n210\nm3: 3≻m\n31≻m\n32≻m\n34≻m\n35≻m\n36≻m\n37≻m\n38≻m\n39≻m\n310\nm4: 7≻m\n41≻m\n42≻m\n43≻m\n46≻m\n44≻m\n45≻m\n48≻m\n49≻m\n410\nm5: 6≻m\n51≻m\n52≻m\n53≻m\n57≻m\n54≻m\n55≻m\n58≻m\n59≻m\n510\nm6: 4≻m\n61≻m\n62≻m\n63≻m\n65≻m\n67≻m\n66≻m\n68≻m\n69≻m\n610\nm7: 5≻m\n71≻m\n72≻m\n73≻m\n74≻m\n77≻m\n76≻m\n78≻m\n79≻m\n710\nm8: 8≻m\n84≻m\n85≻m\n86≻m\n810≻m\n87≻m\n81≻m\n82≻m\n83≻m\n89\nm9: 10≻m\n94≻m\n96≻m\n97≻m\n99≻m\n95≻m\n91≻m\n92≻m\n93≻m\n98\nm10: 9≻m\n104≻m\n105≻m\n107≻m\n108≻m\n106≻m\n101≻m\n102≻m\n103≻m\n1010\nw1: 1≻w\n12≻w\n13≻w\n14≻w\n15≻w\n16≻w\n17≻w\n18≻w\n19≻w\n110\nw2: 1≻w\n22≻w\n23≻w\n24≻w\n25≻w\n26≻w\n27≻w\n28≻w\n29≻w\n210\nw3: 1≻w\n32≻w\n33≻w\n34≻w\n35≻w\n36≻w\n37≻w\n38≻w\n39≻w\n310\nw4: 1≻w\n42≻w\n43≻w\n47≻w\n48≻w\n49≻w\n46≻w\n44≻w\n45≻w\n410\nw5: 1≻w\n52≻w\n53≻w\n56≻w\n58≻w\n59≻w\n57≻w\n54≻w\n55≻w\n510\nw6: 1≻w\n62≻w\n63≻w\n64≻w\n68≻w\n69≻w\n65≻w\n66≻w\n67≻w\n610\nw7: 1≻w\n72≻w\n73≻w\n75≻w\n78≻w\n79≻w\n74≻w\n76≻w\n77≻w\n710\nw8: 2≻w\n810≻w\n88≻w\n87≻w\n81≻w\n83≻w\n84≻w\n85≻w\n86≻w\n89\nw9: 1≻w\n92≻w\n99≻w\n910≻w\n93≻w\n94≻w\n95≻w\n96≻w\n97≻w\n98\nw10: 1≻w\n102≻w\n103≻w\n104≻w\n105≻w\n106≻w\n107≻w\n1010≻w\n108≻w\n109\nThe stable marriages in this instance are:\nx1:{(m1,w1),(m2,w2),(m3,w3),(m4,w7),(m5,w6),(m6,w4),(m7,w5),\n(m8,w8),(m9,w10),(m10,w9)}\nx2:{(m1,w1),(m2,w2),(m3,w3),(m4,w7),(m5,w6),(m6,w5),(m7,w4),\n(m8,w8),(m9,w10),(m10,w9)}\nx3:{(m1,w1),(m2,w2),(m3,w3),(m4,w6),(m5,w7),(m6,w4),(m7,w5),\n(m8,w8),(m9,w10),(m10,w9)}\nx4:{(m1,w1),(m2,w2),(m3,w3),(m4,w6),(m5,w7),(m6,w5),(m7,w4),\n(m8,w8),(m9,w10),(m10,w9)}\nx5:{(m1,w1),(m2,w2),(m3,w3),(m4,w6),(m5,w7),(m6,w5),(m7,w4),\n(m8,w10),(m9,w9),(m10,w8)}\nwhere a pair (mi,wj)means that miandwjare matched.The GGI Stable Marriage Problem 3\nIf one assumes that the dissatisfaction of an individual is e qual to the rank of the partner\nin his/her preference list, then the dissatisfactions indu ced by the previous stable marriages\nare:\nmatching vector of dissatisfactionssum of\ndissatisfactionsmax of\ndissatisfactions\nx1(1,1,1,1,1,1,1,1,1,1,1,2,3,7,7,7,7,3,4,10) 61 10\nx2(1,1,1,1,1,5,5,1,1,1,1,2,3,4,4,7,7,3,4,10) 63 10\nx3(1,1,1,5,5,1,1,1,1,1,1,2,3,7,7,4,4,3,4,10) 63 10\nx4(1,1,1,5,5,5,5,1,1,1,1,2,3,4,4,4,4,3,4,10) 65 10\nx5(1,1,1,5,5,5,5,5,5,5,1,2,3,4,4,4,4,2,3,9) 74 9\nwhere the ithcomponent of the vector is the dissatisfaction of mifori∈ {1,...,10},\nand ofwi−10fori∈ {11,...,20}.\nIn this instance, the matching x4can be considered as inducing a well-balanced vec-\ntor of dissatisfactions. The matchings x1,x2andx3indeed favour more some individuals\n(the men in this case) than others, while matching x5yields quite high dissatisfactions for\nnumerous agents. The matching x4is therefore a good compromise between the utilitarian\nand the egalitarian viewpoints, where the utilitarian viewpoint aims at minimi zing the sum\nof dissatisfactions while the egalitarian viewpoint aims a t minimizing the dissatisfaction of\nthe worst off individual. Both the utilitarian and egalitar ian approaches have been advocated\nfor promoting fairness in the stable marriage problem [Gusfi eld, 1987; Gusfield and Irving,\n1989]. Other approaches aim at treating equally men and wome n, by minimizing the abso-\nlute difference between the total dissatisfactions of the t wo groups ( sex-equal stable mar-\nriage problem [Kato, 1993; McDermid and Irving, 2014]) or by minimizing the maximum\ntotal dissatisfaction between the two groups ( balanced stable marriage problem [Manlove,\n2013]). However, note that, in the instance of Example 1, all these criteria favour either\nx1(utilitarian) or x5(egalitarian, sex-equal, balanced). Finally, there exist s another type of\napproach, that is not based on assigning scores to marriages . In a first step, for each man,\none lists all his possible matches in a stable marriage, in or der of his preferences (this list\nincludes as many elements as there are feasible stable marri ages). In a second step, each\nman is matched with the median woman in the list. This procedu re yields a stable marriage,\nwhich is called median stable marriage [Teo and Sethuraman, 1998; Cheng, 2010]. In the\ninstance of the example, the median stable marriage is x4. Nevertheless, in this article, we\nfocus on determining a fair stable marriage by using a scoring rule .\nIn social choice theory, a scoring rule assigns a score to eac h alternative by summing\nthe scores given by every individual over the alternative. T his summation principle ensures\nthat all individuals contribute equally to the score of an al ternative. An alternative is usually\na candidate in an election, but it can also be an element of a co mbinatorial domain. For\ninstance, in proportional representation problems [Proca ccia et al. , 2008], where one aims\nat electing a committee, every feasible committee is an alte rnative. In the setting of sta-\nble marriage problems, every stable marriage is an alternat ive and the utilitarian approach\nis clearly a scoring rule where each individual evaluates a s table marriage by the rank of\nhis/her match. An interesting extension of the class of scor ing rules is the class of rank de-\npendent scoring rules [Goldsmith et al. , 2014], where, instead of limiting the aggregation to\na summation operation, the scores are aggregated by taking i nto account their ranks in the\nordered list of scores. As emphasized by Goldsmith et al. [2014], rank dependent scoring\nrules can be used to favour fairness by imposing some conditi ons on their parameters. A\nwell known class of rank-dependent scoring rules in inequal ity measurement are the Gen-\neralized Gini Indices (GGI) [Weymark, 1981]. Furthermore, this class of rank depe ndent4 Hugo Gilbert, Olivier Spanjaard\nscoring rules circumvents both the utilitarian and egalita rian criteria. Their optimization\non combinatorial domains have been studied in several setti ngs (often under the name of\nOrdered Weighted Averages ): assignment problems [Lesca et al. , 2018], proportional repre-\nsentation [Elkind and Ismaili, 2015], resource allocation [Heinen et al. , 2015]. To the best\nof our knowledge, the problem of determining a GGI optimal st able marriage has not been\nstudied yet. This is precisely the purpose of the present wor k.\nThe paper is organized as follows. In Section 2, we introduce notations and we formally\ndefine the GGI stable marriage problem studied here. Then, in Section 3, we prove that it\nis NP-hard to determine an optimal stable marriage accordin g to a GGI criterion applied to\nagents’ disutilities. In Section 4, we provide a polynomial time 2-approximation algorithm.\nFinally, in Section 5, we establish a parametrized complexi ty result with respect to a GGI-\nspecific parameter.\n2 The GGI Stable Marriage Problem\nLetM={m1,...,m n}denote the set of men, and W={w1,...,w n}the set of women.\nAs in Example 1, for each mk(resp.wk), a preference relation ≻m\nk(resp.≻w\nk) is defined on\nW(resp.M), wherei≻m\nkj(resp.i≻w\nkj) means that mk(resp.wk) preferswitowj(resp.\nmitomj). We denote by rk(mi,wj)the rank of woman wjin the preference order of man\nmi, and similarly for rk(wj,mi).\nA solution of a stable marriage problem is a matching represe nted by a binary matrix x,\nwherexij= 1means that miis matched with wj. A matching xinduces a matching function\nµxdefined by wj=µx(mi)andmi=µx(wj)ifxij= 1. In a perfect matching (called\nindifferently matching or marriage from now on), every man ( resp. woman) is matched with\na different woman (resp. man). More formally, a matching is d efined by:\n/summationtextn\ni=1xij= 1∀j∈ {1,...,n} (1)/summationtextn\nj=1xij= 1∀i∈ {1,...,n} (2)\nA matching is said to be stable if there exists no man and woman who prefer each\nother to their current partner. More formally, a perfect mat ching is stable if the following\nconstraints hold [Vande Vate, 1989]:\nxij+/summationdisplay\nj′≻m\nijxij′+/summationdisplay\ni′≻w\njixi′j≥1∀(i,j)∈ {1,...,n}2(3)\nThe set of stable marriages, i.e. binary matrices xsuch that constraints 1, 2 and 3 hold,\nis denoted by X. In their seminal paper, Gale and Shapley [1962] states that there always\nexists at least one stable marriage, which can be computed in O(n2).\nThe Gale-Shapley algorithm is based on a sequence of proposa ls from men to women. Each\nman proposes to the women following his preference order, pa using when a women agrees\nto be matched with him but continuing if his proposal is rejec ted. When a woman receives\na proposal, she rejects it if she already has a better proposa l according to her preferences.\nOtherwise, she agrees to hold it for consideration and rejec ts any former proposal that she\nmight had. Such a sequence of proposals always leads to a stab le marriage called man-\noptimal stable marriage and denoted by xm(if the role of men and women is reversed,\nwe obtain the woman-optimal stable marriage denoted by xw). In the man-optimal stable\nmarriage, each man has the best partner, and each woman has th e worst partner, that is\npossible in any stable marriage. Contrarily, in the woman-o ptimal stable marriage, eachThe GGI Stable Marriage Problem 5\nwoman has the best partner, and each man has the worst partner , that is possible in any\nstable marriage.\nTwo important properties of the Gale-Shapley algorithm are that:\n– ifmproposes to w, then there is no stable marriage in which mhas a better match than w.\n– ifmproposes to w, then there is no stable marriage in which whas a worse match than m.\nThese properties justify the notion of preference shortlis ts obtained through the Gale-Shapley\nalgorithm by removing any man mfrom a woman w’s preference list and vice-versa, when\nwreceives a proposal from a man she prefers to m. Note that the shortlists that are obtained\nat the end of the algorithm do not depend on the order in which t he proposals are made.\nExample 2 For instance, with the preferences of Example 1, the Gale-Sh apley algorithm\nleads to the following shortlists:\nm1: 1≻m\n12≻m\n13≻m\n14≻m\n15≻m\n16≻m\n17≻m\n19≻m\n110\nm2: 2≻m\n23≻m\n24≻m\n25≻m\n26≻m\n27≻m\n28≻m\n29≻m\n210\nm3: 3≻m\n34≻m\n35≻m\n36≻m\n37≻m\n310\nm4: 7≻m\n46≻m\n410\nm5: 6≻m\n57≻m\n510\nm6: 4≻m\n65≻m\n610\nm7: 5≻m\n74≻m\n710\nm8: 8≻m\n84≻m\n85≻m\n86≻m\n810≻m\n87\nm9: 10≻m\n94≻m\n96≻m\n97≻m\n99≻m\n95\nm10: 9≻m\n108≻m\n1010\nw1: 1\nw2: 1≻w\n22\nw3: 1≻w\n32≻w\n33\nw4: 1≻w\n42≻w\n43≻w\n47≻w\n48≻w\n49≻w\n46\nw5: 1≻w\n52≻w\n53≻w\n56≻w\n58≻w\n59≻w\n57\nw6: 1≻w\n62≻w\n63≻w\n64≻w\n68≻w\n69≻w\n65\nw7: 1≻w\n72≻w\n73≻w\n75≻w\n78≻w\n79≻w\n74\nw8: 2≻w\n810≻w\n88\nw9: 1≻w\n92≻w\n99≻w\n910\nw10: 1≻w\n102≻w\n103≻w\n104≻w\n105≻w\n106≻w\n107≻w\n1010≻w\n108≻w\n109\nThese shortlists makes it possible to identify some transfo rmations that can be applied\nfrom the man-optimal stable marriage to obtain other stable marriages (more favourable to\nwomen). These transformations are called rotations [Irvin g and Leather, 1986]. A rotation\nis a sequence ρ= (mi0,wi0),...,(mir−1,wir−1)of man-woman pairs such that, for each\nik(0≤k≤r−1), (1)wikis first in mik’s shortlist and (2) wik+1(k+1taken modulo r)\nis second in mik’s shortlist. Such a rotation is said to be exposed in the shor tlists.\nExample 3 Continuing Example 1, there are two rotations exposed in the shortlists, ρ1=\n(4,7),(5,6)andρ2= (6,4),(7,5).\nGiven a rotation, if each mikexchanges his current partner wikforwik+1, then the\nmatching remains stable. Eliminating a rotation ρ= (mi0,wi0),...,(mir−1,wir−1)amounts\nto removing all successors mofmik−1inwik’s shortlist together with the corresponding ap-\npearances of wikin the shortlists of men m. The obtained stable marriage can then be read\nfrom the modified shortlists by matching each man with the firs t woman in his shortlist. In6 Hugo Gilbert, Olivier Spanjaard\nthis new stable marriage, each woman (resp. man) is better of f (resp. worse off) than before\neliminating the rotation.\nOnce an exposed rotation has been identified and eliminated, then one or more rotations\nmay be exposed in the resulting (further reduced) shortlist s. This process may be repeated,\nand once all rotations have been eliminated, we obtain the wo man optimal stable marriage.\nA rotation πis said to be a predecessor of a rotation ρ, denoted by π < ρ , ifρcannot be\nexposed in the men shortlists before πis eliminated. This notion of predecessors makes it\npossible to define what is called the rotation poset (P,≤)wherePis the set of all rotations\nand≤is the precedence relation that we have just mentioned. A clo sed set in a poset (P,≤)\nis a subset RofPsuch that ρ∈R,π < ρ ⇒π∈R.\nThe following theorem is crucial to understand the importan ce of the rotation poset.\nTheorem 1 [Irving and Leather, 1986] The stable marriages of a given st able marriage\ninstance are in one-to-one correspondence with the closed s ubsets of the rotation poset.\nIn this correspondence, each closed subset Rrepresents the stable marriage obtained by\neliminating the rotations in Rstarting from xm.\nThe rotation poset can be represented as a directed acyclic g raph, with the rotations as\nnodes and an arc from πtoρiffπis an immediate predecessor of ρ(i.e.,π < ρ and there is\nno rotation σsuch that π < σ < ρ ). Note that this graph has at most n(n−1)/2nodes, i.e.,\nthere are at most n(n−1)/2rotations [Irving et al. , 1987]. Indeed, there are at most n2−n\npairs that can be involved in rotations (the npairs ofxwcannot be involved in a rotation).\nEach pair belong to at most one rotation and there are at least two pairs in each rotation. We\nwill take advantage of the rotation poset in multiple places in the paper. Importantly, note\nthat the rotation poset (actually a subgraph whose transiti ve closure is the rotation poset) can\nbe generated in O(n2)[Gusfield and Irving, 1989].\nExample 4 For instance, with the preferences of Example 1, the rotatio ns and their imme-\ndiate predecessors are given in the following table.\nRotation New pairs Immediate predecessors\nρ1= (4,7),(5,6) (4,6),(5,7)\nρ2= (6,4),(7,5) (6,5),(7,4)\nρ3= (8,8),(9,10),(10,9)(8,10),(9,9),(10,8)ρ1,ρ2\nρ1\nρ2ρ3\nFig. 1: Rotation poset in Example 1.\nThis rotation poset shows that there are (potentially many) other stable marriages than\nthe man-optimal or woman-optimal stable marriages. These o ther stable marriages are likely\nto be fairer than xmandxwas they are both extreme cases. In order to compute a fair stab leThe GGI Stable Marriage Problem 7\nmarriage, the optimization of several aggregation functio ns has been investigated.\n– Utilitarian approach:/summationtextn\ni=1rk(mi,µx(mi))+/summationtextn\nj=1rk(wj,µx(wj)), which can be min-\nimized in O(n3)[Feder, 1994]).\n– Egalitarian approach: max{rk(p,µx(p)) :p∈ M∪W} , which can also be minimized in\nO(n2)[Gusfield, 1987].\n– Sex-equal stable marriage: |/summationtextn\ni=1rk(mi,µx(mi))−/summationtextn\nj=1rk(wj,µx(wj))|, the mini-\nmization of which is NP-hard [Kato, 1993].\n– Balanced stable marriage: max{/summationtextn\ni=1rk(mi,µx(mi)),/summationtextn\nj=1rk(wj,µx(wj))}, the min-\nimization of which is NP-hard [Manlove, 2013].\nOur contribution differs with previous works on the fair sta ble marriage problem. In-\ndeed, we optimize a generalized Gini index on disutility val ues.\nGiven a matching x, the disutility d(mi,x)(also called dissatisfaction ) of a man miis\ndefined by d(rk(mi,µx(mi))), whered:N→Q+, is a strictly increasing function called\ndisutility function. The disutility values d(wj,x)are defined similarly for women. Every\nstable marriage induces therefore a disutility vector:\nd(x) = (d(m1,x),...,d(mn,x),d(w1,x),...,d(wn,x))\nwithN= 2ncomponents. Note that the use of disutility values (often ca lled weights) is a\ncommon way to extend the traditional framework where the agg regation function is applied\non rank values (see e.g., Teo and Sethuraman [1998]; Gusfield and Irving [1989]). Using a\nunique disutility function for all agents guarantees that t hey all have the same importance in\nthe aggregation operation. Indeed, the disutility values a ssigned to the ranks do not depend\non the agent’s identity. Note that both the egalitarian and t he utilitarian variants of the stable\nmarriage problem remain polynomially solvable if one uses d isutility values.\nExample 5 We come back to Example 1. Let dbe the disutility function defined by d(i) =\n(i−1)2, then the disutility values are given by the matrices dManddWbelow where\ndM[i][j](resp.dW[j][i]) is the disutility of mi(resp.wj) if he (resp. she) is matched with\nwj(resp.mi).\ndM:\n0 1 4 9 16 25 36 49 64 81\n1 0 4 9 16 25 36 49 64 81\n1 4 0 9 16 25 36 49 64 81\n1 4 9 25 36 16 0 49 64 81\n1 4 9 25 36 0 16 49 64 81\n1 4 9 0 16 36 25 49 64 81\n1 4 9 16 0 36 25 49 64 81\n36 49 64 1 4 9 25 0 81 16\n36 49 64 1 25 4 9 81 16 0\n36 49 64 1 4 25 9 16 0 81\ndW:\n0 1 4 9 16 25 36 49 64 81\n0 1 4 9 16 25 36 49 64 81\n0 1 4 9 16 25 36 49 64 81\n0 1 4 49 64 36 9 16 25 81\n0 1 4 49 64 9 36 16 25 81\n0 1 4 9 36 49 64 16 25 81\n0 1 4 36 9 49 64 16 25 81\n16 0 25 36 49 64 9 4 81 1\n0 1 16 25 36 49 64 81 4 9\n0 1 4 9 16 25 36 64 81 49\n\nLetd= (d1,...,dN)denote a disutility vector. The generalized Gini index [Wey mark,\n1981] is defined as follows:\nDefinition 1 Letλ= (λ1,...,λN)be a vector of weights such that λ1≥...≥λN. The\nGGIλ(·)aggregation function induced by λis defined by:\nGGIλ(d) =N/summationdisplay\ni=1λid↓\ni,8 Hugo Gilbert, Olivier Spanjaard\nwhered↓denotes the vector dordered by nonincreasing values, i.e., d↓\n1≥d↓\n2≥...≥d↓\nN.\nThe weights of the GGI aggregation function may be defined in a variety of manner. For\ninstance, the weights initially proposed for the Gini socia l-evaluation function are:\nλi= (2(N−i)+1)/N2∀i∈ {1,...,N} (4)\nExample 6 Coming back to Example 1, if the weights λare defined by Equation 4 and the\ndisutility function is defined by d(i) =i, the GGI values of the different stable marriages\nare (the lower the better):\nmatching x ordered vectors d↓(x) GGIλ(d(x))\nx1(10,7,7,7,7,4,3,3,2,1,1,1,1,1,1,1,1,1,1,1) 4.4525\nx2(10,7,7,5,5,4,4,4,3,3,2,1,1,1,1,1,1,1,1,1) 4.4725\nx3(10,7,7,5,5,4,4,4,3,3,2,1,1,1,1,1,1,1,1,1) 4.4725\nx4(10,5,5,5,5,4,4,4,4,4,3,3,2,1,1,1,1,1,1,1) 4.3925\nx5(9,5,5,5,5,5,5,5,4,4,4,4,3,3,2,2,1,1,1,1) 4.74\nWe thus observe that using a GGI aggregation function makes i t possible to obtain x4as an\noptimal stable marriage.\nThe GGI is also known in multicriteria decision making under the name of ordered\nweighted average [Yager, 1988]. This aggregation function, to minimize, is w ell-known to\nsatisfy the Pigou-Dalton transfer principle if λ1>λ2>...>λ N:\nDefinition 2 An aggregation function Fsatisfies the transfer principle if for any d∈(R+)N\nandε∈(0,dj−di)wheredj> di:\nF(d1,...,di+ε,...,d j−ε,...,d N)< F(d1,...,dN).\nThis condition states that the overall welfare should be imp roved by any transfer of disutility\nfrom a “less happy” agent jto a happier agent igiven that this transfer reduces the gap be-\ntween the disutilities of agent iandj. We can now define the GGI Stable Marriage problem.\nGGI Stable Marriage (GGISM)\nINSTANCE: Two disjoint sets of size n, the men and the women; for each person, a pref-\nerence list containing all the members of the opposite sex; a vector of weight parameters\nλand a disutility function d.\nSOLUTION: A stable marriage x.\nMEASURE: GGIλ(d(x))(to minimize).\n3 Complexity of the GGISM Problem\nThe GGISM problem extends both the egalitarian and the utili tarian approaches to the stable\nmarriage problem. Indeed, if the weights of the GGI operator areλ= (1,...,1), one obtains\nthe sum operation. If the weights are λ= (1,0,...,0), one obtains the max operation.\nWhile both variants are polynomially solvable problems, th e following result states that the\nGGISM problem is NP-hard:\nTheorem 2 The GGISM problem is NP-hard.The GGI Stable Marriage Problem 9\nProof We make a reduction from Minimum 2-Satisfiability, which is s trongly NP-hard\n[Kohli et al. , 1994].\nMinimum 2-Satisfiability (Min 2-SAT) :\nINSTANCE : A setVof variables, a collection Cof disjunctive clauses of at most 2 literals,\nwhere a literal is a variable or a negated variable in V.\nSOLUTION : A truth assignment for V.\nMEASURE : Number of clauses satisfied by the truth assignment (to mini mize).\nTo illustrate the reduction, we will use the following 2-SAT instance:\nV={v1,v2,v3,v4,v5,v6} (5)\nC={(v1∨v2),(¬v2∨¬v4),(¬v1∨v3),(v3∨¬v4),v2,(v5∨v6)} (6)\nAs a preliminary step, note that we can get rid of variables th at are present in only one\nclause. Such a variable is set to true if it is present as a nega tive literal in the clause and to\nfalse otherwise. It can then be removed from the instance. Fu rthermore, we can make sure\nthat there are exactly two literals in each clause (by duplic ating literals). For example, the\ninstance described by Equations 5 and 6 can be modified to:\nV={v1,v2,v3,v4} (7)\nC={(v1∨v2),(¬v2∨¬v4),(¬v1∨v3),(v3∨¬v4),(v2∨v2)} (8)\nIn the following we will denote by nv=|V|the number of variables and by nc=|C|\nthe number of clauses. In the previous example nv= 4 andnc= 5. Furthermore, we will\ndenote by citheithclause in C.\nWe are now going to create an instance of the GGISM problem suc h that:\n–There is a one-to-one correspondence between the stable mar riages and the truth assign-\nments for V.\n–A stable marriage minimizing the GGI of the agent’s disutili ties corresponds to a truth\nassignment of Vminimizing the number of clauses that are satisfied.\nIn order to create a one-to-one correspondence between the s table marriages and the\ntruth assignments for V, we are going to create a rotation ρifor each variable vi∈V. Each\nof these rotations will be exposed in the shortlists from the man-optimal stable marriage for\nthe instance under construction. Additionally, we will ens ure that these rotations will be the\nonly ones of the stable marriage instance. In other words, th e rotation poset will have one\nvertex per variable and no edge, as illustrated in Figure 2.\nWe now give the “meaning” of these rotations. Let’s recall th at in a stable marriage\nthere is a one-to-one correspondence between the closed sub sets of nodes of the rotation\nposet and the stable marriages. Now let xbe a stable marriage corresponding to a closed\nsubsetRof rotations, then the corresponding truth assignment over Vconsists in setting\nvi= 1 ifρi∈Randvi= 0 otherwise. Thus in the generated stable marriage instance,\nthe man-optimal stable marriage (i.e., R=∅) corresponds to a truth assignment where all\nvariables in Vare set to 0 while the woman-optimal stable marriage (i.e., R={ρi|vi∈V})\ncorresponds to a truth assignment where all variables in Vare set to 1.\nWe now describe more precisely the fashion in which rotation sρiare generated. For\neach variable vi, we create a man-woman pair (mij,wij)for each clause cjthat involves10 Hugo Gilbert, Olivier Spanjaard\nρ1\nρ2\n...\nρnv\nFig. 2: Rotation poset of the stable marriage instance gener ated by the reduction.\nvieither as a positive or negative literal. If variable viis present two times in a clause cj,\nthen two man-woman pairs (mij,wij)and(m′\nij,w′\nij)are created. This induces the creation\nof2ncmen and 2ncwomen in the instance. The rotation ρithen involves all the men and\nwomen induced by variable vi. For example, in the instance described by Equations 7 and\n8,ρ2involves men m21,m22,m25,m′\n25and women w21,w22,w25,w′\n25as variable v2is\npresent in c1,c2andc5. Letrdenote the number of times variable viappears in C. The\nrotationρiis then induced by the following patterns in the shortlists o f men{mij|vi∈cj}\nand women {wij|vi∈cj}:\nmij0:wij0≻m\nij0wij1\nmij1:wij1≻m\nij1wij2\n...\nmijr−2:wijr−2≻m\nijr−2wijr−1\nmijr−1:wijr−1≻m\nijr−1wij0wij0:mijr−1≻w\nij0mij0\nwij1:mij0≻w\nij1mij1\n...\nwijr−2:mijr−3≻w\nijr−2mijr−2\nwijr−1:mijr−2≻w\nijr−1mijr−1\nFor instance, rotation ρ2is induced by the following pattern in the shortlists:\nm21:w21≻m\n21w22\nm22:w22≻m\n22w25\nm25:w25≻m\n25w′\n25\nm′\n25:w′\n25≻m′\n25w21w21:m′\n25≻w\n21m21\nw22:m21≻w\n22m22\nw25:m22≻w\n25m25\nw′\n25:m25≻w′\n25m′\n25\nNote that each man mijor woman wijis involved in one and only one rotation, which is\nρi. As a consequence, each man or woman in the generated instanc e has only two possible\nmatches in a stable marriage, namely wijkandwijk+1(modulo the size rof rotation ρi)\nformijk, andmijk−1(modulo r) andmijkforwijk. For simplicity, we will denote by\nrk+(mij)(resp.rk−(mij)) the rank of the best (resp. worst) possible match for mijin a\nstable marriage. Notations rk+(wij)andrk−(wij)are defined similarly for women.\nGiven a stable marriage characterized by a set Rof rotations, it is possible to determine\nif clause cjis satisfied by examining which rotations belong to R. According to the form\nof clause cj, columns “in” and “out” of Table 1 indicate which rotations s hould be included\nor not in Rso thatcjis not satisfied. Assuming that cjinvolves variables viandvk(orThe GGI Stable Marriage Problem 11\nclausecj in out decisive agents\nvi∧vk ρi,ρkmij,mkj\nvi∧¬vkρkρimij,wkj\n¬vi∧vkρiρk wij,mkj\n¬vi∧¬vkρi,ρk wij,wkj\nvi∧vi ρimij,m′\nij\n¬vi∧¬viρi wij,w′\nij\nTable 1: Clause cjis not satisfied iff the rotations of the second (resp. third) column are included (resp. not\nincluded) in R. Consequently, clause cjis not satisfied iff the two agents in the last column are match ed with\ntheir choices of rank rk+(·).\npossibly their negations), it is sufficient to examine the ma tches of two specific agents among\nmij,mkj,wij,wkjto determine if rotations ρiandρkbelong or not to R. These two specific\nagents are called decisive agents of cjin the following. We have indeed ρi∈Riff the rank\nof the match of mijisrk+(mij). Similarly, we have ρi/\\e}atio\\slash∈Riff the rank of the match of\nwijisrk+(wij). Put another way, mij(resp.wij) is a decisive agent of cjifvi(resp.¬vi)\nbelongs to cj. The clause cjis not satisfied iff the two decisive agents are with their mat ch\nof rankrk+(·). The decisive agents according to the form of clause cjare given in the last\ncolumn of Table 1.\nFor illustration, let us return to the 2-SAT instance descri bed by Equations 7 and 8.\nGiven the stable marriage instance generated by the reducti on, and a stable marriage x,\nclausev1∧v2isnotsatisfied iff rk(m11,µx(m11)) =rk+(m11)andrk(m21,µx(m21)) =\nrk+(m21). More generally, it is possible to count the number of clause s that are notsatisfied\nby examining the ranks of the matches of the decisive agents o f each clause.\nWe will soon explain how to use a GGI operator to count the numb er of clauses that\nare not satisfied in the 2-SAT instance. Beforehand, we need t o introduce fictitious agents\nin order to control the positions of the decisive agents in th e ordered vector of disutilities\nfor every stable marriage. More precisely, we introduce fou r fictitious agents mj,m′\nj,wj,\nw′\njper clause cjsuch that mj(resp.m′\nj) is the first choice of wj(resp.w′\nj) and vice-versa.\nThusmj(resp.m′\nj) can only be matched to wj(resp.w′\nj) in a stable marriage, and therefore\nthe fictitious agents will not interfere with the possible ma tches of the other agents.\nThe fictitious agents are placed in the preference lists of th e other agents such that\nrk+(·) = 2j+ 1 andrk−(·) = 2j+ 2 for the two decisive agents of clause cj. Fur-\nthermore, rk+(·) = 1 andrk−(·) = 2 for the remaining (non-decisive) agents. Note that\n2j+1>2asj≥1and therefore the two decisive agents of cjare at positions 2(nc−j)+1\nand2(nc−j)+2 in the permutation that ranks the agents by non-increasing d isutilities.\nTo achieve these properties, we position 2jfictitious agents at the beginning of the\npreference list of the decisive agents of clause cj(e.g.,m1m′\n1...mjm′\njfor a decisive agent\nwij). These agents are positioned just before the two possible m atches of the agent in a\nstable marriage. Regarding the non-decisive agents, their two possible matches in a stable\nmarriage are simply placed at the beginning of their prefere nce lists.\nFor illustration, in the 2-SAT instance described by Equati ons 7 and 8, the preference\nlist of agent w22(who is a decisive agent of c2) is:\nw22:m1≻w\n22m′\n1≻w\n22m2≻w\n22m′\n2≻w\n22m21≻w\n22m22≻w\n22...\nand the preference list of agent m22(who is nota decisive agent of c2) is:\nm22:w22≻m\n22w25≻m\n22...12 Hugo Gilbert, Olivier Spanjaard\nm1:w1...\nm′\n1:w′\n1...\n...\nm5:w5...\nm′\n5:w′\n5...\nm11:w1w′\n1w11w13...\nm13:w1w′\n1w2w′\n2w3w′\n3w13w11...\nm21:w1w′\n1w21w22...\nm22:w22w25...\nm25:w1w′\n1w2w′\n2w3w′\n3w4w′\n4w5w′\n5w25w′\n25...\nm′\n25:w1w′\n1w2w′\n2w3w′\n3w4w′\n4w5w′\n5w′\n25w21...\nm33:w1w′\n1w2w′\n2w3w′\n3w33w34...\nm34:w1w′\n1w2w′\n2w3w′\n3w4w′\n4w34w33...\nm42:w42w44...\nm44:w44w42...w1:m1...\nw′\n1:m′\n1...\n...\nw5:m5...\nw′\n5:m′\n5...\nw11:m13m11...\nw13:m11m13...\nw21:m′\n25m21...\nw22:m1m′\n1m2m′\n2m21m22...\nw25:m22m25...\nw′\n25:m25m′\n25...\nw33:m34m33...\nw34:m33m34...\nw42:m1m′\n1m2m′\n2m44m42...\nw43:m1m′\n1m2m′\n2m3m′\n3m4m′\n4m42m44...\nFig. 3: Preference lists obtained for the min 2-SAT instance of Equations 7 and 8.\nThis construction is illustrated in Figure 3 (where symbols ≻are omitted for readability\nreasons) for the Minimum 2-Satisfiability instance defined b y Equations 7 and 8. The pref-\nerence lists of the agents are only partially given but note t hat they can be completed in any\nconsistent way that would lead to complete and transitive or ders.\nWe now explain how to define the disutility values attributed to each rank, as well as the\nweights of the GGI operator, so that the number of unsatisfied clauses can be inferred from\nthe GGI value of the stable marriage.\nDisutility values and weights of the GGI. We first recall that each clause cjinduces 6 agents\nthat are matched either with their first or second choices and 2 agents (the decisive ones) that\nare matched with their choices of rank 2j+1or2j+2. By construction of the preference\nlists, note that no agent can be matched with a partner that is ranked strictly beyond 2nc+2\nin his/her preference list. Therefore the values of d(i)fori >2nc+2play no role, and can\nbe fixed arbitrarily as long as they are increasing with iand strictly greater than d(2nc+2) .\n– The increasing disutility values for ranks 1 to 2nc+ 2are defined as follows (assuming\nthatnc≥2):\nd(1) = 0\nd(2) = 1\nd(2j+1) =j+1,∀j∈ {1,...,nc}\nd(2j+2) =j+1+n−j\nc,∀j∈ {1,...,nc}The GGI Stable Marriage Problem 13\n– The non-increasing weights of the GGI are defined as follows :\nλ= (nnc+1\nc,nncc,nncc,nnc−1\nc,...,n3\nc,n2\nc,n2\nc,n1\nc/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n2ncweights,0,...,0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n6ncweights).\nWe recall that the 2ncagents with the highest disutility values are the decisive a gents (the\ntwo decisive agents of clause cjare matched with an agent of rank 2j+1or2j+2) and the\n6ncagents with the lowest disutility values are the non-decisive agents (who are matched to\none of their two first choices). Consequently, the weight vec torλattributes a weight 0 in the\nGGI operator to the 6ncnon-decisive agents while, for each clause cj, it attributes a weight\nnj\nc(resp.nj+1\nc) to the most satisfied (resp. least satisfied) of the two decis ive agents of cj.\nAn upper bound on the GGI value is given by ∆u=/summationtextnc\nj=1(nj\nc+nj+1\nc)d(2j+2) . This\nwould correspond to a stable marriage, where for each cj, the two decisive agents of cjare\nboth matched to their choice of rank 2j+ 2. Similarly, a lower bound on the GGI value is\ngiven by ∆l=/summationtextnc\nj=1(nj\nc+nj+1\nc)d(2j+1) (if the two decisive agents of cjare both matched\nto their choice of rank 2j+1). Simple calculations show that ∆l=∆u−nc(1+nc).\nThese bounds are useful for establishing Lemma 1 below, that makes it possible to infer\nthe number of unsatisfied clauses from the GGI value. The lowe r the GGI value, the higher\nthe number of unsatisfied clauses. Hence, minimizing the GGI value amounts to maximizing\nthe number of unsatisfied clauses, which concludes the proof .\nLemma 1 For the GGI stable marriage instance obtained by the method d escribed above,\na stable marriage xcorresponds to a truth assignment on Vfor which the number of unsat-\nisfied clauses is:/floorleftbigg\n∆u−GGIλ(d(x))\nnc+1/floorrightbigg\nProof of Lemma 1 . We wish to show that if a stable marriage xcorresponds to a truth\nassignment on Vwith exactly kunsatisfied clauses then:\n∆u−(k+1)(nc+1)<GGIλ(d(x))≤∆u−k(nc+1),\nfrom which the lemma straightforwardly follows.\nAssume that kclauses{cj1,...,cjk}are unsatisfied for the truth assignment induced by\nx. Then for each cjl, the two decisive agents of cjlare both matched to their choice of rank\n2jl+1. Hence:\nGGIλ(d(x))≤∆u−k/summationdisplay\nl=1(njlc+njl+1\nc)(d(2jl+2)−d(2jl+1))\n=∆u−k/summationdisplay\nl=1(njlc+njl+1\nc)n−jlc=∆u−k(nc+1)\nbecause each decisive agent of clause cjl, forl∈ {1,...,k}, has a disutility of d(2jl+1)\nand notd(2jl+2) .\nNow, let {cj1,...,cjnc−k}denote the satisfied clauses for the truth assignment induce d\nbyx. Then for each cjl, at least one of the two decisive agents of cjlis matched to his/her\nchoice of rank 2jl+2. In the best case (w.r.t. the GGI value), only one of the two is matched14 Hugo Gilbert, Olivier Spanjaard\nto his/her choice of rank 2jl+2, and his/her weight in the GGI aggregation is njl+1\nc because\nhis/her disutility is the highest among the two agents. Henc e:\nGGIλ(d(x))≥∆l+nc−k/summationdisplay\nl=1njl+1\nc(d(2jl+2)−d(2jl+1))\n=∆u−nc(1+nc)+nc−k/summationdisplay\nl=1njl+1\ncn−jlc\n=∆u−nc(1+nc)+(nc−k)nc\n=∆u−(k+1)nc\n> ∆u−(k+1)(nc+1)\nThis concludes the proof of the lemma.\n4 A 2-approximation Algorithm\nWe now present a polynomial time 2-approximation algorithm for the GGI stable marriage\nproblem.\nThe 2-approximation algorithm uses a linear programming fo rmulation of the stable\nmarriage problem, based on the rotation poset [Gusfield and I rving, 1989]. It is indeed well-\nknown that the set of stable marriages can be characterized b y the following set of inequali-\nties where we have one binary variable y(ρ)for each rotation in the rotation poset and:\ny(ρ′)−y(ρ)≤0 (9)\nfor each pair of rotations such that ρprecedes ρ′. Variable y(ρ)is equal to 1if rotation ρ\nis included in the closed set of rotations associated to the s table marriage and 0 otherwise.\nImportantly, note that the extreme points of the polytope de fined by constraints 9 for 0≤\ny(ρ)≤1,∀ρare in one-to-one correspondence with the stable marriages of the instance\n[Gusfield and Irving, 1989]. Furthermore, the stable marria gexcharacterized by variables\ny(ρ)can be inferred by using Equations 10, 11 and 12 below.\nTo explain this point we introduce some notations. Let Γdenote the set of man-woman\npairs included in at least one stable matching. These pairs c an be found by looking at the\npairs that are created and broken by each rotation. Indeed, n ote that for each pair (m,w)∈\nΓ, there exists exactly one rotation, denoted by ρget(m,w), that creates this pair (unless\nthis pair is in xm) and exactly one rotation, denoted by ρbreak(m,w), that breaks this pair\n(unless this pair is in xw). Then, one can compute variables xijcorresponding to a set of\nvariables y(ρ)by using the following equations:\nxij= 1−y(ρbreak(i,j)),∀(i,j)∈Γs.t.xm\nij= 1 (10)\nxij=y(ρget(i,j)),∀(i,j)∈Γs.t.xw\nij= 1 (11)\nxij=y(ρget(i,j))−y(ρbreak(i,j)),∀(i,j)∈Γs.t.xm\nij=xw\nij= 0 (12)\nExample 7 Let us come back to Example 1. The pairs in Γare listed in the left column of\nTable 2. The rotations ρget(m,w)andρbreak(m,w)for each pair (m,w)are given in the\nmiddle and right columns.The GGI Stable Marriage Problem 15\n(m,w)∈Γρget(m,w)ρbreak(m,w)\n(m1,w1)\n(m2,w2)\n(m3,w3)\n(m4,w7) ρ1\n(m4,w6)ρ1\n(m5,w6) ρ1\n(m5,w7)ρ1\n(m6,w4) ρ2\n(m6,w5)ρ2\n(m7,w5) ρ2\n(m7,w4)ρ2\n(m8,w8) ρ3\n(m8,w10)ρ3\n(m9,w10) ρ3\n(m9,w9)ρ3\n(m10,w9) ρ3\n(m10,w8)ρ3\nTable 2: Rotations ρget(m,w)andρbreak(m,w)in Example 1.\nA mathematical programming formulation of the GGISM proble m reads as follows:\nP\n\nmin\nd,x,yGGIλ(d)\ndm\ni=/summationdisplay\n(i,j)∈Γxijd(rk(mi,wj)),∀i∈ {1,...,n}\ndw\nj=/summationdisplay\n(i,j)∈Γxijd(rk(wj,mi)),∀j∈ {1,...,n}\nxij= 1−y(ρbreak(i,j)),∀(i,j)∈Γs.t.xm\nij= 1\nxij=y(ρget(i,j)),∀(i,j)∈Γs.t.xw\nij= 1\nxij=y(ρget(i,j))−y(ρbreak(i,j)),∀(i,j)∈Γs.t.xm\nij=xw\nij= 0\ny(ρ′)−y(ρ)≤0,∀(ρ,ρ′)s.t.ρ < ρ′\ndm\ni≥0,∀i∈ {1,...,n}\ndw\nj≥0,∀j∈ {1,...,n}\nxij≥0,∀(i,j)∈Γ\ny(ρ)∈ {0,1},∀ρ∈P(13)\n(14)\n(15)\n(16)\nwheredm\ni(resp.dw\nj) represents the disutility of mi(resp.wj),d= (dm\n1,...,dmn,dw\n1,...,dwn),\nand as usual:\n–xij= 1(resp. 0) if (mi,wj)is (resp. is not) in the stable marriage x,\n–y(ρ)= 1 (resp. 0) if ρbelongs (resp. does not) to the set of rotations characteriz ingx,\n–Pis the set of all rotations.\nLet us denote by /hatwidePthe linear programming relaxation of Pwherey(ρ)∈ {0,1}is replaced\nby0≤y(ρ)≤1. Importantly, note that variables y(ρ)in an optimal solution to /hatwidePare not\nnecessarily integer because the objective function is non- linear (and therefore there does not\nnecessarily an optimal vertex in the solution polytope).16 Hugo Gilbert, Olivier Spanjaard\nA polynomial time 2-approximation algorithm can be obtaine d by rounding an optimal\nsolution of /hatwideP. The 2-approximation algorithm writes as follows:\nROUNDING ALGORITHM\n1. Solve /hatwidePand let(ˆd,ˆ x,ˆ y)denote an optimal solution to /hatwideP;\n2. For each ρ∈P, sety(ρ)= 1 ifˆy(ρ)≥0.5, andy(ρ)= 0 otherwise;\n3. Return the stable marriage xobtained from yby using constraints 13–15 in P.\nExample 8 Coming back to Example 1, assume that the weights of the GGI op erator are\ndefined by Equation 4 and that the disutility function is defin ed byd(i) =i. Then, an optimal\nsolution(ˆd,ˆ x,ˆ y)to/hatwidePis characterized by ˆy(ρ1) = ˆy(ρ2) = 0.75andˆy(ρ3) = 0 (for a GGI\nvalue of4.3075 ). For this instance, by ROUNDING ALGORITHM , the obtained vector yis\ntherefore y(ρ1) =y(ρ2) = 1 andy(ρ3) = 0 . This corresponds to stable marriage x4,\nwhich is in fact an optimal solution.\nSteps 2 and 3 of the algorithm can obviously be performed in po lynomial time. In step 1,\nsolving/hatwidePcan also be performed in polynomial time by using one of the li nearizations of the\nGGI operator proposed by Ogryczak and ´Sliwi´ nski [2003]. The following lemma ensures\nthat the returned solution is a 2-approximation of an optima l solution of P:\nLemma 2 For any feasible solution (ˆd,ˆ x,ˆ y)of/hatwideP, the feasible solution (d,x,y)ofPob-\ntained by setting\ny(ρ)=/braceleftbigg1ifˆy(ρ)≥0.5,\n0otherwise\nis such that ˆd≥1\n2dwhere≥is taken componentwise.\nProof In order to establish the result stated in the lemma, we intro duce the notion of man\nand woman weights of a rotation. Given a rotation ρ= (mi0,wi0),...,(mir−1,wir−1)we\ndefine the mi-weight of that rotation by:\nωm\ni(ρ) =/braceleftbigg\nd(rk(mik,wik))−d(rk(mik,wi(k+1) modr))ifi∈ {i0,...,ir−1}andi=ik\n0otherwise.\nSimilarly, we define the wj-weight of that rotation by:\nωw\nj(ρ)=/braceleftbigg\nd(rk(wik,mik))−d(rk(wik,mi(k−1)modr))ifj∈ {i0,...,ir−1}andj=ik\n0otherwise.\nNote that a man weight of a rotation will always be negative wh ile a woman weight of a\nrotation will always be positive.\nAssume that ρis a rotation that is exposed in a stable marriage x, and letx′be the stable\nmarriage obtained from xby eliminating ρ. Then:\nd(mi,x′) =d(mi,x)−ωm\ni(ρ), d(wj,x′) =d(wj,x)−ωw\nj(ρ).\nConsequently, if xis the stable marriage obtained from the man-optimal stable marriage xm\nby eliminating rotations ρ1,...,ρt, then:\nd(mi,x) =d(mi,xm)−t/summationdisplay\nk=1ωm\ni(ρk), d(wj,x) =d(wj,xm)−t/summationdisplay\nk=1ωw\nj(ρk).The GGI Stable Marriage Problem 17\nWe now establish the result stated in the lemma. Let (ˆd,ˆ x,ˆ y)denote a feasible solution of\n/hatwideP. The previous equations extend as follows for solutions of /hatwideP:\nˆdm\ni=d(mi,xm)−/summationdisplay\nρ∈Pˆy(ρ)ωm\ni(ρ),ˆdw\nj=d(wj,xm)−/summationdisplay\nρ∈Pˆy(ρ)ωw\nj(ρ). (17)\nNow consider the feasible solution (d,x,y)ofPdefined by y(ρ) = 1 ifˆy(ρ)≥0.5,\nand0otherwise. The feasibility of (d,x,y)comes from the fact that {ρ: ˆy(ρ)≥0.5}is\na closed set of rotations. Indeed, note that constraints 16 e nsures that y(ρ′)≤y(ρ)for all\nρ < ρ′. We have:\nˆdm\ni−1\n2dm\ni=d(mi,xm)−/summationdisplay\nρ∈Pˆy(ρ)ωm\ni(ρ)−1\n2(d(mi,xm)−/summationdisplay\nρ∈Py(ρ)ωm\ni(ρ))\n=1\n2d(mi,xm)−/summationdisplay\nρ∈P(ˆy(ρ)−1\n2y(ρ))ωm\ni(ρ)\n≥1\n2d(mi,xm)≥0\nas0≤(ˆy(ρ)−1\n2y(ρ))for allρ∈Pandωm\ni(ρ)≤0for alli∈ {1,...,n}andρ∈P.\nHence,ˆdm\ni≥1\n2dm\nifor alli∈ {1,...,n}.\nSimilarly, for women we have:\nˆdw\nj−1\n2dw\nj=d(wj,xm)−/summationdisplay\nρ∈Pˆy(ρ)ωw\nj(ρ)−1\n2(d(wj,xm)−/summationdisplay\nρ∈Py(ρ)ωw\nj(ρ))\n=1\n2d(wj,xm)−/summationdisplay\nρ∈P(ˆy(ρ)−1\n2y(ρ))ωw\nj(ρ)\n≥1\n2(d(wj,xm)−/summationdisplay\nρ∈Pωw\nj(ρ))\nas(ˆy(ρ)−1\n2y(ρ))≤0.5for allρ∈Pandωw\nj(ρ)≥0for allj∈ {1,...,n}andρ∈P. Since\neliminating all rotations from xmleads toxw, we have that1\n2(d(wj,xm)−/summationtext\nρ∈Pωw\nj(ρ)) =\n1\n2d(wj,xw). Therefore, ˆdw\nj−1\n2dw\nj≥0and hence, ˆdw\nj≥1\n2dw\njfor allj∈ {1,...,n}.\nBy combining the inequalities obtained for men and women, we obtain that ˆd≥1\n2d,\nwhich concludes the proof.\nWe can now state the main result of this section:\nTheorem 3 ROUNDING ALGORITHM is a polynomial time 2-approximation algorithm for\nthe GGI stable marriage problem, and the bound is tight.\nProof We first recall that all steps of R OUNDING ALGORITHM can be performed in polyno-\nmial time. Furthermore, by Lemma 2, the feasible solution (d,x,y)generated by R OUND -\nINGALGORITHM is such that ˆd≥1\n2d, where(ˆd,ˆ x,ˆ y)is an optimal solution to /hatwideP. Conse-\nquently:\nGGIλ(ˆd)≥1\n2GGIλ(d)\nbecauseGGIλ(d)≤GGIλ(d′)ford≤d′(see e.g. Fodor et al. [1995]) and GGIλ(αd) =\nαGGIλ(d)forα >0.\nFor the tightness of the bound, consider the following insta nce of the stable marriage\nproblem:18 Hugo Gilbert, Olivier Spanjaard\nm1: 1≻m\n12≻m\n13w1: 2≻w\n13≻w\n11\nm2: 2≻m\n23≻m\n21w2: 3≻w\n21≻w\n22\nm3: 3≻m\n31≻m\n32w3: 1≻w\n32≻w\n33\nThere are two rotations ρ1= (1,1),(2,2),(3,3)andρ2= (1,2),(2,3),(3,1), withρ1<\nρ2, which yield three stable marriages:\n–the man-optimal stable marriage xmin which each man is matched with his first choice,\nand which corresponds to eliminating no rotation,\n–the woman-optimal stable marriage xwin which each woman is matched with her first\nchoice, and which corresponds to eliminating both rotation s,\n–a “compromise” stable marriage xcin which each agent is matched with his/her second\nchoice, and which corresponds to eliminating only ρ1.\nWe use the disutility function ddefined by d(1) = 0 ,d(2) = 1 + ǫandd(3) = 2 (with\nǫ >0) and the following GGI weights: λ= (a,b,c,0,0,0)witha≥b≥c >0. The\ndisutility vectors of the three stable marriages are then d(xm) = (0,0,0,2,2,2),d(xw) =\n(2,2,2,0,0,0)andd(xc) = (1+ ǫ,1+ǫ,1+ǫ,1+ǫ,1+ǫ,1+ǫ).\nBy using Equation 17 from the proof of Lemma 2, the value of ˆdm\nifor each man miand\nthe value of ˆdw\njfor each woman wjare written as follows in terms of ˆy(ρ1)andˆy(ρ2):\nˆdm\ni= 0+ ˆy(ρ1)(1+ǫ)+ ˆy(ρ2)(1−ǫ)∀i∈ {1,2,3}\nˆdw\nj= 2−ˆy(ρ1)(1−ǫ)−ˆy(ρ2)(1+ǫ)∀j∈ {1,2,3}\nwhereˆy(ρ1)≥ˆy(ρ2). We see that the three men share the same disutility value, as well as\nthe three women. The GGI value of a feasible solution to /hatwidePis thus completely determined\nby the common disutility of the men or the common disutility o f the women because only\nthe three least satisfied agents are taken into account in λ. Consequently, an optimal solution\nto/hatwidePminimizes max{ˆdm\n1,ˆdw\n1}forˆy(ρ1)≥ˆy(ρ2). Simple calculations make it possible to\nconclude that the only optimal solution to /hatwidePis characterized by ˆy(ρ1) = ˆy(ρ2) = 0.5.\nFor this instance, R OUNDING ALGORITHM returns therefore the woman-optimal stable\nmarriage which has the ordered disutility vector (2,2,2,0,0,0)and a GGI value of 2(a+\nb+c). However, an optimal stable marriage is the “compromise” st able marriage, which\nhas the ordered disutility vector (1+ǫ,1+ǫ,1+ǫ,1+ǫ,1+ǫ,1+ǫ)and a GGI value of\n(1+ǫ)(a+b+c). By taking the limit for ǫgoing to 0, we obtain the tightness of the bound.\nRemark 1 Note that the approach taken in R OUNDING ALGORITHM is valid for any aggre-\ngation criterion Fon dissatisfactions of agents for which the following condi tion holds:\nF(1\n2d(x)) =1\n2F(d(x))\nF(d(x)+r)≥F(d(x))\nwhereris any non-negative vector.\nRemark 2 A general approximation result for the optimization of a gen eralized Gini index in\nmuliobjective optimization problems has been proposed by K asperski and Zieli´ nski [2015].\nFor the GGISM problem, it amounts to compute an optimal stabl e marriage according to\nthe sum of disutilities of pairs (mi,wj), where the disutility of a pair (mi,wj)is defined by\nλ1max{d(rk(mi,wj)),d(rk(wj,mi))}+λ2min{d(rk(mi,wj)),d(rk(wj,mi))}. This can\nbe performed in polynomial time by linear programming. The r eturned solution is a Nλ1-\napproximation, provided/summationtextN\ni=1λi= 1. To obtain a better guarantee than 2, one should\nhaveλ1<2/N. On the contrary, by taking advantage of the specific structu re of the stable\nmarriage problem, our approach yields a 2-approximation whatever weights are used .The GGI Stable Marriage Problem 19\n5 The GGI Stable Marriage Problem with a Bounded Number of Non -zero Weights\nIn this section, we provide an algorithm whose complexity is O(2Kn2K+4)whereK=\nmax{i:λi>0}. Hence, the complexity is polynomial time if Kis assumed to be a con-\nstant, where Kis the number of non-zero weights in the GGI operator. In the p arametrized\ncomplexity terminology [Niedermeier, 2006], this means th at the GGI stable marriage prob-\nlem belongs to class XP for parameter K.\nWe adopt a brute-force approach to solve the problem in O(2Kn2K+4). Let\nt(x) = ((d(m1,x),m1,µx(m1)),...,(d(mn,x),mn,µx(mn)),\n(d(w1,x),w1,µx(w1)),...,(d(wn,x),wn,µx(wn))\ndenote the vectors of triples (d(ai,x),ai,µx(ai))induced by stable marriage x, where\nd(ai,x)is the dissatisfaction of agent aiwhen matched with µx(ai). We denote by T↓(x)\nthe set of vectors t↓(x)that can be obtained from t(x)by sorting the triples in decreasing or-\nder of dissatisfactions. The projection of a vector t↓(x)∈T↓(x)on theKfirst components\nis denoted by t↓\nK(x). We denote by T↓\nK(x)the set{t↓\nK(x) :t↓(x)∈T↓(x)}.\nFor instance, assume that t(x) = ((1,m1,w2),(2,m2,w1),(2,w1,m2),(1,w2,m1)).\nThen the set T↓(x)is\n{((2,m2,w1),(2,w1,m2),(1,m1,w2),(1,w2,m1)),\n((2,w1,m2),(2,m2,w1),(1,m1,w2),(1,w2,m1)),\n((2,m2,w1),(2,w1,m2),(1,w2,m1),(1,m1,w2))\n((2,w1,m2),(2,m2,w1),(1,w2,m1),(1,m1,w2))}\nand the set T↓\n2(x)is{((2,m2,w1),(2,w1,m2)),((2,w1,m2),(2,m2,w1))}.\nThe idea is to enumerate all vectors in T↓\nK=∪x∈XT↓\nK(x)without redundancy. The\npolynomiality of the approach follows from the fact that |T↓\nK| ≤(2n2)Kbecause the num-\nber of distinct triples is upper bounded by 2n2. Note that we have:\nmin\nx∈XGGIλ(d(x)) = min\nt∈T↓\nKGGIλ(t)\nbecauseλi= 0 for alli > K , where, by abuse of notation, we denote by GGIλ(t)the\nvalue of the GGI operator applied to the vector of dissatisfa ctions obtained from t1. Hence\nidentifying an optimal GGI stable marriage will be performe d by finding a vector t∈T↓\nK\nminimizing the GGI operator and computing a corresponding s table marriage.\nExample 9 Coming back to the instance of Example 1, assume that K= 2 and that the\ndisutility function is defined by d(i) =i. Then, our enumeration algorithm would produce\nthe following set T↓\n2:\n1Note that vector t∈T↓\nKis incomplete as it only has Kcomponents, but it is sufficient to apply the GGI\noperator because λi= 0 for alli > K .20 Hugo Gilbert, Olivier Spanjaard\n{((10,w10,m9),(7,w4,m6)),((10,w10,m9),(7,w5,m7)),\n((10,w10,m9),(7,w6,m5)),((10,w10,m9),(7,w7,m4)),\n((10,w10,m9),(5,m4,w6)),((10,w10,m9),(5,m5,w7)),\n((10,w10,m9),(5,m6,w5)),((10,w10,m9),(5,m7,w4)),\n((9,w10,m8),(5,m4,w6)),((9,w10,m8),(5,m5,w7)),\n((9,w10,m8),(5,m6,w5)),((9,w10,m8),(5,m7,w4)),\n((9,w10,m8),(5,m8,w10)),((9,w10,m8),(5,m9,w9)),\n((9,w10,m8),(5,m10,w8))}\nFor this instance, the optimal GGI value is therefore necess arily9λ1+ 5λ2. Note that, in\nmost cases, the optimal GGI value depends on λ(it is not the case here because (9,5)\ndominates componentwise all vectors of dissatisfactions o btained from T↓\n2).\nWe now describe our enumeration algorithm. Algorithm 1 buil ds setT↓\nKby induction\nusing the following formula:\nT↓\n0={()}\nT↓\nk={v◦t:v∈T↓\nk−1andt∈T(v)}whereT(v) ={tk:t∈T↓\nNs.t.(t1,...,tk−1) =v}\nThe aim of Algorithm 2 is to compute T(v), i.e., the set of possible triples for the kthcom-\nponent of a vector in T↓\nkstarting by the (k−1)-vectorv. The idea is to impose restrictions\non the considered stable marriages so that the least satisfie d agents as well as their matches\ncorrespond to the ones in v. For this purpose, we impose mandatory rotations (set INv) and\nforbidden rotations (set OUTv). Note that, each time a rotation is made mandatory (resp. fo r-\nbidden), the set of its ancestors (resp. descendants), deno ted byAnc(ρ)(resp.Desc(ρ)), are\nalso made mandatory (resp. forbidden) so that INv(resp.P\\OUTv) remains a closed set of\nrotations. For each triple (d,a,a′)belonging to v, we ensure that agent ais matched with\nagenta′by making rotation ρget(a,a′)mandatory and ρbreak(a,a′)forbidden (Lines 3–5).\nAdditionally, to ensure that the kleast satisfied agents are indeed those involved in v, we\nput a threshold on the dissatisfactions of the agents in Av=M∪W \\{ a: (d,a,a′)∈v}.\nNote that the set Avis updated in Line 3. Let dmin(v)denote the dissatisfaction of the last\ntriple inv(i.e., the lowest level of dissatisfaction in v). The dissatisfactions of the agents\ninAvshould not be strictly greater than dmin(v). This condition is imposed by using again\nsetsINvandOUTv. More precisely, given a rotation ρ= (mi0,wi0),...,(mir−1,wir−1),\nwe define dwmax(ρ) = max k=0,...,r−1d(wik,mik)the highest dissatisfaction of a woman\ninvolved in ρbeforeρis eliminated, and dm\nmax(ρ) = max k=0,...,r−1d(mik,wik+1)the\nhighest dissatisfaction of a man involved in ρafterρis eliminated. To make sure that the\nagents in Avhave a dissatisfaction lower than or equal to dmin(v), we make mandatory\n(resp. forbidden) any rotation ρ∈P\\OUTv(resp.P\\INv) such that dwmax(ρ)> dmin(v)\n(resp.dm\nmax(ρ)> dmin(v)) (Lines 6–7, resp. Lines 8–9). The enumeration of the triple s\ninT(v)is performed by branching on the gender (man or woman) of the a gent that will\nrealize the kthhighest dissatisfaction. We denote by TW(v)(resp.TM(v)) the set of triples\n(d,a,a′)∈T(v)wherea∈ W (resp.a∈ M ). We have of course TW(v)∪TM(v) =T(v).\nAlgorithm 3 enumerates the triples in TW(v)while Algorithm 4 enumerates the triples in\nTM(v)(Line 10 of Algorithm 2). The validity of the approach follow s from the validity of\nAlgorithms 3 and 4.The GGI Stable Marriage Problem 21\nValidity of the approach. The operations of Algorithms 3 and 4 are similar. They procee d\nin the spirit of the algorithm proposed by Gusfield [1987] for determining a minmax stable\nmarriage. Let xRdenote the stable marriage corresponding to a set Rof rotations. Note\nthat we have built sets INvandOUTvsuch that if R∩INv=INvandR∩OUTv=∅then\nv∈T↓\nk−1(xR). Furthermore, the special case xINv(resp.xP\\OUTv) is the stable marriage\ncompatible with INvandOUTvthat satisfy most the men (resp. women) as it takes as few\n(resp. much) rotations as allowed by sets INvandOUTv. We only explain the operation of\nAlgorithm 3, because the operation of Algorithm 4 is symmetr ic.\nThe aim of Algorithm 3 is to enumerate all triples (d,a,a′)inTW(v). Notably, we\nwill enumerate these triples by nonincreasing values of dby exploring carefully the set of\nstable marriages compatible with sets INvandOUTv. More precisely, at each iteration iof\nthe algorithm (loop while in Line 5) we will consider a stable marriage xicompatible with\nsetsINvandOUTvsuch that all women are always better off in xithan inxi−1fori/\\e}atio\\slash= 0\n(with at least one woman strictly better off). At each iterat ion, the new triples are found by\nlooking at set Withat includes all women in Avwhose dissatisfaction can be ranked in kth\nposition in xi, i.e., whose dissatisfaction is equal to d↓\nk(xi)(Lines 3 and 13)2.\nObviously, for the women, the worst stable marriage compati ble withINvandOUTvis\nxINv. If no woman can be ranked in kthposition w.r.t. stable marriage xINv, then no woman\ncan be ranked in kthposition for anystable marriage compatible with INv. Indeed, elimi-\nnating additional rotations would only increase the dissat isfactions of men and decrease the\ndissatisfactions of women. Otherwise the recurrence is ini tialized with x0=xINvand stable\nmarriage xi+1is obtained from xiby eliminating rotation ρbreak(m,w)(and all required\nancestors) for all woman winWiso that their dissatisfactions are strictly decreased (Lin e\n10). Loopwhile stops if one of the following conditions occurs:\n–ifWi=∅, it means that only men can be ranked in kthposition in xi; as eliminating\nrotations will only improve the situation of women and deter iorate the situation of men,\nwe can safely conclude that all triples in TW(v)have been enumerated;\n–if at least one rotation ρbreak(m,w)does not exist or is forbidden (i.e., (m,w)∈\nxP\\OUTv); indeed, in this case, we can conclude that it is not possibl e to find a triple\ninTW(v)with a dissatisfaction strictly less than the current value d↓\nk(xi)(the boolean\nFlag is then set to True in Line 9).\nComplexity analysis and proof of termination. In Algorithm 3, at every step iof thewhile\nloop, all agents in Wishare the same dissatisfaction level d↓\nk(xi). Furthermore, for all i/\\e}atio\\slash= 0,\nwe have that d↓\nk(xi)< d↓\nk(xi−1). As there are only ndissatisfaction levels (corresponding to\nthenpossible ranks), the while loop necessarily terminates in O(n)iterations. The nested\nfor loop also terminates in O(n)iterations because there can be at most nwomen in Wi.\nAll instructions inside the for loop are in O(1), except the instruction in Line 10 which is\ninO(n2)(the number of rotations is upper bounded by n(n−1)/2). Overall, Algorithm 3\nis inO(n4). The analysis of Algorithm 4 is similar. In Algorithm 2, Line s 4 and 5 are in\nO(n2), hence the for loop in Line 2 is in O((k−1)n2), therefore in O(n3)ask≤2n.\nLines 6–9 are in O(n4). Since we have shown that both calls in Line 10 are in O(n4), the\noverall complexity of Algorithm 2 is O(n4). Finally, the complexity of the three nested for\nloops in Algorithm 1 is O(/summationtextK\nk=1(2n2)k−1(n4+2n2))because:\n– the cardinality of set T↓\n(k−1)in Line 4 is upper bounded by (2n2)k−1(there are at most\n2We recall that d↓\nk(x)denotes the kthcomponent of vector d(x)when sorted by nonincreasing values.22 Hugo Gilbert, Olivier Spanjaard\n2n2triples, and k−1components per vector of triples in T↓\n(k−1));\n– Line 5 is in O(n4);\n– Lines 6–7 are in O(2n2).\nOverall, the complexity of Algorithm 1 thus is O(2Kn2K+4).\nFinal remarks. At the end of Algorithm 1, one obtains a set T↓\nKof vectors of triples. Within\nthis set, one can choose a vector v∗which realizes:\nmin\nv∈T↓\nKGGIλ(v) = min\nx∈XGGIλ(d(x)).\nGiven this vector v∗, any stable marriage x∗such that v∗∈T↓\nK(x∗)verifies\nGGIλ(d(x∗)) = min\nx∈XGGIλ(d(x)).\nGivenv∗, it is easy to compute a stable marriage x∗such that v∗∈T↓\nK(x∗). In particu-\nlar,xINv∗(resp.xP\\OUTv∗) is a best possible stable marriage for men (resp. women) whe re\nsetsINv∗andOUTv∗are generated in the same fashion as in Algorithm 2 (Lines 1–9 ).\nAlgorithm 1: Enumerate\ninput : the GGISM instance and the value of K\noutput:T↓\nK\n1T↓\n0←{()}\n2fork= 1,...,K\n3T↓\nk←∅\n4 forv∈T↓\nk−1\n5 T←NextTriples (v,k)\n6 fort∈T\n7 T↓\nk←T↓\nk∪{v◦t}\n8returnT↓\nK\nAlgorithm 2: NextTriples\ninput : vectorvof imposed triples, index kof the next triple\noutput: setTof possible next triples\n1INv←∅ ;OUTv←∅ ;Av←M∪W\n2fori= 1,...,k−1\n3(a,a′,d) =vi,Av←Av\\{a}\n4INv←INv∪{ρget(a,a′)}∪Anc(ρget(a,a′))\n5OUTv←OUTv∪{ρbreak(a,a′)}∪Desc(ρbreak(a,a′))\n6forρ∈P\\OUTvs.t.dw\nmax(ρ)> dmin(v)\n7INv←INv∪{ρ}∪Anc(ρ)\n8forρ∈P\\INvs.t.dm\nmax(ρ)> dmin(v)\n9OUTv←OUTv∪{ρ}∪Desc(ρ)\n10returnNextWomen (INv,OUTv,k,Av)∪NextMen(INv,OUTv,k,Av)The GGI Stable Marriage Problem 23\nAlgorithm 3: NextWomen\ninput : setINvandOUTvof mandatory and forbidden rotations, index kof the next triple, set Av\noutput:TW(v)\n1Compute xINvandxP\\OUTv\n2T←∅ ;R←INv;i←0;xi←xR\n3Wi←{w∈Av∩W:d(w,xi) =d↓\nk(xi)}\n4Flag←False\n5whileWi/\\e}atio\\slash=∅do\n6 forw∈Wi\n7 letmbe the match of winxi\n8 T←T∪{(d(w,m),w,m)}\n9 if(m,w)∈xP\\OUTvthen Flag←True\n10 elseR←R∪ρbreak(m,w)∪Anc(ρbreak(m,w))\n11 ifFlag then return T\n12i←i+1;xi←xR\n13Wi←{w∈Av∩W:d(w,xi) =d↓\nk(xi)}\n14returnT\nAlgorithm 4: NextMen\ninput : setINvandOUTvof mandatory and forbidden rotations, index kof the next triple, set Av\noutput:TM(v)\n1Compute xINvandxP\\OUTv\n2T←∅ ;R←P\\OUTv;i←0;xi←xR\n3Mi←{m∈Av∩M:d(m,xi) =d↓\nk(xi)}\n4Flag←False\n5whileMi/\\e}atio\\slash=∅do\n6 form∈Mi\n7 letwbe the match of minxi\n8 T←T∪{(d(m,w),m,w)}\n9 if(m,w)∈xINvthen Flag←True\n10 elseR←R\\(ρget(m,w)∪Des(ρget(m,w)))\n11 ifFlag then return T\n12i←i+1;xi←xR\n13Mi←{m∈Av∩M:d(m,xi) =d↓\nk(xi)}\n14returnT\n6 Conclusion\nIn this paper, we have shown that the minimization of a Genera lized Gini Index (GGI) of\nthe dissatisfactions of men and women in a stable marriage pr oblem is an NP-hard problem.\nThen, we have proposed a polynomial time 2-approximation al gorithm for the problem,\nbased on a rounding of the optimal solution to the linear prog ramming relaxation of the\nproblem. Lastly, we have shown that minimizing a GGI of the di ssatisfactions of men and\nwomen in a stable marriage is in the class XP with respect to th e number of strictly positive\nweights in the GGI operator.\nFor future works, following Aziz and Klaus [2017], it could b e worth investigating the\nrandomized version of the GGI stable marriage problem. By randomized , we mean that\nwe consider mixed stable marriages, and not only determinis tic stable marriages. A mixed\nstable marriage is a probability distribution over stable m arriages. This enlargement of the24 Hugo Gilbert, Olivier Spanjaard\nset of feasible solutions could make it possible to enhance t he optimal GGI value (where\nthe GGI operator is applied to the vector of expected dissati sfactions of the agents). Note\nthat the relaxed solution we compute in the first step of the 2- approximation algorithm pro-\nposed in Section 4 can be converted into a mixed stable marria ge by using a trick proposed\nby Teo and Sethuraman [1998]. It turns out that the obtained a pproach returns an optimal\nmarriage for the randomized variant of the GGI stable marria ge problem. A more thorough\ninvestigation of the randomized GGI stable marriage proble m is underway.\nReferences\nHaris Aziz and Bettina Klaus. Random matching under priorit ies: Stability and no envy\nconcepts. arXiv preprint arXiv:1707.01231 , 2017.\nP´ eter Bir´ o. Applications of matching models under prefer ences. Trends in Computational\nSocial Choice , page 345, 2017.\nChristine T Cheng. Understanding the generalized median st able matchings. Algorithmica ,\n58(1):34–51, 2010.\nEdith Elkind and Anisse Ismaili. OWA-based extensions of th e chamberlin–courant rule.\nInInternational Conference on Algorithmic Decision Theory , pages 486–502. Springer,\n2015.\nTom´ as Feder. Network flow and 2-satisfiability. Algorithmica , 11(3):291–319, Mar 1994.\nJanos Fodor, Jean-Luc Marichal, and Marc Roubens. Characte rization of the ordered\nweighted averaging operators. IEEE Trans. on Fuzzy Systems , 3(2):236–240, 1995.\nDavid Gale and Lloyd S Shapley. College admissions and the st ability of marriage. The\nAmerican Mathematical Monthly , 69(1):9–15, 1962.\nJudy Goldsmith, J´ erˆ ome Lang, Nicholas Mattei, and Patric e Perny. V oting with rank depen-\ndent scoring rules. In AAAI , pages 698–704, 2014.\nDan Gusfield and Robert W Irving. The stable marriage problem: structure and algorithms .\nMIT press, 1989.\nDan Gusfield. Three fast algorithms for four problems in stab le marriage. SIAM Journal on\nComputing , 16(1):111–128, 1987.\nTobias Heinen, Nhan-Tam Nguyen, and J¨ org Rothe. Fairness a nd rank-weighted utilitarian-\nism in resource allocation. In International Conference on Algorithmic Decision Theory ,\npages 521–536. Springer, 2015.\nRobert W Irving and Paul Leather. The complexity of counting stable marriages. SIAM\nJournal on Computing , 15(3):655–667, 1986.\nRobert W Irving, Paul Leather, and Dan Gusfield. An efficient a lgorithm for the “optimal”\nstable marriage. Journal of the ACM (JACM) , 34(3):532–543, 1987.\nAdam Kasperski and Paweł Zieli´ nski. Combinatorial optimi zation problems with uncertain\ncosts and the OWA criterion. Theoretical Computer Science , 565:102–112, 2015.\nAkiko Kato. Complexity of the sex-equal stable marriage pro blem. Japan Journal of Indus-\ntrial and Applied Mathematics , 10(1):1, 1993.\nRajeev Kohli, Ramesh Krishnamurti, and Prakash Mirchandan i. The minimum satisfiability\nproblem. SIAM Journal on Discrete Mathematics , 7(2):275–283, 1994.\nJulien Lesca, Michel Minoux, and Patrice Perny. The fair OWA one-to-one assignment\nproblem: NP-hardness and polynomial time special cases. Algorithmica , pages 1–26,\n2018.\nDavid F. Manlove. Algorithmics of Matching Under Preferences . Series on theoretical\ncomputer science. World Scientific, 2013.The GGI Stable Marriage Problem 25\nEric McDermid and Robert W Irving. Sex-equal stable matchin gs: Complexity and exact\nalgorithms. Algorithmica , 68(3):545–570, 2014.\nRolf Niedermeier. Invitation to fixed-parameter algorithms , volume 31 of Oxford Lecture\nSeries in Mathematics and its Applications . Oxford University Press, 2006.\nWlodzimierz Ogryczak and Tomasz ´Sliwi´ nski. On solving linear programs with the ordered\nweighted averaging objective. European J. of Operational Research , 148(1):80–91, 2003.\nAriel D. Procaccia, Jeffrey S. Rosenschein, and Aviv Zohar. On the complexity of achieving\nproportional representation. Social Choice and Welfare , 30(3):353–362, 2008.\nChung-Piaw Teo and Jay Sethuraman. The geometry of fraction al stable matchings and its\napplications. Mathematics of Operations Research , 23(4):874–891, 1998.\nJohn H. Vande Vate. Linear programming brings marital bliss .Operations Research Letters ,\n8(3):147–153, 1989.\nJohn A. Weymark. Generalized gini inequality indices. Mathematical Social Sciences ,\n1(4):409–430, 1981.\nRonald R. Yager. On ordered weighted averaging aggregation operators in multicriteria\ndecision making. IEEE Trans. on Sys., Man, and Cybernetics , 18(1):183–190, 1988."    },    {        "title": "1809.09429v1.Theory_of_damping_in_magnetization_dynamics__dispelling_a_myth_and_pointing_a_way_forward.pdf",        "content": "arXiv:1809.09429v1  [cond-mat.mtrl-sci]  25 Sep 2018Theory of damping in magnetization dynamics, dispelling a m yth and pointing a way\nforward\nD M Edwards\nDepartment of Mathematics, Imperial College London, Londo n SW7 2BZ, United Kingdom\nThere is a widely-held belief amongst theoreticians that th e Gilbert damping parameter αin\nmagnetization dynamics is infinite for a pure metal at T=0. Th e basic error leading to this belief\nis pointed out explicitly and the various methods of calcula tion used are viewed in a unified way\nbased on the Lorentzian lineshape of ferromagnetic resonan ce spectra. A general torque formula\nforαis proposed as a good starting-point for treating inhomogen eous materials such as alloys,\ncompounds and layered structures. Local spin density funct ional theory provides a simple physical\npicture, in terms of a non-uniform precessional cone angle i n ferromagnetic resonance, of how such\ninhomogeneity contributes to the damping. In acomplementa ry many-bodytheory this contribution\nis given by a vertex correction to the torque-torque respons e function.\nThe damping of magnetization dynamics in ferromagnetic metals and a lloys is of critical importance in spintronic\ndevices. Damping largely controls the speed at which a device can ope rate and its energy requirement. In device\nphysics damping is usually treated phenomenologically by means of a Gilb ert term in the Landau-Lifshitz-Gilbert\nequation [1, 2] and many quantum-mechanical calculations of the Gilb ert parameter have been made for specific\nmaterials [3–10]. A reliable treatment of damping in transition metals an d alloys would be an invaluable guide in\nthe search for materials with very low damping [11, 12], as required fo r the future development of devices such\nas magnetic random access memory(MRAM). Most recent work in th is direction is concerned with the important\nintrinsic contribution arising from spin-orbit coupling(SOC) and it is th is which concerns us here. A satisfactory\ntheory should work in the limit of a pure metal but almost all existing ca lculations predict that the Gilbert damping\nparameter αdiverges to infinity for a pure metal at T=0. This would mean that in th e pure metals Fe, Co and Ni\nat low temperature the linewidth in a ferromagnetic resonance (FMR ) experiment would be much too large for the\nresonance to be observed. The prediction or acceptance of infinit e damping has been made by so many authors [3–\n10, 13, 14] over the last forty years that it has acquired the stat us of a myth. A very recent paper [15] repeats it\nonce again. It is noteworthy that no experimentalist seems to have troubled to investigate the problem by work\non high purity metals and dilute alloys at low temperature. The aim of th is article is not only to dispel the myth\nbut to formulate a firm starting-point for future calculations of αin technically important materials such as alloys,\ncompounds and layered structures.\nThe most direct method to investigate damping,both experimentally a nd theoretically, is to study the ferromagnetic\nresonance (FMR) linewidth. In FMR a uniform static magnetic field His applied and the absorption of a transverse\nmicrowave field of angular frequency ωpeaks around the frequency bex//planckover2pi1wherebex= 2µBHis the Zeeman energy.\nForHin the z direction the absorption is determined by the imaginary part o f the dynamical transverse susceptibility\nχ−+(ω). This susceptibility, which must include the effect of SOC, can be calc ulated by standard many-body theory\nusing the Kubo formula or by time-dependent spin-density function al theory (SDFT). In practice the many-body\nmethod is usually based on a tight-binding approximation and employs t he random phase approximation (RPA)\nwith a short-range screened Coulomb interaction. This is then equiv alent to a time-dependent Hartee-Fock mean-\nfield theory. The long-range interaction can also be included if care is taken that it does not enter the exchange\nterms [16, 17]. SDFT is approximated similarly as a time-dependent mea n-field theory in the local spin density\napproximation (LSDA) and the long-range Coulomb interaction pres ents no problem since it is effectively screened\nin the exchange-correlation functional. It is useful to consider bo th these methods in parallel. In a system with\nvarying direction of magnetization SDFT is based on a density matrix o f order 2 [18] rather than just spin and\nparticle densities. χ−+(ω) is then coupled to fifteen other response functions which determ ine the longitudinal spin\nsusceptibility as well as the charge response and mixed charge-spin responses [19, 20]. These last relate to phenomena\nlike the spin-Hall effect. Some of these response functions, includin g the longitudinal spin susceptibility, involve the\nlong-range Coulomb interaction importantly even in the absence of S OC [16, 17, 20]. Costa and Muniz [22], following\nan earlier paper [23], show how SOC produces mode coupling in the RPA m any-body approach. However the long-\nrange Coulomb interaction is left untreated. Their paper is particula rly important for being the first to challenge the\nmyth of infinite damping.\nWe firstdiscussthe caseofaBravaislattice whichisappropriatefor puremetalswith acubic structurelikeFe andNi\nat T=0. In both the approaches described above the dynamical su sceptibility is related to mean-field susceptibilities\nof the general form\nχ0(ω) =N−1/summationdisplay\nkmnMmn(k)fkn−fkm\nEkm−Ekn−/planckover2pi1ω+iη. (1)2\nHereEkmis the energy of the one-electron state with wave-vector kin bandm, calculated in the presence of SOC,\nfkmis the corresponding occupation number, Mmn(k) is a product of matrix elements and ηis a small positive\nconstant which ultimately tends to zero. As in usual time-dependen t perturbation theory equation (1) represents the\nresponse to a perturbing field of angular frequency ωin which transitions occur between occupied and unoccupied\nstates. ”Intraband transitions” with m=nclearly do not occur for ω/negationslash= 0 owing to the cancellation of the Fermi\nfunctions. These transitions between identical states, which are not really transitions at all, can play no role in a\ndynamical process. Hankiewicz et al have made a similar point [24]. How ever, in nearly all calculations of the Gilbert\ndamping parameter α, intraband transitions appear and lead to the infinite damping discus sed above.\nTo dispel a myth effectively it is necessary to see how it has arisen. It is instructive to review, in a unified way, some\nmethods which have been used to calculate α. We start from the Lorentzian form of the FMR lineshape which is\nwell-established experimentally [21] and theoretically [22]. Near the re sonance the dynamical transverse susceptibility\nis dominated by a pole at /planckover2pi1ω=bex+/planckover2pi1∆ωwhere ∆ω∼ξ2,ξbeing the SOC parameter, so that\nχ−+(ω) =−2/angbracketleftSz/angbracketright/N\n/planckover2pi1(ω−∆ω)−bex. (2)\nHereSzis thezcomponent of total spin and Nis the number of atoms in the crystal. Near the resonance the FMR\nabsorption is determined by\nℑ(χ−+(ω)) =−2(/angbracketleftSz/angbracketright/N)ℑ(/planckover2pi1∆ω)\n(/planckover2pi1ω−ℜ(/planckover2pi1∆ω)−bex)2+(ℑ(/planckover2pi1∆ω))2. (3)\nℜ(/planckover2pi1∆ω) corresponds to a shift in the resonance frequency and ℑ(/planckover2pi1∆ω) determines the linewidth, both due to SOC.\nThe Gilbert damping factor αis given by ℑ(/planckover2pi1∆ω)/bex(e.g. [25]). The most direct way to calculate αis a brute-force\nnumerical RPA calculation of ℑ(χ−+(ω)), with SOC included, as a function of ωaround the resonance. Costa and\nMuniz [22] performed such calculations using the tight-binding appro ximation and found perfect Lorentzians from\nwhich they deduced α. Taking a monnolayer of Co as an example they found no tendency fo rαto diverge in the pure\nlimit of sharp electronic states. This method of calculating αis very computer intensive and more economic methods\nexist if one assumes a Lorentzian curve from the outset.\nIt follows immediately from (2) that\nα=ℑ(/planckover2pi1∆ω)/bex=2/angbracketleftSz/angbracketright\nNbexℑ(1\nχ−+(bex//planckover2pi1)). (4)\nThis new formula for αmay be regarded as exact. A full treatment of the transverse su sceptibility includes coupling\nto other modes and leads to a rather complex expression in terms of sixteen mean-field susceptibilities of the form (1)\nwith different sets of matrix elements [20]. There is an enormous simplifi cation in the case of a Bravais lattice if we\ncalculate αonly to second order in the SOC parameter ξ. Following the arguments of [20] it is readily found that\ncoupling of the transverse susceptibility to other modes is then elimin ated and that χ−+in (4) may be replaced by\nthe mean-field susceptibility χ0\n−+. This elimination depends on inversion symmetry, which is a property o f a Bravais\nlattice. Without this symmetry, coupling of the transverse suscep tibility to other modes occurs in general even to\norderξ2, as discussed later. It follows further that to order ξ2\nα= (N∆2/2/angbracketleftSz/angbracketrightbex)ℑ(χ0\n−+(bex//planckover2pi1)) (5)\nwhere ∆ is the exchange splitting in the band structure. It is usually s ufficient to calculate the last factor to first\norder in bexso we may take the unphysical limit bex→0, but with due care as discussed below. Then\nα= (N∆2/2/angbracketleftSz/angbracketright)[∂ωℑ(χ0\n−+(ω)]ω=0 (6)\nwhere the electronic state energiesand matrixelements in ℑ(χ0\n−+(ω) are calculatedwith bex= 0. Beforeproceedingto\nthe static ω→0 limit it is essential not to include contributions from ”intraband tran sitions”, as pointed out after (1).\nThis precaution was not taken in [14], where a similar formula was obtain ed, so the spurious infinite damping for a\npure metal appeared. Sometimes it is preferable to keep the physic al non-zero Zeeman field to remove all danger of\nincluding intraband transitions. This also gives the option of calculatin g the frequency-swept FMR linewidth as a\nfunction of Zeeman field. This has been measured [21] and can be con verted to a frequency dependence of α. Such\na dependence has been discussed by Costa and Muniz [22]. However the low-field limit is usually sufficient and here\nwe take the limit bex→0, with the precaution mentioned above, to compare with other the oretical work. Following\n[14], but excluding intraband terms, we find the following two express ions forαat T=0:\nα= (π∆2/2/angbracketleftSz/angbracketright)/summationdisplay\nk/summationdisplay′\nmn|/angbracketleftkm|S−|kn/angbracketright|2δ(Ekm−EF)δ(Ekn−EF)\n= (πξ2/2/angbracketleftSz/angbracketright)/summationdisplay\nk/summationdisplay′\nmn|/angbracketleftkm|T−|kn/angbracketright|2δ(Ekm−EF)δ(Ekn−EF).(7)3\nHereSSS= (Sx,Sy,Sz) is the total spin operator, S−=Sx−iSy,ξhsois the total spin-orbit interaction, T−= [S−,hso]\nis a torque operator and EFis the Fermi energy. The prime on the sum over bands means m/negationslash=nand the sum over k\nis to be carried out as an integral over the Brillouin zone as usual. As p ointed out these expressions are only correct\nto orderξ2so that in the second expression we must evaluate the electronic st ates and energies with ξ= 0. The prime\non the summation sign may then be omitted since the m=nterms are zero owing to inversion symmetry [10]. The\nresulting expression, which can now be written in terms of one-part icle Green functions if desired, is just the version\nof Kambersky’s torque formula [13] for a Bravais lattice derived in tw o ways by Edwards [20]. It is the mean-field\napproximation to a much more general formula [20], valid for an orde red or disordered system,\nα=−(ξ2/2bex/angbracketleftSz/angbracketright)ℑ[χξ=0\nT(bex//planckover2pi1)]. (8)\nWe shall refer to this as the general torque formula. It is exact to orderξ2and we have left open the option of taking\nthe limit bex→0. Here the torque-torque response function is given by the Four ier transform of a retarded Green\nfunction using the Kubo formula\nχT(ω) =/integraldisplay\n/angbracketleft/angbracketleftT−(t),T+/angbracketright/angbracketrighte−iωtdt. (9)\nThe wide application of (8) is discussed later and we recall that the se cond expression in (7), corresponding to the\nmean-field approximation to χT, is only valid for a Bravais lattice. To evaluate the integral over kin the formula (7)\nnumerically it is usual to replace the delta-functions by Lorentzians of width proportional to an inverse relaxation\ntime parameter τ−1. This broadening of the electron states may be regarded as a crud e representation of the effect\nof impurity and/or phonon scattering. The limit τ−1→0 of a perfect crystal at T=0 leads to a finite value of αbut\nis quite tricky to perform numerically [26]. If we wrongly retain SOC in ca lculating the electron states in the second\nexpression of (7) the diagonal matrix elements are non-zero and le ad to the notorious infinite damping parameter\nα. The only work which deals correctly with αin pure metals is reported in fig.1 of [10] and in [22, 26].(In [26] the\ncaption of fig.1 should read ”with and without SOC included in calculating electronic states”).\nWenowturntothetaskofestablishingafirmbasisforcalculatingthe dampingparameter αintechnicallyimportant\nmaterials, which are typically random alloys or layered structures. T his task is greatly simplified if we are satisfied\nwith calculating αto second order in the SOC parameter ξ. This should be sufficient in nearly all systems of interest.\nAt room temperature the ξ2dependence of αis well-established experimentally in several alloy systems, including\nsome containing Pt with its large SOC [27, 28]. The general torque for mula (8) is a very convenient starting-point. Its\nderivation in Appendix A of [20] is for a completely general ferromag netic material, either ordered or disordered, and\nagain relies only on the universal FMR Lorentzian lineshape. The deriv ation proceeds by comparing an exact relation\nbetween χ−+(ω) andχT(ω) with an expansion of (2) in the limit /planckover2pi1∆ω/(/planckover2pi1ω−bex)→0 followed by /planckover2pi1ω→bex. This\norder of limits is essential and results in the form (8) where χTis evaluated in the absence of SOC. A similar formula\nwas derived by Kambersky [13] in another way where crucially the pre scription ξ= 0 did not become apparent. The\nformula is remarkable for describing the essence of a phenomenon a rising solely from SOC without the need to include\nSOC in the calculation.\nThe calculation of χTin a disordered system is still a very demanding problem. It may be app roached using the\nRPA of standard many-body theory or, less obviously, using time-d ependent LSDA. A diagrammatic RPA treatment\nofχTinvolves a sum of ladder diagrams and the first term, without an inter action line, corresponds to the mean-field\napproximation χ0\nT. The remaining terms constitute a vertex correction and we have s hown above that in a monatomic\nBravais lattice this vanishes. In a disordered system like an alloy, or a metal at finite temperature in a frozen phonon\npicture, this is not the case. However this great simplification persis ts if, in a very crude approximation, the system is\nreplaced, at the outset, by an effective medium with the full transla tional symmetry of the lattice but finite electron\nlifetime. We are then led to the Kambersky-like formula (7) for αwith a Lorentzian broadening of the delta-functions\ndetermined by relaxation times which may be dependent on spin and te mperature. This is the background to a\nrecent calculation of αin bulk Ni at room temperature [26] which is in reasonable agreement w ith experiment. A\nproper treatment of χTin a disordered material must deal simultaneously with the RPA verte x correction and any\nvertex corrections which arise in connection with methods of taking a configurational average, such as the coherent\npotential approximation (CPA). There is a small literature on this pr oblem as applied to χ−+, notχT, in a one-band\nmodel [29, 30]. Santos and Costa [30] find that for dilute non-magne tic impurities the RPA vertex correction is\nparticularly important. However as yet the many-body approach is far from being able to provide reliable results for\nαin real disordered materials. The time-dependent LSDA method see ms more promising. As shown below, it gives\na clear physical picture of the RPA vertex correction and separat es it from the configurational averaging problem.\nIn a FMR experiment the local magnetization vector sweeps out a co ne as it precesses around the Zeeman field\ndirection and in the presence of SOC the cone angle θ(rrr) is a function of position. In the time-dependent LSDA θ(rrr)\nsatisfies an integral equation whose solution is avoided in [14] by tak ing a spatially-independent averaged cone angle4\nθ(rrr). This approximation enforces a uniform precession, as occurs in t he absence of SOC, and removes the possibility\nof coupling between transverse and longitudinal susceptibilities. It is very reasonable for a monatomic Bravais lattice\nwhere the variation of θ(rrr) within a unit cell is largely an artificial consequence of the local appr oximation. In the\ntight-binding framework of [20] it would not be an approximation at all for a monatomic Bravais lattice. However, in\ncompounds, alloys and layered structures, variation of the cone a ngle between different types of atom and different\nlayers may be very important. In the many-body approach the ver tex correction in χTis the difference between\nthe fullχTand the mean-field approximation χ0\nT. Since we have seen that the mean-field approximation works well\nfor a homogeneous system, like a monatomic Bravais lattice, we conc lude that the vertex correction corresponds to\nthe effect of the spatial variation of the cone angle, which can be st udied with the LSDA approach. This will be\ndemonstrated explicitly in a forthcoming publication. This productive interplay between standard many-body theory\nand density-functional theory is quite unusual.\nI would like to acknowledge a useful exchange of e-mails with Filipe Guima res on the subject-matter of this paper.\n[1] L.D. Landau, E.M. Lifshitz and L.P. Pitaevski, Statistical Physics, Part 2 (Oxford: Pergamon 1980).\n[2] T.L. Gilbert, Phys. 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Mat. 46969 (2019)."    },    {        "title": "1809.09801v4.Permutation_invariant_constant_excitation_quantum_codes_for_amplitude_damping.pdf",        "content": "Permutation-invariant constant-excitation quantum codes for\namplitude damping\nYingkai Ouyang1Rui Chao2\n1University of Sheffield, UK\n2University of Southern California, USA\nDecember 2, 2019\nAbstract\nThe increasing interest in using quantum error correcting codes in practical devices has heightened\nthe need for designing quantum error correcting codes that can correct against specialized errors, such\nas that of amplitude damping errors which model photon loss. Although considerable research has been\ndevoted to quantum error correcting codes for amplitude damping, not so much attention has been paid\nto having these codes simultaneously lie within the decoherence free subspace of their underlying phys-\nical system. One common physical system comprises of quantum harmonic oscillators, and constant-\nexcitation quantum codes can be naturally stabilized within them. The purpose of this paper is to give\nconstant-excitation quantum codes that not only correct amplitude damping errors, but are also immune\nagainst permutations of their underlying modes. To construct such quantum codes, we use the nullspace\nof a specially constructed matrix based on integer partitions.\n1 Introduction\nThe ability to manipulate quantum information promises to speed up algorithms such as factoring [1, 2],\nsimulate physical systems more efficiently [3], and unlock the ability to perform cryptographic schemes\nwith unprecedented security [4, 5]. However the inherent fragility of quantum information is a major\nobstacle in realizing the full potential of these quantum schemes. To overcome this, one may rely on\nquantum error correction codes, which offer the possibility of reversing the effects of decoherence [6].\nHowever, if an arbitrary quantum error correction code were used in a physical system, it is invariably\naffected by the underlying system’s natural dynamics. Therein lies the allure of constructing quantum\nerror correction codes within an energy eigenspace of the physical system’s underlying Hamiltonian,\nbecause errors may then be avoided at a fundamental level.\nWe consider here the problem of quantum error correction in quantum harmonic oscillators, which,\nas we shall see, is not just of theoretical interest. In recent years, superconducting qubits have been ex-\ntensively studied, and are considered as one of the leading candidates for realizing quantum information\nin a physical system. In the superconducting electrical circuits that superconducting qubits are based on,\neach superconducting qubit is localized around a cluster of Josephson junctions, and can interact with\nother spatially separated superconducting qubits when coupled with microwave-frequency photons in a\nquantum bus [7, 8, 9]. It is well-known that this quantum bus is in turn just a microwave-frequency elec-\ntrical transmission line [9][Appendix A.3], with a Hamiltonian described by a sum of quantum harmonic\noscillators.\nIf we restrict our attention to using photons of identical frequencies within a quantum bus, its Hamil-\ntonian is up to a constant effectively given by H=åja†\njaj. Here, ajdenotes the lowering operator for\nthej-th mode. Now let jx1i\n\u0001\u0001\u0001\nj xnidenote a quantum state with xjexcitations in the j-th mode, and\nletx1+\u0001\u0001\u0001+xndenote the total excitation number of such a state. Then the eigenspaces of Hare spanned\n1arXiv:1809.09801v4  [quant-ph]  29 Nov 2019by statesjx1i\n\u0001\u0001\u0001\nj xniwith a constant total excitation number. If a quantum code is spanned by states\nwith a constant total excitation number, we call it a constant-excitation quantum code. In this paper,\nwe design constant-excitation quantum codes which can be stabilized by the Hamiltonian of quantum\nharmonic oscillators.\nWe consider two common types of errors that may afflict our physical system modeled by quantum\nharmonic oscillators, particularly in a quantum bus. The first type of errors are amplitude damping\n(AD) errors, which can arise when the system weakly interacts with a zero temperature bosonic and\nMarkovian bath. The second type of errors are permutation errors, which may arise especially during\ntransmission when modes are unexpectedly permuted either spatially or temporally. In this paper, we\nconsider quantum error correction codes that offer protection against not only amplitude damping errors\nbut also permutation errors.\nAmplitude damping errors model energy relaxation in quantum harmonic oscillator systems and\nphoton loss in photonic systems. In this paper, we will consider amplitude damping errors that occur in-\ndependently and identically on every mode. To see how amplitude damping errors may arise in quantum\nharmonic oscillators from the underlying physics, consider the coupling Hamiltonian on the j-th mode\ngiven by\nHint;j=cj(a†\njbj+b†\njaj):\nHere, bjis the lowering operator of the bath that couples to the j-th mode, and so Hint;jcouples each\nunique quantum harmonic oscillator to a unique bath, thereby ensuring that the amplitude damping errors\noccur independently for each mode. There are two reasons why we do not consider couplings between\nthe harmonic oscillators. First the Hamiltonian of an ideal transmission line naturally comprises of\nuncoupled harmonic oscillators. Second, even if there is some spurious linear coupling between the\nharmonic oscillators, it has later been shown that dynamical decoupling can homogenize the system, and\nrender the effective Hamiltonian to be that of a sum of identical uncoupled harmonic oscillators [10].\nIn this paper, we assume that AD errors afflict every mode identically, which can be the case when the\ncoupling strength cjis independent of j, so that cj=c. By assuming that the system and the bath are\ninitially in a product state, and subsequently applying a Born-Markov approximation1, one can show\nthat the noise process can be modeled using the Kraus operators\nAk=¥\nå\nm=ks\u0012m\nk\u0013q\n(1\u0000g)m\u0000kgkjm\u0000kihmj; (1.1)\nwhere kindicates the number of AD errors that afflict a mode, and g=1\u0000cos2(cDt)is the strength of\nthe AD error that corresponds to time Dt[12].\nPermutation errors model the stochastic reordering and coherent exchange of quantum packets as\nwell as out-of-order delivery of packets of information, which plausibly occur due to imperfections in a\ncommunication channel [13]. More precisely, we denote a permutation channel to be a quantum channel\nwith each of its Kraus operators Paproportional to eiqaˆpa=åk\u00150(iqaˆpa)k=k!, where qais the parameter\ncorresponding to the infinitesimal generator iˆpa2, and ˆpa, is any linear combination of operators that\npermute the underlying modes with real coefficients. We also emphasize that we are interested in the\nscenario where qatakes an arbitrary value from the real numbers, so it need not be small.\nAmplitude damping errors are prevalent in bosonic systems. If a single bosonic mode were left\nunprotected against AD errors, the incurred error as quantified by one minus the fidelity is of order g.\nIf a quantum error correction code allows reduction of the order of this error to gt+1, we say that the\nquantum code corrects tAD errors. Unsurprisingly, there has been extensive work on quantum error\ncorrection codes specialized against correcting amplitude damping errors [15, 12, 16, 17, 18, 19, 20, 21,\n22]. Of special note are some previously constructed constant-excitation quantum codes that do offer\nimmunity against the natural dynamics of quantum harmonic oscillators [12, 23, 24]. Chuang, Leung and\nYamamoto restricted their study of bosonic quantum codes to constant-excitation quantum codes, and\nfound that the fidelity after quantum error correction for such codes that correct tAD errors with total\n1See [11] for a detailed exposition.\n2The matrix iˆpais called an infinitesimal generator because it is a bounded operator, and generates the unitary matrix eiqaˆpain\nthe sense consistent with [14, Theorem 1.2].\n2excitation number Ncan be made to be åt\nk=0\u0000N\nk\u0001\ngk(1\u0000g)N\u0000k=1\u0000\u0000N\nt+1\u0001\ngt+1+O(gt+2)[12]. Hence\nfor constant-excitation quantum codes, minimizing Nfor fixed tis the primary goal. In their paper [12],\nChuang, Leung and Yamamoto also found constant-excitation quantum codes correcting 1, 2 and 3 AD\nerrors with total excitation numbers equal to 4, 9 and 16 respectively. Wasilewski and Banaszek later\nintroduced a constant-excitation quantum code with N=3 correcting 1 AD error [23], thereby improving\non the construction of the N=4 code in [12]. The code of Wasilewski and Banaszek is also notably the\nfirst known constant-excitation quantum code that not only corrects amplitude damping errors, but is also\npermutation invariant. Recently, Bergmann and van Loock found constant-excitation quantum codes that\ncan correct any number of AD errors [24]. Namely, their codes can correct tAD errors using N= (t+1)2\nexcitations and are very elegant in the sense that these codes can be encoded simply by using NOON\nstates and beamsplitters. Unfortunately the codes of Bergmann and van Loock are not invariant under\narbitrary permutations, and are hence vulnerable to certain permutation errors.\nApart from specialized quantum codes that correct amplitude damping errors, permutation-invariant\nquantum codes have also been studied in recent years, both with respect to arbitrary errors [25, 26, 27, 28]\nand also amplitude damping errors [23, 27, 13]. Permutation-invariant quantum codes are important\nbecause they are inherently immune to permutation errors. After Ruskai first introduced a 9 qubit\npermutation-invariant quantum code correcting one arbitrary error [25], Pollatsek and Ruskai later im-\nproved this in [26] with a 7 qubit permutation-invariant code that corrects one arbitrary error. Later\nin [27], Ruskai’s 9 qubit permutation-invariant quantum code was generalized to yield permutation-\ninvariant quantum codes correcting tarbitrary errors or tAD errors while encoding a single qubit. In\n[13] and [28], the permutation-invariant codes were generalized in different directions to allow the cor-\nrection of 1 AD error and encoding of a qudit, and correction of arbitrary errors and encoding of a qudit\nrespectively. However, aside from Wasilewski and Banaszek’s quantum code, none of these permutation-\ninvariant codes are also constant-excitation quantum codes.\nThe purpose of this paper is to construct constant-excitation quantum codes that not only correct any\ntAD errors, but are also permutation-invariant (PI). Using the techniques from linear algebra and by\ncounting the sizes of integer partitions, we construct PI constant-excitation quantum codes that correct t\nAD errors for any integer t. For our codes, the total number of modes used is equal to the total excitation\nnumber, so n=N. For example, when the total excitation number Nsatisfies the following inequality,\np\u0012N\nt+1\u0013\n+\u0012t\n2\u0013\n\u0015p(1)+\u0001\u0001\u0001+p(t) (1.2)\nwhere p(t)denotes the number of integer partitions of t, there are corresponding PI constant-excitation\nquantum codes that correct tAD errors. The inequality in (1.2) allows us to easily find code parameters\nfor PI constant-excitation quantum codes.\nAmong the PI constant-excitation quantum codes that we construct, we have codes that correct 2,3,4\nand 5 AD errors using 6,12, 20 and 30 total excitations respectively. These codes are given explicitly\nin Example 1, Example 2, Example 3, Example 4 and Example 5 respectively. We wish to emphasize\nthat these codes are not only permutation-invariant, but also have lower total excitation numbers than the\nconstant-excitation codes of Bergmann and van Loock, which require 9,16,25 and 36 total excitations\nrespectively. In this sense, for small values of t, our constructed codes give the best performance in\nterms of fidelity among the constant-excitation quantum codes. Moreover for large values of t, we\nnumerically find that our constructed PI constant-excitation quantum codes that correct tAD errors have\ntotal excitations N=C(t+1)2where Cis slightly larger than one (see Figure 1). This suggests that our\ncode parameters are asymptotically similar to those of Bergmann and van Loock. The value of this result\nlies in the fact that permutation-invariance can be imbued to constant-excitation quantum codes while\nminimally affecting their output fidelities.\nIn this paper, we construct our PI constant-excitation quantum code using partitions of a well-chosen\ninteger and a real vector given explicitly in (4.3). Independently, we define a matrix Ain (4.1) that\ndepends only on the partitions that label the AD errors that are to be corrected and the partitions that\nlabel the permutation-invariant states of constant-excitation that our code is to be supported on. We\nprove in Theorem 3 that any non-trivial solution to the linear system of equations Ax=0leads to a PI\nconstant-excitation quantum code. Intuitively, the matrix Aquantifies the extent in which AD errors,\n3after acting on Dicke states, can shrink their norms. By obtaining a lower bound on the nullity of A, we\nprove (1.2) in Corollary 6.\n2 Preliminaries and notation\nWe begin by introducing terminology related to vectors of non-negative integers. First define Nto be the\nset of non-negative integers and let nbe a positive integer denoting the number of modes and the total\nexcitation number that will be used for the quantum code. For any integer aand non-negative integer b,\nleta(b)= (a)(a\u00001):::(a\u0000b+1)denote the falling factorial symbol. Here, (a)(0)=1. Let (y1;:::; yn)\ndenote a column vector and (y1;:::; yn)Tdenote a row vector. Define 1uand0uas column vectors of\nlength uwith all components equal to 1 and 0 respectively. For y= (y1;:::; yn), let wt (y) =y1+\u0001\u0001\u0001+yn\ndenote the weight of y. For non-negative integers tsuch that 0\u0014t\u0014n, letKn;t=f(y1;:::; yn)2Nn:\ny1+\u0001\u0001\u0001+yn=tgdenote the set of non-negative vectors of weight t. Also define Kn;t=Kn;0[\u0001\u0001\u0001[Kn;t\nto be the set of non-negative vectors with weights from 0 to t.\nWe now introduce terminology related to the constant-excitaton quantum codes that we will study.\nLet the orthonormal vectors jjiforj2Nspan the Hilbert space of a single bosonic mode, which we\ndenote as H. The quantum codes that we consider in this paper are two-dimensional subspaces of the\nn-mode Hilbert space Hn. Given a vector y= (y1;:::; yn)2Nn, define the computational basis state\njyi=jy1i\n\u0001\u0001\u0001\nj yni2Hn. The weight, or a total excitation number of a computational basis state jyi\nis the weight of y. We say that a quantum code is also a constant-excitation quantum code if it can be\nspanned by linear combinations of states with a constant total excitation number.\nIn this paper, we deal with the matrices A†\nkAkrepeatedly, and hence we evaluate them first.\nProposition 1. For all non-negative integers k, we have A†\nkAk=å¥\nj=k\u0000j\nk\u0001\n(1\u0000g)j\u0000kgkjjihjj.\nWe now require notation for representing AD errors that occur on nmodes. Given a vector k=\n(k1;:::; kn)2Kn;k, letAk=Ak1\n\u0001\u0001\u0001\n Akn. We say that Akhas a weight of k. We can then find that the\ndiagonal matrix elements of åk2Kn;kA†\nkAkin the computational basis jx1i\n\u0001\u0001\u0001\nj xnionly depends on k\nandx1+\u0001\u0001\u0001+xn. The following proposition, which essentially follows the same logic as the equations\nfrom (7.6) to (7.11) in [12], makes this precise.\nProposition 2. Letx= (x1;:::; xn)be a vector of non-negative integers, and let c=x1+\u0001\u0001\u0001+xn. Then\nå\nk2Kn;khxjA†\nkAkjxi= (1\u0000g)c\u0000kgk\u0012x1+\u0001\u0001\u0001+xn\nk\u0013\n:\nProof. Obviously,hxjA†\nkAkjxi=hx1jA†\nk1Ak1jx1i:::hxnjA†\nknAknjxni. Using Proposition 1, we get\nhxjA†\nkAkjxi=n\nÕ\ni=1\u0012xi\nki\u0013\n(1\u0000g)xi\u0000kigki= (1\u0000g)c\u0000kgkn\nÕ\ni=1\u0012xi\nki\u0013\n:\nNote that this equality holds even when xi<kifor some i, because the above equality does show that\nhxjA†\nkAkjxiis zero in this case, which is a simple consequence of the fact that\u0000xi\nki\u0001\n= (xi)(ki)=ki!=\n(xi):::(0):::(xi\u0000ki+1)=ki!=0 whenever xi<ki. The result follows conditioned on the combinatorial\nidentity\nå\nk2Kn;kn\nÕ\ni=1\u0012xi\nki\u0013\n=\u0012x1+\u0001\u0001\u0001+xn\nk\u0013\n; (2.1)\nwhich is stated in [12, (7.10)] for example. We can see that (2.1) holds combinatorially, since the number\nof ways to select kballs from bags with x1;:::; xnballs is sum of the product of the number of ways\nto select kiballs from the ith bag, where the total number of selected balls is k. Alternatively, we can\nalso prove (2.1) algebraically using the method of generating functions. Given any formal power series\n4f(z) =åj\u00150fjzjwhere zis indeterminate, let [zk]f(z) =fk. Then we can write\u0000xi\nki\u0001\n= [zki\ni](1+zi)xifor\nindeterminates zi. Hence, we can rewrite the left hand side of (2.1) as\nå\nk2Kn;k[zk1\n1](1+z1)x1:::[zknn](1+zn)xn= [zk](1+z1)x1:::(1+zn)xn\n= [zk](1+z1)x1+\u0001\u0001\u0001+xn\n=\u0012x1+\u0001\u0001\u0001+xn\nk\u0013\n;\nwhich proves (2.1).\nBecause the quantum codes that we study are invariant under any permutation of the underlying\nmodes, we proceed to define some permutation-invariant states that we will use to construct our quantum\ncodes. To do so, we have to first introduce notation related to integer partitions. Given a positive integer\nn, let p(n)denote the number of its partitions, and denote every partition of nas an n-tuple of non-\nincreasing non-negative integers. For example with 5 =2+2+1, we denote the corresponding partition\nof 5 as (2;2;1;0;0). We denote the set of partitions of nwith P(n). Given tuples x= (x1;:::; xa)and\ny= (y1;:::; yb);let(xjy) = ( x1;:::xa;y1;:::; yb)denote the pasting of tuples xandy. For all positive\nintegers t, let\nP(t) =f(qj0t\u0000k):q2P(k);k=1;:::; t\u00001g[P(t); (2.2)\nand let\np(t) =p(1)+\u0001\u0001\u0001+p(t) =jP(t)j: (2.3)\nGiven any positive integer w, letQbe an arbitrary subset of partitions P(w),i.e Q\u0012P(w). The integers u\nandware such that the number of modes nsatisfies the constraint n=uw. Furthermore, given a positive\ninteger u, letQube the set of vectors in Qmultiplied by ucomponent-wise and extended to length uw,\nand appended by a ones vector, so that we have\nQu=f(uqj0uw\u0000w):q2Qg[f1ng: (2.4)\nWe will see later how our quantum codes can be constructed using linear combinations of states labeled\nby vectors in Qu. For example, the ones vector in Quwill correspond to the permutation-invariant state\nj1i\nuw. We proceed to illustrate this definition with some examples.\n1. When Q=P(1) =f(1)g, we have Q3=f(3;0;0);(1;1;1)g. The set Q3will later correspond to\nthe permutation-invariant basis\n\u001a\nj1;1;1i;j3;0;0i+j0;3;0i+j0;0;3ip\n3\u001b\n: (2.5)\n2. When Q=P(2) =f(2;0);(1;1)g, we have Q3=f(6;0;0;0;0;0);(3;3;0;0;0;0)g. The set Q3will\nlater correspond to the permutation-invariant basis fj1;1;1;1;1;1i;jf1i;jf2igwhere\np\n6jf1i=j(6j05)i+j(0;6j04)i+j(02j6j03)i+j(03j6j02)i+j(04j6;0)i+j(05j6)i\np\n15jf2i=j(3;3j04)i+j(0;3;3j03)i+j(02j3;3j02)i+j(03j3;3;0)i+j(04j3;3)i+j(3j04j3)i\n+j(3;0;3j03)i+j(0;3;0;3j02)i+j(02j3;0;3;0)i+j(03j3;0;3)i+j(3j03j3;0)i+j(0;3j03j3)i\n+j(3;0;0;3;0;0)i+j(0;3;0;0;3;0)i+j(0;0;3;0;0;3)i: (2.6)\nGiven a partition q= (q1;:::; qn)of a positive integer n, we first define eqas the set of all permutations\nofq. For any set of non-negative vectors Q, we defineeQto be a union of all the sets eqfor which q2Q.\nFormally, denoting Snas the symmetric group of order n, we have\neq=f(qp(1);:::; qp(n):p2Sng; (2.7)\neQu=[\nq2Queq: (2.8)\n5For example,\n^(1;1;1;0) =f(1;1;1;0);(1;1;0;1);(1;0;1;1);(0;1;1;1)g:\nNext we definejeqias a uniform superposition over all permutations of the n-mode statejq1i\n\u0001\u0001\u0001\nj qni,\nso that\njeqi=1p\njeqjå\ny2eqjyi: (2.9)\nAs an example using this notation,\nj^(2;2;0;0)i=1p\n6(j(2;2;0;0)i+j(0;2;2;0)i+j(0;0;2;2)i+j(2;0;0;2)i+j(2;0;2;0)i+j(0;2;0;2)i):\nThe codes that we consider lie within the span of these Dicke states jeqiwhere qare partitions of n. In\nparticular, spanfjeqi:q2P(n)gis the space of all permutation-invariant states with a total-excitation\nnumber n.\nLet us denote the distance between any two vectors u= (u1;:::; un)andv= (v1;:::; vn)inNnas\nd(u;v) =n\nå\nj=1juj\u0000vjj: (2.10)\nThis distance, also known as the Manhattan distance, is distinct from the usual Hamming distance. Let\nC\u0012Nn. Then we define the minimum distance of Cto be\nd(C) =min\nu;v2Cfd(u;v):u6=vg: (2.11)\nOur non-standard definition of minimum distance arises because we use a metric induced by the Manhat-\ntan distance as opposed to the Hamming distance. We will later be interested in the minimum distance\nof the setfQuforQ\u0012P(w)for some positive integer w.\n3 Quantum error correction criterion\nHere, we review some of the underlying theory of quantum error correction for AD errors on constant-\nexcitation quantum codes, beginning from the Knill-Laflamme (KL) quantum error correction criterion\n[6] . Given that we wish to correct tAD errors on nmodes, it is both necessary and sufficient to have\nthe KL conditions satisfied for the AD errors Akfor which k1+\u0001\u0001\u0001+kn\u0014t. Just as in [12], for the KL\nquantum error correction criterion to hold for a quantum code with logical codewords j0Liandj1Lionn\nmodes with respect to tAD errors, it suffices to require that the following equations hold.\nh0LjA†\nxAxj0Li=h1LjA†\nxAxj1Li 8x2Kn;t; (3.1)\nh0LjA†\nxAyj0Li=h1LjA†\nxAyj1Li=08distinct x;y2Kn;t; (3.2)\nh0LjA†\nxAyj1Li=08x;y2Kn;t: (3.3)\nIn the usual KL quantum error correction criterion, the right hand side of (3.2) does not have to be\nzero. We have made this restriction just as in [12] to make the construction of our quantum codes\nmore tractable. For convenience, we hereby call the constraints (3.1) the non-deformation quantum\nerror correction criterions, and the constraints (3.2) and (3.3) the orthogonal quantum error correction\ncriterions.\nFor us, the condition (3.2) always holds as long as the logical codewords are linear combinations of\nvectorsjvifor which v2C\u001aNnandd(C)\u00152t+1. To see this, note first that from [12, Theorem 2], as\nlong as vandware distinct vectors from Nnwhere d(v;w)\u00152t+1, then for every xandyinKn;t, we\nhavehvjA†\nxAyjwi=0. Second, when xandyare distinct elements from Kn;t, it is clear that Axjviand\n6Ayjviare either zero or axjuiandayju0irespectively for real numbers ax;ayand vectors u;u0inNn.\nSince u6=u0from the distinctness of xandy, it follows thathvjA†\nyAxjvi=0 for all distinct xandyin\nKn;t. Hence it follows that if jyiis any vector that is a linear combination of basis states jvifor which\nv2Cwith d(C)\u00152t+1, for all distinct xandyinKn;t, we must havehyjA†\nyAxjyi=0.\nWhen the condition (3.2) holds for a quantum code and hiLjA†\nxAxjiLi>0 for all i=0;1 and x2Kn;t,\nthe quantum code is non-degenerate. To see this, note that\nå\nx2Kn;thiLjA†\nxAxjiLijxihxj\nis diagonal with positive diagonal entries. Since this matrix is invertible and hence full rank, Gottesman’s\ndefinition [29, Page 7, line 1], implies that such quantum codes are non-degenerate.\nSince the orthogonality condition (3.3) holds as the logical codewords are linear combinations of\nvectorsjvifor which v2C\u001aNnandd(C)\u00152t+1 and xandyinKn;t[12, Theorem 2], the only\nnon-trivial error correction criterion is the non-deformation condition (3.1). Because quantum codes that\nwe construct are permutation-invariant, it suffices to restrict the error-inducing Kraus operators that arise\nfrom partitions of kwhere k\u0014tandt\u0014n. This is because for any permutation-invariant quantum state\njyiand any Kraus operator B, we have\nhyjB†Bjyi=hyjp†B†pp†Bpjyi=hyj(p†Bp)†p†Bpjyi:\nHence for every partition l= (l1;:::;lk)inP(k), we denote by Al;nthe amplitude damping operator\nonnmodes with respect to lwhere\nAl;n=Al1\n\u0001\u0001\u0001\n Alk\nA\nn\u0000k\n0: (3.4)\nIf the conditions (3.1), (3.2), and (3.3) hold for the constant-excitation quantum code with total\nexcitation number n, then the worst-case fidelity is at least åt\nk=0\u0000n\nk\u0001\ngk(1\u0000g)n\u0000k;as proved in [12]. In\nfact, the entanglement fidelity exhibits the same behavior, as we now illustrate.\nThe entanglement fidelity of a quantum code quantifies how well an entangled state\njyi=j0i\nj0Li+j1i\nj1Lip\n2\nis protected when the half of it which is encoded into a quantum code with logical codewords j0Liand\nj1Liis exposed to noise. If the recovery channel of the quantum code is given by R, its entanglement\nfidelity with respect to AD errors is\nhyjI\n(R\u000eA)(jyihyj)jyi= (h0LjR(A(j0Lih0Lj))j0Li+h1LjR(A(j1Lih1Lj))j1Li)=2;(3.5)\nwhereIis the identity channel on a single qubit, and Ais the quantum channel corresponding to an AD\nchannel that acts independently and identically on every mode in the quantum code. Now we can write\nA=A0+A00whereA0andA00are both quantum operations that induce at most tAD errors and at\nleast t+1 AD errors respectively. Clearly if the quantum code is completely correctible with respect to\nthe quantum operation A0, then Proposition 2 implies that the entanglement fidelity is at least the trace\nof\u0000\nA0(j0Lih0Lj)+A0(j1Lih1Lj)\u0001\n=2;\nwhich is at least åt\nk=0\u0000n\nk\u0001\ngk(1\u0000g)n\u0000k, if the quantum code is a constant-excitation quantum code with\nntotal excitations.\n4 From partitions to quantum codes\nHere we will see how a PI constant-excitation quantum code can be constructed from integer partitions.\nSome of the integer partitions label the AD errors, while the others label the Dicke states that our code is\nsupported on. It is the permutation-invariant property of our code that allows us to restrict our attention\n7to AD errors that are labeled by integer partitions of the numbers from 1 to t. To be more explicit,\nsince the norm of an AD error acting on a permutation-invariant state is equivalent to the norm of a\npermuted AD error acting on the same permutation-invariant state, in studying the non-deformation\nconditions, it suffices to study only the AD errors labeled by integer partitions of the number of AD\nerrors. We label these AD errors with the vectors t1;:::;tp(t)where t1= ((1);0t\u00001),t2= ((2;0);0t\u00002),\nt3= ((1;1);0t\u00002),t4= ((3;0;0);0t\u00003),t5= ((2;1;0);0t\u00003),t6= ((1;1;1);0t\u00003), and so on. We will\nconsider quantum codes supported on Dicke states represented by the partitions of a suitably chosen\ninteger w. We then construct a matrix Awith rows labeled by the AD errors and columns labeled by\nDicke states. In the paragraphs that follow, we will describe the structure of this matrix.\nWe now define the matrix elements of A. They are\nai;j=heqjjA†\nti;nAti;njeqji1\ngwt(ti)(1\u0000g)n\u0000wt(ti); (4.1)\nwhere qjare vectors with weight equal to n. We can arrange these matrix elements into a matrix A, with\nthe rows labeled by the AD errors, and the columns labeled by the quantum code’s basis elements. The\nindices i=1;:::; p(t)label the AD errors, and the indices j=1;:::; clabel the Dicke states that the\nquantum code to be designed will be supported on. Writing down this matrix explicitly, we have\nA=0\nB@a1;1::: a1;c\n... :::...\nap(t);1::: ap(t);c1\nCA=0\nB@\u0000aT\n1\u0000\n...\n\u0000aT\np(t)\u00001\nCA: (4.2)\nWhat is important about the matrix Ais that its properties will be used to design a PI constant-excitation\nquantum code that corrects AD errors. For this to be possible, it is important that Ais independent of g,\nwhich indeed is the case because of the normalization condition in (4.1). Properties of the code will then\nbe inferred from the nullity of A.\nIndependently from the matrix A, we can define basis states for a PI constant-excitation quantum\ncode. Our PI constant-excitation quantum code is thus defined only by the partitions labeling the Dicke\nstates on which it is supported, and a real vector x. We represent the basis states of this quantum code\nin terms of linear combinations of Dicke states labeled by the partitions q1;:::;qcthat all have the same\nweight equal to n, and a non-zero real column vector x= (x1;:::; xc)Tsuch that x1+\u0001\u0001\u0001+xc=0. The\nbasis states of our quantum code are\nj0Li=1px\u0012q\nx+\n1jeq1i+\u0001\u0001\u0001+p\nx+cjeqci\u0013\nj1Li=1px\u0012q\nx\u0000\n1jeq1i+\u0001\u0001\u0001+p\nx\u0000cjeqci\u0013\n(4.3)\nwhere x+\ni=maxfxi;0g,x\u0000\ni=maxf\u0000xi;0gandx=x+\n1+\u0001\u0001\u0001+x+\nc.\nRoughly speaking, the matrix Acan be made to encapsulate the KL quantum error correction criterion\nwith respect to the quantum code that we have defined in (4.3). More precisely, when a certain distance\ncriterion holds and when the nullity of Ais at least one, there are non-trivial solutions of the linear system\nof equations Ax=0 for which x1+\u0001\u0001\u0001+xc=0. This allows the derivation of a PI constant-excitation\nquantum code that corrects tAD errors. This is our main result, and we state it in the following theorem.\nTheorem 3. Let w ;u and t be positive integers and let A be a matrix with matrix elements given by\n(4.1). Let Q =P(w)with Q ube given by (2.4) and eQugiven by (2.8). If d (fQu)\u00152t+1and if the\nnullity of A is at least one, then there exists a permutation-invariant constant-excitation quantum code\nthat corrects t AD errors using uw total excitations. Moreover, such a quantum code can be derived from\n(4.3); the logical codewords (4.3) derived from any non-zero vector xin the nullspace of A will span\nsuch a quantum code.\nBecause bounding the nullity of Ais crucial in demonstrating that the quantum code as defined by\n(4.3) corrects tAD errors, we will proceed to count the number of sets of linearly dependent rows in A\nto obtain such a bound. We use the fact that the rows in Athat correspond to kAD errors are linearly\n8dependent. This arises because of the combinatorial identity in Proposition 2. This idea extends to\ncertain submatrices of Ato demonstrate more linearly dependent rows. The following lemma tells us\nhow some rows of Aare linearly dependent, where aT\nidenotes the i-th row of A.\nLemma 4. Let c idenote the number of ways to permute (tij0n\u0000ni), where n iis the number of components\ninti. For all k =1;:::; t, the sum of the rows of A corresponding to the errors that induce k photon losses\nsum to c p(k\u00001)+1aT\np(k\u00001)+1+\u0001\u0001\u0001+cp(k)aT\np(k)=\u0000n\nk\u0001\n1T\nc.\nProof. Now note that A†\nti;nAti;nis a diagonal matrix. The number of ways to permute qjisjeqjj. Hence\nthe number of elements of the symmetric group that leave qjinvariant is n!=jeqjj. Hence\nå\nx2eqjhxjA†\nti;nAti;njxi=1\nn!=jeqjjå\np2Snhqjjp†A†\nti;nAti;npjqji: (4.4)\nFrom this,\nai;jgwt(ti)(1\u0000g)n\u0000wt(ti)=heqjjA†\nti;nAti;njeqji=1\njeqjjå\nx2eqjhxjA†\nti;nAti;njxi\n=1\nn!å\np2Snhqjjp†A†\nti;nAti;npjqji; (4.5)\nwhere Sndenotes the matrix representation of the symmetric group that permutes the nmodes. Using\nthe definition of ci, it follows that\np(k)\nå\ni=p(k\u00001)+11\nn!=ciå\np2Snhqjjp†A†\nti;nAti;npjqji=å\ny=(y1;:::;yn)2Nn\ny1+\u0001\u0001\u0001+yn=khqjjA†\nyAyjqji=gk(1\u0000g)n\u0000k\u0012n\nk\u0013\n;(4.6)\nwhere the last equality follows from Proposition 2. From this, it follows that\np(k)\nå\ni=p(k\u00001)+1ciai;j=\u0012n\nk\u0013\n; (4.7)\nand hence\np(k)\nå\ni=p(k\u00001)+1ciai=\u0012n\nk\u0013\n1c: (4.8)\nTo better understand the ramification of Lemma 4, we explain the structure of the rows of Ain greater\ndetail. The rows in Aare labeled by integer partitions corresponding to the AD errors. The first row of\nAcorrespond to 1 AD errors. The second row and third corresponds to 2 AD errors with corresponding\npartitions given by (2,0) and (1,1) respectively. The fourth, fifth and sixth rows corresponds to 3 AD\nerrors with corresponding partitions given by (3,0,0) and (2,1,0) and (1,1,1) respectively. Then Lemma\n4 implies the following.\n1. One photon loss: The first row of Ais proportional to a vector of ones.\n2. Two photon losses: A linear combination of the second and third rows of Awith positive integer\ncoefficients is proportional to a vector of ones.\n3. Three photon losses: A linear combination of the fourth, fifth and sixth rows of Awith positive\ninteger coefficients is proportional to a vector of ones.\nCertain subsets of rows in Aare hence linearly dependent according to Lemma 4, namely the rows\nlabeled by elements from fp(k\u00001)+1;:::; p(k)gfor every positive integer k.\nBy employing a different type of counting argument, one can note that different subsets of rows\nwithin Aare linearly dependent. This is given by the following Proposition.\n9Proposition 5. For any h =1;2:::;t\u00001, the row vector aT\n¯p(h)in A is linearly dependent on the rows in\nA where w AD errors occur on at least h modes, for every w =h+1;:::; t.\nProof. The crux of the proof arises from the fact that ciai;jis a non-negative integer with a combinatoric\ninterpretation. Namely, we can interpret qjas a column label in Athat specifies a list of distinguishable\nbins that all together contain nindistinguishable balls. Let wt (ti)denote the sum of the components in\ntheti. Then we can interpret ciai;jas the number of ways of picking wt (ti)balls such that the number\nof balls contributed by individual bins that conform to ti.\nFor simplicity, denote bi;j=ciai;jas entries of a matrix Bwith row vectors bT\ni. Also denote set of rows\nindices of Afor which wAD errors occur on exactly mmodes to be\nIw;m:=fijwt(ti) =w;tiafflicts mmodesg:\nTo establish the proposition, we need to show that for any h=1;2;:::;t\u00001 and w>h, the row vector\nbT\n¯p(h)is a linear combination of bT\ni, where the row indices belong to the set\nIw;h[\u0001\u0001\u0001[ Iw;w:\nA crucial observation is now the following: given any partition qjofn,b¯p(h);jis the number of ways\nof picking hballs from hdifferent bins. There is another way to calculate b¯p(h);jwith an overcounting\nargument, by first considering too many AD errors, and then counting the number of ways to remove\nAD errors to get just the right number and configuration. To be precise, we first pick w>hballs from at\nleast hbins, and then pick hballs from the wselected. Mathematically this reads\n\u0012n\u0000h\nw\u0000h\u0013\nb¯p(h);j=å\ni2[w\nm=hIw;hbi;jdi;h;\nwhere di;his the number of ways of picking hballs out of exactly hbins from those given by ti. Note\nthatdi;hand\u0000n\u0000h\nw\u0000h\u0001\ndo not depend on j. This establishes the lemma.\nBy identifying sets of linearly dependent rows of Aand counting the number of non-intersecting\nlinearly dependent sets, one can obtain a lower bound on the nullity of A, from which a lower bound on\nthe number of basis states can be obtained. This in turns implies that whenever the inequality (1.2) is\nsatisfied, then we have a PI constant-excitation quantum code that corrects tAD errors.\nCorollary 6. Let w and t be positive integers such that p (w)+\u0000t\n2\u0001\n\u0015p(1)+\u0001\u0001\u0001+p(t)and w\u00152. Then\nthere is a permutation-invariant constant-excitation quantum code with w (t+1)total excitations and\nwhich corrects t AD errors.\nProof. Let us construct the matrix Awith columns labeled by the Dicke states labeled by Quand rows\nlabeled by AD errors of weight from 1 to t, where Q=P(w)andu=t+1.\nWe first show that d(eQu)\u00152(t+1), so that the distance criterion in Theorem 3 holds. One can see\nthis for the following reason. The minimum distance of any set of non-negative vectors of fixed length\nis trivially at least 2. Hence the minimum distance of eQis at least 2 u. The minimum distance between\nvectors fromeQand the ones vector is obviously w(u\u00001) + (uw\u0000w) =2w(u\u00001)which is at least 2 u\nwhenever w\u00152.\nLet us denote the rank of Aand the nullity of Aby rank( A) and nullity( A) respectively. Now the\nrank of Ais equal to its row rank, which is the number of its linearly independent rows. We will see\nthat the matrix Ain fact has many linearly dependent rows, and hence its row rank is strictly less than\nthe number of its rows. More precisely, the sets of its rows which correct kAD errors for k=1;:::; t\nare linearly dependent according to Lemma 4. The case for k=1 is trivial, because there is only one\nrow of Athat corresponds to k=1. Now define Lk=fp(k\u00001) +1;:::; p(k)g. When k\u00152, the sets\nLkof labels for the dependent row vectors of Ahave cardinality at least two, and we can eliminate one\ndependent row from each Lk, which leads to an elimination of t\u00001 rows. According to Proposition 5,\nthe row of Awhere hAD errors afflict exactly hmodes is linearly dependent with the rows in Athat\n10afflict wAD errors in at least hmodes, for every w>h. By setting h=2;3;:::, we can eliminate another\nt\u00002;t\u00003;:::rows. The total number of linearly dependent rows is thus at least 1 +\u0001\u0001\u0001+(t\u00001) =\u0000t\n2\u0001\n.\nNow the dimension of the domain of Ais the number of its columns, which is p(w) +1. The rank-\nnullity theorem states that the nullity of Ais precisely p(w)+1\u0000rank(A). From an upper bound of the\nrank of A, we can obtain a lower bound on the nullity of A. We have seen from the previous paragraph\nthat the rank of Ais at most the number of its rows minus\u0000t\n2\u0001\n. Hence the rank-nullity theorem implies\nthat nullity( A)\u0015p(w) +1\u0000(p(t)\u0000\u0000t\n2\u0001\n). It follows that for the nullity of Ato be at least 1, it suffices\nto require p(w) +1\u0000(p(t)\u0000\u0000t\n2\u0001\n)\u00151. Theorem 3 then implies that we can use Ato construct a PI\nconstant-excitation quantum code that corrects tAD errors.\n4.1 Proof of Theorem 3\nFirst we show that the non-deformation condition with respect to the AD error of weight zero holds.\nNotice that for any quantum state jyithat is a superposition of computational basis states each of weight\nk, Proposition 1 implies that hyj(A\nn\n0)†A\nn\n0jyi= (1\u0000g)k:Thus the non-deformation condition for the\nAD error of weight zero trivially holds.\nNow we will demonstrate that the orthogonality conditions of the KL quantum error correction crite-\nrion are satisfied because of the distance criterion imposed. While this has been proved in [12][Theorem\n2], we briefly state the underlying reason for this. An AD error of weight kchanges the weight of com-\nputation basis states by k. The KL quantum error correction criterion involves taking the inner product of\nstates both afflicted by AD errors of weight at most t. Thus, if the vectors underlying the computational\nbasis states form a set of distance at least 2 t+1, all the orthogonal quantum error correction criterions\nwill hold. Moreover, using the simple fact that xi=x+\ni\u0000x\u0000\ni, the non-deformation quantum error correc-\ntion criterion for AD errors of weight from 1 to twith respect to the code (4.3) will be equivalent to the\nconstraints\nc\nå\nj=1xjheqjjA†\nti;nAti;njeqji=h0LjA†\nti;nAti;nj0Li\u0000h1LjA†\nti;nAti;nj1Li=0: (4.9)\nClearly these constraints are equivalent to the system of linear equations Ax=0. But we still have to\nshow is that there is a non-zero xsuch that x1+\u0001\u0001\u0001+xc=0 and Ax=0.\nWe first show that if Ax=0 has a non-trivial solution for x, then x1+\u0001\u0001\u0001+xc=0. To see this, note\nthat Lemma 4 implies that the first row of Ais a vector of ones. Hence Ax=0 implies that 1T\ncx=0\nwhich implies that x1+\u0001\u0001\u0001+xc=0. Therefore if the nullity of Ais at least one, the code as defined\nby (4.3) exists and the non-deformation quantum error correction criterions for AD errors of weights\nfrom 1 to thold. Since we have argued in the previous paragraph how all the orthogonality quantum\nerror correction criterions hold and the non-deformation quantum error correction criterion for the AD\nof weight zero holds, all the KL quantum error correction criterions hold, and the code as defined by\n(4.3) corrects tAD errors.\n5 Explicit code constructions\nIn this section, we demonstrate how one can make use of the results in the previous section to construct\nquantum codes. We illustrate briefly a recipe in which quantum codes may be found. Suppose first that\nwe wish to construct a quantum code that corrects tAD errors. Then we will pick some integer w, and\nsetQ=P(w), so that Qis the set of all integer partitions of 1 ;:::; w. We will next construct the set Qufor\na suitable choice of an integer u. The basis states of our quantum codes are then labeled by the elements\nofQu. With Quand integer partitions labeling the different types of AD errors, we can construct a matrix\nAas given in (4.1). Then we define our quantum codes based on the vectors that we find in the nullspace\nofA.\nExample 1 (Constant energy code correcting 1 AD error [23]) .Consider t=1;w=1;Q=P(w)with\nu=3, so that the number of modes is n=uw=3. Then Qu=f(3;0;0);(1;1;1)g. Obviously d(eQu) =\n114\u00152t+1.\nA=\u0000\na1;1a1;2\u0001\n(5.1)\nwhere\na1;1=h^(3;0;0)jA†\n(1;0;0)A(1;0;0)j^(3;0;0)i=(g(1\u0000g)2)\n=1\n3h(3;0;0)jA†\n(1;0;0)A(1;0;0)j(3;0;0)i=(g(1\u0000g)2): (5.2)\nNote that\na1;2=h(1;1;1)jA†\n(1;0;0)A(1;0;0)j(1;1;1)i=(g(1\u0000g)2) =1: (5.3)\nHence A=\u0000\n1 1\u0001\n. Note that we can obtain the same result for Afrom (4.8) because since the number\nof ways to permute (1;0;0)is 3, 3 A=31T\n2:The vector x= (1;\u00001)clearly lies within the nullspace of A,\nand hence we can derive from (4.3) the quantum code spanned by\nj0Li=1p\n3(j(3;0;0)i+j(0;3;0)i+j(0;0;3)i) (5.4)\nj1Li=j(1;1;1)i: (5.5)\nSince all of the requirements of Theorem 3 are satisfied for t=1, the code spanned by (5.4) and (5.5) is\na constant energy code which also corrects 1 AD error and which is permutation-invariant. This is also\nprecisely Wasilewski and Banaszek’s 3 mode code [23].\nExample 2 (Constant energy code correcting 2 AD errors) .Consider t=2;w=2;Q=P(w)with u=\nt+1=3, so that the number of modes is n=uw=6. Then\nQu=f(6;0;0;0;0;0);(3;3;0;0;0;0);(1;1;1;1;1;1)g: (5.6)\nObviously d(eQu) =6\u00152t+1. Also,\nA=0\n@1 1 1\n5\n21 0\n03\n511\nA: (5.7)\nNow note that A(2\n5;\u00001;3\n5) =0;and hence we can derive from (4.3) the quantum code spanned by\nj0Li=r\n2\n5j^(6;0;0;0;0;0)i+r\n3\n5j1i\n6; (5.8)\nj1Li=j^(3;3;0;0;0;0)i: (5.9)\nExample 3 (Constant energy code correcting 3 AD errors with 12 excitations) .Consider t=3;w=\n3;Q=P(w)with u=t+1=4, so that the number of modes is n=uw=12. Then\nQu=f((12)j011);((8;4)j010);((4;4;4)j09);112g: (5.10)\nObviously d(eQu) =8\u00152t+1. We now proceed to evaluate the matrix elements of A. The first row of A\nis a vectors of ones. By considering only the matrix elements of A†\n2A2, the second row of Ais equal to\nr2= \u000012\n2\u0001\n12;11\u00008\n2\u0001\n+11\u00004\n2\u0001\n2\u000012\n2\u0001 ;\u000011\n2\u0001\u00004\n2\u0001\n\u000012\n3\u0001;0!T\n=\u001211\n2;17\n6;3\n2;0\u0013T\n:\nBy considering only the matrix elements of A†\n1A1\nA†\n1A1, the third row of Ais equal to\nr3= \n0;2\u00008\n1\u0001\u00004\n1\u0001\n2\u000012\n2\u0001;\u000010\n1\u0001\n42\n\u000016\n3\u0001;1!T\n=\u0012\n0;16\n33;8\n11;1\u0013T\n:\n12As one can see, the first, second row and the third row are linearly dependent because\u000012\n1\u0001\nr2+\u000012\n2\u0001\nr3=\u000012\n2\u0001\n112as implied by Lemma 4.\nNext we proceed to evaluate the fourth, fifth and sixth rows of A. By considering only the matrix\nelements of A†\n3A3, the fourth row of Ais equal to\nr4= \u000012\n3\u0001\n12;11\u00008\n3\u0001\n+11\u00004\n3\u0001\n2\u000012\n2\u0001 ;\u000011\n2\u0001\u00004\n3\u0001\n\u000012\n3\u0001;0!T\n= (55=3;5;1;0)T:\nBy considering only the matrix elements of A†\n2A2\nA†\n1A1, the fifth row of Ais equal to\nr5= \n0;\u00008\n2\u0001\u00004\n1\u0001\n+\u00008\n1\u0001\u00004\n2\u0001\n2\u000012\n2\u0001 ;10\u00004\n2\u0001\u00004\n1\u0001\n\u000012\n3\u0001;0!T\n=\u0012\n0;40\n33;12\n11;0\u0013T\n:\nBy considering only the matrix elements of\u0010\nA†\n1A1\u0011\n3\n, the sixth row of Ais equal to\nr6= \n0;0;\u00004\n1\u00013\n3\u000016\n3\u0001;1!T\n=\u0012\n0;0;16\n55;1\u0013T\n:\nClearly, we have\u000012\n1\u0001\nr4+2\u000012\n2\u0001\nr5+\u000012\n3\u0001\nr6=\u000012\n3\u0001\n1T\n12, and hence the first, fourth, fifth and sixth rows are\nlinearly dependent. We now get\nA=0\nBBBBBB@1 1 1 1\n11\n217\n63\n20\n016\n338\n111\n55=3 5 1 0\n040\n3312\n110\n0 016\n5511\nCCCCCCA: (5.11)\nThe matrix rank of Ais 3, and the null space of Ais spanned by (\u000021=32;99=32;\u000055=16;1). Thus we\nhave A(\u000021;99;\u0000110;32) =0. From this we can derive from (4.3) the quantum code spanned by\nj0Li=1p\n131\u0010p\n99j^((8;4)j010)i+p\n32j1i\n12\u0011\n; (5.12)\nj1Li=1p\n131\u0010p\n21j^((12)j011)i+p\n110j^((4;4;4)j09)i\u0011\n: (5.13)\nExample 4 (Constant energy code correcting 4 AD errors) .Consider t=4;w=4;Q=P(w)with u=\nt+1=5, so that the number of modes is n=uw=20. The Dicke states are specified by\nQu=f((20)j019);((15;5)j018);((10;10)j018);((10;5;5)j017);((5;5;5;5)j016);120g: (5.14)\nThen the matrix Ais given by\nA=0\nBBBBBBBBBBBBBBBBB@1 1 1 1 1 1\n19\n223\n49\n213\n42 0\n015\n3810\n1925\n3815\n191\n5793\n412 7 2 0\n0135\n7645\n1975\n3830\n190\n0 0 025\n11425\n571\n969\n4137\n221 11 1 0\n0485\n76120\n1975\n1930\n190\n0105\n19405\n38100\n1960\n190\n0 0 0425\n68450\n570\n0 0 0 0125\n96911\nCCCCCCCCCCCCCCCCCA; (5.15)\n13with rank 5 and nullity 1. The nullspace is spanned by\u000084\n125\u0000456\n125\u0000152\n1251368\n125\u0000969\n1251\u0001\n. From\nthis we can derive from (4.3) the quantum code spanned by\nj0Li=1p\n1577\u0010p\n84j^((20)j019)i+p\n1368j^((10;5;5)j017)i+p\n125j1i\n20\u0011\n; (5.16)\nj1Li=1p\n1577\u0010p\n456j^((15;5)j018)i+p\n152j^((10;10)j018)i+p\n969j^((5;5;5;5)j016)i\u0011\n: (5.17)\nExample 5 (Constant energy code correcting 5 AD errors) .Consider t=5;w=5;Q=P(w)with u=\nt+1=6, so that the number of modes is n=uw=30. The Dicke states are specified by\nQu=f((30)j030);((24;6)j028);((18;12)j028);((18;6;6)j027);\n((12;12;6)j027);((12;6;6;6)j026);((6;6;6;6;6)j026);130g: (5.18)\nThen the matrix Ais given by\nA=0\nBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1 1 1 1 1 1 1 1\n29\n297\n1073\n1061\n1049\n1037\n105\n20\n048\n14572\n14584\n14596\n145108\n14524\n291\n406\n31022\n15518\n15428\n1546\n328\n310\n30\n0336\n145504\n145426\n145456\n145378\n14560\n290\n0 0 0162\n1015216\n101554\n145108\n2031\n1827\n23547\n10237\n210367\n2185\n20\n02104\n1452292\n1451792\n145280\n29180\n2980\n290\n0276\n293366\n145321\n292112\n145243\n29150\n290\n0 0 0729\n1015972\n10151269\n1015270\n2030\n0 0 0 0 096\n101548\n2031\n23751\n51417 312 286 53 27 1 0\n010686\n1451521\n291248\n29606\n29333\n2960\n290\n0276\n293366\n145321\n292112\n145243\n29150\n290\n01196\n2914586\n1451040\n2944 18200\n290\n0 0 01053\n406594\n1451593\n406675\n2030\n0 0 0 0 0312\n1015120\n2030\n0 0 0 0 0 0144\n263911\nCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (5.19)\nwith rank 7 and nullity 1. The nullspace is spanned by\n\u0000\n\u000021505\n31104135575\n3110439875\n15552\u000055825\n3888\u000025375\n25925075\n144\u00002639\n1441\u0001\n:\nFrom this we can derive from (4.3) the quantum code spanned by\nj0Li=1p\n1342629\u0010p\n135575j^((24;6)j028)i+p\n79750j^((18;12)j028)i\n+p\n1096200j^((12;12;6)j027)i+p\n31104j1i\n30\u0011\n; (5.20)\nj1Li=1p\n1342629\u0010p\n21505j^((30)j029)i+p\n446600j^((18;6;6)j027)i\n+p\n304500j^((12;12;6)j027)i+p\n570024j^ ((6;6;6;6;6)j025)i\u0011\n: (5.21)\nTo illustrate the fact that Acan potentially have a nullspace larger than 1, we consider in the following\na 3 AD quantum code with 16 excitations.\nExample 6 (Constant energy code correcting 3 AD errors) .Consider t=3;w=4;Q=P(w)with u=\nt+1=4, so that the number of modes is n=uw=16. Then\nQu=f((16)j015);((12;4)j014);((8;8)j014);((8;4;4)j013);((4;4;4;4)j012);116g: (5.22)\n14Obviously d(eQu) =8\u00152t+1. We now proceed to evaluate the matrix elements of A. The first row of A\nis a vectors of ones. By considering only the matrix elements of A†\n2A2, the second row of Ais equal to\nr2= \u000016\n2\u0001\n16;15\u000012\n2\u0001\n+15\u00004\n2\u0001\n2\u000016\n2\u0001 ;15\u00008\n2\u0001\n\u000016\n2\u0001;\u000015\n2\u0001\u00008\n2\u0001\n+2\u000015\n2\u0001\u00004\n2\u0001\n3\u000016\n3\u0001 ;\u000015\n3\u0001\u00004\n2\u0001\n\u000016\n4\u0001;0!T\n=\u001215\n2;9\n2;7\n2;5\n2;3\n2;0\u0013T\n:\nBy considering only the matrix elements of A†\n1A1\nA†\n1A1, the third row of Ais equal to\nr3= \n0;2\u000012\n1\u0001\u00004\n1\u0001\n2\u000016\n2\u0001;\u00008\n1\u0001\u00008\n1\u0001\n\u000016\n2\u0001;2\u00008\n1\u0001\u00004\n1\u0001\n(14)+\u00004\n1\u0001\u00004\n1\u0001\n(14)\n3\u000016\n3\u0001 ;\u00004\n1\u0001\u00004\n1\u0001\u000014\n2\u0001\n\u000016\n4\u0001;1!T\n=\u0012\n0;2\n5;8\n15;2\n3;4\n5;1\u0013T\n:\nAs one can see, the first, second row and the third row are linearly dependent because\u000016\n1\u0001\nr2+\u000016\n2\u0001\nr3=\u000016\n2\u0001\n116as implied by Lemma 4.\nNext we proceed to evaluate the fourth, fifth and sixth rows of A. By considering only the matrix\nelements of A†\n3A3, the fourth row of Ais equal to\nr4= \u000016\n3\u0001\n16;15\u000012\n3\u0001\n+15\u00004\n3\u0001\n2\u000016\n2\u0001 ;15\u00008\n3\u0001\n\u000016\n3\u0001;\u000015\n2\u0001\u00008\n3\u0001\n+2\u000015\n2\u0001\u00004\n3\u0001\n3\u000016\n3\u0001 ;\u000015\n3\u0001\u00004\n3\u0001\n\u000016\n4\u0001;0!T\n= (35;14;7;4;1;0)T\nBy considering only the matrix elements of A†\n2A2\nA†\n1A1, the fifth row of Ais equal to\nr5=0\n@0;\u000012\n2\u0001\u00004\n1\u0001\n+\u000012\n1\u0001\u00004\n2\u0001\n2\u000016\n2\u0001 ;\u00008\n2\u0001\u00008\n1\u0001\n\u000016\n2\u0001;\u0010\u00008\n2\u0001\u00004\n1\u0001\n+\u00008\n1\u0001\u00004\n2\u0001\u0011\n(14)+\u00004\n2\u0001\u00004\n1\u0001\n(14)\n3\u000016\n3\u0001 ;\u00004\n2\u0001\u00004\n1\u0001\u000014\n2\u0001\n\u000016\n4\u0001;01\nAT\n=\u0012\n0;7\n5;28\n15;23\n15;6\n5;0\u0013T\n:\nBy considering only the matrix elements of\u0010\nA†\n1A1\u0011\n3\n, the sixth row of Ais equal to\nr6= \n0;0;0;3\u00008\n1\u0001\u00004\n1\u0001\u00004\n1\u0001\n3\u000016\n3\u0001;\u00004\n1\u00013\u000013\n1\u0001\n\u000016\n4\u0001;1!T\n=\u0012\n0;0;0;8\n35;16\n35;1\u0013T\n:\nClearly, we have\u000016\n1\u0001\nr4+2\u000016\n2\u0001\nr5+\u000016\n3\u0001\nr6=\u000016\n3\u0001\n1T\n16, and hence the first, fourth, fifth and sixth rows are\nlinearly dependent. We now get\nA=0\nBBBBBB@1 1 1 1 1 1\n15\n29\n27\n25\n23\n20\n02\n58\n152\n34\n51\n35 14 7 4 1 0\n07\n528\n1523\n156\n50\n0 0 08\n3516\n3511\nCCCCCCA: (5.23)\nThe matrix rank of Ais 3, and the null space of Ais spanned by\n0\nBBBBBB@\u000017=12\n115=24\n0\n\u000035=8\n0\n11\nCCCCCCA;0\nBBBBBB@\u00001=3\n4=3\n0\n\u00002\n1\n01\nCCCCCCA;0\nBBBBBB@1=3\n\u00004=3\n1\n0\n0\n01\nCCCCCCA: (5.24)\nThus we trivially have for instance A(1=3;\u00004=3;1;0;0;0) =0. From this we can derive from (4.3) the\nquantum code spanned by\nj0Li=r\n1\n4j^((16)j015)i+r\n3\n4j^((8;4;4)j013)i; (5.25)\nj1Li=j^((12;4)j014)i: (5.26)\n15t N (t+1)2\n1 3 4\n2 6 9\n3 12 16\n4 20 25\n5 30 36\n6 49\u000349\n7 72\u000364\n8 90\u000381\n9 120\u0003100\n10 143\u0003121\nTable 1: Table of code parameters. The first column are values for t, the number of AD errors the quantum\ncode can correct. The second column, Nis the total excitation number for our constant-excitation quantum\ncode that corrects tAD errors and is permutation-invariant. The third column is (t+1)2, which is the\ntotal excitation number of Bergmann and van Loock’s codes [24], which are not permutation-invariant. The\nnumbers for Nmarked with an asterisk are obtained from (1.2) and are likely not to be smallest possible.\nOn the other hand, the numbers for Nmarked without an asterisk have their codes given explicitly in the\nexamples we provided.\nIn Table 5, we present parameters constant-excitation permutation-invariant quantum codes that can\nbe constructed using our methodology for t=1;:::; 10.\n6 Discussions\nIn this paper, we study codes that lie within the decoherence-free subspace of certain Hamiltonians,\nwhile also exhibiting the ability to reverse the effects of some amplitude damping errors. We focus on\nthe Hamiltonian that is a sum of quantum harmonic oscillators of identical frequencies, because it can\ndescribe the quantum bus used by superconducting qubits. Since permutations may unexpectedly occur,\nit is also advantageous for quantum codes to also exhibit permutation-invariance. Here in this paper, we\npresent a method where vectors from the nullspace of a matrix Acan give rise to constant-excitation PI\ncodes that correct AD errors, which are naturally immune to the natural dynamics of both the quantum\nharmonic oscillators and also arbitrary permutations.\nTo label the bases on which our codes are constructed on, we have used the all ones vector along\nwith the set Q=P(w). One might wonder if one could construct quantum codes using our method with\nQa strict subset of P(w). This is useful if certain Dicke states are unphysical to implement for example.\nOne can see from Example 6 that if the initial Qis chosen as a suitable strict subset of P(w), the derived\nquantum code would be exactly the same. In particular, we supply Example 6 to illustrate the fact that\nby choosing Nto be larger than the minimum required to correct terrors, we can have some flexibility\nin choosing which Dicke states our code is to be supported on.\nThe codes considered here are non-degenerate. The effect of different AD errors on the permutation-\ninvariant code are distinct. To see how this happens explicitly, let us consider the example of Wasilewski\nand Banaszek’s excitation 3 code. Consider the AD errors A(1;0;0)andA(0;0;1). Notice that\nA(1;0;0)j0Li=1p\n3p\n3j(2;0;0)iq\ng(1\u0000g)2 (6.1)\nA(0;0;1)j0Li=1p\n3p\n3j(0;0;2)iq\ng(1\u0000g)2: (6.2)\nOne can see that the effect of A(1;0;0)andA(0;0;1)on the logical zero codeword is not the same because\n(6.1) is not equal to (6.2).\n160 20 40 60 80 100 120 140\nnumber of correctible errors, t0.70.80.91.01.11.21.3N/NNOON\nexcitation ratioFigure 1: We denote as NNOON = (t+1)2andNas the number of excitations needed to correct tAD errors\nusing the NOON codes of Bergmann and van Loock codes [24] and our permutation-invariant constant-\nexcitation codes respectively. The first 5 data points corresponding to our explicit code constructions out-\nperform the NOON codes, and the remaining data points suggest that N\u00141:3(t+1)2.\n17Regarding the literature on constant-excitation quantum codes that are not necessarily PI, we have\nalso improved on their construction when 2,3,4 and 5 AD errors are to be corrected. Namely, our codes\nrequire only 6,12,20 and 30 total excitations to correct 2,3,4 and 5 AD errors respectively. In contrast,\nChuang, Leung and Yamamoto constructed codes correcting 2 and 3 AD errors using 9 and 16 total\nexcitations [12], while Bergmann and van Loock constructed codes correcting tAD errors using (t+1)2\ntotal excitations [24]. We construct the best constant-excitation quantum codes that can correct between\n2 to 5 AD errors, as the number of excitations needed to correct terrors for t=2;3;4;5 ist(t+1),\nwhich is less than the (t+1)2total excitations previously needed. Explicit code constructions using\nour method are supplied in Example 2 and Example 3 to correct 2 and 3 AD errors using 6 and 12\ntotal excitations respectively. Our construction can also be seen as a generalization of Wasilewski and\nBanaszek’s constant-excitation PI code correcting 1 AD error [23] to PI constant-excitation quantum\ncodes that can correct an arbitrary number of AD errors. Asymptotically, it also appears that the number\nof excitations needed to correct tAD errors for our codes exhibits the same behavior as Bergmann and\nvan Loock’s codes, because both require O(t2)excitations (see Figure 1).\nBosonic codes with constant excitation number were previously c+onsidered by the authors in [12],\nwhere they established some fundamental properties of such codes. While the quantum codes considered\nhere are indeed a subfamily of the quantum codes considered in [12], they do not follow trivially from\n[12] for the following reasons. First, in [12], quantum codes satisfying the non-deformation Knill-\nLaflamme quantum error correction criterion and certain orthogonality conditions are constructed, with\nexplicit construction algorithms for 1 and 2 AD errors. However for at least 3 AD errors, one could only\nrely on brute force search as there was no systematic way to generate such constant-excitation quantum\ncodes. In contrast, we see in this paper how constant excitation quantum codes for any number of\nAD errors can be generated. Second, to numerically find codes in [12], the authors check if a system of\nlinear equations corresponding to the non-deformation conditions is satisfied. In the language used in this\npaper, they essentially determine the nullity of the matrix A, but it was unclear how one could construct\nthe quantum code using this information. We fill this research gap in this paper by demonstrating how\nthe code can be explicitly constructed from the nullspace of the matrix A, therefore enriching the theory\nof this nascent field. Third, in [12, Eq. (5.3)], the authors give an inequality that is asymptotically\nequivalent to ours in (1.2), which tells us when constant excitation codes exist. However, since arbitrary\nconstant excitation codes were considered, it is a priori unclear if the inequality [12, Eq. (5.3)] holds\nwhen the additional constraint of permutation-invariance on the codes is imposed. Here, we show that\nthis is in fact possible, and slightly improve their inequality. While their inequality states that constant\nexcitation codes exist when\np\u0012N\nt+1\u0013\n\u0015(p(0)+p(1)+:::p(t))+2; (6.3)\nwe say that permutation-invariant constant excitation codes exist when\np\u0012N\nt+1\u0013\n+\u0012t\n2\u0013\n\u0015(p(1)+:::p(t)): (6.4)\nJust like in the case of [12], this analytical bound is not tight, as we demonstrate for small values of t.\nOne limitation of this paper is that our constructed PI constant-excitation quantum codes only correct\nagainst AD errors. It would be advantageous to study when our codes can also correct against a fixed\nnumber of arbitrary errors. Another limitation is that we have only provided a theoretical structure of\nour PI constant-excitation quantum codes; the practicality of implementing our codes has yet to be fully\naddressed. We expect that techniques in preparation of Dicke states and quantum cellular automata to\nbe useful with regards to this issue. However, this lies beyond the scope of the current paper, where\nour focus lies primarily only on the mathematical structure of PI constant-excitation quantum codes that\ncorrect AD errors.\nIn summary, we prove the existence of constant-excitation quantum codes that not only correct any\nnumber of AD errors, but are invariant under any permutations. When certain distance criterion are\nsatisfied, we also provide a new method for obtaining quantum codes by finding vectors that lie within\nthe nullspace of a matrix.\n187 Acknowledgments\nThe author thanks the referees for valuable comments. The author acknowledges support from the Na-\ntional Research Foundation and Ministry of Education, Singapore. This material is based on research\nfunded in part by the Singapore National Research Foundation under NRF Award NRF-NRFF2013-01\nand the U.S. Air Force Office of Scientific Research under AOARD grant FA2386-18-1-4003. This work\nwas supported by the EPSRC (Grant No. EP/M024261/1). This work was also supported by the QCDA\nproject (Grant No. EP/R043825/1)) which has received funding from the QuantERA ERANET Cofund\nin Quantum Technologies implemented within the European Unions Horizon 2020 Programme.\nReferences\n[1] P. W. Shor, “Algorithms for quantum computation: Discrete logarithms and factoring,” in Foun-\ndations of Computer Science, 1994 Proceedings., 35th Annual Symposium on , pp. 124–134, Ieee,\n1994.\n[2] P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quan-\ntum computer,” SIAM review , vol. 41, no. 2, pp. 303–332, 1999.\n[3] S. Lloyd, “Universal quantum simulators,” Science , vol. 273, pp. 1073–1078, 1996.\n[4] C. H. 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A , vol. 90, no. 6, p. 062317, 2014.\n[28] Y . Ouyang, “Permutation-invariant qudit codes from polynomials,” Linear Algebra and its Appli-\ncations , vol. 532, pp. 43 – 59, 2017.\n[29] D. 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H ere, we investigated the in -plane angular - \nand frequenc y- dependen ce of magnetic relaxation of epitaxial Fe /MgO(001)  films  \nwith different capping layers of Pd and Cu. After disentangl ing the parasitic \ncontributions, such as two -magnon scattering  (TMS) , mosaicity,  and field-dragging \neffect, we unambiguously observed  that both local and non- local Gilbert damping are  \nisotropic  in Fe(001) plane , suggest ing that the pure spin current s absorption is \nindependent of Fe magnetization orientation  in the epitaxial  Pd/Fe heterostructure.  \nFirst principles calculation reveal s that the effective spin mixing conductance of Pd/Fe interface is nearly invariant for different magnetization directions in good \nagreement with the experimental observation s. These results offer a valuable insight \ninto the transmission and absorption of pure spin currents, and facilitate us to utilize  \nnext-generation spintronic devices.  \nPACS  number s: 72.25.Mk, 75.78.- n, 76.50.+g \n*Corresponding author  \nE-mail: zhcheng@iphy.ac.cn   \n \n  I. INTRODUCTION \nThe rapid development of spintronic devices inquires deeper understanding of \nthe magnetization relaxation mechanism1-3. The Gilbert damping factor, one of key \nparameter s in spin dynamics , characterizes the energy  transfer from the spin \nsubsystem to the lattice  and governs the magnetization  switching time and the critical \ncurrent density in spin transfer torque  devices4-6. Since the shape of Fermi surface \ndepends  on the orientation of the mag netization direction due to the spin- orbit \ninteraction, an anisotropic Gilbert damping  is expected  in single crystal ultrathin \nfilms7-10. Chen et al. discove red an anisotropic damping in the Fe/GaAs(001) ultrathin \nfilms where an robust interfacial spin -orbit field exists , due to GaAs substrate . The \nmagnitude of damping anisotropy , however, decreases with increasing Fe thickness , \nand disappears when the Fe thickness is larger than 1.9 nm11-13. \nBesides intrinsic Gilbert damping in ferromagnetic materials  (FM) , spin current s \nsink into heavy metal s (HM) or other magnetic layer s importing non -local Gilbert \ndamping in HM/FM bilayer s or spin valve structure  according to  spin pumping  \nmodel14-16. Although anisotropic magnetization relaxation in ferromagnetic \nmultilayers w as observed, it is debated whether the absorption of pure spin currents is \nanisotropic or isotropic  in ferromagnetic multilay ers17-21. This is because the \nfrequency - and angular -dependent ferromagnetic resonance (FMR) linewidth results \nare often  contaminated by parasitic contributions , such as two -magnon  scattering  \n(TMS), mosaicity, and field -dragging effect . Li et al. found that nearly isotropic \nabsorption of pure spin current in Co in Py1-xCux/Cu(5  nm)/Co(5 nm) trilayers using spin pumping technique22. Meanwhile, Baker et al. found an anisotropic absorption of \npure spin currents in Co 50Fe50/Cr/Ni 81Fe19 spin valves with variable Cr thickness, \nwhile the anisotropy is suppressed above the spin diffusion length23. Here, we \ninvestigated spin pumping and clarified the dependence of diverse  magnetic \nrelaxations on Fe  magnetic orientation using Vector Network Analyzer  ferromagnetic \nresonance  (VNA- FMR)  of epitaxial Fe /MgO(001)  films capped by Pd and Cu layers. \nSimple  FM/HM bilayers  would be a more convincing  candidate  to explore  the \nnon-local relaxation mechanism.  Exclu ding the misleading dragging effect and the \ndeceitful extrinsic terms,  we unambiguously observed that both local and non- local \nGilbert damping are isotropic  in Fe(001) plane . The i sotropic non- local Gilbert \ndamping suggest s that the pure spin current s abso rption is independent of Fe \nmagnetization orientation , which is supported by the first principle s calculation.  \nII. EXPERIMENTS  \nSample s were prepared in molecular beam epitaxy  chambers  with a basic \npressure-102 10× mbar24. Prior to deposition, MgO(001) substrate was annealed at 700 ℃  \nfor 2 hours, and then 6 nm Fe film was deposited on a MgO(001) substrate using \nelectron -beam gun , and finally 5 nm Pd w as covered on Fe films . The crystalline \nquality and epitaxial relationship was confirmed by high- resolution transmissio n \nelectron micro scopy (HRTEM), as shown in Fig . 1(a) and (b). It has been revealed \nthat the films were grown with the epitaxial relationship Pd(001)<110>||Fe(001)<100>||MgO(001)<110> (see the inset of Fig . 1(b)). For \ncomparison, Cu(3.5 nm)/Fe (6 nm)/MgO(001)  sample  was also  prepared. In-plane VNA- FMR measurements were performed by facing the sample down on employing \na co-planar waveguide (CPW) and recording the transmission coefficient S 2125-27. All \ndepositions and measurements were performed at room temperature.  \nIII. RESULTS AND DISCUSSI ON \nFig. 2(a) s hows schematically the  stacked sample and the measured \nconfigurat ion.  The representative FMR spectra at fixed frequency 13.4 GHz and \nvarious magnetic field angle s Hϕ are illustrated in Fig. 2(b). T he FMR signal  (the \ntransmission parameter S 21) is a superposition of symmetric and antisymmetric \nLorentzian  functions . The following  equation could be used to extract  the resonance \nfield H r and the resonance line width  H∆:  \n2\n21 0 22 22( / 2)( ) ( / 2)Re ( ) +( ) ( / 2) ( ) ( / 2)r\nrrH HH HSH SL DHH H HH H∆ − ∆= −− +∆ − +∆.            (1)  \nHere, Re S21, S0, H, L and D are the real part of transmission parameter,  the offset, the \nexternal  magnetic  field, the symmetric and antisymmetric magnitude , respectively25-27\n.  \nThe resonance frequency f  is given by Kittel formula28 \n0=2RR\nab f HHγµ\nπ                                                    (2)  \nwith2\n42 cos( ) (3 cos 4 ) / 4 sin ( 45 )R\nar MH M M d H H HH H ϕϕ ϕ ϕ = −++ + − −a, \n42 cos( ) cos 4 sin 2R\nb r MH M M HH H H ϕϕ ϕ ϕ= −+ − and \n02=out\nds\nsKHMMµ− . Here, γ and \n0µ are the gyromagnetic ratio and the vacuum permeability. H , H2, H4 and Ms are the \napplied  magnetic field, the uniaxial and  four-fold magnetic anisotropy field s and \nsaturation magnetization , respectively.  outK is the out -of-plane uniaxial magnetic \nanisotropy  constant. The equilibrium azimuthal angle of magnetization  Mϕis determined by the following equation:  \n42 sin( ) ( / 4)sin 4 ( / 2)cos 2 0r MH M M H HHϕϕ ϕ ϕ −+ + = .                    (3)  \nThe angular dependent FMR measurements were performed by rotating the \nsamples in plane while sweeping the applied magnetic field.  At a fixed frequen cy of \n13.4 GHz, the angular dependence of H r can be derived from Eq. (2) and plotted in \nFig. 2(c) and 2(d) for Fe/MgO(001) sample s capped by  Pd and Cu, respectively . It can \nbe seen clearly that the angular dependence of H r demonstrates a four -fold symmetry  \nand the values of 2=0H Oe, 4=625H Oe and 0 2.0dHµ=  T for Pd/ Fe/MgO(001)  \nand2=0H Oe, 4=625H Oe and 0 1.9dHµ= T for Cu /Fe/MgO(001) , respectivel y. \nCompar ing to the sample with Cu c apping l ayer, Pd/Fe interface modifies the \nout-of-plane uniaxial magnetic anisotropy, and has a negligible contribution to the \nin-plane  uniaxial magnetic anisotropy.  \nIn cont rast to the four -fold symmetry of H r, the angul ar dependence  of H∆for \nthe samples with Pd and Cu capping layers indicates  to be superposition of four-fold \nand quasi -eight -fold contributions , as shown in Fig. 3(a) and 3(b) , respectively . In fact, \nthe quasi -eight -fold broadening  also represent s a four-fold symmetry  with multiple \nextreme value point s. In the case of the sample with Pd capping layer, H∆exhibits \ntwo peaks around the  hard magnetization direction s Fe<1 10>, and the values of H∆ \nfor Fe<100>  and Fe<110 > direction s are almost the same (58 Oe). On the other hand, \na larger difference in the magnitude of  H∆ was observed along these two directions \nof Cu/Fe/MgO(001) sample , i.e. 71 Oe and 4 9 Oe for Fe<100> and Fe<110>  axes, \nrespectively . In order to understand the mechanism of anisotropic magnetic relaxation, we \nmust take both intrinsic and extrinsic contributions into account29-34. H∆ is follow ed \nby the expression32\n: \n_ =mosaicity TMS Gilbert dragging HH H H∆ ∆ +∆ +∆ .                              ( 4) \nThe first term denotes TMS, represent ing that a uniform prerecession magnon ( 0k=) \nis scattered into a degenerate magno n ( 0k≠) due to imperfect  crystal structure.  \nTherefore, the contribution of TMS to the linewidth reli es heavily on the symmetrical \ndistribution  of defects and manifest s anisotropic feature accordingly . The second  term \ndescribes the mo saicity contribution  in a film plane,  which is caused by a slightly  \nspread  of magnetic  parameters on a very large scale. The last term _ Gilbert draggingH∆ is \nthe Gilbert damping contribution with field -dragging . \nIn the case of Fe/ MgO( 001) epitaxial  film, the contribution of TMS to FMR \nlinewidth composes of numerous  two-fold and four -fold TMS channel s31-34, \nj,max 4 j,max j,max 2 j,maxcos ( ) cos 2( )TMS twofold M twofold fourfold M fourfold\njjH ϕϕ ϕϕ ∆ =Γ − +Γ − ∑∑ .          ( 5) \nHere, j,max\ntwofoldϕ  and j,max\nfourfoldϕ  represent angle of the maximum scattering rate  in \ntwo-fold and four -fold scatterings along the direction  j. However, the same values of \nH∆between Fe<100>  and Fe<110>  directions suggest that the TMS can be neglected \nin Pd/Fe/MgO(001 ) epitaxial film. On the other hand, the larger difference in the \nmagnitude of H∆ was observed along these two directions , suggesting  that either \nsignificant TMS contribution or anisotropic Gilbert damping exists in \nCu/Fe/MgO(001) s ample13, 32, 33. \nThe angular dependence of mosaicity contribution can be described as32, 34 =r\nmosaicity H\nHHH ϕϕ∂∆∆∂,                                                ( 6) \nwhere Hϕ∆  represents an in  plane variation  of mosaicity. 0mosaicityH∆=  Oe should \nbe hold along easy magnetization direction s and hard directions where =0r\nHH\nϕ∂\n∂. \nDue to magnetocrystalline anisotropy, magn etization would not always align at  \nthe direction of the applied field when the field  is weaker  than the saturation field. We \nevaluate the field -dragging effect during rotation of the sample or frequency -swept  \nbased on the numerical calculation  using Eq. ( 3). Fig. 4(a) shows  Hϕdependence on \nHϕ at 13.4 GHz. The relation  reveals a conspicuous  dragging effect with a  four-fold \nsymmetry. At 25Hϕ=a, HMϕϕ−  is as high as  12a. Fig. 4(b) sh ows Mϕ \ndependence on f  at various  Hϕ. When  the magnetic  field is applied along Fe<100> or \nFe<110>  directions , the magnetization is always aligned along the applied magnetic \nfield. However, there is a conspicuous  angle between  the magnetization and the \nmagnetic  field with the field along intermediate  axis. Owing to the angle between \nmagnetization and applied field , H∆ corresponding  to Gilbert contributio n with the \nfield-dragging could be disclose d according  to the following equations12, 13 \n_ = [Im( )]Gilbert draggingH χ ∆∆                                              ( 7) \nand 22 2[]Im( )( ) ()RR RR\ne f f ab a a ab s\nRR RR\na b ab e f f ab a bHH H H HH M\nH H HH HH H+ Haχa+=−+ ,                   (8) \nwhere aH and bH are R\naH  and R\nbH in non- resonance condition. The effective \nparameter  effa consist s of the intrinsic  Gilbert damping an d the non- local one driven \nby spin currents . \nGenerally , effa was obtained by the slope of the linear  dependence of H∆ on frequency f  along the directions without field -dragging  28: \n0\n04efffHHπa\nµγ∆ = +∆ ,                                                 (9)  \nwhere 0H∆ is inhomogeneous non- Gilbert linewidth at zero -frequency25-27. Fig. 5 \nshows  H∆  dependence on frequency at various Hϕ. Obviously , H∆ versus  f can \nbe fitted linear ly with  3\n/ 6.0 10Pd Fea−= ×  and 3\n/=4.2 10Cu Fea−×  for magnetic field \nalong easy axes Fe<1 00> or hard axes Fe<1 10> of the samples with Pd and Cu \ncapping layers,  respectively , indicating isotropic damping  (Fig. 5(f) and 5( j)). By \nusing the aforementioned isotropic damping factor s, the contributions  of TMS, \nmosaicity, and field -dragging effect  are separated from the angular  dependence of \nH∆ (Fig. 3 a-b). Table I summar izes the fitted parameters in the two  samples. \nCompared with Cu/Fe  sample , one observes  a significant reduction of mosaicity \nbroadening and a negligible  TMS term in Pd/Fe bilayers. In fact, due to high mobilit y, \nthe capping layer Cu forms nanocrystallites on Fe film, which causes interfacial \ndefects dependence on the crystallographic ax es35-38. The interfacial defects will \nimpact  a four-fold linewidth broadening  due to TM S. In contrast , the  excellent \nepitaxial quality  at Pd/Fe  interface not only ensures a sharp interfacial structure , but \nalso reduces defect density to decrease TMS contribution. Moreover, the mosaicity \ncontribution, indicat ing the fluctuation of the magne tic anisotropy  field, could be \nstrengthen  by the  interfacial  stacking faults. C onsequently , a fully epitaxial structure \ncould significantly decrease the extrinsic contributions, especially TMS and mosaicity  \nterms . \nTaking these contributions  to magnetizatio n relaxation into account, the frequency dependence of H∆ at various directions  can be well reproduced, as shown \nin Fig. 5(f)- (j). For other directions  rather than Fe<1 00> and Fe<1 10>, nonlinear \nrelationship between H∆ and f are evident  and illustrated in Fig . 5(g-i). At =20Hϕa, \nthe H∆ vs f curve brings out a slight bump comparing  to the linear  ones along hard \nor easy ax es. At =27Hϕa, H∆ has a rapid  decrease after H∆ experiencing an \nabrupt enhancement . At =33Hϕa, H∆ decreases more sharply after 11 GHz.  The \nnonlinearity can be ascrib ed to the parasitic contrib utions, such as  TMS, mosaicity, \nand field -dragging effect . It is virtually impossible  to stem from  TMS for the d istorted  \ncurves  because a nonlinear linewidth broadening due to TMS increases as frequency \nincreases, and approach es to saturation  at high frequency31. According to  the \ncalculation  in Fig. 4(b), there is a huge field -dragging effect except the applied \nmagnetic field H  along hard and  easy ax es. The field -dragging will make  H∆ vs f \ndeviate from the linear relationship . As expected , we could effectively fit  the \nexperimental data H∆ vs f using the following equation in association with the \noriginal formula s (7), \n0 [Im( )] HH χ ∆ =∆ +∆ .                                               ( 10) \nEq. (10) converges to the Eq. (9)  with the applied magnetic field along the directions \nwithout field -dragging, i.e. easy  axes Fe<1 00> or hard axes Fe<1 10>13. \nAfter distinguish ing the contributions  of extrinsic terms and field -dragging effect, \nthe Gilbert damping factors effa along various direc tions are  show n in Fig. 6(a). \nAccording to the classical spin pumping model14, precess ional magnetization in FM  \nlayer will pump spins into adjacent  nonmagnetic  metals  across interface. Cu with only s conduction band has a smaller spin- flip probability and a large r spin diffusion length \nthan 500 nm39, therefore, the reference sample Cu/Fe cannot  increase the Gilbert \ndamping due to a capp ing layer Cu. In contrast , Pd-layer  with stron g spin- orbit \ncoupling has a larger spin- flip probability , the injected spin currents are dissipated in \nPd-layer , and enhance the intrinsic Gilbert damping  of Fe film. The enhancement of \nthe Gilbert damping allows us to comprehend the non- local relaxation m echanism.  \nObviously, it can be seen from Fig. 6(a)  that there is no strong relation between the \nnon-local  Gilbert damping and the magnetization orientation in epitaxial film Pd/Fe.  \nThe parameters-3\n/=4.2 10Cu Fea ×  and -3\n/=6.0 10Pd Fea ×  are the Gilbert damping of \nPd/Fe and Cu/Fe , respectively. The non-local Gilbert damping could be evaluated \nusing  the effective spin mixing conductance effg↑↓14 \n// =4B\nPd Fe Cu Fe eff\ns FeggMtµaa aπ↑↓∆ −= .                                     ( 11) \nThe obtained isotropic value  19 2=1.23 10effgm↑↓ −×  is comparable to the literature s40-42.  \nIn order to theoretically  investigate  the dependence of the non- local Gilbert \ndamping  on the magnetization orientation, the first principles calculation was \nperformed  to calculate the  total Gilbert damping of the Pd /Fe/Pd multilayer  on the \nbasis of the scattering theory43-45. The electronic structure of the Pd/Fe  interface was \ncalculated self -consistently using the surface Green’s function technique implemented \nwith the tight- binding linearized muffin -tin orbitals method. Within the atomic sphere \napproximation, the charge and spin densities and the effective Kohn- Sham potentials \nwere evaluat ed inside atomic spheres. The total Gilbert damping was  then calculated \nusing the scattering theory of magnetization dissipation45. We simulate d the room tempe rature via introducing frozen thermal lattice disorder  into a 5x5 lateral \nsupercell43. The root -mean -squared displacement of the atoms is determined by the \nDebye model with the Debye temperature 470 K.  A 28x 28 k-mesh is used to sample \nthe two -dimensional Brillouin zone and five different configurations of disorder have \nbeen calculated for each Fe thickness.  The total Gilbert damping exhibits a linear \ndependence on the length of Fe and the intercept of the linear function can be \nextracted corresponding to the contribution of the spin pumping at the Pd/Fe  \ninterface44. The interfacial  contribution i s converted to the effective spin mixing \nconductance, plotted in Fig. 6( b) as a function of the magnetization orientation. It can \nbe seen that the effective spin mixing conductance  across  Pd/Fe  interface \n19 2=1.29 10effgm↑↓ −×  is independent  of the magnet ization direction , and  is in very \ngood agreement with the experimental value19 21.23 10 m−× . According to the \nElliott- Yafet mechanism  in a nonmagnetic metal , spins  relax indiscriminately energy \nand momentum along all orientation in Pd-layer  since a cubic metal  is expected to \npossess a weak anisotropy of the Elliott -Yafet parameter46. Incidentally , the fitting \nerror will  mislead  an aniso tropic Gilbert damping if ones  use Eq. ( 9) to fit the entire  \ncurves  H∆ vs f. Besides , an epitaxial  magnetic film integrated into a pseudo spin \nvalve could lead to an anisotropic absorption of spin current based on s pin transfer \ntorque mechanism  since it is demanding  to drag magnetization parallel ing to the \napplied field11. \n  I V.  CONCLUSIONS  \nIn summary, a  non-local  Gilbert damping is induced by the spin pumping  in \nPd/Fe bilayers  as spin currents transfer  angular momentum into the Pd- layer . Due to \nstrong magnetocrystalline anisotropy, the field -dragging effect makes the line width \nversus frequency deviate from the linear relationship except magnetic field along hard \nor easy ax es. Extrinsic relaxation , such as TMS  and mosaicity, relies heavil y on \nmagnetization orientation.  Howeve r, an epitaxial interface could significantly \ndecrease  and minimize  the extrinsic contributions, especially TMS and mosaicity. It is \nnoteworthy that an isotropic non- local Gilb ert damping factor is clarified after ruling \nout the misleading field-dragging effect and the deceitful extrinsic contributions. \nMagnetization  orientation has a negligible contribution to the non- local Gilbert \ndamping based on both theoretical and experimental results , manifesting  that the \nabsorption of pure spin currents across  interface Pd(100)[110]/Fe(001)[100]  is \nindependent of Fe magnetization orientation. Our works provide deeper i nsight  into \nthe non- local Gilbert damping mechanism. \n  ACKNOWLEDGMENTS  \nThis work is supported by the National Key Research Program of China (Grant Nos.  \n2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos.  51427801,1187411 ,51671212, and \n11504413) and the Key Research Program of Frontier Sciences, CAS (Grant Nos. \nQYZDJ -SSW -JSC023, KJZD -SW-M01 and ZDYZ2012- 2). The work at Beijing \nNormal University is partly supported by the National Natural Sciences Foundation of \nChina (Grant Nos. 61774017, 61704018, and 11734004), the R ecruitment Program of \nGlobal Youth Experts and the Fundamental Research Funds for the Central Universities (Grant No. 2018EYT03).  REFERENCES  \n1. I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys. 76 (2), 323 -410 (2004).  \n2. K. Ando, S . Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa and E. Saitoh, Phys. Rev. Lett. 101  \n(3), 036601 (2008).  \n3. J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back and T. Jungwirth, Rev. Mod. Phys. 87 (4), \n1213- 1260 (2015).  \n4. V. Kambersky, Czech. J. P hys. 26 (12), 1366- 1383 (1976).  \n5. A. B. Cahaya, A. O. Leon and G. E. W. Bauer, Phys. Rev. B 96 (14) (2017).  \n6. D. Thonig, Y . Kvashnin, O. Eriksson and M. Pereiro, Phys. Rev. Materials 2  (1) (2018).  \n7. D. Steiauf and M. Fähnle, Phys. Rev. B 72 (6) (2005).  \n8. J. Seib, D. Steiauf and M. Fähnle, Phys. 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Rog alev and W. E. Bailey, Phys. \nRev. B 94 (1) (2016).  \n42. J. M. Shaw, H. T. Nembach and T. J. Silva, Phys. Rev. B 85 (5) (2012).  \n43. Y . Liu, A. A. Starikov, Z. Yuan and P . J. Kelly, Phys. Rev. B 84 (1) (2011).  \n44. Y . Liu, Z. Yuan, R. J. Wesselink, A. A. Stari kov and P . J. Kelly, Phys. Rev. Lett. 113 (20), 207202 \n(2014).  \n45. A. A. Starikov, Y . Liu, Z. Yuan and P . J. Kelly, Phys. Rev. B 97 (21) (2018).  \n46. B. Zimmermann, P . Mavropoulos, S. Heers, N. H. Long, S. Blugel and Y . Mokrousov, Phys. Rev. Lett. \n109 (23),  236603 (2012).  \n  FIGURE CAPTIONS  \nFig. 1 ( Color online)  (a) Dark field scanning high -resolution transmission electron \nmicroscopy image and ( b) selected area electron diffraction pattern  of \nPd/Fe/MgO(001). The inset of Fig. 1(b) shows a schematic of the ep itaxial \nrelationship . \nFig. 2 ( Color online)  (a) A schematic illustration of the stacked sample \nPd/Fe/MgO(001). The sample is placed on the CPW for FMR measurement, and \ncould be rotated in plane . (b) Typical  real FMR spectra of Pd/Fe at fixed frequency \n13.4 GHz at various magnetic field angle sHϕ. Magnetic field angle Hϕ dependen ce \nof the resonanc e field H r at a fixed frequency 13.4 GHz  for Pd/Fe (c) and Cu /Fe (d) . \nThe red curves  are fit to Kittel’s formula  (2). (In order to show clearly the tendency , \nwe show the data at 45 225Hϕ−≤≤aa,  the same below ) \nFig. 3 ( Color online)  The measured linewidth H∆ as a function of Hϕat 13.4 GHz \nfor Pd /Fe (a) and Cu/Fe (b). The line width H∆ is superimposed by several  terms, \nsuch as TMS, mosaicity and Gilbert contribution with field- dragging.  \nFig. 4 (Color online)  Field -dragging effect for Pd/Fe. (a) The green line denotes the \nequilibrium direction of magnetization as a function of magnetic field angleHϕ at \n13.4 GHz. T he red line indicates the misalignment between  the magnetization and the \napplied magnetic field according ly. (b) The equilibrium  direction of the magnetization \nin the frequency -swept mode at variousHϕ. \nFig. 5 ( Color online)  Frequency dependence of the resonance  field Hr (a-e) and \nfrequency  dependence of the resonance line width H∆ (f-j) for Pd/Fe at  variousHϕ. The blue solid squares  and curves  in (f) and (j) corresponding  to frequency \ndependence of H∆ at 0Hϕ=a and 45Hϕ=a for Cu/Pd. \nFig. 6 (Color online)  Angular dependent Gilbert damping and first principles \ncalculation. ( a) The opened and solid green squares represent the obtained Gilbert \ndamping for Pd/Fe and Cu/Fe films, respectively. The red and blue lines are guide to \nthe eyes. ( b) The experimental and calculated spin mixing conductance as a function \nof the orientation of the equilibrium magnetization.  \nTable I The fitted  magnetic anisotropy parameters and magnetic relaxation \nparameters in Pd /Fe and Cu/Fe films . \n \n  Fig.1  \n \n \n  \nFig. 2  \n \n  \nFig. 3  \n \n  \nFig. 4  \n \n  \nFig. 5  \n \nFig. 6  \n \n  \n \nTable I  The fitted m agnetic anisotropy parameters and magnetic relaxation \nparameters in Pd /Fe and Cu/Fe films  in Fig. 3.  \nSample   4H(Oe)   2H(Oe)  0 dHµ (T)   effa   100γ<>Γ (710Hz)   ϕ∆(deg.) \nPd/Fe      625       0        2.0     0.0060       0           0.23 \nCu/Fe      625       0        1.9     0.0042      58           1.26 \n \n  "    },    {        "title": "1810.03780v2.The_lifespan_of_solutions_of_semilinear_wave_equations_with_the_scale_invariant_damping_in_one_space_dimension.pdf",        "content": "arXiv:1810.03780v2  [math.AP]  31 May 2019The lifespan of solutions of semilinear wave\nequations with the scale-invariant damping\nin one space dimension\nMasakazu Kato∗, Hiroyuki Takamura†, Kyouhei Wakasa‡\nKeywords: semilinear wave equation, scale-invariant damping, lifespan\nMSC2010: primary 35L71, secondary 35B44\nAbstract\nThe critical constant µ(see (1.1)) of time-decaying damping in the\nscale-invariant case is recently conjectured. It also has b een expected\nthat the lifespan estimate is the same as for the associated s emilinear\nheat equations if the constant is in the “heat-like” domain. In this\npaper, we point out that this is not true if the total integral of the\nsum of initial position and speed vanishes. In such a case, we have\na new type of the lifespan estimates which is closely related to the\nnon-damped case in shifted space dimensions.\n1 Introduction\nWe consider the following initial value problem for semilinear wave equat ions\nwith the scale-invariant damping:\n/braceleftBigg\nvtt−∆v+µ\n1+tvt=|v|pinRn×[0,∞),\nv(x,0) =εf(x), vt(x,0) =εg(x), x∈Rn,(1.1)\n∗College of Liberal Arts, Mathematical Science Research Unit, Muro ran Institute\nof Technology, 27-1, Mizumoto-cho, Muroran, Hokkaido 050-858 5, Japan. email:\nmkato@mmm.muroran-it.ac.jp.\n†Mathematical Institute, Tohoku University, Aoba, Sendai 980-8 578, Japan. e-mail:\nhiroyuki.takamura.a1@tohoku.ac.jp.\n‡Department of Creative Engineering, National Institute of Techn ology, Kushiro\nCollege, 2-32-1 Otanoshike-Nishi, Kushiro-Shi, Hokkaido 084-0916 , Japan. e-mail:\nwakasa@kushiro-ct.ac.jp.\n1wherep >1,µ >0, the initial data ( f,g)∈H1(Rn)×L2(Rn) is of compact\nsupport and ε >0 is “small”. The classification of general damping terms for\nthe linear equation is introduced by Wirth [20, 21, 22]. The scale-invar iant\ncase is critical in the behavior of the solution. For the outline of semilin ear\nequations in other cases, see Introduction of Lai and Takamura [1 0].\nIt is interesting to look for the critical exponent pc(n) such that\n/braceleftbiggp > pc(n) (and may have an upper bound) = ⇒T(ε) =∞,\n1< p≤pc(n) = ⇒T(ε)<∞,\nwhereT(ε) is, the so-called lifespan, the maximal existence time of the en-\nergy solution of (1.1) with arbitrary fixed non-zero data. Then we h ave the\nfollowing conjecture:\n/braceleftbigg\nµ≥µ0(n) =⇒pc(n) =pF(n) (heat-like) ,\n0< µ < µ 0(n) =⇒pc(n) =pS(n+µ) (wave-like) ,(1.2)\nwhere\nµ0(n) :=n2+n+2\nn+2.\nMoreover\npF(n) := 1+2\nn\nis the so-called Fujita exponent which is the critical exponent of the associ-\nated semilinear heat equations vt−∆v=vpand\npS(n) :=n+1+√\nn2+10n−7\n2(n−1)(n/ne}ationslash= 1),:=∞(n= 1)\nis the so-called Strauss exponent which is the critical exponent of t he associ-\nated semilinear wave equations vtt−∆v=|v|p. We note that pS(n) (n/ne}ationslash= 1)\nis a positive root of\nγ(p,n) := 2+( n+1)p−(n−1)p2= 0.\nMoreover, 0 < µ < µ 0(n) is equivalent to pF(n)< pS(n+µ). Concerning the\nconjecture (1.2), D’Abbicco [2] has obtained heat-like existence pa rtially as\nµ≥\n\n5/3 forn= 1,\n3 forn= 2,\nn+2 for n≥3,\nwhile Wakasugi [19] has obtained blow-up for 1 < p≤pF(n) andµ≥1, or\n1< p≤pF(n+µ−1) and 0 < µ <1. We note that his result is the first\nblow-up result for super-Fujita exponents.\n2Making use of the so-called Liouville transform\nu(x,t) = (1+ t)µ/2v(x,t),\none can rewrite (1.1) as\n\n\nutt−∆u+µ(2−µ)\n4(1+t)2u=|u|p\n(1+t)µ(p−1)/2inRn×[0,∞),\nu(x,0) =εf(x), ut(x,0) =ε{µf(x)/2+g(x)}, x∈Rn.\n(1.3)\nDue to this observation, D’Abbicco, Lucente and Reissig [4] have pro ved the\nwave-like part of the conjecture (1.2) for n= 2,3 whenµ= 2. We note that\nthe radial symmetry is assumed for n= 3 in [4]. Moreover D’Abbicco and\nLucente [3] have obtained the wave-like existence part of (1.2) for oddn≥5\nwhenµ= 2 also with radial symmetry. In the case µ= 2, (1.3) is a Cauchy\nproblem for semilinear wave equations with time-dependent coefficien t on\nthe right-hand side. So, the regularity of the solution can be chose n higher,\nsometimes a classical solution is handled. For µ/ne}ationslash= 2, Lai, Takamura and\nWakasa [11] have first studied the wave-like blow-up of the conject ure (1.2)\nwith a loss replacing µbyµ/2 in the sub-critical case. Initiating this, Ikeda\nand Sobajima [5] have obtained the blow-up part of (1.2).\nForthesemilinear wave equationswithscale-invariant dampingandma ss,\nthe global existence of small data and the blow-up behavior were st udied in\n[12] and [13].\nFor the lifespan estimate, one may expect that\nT(ε)∼/braceleftbigg\nCε−(p−1)/{2−n(p−1)}for 1< p < p F(n)\nexp/parenleftbig\nCε−(p−1)/parenrightbig\nforp=pF(n)(1.4)\nfor the heat-like domain µ≥µ0(n) and\nT(ε)∼/braceleftbigg\nCε−2p(p−1)/γ(p,n+µ)for 1< p < p S(n+µ)\nexp/parenleftbig\nCε−p(p−1)/parenrightbig\nforp=pS(n+µ)(1.5)\nfor the wave-like domain 0 < µ < µ 0(n). HereT(ε)∼A(ε,C) stands for the\nfact that there are positive constants, C1andC2, independent of εsatisfying\nA(ε,C1)≤T(ε)≤A(ε,C2). Actually, (1.4) for n= 1 and µ= 2 is obtained\nby Wakasa [18], and(1.5) is obtained by Kato and Sakuraba [8] for n= 3 and\nµ= 2. Also see Lai [9] for the existence part of weaker solution. More over,\nthe upper bound of (1.4) in the sub-critical case is obtained by Waka sugi\n[19]. Also the upper bound of (1.5) is obtained by Ikeda and Sobajima [5 ] in\nthe critical case, later it is reproved by Tu and Li [17], and Tu and Li [1 6] in\nthe sub-critical case.\n3But we have the following fact. For the non-damped case, µ= 0, it is\nknown that (1.5) is true for n≥3, orp >2 andn= 2. The open part\naround this is p=pS(n) forn≥9. Other cases, (1.5) is still true if the total\nintegral of the initial speed vanishes, i.e./integraltext\nRng(x)dx= 0. On the other hand,\nwe have\nT(ε)∼\n\nCε−(p−1)/2forn= 1,\nCε−(p−1)/(3−p)forn= 2 and 1 < p <2,\nCa(ε) for n= 2 and p= 2(1.6)\nif/integraltext\nRng(x)dx/ne}ationslash= 0, where a=a(ε) is a positive number satisfying ε2a2log(1+\na) = 1. We note that (1.6) is smaller than the first line in (1.5) with µ= 0\nin each case. For all the references of the case of µ= 0, see Introduction of\nImai, Kato, Takamura and Wakasa [6].\nOur aim in this paper is to show that the lifespan estimates for (1.3)\nare similar to the ones for the non-damped case even if µis in the heat-like\ndomain by studying the special case of n= 1 and µ= 2≥µ0(1) = 4/3.\nThat is, the result on (1.4) by Wakasa [18] mentioned above is true on ly if/integraltext\nR{f(x)+g(x)}dx/ne}ationslash= 0. More precisely, we shall show that\nT(ε)∼\n\nCε−2p(p−1)/γ(p,3)for 1< p <2,\nCb(ε) for p= 2,\nCε−p(p−1)/(3−p)for 2< p <3,\nexp(Cε−p(p−1)) forp=pF(1) = 3(1.7)\nif/integraltext\nR{f(x)+g(x)}dx= 0, where b=b(ε) is a positive number satisfying\nε2blog(1+b) = 1. (1.8)\nWe note that (1.7) is bigger than (1.4) with n= 1 and µ= 2 in each\ncase. This kind of phenomenon is observed also in two space dimension s for\n1< p≤pF(2) =pS(2+2) = 2 and µ=µ0(2) = 2. Such a result will appear\nin our forthcoming paper [7].\nThis paper is organized as follows. In the next section, we place prec ise\nstatements on (1.7). Section 3, or 4, are devoted to the proof of the lower,\nor upper, bound of the lifespan respectively.\n2 Theorems and preliminaries\nWe shall show (1.7) by establishing the following two theorems.\n4Theorem 2.1 Letn= 1,µ= 2and1< p≤3 =pF(1). Assume that\n(f,g)∈C2\n0(R)×C1\n0(R)satisfies/integraltext\nR{f(x)+g(x)}dx= 0and\nsupp (f,g)⊂ {x∈R:|x| ≤k}, k >1. (2.1)\nThen, there exists a positive constant ε0=ε0(f,g,p,k)such that a classical\nsolutionu∈C2(R×[0,T))of (1.3) exists as far as\nT≤\n\ncε−2p(p−1)/γ(p,3)if1< p <2,\ncb(ε) ifp= 2,\ncε−p(p−1)/(3−p)if2< p <3,\nexp(cε−p(p−1))ifp= 3(2.2)\nfor0< ε≤ε0, wherecis a positive constant independent of εandb(ε)is\ndefined in (1.8).\nTheorem 2.2 Letn= 1,µ= 2and1< p≤3 =pF(1). Assume that\n(f,g)∈C2\n0(R)×C1\n0(R)satisfyf(x)≥0 (/ne}ationslash≡0),f(x)+g(x)≡0and (2.1).\nThen, there exists a positive constant ε1=ε1(f,g,p,k)such that a classical\nsolutionu∈C2(R×[0,T))of (1.3) cannot exist whenever Tsatisfies\nT≥\n\nCε−2p(p−1)/γ(p,3)if1< p <2,\nCb(ε) ifp= 2,\nCε−p(p−1)/(3−p)if2< p <3,\nexp(Cε−p(p−1))ifp= 3\nfor0< ε≤ε1, whereCis a positive constant independent of εandb(ε)is\ndefined in (1.8).\nAs preliminaries for the proofs of the above theorems we list some kn own\nfacts. First, u0is defined by\nu0(x,t) :=1\n2{f(x+t)+f(x−t)}+1\n2/integraldisplayx+t\nx−t{f(y)+g(y)}dy(2.3)\nwith (f,g)∈C2(R)×C1(R) satisfies\n/braceleftbigg\nu0\ntt−u0\nxx= 0 in R×[0,∞),\nu0(x,0) =f(x), u0\nt(x,0) =f(x)+g(x), x∈R.\nIf we assume (2.1) and\n/integraldisplay\nR{f(x)+g(x)}dx= 0,\n5then we have\nsuppu0⊂ {(x,t)∈R×[0,∞) :t−k≤ |x| ≤t+k}.(2.4)\nMoreover, if u∈C(R×[0,∞)) is a solution of\nu(x,t) =εu0(x,t)+L(|u|p)(x,t) for (x,t)∈R×[0,∞),(2.5)\nwhere\nL(F)(x,t) :=1\n2/integraldisplayt\n0/integraldisplayx+t−s\nx−t+sF(y,s)\n(1+s)p−1dyds (2.6)\nforF∈C(R×[0,∞)), thenu∈C2(R×[0,∞)) is the solution to the initial\nvalue problem (1.3). We also note that (2.1) implies\nsuppu⊂ {(x,t)∈R×[0,∞) :|x| ≤t+k}. (2.7)\nWe define the L∞norm ofVby\n/bardblV/bardbl0:= sup\n(x,t)∈R×[0,T]|V(x,t)|. (2.8)\nLetr=|x|. Forr,t≥0, we define the following weighted functions:\nw(r,t) :=\n\n1 if p >2,\n{logτ+(r,t)}−1ifp= 2,\nτ+(r,t)p−2if 1< p <2,(2.9)\nwhere we set\nτ+(r,t) :=t+r+2k\nk.\nFor these weighted functions, we denote a weighted L∞norm ofVby\n/bardblV/bardbl:= sup\n(x,t)∈R×[0,T]{w(|x|,t)|V(x,t)|}. (2.10)\nFinally, we shall show some useful representations for L. It is trivial that\n1+s≥(2k+s)/2kis valid for s≥0 andk >1. Setting s= (α+β)/2≥0\nwithα≥0,β≥ −k, we have\n1+s≥α+2k\n4k,or≥β+2k\n4k.\nThus, for 0 ≤θ≤1, we get\n1\n1+s≤4\n{(α+2k)/k}θ{(β+2k)/k}1−θ. (2.11)\n6LetF=F(|x|,t)∈C(R×[0,T]) and\nsuppF⊂ {(x,t)∈R×[0,T] :|x| ≤t+k}.\nFrom (2.6), we obtain\n|L(F)(x,t)| ≤1\n2/integraldisplayt\n0ds/integraldisplayr+t−s\nr−t+s|F(|y|,s)|\n(1+s)p−1dy\n=:L1(F)(r,t)+L2(F)(r,t),\nwhere\nL1(F)(r,t) :=1\n2/integraldisplayt\n0ds/integraldisplayr+t−s\n|r−t+s||F(|y|,s)|\n(1+s)p−1dy\nand\nL2(F)(r,t) :=1\n2/integraldisplay(t−r)+\n0ds/integraldisplayt−r−s\nr−t+s|F(|y|,s)|\n(1+s)p−1dy\n=/integraldisplay(t−r)+\n0ds/integraldisplayt−r−s\n0|F(|y|,s)|\n(1+s)p−1dy.\nHere we write ( a)+= max(a,0) fora∈R. Changing the variables by\nα=s+y,β=s−yand making use of (2.11), we have\nL1(F)(r,t)\n≤/integraldisplayt−r\n−kdβ/integraldisplayt+r\n|t−r|4p−2|F((α−β)/2,(α+β)/2)|\n{(α+2k)/k}θ(p−1){(β+2k)/k}(1−θ)(p−1)dα\n≤/integraldisplayt+r\n−kdβ/integraldisplayt+r\nβ4p−2|F((α−β)/2,(α+β)/2)|\n{(α+2k)/k}θ(p−1){(β+2k)/k}(1−θ)(p−1)dα.(2.12)\nSimilarly it follows from (2.11) that\nL2(F)(r,t)\n≤/integraldisplayt−r\n−kdβ/integraldisplayt−r\n|β|2−14p−1|F((α−β)/2,(α+β)/2)|\n{(α+2k)/k}θ(p−1){(β+2k)/k}(1−θ)(p−1)dα\n≤/integraldisplayt+r\n−kdβ/integraldisplayt+r\nβ2−14p−1|F((α−β)/2,(α+β)/2)|\n{(α+2k)/k}θ(p−1){(β+2k)/k}(1−θ)(p−1)dα.(2.13)\nTherefore, we obtain by (2.12) and (2.13) that\n|L(F)(x,t)|\n≤/integraldisplayt+r\n−kdβ/integraldisplayt+r\nβ4p−1|F((α−β)/2,(α+β)/2)|\n{(α+2k)/k}θ(p−1){(β+2k)/k}(1−θ)(p−1)dα.(2.14)\n73 Proof of Theorem2.1\nFirst of all, we prove an estimate for the linear part of the solution fr om\n(2.5).\nLemma 3.1 Letu0be as in(2.3). Assume that the assumptions in Theorem\n2.1 are fulfilled. Then, there exists a positive constant C0such that\n/bardblu0/bardbl0≤C0. (3.1)\nProof.It follows from (2.3) and (2.4) that\n|u0(x,t)| ≤ /bardblf/bardblL∞(R)+/bardblf+g/bardblL1(R).\nTherefore, due to (2.8), we obtain (3.1). This completes the proof .✷\nNext, we prove an a-priori estimate for a linear integral operator related\nwith the right-hand side of (1.3).\nLemma 3.2 LetLbe the linear integral operator defined by (2.6). Assume\nthatV0∈C(R×[0,T])withsuppV0⊂ {(x,t)∈R×[0,T] :t−k≤ |x| ≤\nt+k}. Then, there exists a positive constant C1independent of Tandksuch\nthat\n/bardblL(|V0|p)/bardbl ≤C1k2/bardblV0/bardblp\n0. (3.2)\nProof.We note that (3.2) follows from the following basic estimates:\n|L(χt−k≤r≤t+k)(x,t)| ≤C1k2w(r,t)−1, (3.3)\nwhereχAis a characteristic function of a set A.\nFrom now on to the end of this section, Cstands for a positive constant\nindependent of Tandk, and may change from line to line. It is easy to show\n(3.3) by (2.14) with θ= 1 and (2.9). Actually we have that\n|L(χt−k≤r≤t+k)(x,t)| ≤C/integraldisplayk\n−kdβ/integraldisplayt+r\n−kdα\n{(α+2k)/k}p−1\n≤Ck2×\n\n1 if p >2,\nlogτ+(r,t) ifp= 2,\nτ+(r,t)2−pif 1< p <2\n≤Ck2w(r,t)−1.\nThis completes the proof. ✷\nThe following lemma contains one of the most essential estimates.\n8Lemma 3.3 LetLbe the linear integral operator defined by (2.6). Assume\nthatV∈C(R×[0,T])withsuppV⊂ {(x,t)∈R×[0,T] :|x| ≤t+k}.\nThen, there exists a positive constant C2independent of Tsuch that\n/bardblL(|V|p)/bardbl ≤C2k2/bardblV/bardblpD(T), (3.4)\nwhereD(T)is defined by\nD(T) :=\n\nlogTkifp= 3,\nT3−p\nk if2< p <3,\nTklogTkifp= 2,\nTγ(p,3)/2\nk if1< p <2(3.5)\nwithTk:= (T+2k)/k.\nProof.We note that (3.4) follows from the following basic estimates:\n|L(w−p)(x,t)| ≤C2k2D(T)w(r,t)−1.\nWe divide the proof into three cases.\n(i) Case of 2 < p≤3.\nIt follows from (2.9), (2.14) with θ= 1 and (3.5) that\n|L(w−p)(x,t)| ≤C/integraldisplayt+r\n−kdβ/integraldisplayt+r\nβdα\n{(α+2k)/k}p−1\n≤Ck/integraldisplayt+r\n−k{(β+2k)/k}2−pdβ\n≤Ck2×/braceleftbigg\nlogτ+(r,t) (p= 3)\nτ+(r,t)3−p(2< p <3)\n≤Ck2D(T)w(r,t)−1.\nHere we have used by (2.7) that\nτ+(r,t)≤2t+3k\nk≤2TkandTk≥2.\nFrom now on, we will employ this estimate at the end of each case.\n(ii) Case of p= 2.\nIt follows from (2.14) with θ= 1/2, (2.9) and (3.5) that\n|L(w−p)(x,t)|\n≤C/integraldisplayt+r\n−kdβ/integraldisplayt+r\nβlog2{(α+2k)/k}\n{(α+2k)/k}1/2{(β+2k)/k}1/2dα\n≤Clog2τ+(r,t)/integraldisplayt+r\n−k/parenleftbiggβ+2k\nk/parenrightbigg−1/2\ndβ/integraldisplayt+r\n−k/parenleftbiggα+2k\nk/parenrightbigg−1/2\ndα\n≤Ck2τ+(r,t)log2τ+(r,t)\n≤Ck2D(T)w(r,t)−1.\n9(iii) Case of 1 < p <2.\nSimilarly to the above, it follows from (2.14) with θ= 1 and (2.9) that\n|L(w−p)(x,t)| ≤C/integraldisplayt+r\n−kdβ/integraldisplayt+r\n−k/parenleftbiggα+2k\nk/parenrightbigg(2−p)p−(p−1)\ndα\n≤Ck2τ+(r,t)−p2+p+3\n≤Ck2D(T)w(r,t)−1.\nThe proof is now completed. ✷\nFinally, we state an a-priori estimate of mixed type.\nLemma 3.4 LetLbe the linearintegraloperator definedby (2.6), andV,D(T)\nbe as in Lemma 3.3. Assume that V0∈C(R×[0,T])with\nsuppV0⊂ {(x,t)∈R×[0,T] :t−k≤ |x| ≤t+k}.\nThen, there exists a positive constant C3independent of Tandksuch that\n/bardblL(|V0|p−1|V|)/bardbl ≤C3k2/bardblV0/bardblp−1\n0/bardblV/bardblD(T)1/p.\nProof.Similarly to the proof of Lemma 3.2, we shall show\n|L(χt−k≤r≤t+kw−1)(x,t)| ≤C3k2w(r,t)−1D(T)1/p. (3.6)\n(i) Case of 2 < p≤3.\nSincew(r,t) = 1, (3.6) is established by the estimates for 2 < p≤3 in\nLemma 3.3 and 1 ≤D(T)1/p.\n(ii) Case of p= 2.\nIt follows from (2.14) with θ= 1 and (2.9) that\n|L(χt−k≤r≤t+kw−1)(x,t)| ≤C/integraldisplayk\n−kdβ/integraldisplayt+r\n−klog{(α+2k)/k}\n(α+2k)/kdα\n≤Ck2log2τ+(r,t)\n≤Ck2logTk·w(r,t)−1.\nSince log Tk≤D(T)1/2, we obtain (3.6).\n(iii) Case of 1 < p <2.\nIt follows from (2.14) with θ= 1 that\n|L(χt−k≤r≤t+kw−1)(x,t)| ≤C/integraldisplayk\n−kdβ/integraldisplayt+r\n−k/parenleftbiggα+2k\nk/parenrightbigg2−p−(p−1)\ndα\n≤Ck2T2−p\nkw(r,t)−1.\n10Since 2−p≤γ(p,3)/2p, we obtain (3.6).\nThe proof is now completed. ✷\nProof of Theorem 2.1. We consider the following integral equation.\nU=L(|εu0+U|p) inR×[0,T]. (3.7)\nSuppose we have a solution U=U(x,t) of (3.7). Then, by putting u=\nU+εu0, we obtain a solution of (2.5) and its lifespan is the same as that\nofU. Thus, our aim here is to construct a solution of (3.7) in the Banach\nspace,\nX:={U(x,t)∈C(R×[0,T]) : supp U⊂ {(x,t) :|x| ≤t+k}}\nwhich is equipped with the norm (2.10).\nDefine a sequence of functions {Ul} ⊂Xby\nU1= 0, Ul=L(|εu0+Ul−1|p) forl≥2\nand set\nM0:= 2p−1C1k2Cp\n0,\nC4:= (22(p+1)p)pmax{C2k2Mp−1\n0,(C3k2Cp−1\n0)p},\nwhereCi(0≤i≤3) are positive constants given in Lemma 3.1, Lemma 3.2,\nLemma 3.3 and Lemma 3.4. Then, analogously to the proof of Theorem 1 in\n[6], we see that {Ul}is a Cauchy sequence in Xprovided that the inequality\nC4εp(p−1)D(T)≤1 (3.8)\nholds. Since Xis complete, there exists a function Usuch that Ulconverges\ntoUinX. Therefore Usatisfies (3.7).\nNote that (2.2) follows from (3.8). We shall show this fact only in the\ncase ofp= 2 since the other cases can be proved similarly. By definition of\nbin (1.8), we know that b(ε) is decreasing in εand lim\nε→0+0b(ε) =∞. Let us\nfixε0>0 as\n1< C5b(ε0), (3.9)\nwhereC5= min/braceleftbig\n2−1,(3C4)−1/bracerightbig\n. For 0< ε≤ε0, we take Tto satisfy\n1≤T < C 5b(ε). (3.10)\n11Sincek >1, it follows from (3.5) and (3.10) that\nC4ε2D(T)≤C4ε2(3T)log(2T+1)\n≤3C4C5ε2b(ε)log(2C5b(ε)+1)\n≤b(ε)ε2log(b(ε)+1) = 1 .\nHence, if we assume (3.9) and (3.10), then (3.8) holds. Therefore ( 2.2) in the\ncasep= 2 is obtained for 0 < ε≤ε0. This completes the proof of Theorem\n2.1. ✷\n4 Proof of Theorem2.2\nIn order to obtain an upper bound of the lifespan, we shall take a loo k on\nthe ordinary differential inequality for\nF(t) :=/integraldisplay\nRu(x,t)dx\nand shall follow the arguments in Section 5 of Takamura [14]. The equa tion\nin (1.3) with µ= 2 and (2.7) imply that\nF′′(t) =1\n(1+t)p−1/integraldisplay\nR|u(x,t)|pdxfort≥0. (4.1)\nHence, H¨ older’s inequality and (2.7) yield that\nF′′(t)≥2−(p−1)(t+k)−2(p−1)|F(t)|pfort≥0. (4.2)\nDue to the assumption on the initial data in Theorem 2.2,\nf(x)≥0 (/ne}ationslash≡0), f(x)+g(x)≡0,\nwe have\nF(0)>0, F′(0) = 0. (4.3)\nNeglecting the nonlinear term in (2.5), from (2.3) and (2.1), we also ob tain\nthe following point-wise estimate.\nu(x,t)≥1\n2f(x−t)εforx+t≥kand−k≤x−t≤k.(4.4)\nFirst, we shall handle the sub-critical case. In such a case, the fo llowing\nbasic lemma is useful.\n12Lemma 4.1 ([14]) Letp >1,a >0,q >0satisfy\nM:=p−1\n2a−q\n2+1>0. (4.5)\nAssume that F∈C2([0,T))satisfies\nF(t)≥Atafort≥T0, (4.6)\nF′′(t)≥B(t+k)−q|F(t)|pfort≥0, (4.7)\nF(0)>0, F′(0) = 0, (4.8)\nwhereA,B,k,T 0are positive constants. Moreover, assume that there is a\nt0>0such that\nF(t0)≥2F(0). (4.9)\nThen, there exists a positive constant C∗=C∗(p,a,q,B)such that\nT <22/MT1 (4.10)\nholds provided\nT1:= max{T0,t0,k} ≥C∗A−(p−1)/(2M). (4.11)\nThis is exactly Lemma 2.2 in [14], so that we shall omit the proof here.\nWe already have (4.7) and (4.8), so that the key estimate is (4.6) whic h is\nexpected better than a constant F(0) trivially follows from (4.7).\nFrom now on to the end of this section, Cstands for a positive constant\nindependent of ε, and may change from line to line. It follows from (4.1) and\n(4.4) that\nF′′(t)≥1\n(1+t)p−1/integraldisplayt+k\nt−k|u(x,t)|pdx≥Cεpt1−pfott≥k.\nSince (4.7) and (4.8) imply F(t)>0 andF′(t)≥0 fort≥0, integrating this\ninequality twice in t, we obtain\nF(t)≥Cεp×\n\nt3−pif 1< p <2,\ntlogt\n2kifp= 2,\nt ifp >2fort≥4k. (4.12)\n(i) Case of 1 < p <2.\nAccording to (4.12), one can apply Lemma 4.1 to our situation with\nA=Cεp, a= 3−p >0, B= 2−(p−1), q= 2(p−1).\n13In this case, the blow-up condition (4.5) is satisfied by\n2M= (p−1)(3−p)−2(p−1)+2 =γ(p,3)\n2>0.\nNext we fix t0to satisfy (4.9). Due to (4.12), it is\nF(t0)≥Cεpt3−p\n0= 2F(0) = 2/bardblf/bardblL1(R)ε,\nnamely\nt0=Cε−(p−1)/(3−p).\nHence, setting\nT0=C∗A−(p−1)/(2M)=Cε−2p(p−1)/γ(p,3),\nwe have a fact that there exists an ε1=ε1(f,g,p,k)>0 such that\nT1:= max{T0,t0,k}=T0=Cε−2p(p−1)/γ(p,3)≥4k\nholds for 0 < ε≤ε1because of\n1\n3−p<2p\nγ(p,3)⇐⇒p >1.\nTherefore, from (4.10), we obtain T <22/MT1=Cε−2p(p−1)/γ(p,3)as desired.\n(ii) Case of 2 < p <3.\nAccording to (4.12), one can apply Lemma 4.1 to our situation with\nA=Cεp, a= 1, B= 2−(p−1), q= 2(p−1).\nIn this case, the blow-up condition (4.5) is satisfied by\n2M=p−1−2(p−1)+2 = 3 −p >0.\nNext we fix t0to satisfy (4.9). Due to (4.12), it is\nF(t0)≥Cεpt0= 2F(0) = 2/bardblf/bardblL1(R)ε,\nnamely\nt0=Cε−(p−1).\nHence, setting\nT0=C∗A−(p−1)/(2M)=Cε−p(p−1)/(3−p),\n14we have a fact that there exists an ε1=ε1(f,g,p,k)>0 such that\nT1:= max{T0,t0,k}=T0=Cε−p(p−1)/(3−p)≥4k\nholds for 0 < ε≤ε1because of\n1<p\n3−p⇐⇒p >3\n2.\nTherefore we obtain T <22/MT1=Cε−p(p−1)/(3−p)as desired.\n(iii) Case of p= 2.\nNeglecting the logarithmic term in (4.12), similarly to the case of 2 < p <\n3, one can apply Lemma 4.1 to our situation with\nA=Cε2, a= 1, B= 2−1, q= 2,2M= 1.\nWe shall fix a T0as follows. In order to establish (4.11) in Lemma 4.1, we\nhave to assume that T0≥C∗A−1namely\nA≥C∗T−1\n0.\nOn the other hand, (4.6) in Lemma 4.1 can be established by (4.12) as f ar as\nCε2logT0\n2k≥A.\nHenceT0must satisfy\nε2T0logT0\n2k≥C∗∗, (4.13)\nwhereC∗∗isapositive constantindependent of ε. Hereweidentify aconstant\nCasC∗∗to fixT0. Recall the definition of b(ε) in (1.8) and the fact that b(ε)\nis monotonously decreasing in εand lim ε→0+0b(ε) =∞. IfC∗∗≥1, then we\nsetT0= 4kC∗∗b(ε). Taking εsmall to satisfy C∗∗b(ε)≥1, we have\nε2T0logT0\n2k≥4kC∗∗ε2b(ε)log{1+C∗∗b(ε)} ≥4kC∗∗.\nTherefore (4.13) holds if C∗∗≥1 byk >1. On the other hand, if C∗∗<1,\nthen we set T0= 4kb(ε). In this case, taking εsmall to satisfy b(ε)≥1, we\nhave\nε2T0logT0\n2k≥4kε2b(ε)log{1+b(ε)}= 4k,\nso that (4.13) holds by 4 k >1> C∗∗. In this way one can say that our\nsituation can be applicable to Lemma 4.1 with T0=Cb(ε) for small εexcept\nfort0in (4.9).\n15In this case, (4.9) follows from (4.12) and\nF(t0)≥Cε2t0logt0\n2k= 2F(0) = 2/bardblf/bardblL1(R)ε,\nnamely\nεt0logt0\n2k=C.\nComparing this equality with (4.13), we know that there exists an ε1=\nε1(f,g,k)>0 such that\nT1:= max{T0,t0,k}=T0=Cb(ε)≥4k\nholds for 0 < ε≤ε1. Therefore we obtain T <22/MT1=Cb(ε) as desired.\n(iv) Case of p=pF(1) = 3\nEven in this case, (4.12) is still valid. But a= 1 and p= 3 yield M= 0\nin Lemma 4.1. So we need a critical version of the lemma, which is a varian t\nof Lemma 2.1 in Takamura and Wakasa [15] with a slightly different initial\ncondition. Onecanreadilyshowitbysmall modification. Hereweshalla void\nto employ it, and shall make use of (4.7) and (4.12) only to give a simple\nproof by means of “slicing method” of the blow-up domain introduced in\nAgemi, Kurokawa and Takamura [1].\nForj∈N∪{0}, define\naj:=j/summationdisplay\ni=01\n2iandK:= 4k.\nAssume presumably\nF(t)≥Djtlogbjt\najKfort≥ajK, (4.14)\nwhere each bjandDjare positive constants. We note that (4.14) with j= 0\nis true by (4.12) if we set b0= 0 and D0=Cε3. Plugging (4.14) into the\nright hand side of (4.2) with a restriction to the interval [ ajK,∞), we obtain\nthat\nF′′(t)≥2−6D3\njt−1log3bjt\najKfort≥ajK\nwhich yields that\nF′(t)≥2−6D3\nj·1\n3bj+1log3bj+1t\najKfort≥ajK.\n16Integrating this inequality and diminishing the interval to make use of\n/integraldisplayt\najKlog3bj+1s\najKds≥/integraldisplayt\najt/aj+1log3bj+1s\najKdsfort≥aj+1K,\nwe obtain that\nF(t)≥2−6D3\nj·1\n3bj+1/parenleftbigg\n1−aj\naj+1/parenrightbigg\ntlog3bj+1t\naj+1Kfort≥aj+1K.\nThus, due to\n1−aj\naj+1=1\n2j+1aj+1≥1\n2j+2,\n(4.14) inductively holds if the sequence {bj}is defined by\nbj+1= 3bj+1, b0= 0 for j∈N∪{0} (4.15)\nand{Dj}is defined by\nDj+1:=D3\nj\n2j+8(3bj+1), D0:=Cε3forj∈N∪{0}.(4.16)\nIt is easy to see that (4.15) gives us\nbj=3j−1\n2forj∈N∪{0}. (4.17)\nFromnow on, let us look for a suitable lower bound of Djby (4.16). Since\n3bj+1 =bj+1≤3j+1\n2forj∈N∪{0}\nby (4.17), we have\nlogDj+1≥3logDj−(2j+8)log3 for j∈N∪{0}\nwhich yields\nlogDj≥3j−1logD0−log3j−1/summationdisplay\ni=03j−1−i(2i+8) for j∈N.\nHence, it follows from\nS:= lim\nj→∞j−1/summationdisplay\ni=02i+8\n3i>0\n17by d’Alembert criterion that\nDj≥/parenleftbiggD0\n3S/parenrightbigg3j−1\nforj∈N.\nTherefore, together with (4.14), we have\nF(t)≥/parenleftbiggD0\n3S/parenrightbigg3j−1\ntlog(3j−1)/2t\n2K=3S\nD0t/parenleftbigg\nlog−1/2t\n2K/parenrightbigg\nI(t)3j\nfort≥2Kandj≥1, where we set\nI(t) :=D0\n3Slog1/2t\n2K.\nThis inequality means that\nlim\nj→∞F(t1) =∞\nif there exists a t1≥2Ksuch that I(t1)>1. 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Methods Appl. Sci., 27(2004), 101-124.\n[21] J.Wirth, Wave equations with time-dependent dissipation. I. Non-\neffective dissipation , J. Differential Equations, 222(2006), 487-514.\n[22] J.Wirth, Wave equations with time-dependent dissipation. II. Effect ive\ndissipation , J. Differential Equations, 232(2007), 74-103.\n20"    },    {        "title": "1810.04973v1.Propagating_spin_waves_in_nanometer_thick_yttrium_iron_garnet_films__Dependence_on_wave_vector__magnetic_field_strength_and_angle.pdf",        "content": "Propagating spin waves in nanometer-thick yttrium iron garnet \flms: Dependence on\nwave vector, magnetic \feld strength and angle\nHuajun Qin,1,\u0003Sampo J. H am al ainen,1Kristian Arjas,1Jorn Witteveen,1and Sebastiaan van Dijken1,y\n1NanoSpin, Department of Applied Physics, Aalto University School of Science, P.O. Box 15100, FI-00076 Aalto, Finland\n(Dated: October 12, 2018)\nWe present a comprehensive investigation of propagating spin waves in nanometer-thick yttrium\niron garnet (YIG) \flms. We use broadband spin-wave spectroscopy with integrated coplanar waveg-\nuides (CPWs) and microstrip antennas on top of continuous and patterned YIG \flms to characterize\nspin waves with wave vectors up to 10 rad/ \u0016m. All \flms are grown by pulsed laser deposition. From\nspin-wave transmission spectra, parameters such as the Gilbert damping constant, spin-wave dis-\npersion relation, group velocity, relaxation time, and decay length are derived and their dependence\non magnetic bias \feld strength and angle is systematically gauged. For a 40-nm-thick YIG \flm, we\nobtain a damping constant of 3 :5\u000210\u00004and a maximum decay length of 1.2 mm. Our experiments\nreveal a strong variation of spin-wave parameters with magnetic bias \feld and wave vector. Spin-\nwave properties change considerably up to a magnetic bias \feld of about 30 mT and above a \feld\nangle of\u0012H= 20\u000e, where\u0012H= 0\u000ecorresponds to the Damon-Eshbach con\fguration.\nPACS numbers:\nI. INTRODUCTION\nMagnonics aims at the exploitation of spin waves for\ninformation transport, storage, and processing1{7. For\npractical devices, it is essential that spin waves propa-\ngate over long distances in thin \flms. Because of its ul-\ntralow damping constant, ferrimagnetic YIG is a promis-\ning material. Bulk crystals and \u0016m-thick YIG \flms ex-\nhibit a Gilbert damping constant \u000b\u00193\u000210\u00005at GHz\nfrequencies. In recent years, nm-thick YIG \flms with\nultralow damping parameters have also been prepared\nsuccessfully. High-quality YIG \flms have been grown on\nGd3Ga5O12(GGG) single-crystal substrates using liquid\nphase epitaxy8{11, magnetron sputtering12{15, and pulsed\nlaser deposition (PLD)16{25. For thin YIG \flms, damp-\ning constants approaching the value of bulk crystals have\nbeen reported21,22. Meanwhile, YIG-based magnonic\ndevices such as logic gates, transistors, and multiplex-\ners have been demonstrated26{30. Spin-wave transmis-\nsion in nm-thick YIG \flms24,31{36and the excitation of\nshort-wavelength spin waves have been investigated as\nwell37{40. To advance YIG-magnonics further, knowledge\non the transport of spin waves in nm-thick YIG \flms and\nits dependence on wave vector and external magnetic bias\n\feld is essential.\nIn this paper, we present a broadband spin-wave spec-\ntroscopy study of PLD-grown YIG \flms with a thickness\nof 35 nm and 40 nm. Spin-wave transmission spectra are\nrecorded by patterning CPWs and microstrip antennas\non top of continuous and patterned YIG \flms. CPWs are\nused because they generate spin waves with well-de\fned\nwave vectors. This enables extraction of key parame-\nters such as the Gilbert damping constant ( \u000b), the spin-\nwave dispersion relation, group velocity ( \u001dg), relaxation\ntime (\u001c), and decay length ( ld). For a 40 nm YIG \flm,\nwe \fnd\u000b\u00193:5\u000210\u00004and a maximum group velocity\nand decay length of 3.0 km/s and 1.2 mm, respectively.\nWe show that spin-wave properties vary strongly withwave vector up to an in-plane external magnetic bias \feld\n\u00160Hext= 30 mT and below a \feld angle \u0012H= 20\u000e(\u0012H\n= 0 corresponds to the Damon-Eshbach geometry). Be-\nyond these \feld parameters, the dependence of spin-wave\nproperties on wave vector weakens. We demonstrate also\nthat broadband spectroscopy with integrated CPWs and\nmicrostrip antennas provide similar spin-wave parame-\nters.\nThe paper is organized as follows. In Sec. II,\nwe describe the PLD process, broadband spin-wave\nspectroscopy setup, and simulations of the CPW- and\nmicrostrip-antenna excitation spectra. In Sec. III, we\npresent vector network analyzer ferromagnetic resonance\n(VNA-FMR) results and broadband spin-wave transmis-\nsion spectra for CPWs. In Sec. IV, we \ft the ex-\nperimental data and extract parameters of propagating\nspin waves. Spin-wave transmission measurements using\nCPWs and microstrip antennas are compared in Sec. V.\nSection VI summarizes the paper.\nII. EXPERIMENT\nA. PLD of YIG thin \flms\nYIG \flms with a thickness of 35 nm and 40 nm were\ngrown on single-crystal GGG(111) substrates using PLD.\nPrior to loading into the PLD vacuum chamber, the\nsubstrates were ultrasonically cleaned in acetone, iso-\npropanol, and distilled water. The substrates were \frst\ndegassed at 550\u000eC for 15 minutes and then heated to\n800\u000eC at a rate of 5\u000eC per minute in an O 2pressure of\n0.13 mbar. YIG \flms were deposited under these condi-\ntions by ablation from a stoichiometric target using an\nexcimer laser with a pulse repetition rate of 2 Hz and\na \ruence of 1.8 J/cm2. After deposition, the YIG \flms\nwere \frst annealed at 730\u000eC for 10 minutes in 13 mbar\nO2before cooling down to room temperature at a rate ofarXiv:1810.04973v1  [cond-mat.mes-hall]  11 Oct 20182\n-1.0 -0.5 0.0 0.5 1.0-100-50050100\n49 50 51 52 53300 400 500 6000.00.51.0Ms (kA/m)\nMagnetic field (mT)GGG (444)Intensity (a.u.)\n2oYIG (444)\nM/Ms\nTemperature (K)(a) (b)\nFIG. 1: (a) XRD \u0012\u00002\u0012scan of the (444) re\rections from a\nPLD-grown YIG \flm on a GGG(111) substrate. The period\nof Laue oscillations surrounding the (444) peaks corresponds\nto a \flm thickness of 40 nm. (b) Room temperature VSM\nhysteresis loop of the same \flm. The inset shows how the\nYIG saturation magnetization varies with temperature.\n\u00003\u000eC per minute.\nB. Structural and magnetic characterization\nThe crystal structure of our YIG \flms was inspected\nby high-resolution X-ray di\u000braction (XRD) on a Rigaku\nSmartLab system. Figure 1(a) shows a XRD \u0012\u00002\u0012scan\nof a 40-nm-thick YIG \flm on GGG(111). Clear (444) \flm\nand substrate peaks are surrounded by Laue oscillations,\nsignifying epitaxial and smooth \flm growth. We used a\nvibrating sample magnetometer (VSM) in a PPMS Dy-\nnacool system from Quantum Design to characterize the\nmagnetic properties. Figure 1(b) depicts a VSM hystere-\nsis loop of a 40-nm-thick YIG \flm. At room temperature,\nthe coercive \feld of the YIG \flm is only 0.1 mT and the\nsaturation magnetization ( Ms) is 115 kA/m. The evo-\nlution ofMswith temperature is shown in the inset of\nFig. 1(b). From these data, we derive a Curie temper-\nature (TC) of around 500 K. The values of MsandTC\nare similar to those obtained in previous studies on nm-\nthick YIG \flms14,22,23and about 10% smaller compared\nto values of YIG bulk crystals ( Ms= 139 kA/m, TC=\n559 K). Minor o\u000b-stoichiometries in the YIG \flm might\nbe the reason for the small discrepancy41.\nC. Broadband spin-wave spectroscopy\nVNA-FMR and spin-wave transmission measurements\nwere performed using a two-port VNA and a microwave\nprobing station with a quadrupole electromagnet. In\nVNA-FMR experiments, the YIG \flm was placed face-\ndown onto a prepatterned CPW on a GaAs substrate.\nThe signal line and ground lines of this CPW had a\nwidth of 50 \u0016m and 800 \u0016m, respectively, and were sep-\narated by 30 \u0016m. Broadband spin-wave spectroscopyin transmission geometry was conducted by contacting\ntwo integrated CPWs or microstrip antennas on top of\na continuous YIG \flm or YIG waveguide. Most of the\nexperiments were performed with CPWs consisting of 2\n\u0016m-wide signal and ground lines with a separation of 1.6\n\u0016m. For comparison measurements, we used CPWs and\nmicrostrip antennas with 4- \u0016m-wide signal lines. All an-\ntenna structures were fabricated by electron-beam lithog-\nraphy and were composed of 3-nm Ta and 120-nm Au.\nA microwave current provided by the VNA was used to\ngenerate a rf magnetic \feld around one of the CPWs\nor microstrip antennas. We used CST microwave studio\nsoftware to simulate the excitation spectra of the antenna\nstructures (see next section).\nSpin waves that are excited by a rf magnetic \feld pro-\nduce an inductive voltage across a nearby antenna. At\nthe exciting CPW or microstrip antenna, this voltage is\ngiven by42:\nVind/Z\n\u001f(!;k)j\u001a(k)j2dk; (1)\nwhere\u001f(!;k) is the magnetic susceptibility and j\u001a(k)j2\nis the spin-wave excitation spectrum. Propagating spin\nwaves arriving at the receiving CPW or microstrip an-\ntenna produce an inductive voltage:\nVind/Z\n\u001f(!;k)j\u001a(k)j2exp(\u0000i(ks+ \b 0))dk; (2)\nwheresis the propagation distance and \b 0is the initial\nphase of the spin waves. In the experiments, we used the\n\frst and second port of the VNA to measure these induc-\ntive voltages by recording the S12scattering parameter.\nD. Simulations of CPW and microstrip antenna\nexcitation spectra\nWe used CST microwave studio software to simulate\nthe spin-wave excitation spectra of the di\u000berent antenna\nstructures43. This commercial solver of Maxwell's equa-\ntions uses a \fnite integration method to calculate the rf\nmagnetic \feld \u00160hrfand its in-plane ( \u00160hrf\nx,\u00160hrf\ny) and\nout-of-plane ( \u00160hrf\nz) components. Since the excitation\n\feld along the CPW or antenna ( \u00160hrf\nx) is nearly uni-\nform and\u00160hrf\nzis much smaller than \u00160hrf\ny, we Fourier-\ntransformed only the latter component. Figure 2 depicts\nseveral CPW and antenna con\fgurations used in the ex-\nperiments together with their simulated spin-wave exci-\ntation spectra. The large prepatterned CPW on a GaAs\nsubstrate (Fig. 2(a)), which is used for VNA-FMR mea-\nsurements, mainly excites spin waves with k\u00190 rad/\u0016m\n(Fig. 2(d)). The excitation spectrum of the smaller in-\ntegrated CPW with a 2- \u0016m-wide signal line (Fig. 2(b))\nincludes one main spin-wave mode with wave vector k1\n= 0.76 rad/ \u0016m and several high-order modes k2\u0000k7\n(Fig. 2(e)). The 4- \u0016m-wide microstrip antenna (Fig.\n2(c)) mainly excites spin waves with k1ranging from 03\n(a)\nGGS\n02468 1 0 1 20.00.51.0\n-20 -10 0 10 20Amplitude (Normalized)\nWave vector (rad/ m)0Hy (a. u.)\ny (m)S\n02468 1 0 1 20.00.51.0-100 -50 0 50 100Amplitude (Normalized)\nWave vector (rad/ m)GG0Hy (a. u.)\ny (m)S\n02468 1 0 1 20.00.51.0 -10 -5 0 5 10Amplitude (Normalized)\nWave vector (rad/ m)G G0Hy (a. u.)\ny (m)S\n(b) (c)\n(d) (e) (f)\nk1\nk2k3k4k5k6k7k1\nk2k3xyz\nxyz\nx yz\nFIG. 2: (a-c) Schematic illustrations of several measurement con\fgurations used in this study. (a) VNA-FMR measurements\nare performed by placing the YIG/GGG sample face-down onto a CPW. The CPW consists of a 50 \u0016m-wide signal line and\ntwo 800\u0016m-wide ground lines. The gap between the signal and ground lines is 30 \u0016m. (b-c) Spin-wave transmission through\nthe YIG \flm is characterized by patterning two CPWs (b) or two microstrip antennas (c) on top of a YIG \flm. The signal and\nground lines of the CPWs in (b) are 2 \u0016m wide and separated by 1.6 \u0016m gaps. The microstrip antennas, which are marked\nby red arrows in (c), are 4 \u0016m wide. (d-f) Simulated spin-wave excitation spectra of the di\u000berent antenna structures. The\nin-plane rf magnetic \felds ( \u00160hrf\ny) that are produced by passing a microwave current through the CPWs in (a) and (b) or the\nmicrostrip antenna in (c) are shown in the insets.\nto 1.5 rad/\u0016m and some higher order modes at k2\u00192:0\nrad/\u0016m andk3\u00193:8 rad/\u0016m (Fig. 2(f)). The insets of\nFigs. 2(d-f) show the simulated rf magnetic \felds \u00160hrf\ny\nalong they-axis for each antenna structure.\nIII. RESULTS\nA. VNA-FMR\nWe recorded FMR spectra for various in-plane exter-\nnal magnetic bias \felds by measuring the S12scatter-\ning parameter on a 40-nm-thick YIG \flm. As an ex-\nample, the imaginary part of S12recorded with a mag-\nnetic bias \feld \u00160Hext= 80 mT is shown in Fig. 3(a).\nThe spectrum was subtracted a reference measured at\na bias \feld of 200 mT for enhancing signal-to-noise ra-\ntio. The resonance at f= 4:432 GHz is \ftted by a\nLorentzian function. From similar data taken at other\nbias \felds, we extracted the \feld-dependence of FMR\nfrequency and the evolution of resonance linewidth (\u0001 f)\nwith frequency. Figures 3(b) and 3(c) summarize our\nresults. Fitting the data of Fig. 3(b) to the Kittel for-\nmulafres=\r\u00160\n2\u0019p\nHext(Hext+Meff) using\r=2\u0019= 28\nGHz/T, we \fnd Meff= 184 kA/m. The measured\nvalue ofMeffis comparable to those of other PLD-grown\nYIG thin \flms23,24, but it is large compared to Ms(115\nkA/m). Since Meff=Ms-Hani, this means that the\nanisotropy \feld Hani=\u000069 kA/m in our \flm. The\nnegative anisotropy \feld is caused by a lattice mismatch\nbetween the YIG \flm and GGG substrate23. Fitting the\n4.42 4.44 3456681012\n05 0 1 0 0 1 5 00246Im S12\nFrequency (GHz) Frequency (GHz)\nFrequency (GHz)\n0Hext (mT)(a) (b)\n= 3.5 × 10-4f(c)FIG. 3: (a) Imaginary part of the S12scattering parameter\nshowing FMR for an in-plane external magnetic bias \feld of\n80 mT along the CPW. The orange line is a Lorentzian func-\ntion \ft. (b) FMR frequency as a function of external magnetic\nbias \feld. The orange line represents a \ft to the experimen-\ntal data using the Kittel formula. (c) Dependence of FMR\nlinewidth (\u0001 f) on resonance frequency. From a linear \ft to\nthe data, we derive \u000b= 3:5\u000210\u00004.\ndata of Fig. 3(c) using \u0001 f= 2\u000bf+\u001dg\u0001kgives a Gilbert\ndamping constant \u000b= 3:5\u000210\u00004, which is comparable to\nother experiments on PLD-grown \flms17,18,20. In the \ft-\nting formula, \u001dgand \u0001kare the spin-wave group velocity\nand excitation-spectrum width, respectively44.4\n1.8 2.1 2.4 2.7 3.0 10 20 30 40 501234\n0Hext (mT)Frequency (GHz)Im S12k7k5k4 k6k3k2\nFrequency (GHz)k1\n-60 -30 0 30 601.82.12.42.73.03.3\nH (O)Frequency (GHz)(a) (b) (c)\nk1k7\nk1k7\nCPW 1\nCPW 2\n0Hext45 m\nFIG. 4: (a) Spin-wave transmission spectrum (imaginary part of S12) recorded on a 40-nm-thick YIG waveguide with an\nexternal magnetic bias \feld \u00160Hext= 15:5 mT along the CPWs. The inset shows a top-view schematic of the measurement\ngeometry. (b) 2D map of spin-wave transmission spectra measured as a function of magnetic bias \feld strength. (c) Angular\ndependence of spin-wave transmission spectra for a constant bias \feld of 15.5 mT. The \feld angle \u0012H= 0\u000ecorresponds to the\nDamon-Eshbach con\fguration.\nB. Propagating spin waves\nWe measured spin-wave transmission spectra on a 40-\nnm-thick YIG \flm. The measurement geometry con-\nsisted of two CPWs on top of YIG waveguides with 45\u000e\nedges (see the inset of Fig. 4(a)). The CPW parame-\nters were identical to those in Fig. 2(b) and their sig-\nnal lines were separated by 45 \u0016m. During broadband\nspin-wave spectroscopy, spin waves with characteristic\nwave vectors ki(i= 1, 2...) were excited by passing\na rf current through one of the CPWs. After propaga-\ntion through the YIG \flm, the other CPW inductively\ndetected the spin waves. Figure 4(a) shows the imagi-\nnary part of the S12scattering parameter for an external\nmagnetic bias \feld \u00160Hext= 15:5 mT parallel to the\nCPWs (Damon-Eshbach con\fguration). The graph con-\ntains seven envelope-type peaks ( k1\u0000k7) with clear pe-\nriodic oscillations. The peak intensities decrease with\nfrequency because of reductions in the excitation e\u000e-\nciency and spin-wave decay length. The oscillations sig-\nnify spin-wave propagation between the CPWs44. Fig-\nure 4(b) shows a 2D representation of spin-wave trans-\nmission spectra recorded at di\u000berent bias \felds. As the\n\feld strengthens, the frequency gaps between spin-wave\nmodes become smaller. Figure 4(c) depicts the angu-\nlar dependence of S12spectra at a constant magnetic\nbias \feld of 15.5 mT. In this measurement, the in-plane\nmagnetic bias \feld was rotated from -72\u000eto 72\u000e, where\n\u0012H= 0\u000ecorresponds to the Damon-Eshbach con\fgura-\ntion. As the magnetization rotates towards the wave vec-\ntor of propagating spin waves, the frequency and inten-\nsity of thek1\u0000k7modes drop. The frequency evolutions\nof the spin-wave modes in Figs. 4(b) and 4(c) are ex-\nplained by a \rattening of the dispersion relation with\nincreasing magnetic bias \feld strength and angle.\n1 . 71 . 81 . 92 . 02 . 12 . 22 . 3-101Im S12 (Normalized)  Fit\n  Exp.\nFrequency (GHz)k1\nk2FIG. 5: A \ft to the spectrum for \u00160Hext= 15:5 mT and\n\u0012H= 0\u000e(blue squares) using Eq. 3 (orange line).\nIV. DISCUSSION\nA. Fitting of spin-wave transmission spectra\nWe used Eq. 2 to \ft spin-wave transmission spectra.\nIn this equation, \u001f(!;k) is described by a Lorentzian\nfunction, while the excitation spectrum j\u001a(k)j2is ap-\nproximated by a Gaussian function (see Fig. 2(e)). For\nDamon-Eshbach spin waves with kd\u001c1, the wave vec-\ntor is given by k=2\nd(2\u0019f)2\u0000(2\u0019fres)2\n(\r\u00160Ms)2, wheredis the \flm\nthickness. Based on these approximations, we rewrite\nEq. 2 as:\nImS 12/\u0001f\n(f\u0000fres)2+ (\u0001f)2\u0002e\u00004ln2(k\u0000k0)2=\u0001k2\n\u0002sin(ks+ \b);(3)5\n02468 1 0 1 201234\n0369 1 2 1 51.52.02.53.040 mT 15.5 mTFrequency (GHz)\nWave vector (rad/ m)1 mT\n9070604836H = 0Frequency (GHz)\nWave vector (rad/ m)18(a) (b)\nFIG. 6: Spin-wave dispersion relations for di\u000berent external\nmagnetic bias \felds (a) and \feld angles (b). In (a) \u0012H= 0\u000e\nand in (b)\u00160Hext= 15.5 mT. The colored lines represent \fts\nto the disperion relations using Eq. 4.\nwhere \u0001fis theS12envelope width, \u0001 kis the width of\nthe spin-wave excitation spectrum, \b is the initial phase,\nandsis the propagation distance. Figure 5 shows a \ft-\nting result for a spin-wave transmission spectrum with\n\u00160Hext= 15.5 mT and \u0012H= 0\u000e. As input parameters,\nwe usedfres= 1.75 GHz, d= 40 nm,s= 45\u0016m, and\nMeff= 184 kA/m, which are either determined by ge-\nometry or extracted from measurements. \u0001 f, \u0001k,k0\nare \ftting parameters. For the k1peak, we obtained\nthe best \ft for \u0001 f= 0.25 GHz, \u0001 k= 0.6 rad/\u0016m, and\nk1= 0:72 rad/\u0016m. Thek2peak was \ftted with k2= 1:87\nrad/\u0016m. The values of \u0001 k,k1, andk2are in good agree-\nment with the simulated excitation spectrum of the CPW\n(Fig. 2(e)) and \u0001 fmatches the width of the envelope\npeak in the experimental S12spectrum.\nB. Spin-wave dispersion relations\nWe extracted spin-wave dispersion relations for di\u000ber-\nent magnetic bias \felds and \feld angles by \ftting the S12\ntransmission spectra shown in Figs. 4(b) and 4(c). The\nsymbols in Fig. 6 summarize the results. We also cal-\nculated the dispersion relations using the Kalinikos and\nSlavin formula45:\nf=\r\u00160\n2\u0019\u0014\nHext\u0000\nHext+Meff\u0002\n1\u0000Fsin2\u0012H\n+Meff\nHextF(1\u0000F) cos2\u0012H\u0003\u0001\u00151=2\n;(4)\nwithF= 1\u00001\u0000exp(\u0000kd)\nkd. The calculated dispersion re-\nlations for\r=2\u0019= 28 GHz/T, Meff= 184 kA/m, and d\n= 40 nm are shown as lines in Fig. 6.\nThe dispersion curves \ratten with increasing magnetic\nbias \feld. For instance, at \u00160Hext= 1 mT, the frequency\nof propagating spin waves changes from 0.5 GHz to 2.4\nGHz for wave vectors ranging from 0 to 10 rad/ \u0016m. At\n\u00160Hext= 40 mT, the frequency evolution with wave vec-\n0 1 02 03 04 05 00.51.01.52.02.53.0\n02 0 4 0 6 00.30.60.91.21.5g (k1)\ng (k2)\ng (k3)Group velocity g (km/s)\next (mT)\nGroup velocity g (km/s)\nH (o)g (k1)\ng (k2)\ng (k3)(a) (b)FIG. 7: Spin-wave group velocity \u001dgofk1\u0000k3modes as a\nfunction of external magnetic bias \feld (a) and \feld angle (b).\nIn (a)\u0012H= 0\u000eand in (b)\u00160Hext= 15.5 mT.\ntor is reduced to 3 \u00003:7 GHz. This magnetic-\feld de-\npendence of the dispersion relation narrows the spin-wave\ntransmission bands in Fig. 4(b) at large \u00160Hext.\nThe angular dependence of the spin-wave dispersion\ncurves in Fig. 6(b) is explained by in-plane magneti-\nzation rotation from M?k(\u0012H= 0\u000e) towardsMkk\n(\u0012H= 90\u000e). At\u0012H= 0\u000e, dispersive Damon-Eshbach spin\nwaves with positive group velocity propagate between the\nCPWs. The character of excited spin waves changes\ngradually with increasing \u0012Huntil it has fully trans-\nformed into a backward-volume magnetostatic mode at\n\u0012H= 90\u000e. This mode is only weakly dispersive and ex-\nhibits a negative group velocity.\nC. Group velocity\nThe phase relation between signals from the two CPWs\nis given by \b = k\u0002s31,44. Since the phase shift between\ntwo neighboring maxima ( \u000ef) in broadband spin-wave\ntransmission spectra corresponds to 2 \u0019, the group veloc-\nity can be written as:\n\u001dg=@!\n@k=2\u0019\u000ef\n2\u0019=s=\u000ef\u0002s; (5)\nwheres= 45\u0016m in our experiments. Using this equa-\ntion, we extracted the spin-wave group velocity for wave\nvectorsk1\u0000k3from the transmission spectra shown in\nFigs. 4(b) and 4(c). Figure 7 summarizes the variation\nof\u001dgwith external magnetic bias \feld and \feld angle.\nFor weak bias \felds ( \u00160Hext<30 mT), the group ve-\nlocity decreases swiftly, especially if kis small. For in-\nstance,\u001dg(k1) reduces from 3.0 km/s to 1.0 km/s in the\n0\u000030 mT \feld range, while \u001dg(k3) only changes from\n1.2 km/s to 0.8 km/s. At larger external magnetic bias\n\felds,\u001dgdecreases more slowly for all wave vectors. Fig-\nure 7(b) shows how \u001dgvaries as a function of \feld angle\nat\u00160Hext= 15.5 mT. For all wave vectors, the group\nvelocity is largest in the Damon-Eshbach con\fguration\n(\u0012H= 0\u000e). At larger \feld angles, \u001dgdecreases and its6\n0 1 02 03 04 05 0200400600\n02 0 4 0 6 0200250300Relaxation time  (ns)\next (mT) (k1)\n (k2)\nk3)\nRelaxation time (ns)\nH (o)  (k1)\n  (k2)\n  (k3)(a) (b)\nFIG. 8: Spin-wave relaxation time \u001cofk1\u0000k3modes as a\nfunction of external magnetic bias \feld (a) and \feld angle (b).\nIn (a)\u0012H= 0\u000eand in (b)\u00160Hext= 15.5 mT.\n0 1 02 03 04 05 0030060090012001500\n0 1 02 03 04 05 06 00100200300400Decay length ld (m)\n0Hext (mT) ld (k1)\n ld (k2)\n ld (k3)\nDecay length ld (m)\no ld (k1)\n ld (k2)\n ld (k3)(a) (b)\nFIG. 9: Spin-wave decay length ldofk1\u0000k3modes as a\nfunction of external magnetic bias \feld (a) and \feld angle\n(b). In (a) \u0012H= 0\u000eand in (b)\u00160Hext= 15.5 mT.\ndependence on wave vector diminishes. Variations of the\nspin-wave group velocity with wave vector and magnetic-\n\feld strength or angle are explained by a \rattening of the\ndispersion relations, as illustrated by the data in Fig. 6.\nD. Spin-wave relaxation time and decay length\nWe now discuss the relaxation time ( \u001c) and decay\nlength (ld) of spin waves in our YIG \flms. Following\nRef. 46, the relaxation time is estimated by \u001c= 1=2\u0019\u000bf.\nUsing\u000b= 3:5\u000210\u00004and spin-wave transmission data\nfrom Fig. 4, we determined \u001cfor wave vectors k1\u0000k3.\nThe dependence of \u001con external magnetic bias \feld and\n\feld angle is shown in Fig. 8. The maximum spin-wave\nrelaxation time in our 40-nm-thick YIG \flms is approx-\nimately 500 ns. Resembling the spin-wave group veloc-\nity,\u001cis largest for small wave vectors and it decreases\nwith increasing bias \feld (Fig. 8(a)). In contrast to \u001dg,\nthe spin-wave relaxation time is smallest in the Damon-\nEshbach con\fguration ( \u0012H= 0\u000e) and it evolves more\nstrongly with increasing \u0012Hifkis large (Fig. 8(b)). This\nresult is explained by \u001c/1=fand a lowering of thespin-wave frequency if the in-plane bias \feld rotates the\nmagnetization towards k(see Fig. 4(c)).\nThe spin-wave decay length is calculated using ld=\n\u001dg\u0002\u001cand data from Figs. 7 and 8. Figure 9(a) shows\nthe dependence of ldon\u00160Hextfor wave vectors k1\u0000k3.\nThe largest spin-wave decay length in our 40-nm-thick\nYIG \flms is 1.2 mm, which we measured for k1= 0:72\nrad/\u0016m and\u00160Hext= 2 mT. The decay length decreases\nwith magnetic bias \feld to about 100 \u0016m at\u00160Hext=\n50 mT. Figure 9(b) depicts the dependence of ldon the\ndirection of a 15.5 mT bias \feld. The spin-wave decay\nlength decreases substantially with \u0012Hfor smallk, but\nits angular dependence weakens for larger wave vectors.\nThe decay of propagating spin waves between the ex-\nciting and detecting CPW in the broadband spectroscopy\nmeasurement geometry is given by exp( \u0000s=ld)46. Based\non the results of Fig. 9, one would thus expect the in-\ntensity of spin waves to drop with increasing wave vector\nand in-plane bias \feld strength or angle. The spin-wave\ntransmission spectra of Fig. 4 con\frm this behavior.\nE. CPWs versus microstrip antennas\nFinally, we compare broadband spin-wave spec-\ntroscopy measurements on YIG thin \flms using CPWs\nand microstrip antennas. In these experiments, the\nCPWs and antenna structures have 4- \u0016m-wide signal\nlines and they were patterned onto the same 35-nm-thick\nYIG \flm. For comparison, we also recorded transmission\nspectra on 50- \u0016m wide YIG waveguides. The separation\ndistance (s) between the CPWs or microstrip antennas\nwas set to 110 \u0016m or 220\u0016m. Schematics of the di\u000berent\nmeasurement geometries are depicted on the sides of Fig.\n10. Transmission spectra that were obtained for Damon-\nEshbach spin waves in each con\fguration are also shown.\nIn all measurements, we used an in-plane external mag-\nnetic bias \feld of 10 mT. The plots focus on phase os-\ncillations in the \frst-order excitation at k1(higher-order\nexcitations were measured also, but are not shown). The\ndi\u000berently shaped outline of the S12peak for two CPWs\n(left) or two microstrip antennas (right) mimics the pro-\n\fle of their excitation spectra (Fig. 2). As expected from\n\u000ef=\u001dg=s, the period of frequency oscillations ( \u000ef) be-\ncomes smaller if the separation between antennas ( s) is\nenhanced (Figs. 10(c) and 10(f)).\nWe \ftted the spin-wave transmission spectra obtained\nwith CPWs (Figs. 10(a)-(c)) using the same procedure as\ndescribed earlier. Good agreements between experimen-\ntal data (blue squares) and calculations (orange lines)\nwere obtained by inserting Meff= 190\u00064 kA/m, \u0001f=\n0.18 GHz,k= 0.34 rad/ \u0016m, and \u0001k= 0.33 rad/ \u0016m into\nEq. 3. To \ft S12spectra measured by microstrip anten-\nnas, we approximated the wave vector of the excitation\nask=2\nd(2\u0019f)2\u0000(2\u0019fres)2\n(\r\u00160Meff)2H(f\u0000fres), whereHis a Heav-\niside step function47. The best results were achieved for\nMeff= 178\u00062 kA/m, \u0001 f= 0.25 GHz, k= 0 rad/\u0016m\nand \u0001k= 0.65 rad/ \u0016m. From the data comparison in7\n-101\n-101\n1.3 1.4 1.5-101-101\n-101\n1.3 1.4 1.5-101\nFrequency (GHz) Frequency (GHz)110 m\n220 mCPW1\nCPW2\n0Hext\n110 m110 mantenna2\n0Hextantenna1\n220 m(a) (d)\n(b) (e)\n(c) (f)Im S12 (Normalized)\nIm S12(Normalized)110 m\n220 m\nFIG. 10: (a)-(c) Spin-wave transmission spectra measured using CPWs on a continuous YIG \flm (a) and 50- \u0016m-wide YIG\nwaveguides ((b) and (c)). The YIG \flm and waveguides are 35 nm thick and the CPWs are separated by 110 \u0016m ((a) and\n(b)) and 220 \u0016m (c). (d)-(f) Spin-wave transmission spectra measured using microstrip antennas on the same YIG \flm and\nwaveguides. The signal lines of the CPWs and microstrip antennas are 4 \u0016m wide. The orange lines represent \fts to the\nexperimental data using Eq. 3. The measurement geometry for each spectrum is illustrated next to the graphs. In the\nschematics, the green areas depict a continuous YIG \flm or waveguide.\nFig. 10, we conclude that broadband spin-wave spec-\ntroscopy measurements with CPWs and microstrip an-\ntennas yield similar results for Meff. We also note that\ntheS12peak width (\u0001 f) obtained from measurements\non continuous YIG \flms and YIG waveguides are nearly\nidentical (\u0001 f= 0:18 GHz for CPWs, \u0001 f= 0:22 GHz for\nantennas). Thus, patterning of the YIG \flm into waveg-\nuides does not deteriorate the Gilbert damping constant.\nFrom the oscillation periods ( \u000ef) in the transmission\nspectra of Fig. 10, we extracted the properties of prop-\nagating spin waves. By averaging \u000efover the same fre-\nquency range in spectra measured by CPWs and mi-\ncrostrip antennas, we obtained \u001dg= 1:67 km/s and\n\u001dg= 1:53 km/s, respectively. The spin-wave relax-\nation time was determined as \u001c= 225 ns (CPW) and\n\u001c= 237 ns (antenna) and the decay length was extracted\nasld= 375\u0016m (CPW) and ld= 363\u0016m (antenna).\nThese results clearly demonstrate that broadband spin-\nwave spectroscopy measurements on YIG thin \flms us-\ning CPWs or microstrip antennas provide comparable\nresults.\nV. SUMMARY\nIn conclusion, we prepared 35 \u000040 nm thick epitaxial\nYIG \flms with a Gilbert damping constant \u000b= 3:5\u000210\u00004on GGG(111) substrates using PLD. The dependence of\nspin-wave transmission on the strength and angle of an\nin-plane magnetic bias \feld was systematically gauged.\nWe used the measurements to demonstrate strong tun-\ning of the spin-wave group velocity ( \u001dg), relaxation time\n(\u001c), and decay length ( ld) up to a \feld strength of about\n30 mT and above a \feld angle of 20\u000e. Maximum val-\nues of\u001dg= 3:0 km/s,\u001c= 500 ns, and ld= 1:2 mm\nwere extracted for Damon-Eshbach spin waves with k1\n= 0.72 rad/ \u0016m. Moreover, we demonstrated that broad-\nband spin-wave spectroscopy performed with integrated\nCPWs and microstrip antennas yield similar results.\nVI. ACKNOWLEDGEMENTS\nThis work was supported by the European Re-\nsearch Council (Grant Nos. ERC-2012-StG 307502-E-\nCONTROL and ERC-PoC-2018 812841-POWERSPIN).\nS.J.H. acknowledges \fnancial support from the V ais al a\nFoundation. Lithography was performed at the Mi-\ncronova Nanofabrication Centre, supported by Aalto\nUniversity. We also acknowledge the computational re-\nsources provided by the Aalto Science-IT project.\n\u0003Electronic address: huajun.qin@aalto.\f\nyElectronic address: sebastiaan.van.dijken@aalto.\f1A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys.\nD: Appl. Phys. 43, 264002 (2010).8\n2V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. 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Lett. 109, 012403 (2016)."    },    {        "title": "1810.06471v1.Localized_spin_waves_in_isolated__kπ__skyrmions.pdf",        "content": "Localized spin waves in isolated k\u0019skyrmions\nLevente Rózsa,1,\u0003Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: October 16, 2018)\nThelocalizedmagnonmodesofisolated k\u0019skyrmionsonafield-polarizedbackgroundareanalyzed\nbasedontheLandau–Lifshitz–Gilbertequationwithinthetermsofanatomisticclassicalspinmodel,\nwith system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion\norderka higher number of excitation modes are found, including modes with nodes in the radial\neigenfunctions. Itisshownthatatlowfields 2\u0019and 3\u0019skyrmionsaredestroyedviaaburstinstability\nconnected to a breathing mode, while 1\u0019skyrmions undergo an elliptic instability. At high fields all\nk\u0019skyrmions collapse due to the instability of a breathing mode. The effective damping parameters\nof the spin waves are calculated in the low Gilbert damping limit, and they are found to diverge\nin the case of the lowest-lying modes at the burst and collapse instabilities, but not at the elliptic\ninstability. It is shown that the breathing modes of k\u0019skyrmions may become overdamped at higher\nGilbert damping values.\nI. INTRODUCTION\nMagneticskyrmionsarelocalizedparticle-likespincon-\nfigurations [1], which have become the focus of intense\nresearch activities over the last years due to their promis-\ning applications in spintronic devices [2–5]. While their\nparticle-like properties make them suitable to be used\nas bits of information, the collective excitations of the\nspins constituting the magnetic skyrmion, known as spin\nwaves or magnons, open possible applications in the field\nof magnonics [6].\nThese spin wave modes were first investigated theo-\nretically [7–10] and experimentally [11–13] in skyrmion\nlattice phases, where the interactions between the\nskyrmions lead to the formation of magnon bands. If\na skyrmion is confined in a finite-sized nanoelement, it\nwill possess discrete excitation frequencies [14–17]. Al-\nthough such geometries have also been successfully ap-\nplied to the time-resolved imaging of the dynamical mo-\ntion of magnetic bubble domains [18, 19], in such a case\nit is not possible to distinguish between the excitations\nof the particle-like object itself and spin waves forming\nat the edges of the sample [14]. In order to rule out\nboundary effects, the excitations of isolated skyrmions\nhave to be investigated, as was performed theoretically\nin Refs. [20–23]. It was suggested recently [24] that\nthe experimentally determined excitation frequencies in\nthe Ir/Fe/Co/Pt multilayer system may be identified as\nspin wave modes of isolated skyrmions, rather than as\nmagnons stemming from an ordered skyrmion lattice.\nIn most investigations skyrmions correspond to sim-\nple domains with the magnetization in their core point-\ning opposite to the collinear background. However, it\nwas shown already in Ref. [25] that the Dzyaloshinsky–\nMoriya interaction [26, 27] responsible for their stabiliza-\ntion may also lead to the formation of structures where\nthe direction of the magnetization rotates multiple times\n\u0003rozsa.levente@physnet.uni-hamburg.debetween the center of the structure and the collinear re-\ngion. Such target states or k\u0019skyrmions, where kis the\nnumber of sign changes of the out-of-plane magnetization\nwhen moving along the radial direction, have also been\ninvestigated in constricted geometries [28–32]. The ex-\nperimental observation of localized spin structures with\nmultiple rotations has been mainly restricted to systems\nwith negligible Dzyaloshinsky–Moriya interaction so far\n[19, 33, 34], where the formation of domain structures is\nattributed to the magnetostatic dipolar interaction.\nThe collapse of isolated k\u0019skyrmions and their cre-\nationinnanodotsbyswitchingtheexternalfielddirection\nwas recently investigated in Ref. [35]. It was found that\nduring the creation process the skyrmions display signif-\nicant size oscillations resembling breathing eigenmodes.\nIn Ref. [25], the stability of k\u0019skyrmions was studied\nin a system with a ferromagnetic ground state, and it\nwas found that applying the external field opposite to\nthe background magnetization leads to a divergence of\nthe skyrmion radius at a critical field value, a so-called\nburst instability. This instability can be attributed to a\nsign change of one of the eigenvalues of the energy func-\ntional expanded around the k\u0019skyrmion configuration,\nintrinsically related to the dynamics of the system. How-\never, the spin wave frequencies of isolated k\u0019skyrmions\nremain unexplored.\nBesides the excitation frequencies themselves, the life-\ntime of spin waves is also of crucial importance in\nmagnonics applications. This is primarily influenced by\nthe Gilbert damping parameter \u000b[36], the value of which\ncan be determined experimentally based on resonance\nlineshapes measured in the collinear state [11, 19, 24].\nIt was demonstrated recently [23] that the noncollinear\nspin structure drastically influences the effective damp-\ning parameter acting on the spin waves, leading to mode-\ndependent and enhanced values compared to the Gilbert\ndamping parameter. This effect was discussed through\nthe example of the 1\u0019skyrmion in Ref. [23], but it is also\nexpected to be observable for k\u0019skyrmions with higher\norderk.\nHere the localized spin wave frequencies of isolated k\u0019arXiv:1810.06471v1  [cond-mat.mes-hall]  15 Oct 20182\nskyrmions are investigated in a classical atomistic spin\nmodel. The parameters in the Hamiltonian represent the\nPd/Fe/Ir(111) model-type system, where the properties\nof skyrmions have been studied in detail both from the\nexperimental [37, 38] and from the theoretical [35, 39–\n41] side. The paper is organized as follows. The classical\natomistic spin Hamiltonian and the method of calculat-\ning the eigenmodes is introduced in Sec. IIA, while the\nangular momentum and nodal quantum numbers charac-\nterizing the excitations are defined in Sec. IIB within the\nframework of the corresponding micromagnetic model.\nEigenfrequencies equal to or approaching zero are dis-\ncussed in Sec. IIC, and the effective damping param-\neters are introduced in Sec. IID. The eigenmodes of k\u0019\nskyrmions with k= 1;2;3are compared in Sec. IIIA, the\ninstabilities occurring at low and high field values are dis-\ncussed in connection to magnons with vanishing frequen-\ncies in Sec. IIIB, and the effective damping parameters\nof the different modes are calculated for vanishing and\nhigher values of the Gilbert damping in Secs. IIIC and\nIIID, respectively. A summary is given in Sec. IV.\nII. METHODS\nA. Atomistic model\nThe system is described by the classical atomistic\nmodel Hamiltonian\nH=\u00001\n2X\nhi;jiJSiSj\u00001\n2X\nhi;jiDij(Si\u0002Sj)\n\u0000X\niK(Sz\ni)2\u0000X\ni\u0016sBSi; (1)\nwith theSiunit vectors representing the spins in\na single-layer triangular lattice; J,Dij, andKde-\nnoting nearest-neighbor Heisenberg and Dzyaloshinsky–\nMoriya exchange interactions and on-site magnetocrys-\ntalline anisotropy, respectively; while \u0016sandBstand\nfor the spin magnetic moment and the external mag-\nnetic field. The numerical values of the parameters are\ntaken from Ref. [35], being J= 5:72meV;D=jDijj=\n1:52meV;K= 0:4meV, and\u0016s= 3\u0016B, describing the\nPd/Fe/Ir(111) system. The energy parameters were de-\ntermined based on measuring the field-dependence of 1\u0019\nskyrmion profiles in the system by spin-polarized scan-\nning tunneling microscopy in Ref. [38].\nDuring the calculations the external field Bis ori-\nented along the out-of-plane zdirection. The equilib-\nriumk\u0019skyrmion structures are determined from a rea-\nsonable initial configuration by iteratively rotating the\nspinsSitowards the direction of the effective magnetic\nfieldBeff\ni=\u00001\n\u0016s@H\n@Si. The iteration is performed un-\ntil the torque acting on the spins, Ti=\u0000Si\u0002Beff\ni,\nbecomes smaller at every lattice site than a predefinedvalue, generally chosen to be 10\u00008meV=\u0016B. The calcula-\ntions are performed on a lattice with periodic boundary\nconditions, with system sizes up to 256\u0002256for the\nlargestk\u0019skyrmions in order to avoid edge effects and\nenable the accurate modeling of isolated skyrmions.\nOnce the equilibrium configuration S(0)\niis determined,\nthe spins are rotated to a local coordinate system ~Si=\nRiSiusing the rotational matrices Ri. In the local coor-\ndinatesystemtheequilibriumspindirectionsarepointing\nalong the local zaxis, ~S(0)\ni= (0;0;1). The Hamiltonian\nin Eq. (1) is expanded up to second-order terms in the\nsmall variables ~Sx\ni;~Sy\nias (cf. Ref. [23])\nH\u0019H0+1\n2\u0010\n~S?\u0011T\nHSW~S?\n=H0+1\n2\u0002~Sx~Sy\u0003\u0014A1A2\nAy\n2A3\u0015\u0014~Sx\n~Sy\u0015\n:(2)\nThematrix products areunderstoodtorunoverlattice\nsite indices i, with the matrix components reading\nA1;ij=\u0000~Jxx\nij+\u000eij X\nk~Jzz\nik\u00002~Kxx\ni+ 2~Kzz\ni+\u0016s~Bz\ni!\n;(3)\nA2;ij=\u0000~Jxy\nij\u0000\u000eij2~Kxy\ni; (4)\nA3;ij=\u0000~Jyy\nij+\u000eij X\nk~Jzz\nik\u00002~Kyy\ni+ 2~Kzz\ni+\u0016s~Bz\ni!\n:(5)\nThe energy terms in the Hamiltonian are ro-\ntated to the local coordinate system via ~Jij=\nRi[JI\u0000Dij\u0002]RT\nj;~Ki=RiKRT\nj;and ~Bi=RiB,\nwhereIis the 3\u00023identity matrix, Dij\u0002is the ma-\ntrix describing the vector product with Dij, andKis\nthe anisotropy matrix with the only nonzero element be-\ningKzz=K.\nThe spin wave frequencies are obtained from the lin-\nearized Landau–Lifshitz–Gilbert equation [36, 42]\n@t~S?=\r0\n\u0016s(\u0000i\u001by\u0000\u000b)HSW~S?=DSW~S?;(6)\nwith\u001by=\u0014\n0\u0000iIs\niIs0\u0015\nthe Pauli matrix in Cartesian\ncomponents and acting as the identity matrix Isin the\nlattice site summations. The symbol \r0denotes the gyro-\nmagnetic ratio \r=ge\n2mdivided by a factor of 1+\u000b2, with\ngthe electron gfactor,ethe elementary charge, mthe\nelectron’s mass, and \u000bthe Gilbert damping parameter.\nEquation (6) is rewritten as an eigenvalue equation by\nassuming the time dependence ~S?(t) =e\u0000i!qt~S?\nqand\nperforming the replacement @t!\u0000i!q.\nSince thek\u0019skyrmions represent local energy minima,\nHSWin Eq. (2) is a positive semidefinite matrix. For\n\u000b= 0the!qfrequenciesof DSWarerealandtheyalways\noccurin\u0006!qpairsonthesubspacewhere HSWisstrictly\npositive, for details see, e.g., Ref. [23]. In the following,\nwe will only treat the solutions with Re !q>0, but their3\nRe!q<0pairs are also necessary for constructing real-\nvalued eigenvectors of Eq. (6). The zero eigenvalues are\ndiscussed in Sec. IIC.\nAs is known from previous calculations for 1\u0019\nskyrmions [21–23], the localized excitation modes of k\u0019\nskyrmions are found below the ferromagnetic resonance\nfrequency!FMR =\r\n\u0016s(2K+\u0016sB). During the numerical\nsolution of Eq. (6) these lowest-lying eigenmodes of the\nsparse matrix DSWare determined, as implemented in\nthemontecrystal atomistic spin simulation program\n[43].\nB. Micromagnetic model\nThe atomistic model described in the previous Sec-\ntion enables the treatment of noncollinear spin structures\nwhere the direction of the spins significantly differs be-\ntween neighboring lattice sites. This is especially impor-\ntant when discussing the collapse of k\u0019skyrmions on the\nlattice as was performed in Ref. [35]. Here we will dis-\ncuss the micromagnetic model which on the one hand is\napplicable only if the characteristic length scale of non-\ncollinear structures is significantly larger than the lattice\nconstant, but on the other hand enables a simple classi-\nfication of the spin wave modes.\nThe free energy functional of the micromagnetic model\nis defined as\nH=Z\nAX\n\u000b=x;y;z(rS\u000b)2+K(Sz)2\u0000MBSz\n+D(Sz@xSx\u0000Sx@xSz+Sz@ySy\u0000Sy@ySz)dr;\n(7)\nwhere for the Pd/Fe/Ir(111) system the following pa-\nrameter values were used: A= 2:0pJ/m is the ex-\nchange stiffness,D=\u00003:9mJ/m2is the Dzyaloshinsky–\nMoriya interaction describing right-handed rotation [39],\nK=\u00002:5MJ/m3is the easy-axis anisotropy, and M=\n1:1MA/m is the saturation magnetization.\nThe equilibrium spin structure S(0)=\n(sin \u0002 0cos \b 0;sin \u0002 0sin \b 0;cos \u0002 0)ofk\u0019skyrmions will\nbe cylindrically symmetric, given by \b0(r;') ='+\u0019\ndue to the right-handed rotational sense and\n\u00020(r;') = \u0002 0(r), which is the solution of the\nEuler–Lagrange equation\nA\u0012\n@2\nr\u00020+1\nr@r\u00020\u00001\nr2sin \u0002 0cos \u0002 0\u0013\n+jDj1\nrsin2\u00020\n+Ksin \u0002 0cos \u0002 0\u00001\n2MBsin \u0002 0= 0: (8)\nThe skyrmion order kis encapsulated in the bound-\nary conditions \u00020(0) =k\u0019;\u00020(1) = 0. Equation (8) is\nsolved numerically in a finite interval r2[0;R]signifi-\ncantly larger than the equilibrium k\u0019skyrmion size. A\nfirst approximation to the spin structure is constructed\nbased on the corresponding initial value problem usingthe shooting method [25], then iteratively optimizing the\nstructure using a finite-difference discretization.\nThe spin wave Hamiltonian may be determined anal-\nogously to Eq. (2), by using the local coordinate system\n\u0002 = \u0002 0+~Sx;\b = \b 0+1\nsin \u0002 0~Sy. ThematricesinEqs.(3)-\n(5) are replaced by the operators\nA1=\u00002A\u0012\nr2\u00001\nr2cos 2\u0002 0\u0013\n\u00002jDj1\nrsin 2\u0002 0\n\u00002Kcos 2\u0002 0+MBcos \u0002 0; (9)\nA2= 4A1\nr2cos \u0002 0@'\u00002jDj1\nrsin \u0002 0@'; (10)\nA3=\u00002A\u001a\nr2+\u0014\n(@r\u00020)2\u00001\nr2cos2\u00020\u0015\u001b\n\u00002jDj\u0012\n@r\u00020+1\nrsin \u0002 0cos \u0002 0\u0013\n\u00002Kcos2\u00020+MBcos \u0002 0: (11)\nDue to the cylindrical symmetry of the structure, the\nsolutions of Eq. (6) are sought in the form ~S?(r;';t ) =\ne\u0000i!n;mteim'~S?\nn;m(r), performing the replacements @t!\n\u0000i!n;mand@'!im. For each angular momentum\nquantum number m, an infinite number of solutions in-\ndexed bynmay be found, but only a few of these are\nlocated below !FMR =\r\nM(\u00002K+MB), hence repre-\nsenting localized spin wave modes of the k\u0019skyrmions.\nThe different nquantum numbers typically denote solu-\ntions with different numbers of nodes, analogously to the\nquantum-mechanical eigenstates of a particle in a box.\nBecause of the property HSW(m) =H\u0003\nSW(\u0000m)and\nHSWbeing self-adjoint, the eigenvalues of HSW(m)\nandHSW(\u0000m)coincide, leading to a double degeneracy\napart from the m= 0modes. The\u0006!qeigenvalue pairs\nofDSWdiscussed in Sec. IIA for the atomistic model at\n\u000b= 0in this case can be written as !n;m=\u0000!n;\u0000m.\nHowever, considering only the modes with Re !n;m>0,\none has!n;m6=!n;\u0000mindicating nonreciprocity or an\nenergy difference between clockwise ( m < 0) and coun-\nterclockwise ( m> 0) rotating modes [17, 23].\nFor finding the eigenvectors and eigenvalues of the\nmicromagnetic model, Eq. (6) is solved using a finite-\ndifference method on the r2[0;R]interval. For treat-\ning the Laplacian r2in Eqs. (9) and (11) the improved\ndiscretization scheme suggested in Ref. [44] was applied,\nwhich enables a more accurate treatment of modes with\neigenvalues converging to zero in the infinite and contin-\nuous micromagnetic limit.\nThe spin wave modes of the atomistic model discussed\nin Sec. IIA were assigned the (n;m)quantum numbers,\nwhich are strictly speaking only applicable in the mi-\ncromagnetic limit with perfect cylindrical symmetry, by\nvisualizingthe real-spacestructureofthe numericallyob-\ntained eigenvectors.4\nC. Goldstone modes and instabilities\nSince the translation of the k\u0019skyrmions on the\ncollinear background in the plane costs no energy, the\nspin wave Hamiltonian HSWpossesses two eigenvectors\nbelonging to zero eigenvalue, representing the Goldstone\nmodes of the system. Within the micromagnetic descrip-\ntion of Sec. IIB, these may be expressed analytically as\n[21–23]\n\u0010\n~Sx;~Sy\u0011\n=e\u0000i'\u0012\n\u0000@r\u00020;i1\nrsin \u0002 0\u0013\n;(12)\n\u0010\n~Sx;~Sy\u0011\n=ei'\u0012\n\u0000@r\u00020;\u0000i1\nrsin \u0002 0\u0013\n:(13)\nEquations (12) and (13) represent eigenvectors of the\ndynamical matrix DSWas well. From Eqs. (2) and (6) it\nfollows that the eigenvectors of HSWandDSWbelong-\ning to zero eigenvalue must coincide, HSW~S?=0,\nDSW~S?=0, because (\u0000i\u001by\u0000\u000b)in Eq. (6) is an in-\nvertible matrix. Because from the solutions of the equa-\ntion of motion (6) we will only keep the ones satisfying\nRe!n;m>0, the eigenvectors from Eqs. (12) and (13)\nwill be denoted as the single spin wave mode !0;\u00001= 0.\nSince the eigenvectors and eigenvalues are determined\nnumerically in a finite system by using a discretization\nprocedure, the Goldstone modes will possess a small fi-\nnite frequency. However, these will not be presented\nin Sec. IIIA together with the other frequencies since\nthey represent a numerical artifact. For the 1\u0019and\n3\u0019skyrmions the !0;1eigenmode has a positive fre-\nquency and an eigenvector clearly distinguishable from\nthat of the !0;\u00001translational mode. However, for the\n2\u0019skyrmion both the !0;\u00001and the!0;1eigenfrequen-\ncies ofDSWare very close to zero, and the correspond-\ning eigenvectors converge to Eqs. (12) and (13) as the\ndiscretization is refined and the system size is increased.\nThis can occur because DSWis not self-adjoint and its\neigenvectors are generally not orthogonal. In contrast,\nthe eigenvectors of HSWremain orthogonal, with only a\nsingle pair of them taking the form of Eqs. (12) and (13).\nIn contrast to the Goldstone modes with always zero\nenergy, the sign change of another eigenvalue of HSW\nindicates that the isolated k\u0019skyrmion is transformed\nfrom a stable local energy minimum into an unstable\nsaddle point, leading to its disappearance from the sys-\ntem. Such instabilities were determined by calculating\nthe lowest-lying eigenvalues of HSWin Eq. (2). Due\nto the connection between the HSWandDSWmatrices\nexpressed in Eq. (6), at least one of the precession fre-\nquencies!qwill also approach zero at such an instability\npoint.\nD. Effective damping parameters\nFor finite values of the Gilbert damping \u000b, the spin\nwaves in the system will decay over time as the systemrelaxes to the equilibrium state during the time evolu-\ntion described by the Landau–Lifshitz–Gilbert equation.\nThe speed of the relaxation can be characterized by the\neffective damping parameter, which for a given mode q\nis defined as\n\u000bq;eff=\f\f\f\fIm!q\nRe!q\f\f\f\f: (14)\nAs discussed in detail in Ref. [23], \u000bq;effis mode-\ndependent and can be significantly higher than the\nGilbert damping parameter \u000bdue to the elliptic polar-\nization of spin waves, which can primarily be attributed\nto the noncollinear spin structure of the k\u0019skyrmions.\nFor\u000b\u001c1,\u000bq;effmay be expressed as\n\u000bq;eff\n\u000b=X\ni\f\f\f~S(0);x\nq;i\f\f\f2\n+\f\f\f~S(0);y\nq;i\f\f\f2\nX\ni2Imh\u0010\n~S(0);x\nq;i\u0011\u0003~S(0);y\nq;ii;(15)\nwheretheeigenvectorsinEq.(15)arecalculatedat \u000b= 0\nfrom Eq. (6). Equation (15) may also be expressed by\nthe axes of the polarization ellipse of the spins in mode\nq, see Ref. [23] for details.\nForhighervaluesof \u000b, thecomplexfrequencies !qhave\nto be determined from Eq. (6), while the effective damp-\ning parameters can be calculated from Eq. (14). Also for\nfinite values of \u000bfor each frequency with Re !q>0there\nexists a pair with Re !q0<0such that!q0=\u0000!\u0003\nq[23].\nThe spin waves will be circularly polarized if A1=A3\nandAy\n2=\u0000A2in Eq. (2), in which case the dependence\nof!qon\u000bmay simply be expressed by the undamped\nfrequency!(0)\nqas\nRe!q(\u000b) =1\n1 +\u000b2!(0)\nq; (16)\njIm!q(\u000b)j=\u000b\n1 +\u000b2!(0)\nq: (17)\nThese relations are known for uniaxial ferromagnets;\nsee, e.g., Ref. [45]. In the elliptically polarized modes of\nnoncollinear structures, such as k\u0019skyrmions, a devia-\ntion from Eqs. (16)-(17) is expected.\nIII. RESULTS\nA. Eigenmodes\nThe frequencies of the localized spin wave modes of the\n1\u0019,2\u0019, and 3\u0019skyrmion, calculated from the atomistic\nmodel for\u000b= 0as described in Sec. IIA, are shown in\nFig. 1. For the 1\u0019skyrmion six localized modes can be\nobserved below the FMR frequency of the field-polarized\nbackground in Fig. 1(a), four of which are clockwise ro-\ntating modes ( m < 0), one is a gyration mode rotating\ncounterclockwise ( m= 1), while the final one is a breath-\ning mode (m= 0). The excitation frequencies show good5\n0.7 0.8 0.9 1.0 1.1 1.20255075100125150175\n(a)\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175\n(b)\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175\n(c)\nFIG. 1. Frequencies of localized spin wave modes at \u000b= 0for\n(a) the 1\u0019, (b) the 2\u0019, and (c) the 3\u0019skyrmion. Selected spin\nwave modes are visualized in contour plots of the out-of-plane\nspin component and denoted by open symbols connected by\nlines in the figure, the remaining modes are denoted by con-\nnected dots.quantitative agreement with the ones calculated from the\nmicromagnetic model for the same system in Ref. [23].\nCompared to Ref. [21], the additional appearance of the\neigenmodes with m= 1;\u00004;\u00005can be attributed to the\nfinite value of the anisotropy parameter Kin the present\ncase. Increasing the anisotropy value makes it possible\nto stabilize the skyrmions at lower field values, down to\nzerofieldatthecriticalvalueinthemicromagneticmodel\njKcj=\u00192D2\n16A, where the transition from the spin spiral\nto the ferromagnetic ground state occurs at zero exter-\nnal field [46]. Since the excitation frequencies decrease\nat lower field values as shown in Fig. 1(a), this favors\nthe appearance of further modes. Simultaneously, the\nFMR frequency increases with K, meaning that modes\nwith higher frequencies become observable for larger uni-\naxial anisotropy. For each angular momentum quantum\nnumberm, only a single mode ( n= 0) appears.\nIn the case of the 2\u0019skyrmion an increased number of\neigenmodes may be seen in Fig. 1(b). This can mainly be\nattributed to the appearance of spin waves with higher\nangular momentum quantum numbers both for clockwise\n(up tom=\u000017) and counterclockwise (up to m= 12)\nrotational directions. Furthermore, in this case modes\nwithn= 1node in the eigenfunction can be observed\nas well. The same trend continues in the case of 3\u0019\nskyrmionsinFig.1(c), thelargenumberofinternaleigen-\nmodes can be attributed to angular momentum quantum\nnumbers ranging from m=\u000022tom= 16, as well as to\nspin wave eigenvectors with up to n= 2nodes. The dif-\nferent rotational directions and numbers of nodes are il-\nlustrated in Supplemental Videos 1-4 [47] via the square-\nshaped modes ( n= 0;1,m=\u00064) of the 3\u0019skyrmion at\nB= 0:825T.\nThe increase of possible angular momentum quantum\nnumbers for higher skyrmion order kas well as for de-\ncreasing magnetic field Bmay be qualitatively explained\nby an increase in the skyrmion size. Modes with a given\nvalue ofmindicate a total of jmjmodulation periods\nalong the perimeter of the skyrmion; for larger skyrmion\nsizes this corresponds to a modulation on a longer length\nscale, which has a smaller cost in exchange energy.\nThe breathing modes of the 3\u0019skyrmion with dif-\nferent numbers of nodes are visualized in Fig. 2 at\nB= 1T. The results shown in Fig. 2 are obtained\nfrom the micromagnetic model in Sec. IIB, which is in\ngood quantitative agreement with the atomistic calcu-\nlations at the given field. All the eigenmodes display\nthree peaks of various heights, while they decay expo-\nnentially outside the 3\u0019skyrmion. As can be seen in\nFig. 2, the peaks are localized roughly around the re-\ngions where the spins are lying in-plane, indicated by\nthe domain walls (DW) between pairs of dashed lines.\nThe widths of the domain walls were determined by ap-\nproximating the 3\u0019skyrmion profile with linear func-\ntions close to the inflection points rj;\u00020;j;j= 1;2;3\nwhere the spins are lying in-plane, and calculating where\nthese linear functions intersect integer multiples of \u0019in6\n0 10 20 30 40 50-2-023\n-0.100.000.10\nFIG. 2. Comparison between the 3\u0019skyrmion profile (left\nvertical axis) and the eigenvectors of the breathing modes\n(m= 0) with different numbers of nodes n= 0;1;2(right\nvertical axis). The calculations were performed using the mi-\ncromagnetic model described in Sec. IIB at B= 1T, the\nlattice constant is a= 0:271nm. Double arrows between ver-\ntical dashed lines indicate the extensions of the domain walls\nin the structure.\n\u00020. Thus, the domain walls are located between the in-\nnerRin;j=rj+[@r\u00020(rj)]\u00001[(4\u0000j)\u0019\u0000\u00020;j]and outer\nRout;j=rj+ [@r\u00020(rj)]\u00001[(3\u0000j)\u0019\u0000\u00020;j]radii. Such\na description was used to calculate the skyrmion radius\nin, e.g., Ref. [46], and it was also applied for calculating\nthe widths of planar domain walls [48].\nThe nodes of the eigenmodes are located roughly be-\ntween these domain walls, meaning that typically excita-\ntion modes with n= 0;:::;k\u00001nodes may be observed\nink\u0019skyrmions, in agreement with the results in Fig. 1.\nA higher number of nodes would require splitting a single\npeak into multiple peaks, the energy cost of which gen-\nerally exceeds the FMR frequency, thereby making these\nmodes unobservable. The sign changes in the ~Sx\nn;meigen-\nvectors mean that the different modes can be imagined\nas the domain walls breathing in the same phase or in\nopposite phase, as can be seen in Supplemental Videos\n5-7 [47]. Note that eigenmodes with higher nquantum\nnumbers may also be observed for skyrmions confined in\nnanodots [14–16] where the peaks of the eigenmodes may\nalso be localized at the edge of the sample, in contrast\nto the present case where isolated k\u0019skyrmions are dis-\ncussed on an infinite collinear background.\nItisalsoworthnotingthatthelowest-lyingnonzerogy-\nration mode is n= 0;m= 1for the 1\u0019and3\u0019skyrmions,\nwhile it isn= 1;m= 1for the 2\u0019skyrmion, see Fig. 1.\nAs already mentioned in Sec. IIC, numerical calculations\nfor the 2\u0019skyrmion indicate both in the atomistic and\nthemicromagneticcasethatbyincreasingthesystemsize\nor refining the discretization the eigenvectors of both the\nn= 0;m=\u00001and then= 0;m= 1modes ofDSW\nin Eq. (6) converge to the same eigenvectors in Eqs. (12)\nand(13)and 0eigenvalue, whichcorrespondtothetrans-\nlational Goldstone mode in the infinite system. This dif-ference can probably be attributed to the deviation in\nthe value of the topological charge, being finite for 1\u0019\nand3\u0019skyrmions but zero for the 2\u0019skyrmion [35].\nB. Instabilities\nSkyrmions with different order kdeviate in their low-\nfield behavior. Since the considered Pd/Fe/Ir(111) sys-\ntem has a spin spiral ground state [38], decreasing the\nmagnetic field value will make the formation of domain\nwallsenergeticallypreferableinthesystem. Inthecaseof\nthe1\u0019skyrmion this means that the lowest-lying eigen-\nmode ofHSWin Eq. (2), which is an elliptic mode with\nm=\u00062, changes sign from positive to negative, occur-\nring between B= 0:650T andB= 0:625T in the present\nsystem. This is indicated in Fig. 1(a) by the fact that the\nfrequency of the n= 0;m=\u00002eigenmode of DSWin\nEq. (6) converges to zero. This leads to an elongation of\nthe skyrmion into a spin spiral segment which gradually\nfills the ferromagnetic background, a so-called strip-out\nor elliptic instability already discussed in previous publi-\ncations [21, 46]. In contrast, for the 2\u0019and3\u0019skyrmions\nthe lowest-lying eigenmode of HSWis a breathing mode\nwithm= 0, which tends to zero between B= 0:800T\nandB= 0:775Tforbothskyrmions. Thisisindicatedby\nthe lowest-lying n= 0;m= 0mode ofDSWin Fig. 1(b)\nfor the 2\u0019skyrmion, which is the second lowest after the\nn= 0;m= 1mode for the 3\u0019skyrmion in Fig. 1(c). This\nmeans that the radius of the outer two rings of 2\u0019and\n3\u0019skyrmions diverges at a finite field value, leading to a\nburst instability. Such a type of instability was already\nshown to occur in Ref. [25] in the case of a ferromagnetic\nground state at negative field values, in which case it also\naffects 1\u0019skyrmions.\nAt the burst instability, modes with n= 0and all\nangular momentum quantum numbers mappear to ap-\nproach zero because of the drastic increase in skyrmion\nradius decreasing the frequency of these modes as dis-\ncussed in Sec. IIIA. A similar effect was observed for the\n1\u0019skyrmion in Ref. [22] when the critical value of the\nDzyaloshinsky–Moriya interaction, jDcj=4\n\u0019p\nAjKj, was\napproached at zero external field from the direction of\nthe ferromagnetic ground state. In contrast, the ellip-\ntic instability only seems to affect the n= 0;m=\u00002\nmode, while other mvalues and the nonreciprocity are\napparently weakly influenced.\nIn the atomistic model, skyrmions collapse when their\ncharacteristic size becomes comparable to the lattice\nconstant. For the 1\u0019,2\u0019, and 3\u0019skyrmions the col-\nlapse of the innermost ring occurs at Bc;1\u0019\u00194:495T,\nBc;2\u0019\u00191:175T, andBc;3\u0019\u00191:155T, respectively [35].\nAs can be seen in Figs. 1(b), 1(c), and 3, this instabil-\nity is again signaled by the n= 0;m= 0eigenfrequency\ngoing to zero, but in contrast to the burst instability,\nthe other excitation frequencies keep increasing with the\nfield in this regime. Figure 3 demonstrates that close to\nthe collapse field the excitation frequency may be well7\n4.45 4.46 4.47 4.48 4.49 4.50020406080100\nFIG. 3. Frequency of the breathing mode n= 0;m = 0of\nthe 1\u0019skyrmion close to the collapse field. Calculation data\nare shown by open symbols, red line denotes the power-law\nfitf0;0=Af(Bc;1\u0019\u0000B)\ff.\napproximated by the power law f0;0=Af(Bc;1\u0019\u0000B)\ff,\nwithAf= 175:6GHz\nT\ff,Bc;1\u0019= 4:4957T, and\ff= 0:23.\nC. Effective damping parameters in the limit of\nlow\u000b\nThe effective damping parameters \u000bn;m;effwere first\ncalculated from the eigenvectors obtained at \u000b= 0fol-\nlowing Eq. (15). The results for the 1\u0019,2\u0019, and 3\u0019\nskyrmions are summarized in Fig. 4. As discussed in\nRef. [23], the \u000bn;m;effvalues are always larger than the\nGilbert damping \u000b, and they tend to decrease with in-\ncreasing angular momentum quantum number jmjand\nmagnetic field B. The spin wave possessing the high-\nest effective damping is the n= 0;m= 0breathing\nmode both for the 1\u0019and 2\u0019skyrmion, but it is the\nn= 0;m= 1gyration mode for the 3\u0019skyrmion for a\nlarge part of the external field range where the struc-\nture is stable. Excitation pairs with quantum num-\nbersn;\u0006mtend to decay with similar \u000bn;m;effvalues to\neach other, with \u000bn;jmj;eff< \u000bn;\u0000jmj;eff, where clockwise\nmodes (m < 0) have lower frequencies and higher effec-\ntive damping due to the nonreciprocity.\nThe effective damping parameters drastically increase\nand for the lowest-lying modes apparently diverge close\nto the burst instability, while no such sign of nonan-\nalytical behavior can be observed in the case of the\n1\u0019skyrmion with the elliptic instability. For the same\nn;mmode, the effective damping parameter tends to in-\ncrease with skyrmion order kaway from the critical field\nregimes; for example, for the n= 0;m= 0mode at\nB= 1:00T one finds \u000b0;0;eff;1\u0019= 2:04,\u000b0;0;eff;2\u0019= 5:87,\nand\u000b0;0;eff;3\u0019= 10:09.\nClose to the collapse field, the effective damping pa-\nrameter of the n= 0;m= 0breathing mode tends to\n0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.5\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100\n0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100FIG.4. Effectivedamping parameters calculatedaccordingto\nEq. (15) for the eigenmodes of the (a) 1\u0019, (b) 2\u0019, and (c) 3\u0019\nskyrmions, plotted on a logarithmic scale. The corresponding\nexcitation frequencies are shown in Fig. 1.\ndiverge as shown in Figs. 4(b), 4(c), and 5 for the 2\u0019,\n3\u0019, and 1\u0019skyrmions, respectively. Similarly to the\neigenfrequency converging to zero in Fig. 3, the criti-\ncal behavior of the effective damping may be approxi-\nmated by a power-law fit \u000b0;0;eff=A\u000b(Bc;1\u0019\u0000B)\u0000\f\u000b\nas shown in Fig. 5, this time with a negative exponent8\n4.45 4.46 4.47 4.48 4.49 4.50024681012\nFIG. 5. Effective damping parameter \u000b0;0;effof the breathing\nmoden= 0;m= 0of the 1\u0019skyrmion close to the collapse\nfield. The corresponding excitation frequencies are shown in\nFig. 3. Calculation data are shown by open symbols, red line\ndenotes the power-law fit \u000b0;0;eff=A\u000b(Bc;1\u0019\u0000B)\u0000\f\u000b.\ndue to the divergence. The fitting yields the parameters\nA\u000b= 0:96T\f\u000b,Bc;1\u0019= 4:4957T, and\f\u000b= 0:23. Natu-\nrally, the critical field values agree between the two fits,\nbut interestingly one also finds \ff=\f\u000bup to two digits\nprecision. Rearranging Eq. (14) yields\n\u000b0;0;eff\n\u000bRe!0;0=1\n\u000bjIm!0;0j; (18)\nwhere the left-hand side is proportional to\n(Bc;1\u0019\u0000B)\ff\u0000\f\u000bwhich is approximately constant\ndue to the exponents canceling. This indicates that\nwhile Re!0;0diverges close to the collapse field,\njIm!0;0j=\u000bremains almost constant at low \u000bvalues.\nD. Damping for higher \u000bvalues\nDue tothe divergences oftheeffective damping param-\neters found at the burst instability and collapse fields, it\nis worthwhile to investigate the consequences of using a\nfinite\u000bvalue in Eq. (6), in contrast to relying on Eq. (15)\nwhich is determined from the eigenvectors at \u000b= 0. The\n\u000bdependence of the real and imaginary parts of the !0;0\nbreathingmodefrequencyofthe 1\u0019skyrmionisdisplayed\nin Fig. 6, at a field value of B= 1T far from the el-\nliptic and collapse instabilities. As shown in Fig. 6(a),\nunlike circularly polarized modes described by Eq. (16)\nwhere Re!qdecreases smoothly and equals half of the\nundamped value at \u000b= 1, the Re!0;0value for the ellip-\ntically polarized eigenmode displays a much faster decay\nand reaches exactly zero at around \u000b\u00190:58. According\ntoEq.(14), thisindicatesthatthecorrespondingeffective\ndamping parameter \u000b0;0;effdiverges at this point.\nSince the real part of the frequency disappears, the\n!q0=\u0000!\u0003\nqrelation connecting Re !q>0and Re!q0<0\n0.0 0.2 0.4 0.6 0.8 1.00102030405060\n0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.6FIG. 6. (a) Frequency f0;0=Re!0;0=2\u0019and (b) inverse\nlifetimejIm!0;0jof then= 0;m = 0breathing mode of\nthe 1\u0019skyrmion at B= 1T as a function of the Gilbert\ndamping parameter \u000b. The solutions of Eq. (6) for the ellip-\ntically polarized eigenmode of the 1\u0019skyrmion are compared\nto Eqs. (16)-(17) which are only valid for circularly polarized\nmodes.\nsolutions of Eq. (6) discussed in Sec. IID no longer holds,\nand two different purely imaginary eigenfrequencies are\nfound in this regime as shown in Fig. 6(b). This is analo-\ngous to overdamping in a classical linear harmonic oscil-\nlator, meaningthatthepurelyprecessionalfirst-orderdif-\nferential equation describing circularly polarized modes\nis transformed into two coupled first-order differential\nequations [23] with an effective mass term for the breath-\ning mode of k\u0019skyrmions. This implies that when per-\nforming spin dynamics simulations based on the Landau–\nLifshitz–Gilbert equation, the value of the Gilbert damp-\ning parameter has to be chosen carefully if the fastest\nrelaxation to the equilibrium spin structure is required.\nThe high effective damping of the breathing mode in the\n\u000b\u001c1limit (cf. Fig. 4(a)) ensures that the inverse\nlifetime of the elliptically polarized excitations remains\nlarger for a wide range of \u000bvalues in Fig. 6(b) than what\nwould be expected for circularly polarized modes based9\n0.85 0.90 0.95 1.00 1.05 1.10 1.1505101520\n0.85 0.90 0.95 1.00 1.05 1.10 1.150.000.020.040.060.080.10\nFIG. 7. (a) Frequency f0;0=Re!0;0=2\u0019and (b) inverse\nlifetimejIm!0;0jof then= 0;m= 0breathing mode of the\n2\u0019skyrmion at \u000b= 0:1as a function of the external magnetic\nfieldB. The solutions of Eq. (6) for the elliptically polarized\neigenmode of the 2\u0019skyrmion are compared to Eqs. (16)-(17)\nwhich are only valid for circularly polarized modes.\non Eq. (17). Note that contrary to Sec. IIIB, Re !0;0be-\ncomingzeroinFig.6(a)doesnotindicateaninstabilityof\nthe system, since stability is determined by the eigenval-\nues of the matrix HSWin Eq. (2) which are independent\nof\u000b.\nSince the disappearance of Re !0;0and the bifurcation\nof Im!0;0occurs as the excitation frequency becomes\nsmaller, it is expected that such an effect may also be ob-\nservedatafixed \u000bvalueastheexternalfieldisdecreased.\nThis is illustrated for the n= 0;m= 0breathing mode\nof the 2\u0019skyrmion in Fig. 7 at \u000b= 0:1. For this interme-\ndiate value of the damping, the breathing mode becomes\noverdamped around B= 0:875T, which is significantly\nhigher than the burst instability between B= 0:775T\nandB= 0:800T (cf. Fig. 1(b) and the circularly polar-\nized approximation in Fig. 7(a)). This means that the\nlowest-lying breathing mode of the 2\u0019skyrmion cannot\nbe excited below this external field value. In Fig. 7(b) it\ncan be observed that contrary to the circularly polarizedapproximation Eq. (17) following the field dependence of\nthe frequency, for the actual elliptically polarized eigen-\nmodejIm!0;0jisalmostconstantforallfieldvaluesabove\nthebifurcationpoint. Althoughasimilarobservationwas\nmade at the end of Sec. IIIC as the system approached\nthe collapse field at \u000b= 0, it is to be emphasized again\nthat no instability occurs where Re !0;0disappears in\nFig. 7(a).\nIV. CONCLUSION\nIn summary, the localized spin wave modes of k\u0019\nskyrmions were investigated in an atomistic spin model,\nwith parameters based on the Pd/Fe/Ir(111) system. It\nwas found that the number of observable modes increases\nwith skyrmion order k, firstly because of excitations with\nhigher angular momentum quantum numbers mforming\nalong the larger perimeter of the skyrmion, secondly be-\ncause of nodes appearing between the multiple domain\nwalls. It was found that the 2\u0019and3\u0019skyrmions un-\ndergo a burst instability at low fields, in contrast to the\nelliptic instability of the 1\u0019skyrmion. At high field val-\nues the innermost ring of the structure collapses in all\ncases, connected to an instability of a breathing mode.\nThe effective damping parameters of the excitation\nmodes were determined, and it was found that for the\nsamen;mmode they tend to increase with skyrmion\norderk. The effective damping parameter of the n=\n0;m= 0breathing mode diverges at the burst and\ncollapse instabilities, but no such effect was observed\nin case of the elliptic instability. For higher values of\nthe Gilbert damping parameter \u000ba deviation from the\nbehavior of circularly polarized modes has been found,\nwith the breathing modes becoming overdamped. It was\ndemonstrated that such an overdamping may be observ-\nable in 2\u0019and3\u0019skyrmions for intermediate values of\nthe damping significantly above the burst instability field\nwhere the structures themselves disappear from the sys-\ntem.\nThe results presented here may motivate further ex-\nperimental and theoretical studies on k\u0019skyrmions, of-\nfering a wider selection of localized excitations compared\nto the 1\u0019skyrmion, thereby opening further possibilities\nin magnonics applications.\nACKNOWLEDGMENTS\nThe authors would like to thank A. Siemens for fruit-\nful discussions. Financial support for this work from the\nAlexander von Humboldt Foundation, from the Deutsche\nForschungsgemeinschaft via SFB 668, from the European\nUnion via the Horizon 2020 research and innovation pro-\ngram under Grant Agreement No. 665095 (MAGicSky),\nand from the National Research, Development and Inno-\nvation Office of Hungary under Project No. K115575 is\ngratefully acknowledged.10\n[1] A. N. Bogdanov and D. A. Yablonski ˘i, Sov. Phys. JETP\n68, 101 (1989).\n[2] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8,\n152 (2013).\n[3] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. B.\nJungfleisch, J. E. Pearson, X. Cheng, O. Heinonen, K. L.\nWang, Y. Zhou, A. Hoffmann, and S. G. E. te Velthuis,\nNat. Phys. 13, 162 (2017).\n[4] P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von\nBergmann, and R. Wiesendanger, Nat. Nanotechnol. 12,\n123 (2017).\n[5] F. Büttner, I. Lemesh, M. Schneider, B. Pfau, C. M.\nGünther, P. Hessing, J. Geilhufe, L. Caretta, D. Engel,\nB. Krüger, J. 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Schäfer, Magnetic Domains (Springer,\nBerlin, 1998)."    },    {        "title": "1810.06875v1.Spin_wave_induced_lateral_temperature_gradient_in_a_YIG_thin_film_GGG_system_excited_in_an_ESR_cavity.pdf",        "content": "1  Spin-wave-induced lateral temperature gradient in a YIG thin film/GGG system excited in an ESR cavity Ei Shigematsu, Yuichiro Ando, Sergey Dushenko, Teruya Shinjo, and Masashi Shiraishi Department of Electronic Science and Engineering, Kyoto University, 615-8510, Kyoto, Japan  Lateral thermal gradient of an yttrium iron garnet (YIG) film under the microwave application in the cavity of the electron spin resonance system (ESR) was measured at room temperature by fabricating a Cu/Sb thermocouple onto it. To date, thermal transport in YIG films caused by the Damon-Eshbach mode (DEM)—the unidirectional spin-wave heat conveyer effect—was demonstrated only by the excitation using coplanar waveguides. Here we show that effect exists even under YIG excitation using the ESR cavity—tool often employed to realize spin pumping. The temperature difference observed around the ferromagnetic resonance (FMR) field under the 4 mW microwave power peaked at 13 mK. The observed thermoelectric signal indicates the imbalance of the population between the DEMs that propagate near the top and bottom surfaces of the YIG film. We attribute the DEM population imbalance to the different magnetic damping near the top and bottom YIG surfaces. Additionally, the spin wave dynamics of the system were investigated using the micromagnetic simulations. The micromagnetic simulations confirmed the existence of the DEM imbalance in the system with the increased Gilbert damping at one of the YIG interfaces. The reported results are indispensable for the quantitative estimation of the electromotive force in the spin-charge conversion experiments using ESR cavities.   2  Spin caloritronics1—young but quickly developing spintronics field—is in pursuit of the comprehensive understanding of the connection between heat and spin currents. Following the discovery of the spin Seebeck effect2, plenty of experimental demonstrations of spin caloritronic phenomena have been reported, such as the spin-dependent Seebeck effect3 and the spin Peltier effect4. Especially, the heat transport via spin waves and spin-phonon interaction has attracted attention after the unidirectional spin-wave heat conveyer effect was reported5,6. In contrast to the conventional case of the heat transport against the temperature gradient, Damon-Eshbach mode (DEM) spin wave7,8 induces heat transport in the direction of the thermal gradient. Apart from this surprising achievement, the unidirectional spin-wave heat conveyer effect has important implications in the field of the dynamical spin injection, also known as spin pumping, since they often occur simultaneously in a studied system. Spin pumping9,10 is a method of generation of the pure spin current in the material due to the coupling of the interface spins to the precession magnetization of the adjacent ferromagnet layer. It quickly gained popularity as a spin injection method that can easily be used in any bilayer system consisting of nonmagnetic/ferromagnetic material11,12 whereas the electrical spin injection needs more elaborate surface treatment, formation of tunnel barriers and nanofabrication13–15. Using the spin pumping technique, the spin-to-charge conversion-related properties of various heavy metals11,16, semimetals17–19, semiconductors20–22 and even two-dimensional materials23–25 were unveiled, along with the spin transport properties of the materials 24,26,27. The DEM is the surface spin wave that is excited under the conditions close to the ferromagnetic resonance (FMR) and propagates in the opposite directions on the top and bottom surfaces28,29. When DEM reaches the end of the sample, its energy is damped as the heat, raising the temperature near the sample edge. In the case of the uniform excitation across the ferromagnet, the population of the DEM on top and bottom surfaces is the same, 3  and the net quantity of the transported heat cancels out. However, when the equivalence of the population of the two DEM propagating in the opposite directions is broken, the unidirectional thermal transport takes place. In the previous studies of the unidirectional spin-wave heat conveyer effect5,30, such inequivalence was shown to be present in case of the DEM excitation using the microstrip lines waveguides. In that case, the bottom surface of the ferromagnet is located closer to the microstrip line than the top surface, thus difference in the intensity of the microwave AC magnetic field causes population difference of the two DEM spin waves. Induced unidirectional heat transport happens in the direction of the propagation of the dominant DEM. Importantly, direction of the propagation of the dominant DEM (wave vector k) can be reversed by reversing the direction of the external magnetic field. Thus, voltage generated due to thermal effects (for example, the Seebeck effect) also reversed with the direction of the magnetic field. Incidentally, the spin pumping and spin-charge conversion experiments rely heavily on the reversal of the magnetic field to exclude non-magnetic spurious effects, including the thermal ones: sign reversal of the generated electromotive force with the external magnetic field usually taken as a proof of its spin-charge conversion origin. Thus, to confirm the origin of the electromotive force in the spin-charge conversion experiments, it is crucial to precisely estimate the unidirectional heat transfer induced by the DEM. While there were a few experimental5,30 and theoretical31 studies on the unidirectional heat transfer effect under the microwave excitation using wave guides, there were no such reports in the microwave cavities. In contrast, broad variety of the spin pumping and spin-charge conversion experiments are carried out using the TE011 cavity of the electron spin resonance (ESR) systems11,32,33. Our study fills the experimental gap, and reports the observation of the heat transfer by the DEM in the TE011 ESR cavity. We also performed the micromagnetic simulations, and discuss the origin of the DEM imbalance 4  observed experimentally. We now proceed to the experimental details and results. The 1.2-µm-thick YIG film was grown by liquid phase epitaxy on top of the GGG substrate and is available commercially (Granopt, Japan). We fabricated thermocouple on top of the YIG surface to measure temperature difference generated due to the heat transport by the DEM. While there are many types of thermocouples available commercially, the most common ones (types E, J, K, T) use ferromagnetic metals nickel (Ni) and iron (Fe), or their alloys, which may exhibit ferromagnetism due to the insufficient uniformity of the alloy. In the spin pumping experiments the lateral static magnetic field is applied in plane of the samples under the ferromagnetic resonance condition. In this geometry, the anomalous Hall effect in the thermocouple may be induced by the heating of the YIG film, which would add up to the electromotive force generated by the lateral thermal gradient of YIG film and prevent its quantitative estimation. To realize a thermocouple comprised of nonmagnetic metals, we focus on the combination of copper (Cu) and antimony (Sb), and use Cu wiring to make an electrical contact to the sample. First, we formed a 50-nm-thick SiO2 insulating layer on top of YIG to exclude the influence of spin pumping, which was shown to decrease exponentially with the thickness of the tunnel barrier34 . On top of it, 50-nm-thick Sb layer was deposited by resistance heating deposition. Finally, the third layer consisting of two Cu pads separated by a 1 mm gap was deposited. The sample with the formed thermocouple was set in the Seebeck effect measurement system (Fig.1(a)). Room temperature acted as a baseline level, while the hot and cold heat sinks—the temperature of which was controlled by the Peltier elements—were attached to the opposite sides of the sample. Lateral temperature difference and thermoelectric electromotive force were monitored simultaneously. For the ferromagnetic resonance measurements, the sample was mounted into the TE011 cavity of the ESR system (JEOL JES-FA200) at room temperature. The DC and AC magnetic fields were applied in 5  plane of the sample in DEM geometry as shown in Fig.1(b). The frequency of the AC magnetic field was set to 9.12 GHz, and applied microwave power was set to 4 mW. An estimated value of AC magnetic field applied to the sample was 2.2 µT. The DC magnetic field was swept through the FMR field of the YIG film, while the microwave absorption spectrum and the electromotive force between Cu electrodes were measured simultaneously. Since we used a bipolar electromagnet, measurements in 0° and 180° DC magnetic field are carried out without rotating the sample position Figure 1(a) and Figure 2 show the schematic layout and the detected thermoelectric electromotive force in the Seebeck effect measurement of the Cu/Sb thermocouple fabricated on top of the the YIG/GGG sample. We follow the conventional definition of the Seebeck coefficient S:  Δ𝑉=−𝑆Δ𝑇. (1) where ΔV and ΔT are the thermoelectric electromotive force and the temperature difference, respectively. From the linear fitting (black solid line in Fig. 2), the Seebeck coefficient of the fabricated sample was determined to be +15 nV/mK. This result is comparable to the Seebeck coefficient of amorphous Sb film reported in the literature35.  Next, the sample was placed in the cavity of the ESR system for the measurement of the magnetic-field-dependent heat transport induced by the DEM. The ferromagnetic resonance measurements with simultaneous detection of the electromotive force and FMR spectra were carried for the opposite directions of the DC magnetic field 0° and 180°. The wave vector k of the DEM is parallel to the cross product of the DC component of the magnetization of the YIG film M and the normal vector to the surface n. Direction of the k determines the direction ζΔT of the generated temperature difference ΔT on the propagation surface5,7,  k // ζΔT // M × n (2) 6  Therefore, we extracted the magnetization-dependent component of the observed thermoelectric signal by subtracting the signals measured at 0° and 180° direction of external magnetic field (V0° and V180°, correspondingly). Figure 3 shows the thermoelectric signal generated by the DEM, which is given by Vm = (V0° - V180°)/2, and the FMR spectra at the DC magnetic fields of 0° and 180°. The coincidence of the two FMR spectra confirms the identical resonance conditions for the opposite directions of the DC magnetic field. Interestingly, the DEM thermoelectric signal shows reversal of the polarity when approaching the FMR condition. Following the results of the Seebeck effect measurement of the sample, positive Vm signal corresponds to +y direction of the thermal gradient ζΔT, which is due to the DEM at the GGG/YIG interface, and negative Vm to -y direction, which is due to the DEM at the SiO2/YIG interface, respectively. The amplitude of the negative peak of the thermoelectric signal was measured to be -190 nV. Using the Seebeck coefficient of the sample, the estimated temperature difference between the Cu pads separated by the 1 mm gap (-y direction) is 13 mK. Figure 4 shows the schematic layout of the DEM excitation in our measuring geometry for 0° direction of the external magnetic field. The thermal gradient direction ζΔT of -y (+y) suggests the contribution of the DEM from the YIG interface with the SiO2 (the GGG) film. Note that the uniformly excited DEMs in the thin ferromagnetic film has the same population of the +k and -k modes, thus they transfer the equal amount of heat in the opposite directions and the induced temperature differences by the two modes cancel each other out. Therefore, the negative peak of the thermoelectric electromotive force at the DC magnetic field close to the FMR condition signifies that the magnitude of the DEM at the SiO2/YIG interface is superior to that on the GGG/YIG interface. The previous analytical magnetostatic studies of the DEM assumed that the ferromagnetic film was placed in the vacuum and did not treat the symmetry breaking of the top and bottom sides of the film7. Furthermore, the influence of the Gilbert damping on the 7  DEM propagation and damping was not considered. We carried out numerical micromagnetic simulations that evaluate effect of the symmetry breaking in our SiO2/YIG/GGG system on the DEM population using program MuMax3 36. GGG is known as a paramagnetic material with substantially large magnetization. Influence of the GGG layer attached to the YIG interface on the Gilbert damping of the surface YIG layer was already pointed out30. Thus, in the micromagnetic simulations, we set the Gilbert damping parameter α of one marginal layer next to the YIG/GGG interface (we refer to it as the bottom layer) larger than the other layers. The Gilbert damping parameter of the bottom layer was 0.02 and that of the other layers was 0.002 (Fig. 5(a)). As for the other magnetic parameters, we use those of permalloy presented in the specification paper of MuMax3 36, as a simple magnetic thin film model. The saturation magnetization and the exchange stiffness were set to be 8.6×105 A/m and 1.3×10-11 J/m, respectively. At the beginning of the simulation, the DC magnetic field was set, and the magnetization of the whole system was relaxed. Following that, the AC excitation of the magnetic field was applied, and—after the magnetization precession reached the steady state—we extracted the z component of the magnetization of each spin cell in the slice of x = 25 (where coordinate represents layer number in that direction). The magnetization motion in the slice consists of a non-time-dependent bias, standing waves, and traveling waves along the y direction. We can evaluate the DEM by extracting a portion of the traveling waves. For this purpose, the Fourier transform was implemented:  𝑚!𝑓,𝑘!2𝜋=W(𝑡)∙W(𝑦)∙𝑚!𝑡,𝑦𝑒!!!!!\"!!!!!!d𝑡d𝑦!!\"#!!\"#!!\"#!!\"# (3) where mz, W, f, and ky denote the z component of the normalized magnetization, the hamming window function, the excitation frequency, and the wavenumber of the magnetization in y direction, respectively. As we extracted the data in the finite range of [[ymin, ymax],[tmin, tmax]], we applied the hamming function to reduce obstructive sub lobes in the resulting Fourier spectra. We focus on the Fourier spectra in the frequency of the excitation f0, which was set 8  to 15 GHz. When 𝑚!𝑓!,!!!! is deconvoluted into 𝑚!𝑓!,!!!!=𝐴+𝐵j and 𝑚!−𝑓!,!!!!!=𝐶+𝐷j, the absolute amplitude with wavenumber k leads to 𝑚!!\"#!!!!=𝐴+𝐶!+−𝐵+𝐷!. Then we obtain the wave distribution as a function of the wavenumber ky/2π. A plot of the amplitude mzabs vs. wavenumber at different DC magnetic fields is shown in Fig. 5(b). The extracted plot at the magnetic field of 325.6 mT is also shown in Fig. 5(c). The clear break of the symmetry can be seen between the wavenumber spectra in the top (red line) and bottom layers (green line). The peak height in the top layer was superior to that in the bottom layer. Figure 5(d) shows the absolute amplitude mzabs in the central layer (top box), which is analogous to the FMR spectrum detected experimentally; the subtraction of the mzabs between the top and bottom layers (middle box), which characterizes the imbalance of the DEM between them; and the wave number (bottom box) corresponding to the maximum mzabs in the top (red filled circles) and bottom (green filled circles) layers. The peak of the DEM in simulated spectra appeared at the lower magnetic field than the FMR, in agreement with theoretical and experimental results in the literature37. The DEM modes in the top and bottom layers had opposite sign of the wave number (Fig. 4(d) bottom), i.e. the propagation direction of the DEM, and were consistent with DEM propagation direction in the previous studies5,30. Thus, the numerical simulation showed the imbalance between the DEM in the top and bottom layers due to the difference in damping constant. Finally, we performed MuMax3 simulations using parameters close to the experimental values. The AC magnetic field excitation frequency was set to f0 = 9.12 GHz. The saturation magnetization of the YIG layer was set to be 1.275×105 A/m, and the exchange stiffness to 3.7×10-12 J/m38. The calculated geometry is illustrated in Fig. 6(a). The Gilbert damping was 0.01 and 0.001 in the bottom layer and bulk, respectively. As in the case of permalloy, the DEM amplitudes in the top and bottom layers showed a clear difference (Fig. 6(b)). The difference in the amplitude between the two DEM propagating in the opposite directions peaked around the 9  FMR resonance field (Fig. 6(c)). Thus, numerical simulation of YIG also indicated that the top-layer DEM is dominant over the bottom-layer DEM due to the break of the reflection symmetry because of the different magnetic damping at the surfaces. This result explains the dominance of the heat generated by the top DEM observed in the experiment. We note that micromagnetic calculations for the full-size sample are necessary for the precise quantitative simulation of the experimental results, which was limited by the computational power in this study. Additionally, the quantitative determination of the heat drift velocity—a key parameter in the thermal distribution induced by the DEM imbalance—needs a more elaborate analysis of the spin-phonon interaction, which is left for the further study. However, our experimental and numerical results clearly show that reflection symmetry breaking between the two YIG surfaces by the magnetic damping at the interfaces induces the imbalanced DEM population and the unidirectional heat transfer.  In conclusion, we observed the unidirectional spin-wave heat conveyer effect in a 1.2-µm-thick YIG film under the uniform microwave excitation in the ESR cavity. The origin of the DEM imbalance that led to the heat transport is explained by the increased Gilbert damping at one of the YIG interfaces. The micromagnetic simulations confirmed the existence of the DEM imbalance in such system. Our study fills the experimental gap that existed in the literature on the unidirectional spin-wave heat conveyer effect generated in ESR cavity. The reported results are indispensable for the quantitative estimation of the electromotive force in the spin-charge conversion experiments using ESR cavities. Supplementary Material The supplementary material includes the following information, microwave power dependence of the detected electromotive force, dependence of the detected electromotive force on the direction and speed of the DC magnetic field sweep, reproducibility of the results, and details of the MuMax3 calculation. 10  Acknowledgements E.S. acknowledges the financial support from the JSPS Research Fellowship for Young Researchers and JSPS KAKENHI Grant No. 17J09520. This work was supported in part by MEXT (Innovative Area “Nano Spin Conversion Science” KAKENHI No. 26103003), Grant-in-Aid for Scientific Research (S) No. 16H06330, and Grant-in-Aid for Young Scientists (A) No. 16H06089. S.D. acknowledges support by JSPS Postdoctoral Fellowship and JSPS KAKENHI Grant No. 16F16064. The authors thank T. Takenobu and K. Kanahashi for the informative advice regarding the Seebeck coefficient measurement.    11  References 1 G.E. W Bauer, E. Saitoh, and B.J. van Wees, Nat. Mater. 11, 391 (2012). 2 K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 3 A. Slachter, F.L. Bakker, J.-P. Adam, and B.J. van Wees, Nat. Phys. 6, 879 (2010). 4 J. Flipse, F.K. Dejene, D. Wagenaar, G.E.W. Bauer, J. Ben Youssef, and B.J. Van Wees, Phys. Rev. Lett. 113, 027601 (2014). 5 T. An, V.I. Vasyuchka, K. Uchida, A. V Chumak, K. Yamaguchi, K. Harii, J. Ohe, M.B. Jungfleisch, Y. Kajiwara, H. Adachi, B. 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The fabricated sample was attached to the two Peltier elements that controlled the temperature difference between the edges of the sample. (b) A schematic image of the measurement of the DEM heat transfer under the FMR excitation in the ESR TE011 cavity.     \n Fig. 2. The thermoelectric electromotive force dependence on the applied temperature gradient for the Sb/Cu thermocouple fabricated on top of the YIG film. The black solid line is a linear fitting.   \n0100200300400500600-8000-6000-4000-20000ΔV\t\t[nV]ΔT\t\t[m K ]15      \n Fig. 3. Two lines are the electromotive forces observed under the microwave excitation of 4 mW at the DC magnetic fields of 0° and 180° A line in the middle box is the halved subtraction of the electromotive force (Vm). Two overlapped lines in the lower box are FMR spectra measured for 0° (red) and 180° (green) direction of the DC magnetic field.     SiO2faceGGGface-5. 0x10-70. 0\tV0°\tV180°EMF\t [V]200 nV\n-2. 0x10-70. 02. 0x10-7\t(V0°-V180°)/2Vm\t[V ]200 nV\n246. 0246. 5247. 0247. 5\t0°\t180°FMR\tspectrum\t[arb. \tuni t]Magneti c\tFi el d\t[mT]16   Fig. 4. A cross-sectional illustration of the thermal gradient generation by the DEM under the application of the external magnetic field in 0° direction. Direction of the k wave vector of the DEM is locked to the direction of the cross product of the YIG magnetization M and surface normal vector n.   \n  Fig. 5. (a) A schematic illustration of the structure used in micromagnetic simulation. The magnetic film had 31 layers in the z direction, and each layer consisted of 51 × 51 unit cells. Gilbert damping parameter was set to be 0.02 in the bottom layer, and 0.002 in the other layers. The directions of the DC and AC magnetic fields are indicated by the arrows. (b) The waterfall plot of the absolute amplitude of mz component in the x = 25 slice with respect to ky/2π and the DC magnetic field. The red and green lines indicate the top and bottom layer, respectively. (c) The absolute amplitude of mz component in the x = 25 slice in the top and bottom layer at the DC magnetic field 325.6 mT. The wave form of the top (bottom) layer is biased to the -ky/2π  (+ky/2π) direction, indicating the propagation direction of the DEM. The \n(a)(c)(d)\n(b)\nAbsolute amplitude [arb. unit]0. 00. 20. 40. 6\n-0. 020-0. 015-0. 010-0. 0050. 000\n300310320330340350-0. 004-0. 0020. 0000. 0020. 004FMR\t[arb. \tuni t]DEM\t[arb. \tuni t]\tz\t=\t30\t(Top)\tz\t=\t0\t(Bottom)ky/2π\t[1/51cel l ]Magneti c\tFi l ed\t[mT]-0. 10-0. 050. 000. 050. 100. 000. 020. 040. 060. 080. 10Absol ute\tampl i tude\t[arb. \tuni t]ky/2π  [1/51cel l ]\tz\t=\t30\t\t\t\t\t\t\t\t\t(T op)\tz\t=\t0\t\t\t\t\t\t\t\t\t(B ottom )325. 6\tmT17  amplitude of the main lobe in the top layer is higher than that in the bottom layer. (d) The upper box: the FMR intensity represented by the absolute amplitude of mz in the z = 15 layer). The middle box: difference in the absolute amplitude of mz between the bottom and top layers, indicating the DEM imbalance and heat transport amplitude. The lower box: the wave numbers corresponding to the maximum of the absolute amplitude of mz in top (red) and bottom (green) layers.   18    \n Fig. 6. (a) A schematic illustration of the structure used in micromagnetic simulation. The magnetic film had 21 layers in the z direction, and each layer consisted of 101 × 301 unit cells. Gilbert damping was set to be 0.01 in the bottom layer, and 0.001 in the others. (b) The absolute amplitude of mz component of the magnetization in the x = 50 slice in the top (red) and bottom (green) layers. The DC magnetic field was set to 251 (left box) and 253 mT (right box). (c) The upper box: The FMR intensity (the absolute amplitude of the z = 15 layer). The lower box: the DEM imbalance between bottom and top layers calculated from the micromagnetic simulation. Experimental measurements indicated the similar dominance of the top DEM from the observed heat transport.  (a)(c)(b)\n-1E-0401E-048. 4308. 4328. 434\n-1E-0401E-043. 1163. 1183. 120\n0.001Absol ute\tampl i tude\t[arb. \tuni t]ky/2π\t[1/301\tcel l ]\tz\t=\t20\t(Top)\tz\t=\t0\t(Bottom)251\tmT0246810\n246248250252254-8x10-4-6x10-4-4x10-4-2x10-40FMR\t[arb. \tuni t]\nMagneti c\tFi el d\t[mT]DEM\t[arb. \tuni t]253\tmTky/2π\t[1/301\tcel l ]\tz\t=\t20\t(Top)\tz\t=\t0\t(Bottom)0.001-1x10-40   1x10-4-1x10-40   1x10-4"    },    {        "title": "1810.07020v4.Superfluid_spin_transport_in_ferro__and_antiferromagnets.pdf",        "content": "Super\ruid spin transport in ferro- and antiferromagnets\nE. B. Sonin\nRacah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel\n(Dated: March 25, 2019)\nThis paper focuses on spin super\ruid transport, observation of which was recently reported in\nantiferromagnet Cr 2O3[Yuan et al. , Sci. Adv. 4, eaat1098 (2018)]. This paper analyzes the role of\ndissipation in transformation of spin current injected with incoherent magnons to a super\ruid spin\ncurrent near the interface where spin is injected. The Gilbert damping parameter in the Landau{\nLifshitz{Gilbert theory does not describe dissipation properly, and the dissipation parameters are\ncalculated from the Boltzmann equation for magnons scattered by defects. The two-\ruid theory is\ndeveloped similar to the two-\ruid theory for super\ruids. This theory shows that the in\ruence of\ntemperature variation in bulk on the super\ruid spin transport (bulk Seebeck e\u000bect) is weak at low\ntemperatures. The scenario that the results of Yuan et al. are connected with the Seebeck e\u000bect at\nthe interface between the spin detector and the sample is also discussed.\nThe Landau criterion for an antiferromagnet put in a magnetic \feld is derived from the spectrum\nof collective spin modes. The Landau instability starts in the gapped mode earlier than in the\nGoldstone gapless mode, in contrast to easy-plane ferromagnets where the Goldstone mode becomes\nunstable. The structure of the magnetic vortex in the geometry of the experiment is determined.\nThe vortex core has the skyrmion structure with \fnite magnetization component normal to the\nmagnetic \feld. This magnetization creates stray magnetic \felds around the exit point of the vortex\nline from the sample, which can be used for experimental detection of vortices.\nI. INTRODUCTION\nThe concept of spin super\ruidity is based on the anal-\nogy of the equations of magnetodynamics with the equa-\ntions of super\ruid hydrodynamics.1. The analogy led to\nthe suggestion that in magnetically ordered media persis-\ntent spin currents are possible, which are able to trans-\nport spin on macroscopical distances without essential\nlosses.2\nThe phenomenon of spin super\ruidity has been dis-\ncussed for several decades.2{15We de\fne the term super-\n\ruidity in its original meaning known from the times of\nKamerlingh Onnes and Kapitza: transport of some phys-\nical quantity (mass, charge, or spin) over macroscopical\ndistances without essential dissipation. This requires a\nconstant or slowly varying phase gradient at macroscopic\nscale with the total phase variation along the macroscopic\nsample equal to 2 \u0019multiplied by a very large number.\nSpin super\ruidity assumes the existence of spin current\nproportional to the gradient of the phase (spin super-\ncurrent). In magnetically ordered media the phase is an\nangle of rotation in spin space around some axis (further\nin the paper the axis z). In contrast to the dissipative\nspin-di\u000busion current proportional to the gradient of spin\ndensity, the spin supercurrent is not accompanied by dis-\nsipation.\nSpin super\ruidity require special topology of the order\nparameter space. This topology is realized at the pres-\nence of the easy-plane magnetic anisotropy, which con-\n\fnes the magnetization of the ferromagnet or sublattice\nmagnetizations of the antiferromagnet in an easy plane.\nIn this case one may expect that the current state is sta-\nble with respect to phase slips, which lead to relaxation of\nthe supercurrent. In the phase slip event a vortex with\n2\u0019phase variation around it crosses streamlines of thesupercurrent decreasing the total phase variation across\nstreamlines by 2 \u0019. The concept of the phase slip was\nintroduced by Anderson16for super\ruid4He and later\nwas used in studying spin super\ruidity.2,3\nPhase slips are suppressed by energetic barriers for vor-\ntex expansion. But these barriers disappear when phase\ngradients reach critical values determined by the Landau\ncriterion. The physical meaning of the Landau criterion\nis straightforward: the current state becomes unstable\nwhen there are elementary excitations with negative en-\nergy. So, to check the Landau criterion one must know\nthe full spectrum of collective modes.\nSometimes any presence of spin current proportional\nto the phase gradient is considered as a manifestation\nof spin super\ruidity.17,18However, spin current propor-\ntional to the spin phase gradient is ubiquitous and ex-\nists in any spin wave or domain wall, also in the ground\nstate of disordered magnetic media. In all these cases\nthe total variation of the phase is smaller, or on the or-\nder of\u0019. Connecting these cases with spin super\ruid-\nity makes this phenomenon trivial and already observed\nin old experiments on spin waves in the middle of the\n20th Century. One may call the supercurrent produced\nby the total phase variation of the order or less than\n2\u0019microscopical supercurrent, in contrast to persistent\nmacroscopical supercurrents able to transport spin over\nmacroscopical distances.\nThe analogy with usual super\ruids is exact only if\nthe spin space is invariant with respect to spin rotation\naround the hard axis normal to the easy plane. Then\nthere is the conservation law for the spin component\nalong the hard axis. In reality this invariance is bro-\nken by in-plane anisotropy. But this anisotropy is usu-\nally weak, because it originates from the spin-orbit in-\nteraction, which is relativistically small compared to thearXiv:1810.07020v4  [cond-mat.mes-hall]  22 Mar 20192\nexchange interaction, i.e., inversely proportional to the\nspeed of light.19Macroscopical spin supercurrents are\nstill possible if the energy of supercurrents exceeds the\nin-plane anisotropy energy. Thus, one cannot observe\nmacroscopical spin supercurrents not only at large cur-\nrents as in usual super\ruids, but also at small currents.2\nFrom the time when the concept of spin super\ruidity\n(in our de\fnition of this term) was suggested2, it was\ndebated about whether the super\ruid spin current is a\n\\real\" transport current. As a response to these con-\ncerns, in Ref 2 a Gedanken (at that time) experiment\nfor demonstration of reality of super\ruid spin transport\nwas proposed. The spin is injected to one side of a mag-\nnetically ordered layer of thickness dand spin accumula-\ntion is checked at another side. If the layer is not spin-\nsuper\ruid, then the spin is transported by spin di\u000busion.\nThe spin current and the spin density exponentially de-\ncay at the distance of the spin di\u000busion length, and the\ndensity of spin accumulated at the other side decreases\nexponentially with growing distance d. However, if the\nconditions for spin super\ruidity are realized in the layer,\nthen the super\ruid spin current decays much slower, and\nthe accumulated spin density at the side opposite to the\nside where the spin is injected is inversely proportional\ntod+C, whereCis some constant.\nThe interest to long-distance spin transport, especially\nto spin super\ruid transport, revived recently. Takei and\nTserkovnyak7carried out a microscopic analysis of in-\njection of spin to and ejection of spin out of the spin-\nsuper\ruid medium in an easy-plane ferromagnet justify-\ning the aforementioned scheme of super\ruid spin trans-\nport. Takei et al.8extended this analysis to easy-plane\nantiferromagnets. Finally Yuan et al.20were able to real-\nize the suggested experiment in antiferromagnetic Cr 2O3\nobserving spin accumulation inversely proportional to the\ndistance from the interface where spin was injected into\nCr2O3.\nPreviously Borovik-Romanov et al.21reported evi-\ndence of spin super\ruidity in the Bphase of super\ruid\n3He. They detected phase slips in a channel with su-\nper\ruid spin current close to its critical value. It was\nimportant evidence that persistent spin currents are pos-\nsible. But real long-distance transportation of spin by\nthese currents was not demonstrated. Moreover, it is\nimpossible to do in the nonequilibrium magnon Bose{\nEinstein condensate, which was realized in the Bphase\nof3He super\ruid6and in yttrium-iron-garnet magnetic\n\flms.22The nonequilibrium magnon Bose{Einstein con-\ndensate requires pumping of spin in the whole bulk for\nits existence. In the geometry of the aforementioned spin\ntransport experiment this would mean that spin is per-\nmanently pumped not only by a distant injector but also\nall the way up the place where its accumulation is probed.\nThus, the spin detector measures not only spin coming\nfrom a distant injector but also spin pumped close to\nthe detector. Therefore, the experiment does not prove\nthe existence of long-distance spin super\ruid transport.\nThere were also reports on experimental detection ofspin super\ruidity in magnetically ordered solids17,18, but\nthey addressed microscopical spin supercurrent.23As ex-\nplained above, \\super\ruidity\" connected with such cur-\nrents was well proved by numerous old experiments on\nspin waves and does not need new experimental con\fr-\nmations. The work of Yuan et al.20was the \frst report\non long-distance super\ruid spin transport with spin ac-\ncumulation decreasing with distance from the injector as\nexpected from the theory. Long distance super\ruid spin\ntransport was also recently reported in a graphene quan-\ntum antiferromagnet.24\nThe experiment on super\ruid spin transport20has put\nto rest another old dispute about the spin super\ruidity\nconcept. At studying spin super\ruidity in the Bphase\nof super\ruid3He, it was believed4that spin super\ruidity\nis possible only if there are mobile carriers of spin and\na counter\row of carriers with opposite spins transports\nspin. If so, then spin super\ruidity is impossible in insu-\nlators. Moreover, Shi et al.25argued that it is a critical\n\raw of spin-current de\fnition if it predicts spin currents\nin insulators. Since Cr 2O3is an insulator the experiment\nof Yuan et al.20rules out this presumption.\nBoosted by the super\ruid spin transport experiment20\nthis paper addresses some issues deserving further inves-\ntigation. It is especially needed because Lebrun et al.26\nmade an experiment in an antiferromagnetic iron oxide\nsimilar to that of Yuan et al.20and observed similar de-\npendence of spin accumulation on the distance from the\ninjector. However, Lebrun et al.26explain it not by spin\ntransport from the distant injector but by the Seebeck\ne\u000bect at the detector, which is warmed by the heat \row\nfrom the injector. We shall compare these two interpre-\ntations in Sec. VIII.\nWe analyzed the role of dissipation in the super\ruid\nspin transport. A widely used approach to address dis-\nsipation in magnetically ordered solids is the Landau{\nLifshitz{Gilbert (LLG) theory with the Gilbert damp-\ning parameter. But we came to the conclusion that\nthe Gilbert damping does not provide a proper descrip-\ntion of dissipation processes in easy-plane ferromagnets.\nThe Gilbert damping is described by a single parame-\nter, which scales alldissipation processes independently\nfrom whether they do violate the spin conservation law,\nor do not. Meanwhile, the processes violating the spin\nconservation law, the Bloch spin relaxation in particular,\noriginate from spin-orbit interaction and must be rela-\ntivistically small as explained above. This requires the\npresence of a small factor in the intensity of the Bloch\nspin relaxation, which is absent in the Gilbert damping\napproach. So we determined the dissipation parameters\nfrom the Boltzmann equation for magnons scattered by\ndefects. Dissipation is possible only in the presence of\nthermal magnons, and we developed the two-\ruid theory\nfor easy-plane ferromagnets similar to that in super\ruid\nhydrodynamics for the clamped regime, when the gas of\nquasiparticles cannot freely drift without dissipation in\nthe laboratory frame.\nAs mentioned above, to check the Landau criterion for3\nsuper\ruidity, one must calculate the spectrum of collec-\ntive modes and check whether some modes have nega-\ntive energies. The Landau critical gradient is determined\nby easy-plane crystal anisotropy and was known qualita-\ntively both for ferro- and antiferromagnets long ago.2For\neasy-plane ferromagnets the Landau critical gradient was\nrecently determined quantitatively from the spin-wave\nspectrum in the analysis of ferromagnetic spin-1 BEC\nof cold atoms.15But Cr 2O3, which was investigated in\nthe experiment,20has no crystal easy-plane anisotropy,\nand an \\easy plane\" necessary for spin super\ruidity is\nproduced by an external magnetic \feld. The magnetic\n\feld should exceed the spin-\rop \feld, above which mag-\nnetizations of sublattices in antiferromagnet are kept in\na plane normal to the magnetic \feld. We analyze the\nmagnon spectrum in the spin current states in this situ-\nation. The analysis has shown that the Landau critical\ngradient is determined by the gapped mode, but not by\nthe Goldstone gapless mode as in the cases of easy-plane\nferromagnets.\nWithin the two-\ruid theory the role of spatial temper-\nature variation was investigated. This variation produces\nthe bulk Seebeck e\u000bect. But the e\u000bect is weak because it\nis proportional not to the temperature gradient, but to a\nhigher (third) spatial derivative of the temperature.\nThe transient processes near the interface through\nwhich spin is injected were also discussed. Conversion\nfrom spin current of incoherent thermal magnons to co-\nherent (super\ruid) spin transport is among these pro-\ncesses. The width of the transient layer (healing length),\nwhere formation of the super\ruid spin current occurs,\ncan be determined by di\u000berent scales at di\u000berent condi-\ntion. But at low temperatures it is apparently not less\nthan the magnon mean-free-path.\nIn reality the decay of super\ruid currents starts at val-\nues less than the Landau critical value via phase slips\nproduced by magnetic vortices. The di\u000berence in the\nspectrum of collective modes in ferro- and antiferromag-\nnets leads to the di\u000berence in the structure of magnetic\nvortices. In the past magnetic vortices were investi-\ngated mostly in ferromagnets (see Ref. 15 and references\ntherein). The present work analyzes a vortex in an anti-\nferromagnet. The vortex core has a structure of skyrmion\nwith sublattice magnetizations deviated from the direc-\ntion normal to the magnetic \feld. At the same time\ninside the core the total magnetization has a component\nnormal to the magnetic \feld. In the geometry of the\nCr2O3experiment this transverse magnetization creates\nsurface magnetic charges at the point of the exit of the\nvortex line from the sample. Dipole stray magnetic \felds\nproduced by these charges hopefully can be used for de-\ntection of magnetic vortices experimentally.\nSection II reminds the phenomenological model of\nRef. 2 describing the spin di\u000busion and super\ruid spin\ntransport. Section III reproduces the derivation of the\nspectrum of the collective spin mode and the Landau\ncriterion in a spin current state of an easy-plane ferro-\nmagnet known before15. This is necessary for compari-son with the spectrum of the collective spin modes and\nthe Landau criterion in a spin current state of an easy-\nplane antiferromagnet derived in Sec. IV. Thus, Sec. III,\nas well as Sec. II, do not contain new results, but were\nadded to the paper to make it self-su\u000ecient and more\nreadable. In Sec. V we address two-\ruid e\u000bects and dis-\nsipation parameters (spin di\u000busion and second viscosity\ncoe\u000ecients) deriving them from the Boltzmann equation\nfor magnons. The section also estimates the bulk See-\nbeck e\u000bect and shows that it is weak. Section VI ana-\nlyzes the transient layer near the interface through which\nspin is injected and where the bulk super\ruid spin cur-\nrent is formed. Various scales determining the width of\nthis layer (healing length) are discussed. In Sec. VII the\nskyrmion structure of the magnetic vortex in an anti-\nferromagnets is investigated. The concluding Sec. VIII\nsummarizes the results of the work and presents some\nnumerical estimations for the antiferromagnetic Cr 2O3\ninvestigated in the experiment. The Appendix analyzes\ndissipation in the LLG theory with the Gilbert damping.\nIt is argued that this theory predicts dissipation coe\u000e-\ncients incompatible with the spin conservation law.\nII. SUPERFLUID SPIN TRANSPORT VS SPIN\nDIFFUSION\nHere we remind the simple phenomenological model of\nspin transport suggested in Ref. 2 (see also more recent\nRefs. 5, 7, and 8). The equations of magnetodynamics\nare\ndMz\ndt=\u0000r\u0001J\u0000M0\nz\nT1; (1)\nd'\ndt=\u0000\rM0\nz\n\u001f+\u0010r2': (2)\nHere\u001fis the magnetic susceptibility along the axis z,\n'is the angle of rotation (spin phase) in the spin space\naround the axis z, andM0\nz=Mz\u0000\u001fHis a nonequilib-\nrium part of the magnetization density along the mag-\nnetic \feldHparallel to the axis z. The time T1is the\nBloch time of the longitudinal spin relaxation. The term\n/r2'in Eq. (2) is an analog of the second viscosity in\nsuper\ruid hydrodynamic.27,28The magnetization density\nMzand the magnetization current Jdi\u000ber from the spin\ndensity and the spin current by sign and by the gyromag-\nnetic factor \r. Nevertheless, we shall call the current J\nthe spin current to stress its connection with spin trans-\nport. The total spin current J=Js+Jdconsists of the\nsuper\ruid spin current\nJs=Ar'; (3)\nand the spin di\u000busion current\nJd=\u0000DrMz: (4)4\nJzxJLzxSpin injection\nSpin injectionSpin injectionMedium withoutspin superfluidityMedium withspin superfluiditymxzmz\n0\na)\nb)Spin detection\ndyxzPtPtCr2O3c)H\nc)PtPtCr2O3Hzdxy\nFIG. 1. Long distance spin transport. (a) Spin injection to\na spin-nonsuper\ruid medium. (b) Spin injection to a spin-\nsuper\ruid medium. (c) Geometry of the experiment by Yuan\net al.20. Spin is injected from the left Pt wire and \rows along\nthe Cr 2O3\flm to the right Pt wire, which serves as a detector.\nThe arrowed dashed line shows a spin-current streamline. In\ncontrast to (a) and (b), the spin current is directed along\nthe same axis zas a magnetization parallel to the external\nmagnetic \feld H.\nThe pair of the hydrodynamical variables ( Mz;') is a\npair of conjugate Hamiltonian variables analogous to\nthe pair \\particle density{super\ruid phase\" in super\ruid\nhydrodynamics.1\nThere are two kinds of spin transport illustrated in\nFig. 1. In the absence of spin super\ruidity ( A= 0) there\nis no super\ruid current. Equation (2) is not relevant, and\nEq. (1) describes pure spin di\u000busion [Fig. 1(a)]. Its solu-\ntion, with the boundary condition that the spin current\nJ0is injected at the interface x= 0, is\nJ=Jd=J0e\u0000x=L d; M0\nz=J0r\nT1\nDe\u0000x=L d;(5)\nwhere\nLd=p\nDT1 (6)\nis the spin-di\u000busion length. Thus the e\u000bect of spin injec-\ntion exponentially decays at the scale of the spin-di\u000busion\nlength.However, if spin super\ruidity is possible ( A6= 0), the\nspin precession equation (2) becomes relevant. As a re-\nsult of it, in a stationary state the magnetization M0\nz\ncannot vary in space (Fig. 1b) since according to Eq. (2)\nthe gradient rM0\nzis accompanied by the linear in time\ngrowth of the gradient r'. The requirement of constant\nin space magnetization Mzis similar to the requirement\nof constant in space chemical potential in super\ruids, or\nthe electrochemical potential in superconductors. As a\nconsequence of this requirement, spin di\u000busion current is\nimpossible in the bulk since it is simply \\short-circuited\"\nby the super\ruid spin current. Only in AC processes\nthe oscillating spin injection can produce an oscillating\nbulk spin di\u000busion current coexisting with an oscillating\nsuper\ruid spin current.\nIn the super\ruid spin transport the spin current can\nreach the other boundary opposite to the boundary where\nspin is injected. We locate it at the plane x=d. As a\nboundary condition at x=d, one can use a phenomeno-\nlogical relation connecting the spin current with the mag-\nnetization: Js(d) =M0\nzvd, wherevdis a phenomenologi-\ncal constant. This boundary condition was derived from\nthe microscopic theory by Takei and Tserkovnyak7. To-\ngether with the boundary condition Js(0) =J0atx= 0\nthis yields the solution of Eqs. (1) and (2):\nM0\nz=T1\nd+vdT1J0; Js(x) =J0\u0012\n1\u0000x\nd+vdT1\u0013\n:(7)\nThus, the spin accumulated at large distance dfrom the\nspin injector slowly decreases as the inverse distance 1 =d\n[Fig. 1(b)], in contrast to the exponential decay /e\u0000d=Ld\nin the spin di\u000busion transport [Fig. 1(a)].\nIn Figs. 1(a) and 1(b) the spin \rows along the axis\nx, while the magnetization and the magnetic \feld are\ndirected along the axis z. In the geometry of the experi-\nment of Yuan et al.20the spin \rows along the magnetiza-\ntion axiszparallel to the magnetic \feld. This geometry is\nshown in Fig. 1c. The di\u000berence between two geometries\nis not essential if spin-orbit coupling is ignored. In this\nsection we chose the geometry with di\u000berent directions of\nthe spin current and the magnetization in order to stress\nthe possibility of the independent choice of axes in the\nspin and the con\fgurational spaces. But in Sec. VII ad-\ndressing a vortex in an antiferromagnet we shall switch\nto the geometry of the experiment because in this case\nthe di\u000berence between geometries is important.\nWithout dissipation-connected terms, the phenomeno-\nlogical theory of this section directly follows from the\nLLG theory. For ferromagnets the LLG equation is\ndM\ndt=\r[Heff\u0002M]; (8)\nwhere\nHeff=\u0000\u000eH\n\u000eM=\u0000@H\n@M+rj@H\n@rjM(9)\nis the e\u000bective \feld determined by the functional deriva-\ntive of the Hamiltonian H. For a ferromagnet with uni-5\naxial anisotropy the Hamiltonian is\nH=GM2\nz\n2+AriM\u0001riM\u0000MzH: (10)\nHereHis an external constant magnetic \feld parallel to\nthe axisz, and the exchange constant Adetermines sti\u000b-\nness with respect to deformations of the magnetization\n\feld. In the case of easy-plane anisotropy the anisotropy\nparameter Gis positive and coincides with the inverse\nsusceptibility: G= 1=\u001f.\nSince the absolute value Mof the magnetization is\na constant, one can describe the 3D magnetization vec-\ntorMonly by two Hamiltonian conjugate variables: the\nmagnetization zcomponent Mzand the angle 'of rota-\ntion around the zaxis. Then the LLG theory yields two\nequations\n_Mz=\u0000r\u0001Js; (11)\n_'=\u0000\r\u0016; (12)\nwith the Hamiltonian in new variables\nH=M2\nz\n2\u001f+AM2\n?r'2\n2+AM2(rMz)2\n2M2\n?\u0000MzH: (13)\nHereM?=p\nM2\u0000M2z, and the spin \\chemical poten-\ntial\" and the super\ruid spin current are\n\u0016=\u000eH\n\u000eMz=@H\n@Mz\u0000rj@H\n@rjMz;Js=\r@H\n@r':(14)\nAfter substitution of explicit expressions for functional\nderivatives of the Hamiltonian (13) the equations become\n_Mz\n\r=\u0000r\u0001(AM2\n?r'); (15)\n_'\n\r=\u0000Mz\u00141\n\u001f\u0000A(r')2\u0000AM2(rMz)2\nM4\n?\u0015\n+AM2\nM2\n?r2Mz+H: (16)\nThe equations (1) and (2) without dissipation terms fol-\nlow from Eqs. (15) and (16) after linearization with re-\nspect to small gradients r'and nonequilibrium magne-\ntizationM0\nz=Mz\u0000\u001fHand ignoring the dependence\nof the spin chemical potential \u0016onrMz. ThenA=\n\rAM2\n?, andM?is determined by its valuep\nM2\u0000\u001f2H2\nin the equilibrium.\nIII. COLLECTIVE MODES AND THE LANDAU\nCRITERION IN EASY-PLANE FERROMAGNETS\nTo check the Landau criterion one should know the\nspectrum of collective modes. In an easy-plane ferromag-\nnet the collective modes (spin waves) are determined byEqs. (15) and (16) linearized with respect to weak pertur-\nbations of stationary states. Further the angle variable \u0012\nwill be introduced instead of the variable Mz=Msin\u0012.\nLet us consider a current state with constant gradient\nK=r'and constant magnetization\nMz=Msin\u0012=\u001fH\n1\u0000\u001fAK2: (17)\nTo derive the spectrum of collective modes, we consider\nweak perturbations \u0002 and \b of this state: \u0012!\u0012+ \u0002,\n'!'+ \b. Equations (15) and (16) after linearization\nare:\n_\u0002\u00002\rMzAK\u0001r\u0002 =\u0000\rAM cos\u0012r2\b;\n_\b\u00002\rMzAK\u0001r\b =\n\u0000\rMcos\u0012\n\u001f\u0000\n1\u0000\u001fAK2\u0001\n\u0002 +\rAM cos\u0012r2\u0002:(18)\nFor plane waves/eik\u0001r\u0000i!tthese equations describe the\ngapless Goldstone mode with the spectrum:13,15\n(!+w\u0001k)2= ~c2\nsk2: (19)\nHere\n~cs=r\u001f\n~\u001fcs; (20)\n~\u001f=\u001f\n1\u0000\u001fA\u0010\nK2\u0000M2k2\nM2\n?\u0011; (21)\nand\ncs=\rM?s\nA\n\u001f(22)\nis the spin-wave velocity in the ground state without any\nspin current. In this state the spectrum becomes\n!=csks\n1 +\u001fAM2k2\nM2\n?: (23)\nThe velocity\nw= 2\rMzAK; (24)\ncan be called Doppler velocity because its e\u000bect on the\nmode frequency is similar to the e\u000bect of the mass ve-\nlocity on the mode frequency in a Galilean invariant\n\ruid (Doppler e\u000bect). But our system is not Galilean\ninvariant,13and the gradient Kis present also in the\nright-hand side of the dispersion relation (19).\nIn the long-wavelength hydrodynamical limit magnons\nhave the sound-like spectrum linear in k. Quadratic cor-\nrections/k2become important at k\u0018M?=Mp\u001fA[see\nEq. (23)]. These corrections emerge from the terms in6\nthe Hamiltonian, which depend on rMz. So the hydro-\ndynamical approach is valid at scales exceeding\n\u00180=M\nM?p\n\u001fA; (25)\nwhich can be called the coherence length, in analogy with\nthe coherence length in the Gross{Pitaevskii theory for\nBEC. Also in analogy with BEC, the coherence length\ndiverges at M?!0, i.e., at the second-order phase tran-\nsition from the easy-plane to the easy-axis anisotropy.\nThe same scale determines the Landau critical gradient\nand the vortex core radius. Telling about hydrodynamics\nwe bear in mind hydrodynamics of a perfect \ruid without\ndissipation. Later in this paper we shall discuss hydro-\ndynamics with dissipation. In this case the condition\nk\u001c1=\u00180is not su\u000ecient, and an additional restriction\non using hydrodynamics is determined by the mean-free\npath of magnons.\nAccording to the Landau criterion, the current state\nbecomes unstable at small kwhenkis parallel to wand\nthe frequency !becomes negative. This happens at the\ngradientKequal to the Landau critical gradient\nKc=M?p4M2\u00003M?1p\u001fA\u00181\n\u00180: (26)\nSpin super\ruidity becomes impossible at the phase tran-\nsition to the easy-axis anisotropy ( M?= 0). In the oppo-\nsite limit of small Mz\u001cMthe pseudo-Doppler e\u000bect is\nnot important, and the Landau critical gradient Kcis de-\ntermined from the condition that the spin-wave velocity\n~csvanishes at small k:\nKc=1p\u001fA=\rM\n\u001fcs: (27)\nExpanding the Hamiltonian (13) with respect to weak\nperturbations \u0002 and \b up to the second order one obtains\nthe energy of the spin wave mode per unit volume,\nEsw=M?!(k)\n\rp~\u001fAkj\u0002kj2; (28)\nwherej\u0002kj2is the squared perturbation of the angle \u0012\nwith the wave vector kaveraged over the wave period.\nIn the quantum theory the energy density Eswcorre-\nsponds to the magnon density\nn(k)\nV=Esw\n~!(k)=M?j\u0002kj2\n~\rp~\u001fAk; (29)\nwheren(k) is the number of magnons in the plane-wave\nmode with the wave vector kandVis the volume of the\nsample. Summing over the whole kspace, the averaged\nsquared perturbation is\nh\u00022i=X\nkj\u0002kj2=~\rp\nA\nM?Zp\n~\u001fn(k)kd3k\n(2\u0019)3:(30)Further we proceed within the hydrodynamical ap-\nproach neglecting quadratic corrections to the spectrum.\nThere are quadratic in spin-wave amplitudes corrections\nto the spin super\ruid current and to the spin chemical\npotential:\nJsjsw=\u0000\rM?A(M?h\u00022iK+ 2Mzh\u0002r\bi);(31)\n\u0016jsw=\u0000A(Mzh(r\b)2i+ 2M?K\u0001h\u0002r\bi):(32)\nUsing Eq. (30) and the relation\nr\b =\u0002p\u001fAk\nk; (33)\nwhich follows from the equations of motion (18), one ob-\ntains:\nJsjsw=\u0000\u001f2~c3\ns\n\rM2\n?Z\nn(k)\u0012\nK+2\rMz\n\u001fcsk\nk\u0013\nkd3k\n(2\u0019)3;\n(34)\n\u0016jsw=\u0000\u001f~c2\ns\n\rM2\n?Z\nn(k)\u0012\rMz\n\u001fcs+2K\u0001k\nk\u0013\nkd3k\n(2\u0019)3:\n(35)\nIV. COLLECTIVE MODES AND THE LANDAU\nCRITERION IN ANTIFERROMAGNETS\nFor ferromagnetic state of localized spins the deriva-\ntion of the LLG theory from the microscopic Heisenberg\nmodel was straightforward.29The quantum theory of the\nantiferromagnetic state even for the simplest case of a\ntwo-sublattice antiferromagnet, which was widely used\nfor Cr 2O3, is more di\u000ecult. This is because the state\nwith constant magnetizations of two sublattices is not\na well de\fned quantum-mechanical eigenstate.29Never-\ntheless, long time ago it was widely accepted to ignore\nthis complication and to describe the long-wavelength dy-\nnamics by the LLG theory for two sublattices coupled via\nexchange interaction:30\ndMi\ndt=\r[Hi\u0002Mi]; (36)\nwhere the subscript i= 1;2 points out to which sublattice\nthe magnetization Mibelongs, and\nHi=\u0000\u000eH\n\u000eMi=\u0000@H\n@Mi+rj@H\n@rjMi(37)\nis the e\u000bective \feld for the ith sublattice determined by\nthe functional derivative of the Hamiltonian H. For an\nisotropic antiferromagnet the Hamiltonian is\nH=M1\u0001M2\n\u001f+A(riM1\u0001riM1+riM2\u0001riM2)\n2\n+A12rjM1\u0001rjM2\u0000H\u0001(M1+M2):(38)7\nIn the uniform ground state without the magnetic \feld H\nthe two magnetizations are antiparallel, M2=\u0000M1, and\nthe total magnetization M1+M2vanishes. At H6= 0 the\nsublattice magnetizations are canted, and in the uniform\nground state the total magnetization is parallel to H:\nm=M1+M2=\u001fH: (39)\nThe \frst term in the Hamiltonian (38), which determines\nthe susceptibility \u001f, originates from the exchange inter-\naction between spins of two sublattices. This is the sus-\nceptibility normal to the staggered magnetization (anti-\nferromagnetic vector) L=M1\u0000M2. Since in the LLG\ntheory absolute values of magnetizations M1andM2are\n\fxed the susceptibility parallel to Lvanishes.\nIn the uniform state only the uniform exchange en-\nergy/1=\u001fand the Zeeman energy (the \frst and the\nlast terms) are present in the Hamiltonian, which can be\nrewritten as\nH=\u0000L2\u0000m2\n4\u001f\u0000H\u0001m=\u0000M2\n\u001f+m2\n2\u001f\u0000mHm;(40)\nwhereHm= (H\u0001m)=mis the projection of the mag-\nnetic \feld on the direction of the total magnetization m.\nMinimizing the Hamiltonian with respect to the absolute\nvalue of m(at it \fxed direction, i.e., at \fxed Hm) one\nobtains\nH=\u0000M2\n\u001f\u0000\u001fH2\nm\n2=\u0000M2\n\u001f\u0000\u001fH2\n2+\u001fH2\nL\n2;(41)\nwhereHL= (H\u0001L)=Lis the projection of the magnetic\n\feld on the staggered magnetization L. The \frst two\nterms are constant, while the last term plays the role of\nthe easy-plane anisotropy energy con\fning Lin the plane\nnormal to H. 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sha1_base64=\"iO+t6s0e5FdT3FzK2f3PzRc30JA=\">AAAB/XicbVBNSwMxEJ31s9avqkcvwSJ4KrtF0GPRixehgv2AdilJmm1Dk+ySZIWyFH+DVz17E6/+Fo/+E9N2D7b1wcDjvRlm5pFEcGN9/9tbW9/Y3Nou7BR39/YPDktHx00Tp5qyBo1FrNsEGya4Yg3LrWDtRDMsiWAtMrqd+q0npg2P1aMdJyyUeKB4xCm2Tmp1iUT3vWqvVPYr/gxolQQ5KUOOeq/00+3HNJVMWSqwMZ3AT2yYYW05FWxS7KaGJZiO8IB1HFVYMhNms3Mn6NwpfRTF2pWyaKb+nciwNGYsieuU2A7NsjcV//M6qY2uw4yrJLVM0fmiKBXIxmj6O+pzzagVY0cw1dzdiugQa0ytS2hhC5ETl0mwnMAqaVYrgV8JHi7LtZs8nQKcwhlcQABXUIM7qEMDKIzgBV7hzXv23r0P73PeuublMyewAO/rFxoXlX8=</latexit>\nFIG. 2. Angle variables \u0012and\u00120for the case when the both\nmagnetizations are in the plane xz('0='= 0).\nWe introduce the pairs of angle variables \u0012i,'ideter-\nmining directions of the sublattice magnetizations:\nMix=Mcos\u0012icos'i; Miy=Mcos\u0012isin'i;\nMiz=Msin\u0012i:(43)\nThe equations of motion in the angle variables are\ncos\u0012i_\u0012i\n\r=1\nM\u0012@H\n@'i\u0000r@H\n@r'i\u0013\n;\ncos\u0012i_'i\n\r=\u00001\nM\u0012@H\n@\u0012i\u0000r@H\n@r\u0012i\u0013\n: (44)\nIn the further analysis it is convenient to use other angle\nvariables:\n\u00120=\u0019+\u00121\u0000\u00122\n2; \u0012=\u0019\u0000\u00121\u0000\u00122\n2;\n'0='1+'2\n2; '='1\u0000'2\n2: (45)\nIn these variables the Hamiltonian becomes\nH=\u0000M2\n\u001f(cos 2\u00120cos2'\u0000cos 2\u0012sin2')\u00002HM cos\u0012sin\u00120\n+AM2[(1 + cos 2\u00120cos 2\u0012)r'2\n0+r'2\n2\u0000sin 2\u00120sin 2\u0012r'0\u0001r'+r\u00122\n0+r\u00122]\n+A12M2f(cos 2\u0012sin2'+ cos 2\u00120cos2')(r\u00122\n0\u0000r\u00122)\u0000cos 2\u00120+ cos 2\u0012\n2cos 2'(r'2\n0\u0000r'2)\n\u0000sin 2'[sin 2\u0012(r\u00120\u0001r'0+r\u0012\u0001r') + sin 2\u00120(r\u0012\u0001r'0+r\u00120\u0001r')]g: (46)\nThe polar angles \u0012for the staggered magnetization Land\nthe canting angle \u00120are shown in Fig. 2 for the case when\nthe both magnetizations are in the plane xz('0='=\n0).\nIn the uniform ground state \u0012= 0,'= 0,mz=\n2Msin\u00120=\u001fH, while the angle '0is an arbitrary con-stant. Since we consider \felds Hweak compared to the\nexchange \feld, \u00120is always small. In the state with con-\nstant current K=r'0the magnetization along the\nmagnetic \feld is\nmz=\u001fH\n1\u0000\u001fA\u0000K2=2; (47)8\nwhereA\u0006=A\u0006A12.\nIn a weakly perturbed current state small but nonzero\n\u0012and'appear. Also the angles \u00120and'0di\u000ber from\ntheir values in the stationary current state: \u00120!\u00120+ \u0002,\n'0!'0+ \b. Linearization of the nonlinear equations\nof motion with respect to weak perturbations \u0002, \b, \u0012,\nand'yields decoupled linear equations for two pairs of\nvariables (\u0002 ;\b) and (\u0012;'):\n_\u0002\n\r\u0000A\u0000mzK\u0001r\u0002 =\u0000A\u0000M?r2\b;\n_\b\n\r\u0000A\u0000mzK\u0001r\b =\u0000\u0012\n1\u0000\u001fA\u0000K2\n2\u00132M?\n\u001f\u0002\n+(A+A12cos 2\u00120)\ncos\u00120Mr2\u0002;(48)\n_\u0012\n\r\u0000A+mzK\u0001r\u0012\n=\u00002M?\n\u001f\u0000\n1 +\u001fA12K2\u0001\n'+A+M?r2';\n_'\n\r\u0000A+mzcos\u00120K\u0001r'\n=m2\nz\n2\u001fM?(1 +\u001fA12K2)\u0012\u0000A\u0000K2M?\u0012\n\u0000A\u0000A12cos 2\u00120\ncos\u00120Mr2\u0012: (49)\nFor plane waves/eik\u0001r\u0000i!tEq. (48) describes the gapless\nGoldstone mode with the spectrum:\n(!+\rmzA\u0000K\u0001k)2\n=c2\ns\u0014\n1\u0000\u001fA\u0000K2\n2+\u001f(A+A12cos 2\u00120)k2\n2 cos2\u00120\u0015\nk2:(50)\nHere\ncs=\rM?s\n2A\u0000\n\u001f(51)\nis the spin-wave velocity in the ground state without spin\ncurrent. Apart from quadratic corrections k2to the fre-\nquency, the gapless mode in an antiferromagnet does not\ndi\u000ber from that in a ferromagnet, if one replaces in all\nexpressions for the ferromagnet AbyA\u0000=2 and the pa-\nrameterMby 2M.\nEquation (49) describes the gapped mode with the\nspectrum\n(!+\rmzA+K\u0001k)2=\u0012\n1 +\u001fA12K2+\u001fA+k2\n2\u0013\n\u0002\u0014(1 +\u001fA12K2)\r2m2\nz\n\u001f2\u0000c2\nsK2\n+2\r2M2(A\u0000A12cos 2\u00120)k2\n\u001f\u0015\n:(52)Without spin current and neglecting the term /A+k2\nthe spectrum is\n!=s\n\r2m2z\n\u001f2+c2sk2: (53)\nThis spectrum determines a new correlation length\n\u0018=M\nHs\n2A\u0000\n\u001f=cs\n\rH; (54)\nwhich is connected with the easy-plane anisotropy energy\n(42) and determines the wave vector k= 1=\u0018at which\nthe gap and the kdependent frequency become equal.\nApplying the Landau criterion to the gapless mode one\nobtains the critical gradientp\n2=\u001fA\u0000similar to the value\n(27) obtained for a ferromagnet. But in contrast to a fer-\nromagnet where the susceptibility \u001fis connected with\nweak anisotropy energy, in an antiferromagnet the sus-\nceptibility\u001fis determined by a much larger exchange\nenergy and is rather small. As a result, in an antiferro-\nmagnet the gapless Goldstone mode becomes unstable at\nthe very high value of K. But at much lower values of\nKthe gapped mode becomes unstable. According to the\nspectrum (52), the gap in the spectrum vanishes at the\ncritical gradient\nKc=1\n\u0018=\rH\ncs=\rmz\n\u001fcs: (55)\nV. TWO-FLUID EFFECTS AND DISSIPATION\nFROM THE BOLTZMANN EQUATION FOR\nMAGNONS\nKnowledge of the spectrum of collective modes allows\nto derive the dynamical equations at \fnite temperatures\ntaking into account the presence of thermal magnons.\nFurther we follow the procedure of the derivation of the\ntwo-\ruid hydrodynamics in super\ruids.27We address the\nhydrodynamical limit when all parameters ( Mz,K,T)\nof the system slowly vary in space and time.\nWe shall focus on ferromagnets. The equilibrium\nPlanck distribution of magnons in a ferromagnet with\na small spin current /Kis\nnK=1\ne~!(k)=T\u00001\u0019n0(!0)\u00002\u001fc2\nsMz\n\rM2\n?@n0(!0)\n@!0K\u0001k;\n(56)\nwhere!0=cskand\nn0(!0) =1\ne~!0=T\u00001(57)\nis the Planck distribution in the state without spin cur-\nrent.\nIn the theory of super\ruidity the Plank distribution\nof phonons in general depends not only on density and\nsuper\ruid velocity (analogs of our MzandK) but also on9\nthe normal velocity, which characterizes a possible drift of\nthe gas of quasiparticles with respect to the laboratory\nframe of coordinates. This drift is possible because of\nthe Galilean invariance of super\ruids. In our case the\nGalilean invariance is broken by possible interaction of\nmagnons with defects, and in the equilibrium the drift of\nthe quasiparticle gas is impossible. The case of broken\nGalilean invariance, when the normal velocity vanishes,\nwas also investigated for super\ruids in porous media or\nin very thin channels, when the Galilean invariance is\nbroken by interaction with channel walls. It was called\nthe clamped regime.31,32\nSubstituting the Planck distribution (56) into Eqs. (34)\nand (35) one obtains the contribution of equilibrium\nmagnons to the spin current and the spin chemical po-\ntential:\nJsjeq=\r@\n@K=\u0000\u00192\u001f2T4\n30\rM2\n?~3csK\u0012\n1 +16M2\nz\n3M2\n?\u0013\n;(58)\n\u0016jeq=@\n@Mz=\u00192MzT4\n30~3c3sM2\n?; (59)\nwhere\n\n =TZ\nln(1\u0000e\u0000~!(k)=T)d3k\n(2\u0019)3: (60)\nis the thermodynamical potential for the magnon Bose-\ngas. The contribution (58) decreases the super\ruid spin\ncurrent at \fxed phase gradient K, similarly to the de-\ncrease of the mass super\ruid current after replacing the\ntotal mass density by the lesser super\ruid density.\nYuan et al.20used in their experiment very thin \flm at\nlow temperature, when de Broglie wavelength of magnons\nexceeds \flm thickness, and it is useful to give also the\ntwo-\ruid corrections for a two-dimensional case. Repeat-\ning our calculations after replacing integralsR\nd3k=(2\u0019)3\nby integrals WR\nd2k=(2\u0019)2, one obtains:\nJsjeq=\u0000\u0010(3)\u001f2T3\n\u0019W\rM2\n?~2K\u0012\n1 +6M2\nz\nM2\n?\u0013\n; (61)\n\u0016jeq=\u0010(3)MzT3\n\u0019W~2c2sM2\n?; (62)\nwhere the value of the Riemann zeta function \u0010(3) is\n1.202 andWis the \flm thickness.\nThe next step in derivation of the two-\ruid theory at \f-\nnite temperatures is the analysis of dissipation. A widely\nused approach of studying dissipation in magnetically or-\nder systems is the LLG theory with the Gilbert damp-\ning term added. However, this approach is incompatible\nwith the spin conservation law. This law, although being\napproximate, plays a key role in the problem of spin su-\nper\ruidity. Therefore, we derived dissipation parameters\nfrom the Boltzmann equation for magnons postponing\ndiscussion of the LLG theory with the Gilbert damping\nto the Appendix.Dissipation is connected with nonequilibrium correc-\ntions to the magnon distribution. At low temperatures\nthe number of magnons is small, and magnon-magnon\ninteraction is weak. Then the main source of dissipa-\ntion is scattering of magnons by defects. The Boltzmann\nequation with the collision term in the relaxation-time\napproximation is\n_n+@!\n@k\u0001rn\u0000r!\u0001@n\n@k=\u0000n\u0000nK\n\u001c: (63)\nIf parameters, which determine the magnon distribution\nfunctionn, vary slowly in space and time one can substi-\ntute the equilibrium Planck distribution nKinto the left-\nhand side of the Boltzmann equation (63). This yields:\n@n0\n@!_!+@n0\n@T\u0012\n_T+@!\n@k\u0001rT\u0013\n=\u0000n\u0000n0\n\u001c; (64)\nWe consider small gradients Kwhen the di\u000berence be-\ntweennKandn0is not important. But weak depen-\ndence of!onKis important at calculation of _ !. One\ncan see that at the constant temperature Tin any sta-\ntionary state the left-hand side vanishes, and there is\nno nonequilibrium correction to the magnon distribution.\nCorrespondingly, there is no dissipation. This is one more\nillustration that stationary super\ruid currents do not de-\ncay.\nIn nonstationary cases time derivatives are determined\nby the equations of motions. The equations of motion for\nMzandKare not su\u000ecient, and the equation of heat bal-\nance is needed for \fnding _T. In general the heat balance\nequation is rather complicated since it must take into ac-\ncount interaction of magnons with other subsystems, e.\ng., phonons. Instead of it we consider a simpler case,\nwhen magnons are not important in the heat balance,\ni.e., the temperature does not depend on magnon pro-\ncesses. In other words we consider the isothermal regime\nwhen _T= 0. But we allow slow temperature variation in\nspace.\nThe temporal variation of the frequency !emerges\nfrom slow temporal variation of MzandK, and at small\nK\n_!=@!\n@Mz_Mz+@!\n@K_K=\u0000Mz\nM2\n?\u0012\ncsk_Mz+2\u001fc2\ns\n\rk\u0001_K\u0013\n:\n(65)\nThe partial derivatives @!=@Mzand@!=@Kwere deter-\nmined from the spectrum (19), while the time derivatives\nofMzandKwere found from the linearized equations\n(15) and (16) assuming that r'=Kis small and ig-\nnoring gradients of Mzin the right-hand side of Eq. (16),\nwhich are beyond the hydrodynamical limit. Then\n_!=Mz\nM2\n?c2\ns\u0014\u001f\n\rcskr\u0001K+ 2(k\u0001r)Mz\u0015\n: (66)\nEventually the nonequilibrium correction to the magnon10\ndistribution function is\nn0=n\u0000n0=\u0000Mz\nM2\n?cs\u0014\u001fcs\n\rkr\u0001K\n+2(k\u0001r)Mz\u0000M2\n?\nMzT(k\u0001r)T\u0015\n\u001c@n0\n@k(67)\nSubstituting n0into Eqs. (34) and (35) one obtains dis-\nsipation terms in the spin current and the spin chemical\npotential:\nJd=\u0000D\u0012\nrMz\u00001\n2TM2\n?\nMzrT\u0013\n; (68)\n\u0016d=\u0000\u0010\n\rr\u0001K; (69)\nwhere\nD=\u00002\u001f~c3\ns\n3\u00192M2\nz\nM4\n?Z\n\u001c@n0\n@kk4dk;\n\u0010=\u0000\u001f~c3\ns\n2\u00192M2\nz\nM4\n?Z\n\u001c@n0\n@kk4dk: (70)\nIn addition to the spin di\u000busion current, the dissipative\nspin current Jdcontains also the current proportional\nto the temperature gradient. This is the bulk Seebeck\ne\u000bect. Estimation of the integral in these expressions\nrequires knowledge of possible dependence of the relax-\nation time \u001con the energy. Under the assumption that\n\u001cis independent from the energy,\nD=8\u00192\u001c\r2T4M2\nz\n45~3c3sM2\n?; \u0010=2\u00192\u001c\r2T4M2\nz\n15~3c3sM2\n?; (71)\nor for the two-dimensional case,\nD=16\u0010(3)\u001c\r2T3M2\nz\n3\u0019W~2c2sM2\n?; \u0010=4\u0010(3)\u001c\r2T3M2\nz\n\u0019W~2c2sM2\n?:(72)\nAlthough in antiferromagnets the Landau critical gra-\ndient is connected with the gapped mode, at small phase\ngradients the gapless Goldstone mode has lesser energy,\nand at low temperatures most of magnons belong to this\nmode. Since the Goldstone modes in ferromagnets and\nantiferromagnets are similar, our estimation of dissipa-\ntion coe\u000ecients for ferromagnets is valid also for antifer-\nromagnets after replacing AbyA\u0000=2 andMby 2M.\nThe microscopic analysis of this section agrees with\nthe following phenomenological equations similar to the\nhydrodynamical equations for super\ruids in the clamped\nregime:\n_Mz=\u0000r\u0001Js\u0000@R\n@\u0016+r@R\n@r\u0016; (73)\n_'=\u0000\r\u0016+@R\n@(r\u0001Js); (74)where the spin chemical potential and the super\ruid spin\ncurrent,\n\u0016=\u000eF\n\u000eMz;Js=\r@F\n@r'; (75)\nare determined by derivatives of the free energy\nF=H+ \n\u0000TS: (76)\nThe spin conservation law forbids the term @R=@\u0016 in the\ncontinuity equation (73), because it is not a divergence of\nsome current. Thus, the dissipation function is compati-\nble with the spin conservation law if it depends only on\nthe gradient of the spin chemical potential \u0016, but not on\n\u0016itself. This does not take place in the LLG theory with\nthe Gilbert damping discussed in the Appendix. The\nanalysis of this section assumed the spin conservation\nlaw and corresponded to the dissipation function\nR=\u001fD\n2r\u00162\u0000D\n2TM2\n?\nMzr\u0016\u0001rT+\u0010\n2\rAM2\n?(r\u0001Js)2:\n(77)\nIn general the dissipation function contains also the term\n/rT2responsible for the thermal conductivity. But it\nis important only for the heat balance equation, which\nwas not considered here.\nIf the temperature does not vary in space, then the only\ntemperature e\u000bect is a correction to the spin chemical\npotential. This does not a\u000bect the basic feature of super-\n\ruid spin transport: there is no gradient of the chemical\npotential in a stationary current state, and all dissipation\nprocesses are not e\u000bective except for the relativistically\nsmall spin Bloch relaxation. If there is spatial variation of\ntemperature, then the spin chemical potential also varies\nin space. One can \fnd its gradient by exclusion of r\u0001Js\nfrom Eqs. (73) and (74):\nr\u0016=rMz\n\u001f=\u0000D\u0010\n2\r2AMzTr(r2T): (78)\nNote that the spin chemical potential gradient is propor-\ntional not to the \frst but to the third spatial derivative\nof the temperature. The constant temperature gradi-\nent does not produce spatial variation of the chemical\npotential. This is an analog of the absence of thermo-\nelectric e\u000bects proportional to the temperature gradients\nin superconductors.33Naturally the e\u000bect produced by\nhigher derivatives of the temperature is weaker than pro-\nduced by the \frst derivative.\nThe nonuniform correction to the spin chemical po-\ntential strongly depends on temperature. Assuming the\nT4dependence of the dissipation parameters Dand\u0010in\nEq. (71) the coe\u000ecient before the temperature-gradient\nterm in Eq. (78) is proportional to T8. Now the spin dif-\nfusion current\u0000\u001fDr\u0016does not disappear in the equa-\ntion (73) of continuity for the spin, but it is proportional\ntoT12.\nEarlier Zhang and Zhang34used the Boltzmann equa-\ntion for derivation of the spin di\u000busion coe\u000ecient and11\nthe Bloch relaxation time in an isotropic ferromagnet in\na constant magnetic \feld. We derived the spin di\u000busion\nand the second viscosity coe\u000ecients in an easy-plane fer-\nromagnet with di\u000berent spin-wave spectrum. Two-\ruid\ne\u000bects in easy-plane ferromagnets were investigated by\nFlebus et al.35. They solved the Boltzmann equation\nusing the equilibrium magnon distribution function with\nnonzero chemical potential of magnon (do not confuse it\nwith the spin chemical potential introduced in the present\npaper). In contrast, we assumed complete thermaliza-\ntion of the magnon distribution when the magnon chem-\nical potential vanishes. The thermalization assumption\nis questionable in the transient layer near the interface\nthrough which spin is injected, and in this layer the ap-\nproach Flebus et al.35may become justi\fed. The tran-\nsient layer is discussed in the next section.\nVI. TRANSIENT (HEALING) LAYER NEAR\nTHE INTERFACE INJECTING SPIN\nInjection of spin from a medium without spin super\ru-\nidity to a medium with spin super\ruidity may produce\nnot only a super\ruid spin current but also a spin cur-\nrent of incoherent magnons. But at some distance from\nthe interface between two media, which will be called the\nconversion healing length, the spin current of incoherent\nmagnons (spin di\u000busion current) must inevitably trans-\nform to super\ruid spin current, as we shall show now.\nWe return back to Eqs. (1) and (2) but now we neglect\nthe relativistically small Bloch spin relaxation (the term\n/1=T1). In Sec. II we considered the stationary solution\nof the these equations with constant magnetization and\nabsent spin di\u000busion current. But it is not the only sta-\ntionary solution. Another solution is an evanescent mode\nM0\nz/r'/e\u0000x=\u0015, where\n\u0015=s\n\u001fD\u0010\n\rA(79)\nis the conversion healing length. We look for superposi-\ntion of two solutions, which satis\fes the condition that\nthe injected current J0transforms to the spin di\u000busion\ncurrent, while the super\ruid current vanishes at x= 0:\nJ0=\u0000DrxM0\nz(0);rx'(0) = 0: (80)\nThis superposition is\nM0\nz(x) =M0\nz+\u0015J0\nDe\u0000x=\u0015;rx'(x) =J0\nA(1\u0000e\u0000x=\u0015);\n(81)\nwhereM0\nzin the right-hand side is a constant magneti-\nzation far from the interface x= 0. Thus, at the length\n\u0015the spin di\u000busion current Jddrops from J0to zero,\nwhile the super\ruid spin current grows from zero to J0\nand remains at larger distances constant.\nAs pointed out in the end of Sec. II, the phenomenolog-\nical equations (1) and (2) were derived assuming that thespin chemical potential \u0016=M0\nz=\u001f\u0000Hdoes not depend\non gradients rMz. However, the dissipation coe\u000ecients\nDand\u0010decrease very sharply with temperature, and\nthe conversion healing length eventually becomes much\nsmaller than the scale \u00180[see Eq. (25)], when the depen-\ndence of the free energy and the spin chemical potential\non the gradients rMzbecomes important. But in fact\naddingrMz-dependent terms into the expression for \u0016,\n\u0016=Mz\n\u001f\u0000H\u0000AM2r2Mz\nM2\n?; (82)\ndoes not a\u000bect the expression (79) for the healing length.\nThe generalization of the analysis reduces to replacing of\nM0\nzin Eqs. (1), (2), and (81) by \u001f\u0016.\nTransformation of the injected incoherent magnon spin\ncurrent to the super\ruid spin current is not the only tran-\nsient process near the interface between media with and\nwithout spin super\ruidity. Even in the absence of spin\ncurrent the interface may a\u000bect the equilibrium mag-\nnetic structure. For example, the interface can induce\nanisotropy di\u000berent from easy-plane anisotropy in the\nbulk. Then the crossover from surface to bulk anisotropy\noccurs at the healing length of the order of the correla-\ntion length \u00180determined by Eq. (25) in ferromagnets, or\nthe correlation length \u0018determined by Eq. (54) in anti-\nferromagnets. The similar healing length was suggested\nfor ferromagnets by Takei and Tserkovnyak7and for an-\ntiferromagnets by Takei et al.8although using di\u000berent\narguments.\nThe expression (79) for \u0015was derived within hydrody-\nnamics with dissipation. At distances shorter than the\nmean-free path incoherent magnons are in the ballistic\nregime and cannot converge to the super\ruid current,\nsince conversion is impossible without dissipation. Alto-\ngether this means that the real healing length at which\nthe bulk super\ruid spin current state is formed cannot\nbe less than the longest from three scales: \u0015,\u00180, and the\nmagnon mean-free path cs\u001c. Apparently at low tempera-\ntures and weak magnetization Mzthe latter is the longest\none from three scales. However, close to the phase transi-\ntion to the easy-axis anisotropy ( Mz=M) the coherence\nlength\u00180diverges and becomes the longest scale.\nSolving the Boltzmann equation we assumed complete\nthermalization of the magnon distribution. At low tem-\nperatures when magnon-magnon interaction is weak the\nlength at which thermalization occurs essentially exceeds\nthe mean-free path on defects. It could be that the heal-\ning length would grow up to the thermalization length.\nThis requires a further analysis.\nVII. MAGNETIC VORTEX IN AN\nEASY-PLANE ANTIFERROMAGNET\nLet us consider structure of an axisymmetric vortex in\nan antiferromagnet with one quantum of circulation of\nthe angle'0of rotation around the vortex axis. Now\nwe consider the geometry of the experiment20when the12\nPtPtzxyCr2O3H\n\u0001\u0002\u0001\u0002\u0001\u0002\u0002\u0003\u0001\u0002\u0001\u0004\u0001\u0003\u0002\u0004\u0002\u0003\u0003\u0004\u0002\u0004\u0001\u0006\u0001\u0002\u0002\u0006\u0001\u0002\u0004\u0006\u0001\u0002\u0003\u0006\u0003\u0004\u0001\u0002\u0001\u0002\u0001\u0002\u0001\u0002\u0001\u0002\u0001\u0002\u0004\u0006\u0001\u0002\u0002\u0006\u0001\u0002\u0001\u0006\u0002\u0004\u0005\u0006\u0002\u0003\u0002\u0004\u0001\u0003\u0001\u0004\u0001\u0002\u0001\u0003\u0001\u0004\u0003\u0001a)\nb)\nFIG. 3. Precession of magnetization maround the direction\nof the magnetic \feld Halong the path around the vortex\naxis. (a) The geometry of the experiment20with the magnetic\n\feld (the axis z) in the plane of the Cr 2O3\flm. The vortex\naxis is normal to the \flm (the axis y). (b) Precession of the\nmagnetization mis shown in the plane xz(the plane of the\n\flm). The path around the vortex axis (dashed lines) is inside\nthe vortex core where the total magnetization is not parallel\ntoH(\u0012 6= 0).\nmagnetic \feld H(the axisz) is in the \flm plane. The\nvortex axis is the axis ynormal to the \flm plane (Fig, 3a).\nThe azimuthal component of the angle '0gradient is\nr'0=1\nr: (83)\nAt the same time '= 0 and\u00120is small. Then the Hamil-\ntonian (46) transforms to\nH=2M2\n\u001f\u00122\n0\u00002HM cos\u0012\u00120+A\u0000M2\u0012cos2\u0012\nr2+r\u00122\u0013\n:\n(84)\nMinimization with respect to small \u00120yields\n\u00120=\u001fHcos\u0012\n2M; (85)\nand \fnally the Hamiltonian is\nH=\u0000\u001fH2cos2\u0012\n2+A\u0000M2\u0012cos2\u0012\nr2+r\u00122\u0013\n:(86)The Euler{Lagrange equation for this Hamiltonian de-\nscribes the vortex structure in polar coordinates:\nd2\u0012\ndr2+1\nrd\u0012\ndr\u0000sin 2\u0012\n2\u00121\n\u00182\u00001\nr2\u0013\n= 0; (87)\nwhere the correlation length \u0018is given by Eq. (54) and\ndetermines the size of the vortex core.\nThe vortex core has a structure of a skyrmion, in which\nthe total weak magnetization deviates from the direction\nof the magnetic \feld H(\u00126= 0). The component of\nmagnetization transverse to the magnetic \feld is\nm?=\rHsin 2\u0012\n2: (88)\nThe transverse magnetization creates stray magnetic\n\felds at the exit of the vortex line from the sample. Fig-\nure 3 shows variation of the magnetization inside the core\nalong the path around the vortex axis parallel to the axis\ny. Along the path the magnetization mrevolves around\nthe direction of the magnetic \feld forming a cone. The\nprecession in space creates an oscillating ycomponent\nof magnetization my=m?(r) sin\u001e, where\u001eis the az-\nimuthal angle at the circular path around the vortex line.\nThis produces surface magnetic charges 4 \u0019myat the exit\nof the vortex to the boundary separating the sample from\nthe vacuum. These charges generate the curl-free stray\n\feldh=r . At distances from the vortex exit point\nmuch larger that the core radius the stray \feld is a dipole\n\feld with the scalar potential\n (R) =\u0019\u001fH\n2(R\u0001n)\nR3Z1\n0sin 2\u0012(r)r2dr\n= 1:2\u0019\u001fH\u00183(R\u0001n)\nR3=1:2\u0019\u001fc3\ns\n\r3H2(R\u0001n)\nR3: (89)\nHereR(x;y;z ) is the position vector with the origin in\nthe vortex exit point and nis a unit vector in the plane\nxzalong which the surface charge is maximal ( \u001e=\u0019=2).\nIn our model the direction of nis arbitrary, but it will\nbe \fxed by spin-orbit interaction or crystal magnetic\nanisotropy violating invariance with respect to rotations\naround the axis z. These interactions were ignored in our\nmodel. In principle, the stray \feld can be used for detec-\ntion of vortices nucleated at spin currents approaching\nthe critical value.\nVIII. DISCUSSION AND SUMMARY\nThe paper analyzes the long-distance super\ruid spin\ntransport. The super\ruid spin transport does not require\na gradient of the spin chemical potential (as the electron\nsupercurrent in superconductors does not require a gra-\ndient of the electrochemical potential). As result of it,\nmechanisms of dissipation are suppressed except for weak\nBloch spin relaxation. Other dissipation mechanisms af-\nfect the spin transport only at the transient (healing)13\nlayer close to the interface through which spin is injected,\nor in nonstationary processes.\nThe paper calculates the Landau critical spin phase\ngradient in a two-sublattice antiferromagnet when the\neasy-plane topology of the magnetic order parameter is\nprovided not by crystal magnetic anisotropy but by an\nexternal magnetic \feld. This was the case realized in\nthe experiment by Yuan et al.20. For this goal it was\nnecessary to derive the spectrum of collective modes (spin\nwaves) in spin current states. The Landau instability\ndestroying spin super\ruidity sets on not in the Goldstone\ngapless mode as in easy-plane ferromagnets but in the\ngapped mode, despite that at small spin currents the\nlatter has energy larger than the Goldstone mode.\nThe paper analyzes dissipation processes determining\ndissipation parameters (spin di\u000busion and second viscos-\nity coe\u000ecients) by solving the Boltzmann equation for\nmagnons scattered by defects. The two-\ruid theory sim-\nilar to the super\ruid two-\ruid hydrodynamics was sug-\ngested. It is argued that the LLG theory with the Gilbert\ndamping parameter is not able to properly describe dissi-\npation in easy-plane magnetic insulators. Describing the\nwhole dissipation by a single Gilbert parameter one can-\nnot di\u000berentiate between strong processes connected with\nhigh exchange energy (e.g., spin di\u000busion) and weak pro-\ncesses connected with spin-orbit interaction (Bloch spin\nrelaxation), which violate the spin conservation law.\nThe formation of the super\ruid spin current in the\ntransient (healing) layer near the interface through which\nspin is injected was investigated. The width of this layer\n(healing length) is determined by processes of dissipation,\nand at low temperatures can reach the scale of relevant\nmean-free paths of magnons including those at which the\nmagnon distribution is thermalized.\nThe structure of the magnetic vortex in the geometry\nof the experiment on Cr 2O3is investigated. In the vortex\ncore there is a magnetization along the vortex line, which\nis normal to the magnetic \feld. This magnetization pro-\nduces magnetic charges at the exit of the vortex line from\nthe sample. The magnetic charges create a stray dipole\nmagnetic \feld, which probably can be used for detection\nof vortices.\nWithin the developed two-\ruid theory the paper ad-\ndresses the role of the temperature variation in space on\nthe super\ruid spin transport. This is important because\nin the experiment of Yuan et al.20the spin is created\nin the Pt injector by heating (the Seebeck e\u000bect). Thus\nthe spin current to the detector is inevitably accompa-\nnied by heat \row. The temperature variation produces\nthe bulk Seebeck e\u000bect, which is estimated to be rather\nweak at low temperatures. However, it was argued26that\nprobably Yuan et al.20detected a signal not from spin\ncoming from the injector but from spin produced by the\nSeebeck e\u000bect at the interface between the heated anti-\nferromagnet and the Pt detector. Such e\u000bect has already\nbeen observed for antiferromagnet Cr 2O3.36If true, then\nYuan et al.20observed not long-distance spin transport\nbut long-distance heat transport. It is not supported bythe fact that Yuan et al. observed a threshold for super-\n\ruid spin transport at low intensity of injection, when ac-\ncording to the theory5violation of the approximate spin\nconservation law becomes essential. Investigation of su-\nper\ruid spin transport at low-intensity injection is more\ndi\u000ecult both for theory and experiment. But the exis-\ntence of the threshold is supported by extrapolation of\nthe detected signals from high-intensity to low-intensity\ninjection. According to the experiment, the signal at the\ndetector is not simply proportional to the squared elec-\ntric current j2responsible for the Joule heating in the\ninjector, but to j2+a. The o\u000bset ais evidence of the\nthreshold, in the analogy with the o\u000bset of IVcurves in\nthe mixed state of type II superconductors determining\nthe critical current for vortex deepening. With all that\nsaid, the heat-transport interpretation cannot be ruled\nout and deserves further investigation. According to this\ninterpretation, one can see the signal observed by Yuan\net al.20at the detector even if the Pt injector is replaced\nby a heater, which produces the same heat but no spin.\nAn experimental check of this prediction would con\frm\nor reject the heat-transport interpretation.\nLet us make some numerical estimations for Cr 2O3us-\ning the formulas of the present paper. It follows from\nneutron scattering data37that the spin-wave velocity is\ncs= 8\u0002105cm/sec. According to Foner38, the magne-\ntization of sublattices is M= 590 G and the magnetic\nsusceptibility is \u001f= 1:2\u000210\u00004. Then the total magne-\ntizationmz=\u001fHin the magnetic \feld H= 9 T used\nin the experiment is about 10 G, and the canting an-\ngle\u00120=mz=2M\u00190:01 is small as was assumed in our\nanalysis. The correlation length (54), which determines\nvortex core radius, is about \u0018\u00190:5\u000210\u00006cm. The stray\nmagnetic \feld produced by magnetic charges at the exit\nof the vortex line from the sample is 10( \u00183=R3) G, where\nRis the distance from the vortex exit point. The task to\ndetect such \felds does not look easy, but it is hopefully\npossible with modern experimental techniques.\nACKNOWLEDGMENTS\nI thank Eugene Golovenchits, Wei Han, Mathias Kl aui,\nRomain Lebrun, Allures Qaiumzadeh, Victoria Sanina,\nSo Takei, and Yaroslav Tserkovnyak for fruitful discus-\nsions and comments.\nAppendix: Dissipation in the LLG theory\nFor ferromagnets the LLG equation taking into ac-\ncount dissipation is\ndM\ndt=\r[Heff\u0002M] +\u000b\nM\u0014\nM\u0002dM\ndt\u0015\n; (A.1)\nwhere\u000bis the dimensionless Gilbert damping parameter.\nFor small\u000bthis equation is identical to the equation with14\nthe Landau{Lifshitz damping term:\n1\n\rdM\ndt=\u0014\nM\u0002\u000eH\n\u000eM\u0015\n+\u000b\nM\u0014\nM\u0002\u0014\nM\u0002\u000eH\n\u000eM\u0015\u0015\n:\n(A.2)\nTransforming the vector LLG equation to the equations\nfor two Hamiltonian conjugate variables, the zcompo-\nnentMzof magnetization and the angle 'of rotation\naround the zaxis, one obtains Eqs. (73) and (74) without\nthe term r(@R=@r\u0016) and with the dissipation function\nR=\u000b\rM2\n?\n2M\u00162+\u000bM\n2M2\n?(r\u0001Js)2; (A.3)\nwhich depends on the spin chemical potential \u0016itself,\nbut not on its gradient. Meanwhile, according to the\ntwo-\ruid theory of Sec. V, the r\u0016-dependent term in the\ndissipation function was responsible for the spin-di\u000busion\nterm in the continuity equation for Mz. Indeed, at deriva-\ntion of the continuity equation (1) from the LLG theory\nunder the assumption that \u0016\u0019M0\nz=\u001f=Mz=\u001f\u0000Hthe\nspin di\u000busion term /Ddoes not appear. The term\ndoes appear only if \u0016in the dissipation function (A.3)\nis determined by the more general expression (82) taking\ninto account the dependence on rMz. Then one obtains\nEqs. (1) and (2) with the equal spin di\u000busion and spin\nsecond viscosity coe\u000ecients\nD=\u0010=\u000b\rMA; (A.4)\nand the inverse Bloch relaxation time\n1\nT1=\u000b\rM2\n?\n\u001fM: (A.5)\nThe outcome looks bizarre. The spin di\u000busion emerges\nfrom the\u0016-dependent term in the dissipation function,\nwhich is incompatible with the spin conservation law, asif the spin di\u000busion is forbidden by the spin conserva-\ntion law. Evidently this conclusion is physically incor-\nrect. Moreover, in the analogy of magnetodynamics and\nsuper\ruid hydrodynamics the magnetization Mzcorre-\nsponds to the \ruid density. In hydrodynamics the \ruid\ndensity gradients are usually not taken into account in\nthe Hamiltonian and in the chemical potential since they\nbecome important only at small scales beyond the hydro-\ndynamical approach. This does not rule out the di\u000busion\nprocess. Similarly, one should expect that it is possible\nto ignore the magnetization gradients in the spin chemi-\ncal potential either. It is strange that the spin di\u000busion\nbecomes impossible in the hydrodynamical limit.\nAccording to the Noether theorem the total magnetiza-\ntion along the axis zis conserved if the Hamiltonian is in-\nvariant with respect to rotations around the axis zin the\nspin space. The Landau{Lifshitz theory of magnetism19\nis based on the idea that the spin-orbit interaction, which\nbreaks rotational symmetry in the spin space and there-\nfore violates the spin conservation law, is relativistically\nsmall compared to the exchange interaction because the\nformer is inversely proportional to the speed of light. So,\nalthough the spin conservation law is not exact, it is a\ngood approximation (see Sec. I). Then the spin Bloch\nrelaxation term/1=T1, which violates the spin conser-\nvation law, must be proportional to a small parameter\ninversely proportional to the speed of light and cannot\nbe determined by the same Gilbert parameter as other\ndissipation terms, which do not violate the spin conser-\nvation law\nThe insu\u000eciency of the LLG theory for description of\ndissipation was discussed before, but mostly at higher\ntemperatures. It was suggested to replace of the LLG\nequation by the Landau{Lifshitz{Bloch equation, in\nwhich the Bloch longitudinal spin relaxation is present\nexplicitly (see, e.g., Ref. 39 and references to earlier works\ntherein). Our analysis shows that the problem exists also\nat low temperatures.\n1B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898\n(1969).\n2E. B. Sonin, Zh. Eksp. Teor. Fiz. 74, 2097 (1978), [Sov.\nPhys.{JETP, 47, 1091 (1978)].\n3E. B. Sonin, Usp. Fiz. Nauk 137, 267 (1982), [Sov. Phys.{\nUsp., 25, 409 (1982)].\n4Y. 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B 90, 104428 (2014)."    },    {        "title": "1810.07384v2.Perpendicularly_magnetized_YIG_films_with_small_Gilbert_damping_constant_and_anomalous_spin_transport_properties.pdf",        "content": "Perpendicularly magnetized  YIG films with small  Gilbert \ndamping constant and anomalous spin transport properties  \nQianbiao  Liu1, 2, Kangkang  Meng1*, Zedong  Xu3, Tao Zhu4, Xiao guang Xu1, Jun \nMiao1 & Yong Jiang1* \n \n1. Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and \nTechnology Beijing, Beijing 100083, China  \n2. Applied and Engineering physics, Cornell University, Ithaca, NY 14853, USA  \n3. Department of Physics, South University of Science and Technology of China , Shenzhen 518055, \nChina  \n4. Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China  \nEmail: kkmeng@ustb.edu.cn ; yjiang@ustb.edu.cn  \n \nAbstract: The Y 3Fe5O12 (YIG) films with perpendicular magnetic anisotropy (PMA) \nhave recently attracted a great deal of attention for spintronics  applications. Here, w e \nreport the induced PMA in the ultrathin YIG films grown on \n(Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12 (SGGG) substrate s by epitaxial strain without \npreprocessing. Reciprocal space mapping shows  that the film s are  lattice -matched to \nthe substrate s without strain relaxation. Through ferromagnetic resonance and \npolarized neutron reflectometry measurements, we find that these YIG films have \nultra-low Gilbert damping constant  (α < 1×10-5) with a magnetic dead layer as thin as \nabout 0.3 nm  at the YIG/SGGG interfaces. Moreover, the transport behavior of the \nPt/YIG/SGGG films reveals an enhancement of spin mixing conductance and a large \nnon-monotonic magnetic field dependence of anomalous Hall effect as compared with \nthe Pt/YIG/Gd 3Ga5O12 (GGG) films. The non- monotonic anomalous Hall signal is \nextracted in the temperature range from 150 to 350 K, which has been ascribed to the possible non -collinear magnetic order at the Pt/YIG interface induced by uniaxial \nstrain.  \n \nThe spin transport in ferrim agnetic insulator (FMI)  based devices has received \nconsiderable interest due to its free of current -induced Joule heating and beneficial for \nlow-power spintronic s applications  [1, 2]. Especially, the high-quality Y3Fe5O12 (YIG)  \nfilm as a widely  studied FMI  has low damping  constant,  low magnetostriction and \nsmall magnetocrystalline anisotropy,  making it a key material for magnonics  and spin \ncaloritronics . Though the magnon s can carr y information over distances as long as \nmillimeters  in YIG film , there remain s a challenge to control its magnetic anisotropy \nwhile maintaining the low damping constant [3] , especially for  the thin film with \nperpendicular  magnetic anisotropy (PMA) , which  is very useful for spin polarizers,  \nspin-torque oscillators, magneto -optical d evices  and m agnon valve s [4-7]. In addition, \nthe spin- orbit torque (SOT) induced magnetization switching with low current \ndensities has been realized in non -magnetic heavy metal  (HM)/FMI heterostructures , \npaving the road towards  ultralow -dissipation SOT de vices based on FMI s [8-10]. \nFurthermore, p revious theoretical studies have pointed that  the current density will \nbecome much smaller if the domain structures were topologically protected (chiral) [11].  However,  most FMI  films  favor in-plane easy axis dominated by shape \nanisotropy , and the investigation is eclipsed as compared with ferromagnetic materials \nwhich show abundant and interesting domain structures such as chiral domain walls and magnetic skyrmions et al.  [12-17]. Recently, the interface- induc ed chiral domain walls  have been observed in centrosymmetric oxides Tm 3Fe5O12 (TmIG) thin films, \nand the  domain walls  can be propelled by spin current from an adjacent platinum \nlayer  [18]. Similar with the TmIG films, the possible chiral  magnetic structures are \nalso expected in the YIG films with lower damping  constan t, which  would further \nimprove the chiral domain walls’ motion speed.  \nRecently, several ways  have been reported to attain the perpendicular ly \nmagnetized YIG films , one of which is utiliz ing the lattice distortion and \nmagnetoelastic effect induced by  epitaxial strain [1 9-22]. It is noted that  the strain \ncontrol can not only enable the field -free magnetization switching but also assist the \nstabilization of the non- collinear  magnetic textures  in a broad range of magnetic field \nand temperature. Therefore, abundant and interesting physical phenomena  would \nemerge in epitaxial grown YIG films with PMA. However, either varying the buffer \nlayer or doping would increase the Gilbert damping constant of YIG, which will  \naffect the efficiency of the SOT induced magnetization switching  [20, 21]. On the \nother hand, these preprocessing would lead to a more complicate magnetic structures  \nand impede  the further discussion of spin transport properties such as possible \ntopological Hall effect  (THE).  \nIn this work, we realized the PMA of ultrathin YIG films deposited on SGGG \nsubstrates  due to epitaxial strain . Through ferromagnetic resonance (FMR) and \npolarized neutron reflectometry (PNR) measurements, we have found that the YIG \nfilms had small Gilbert damping constant with a magnetic dead layer as thin as about \n0.3 nm  at the YIG/SGGG interfaces. Moreover, we have carried out the transport measurements of the Pt/YIG/SGGG films and observed a large non -monotonic \nmagnet ic field dependence of the anomalous Hall resistivity, which did not exis t in \nthe compared Pt/YIG/GGG films. The non -monotonic anomalous Hall signal was \nextracted in the temperature range from 150 to 350 K, and we ascribed it  to the \npossible non -collinear magnetic order at the Pt/YIG interfaces induced by uniaxial \nstrain.  \n \nResults  \nStructural and magnetic characterization.  The epitaxial YIG films with varying \nthickness from 3 to 90 nm  were grown on the [111] -oriented GGG substrate s (lattice \nparameter a = 1.237 nm) and SGGG substrates  (lattice parameter a = 1.248 nm)  \nrespectively by pulsed laser deposition technique  (see methods). After the deposition, \nwe have investigated the surface morphology of the two kinds of films using atomic \nforce microscopy (AFM) as shown in Fig. 1 ( a), and the two films have a  similar and \nsmall surface roughness ~0.1 nm. Fig. 1 ( b) shows the  enlarged  XRD ω-2θ scan \nspectra of the YIG (40 nm) thin film s grow n on the two different substrates (more \ndetails are shown in the Supplementary Note 1 ), and they  all show predominant (444) \ndiffraction peaks without any other diffraction peaks, excluding impurity phases or other crystallographic orientation s and indicat ing the single -phase  nature. According \nto the (444) diffraction pe ak position and the reciprocal space map of the (642) \nreflection of a 40 -nm-thick YIG film grown on SGGG as shown in Fig. 1(c), we have \nfound that the lattice constant of SGGG  (~1.248 nm) substrate was larger than the YIG layer  (~1.236 nm). We quantify thi s biaxial strain as ξ  = (aOP - aIP)/aIP, where a OP \nand aIP represent the pseudo  cubic lattice constant calculated from the ou t-of-plane  \nlattice constant d(4 4 4) OP and in-plane lattice constant d(1 1 0) IP, respectively, \nfollowing the equation of  \n2 2 2lkhad\n++= , with h, k, and l  standing for the Miller \nindices of the crystal planes . It indicates that the SGGG substrate provides a tensile \nstress ( ξ ~ 0.84%) [21]. At the same time, the magnetic properties of the YIG films \ngrown on the two different substrates were measured via VSM magnetometry at room \ntemperature. According to the magnetic field ( H) dependence of the magnetization  (M) \nas shown in Fig. 1 (d), the magnetic anisotropy of the YIG film grown on SGGG \nsubstrate has been modulated by strain, while the two films have similar in -plane \nM-H curves.  \nTo further investigate the quality of the YIG films  grown on SGGG substrates  \nand exclude  the possibility of the strain induced large stoichiometry  and lattice \nmismatch, compositional analyse s were carried out using x -ray photoelectron \nspectroscopy (XPS) and PNR. As shown in Fig. 2 (a), the difference of binding \nenergy between the 2p 3/2 peak and the satellite peak is about 8.0 eV, and the Fe ions \nare determined to be in the 3+ valence state.  It is found that there is  no obvious \ndifference for Fe elements in the YIG films grown on GGG and SGGG substrates. \nThe Y 3 d spectrums show a small energy shift as shown in Fig. 2 (b) and the binding \nenergy shift may be related to the lattice strain and the variation of bond length  [21]. \nTherefore, the stoichiometry of the YIG  surface has not been dramatically modified \nwith the strain control.  Furthermore, we have performed the PNR meas urement  to probe the depth dependent struc ture and magnetic information of YIG films grown on \nSGGG substrates. The PNR  signals and scattering length density (SLD) profiles  for \nYIG (12.8 nm)/SGGG films by applying an in- plane magnetic field of 900 mT at \nroom temperature are shown in Fig. 2 ( c) and ( d), respectively.  In Fig. 2(c), R++ and \nR-- are the nonspin -flip reflectivities, where the spin polarizations are the same for the \nincoming and reflected neutrons. The inset of Fig. 2(c) shows the experimental and  \nsimulated spin -asymmetry (SA), defined as SA = ( R++ – R--)/(R++ + R--), as a function \nof scattering vector Q. A reasonable fitting was obtained with a three- layer model for \nthe single YIG film, containing the interface layer , main YIG layer and surface layer. \nThe nuclear SLD and  magnetic SLD are directly proportional to the nuclear scattering \npotential and the magnetization , respectively . Then, the depth- resolved structural and \nmagnetic SLD profiles delivered by fitting are s hown in Fig. 2(d) . The Z -axis \nrepresents the distance for the vertical direction of the film, where Z  = 0 indicates the \nposition at the YIG/SGGG interface. It is obvious that  there is few Gd diffusion into \nthe YIG film, and the dead layer  (0.3 nm ) is much thinner than the reported values \n(5-10 nm) between YIG  (or T mIG) and substrates [23 -25]. The net magnetization of \nYIG is 3.36 μB (～140  emu/cm3), which is similar with that of bulk YIG [2 6]. The \nPNR results also showed that besides the YIG/ SGGG interface region, there is also  \n1.51- nm-thick  nonmagnetic surface layer, which may be  Y2O3 and is likely to be \nextremely important in magnetic proximity effect [ 23]. \n Dynamical characterization and spin transport properties.  To quantitatively \ndetermine  the magnetic anisotropy and dynamic properties of the YIG films, the FMR \nspectra were measured at room temperature using an electron paramagnetic resonance \nspectrometer with rotating the films. Fig. 3(a) shows the geometric configuration of the angle reso lved FMR measurements. We use the FMR absorption line shape to \nextract the resonance field  (H\nres) and peak -to-peak linewidth ( ΔHpp) at different θ  for \nthe 40 -nm-thick YIG fil ms grown on GGG and SGGG substrates, respectively.  The \ndetails for 3 -nm-thick  YIG film are show n in the Supp lementary  Note 2 . According to \nthe angle  dependence of H res as shown in Fig. 3(b), one can find that as compared \nwith the YIG films grown on GGG substrate s, the minimum Hres of the 40- nm-thick \nYIG film grown on SGGG substrate increases with varying θ from 0° to 90° .On the \nother hand,  according to the frequency dependence of Hres for the YIG (40 nm)  films \nwith applying H  in the XY  plane as shown in Fig. 3(c), in contrast to the YIG/GGG \nfilms, the H res in YIG/SGGG films could not be fitted by the in-plane magnetic \nanisotropy Kittel formula  21)] 4 ( )[2(/\neff res res πM H Hπγ/ f + = . All these results \nindicate that the easy axis of YIG  (40 nm) /SGGG films  lies out -of-plane. The angle  \ndependent  ΔHpp for the  two films are  also compared as shown in Fig. 3(d) , the \n40-nm-thick YIG film grown on SGGG substrate has  an optimal value of Δ Hpp as low \nas 0.4 mT at θ =64°, and the corresponding FMR absorption line and Lorentz fitting \ncurve are shown in Fig. 3(e).  Generally , the ΔHpp is expected to be minimum \n(maximum) along  magnetic easy (hard) axis, which is basically coincident with the \nangle dependent  ΔHpp for the YIG films grown on GGG substrates.  However, as shown in Fig. 3(d), the  ΔHpp for the YIG/SGGG films  shows an anomalous variation. \nThe lowest  ΔHpp at θ=64° could be ascribed to the high YIG film quality and  ultrathin  \nmagnetic dead layer at the YIG/SGGG interface.  It should be noted that , as compared \nwith YIG/GGG films , the Δ Hpp is independent on the frequency from 5 GHz to 14 \nGHz as shown in Fig. 3(f). Then, w e have calculate d the Gilbert damping constant α  \nof the  YIG (40 nm)/SGGG films by extracting the Δ Hpp at each frequency as shown in \nFig. 3(f). The obtained α  is smaller tha n 1 × 10−5, which  is one order of magnitude \nlower than t he report in  Ref. [20]  and would  open new perspectives for the \nmagnetization dynamics.  According to the theor etical theme, the ΔHpp consists of \nthree parts: Gilbert damping, two magnons scattering relaxation process and \ninhomogeneities, in which both the Gilbert damping and the two magnons scattering \nrelaxation process depend on frequency. Therefore, the large Δ Hpp in the YIG/SGGG \nfilms  mainly  stems from the inhomogeneities, w hich will be discussed next with the \nhelp of the transport measurements. All of the above results have proven that the  \nultrathin YIG films grown on SGGG substrate s have not only evident  PMA  but also \nultra-low Gilbert damping constant.  \nFurthermore, we have also investigated the spin transport properties for  the high \nquality YIG film s grown on SGGG substrate s, which  are basically sensitive to  the \nmagnetic details of YIG. The magnetoresistance (MR) has been proved as a powerful \ntool to effectively explore magnetic information originating from the interfaces [ 27]. \nThe temperature dependent spin Hall magnetoresistance (SMR) of the Pt (5 nm)/YIG \n(3 nm) films grown on the two different substrates were measured using a small and non-perturbative current densit y (~ 106 A/cm2), and the s ketches of the measurement  \nis shown in Fig. 4 (a). The β scan of the longitudinal  MR, which is defined as \nMR=ΔρXX/ρXX(0)=[ρXX(β) -ρXX(0)]/ρXX(0) in the YZ  plane  for the two films  under a 3 T \nfield (enough to saturate the magnetization  of YIG ), shows cos2β behavior s with \nvarying temperature  for the Pt/YIG/GGG and Pt/YIG/SGGG films as shown in Fig. 4 \n(b) and (c), respectively. T he SMR of  the Pt/YIG /SGGG films  is larger than that of \nthe Pt/YIG /GGG films with the same thickness of YIG  at room temperature,  \nindica ting an enhanced  spin mixing conductance ( G↑↓) in the Pt/YIG /SGGG films. \nHere, it should be noted that the spin transport properties for the Pt layers ar e \nexpected to be the same because of the similar resistivity and film s quality . Therefore, \nthe SGGG substrate not only induces the PMA but also enhances G ↑↓ at the Pt/YIG \ninterface. Then, we have also investigated the field dependent Hall resistivities  in the \nPt/YIG/SGGG films at the temperature range from 260 to 350 K  as shown in Fig. 4(d).  \nThough the conduction electrons cannot penetrate into the FMI layer, the possible \nanomalous Hall effect (AHE) at the HM/FMI interface is proposed to emerge, and the \ntotal Hall resistivity can usually be expressed as the sum of various contributions  [28, \n29]: \nS-A S H ρ  ρ H R ρ + + =0 ,         (1)  \nwhere R0 is the normal Hall coefficient, ρ S the transverse manifestation of SMR, and \nρS-A the spin Hall anomalous Hall effect (SAHE) resistivity. Notably, the external field \nis applied out -of-plane, and  ρs (~Δρ1mxmy) can be neglected [ 29]. Interestingly, the \nfilm grown on SGGG substrate shows a bump and dip feature during the hysteretic measurements in the temperature range from 260 to 350 K. In the following \ndiscussion, we term the part of extra anomalous signals as the anomalous SAHE resistivity ( ρ\nA-S-A). The ρ A-S-A signals clearly coexist with the large background of \nnormal Hall effect. Notably, the broken (space) inversion symmetry with strong \nspin-orbit coupling (SOC)  will induce the Dzyaloshinskii -Moriya interaction (DMI) . \nIf the DMI could be compared with the Heisenberg exchange interaction and the \nmagnetic anisotropy  that were controlled by st rain, it c ould stabilize non-collinear \nmagnetic textures  such as skyrmions, producing  a fictitious magnetic field and the \nTHE . The ρA-S-A signals  indicate that a chiral spin texture may exist, which  is similar \nwith B20-type compounds Mn 3Si and Mn 3Ge [ 30,31]. To more clearly demonstrate \nthe origin of the anomalous  signals, we have subtracted the normal Hall term , and the \ntemperature dependence of ( ρS-A + ρ A-S-A) has been shown in Fig. 4 (e). Then, we can \nfurther discern the peak and hump structure s in the temperature range from 260 to 350 \nK. The SAHE  contribution ρS-A can be expressed  as 𝜌𝑆−𝐴=𝛥𝜌2𝑚𝑍 [32, 33],\n where  \n𝛥𝜌2 is the coefficient depending on the imaginary part of G ↑↓, and mz is the unit \nvector of the magnetization  orientation along  the Z direction . The extracted Hall \nresist ivity ρA-S-A has been shown in Fig. 4 (f), and the temperature dependence of the \nlargest ρA-S-A (𝜌𝐴−𝑆−𝐴Max) in all the films have been shown in Fig. 4 (g). Finite values of \n𝜌𝐴−𝑆−𝐴Max exist  in the temperature range from 150 to 350 K , which is much d ifferen t \nfrom that in B20 -type bulk chiral magnets which are subjected to low temperature and \nlarge magnetic field [34]. The large non -monotonic magnetic field dependence of anomalous Hall resistivity  could not stem from the We yl points, and the more detailed \ndiscussion was shown in the Supplementary Note 3.  \nTo further discuss the origin of the anomalous transport signals, we have \ninvestigated the small field dependence of the Hall resistances for Pt  (5 nm) /YIG (40 \nnm)/SGGG films as shown in Fig. 5(a). The out-of-plane hysteresis loop  of \nPt/YIG/SGGG is not central symmetry, which indicates the existence of an  internal \nfield leading to opposite  velocities  of up to down and down to domain walls in the  \npresence of current along the +X  direction. The large field dependences of the Hall \nresistances are shown in Fig. 5(b), which could not be described by Equation (1). \nThere are large variations for the Hall signals when the external magnetic field  is \nlower than the saturation field ( Bs) of YIG film (~50 mT at 300 K and ~150 mT at 50 \nK). More interestingly, we have firstly applied a large out -of-plane external magnetic \nfield of +0.8 T ( -0.8 T) above Bs to saturate the out -of-plane magnetization \ncomp onent MZ > 0 ( MZ < 0), then decreased the field to zero, finally the Hall \nresistances were measured in the small field range ( ± 400 Oe), from which  we could \nfind that the shape was reversed as shown in Fig. 5(c). Here, we infer that the magnetic structures  at the Pt/YIG interface grown on SGGG substrate could not be a \nsimple linear magnetic order.  Theoretically , an additional chirality -driven Hall effect \nmight be present in the ferromagnetic regime due to spin canting [3 5-38]. It has been \nfound that  the str ain from an insulating substrate could produce a tetragonal distortion, \nwhich would drive an orbital selection, modifying  the electronic properties and the \nmagnetic ordering of manganites. For A\n1-xBxMnO 3 perovskites, a compressive strain makes the ferromagnetic configuration relatively more stable than the \nantiferromagnetic state [3 9]. On the other hand, the strain would induce the spin \ncanting [ 40]. A variety of experiments and theories have reported that the ion \nsubstitute, defect and magnetoelast ic interaction would cant the magnetization of YIG \n[41-43]. Therefore, if we could modify the magnetic order by epitaxial strain, the \nnon-collinear magnetic structure is expected to emerge in the YIG film.  For YIG \ncrystalline structure, the two Fe sites ar e located on the octahedrally coordinated 16(a) \nsite and the tetrahedrally coordinated 24(d) site, align ing antiparallel with each other  \n[44]. According to the XRD and RSM results, the tensile strain due to SGGG \nsubstrate would result in the distortion ang le of the facets of the YIG unit cell smaller \nthan 90 ° [45]. Therefore, the magneti zations  of Fe at two sublattice s should  be \ndiscussed separately rather than as a whole. Then, t he anomalous  signals  of \nPt/YIG/SGGG films  could be  ascribed to the emergence o f four different Fe3+ \nmagnetic orientation s in strained Pt/YIG films, which are shown in Fig. 5(d).  For \nbetter to understand our results, w e assume that, in analogy with ρ S, the ρA-S-A is larger \nthan  ρA-S and scales linearly with m ymz and mxmz. With applying a large external field \nH along  Z axis, the uncompensated magnetic moment at the tetrahedrally coordinated \n24(d) is along with the external fields H  direction for |H | > Bs, and the magnetic \nmoment tends to be along A (-A) axis  when the external fields is swept  from 0.8 T \n(-0.8 T) to 0 T. Then,  if the Hall resistance  was measured at small out -of-plane field , \nthe uncompensated magnetic moment  would switch from A (-A) axis to B (-B) axis. In \nthis case, the ρ A-S-A that scales with Δ ρ3(mymz+mxmz) would change the sign because the mz is switched from the Z  axis to - Z axis as shown in Fig. 5(c). However, there is \nstill some problem that needs to be further clarified. There are no anomalous  signals  \nin Pt/YIG/GGG films that could be ascribed to the weak strength of Δρ3 or the strong \nmagnetic anisotropy . It is still valued for further discussion of the origin of Δ ρ3 that \nwhether it could stem from the skrymions et al ., but until now we have not observed \nany chiral domain structures in Pt/YIG/SGGG films through the Lorentz transmission \nelectron microscopy. Therefore, we hope that future work would  involve more \ndetailed magnetic microscopy imaging and microstructure analysis, which can further elucidate the real microscopic origin of the large non -monotonic magnetic field \ndependence of anomalous Hall resistivity.   \n \nConclusion  \nIn conclusion, the YIG film with PMA could be realized using both epitaxial strain \nand growth -induced anisotropies. These YIG films grown on SGGG substrates had \nlow G ilbert damping constants (<1 ×10\n-5) with a magnetic dead layer as thin as about \n0.3 nm  at the YIG/SGGG interface. Moreover, we observe d a large non -monotonic \nmagnetic field dependence of anomalous Hall resistivity in Pt/YIG/SGGG films, \nwhich did not exist in Pt/YIG/GGG films. The non -monotonic anomalous portion of \nthe Hall signal was extracted in the temperature range from 150 to 350 K and w e \nascribed it to the possible non -collinear magnetic order at the Pt/YIG interface \ninduced by uniaxial strain. The present work not only demonstrate that the strain \ncontrol can effectively tune the electromagnetic properties of FMI but also open up the exp loration of non -collinear spin texture for fundamental physics and magnetic \nstorage technologies based on FMI.  \n \nMethods  \nSample preparation.  The epitaxial YIG films with varying thickness from 3 to 90 \nnm were grown on the [111] -oriented GGG substrate s (lattice parameter a =1.237 nm) \nand SGGG substrates  (lattice parameter a  =1.248 nm)  respectively by pulsed laser \ndeposition technique . The growth  temperature was TS =780 ℃ and the oxyg  \npressure  was varied from  10 to 50 Pa . Then,  the films were annealed at 780℃ for 30 \nmin at the oxygen pressure of  200 Pa . The Pt (5nm) layer was deposited on  the top of  \nYIG films  at room temperature by magnetron sputtering. After the deposition, the \nelectron beam lithography and Ar ion milling were used to pattern Hall bars, and a lift-off process was used to form contact electrodes . The size of all the Hall bars is 20 \nμm×120 μm. \nStructural and magnetic characterization. The s urface morphology was measured \nby AFM (Bruke Dimension Icon).  Magnetization measurements were carried out \nusing a Physical Property Measurement System (PPMS) VSM.  A detailed \ninvestigation of the magnetic information  of Y IG was investigated by  PNR at the \nSpallation Neutron Source of China.   \nFerromagnetic resonance measurements. The measurement setup is depicted in Fig. \n3(a). For FMR measurements, the DC magnetic field was modulated with an AC field. \nThe transmitted signal was detected by a lock -in amplifier. We observed the FMR spectrum of the sample by sweeping the external magnetic field. The data obtained \nwere then fitted to a sum of symmetric and antisymmetric Lorentzian functions to \nextract the linewidth.  \nSpin transport measurements . The measurements  were carried out  using PPMS \nDynaCool.  \n \nAcknowledgments  \nThe authors thanks Prof. L. Q. Yan and Y. Sun for the technical assistant in \nferromagnetic resonance  measurement . This work was partially supported by the \nNational Science Foundation of China (Grant Nos. 51971027, 51927802, 51971023 , \n51731003, 51671019, 51602022, 61674013, 51602025), and the Fundamental Research Funds for the Central Universities (FRF- TP-19-001A3).  \n References  \n[1] Wu, M.-Z. & Hoffmann , A. Recent advances in magnetic insulators from \nspintronics to microwave applications. Academic Press , New York, 64 , 408 \n(2013) . \n[2] Maekawa,  S. Concepts in spin electronics. Oxford Univ., ( 2006) . \n[3] Neusser, S. & Grundler, D. Magnonics: spin waves on the nanoscale. Adv. 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N. & Saslow W. M. Defect interactions and canting in ferromagnets. Phys. Rev. B 38, 11718 (1988).  \n[42] Rosencwaig  A. Localized canting model for substituted ferrimagnets. I. Singly \nsubstituted YIG systems. Can. J. Phys. 48, 2857- 2867(1970).  \n[43] AULD B. A. Nonlinear magnetoelastic interactions. Proceedings of the IEEE, 53,  \n1517- 1533 (1965).  \n[44] Ching W. Y., Gu Z. & Xu Y N. Th eoretical calculation of the optical properties \nof Y\n3Fe5O12. J. Appl. Phys. 89,  6883- 6885 (2001).  [45] Baena A., Brey L. & Calder ón M. J. Effect of strain on the orbital and magnetic \nordering of manganite thin films and their interface with an insulator. Phys. Rev. \nB 83, 064424 (2011).  \n   \nFigure Captions  \n \nFig. 1 Structural  and magnetic  properties of YIG films.  (a) AFM images of the  \nYIG films  grown on  the two  substrates (scale bar, 1 μ m). (b) XRD ω-2θ scans of the \ntwo different YIG films  grown on  the two substrates . (c) High -resolution XRD \nreciprocal space map of t he YIG film deposited on the SGGG substrate. (d) Field \ndependence of the normalized magnetization of the YIG films  grown  on the two \ndifferent substrates . \n \n \nFig. 2 Structural  and magnetic  properties of YIG films.  Room temperature XPS \nspectra of (a) Fe 2p and (b) Y 3d for YIG films grown  on the two substrates . (c) P NR \nsignals (with a 900 mT in -plane field) for the spin -polarized R++ and R-- channels. \nInset: The experimental and simulated SA  as a function of scattering vector Q.  (d) \nSLD profiles of the  YIG/SGGG films.  The nuclear SLD and  magnetic SLD is directly \nproportional to the nuclear scattering potential and the magnetization , respectively.  \n \n \n \n \nFig. 3 Dynamical properties of YIG films . (a) The geometric configuration of the \nangle dependent FMR measurement. (b) The angle  dependence of the H res for the YIG \nfilms on GGG and SGGG substrates. (c) The frequency dependence of the H res for \nYIG films grown on GGG and S GGG substrates. (d) The ang le dependence of Δ Hpp \nfor the YIG films on GGG and SGGG substrates. (e) FMR spectrum of the \n40-nm-thick YIG film grown on SGGG substrate with 9.46 GHz at θ =64°. (f) The \nfrequency dependence of Δ Hpp for the 40 -nm-thick YIG films grown on GGG and \nSGGG substr ates.  \n \nFig. 4 Spin transport properties  of Pt/YIG (3nm) films . (a) The definition of the \nangle, the axes and the measurement configurations. ( b) and ( c) Longitudinal MR at \ndifferent temperatures in  Pt/YIG/GGG and Pt/YIG/SGGG films respectively (The \napplied magnetic field is 3 T). (d) Total Hall resistivities vs  H for Pt/YIG/SGGG films \nin the temperature range from 260 to 300 K. (e)  (ρS-A+ρA-S-A) vs H for two films in the \ntemperature range from 260 to 300 K. (f) ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. \nInset: ρS-A and ρS-A + ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. (g) Temperature \ndependence of the 𝜌𝐴−𝑆−𝐴𝑀𝑎𝑥. \n \n \n \nFigure 5 S pin transport properties  of Pt/YIG ( 40 nm) films . (a) and (b) The Hall \nresistances vs H for the Pt/YIG/SGGG films in the temperature range from 50 to 300 \nK in small  and large magnetic field range, respectively. (c) The Hall resistances vs  H \nat small magnetic field range after sweeping a large out -of-plane  magnetic field +0.8 \nT (black line) and - 0.8 T (red line) to zero.  (d) An illustration of the orientations of the \nmagnetizations Fe ( a) and Fe ( d) in YIG films with the normal in -plane magnetic \nanisotropy (IMA), the ideal  strain induced PMA and the actual magnetic anisotropy \ngrown on SGGG in our work.  \n"    },    {        "title": "1810.08487v1.Magnon_properties_of_random_alloys.pdf",        "content": "Magnon properties of random alloys\nFan Pan,1, 2Anna Delin,1, 2, 3Anders Bergman,3and Lars Bergqvist1, 2\n1Department of Applied Physics, School of Engineering Sciences,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, Sweden\n2SeRC (Swedish e-Science Research Center), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden\n3Department of Physics and Astronomy, Materials Theory Division,\nUppsala University, Box 516, SE-75120 Uppsala, Sweden\n(Dated: November 15, 2021)\nWe study magnon properties in terms of spin sti\u000bness, Curie temperatures and magnon spectrum\nof Fe-Ni, Co-Ni and Fe-Co random alloys using a combination of electronic structure calculations\nand atomistic spin dynamics simulations. In\ruence of the disorder are studied in detail by use of\nlarge supercells with random atomic arrangement. It is found that disorder a\u000bects the magnon\nspectrum in vastly di\u000berent ways depending on the system. Speci\fcally, it is more pronounced in\nFe-Ni alloys compared to Fe-Co alloys. In particular, the magnon spectrum at room temperature\nin Permalloy (Fe 20Ni80) is found to be rather di\u000buse in a large energy interval while in Fe 75Co25it\nforms sharp branches. Fe-Co alloys are very interesting from a technological point of view due to\nthe combination of large Curie temperatures and very low calculated Gilbert damping of \u00180.0007\nat room temperature for Co concentrations around 20{30%.\nI. INTRODUCTION\nThere has been a growing interest in disordered\nmagnetic materials in the last few decades in the\nform of transition metal alloys and diluted magnetic\nsemiconductors1{12. A central motivation for many stud-\nies is the potential of these materials in spintronics and\nmagnonics applications. Magnon excitations are com-\nmonly studied experimentally using inelastic neutron\nscattering suitable for bulk systems such as Co13or spin\npolarized electron loss spectroscopy (SPEELS) for low\ndimensional magnets such as Co 8/Cu00114. Theoreti-\ncally, the simplest approach for calculating magnon spec-\ntrum for elements and compounds is through linear spin\nwave theory of the Heisenberg Hamiltonian. However,\nfor accurate studies of alloys, both the treatment of dis-\norder and thermal e\u000bects needs to be handled reliable.\nMagnons in disordered magnets, either random alloys or\ndiluted, are more complicated than for ordered systems\nfor a number of reasons. Due to broken translational\nsymmetry, perfect magnon modes with in\fnite life time\nas in ordered magnets are absent but at certain condi-\ntions, one may still expect well de\fned magnon modes\nbut with a \fnite lifetime due to disorder. The damping\nand formation of these modes are of great interest both\ntheoretically and for applications.\nPrevious studies of magnon properties of disordered\nmagnets have been focused on diluted magnets and the\ne\u000bect of dilution on the magnon spectrum and spin\nsti\u000bness15{18. The main \fndings from these studies are\nthat the region with well de\fned magnon modes are de-\ncreasing with dilution and properties are strongly dimen-\nsionality dependent. Surprisingly, there are only very few\npublished studies of magnons in random alloys with full\nconcentration of magnetic elements19{21, such as Fe-Co\nalloys22,23. The aim for the present study is to introduce\na simple methodology for theoretical studies of magnons\nin disordered materials. We are using this methodologyto investigate magnon and other \fnite temperature prop-\nerties, i.e. spin sti\u000bness, Cure temperatures and Gilbert\ndamping for bulk transition metal alloys that hopefully\nwill stimulate experiments in the new generation of neu-\ntron scattering facilities currently in construction.\nThe paper is organized as follows: In Section II we\nintroduce the methodology and give the details of the\ncalculations, in Section III we present our \fndings and\nin Section IV we give a summary and provide an outlook.\nII. FORMALISM\nA. Spin excitations in solids\nA magnetic solid at \fnite temperature displays two dif-\nferent kinds of magnetic excitations, namely spin wave\nexcitations (magnons) and electron-hole pair excitations\n(Stoner). The magnon excitations are responsible for\ntransversal \ructuations while Stoner excitations cause\nlongitudinal changes of the moments. At low temper-\natures and in particular for bulk materials, as in this\nstudy, the magnon excitations dominate and as a \frst\napproximation the Stoner excitations can be neglected.\nHowever, it is worth noting that they may play an im-\nportant role at high temperatures and also for certain\nmaterials with induced magnetic moments. Longitudinal\n\ructuations can however be modelled in a more advanced\ntreatment24.\nThe low energy spin excitation in a form of a magnon\nis characterized by the wave vector qwithin the Brillouin\nzone and for a cubic, ordered, material the magnon en-\nergyE(q) = ~!(q)\u0019Dq2, whereDis the spin wave\nsti\u000bness25\nD=2\n3MX\njJ0jR2\noj; (1)arXiv:1810.08487v1  [cond-mat.mtrl-sci]  19 Oct 20182\nandMis the magnetization, Jijis the exchange inter-\nactions between magnetic moments mat sitesiandj\nconnected with position vector R. In the case of disor-\nder, the spin wave sti\u000bness Dis obtained in by averaging\nover allNatoms in the system as\nD=2\n3M1\nNX\nnX\njJnjR2\nnj; (2)\nB. Atomistic spin dynamics\nThe dynamics of a magnetic material at \fnite tem-\nperature and thus the magnetic excitations, is conve-\nniently modelled through atomistic spin dynamics (ASD)\nsimulations26. Within ASD, the temporal evolution of\nthe atomic moments mat \fnite temperature is governed\nby Langevin dynamics, through coupled stochastic dif-\nferential equations, the Landau-Lifshitz-Gilbert (LLG)\nequations, here written in the Landau-Lifshitz form,\ndmi\ndt=\u0000\r\n(1 +\u000b2)mi\u0002[Bi+bi(t)] (3)\n\u0000\r\u000b\nm(1 +\u000b2)mi\u0002fmi\u0002[Bi+bi(t)]g;\nwhere\ris the electron gyromagnetic ratio and \u000bis the\nGilbert damping parameter. The latter can either be\ntaken from experiments using ferromagnetic resonance\n(FMR) or calculated from \frst-principles. The e\u000bective\ninteraction \feld Biexperienced by each atomic moment\niis given by\nBi=\u0000@H\n@mi: (4)\nwhereHis the spin Hamiltonian governing the interac-\ntions between the magnetic moments. We are employing\nthe semi-classical Heisenberg model,\nH=\u0000P\nijJijmi\u0001mj, where the exchange interactions\nare parametrized from \frst-principles calculations. The\ne\u000bective interaction \feld is complemented with a stochas-\ntic \feld bithat is modeled with uncorrelated white noise\nwith a temperature dependent variance26.\nC. Magnon dispersion\nWe are employing two di\u000berent complementary meth-\nods for calculating the magnon spectrum, 1) the adia-\nbatic magnon spectrum (AMS) valid for the ground state\nand 2) from ASD simulations through the dynamical\nstructure factor at \fnite temperatures and damping.\n1. Adiabatic magnon spectrum\nThe adiabatic magnon spectrum is directly connected\nto the real-space exchange interactions JijthroughFourier transformation27,28. LetJ\u000b\f(q) denote the\nFourier transform of the exchange interaction between\nchemical type \u000band\fwith a wave-vector qlying in the\nBrillouin zone (BZ). J\u000b\f(q) is calculated as\nJ\u000b\f(q) =X\nj6=0J\u000b\f\n0jexp(iq\u0001R0j): (5)\nIn the spirit of virtual crystal approximation (VCA), it\nis tempting to perform a chemical average of the Fourier\ntransformed exchange interactions, i.e.\n~J(q) =J11(q)x2\n1+J12(q)x1x2+J21(q)x1x2+J22(q)x2\n2\n(6)\nin the case of binary alloy and where x1andx2are the\nconcentration of each chemical type. In such a case,\nthe \"e\u000bective\" magnon energy ( ~=1) for each wavevec-\ntorqcan then be adapted to the expression valid for one\natom/cell of ordered systems27,28\n~!(q) =4\n~M\u0010\n~J(0)\u0000~J(q)\u0011\n; (7)\nwhere ~Mis the saturation magnetization. However,\nthis treatment of the disorder is over-simpli\fed and\ndoes not reproduce experimentally observed excitations.\nAnalogous to multi-sublattice ordered systems, where N\nmagnon branches appear in the spectrum ( Nis the num-\nber of sublattices), chemically disordered systems con-\ntainingKchemical components will exhibit Kmagnon\nbranches. More speci\fcally, in the case of a binary al-\nloy (K=2), the magnon energies at each wave-vector q\nwill be given by the eigenvalues of the following 2 \u00022\ndynamical matrix\n!(q) = 4Eig (J11(0)\u0000J11(q))x1+J12(0)x2\nM1\u0000J12(q)x2\nM1\n\u0000J21(q)x1\nM2(J22(0)\u0000J22(q))x2+J21(0)x1\nM2!\n:\n(8)\n2. Dynamical structure factor\nThe magnon dispersion at \fnite temperatures are di-\nrectly accessible in ASD through the dynamical structure\nfactorS(q;!)29{31. The key ingredient is the measure-\nment of the time and space correlation function\nC\u0016\u0017(r;t) =1\nNX\ni;jwhere\nri\u0000rj=rhm\u0016\ni(t)m\u0017\nj(0)i\u0000hm\u0016\ni(t)ihm\u0017\nj(0)i:\n(9)\nThe correlation function de\fned in Eqn. (9) describes\nhow the magnetic order evolves both in space ( \u0016;\u0017de-\nnotes carteisian components) and over time. The per-\nhaps most valuable application of C(r;t) is obtained by3\na Fourier transform over space and time to give the dy-\nnamical structure factor\nS\u0016\u0017(q;!) =1p\n2\u0019NX\nreiq\u0001rZ1\n\u00001ei!tC\u0016\u0017(r;t)dt:(10)\nThe magnon energies are determined by the peak val-\nues ofS(q;!) at wavevector q. In contrast to the adi-\nabatic treatment, temperature e\u000bects from the Gilbert\ndamping processes are included that give rise to inten-\nsity variation of the available energies. In the present\nstudy, we have not included longitudinal \ructuations of\nthe magnetic moment. Such \ructuations give rise to\nStoner excitations and an additional damping mechanism\nfor magnons so-called Landau damping.\nD. Details of calculation\nAll \frst-principles calculations in this study was\nperformed using a multiple-scattering (Korringa-Kohn-\nRostoker, KKR) implementation of the density func-\ntional theory (DFT) as implemented in the SPR-KKR\nsoftware32,33. The generalized gradient approxima-\ntion (GGA) using the Perdew-Burke-Enzerhof (PBE)\nparametrization was used as exchange-correlation for\nthe volume relaxation while all other calculations em-\nployed the local spin density approximation (LDA). The\ncalculations are fully relativistic employing the atomic\nsphere approximation with a basis set consisting of spdf-\norbitals. The coherent potential approximation (CPA)\nwas employed for treating the disorder. In order to study\nthe magnetic excitations and \fnite temperature proper-\nties, the total energies from the electronic structure cal-\nculations are mapped onto an e\u000bective Heisenberg Hamil-\ntonian generalized to random alloys.\nThe magnetic exchange interactions were obtained\nfrom the magnetic force theorem using the formalism\nof Lichtenstein, Katsnelson, Antropov and Gubanov\n(LKAG)34,35. Gilbert damping was calculated using the\nlinear response formalism of the torque-torque correla-\ntion method as described in Ref.[36]. The alloy-analogy\nmodel within CPA37was employed for the \fnite tempera-\nture damping where both atomic displacements and spin\n\ructuations from Monte Carlo data were included. The\natomistic simulations, either the Monte Carlo or atom-\nistic spin dynamics, were performed using the UppASD\nsoftware26,38. Here the disorder is instead treated by\nusing a large supercell in which each site is chemically\nrandomly occupied according to the concentration. We\nare using large supercells consisting of between 110592\natoms (for the calculation of the spin sti\u000bness and AMS)\nand 512000 atoms (for the calculation of the dynamical\nstructure factor), such that most of the local environ-\nment con\fgurations from a central atom exist within the\nsupercell. The spin sti\u000bness was calculated for each in-\ndividual atom in the supercell and the \fnal result was\nobtained by performing an average over all atoms.III. RESULTS\nA. Electronic band structure\nFIG. 1. Electronic band structure in terms of the Bloch spec-\ntral function of a) ordered Fe-Co compound in the B2 struc-\nture and b) Fe 50Co50random alloy in the bcc structure.\nBefore describing in details how the magnetic proper-\nties are a\u000bected by chemical disorder, we \frst look into\nthe electronic band structure. For ordered elements and\ncompounds, the electron bands are well de\fned with no\nassociated broadening as function of energy and wave\nvector within the LDA/PBE treatment. This corre-\nsponds to electrons having in\fnite lifetime. A typical\nelectron band structure of an ordered Fe-Co compound is\ndisplayed in Fig. 1a). However, if the system has chemical\ndisorder (or if the \fnite lifetime of the quasi-particles are\ntaken into account) the bands become \"fuzzy\" and obtain\na \fnite broadening, with a line width inversely propor-\ntional to the lifetime. The broadening is however not\nuniform over the considered energy range. For random\nalloys, the electron band structure is conveniently ob-\ntained through the Bloch spectral function within CPA,\nas demonstrated in Fig. 1b) where the electron band\nstructure of Fe 50Co50random alloy in the body centered\ncubic (bcc) lattice is displayed. For this particular system\nand concentration, the disorder is most visible around\nthe Fermi level . This a\u000bects many properties such as4\nthe Gilbert damping (see Section III D).\nB. Spin sti\u000bness\nFIG. 2. Calculated spin sti\u000bness Din (meV \u0017A2) for the ran-\ndom alloys Fe 1\u0000xNix, Co 1\u0000xNixand Fe 1\u0000xCox.\nThe calculated values of the spin sti\u000bness Dare shown\nin Fig. 2 for Fe-Ni, Co-Ni and Fe-Co random alloys. For\nthe Fe-Ni alloys, Dis monotonously increasing with the\nNi concentration. This has a rather simple explanation.\nPure Fe in the face-centered cubic (fcc) lattice at the\nhere considered volumes possess rather complicated non-\ncollinear magnetic structures39which translates into a\nvanishing spin sti\u000bness. Overall, the magnetic properties\nin Fe-Ni alloys are rather sensitive to volume changes.\nEven at Invar concentration (Fe 65Ni35) it is possible to\nkill the ferromagnetic order by applying pressure, and in\nthis way obtain a vanishing spin sti\u000bness.\nThe Co-Ni alloys behave di\u000berently. Here the val-\nues of the spin sti\u000bness are rather constant throughout\nthe whole concentration range and therefore the magnon\nproperties are not expected to change much. This is per-\nhaps not so unexpected since both elemental Co and Ni\nare stable in the fcc lattice.\nThe spin sti\u000bness of the Fe-Co alloys in the bcc lattice\nshows a more interesting behaviour. At low concentra-\ntions of Co ( x <0:2), the spin sti\u000bness is similar as for\nelemental Fe while it increases for higher concentrations\nof Co. At the phase boundary around x= 0:7, the spin\nsti\u000bness is approximately twice as large as that of Fe.\nThis suggests that the are ample possibilities for tuning\nthe magnetic properties in this system.\nC. Curie temperatures\nOur computed Curie temperatures are shown in Fig. 3.\nTwo di\u000berent approaches have been used, the mean \feld\napproximation (MFA) and the random phase approxima-\ntion (RPA). In principle, MFA corresponds to the arith-\nmetric average of the exchange interactions and RPA tothe harmonic average. It can be shown40,41that for fer-\nromagnetic interactions TMFA\nc> Tc> TRPA\nc, whereTc\nis the \"true\" value (which can be obtained from Monte\nCarlo). The two di\u000berent methods (MFA and RPA) then\nset the upper and lower bounds of Tc. Of the three con-\nsidered alloy systems, the Fe-Ni system displays the low-\nest values of Tcwhile Fe-Co the highest with values peak-\ning around 1500 K for Co concentrations around 0.5.\nFIG. 3. Calculated Curie temperatures for the random alloys\na) Fe 1\u0000xNix), b) Co 1\u0000xNixand c) Fe 1\u0000xCox. MFA denotes\nvalues from mean \feld approximation and RPA from random\nphase approximation.\nD. Gilbert damping\nFIG. 4. Calculated Gilbert damping at T= 300 K for the ran-\ndom alloys Fe 1\u0000xNix(red), Co 1\u0000xNix(green) and Fe 1\u0000xCox\n(black).\nGilbert damping in magnetic materials determines the\nrate of dissipative energy processes with the surround-\nings. Very often a low damping is wanted in order to min-\nimize energy losses but equally important is the ability to\ntune the damping. This can be achieved by, e.g., impurity\ndoping42or by varying the alloy composition. The latter\nis pursued here. For both fcc alloy systems considered5\nin this study, i.e. Fe-Ni and Co-Ni, the Gilbert damping\nincreases with Ni concentration. The Gilbert damping\nfor Fe-Ni is consistently lower than the one seen in Co-\nNi. For elemental Ni (o\u000b scale), we obtain \u000b= 0:013,\nwhich is in the same range as reported previously36,43.\nWorth noting is that the damping is one order of mag-\nnitude smaller in elemental Co and Fe. What is perhaps\nmost remarkable however is the very low damping found\nin certain Fe-Co alloys, in which it is even lower than\nfor elemental Fe. This behaviour is due to variation of\nthe density of states and was explained in detail in pre-\nvious studies36,44. The experimental values reported in\nRef.[44] are in good agreement with our calculated values\npresented here.\nE. Magnon properties\nFIG. 5. Magnon spectrum of permalloy (Fe 20Ni80). The\nthin red line denotes e\u000bective adiabatic spectrum, Eq.(7), and\nthick black lines full adiabatic treatment, Eq.(8). Blue (green)\npoints denote peak position at each wavevector of the dynami-\ncal structure factor from atomistic spin dynamics calculations\natT= 10 K ( T= 300 K) using the calculated Gilbert damp-\ning.\nOverall, we \fnd that the main features of the magnon\nspectra are quite similar in all systems we have consid-\nered here. We therefore choose in this section to present\nresults only for two systems of particular technological\ninterest: i) permalloy (Fe 20Ni80) in the fcc lattice and ii)\nFe75Co25in the bcc lattice chosen due to its large mag-\nnetic moment and low damping.\nIn Fig. 5, the calculated magnon spectrum of Py is dis-\nplayed using a variety of di\u000berent tools as described in\nSection. II C. Both the thin red line and the bold black\nlines are from adiabatic calculations, Eqs. (7) and (8), re-\nspectively. Between the two, the spectrum in black lines\nis expected to hold which is clear from comparison with\ndynamical structure factor from atomistic spin dynam-\nics calculations as indicated in squares for two di\u000berent\ntemperatures, namely T= 10 K and T= 300 K. It is\nFIG. 6. Magnon density of states of permalloy (Fe 20Ni80)\nfrom AMS and atomistic spin dynamics simulations at T=\n10 K and T= 300 K.\nimportant to remember that AMS only re\rects the ex-\nchange interactions and the chemical disorder of the sys-\ntem. Temperature e\u000bects in the form of transversal \ruc-\ntuations and damping are however included in the ASD\nsimulations where the calculated Gilbert damping at T=\n300 K was employed. The curvature around the \u0000 point\nis the spin sti\u000bness and by inspection it is clear that the\nspectrum softens drastically at room temperature com-\npared to the low temperature data. This temperature\ndependence of the sti\u000bness was also analyzed in a re-\ncent study20. At higher energies, due to a combination\nof thermal \ructuations, disorder and damping processes\nthe spectrum broadens which is much clearly shown in\nFig. 6 where the magnon density of states (MDOS) is\ndisplayed.\nFIG. 7. Magnon spectrum of Fe 75Co25. The thin red line\ndenotes e\u000bective adiabatic spectrum, Eq. (7), and thick black\nlines full adiabatic treatment, Eq. (8). Blue (red) points de-\nnote peak position at each wavevector of the dynamical struc-\nture factor from atomistic spin dynamics calculations at T=\n0 K (T= 300 K) using the calculated Gilbert damping.\nThe MDOS obtained from AMS has two distinct peaks6\nwhich we can denote \"acoustic\" and \"optical\" branch in\nanalogy to phonons. The two branches are separated by\na small gap. However, even at low temperature ( T= 10\nK), the MDOS as obtained by ASD simulations is broad-\nened enough such that the two branches overlap. The\npeak positions are however almost identical. Although\none need to keep in mind that AMS is using a simpli\fed\ntreatment of disorder, namely VCA, while ASD simula-\ntions are treating the disorder much more accurately by\nusing a large random supercell. Increasing the temper-\nature to 300 K softens the spectrum almost uniformly.\nThis \fnding has been used to describe the low tempera-\nture dependence of MDOS with a quasiharmonic approx-\nimation in Refs. [45 and 46].\nThe calculated magnon spectrum for Fe 75Co25, dis-\nplayed in Fig. 7, is quite di\u000berent from the one of Py.\nFirst of all, given its much higher Curie temperature the\ndi\u000berence of the spectrum between T= 10 K and T=\n300 K is minimal. Secondly, since Fe and Co atoms are\nrather similar chemically, both in terms of magnetic mo-\nments, 2.5\u0016Band 1.8\u0016B, respectively and the exchange\ninteractions are of similar magnitude, the spectrum has\nmuch less disorder broadening.\nF. Magnon lifetimes, ordered versus disordered\nTo further quantify the e\u000bects of disorder on magnon\nproperties, in this section we compare ordered system\nwith disordered system having the same composition.\nMore speci\fcally, we compare Fe 50Co50which exists both\nin ordered structure, B2, or as a disordered random\nalloy in bcc structure. Both the magnetic moments\n(Ms\u00192:2\u0016B) and Curie temperatures ( \u00191400 K) are\nrather similar between the two structures. The calculated\nGilbert damping at room temperature is however lower\nfor the ordered B2 structure, 0.0007 vs 0.0011 for the ran-\ndom alloy. It is worth noting that the damping for the B2\nis remarkable low for a metallic compound. In Fig. 8, the\nmagnon spectrum from ASD simulations at T= 300 K is\nshown for the both compounds, together with the AMS\nspectrum as reference. Due to the lack of disorder in the\nB2 structure, the magnon states are very well de\fned\nthroughout the whole Brillouin zone. It is immediately\nclear that the disorder of the random alloy broadens the\nmagnon states a\u000becting the magnon lifetimes, similar as\nfound for the electron bands in Fig. 1. However, even for\nthe random alloy there are relatively well de\fned magnon\nstates throughout the Brillouin zone, in contrast to the\nFe-Ni alloys where the magnon states away from the \u0000-\npoint are very di\u000buse.\nIn Fig. 9, the magnon DOS is displayed for the two\ncompounds. As also clear from the spectrum, in the B2\nstructure the magnon states are divided in two distinct\nbranches, \"acoustic\" and \"optical\" with small tempera-\nture dependence of the peak positions. The width of the\npeak is inversely related to the magnon lifetime. From\ninspection, the width for the B2 structure at T= 300 K isslightly larger than at T= 10 K and thus giving shorter\nmagnon lifetimes. However, the magnon lifetimes will\nalso have a wave-vector dependence and a more involved\nanalysis is needed. In the random alloy, the \"acoustic\"\nmagnon branch is roughly located at the same energies\nas in the B2 structure, while the \"optical\" branch is sig-\nni\fcantly broadened in comparison.\nThe most elaborate way to determine magnon life-\ntimes theoretically is through time dependent density\nfunctional theory and linear response47. Due to the com-\nplexity of such calculations, it has so far only been ap-\nplied to elemental systems and not in alloys. Here, we\ntherefore use an alternative simpli\fed method to obtain\nthe wave vector dependent magnon lifetimes \u001c(q). By\n\ftting the dynamical structure factor for each wave vec-\ntor with a Lorentzian and determine the full width half\nmaximum (FWHM), \u0001( q), the corresponding magnon\nlifetime is obtained through the relation \u001c(q) =2~\n\u0001(q).\nIt is worth stressing that this approach only takes into\naccount decay through Gilbert damping mechanism and\nnot via Landau damping corresponding to electron-hole\npair excitations within the Stoner continuum. However,\nfor bulk materials as in this study it should be a rea-\nsonable good approximation, at least for the \"acoustic\"\nmagnon branch.\nFIG. 8. Magnon spectrum from atomistic spin dynamics sim-\nulations at T= 300 K of Fe 50Co50in the ordered B2 structure\n(top) and as a random alloy (bottom). The white line is the\ncorresponding spectrum as obtained from AMS.7\nFIG. 9. Magnon density of states of Fe 50Co50in ordered B2\nstructure (top) and random alloy (bottom) from AMS and\natomistic spin dynamics simulations at T= 10 K and T=\n300 K.\nIn Fig. 10, the calculated magnon lifetimes in speci\fc\ndirections of the Brillouin zone are displayed for both the\nordered B2 structure and as random alloy. Due to the\nsensitivity of the \ftting, the calculated lifetimes for each\nwave vector have an associated error bar that is of the\norder of the variation between neighbouring wave vectors\nvalues. Nevertheless, it is clear that on average the or-\ndered B2 structure has longer magnon lifetimes compared\nto the random alloy. For the speci\fc directions in Fig. 10,\nthe average magnon lifetime in B2 is approximately three\ntimes larger than for the random alloy (0.6 ps vs 0.2 ps).\nThis behaviour is in line with what is normally expected\nfrom disordered systems, i.e. that increased disorder in-\ncreases the scattering rates which e\u000bectively gives shorter\nquasi-particle lifetimes. A direct comparison can be made\nwith the broadening of the electron bands in the spectral\nfunctions shown in Fig. 1.\nIV. SUMMARY\nWe have presented magnon and \fnite temperature\nproperties of random alloys using a combination of elec-\ntronic structure calculations and atomistic spin dynamics\nsimulations. Disorder is seen to have a pronounced e\u000bect\non the magnon properties causing additional scattering\nand damping of magnon modes. However, the degree of\nmagnon scattering and damping depends sensitively on\nthe chemical composition of the alloy and also on the\nrelative concentration of the constituent atomic species,\nprompting for material speci\fc studies. For example,the magnon spectrum of permalloy (Fe 20Ni80) is much\nmore a\u000bected by disorder causing di\u000buse spectra in most\nof the Brillouin zone than that of Fe 75Co25where well-\nde\fned magnon states exist everywhere. Similarly, we\ncompared the magnon properties of Fe 50Co50, both as\nordered structure and as random alloy. We found a dis-\ntinct di\u000berence in the magnon density of states between\nthe two as well as observing shorter magnon lifetimes in\nthe random alloy. We hope that the present study will\nmotivate new experiments in the next generation of neu-\ntron scattering facilities. 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B\n84, 174418 (2011)."    },    {        "title": "1810.10595v4.Nearly_isotropic_spin_pumping_related_Gilbert_damping_in_Pt_Ni___81__Fe___19___Pt.pdf",        "content": "Nearly isotropic spin-pumping related Gilbert damping in Pt/Ni 81Fe19/Pt\nW. Cao,1,\u0003L. Yang,1S. Au\u000bret,2and W.E. Bailey1, 2,y\n1Materials Science and Engineering, Department of Applied Physics and Applied Mathematics,\nColumbia University, New York, New York 10027, USA\n2SPINTEC, Universit \u0013eGrenoble Alpes/CEA/CNRS, F-38000 Grenoble, France\n(Dated: July 12, 2021)\nA recent theory by Chen and Zhang [Phys. Rev. Lett. 114, 126602 (2015)] predicts strongly\nanisotropic damping due to interfacial spin-orbit coupling in ultrathin magnetic \flms. Interfacial\nGilbert-type relaxation, due to the spin pumping e\u000bect, is predicted to be signi\fcantly larger for\nmagnetization oriented parallel to compared with perpendicular to the \flm plane. Here, we have\nmeasured the anisotropy in the Pt/Ni 81Fe19/Pt system via variable-frequency, swept-\feld ferromag-\nnetic resonance (FMR). We \fnd a very small anisotropy of enhanced Gilbert damping with sign\nopposite to the prediction from the Rashba e\u000bect at the FM/Pt interface. The results are contrary\nto the predicted anisotropy and suggest that a mechanism separate from Rashba spin-orbit coupling\ncauses the rapid onset of spin-current absorption in Pt.\nINTRODUCTION\nThe spin-transport properties of Pt have been studied\nintensively. Pt exhibits e\u000ecient, reciprocal conversion\nof charge to spin currents through the spin Hall e\u000bect\n(SHE)[1{4]. It is typically used as detection layer for\nspin current evaluated in novel con\fgurations[5{7]. Even\nso, consensus has not yet been reached on the experi-\nmental parameters which characterize its spin transport.\nThe spin Hall angle of Pt, the spin di\u000busion length of Pt,\nand the spin mixing conductance of Pt at di\u000berent inter-\nfaces di\u000ber by as much as an order of magnitude when\nevaluated by di\u000berent techniques[2, 3, 8{12].\nRecently, Chen and Zhang [13, 14] (hereafter CZ) have\nproposed that interfacial spin-orbit coupling (SOC) is\na missing ingredient which can bring the measurements\ninto greater agreement with each other. Measurements of\nspin-pumping-related damping, particularly, report spin\ndi\u000busion lengths which are much shorter than those es-\ntimated through other techniques[15, 16]. The introduc-\ntion of Rashba SOC at the FM/Pt interface leads to\ninterfacial spin-memory loss, with discontinuous loss of\nspin current incident to the FM/Pt interface. The model\nsuggests that the small saturation length of damping en-\nhancement re\rects an interfacial discontinuity, while the\ninverse spin Hall e\u000bect (ISHE) measurements re\rect the\nbulk absorption in the Pt layer[15, 16].\nThe CZ model predicts a strong anisotropy of the en-\nhanced damping due to spin pumping, as measured in\nferromagnetic resonance (FMR). The damping enhance-\nment for time-averaged magnetization lying in the \flm\nplane ( pc-FMR, or parallel condition) is predicted to be\nsigni\fcantly larger than that for magnetization oriented\nnormal to the \flm plane ( nc-FMR, or normal condition).\nThe predicted anisotropy can be as large as 30%, with\npc-FMR damping exceeding nc-FMR damping, as will be\nshown shortly.\nIn this paper, we have measured the anisotropy of the\nenhanced damping due to the addition of Pt in symmet-ric Pt/Ni 81Fe19(Py)/Pt structures. We \fnd that the\nanisotropy is very weak, less than 5%, and with the op-\nposite sign from that predicted in [13].\nTHEORY\nWe \frst quantify the CZ-model prediction for\nanisotropic damping due to the Rashba e\u000bect at the\nFM/Pt interface. In the theory, the spin-memory loss\nfor spin current polarized perpendicular to the interfa-\ncial plane is always larger than that for spin current po-\nlarized in the interfacial plane. The pumped spin po-\nlarization\u001b=m\u0002_mis always perpendicular to the\ntime-averaged or static magnetization hmit'm. For\nnc-FMR, the polarization \u001bof pumped spin current is\nalways in the interfacial plane, but for pc-FMR, is nearly\nequally in-plane and out-of-plane. A greater damping\nenhancement is predicted in the pccondition than in the\nnccondition, \u0001 \u000bpc>\u0001\u000bnc:\n\u0001\u000bnc=Kh1 + 4\u0011\u0018(tPt)\n1 +\u0018(tPt)i\n(1)\n\u0001\u000bpc=Kh1 + 6\u0011\u0018(tPt)\n1 +\u0018(tPt)+\u0011\n2[1 +\u0018(tPt)]2i\n(2)\n\u0018(tPt) =\u0018(1)\u0002coth(tPt=\u0015sd) (3)\nwhere the constant of proportionality K is the same for\nboth conditions and the dimensionless parameters, \u0011and\n\u0018, are always real and positive. The Rashba parameter\n\u0011= (\u000bRkF=EF)2(4)\nis proportional to the square of the Rashba coe\u000ecient\n\u000bR, de\fned as the strength of the Rashba potential,arXiv:1810.10595v4  [cond-mat.mtrl-sci]  22 Feb 20192\nFIG. 1. Frequency-dependent half-power FMR linewidth\n\u0001H1=2(!) of the reference sample Py(5 nm) (black) and sym-\nmetric trilayer samples Pt(t)/Py(5 nm)/Pt(t) (colored). (a)\npc-FMR measurements. (b) nc-FMR measurements. Solid\nlines are linear \fts to extract Gilbert damping \u000b. (Inset):\ninhomogeneous broadening \u0001 H0inpc-FMR (blue) and nc-\nFMR (red).\nV(r) =\u000bR\u000e(z)(^k\u0002^z)\u0001\u001b, where\u000e(z) is a delta function\nlocalizing the e\u000bect to the interface at z= 0 (\flm plane\nisxy),kFis the Fermi wavenumber, and EFis the Fermi\nenergy. The back\row factor \u0018is a function of Pt layer\nthickness, where the back\row fraction at in\fnitely large\nPt thickness de\fned as \u000f=\u0018(1)=[1 +\u0018(1)].\u000f= 0 (1)\nrefers to zero (complete) back\row of spin current across\nthe interface. \u0015sdis the spin di\u000busion length in the Pt\nlayer.\nTo quantify the anisotropy of the damping, we de\fne\nQ:\nQ\u0011(\u0001\u000bpc\u0000\u0001\u000bnc)=\u0001\u000bnc (5)\nas an anisotropy factor , the fractional di\u000berence be-\ntween the enhanced damping in pc and nc conditions.\nPositive Q (Q >0) is predicted by the CZ model. A\nspin-memory loss \u000efactor of 0.9\u00060.1, corresponding\nto nearly complete relaxation of spin current at the in-\nterface with Pt, was measured through current perpen-\ndicular to plane-magnetoresistance (CPP-GMR)[8] Ac-\ncording to the theory[13, 14], the spin-memory loss can\nbe related to the Rashba parameter by \u000e= 2\u0011, so we\ntake\u0011\u00180:45. The e\u000bect of variable \u0011 < 0:45 will be\nshown in Figure 3. To evaluate the thickness dependent\nback\row\u0018(tPt), we assume \u0015Pt\nsd= 14 nm, which is asso-\nciated with the absorption of the spin current in the bulk\nof Pt layer, as found from CPP-GMR measurements[8]\nand cited in [13]. Note that this \u0015Pt\nsdis longer than that\nused sometimes to \ft FMR data[15, 16]; Rashba interfa-\ncial coupling in the CZ model brings the onset thickness\ndown. The calculated anisotropy factor Q should then\nFIG. 2. Pt thickness dependence of Gilbert damping \u000b=\n\u000b(tPt) inpc-FMR (blue) and nc-FMR (red). \u000b0refers to the\nreference sample ( tPt= 0). (Inset): Damping enhancement\n\u0001\u000b(tPt) =\u000b(tPt)\u0000\u000b0due to the addition of Pt layers in\npc-FMR (blue) and nc-FMR (red). Dashed lines refer to cal-\nculated \u0001\u000bncusing Equation 1 by assuming \u0015Pt\nsd= 14 nm\nand\u000f= 10%. The red dashed line ( \u0011= 0:15) shows a similar\ncurvature with experiments; The black dashed line ( \u0011\u00150:25)\nshows a curvature with the opposite sign.\nbe as large as 0.3, indicating that \u0001 \u000bpcis 30% greater\nthan \u0001\u000bnc(see Results for details).\nEXPERIMENT\nIn this paper, we present measurements of the\nanisotropy of damping in the symmetric Pt( tPt)/Py(5\nnm)/Pt(tPt) system, where \\Py\"=Ni 81Fe19. Because\nthe Py thickness is much thicker than its spin coher-\nence length[17], we expect that spin-pumping-related\ndamping at the two Py/Pt interfaces will sum. The\nfull deposited stack is Ta(5 nm)/Cu(5 nm)/Pt( tPt)/Py(5\nnm)/Pt(tPt)/Al 2O3(3 nm),tPt= 1{10 nm, deposited\nvia DC magnetron sputtering under computer control on\nion-cleaned Si/SiO 2substrates at ambient temperature.\nThe deposition rates were 0.14 nm/s for Py and 0.07\nnm/s for Pt. Heterostructures deposited identically, in\nthe same deposition chamber, have been shown to exhibit\nboth robust spin pumping e\u000bects, as measured through\nFMR linewidth[18, 19], and robust Rashba e\u000bects (in\nCo/Pt), as measured through Kerr microscopy[20, 21].\nThe stack without Pt layers was also deposited as the ref-\nerence sample. The \flms were characterized using vari-\nable frequency FMR on a coplanar waveguide (CPW)\nwith center conductor width of 300 \u0016m. The bias mag-\nnetic \feld was applied both in the \flm plane ( pc) and\nperpendicular to the plane ( nc), as previously shown in\n[22]. The nc-FMR measurements require precise align-\nment of the \feld with respect to the \flm normal. Here,3\nFIG. 3. Anisotropy factor Q for spin-pumping enhanced damping, de\fned in Equation 5. Solid lines are calculations using the\nCZ theory[13], Equations 1{3, for variable Rashba parameter 0 :01\u0014\u0011\u00140:45.\u0015Pt\nsdis set to be 14 nm. Back\row fraction \u000fis\nset to be 10% in (a) and 40% in (b). Black triangles, duplicate in (a) and (b), show the experimental values from Figure 2.\nsamples were aligned by rotation on two axes to maxi-\nmize the resonance \feld at 3 GHz.\nRESULTS AND ANALYSIS\nFigure 1 shows frequency-dependent half-power\nlinewidth \u0001 H1=2(!) in pc- and nc-FMR. The measure-\nments were taken at frequencies from 3 GHz to a cut-o\u000b\nfrequency above which the signal-to-noise ratio becomes\ntoo small for reliable measurement of linewidth. The\ncuto\u000b ranged from 12{14 GHz for the samples with Pt\n(linewidth\u0018200{300 G) to above 20 GHz for tPt= 0.\nSolid lines stand for linear regression of the variable-\nfrequency FMR linewidth \u0001 H1=2= \u0001H0+2\u000b!=\r , where\n\u0001H1=2is the full-width at half-maximum, \u0001 H0is the in-\nhomogeneous broadening, \u000bis the Gilbert damping, !\nis the resonance frequency and \ris the gyromagnetic ra-\ntio. The \fts show good linearity with frequency !=2\u0019for\nall experimental linewidths \u0001 H1=2(!). The inset sum-\nmarizes inhomogeneous broadening \u0001 H0inpc- and nc-\nFMR; its errorbar is \u00182 Oe.\nIn Figure 2, we plot Pt thickness dependence of damp-\ning parameters \u000b(tPt) extracted from the linear \fts in\nFigure 1, for both pc-FMR and nc-FMR measurements.\nStandard deviation errors in the \fts for \u000bare\u00183\u000210\u00004.\nThe Gilbert damping \u000bsaturates quickly as a function\noftPtin both pc and nc conditions, with 90% of the ef-\nfect realized with Pt(3 nm). The inset shows the damp-\ning enhancement \u0001 \u000bdue to the addition of Pt layers\u0001\u000b=\u000b\u0000\u000b0, normalized to the Gilbert damping \u000b0of\nthe reference sample without Pt layers. The Pt thickness\ndependence of \u0001 \u000bmatches our previous study on Py/Pt\nheterostructures[19] reasonably; the saturation value of\n\u0001\u000bPt=Py=Pt is 1.7x larger than that measured for the\nsingle interface \u0001 \u000bPy=Pt [19] (2x expected). The dashed\nlines in the inset refer to calculated \u0001 \u000bncusing Equation\n1 (assuming \u0015Pt\nsd= 14 nm and \u000f= 10%).\u0011= 0:25 shows\na threshold of Pt thickness dependence. When \u0011>0:25,\nthe curvature of \u0001 \u000b(tPt) will have the opposite sign to\nthat observed in experiments, so \u0011= 0:25 is the maxi-\nmum which can qualitatively reproduce the Pt thickness\ndependence of the damping.\nAs shown in Figure 2 inset, the damping enhancement\ndue to the addition of Pt layers is slightly larger in the\nncgeometry than in the pcgeometry: \u0001 \u000bnc>\u0001\u000bpc.\nThis is opposite to the prediction of the model in [13].\nThe anisotropy factor Q\u0011(\u0001\u000bpc\u0000\u0001\u000bnc)=\u0001\u000bncfor the\nmodel (Q>0) and the experiment (Q <0) are shown to-\ngether in Figure 3 (a) and (b). The magnitude of Q\nfor the experiment is also quite small, with -0.05 <Q<0.\nThis very weak anisotropy, or near isotropy, of the spin-\npumping damping is contrary to the prediction in [13],\nand is the central result of our paper.\nThe two panels (a) and (b), which present the same\nexperimental data (triangles), consider di\u000berent model\nparameters, corresponding to negligible back\row ( \u000f=\n0:1, panel a) and moderate back\row ( \u000f= 0:4, panel b)\nfor a range of Rashba couplings 0 :01\u0014\u0011\u00140:45. A spin\ndi\u000busion length \u0015sd= 14 nm for Pt[8] was assumed in all4\ncases.\nThe choice of back\row fraction \u000f= 0:1 or 0:4 and the\nchoice of spin di\u000busion length of Pt \u0015sd= 14 nm follow\nthe CZ paper[13] for better evaluation of their theory.\nFor good spin sinks like Pt, the back\row fraction is usu-\nally quite small. If \u000f= 0, then there will be no spin\nback\row. In this limit, \u0001 \u000bpc, \u0001\u000bncand the Q factor\nwill be independent of Pt thickness.\nIn the case of a short spin di\u000busion length of Pt, e.g.,\n\u0015sd= 3 nm, the anisotropy Q as a function of Pt thick-\nness decreases more quickly for ultrathin Pt, closer to\nour experimental observations. However, we note that\nthe CZ theory requires a long spin di\u000busion length in or-\nder to reconcile di\u000berent experiments, particularly CPP-\nGMR with spin pumping, and is not relevant to evaluate\nthe theory in this limit.\nLeaving apart the question of the sign of Q, we can see\nthat the observed absolute magnitude is lower than that\npredicted for \u0011= 0:05 for small back\row and 0.01 for\nmoderate back\row. According to ref [13], a minimum\nlevel for the theory to describe the system with strong\ninterfacial SOC is \u0011= 0:3.\nDISCUSSION\nHere, we discuss extrinsic e\u000bects which may result in\na discrepancy between the CZ model (Q \u0018+0.3) and our\nexperimental result (-0.05 <Q<0). A possible role of two-\nmagnon scattering[23, 24], known to be an anisotropic\ncontribution to linewidth \u0001 H1=2, must be considered.\nTwo-magnon scattering is present for pc-FMR and nearly\nabsent for nc-FMR. This mechanism does not seem to\nplay an important role in the results presented. It is\ndi\u000ecult to locate a two-magnon scattering contribution\nto linewidth in the pure Py \flm: Figure 1 shows highly\nlinear \u0001H1=2(!), without o\u000bset, over the full range to\n!=2\u0019= 20 GHz, thereby re\recting Gilbert-type damp-\ning. The damping for this \flm is much smaller than\nthat added by the Pt layers. If the introduction of Pt\nadds some two-magnon linewidth, eventually mistaken\nfor intrinsic Gilbert damping \u000b, this could only produce\na measurement of Q >0, which was not observed.\nOne may also ask whether the samples are appropriate\nto test the theory. The \frst question regards sample qual-\nity. The Rashba Hamiltonian models a very abrupt inter-\nface. Samples deposited identically, in the same deposi-\ntion chamber, have exhibited strong Rashba e\u000bects, so we\nexpect the samples to be generally appropriate in terms\nof quality. Intermixing of Pt in Ni 81Fe19(Py)/Pt[25] may\nplay a greater role than it does in Co/Pt[26], although\ndefocused TEM images have shown fairly well-de\fned in-\nterfaces for our samples[27].\nA second question might be about the magnitude of\nthe Rashba parameter \u0011in the materials systems of in-\nterest. Our observation of nearly isotropic damping isconsistent with the theory, within experimental error and\napart from the opposite sign, if the Rashba parameter \u0011is\nvery low and the back\row fraction \u000fis very low. Ab-initio\ncalculations for (epitaxial) Co/Pt in the ref[28] have in-\ndicated\u0011= 0.02{0.03, lower than the values of \u0011\u00180.45\nassumed in [13, 14] to treat interfacial spin-memory loss.\nThe origin of the small, negative Q observed here is un-\nclear. A recent paper has reported that \u0001 \u000bpcis smaller\nthan \u0001\u000bncin the YIG/Pt system via single-frequency,\nvariable-angle measurements[7], which is contrary to the\nCZ model prediction as well. It is also possible that a\nfew monolayers of Pt next to the Py/Pt interfaces are\nmagnetized in the samples[19], and this may have an un-\nknown e\u000bect on the sign, not taken into account in the\ntheory.\nCONCLUSIONS\nIn summary, we have experimentally demonstrated\nthat in Pt/Py/Pt trilayers the interfacial damping at-\ntributed to spin pumping is nearly isotropic, with an\nanisotropy between \flm-parallel and \flm-normal mea-\nsurements of <5%. The nearly isotropic character of the\ne\u000bect is more compatible with conventional descriptions\nof spin pumping than with the Rashba spin-memory loss\nmodel predicted in [13].\nACKNOWLEDGEMENTS\nWe acknowledge support from the US NSF-DMR-\n1411160 and the Nanosciences Foundation, Grenoble.\n\u0003wc2476@columbia.edu\nyweb54@columbia.edu\n[1] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Ap-\nplied Physics Letters 88, 182509 (2006).\n[2] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\nS. D. Bader, and A. Ho\u000bmann, Phys. Rev. Lett. 104,\n046601 (2010).\n[3] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n[4] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel,\nand F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014).\n[5] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n[6] D. Ellsworth, L. Lu, J. Lan, H. Chang, P. Li, Z. Wang,\nJ. Hu, B. Johnson, Y. Bian, J. Xiao, R. Wu, and M. Wu,\nNature Physics 12, 861 (2016).\n[7] H. Zhou, X. Fan, L. Ma, Q. Zhang, L. Cui, S. Zhou, Y. S.\nGui, C.-M. Hu, and D. Xue, Phys. Rev. B 94, 134421\n(2016).\n[8] H. Kurt, R. Loloee, K. Eid, W. P. Pratt, and J. Bass,\nApplied Physics Letters 81, 4787 (2002).5\n[9] L. Vila, T. Kimura, and Y. Otani, Phys. Rev. Lett. 99,\n226604 (2007).\n[10] K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda,\nS. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601\n(2008).\n[11] A. Azevedo, L. H. Vilela-Le~ ao, R. 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Lett. 109, 127202 (2012).[18] A. Ghosh, J. F. Sierra, S. Au\u000bret, U. Ebels, and W. E.\nBailey, Applied Physics Letters 98, 052508 (2011).\n[19] M. Caminale, A. Ghosh, S. Au\u000bret, U. Ebels, K. Ollefs,\nF. Wilhelm, A. Rogalev, and W. E. Bailey, Phys. Rev.\nB94, 014414 (2016).\n[20] I. M. Miron, G. Gaudin, S. Au\u000bret, B. Rodmacq,\nA. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella,\nNature Materials 9, 230 (2010).\n[21] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-\nPrejbeanu, S. Au\u000bret, B. Rodmacq, S. Pizzini, J. Vogel,\nM. Bon\fm, A. Schuhl, and G. Gaudin, Nature Materials\n10, 419 (2011).\n[22] H. Yang, Y. Li, and W. E. Bailey, Applied Physics Let-\nters108, 242404 (2016).\n[23] R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999).\n[24] R. McMichael and P. Krivosik, IEEE Transactions on\nMagnetics 40, 2 (2004).\n[25] T. Golod, A. Rydh, and V. M. Krasnov, Journal of Ap-\nplied Physics 110, 033909 (2011).\n[26] G. Bertero and R. Sinclair, Journal of Magnetism and\nMagnetic Materials 134, 173 (1994).\n[27] W. E. Bailey, A. Ghosh, S. Au\u000bret, E. Gautier, U. Ebels,\nF. Wilhelm, and A. Rogalev, Phys. Rev. B 86, 144403\n(2012).\n[28] S. Grytsyuk, A. Belabbes, P. M. Haney, H.-W. Lee, K.-J.\nLee, M. D. Stiles, U. Schwingenschl ogl, and A. Manchon,\nPhys. Rev. B 93, 174421 (2016)."    },    {        "title": "1810.11016v2.Time_retarded_damping_and_magnetic_inertia_in_the_Landau_Lifshitz_Gilbert_equation_self_consistently_coupled_to_electronic_time_dependent_nonequilibrium_Green_functions.pdf",        "content": "Time-retarded damping and magnetic inertia in the Landau-Lifshitz-Gilbert equation\nself-consistently coupled to electronic time-dependent nonequilibrium Green functions\nUtkarsh Bajpai and Branislav K. Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\nThe conventional Landau-Lifshitz-Gilbert (LLG) equation is a widely used tool to describe dy-\nnamics of local magnetic moments, viewed as classical vectors of fixed length, with their change\nassumed to take place simultaneously with the cause. Here we demonstrate that recently devel-\noped [M. D. Petrovi´ c et al. , Phys. Rev. Applied 10, 054038 (2018)] self-consistent coupling of\nthe LLG equation to time-dependent quantum-mechanical description of electrons—where nonequi-\nlibrium spin density from time-dependent nonequilibrium Green function (TDNEGF) calculations\nis inserted within a torque term into the LLG equation while local magnetic moments evolved by\nthe LLG equation introduce time-dependent potential in the quantum Hamiltonian of electrons—\nmicroscopically generates time-retarded damping in the LLG equation described by a memory kernel\nwhich is also spatially dependent. For sufficiently slow dynamics of local magnetic moments on the\nmemory time scale, the kernel can be expanded into power series to extract the Gilbert damping\n(proportional to first time derivative of magnetization) and magnetic inertia (proportional to second\ntime derivative of magnetization) terms whose parameters, however, are time-dependent in contrast\nto time-independent parameters used in the conventional LLG equation. We use examples of single\nor multiple local magnetic moments precessing in an external magnetic field, as well as field-driven\nmotion of a magnetic domain wall (DW), to quantify the difference in their time evolution computed\nfrom conventional LLG equation vs. TDNEGF+LLG quantum-classical hybrid approach. The faster\nDW motion predicted by TDNEGF+LLG approach reveals that important quantum effects, stem-\nming essentially from a finite amount of time which it takes for conduction electron spin to react\nto the motion of classical local magnetic moments, are missing from conventional classical micro-\nmagnetics simulations. We also demonstrate large discrepancy between TDNEGF+LLG-computed\nnumerically exact and, therefore, nonperturbative result for charge current pumped by a moving\nDW and the same quantity computed by perturbative spin motive force formula combined with the\nconventional LLG equation.\nI. INTRODUCTION\nThe conventional Landau-Lifshitz-Gilbert (LLG)\nequation [1–3] is the cornerstone of numerical micro-\nmagnetics [4] and atomistic spin dynamics [5] where one\nsimulates the classical time evolution of many magnetic\nunits coupled by exchange or magnetostatic interactions.\nThe LLG equation\n∂m(r,t)\n∂t=−gm(r,t)×Beff(r,t)+λGm(r,t)×∂m(r,t)\n∂t,\n(1)\ndescribes time evolution of m(r,t) as the unit vector\n|m|= 1 of constant length representing the direction\nof the local magnetization. Here gis the gyromag-\nnetic ratio and Beffis the sum of an external mag-\nnetic field and effective magnetic fields due to magnetic\nanisotropy and exchange coupling (additional stochastic\nmagnetic field can contribute to Beffto take into ac-\ncount finite temperature effects [6]). The second term\non the right-hand side of Eq. (1) is introduced phe-\nnomenologically to break the time-inversion symmetry,\nthereby generating a damping mechanism. The conven-\ntional intrinsic Gilbert damping λGis assumed to be ma-\nterials specific and, therefore, time-independent parame-\n∗bnikolic@udel.eduter. It is typically computed using the so-called breath-\ning Fermi surface [7] or torque-torque correlation formu-\nlas [8] within single-particle quantum-mechanical frame-\nwork (additional many-body processes have to be taken\ninto account to make λGfinite in the clean limit at low\ntemperatures [9]). In the original form, Eq. (1) is written\nfor a bulk material as a highly nonlinear partial differen-\ntial equation. It can also be re-written for a macrospin or\na lattice of atomic spins leading to a system of nonlinear\nordinary differential equations [5].\nIn the case of conducting ferromagnets, LLG equation\nhas to be extended by including additional terms, such as:\n(i) spin-transfer torque [10] T∝/angbracketleftˆs/angbracketright×mdue to injected\nelectrons generating nonequilibrium spin density /angbracketleftˆs/angbracketrightthat\nis noncollinear to local magnetization; ( ii) additional\nGilbert damping, ( g↑↓/4π)m×∂m/∂t, due to pumping\nof spin currents by the dynamics of m(t) whereg↑↓is the\nso-called spin-mixing conductance [11]; ( iii) additional\nnonlocal Gilbert damping [12–18], m×(D·∂m/∂t), due\nto spin pumping by noncollinear magnetic textures where\nDαβ=η/summationtext\ni(m×∂im)α(m×∂im)βis the 3×3 damping\ntensor,∂i=∂/∂iandα,β,i∈{x,y,z}; and ( iv) mag-\nnetic inertia [19–25], Im×∂2m/∂t2, of relevance to ul-\ntrafast magnetization dynamics. Like the original Gilbert\ndamping parameter λGin Eq. (1), T,g↑↓,DandIrequire\nmicroscopic quantum-mechanical calculations which are\noften combined [8, 9, 26–31] with first-principles Hamil-\ntonians of realistic materials.\nFurthermore, generalizations of LLG equation havearXiv:1810.11016v2  [cond-mat.mes-hall]  6 Dec 20182\nbeen considered to take into account the retardation ef-\nfects [32, 33]\n∂m(r,t)\n∂t=\u0002t\n0dt/prime\u0002\nd3r/primeΓ(r,t;r/prime,t/prime)m(r/prime,t/prime)\n×/bracketleftbigg\n−gBeff(r/prime,t/prime) +λG∂m(r/prime,t/prime)\n∂t/prime/bracketrightbigg\n,(2)\nby introducing a memory kernel Γ( r,t;r/prime,t/prime). The mem-\nory kernel models space-time correlation between local\nmagnetic moments, i.e., the fact that the cause for the\nchange of local magnetization occurs at time t−t/primeand\nat position r−r/primewhile the effect at position ris vis-\nible at the later time t. It has been specified phe-\nnomenologically, such as the sum of an instantaneous\nand time-dependent part which exponentially decays on\na characteristic time scale defining the strength of mem-\nory [32, 33]. It is also often simplified [33] by considering\ntime-retardation only, Γ( r,t;r/prime,t/prime)→Γ(t,t/prime), so that any\nspace-retardation effects are included only through the\neffective field Beff(r,t/prime).\nThe time-retardation described by Γ( t,t/prime) is a damping\nmechanism in addition to well-established mechanisms—\nthe combined effects of spin-orbit coupling and electron-\nphonon interaction [7, 8]—which govern λGin Eq. (1).\nHowever, the magnitude of Γ( t,t/prime) cannot be deduced\nfrom purely phenomenological considerations [32, 33].\nInstead, the introduction of the memory kernel Γ( t,t/prime)\ncan be justified microscopically [6, 22, 34–37] by us-\ningquantum-classical hybrid approaches, where time-\ndependent quantum formalism is used to compute /angbracketleftˆs/angbracketright(t)\nwhich is then fed into the LLG equation, while in turn, lo-\ncal magnetization from the LLG equation generates time-\ndependent field in the quantum Hamiltonian of electrons.\nAlthough electron dynamics is assumed to be much faster\nthan that of local magnetic moments, it still takes fi-\nnite time for electron spin to react to new position of\nm(r,t). This is the fundamental reason for time-retarded\ndamping effects encoded by Eq. (2), which are present\neven if the intrinsic Gilbert damping in Eq. (1) is van-\nishingly small due to small spin-orbit coupling (nonzero\nλGrequires spin-orbit coupling [7–9] and scales quadrati-\ncally with it [38]). Since classical micromagnetics simula-\ntions typically use only the conventional intrinsic Gilbert\ndamping term in Eq. (1), while not considering explicitly\nthe flow of conduction electrons in the presence of mag-\nnetization dynamics, the question arises about the mag-\nnitude of neglected effects like time-retarded damping in\nstandard simulations of magnetic-field- or current-driven\ndynamics of noncollinear magnetic textures such as mag-\nnetic domain walls (DWs) [39–47] and skyrmions [48, 49].\nAlthough quantum-classical approaches which auto-\nmatically include time-retardation effects have been dis-\ncussed previously [6, 22, 34–37], they have been focused\non the simples examples where one or two local magnetic\nmoments (pertinent to, e.g., magnetic molecules) inter-\nact with either closed electronic quantum system [22, 35]\n(i.e., not attached to macroscopic reservoirs to allow elec-\ntron spin and charge currents to flow into and from an\nFIG. 1. Schematic view of two-terminal devices where an infi-\nnite 1D TB chain, describing electrons quantum-mechanically,\nis attached to two macroscopic reservoirs while its middle part\nhosts: (a) single local magnetic moment, initially oriented in\nthe +x-direction, placed in an external magnetic field point-\ning along the + z-direction; (b) 11 local magnetic moments\n(illustration shows 7 of them), initially oriented in the + x-\ndirection, placed in an external magnetic field pointing along\nthe +z-direction; (c) three-site-wide head-to-head magnetic\nDW whose motion is driven by an external magnetic field\npointing in the + x-direction. Electrons within 1D TB chain\nand classical local magnetic moments interact via the s-dex-\nchange coupling of strength Jsd, and classical local magnetic\nmoments within the DW in (c) additionally interact with each\nother via the Heisenberg exchange coupling of strength J.\nexternal circuit), or open electronic quantum system but\nemploying approximations [6, 34, 36, 37] to obtain analyt-\nical solution. Thus, these approaches are not suitable for\nsimulations of spintronic devices containing large number\nof noncollinear local magnetic moments.\nHere we employ recently developed [50] numerically\nexact and, therefore, nonperturbative algorithm combin-\ning time-dependent nonequilibrium Green function for-\nmalism [51, 52] with the conventional LLG Eq. (1) (TD-\nNEGF+LLG) to demonstrate how it effectively generates\ntime-retardation effects, whose memory kernel can be ex-\nplicitly extracted in terms of TDNEGFs only in some lim-\nits (such as weak electron-spin/local-magnetic-moment\ninteraction and weak coupling of the active region to\nmacroscopic reservoirs). The paper is organized as fol-\nlows. Section II A introduces model quantum Hamilto-\nnian for electronic subsystem and classical Hamiltonian\nfor the subsystem comprised of local magnetic moments.\nIn Sec. II B, we show how the nonequilibrium expectation3\nvalue of spin density\n/angbracketleftˆs/angbracketrighti(t) =~\n2Tr [(\u001aneq(t)−\u001aeq)|i/angbracketright/angbracketlefti|⊗\u001b], (3)\ninserted into the LLG Eq. (1) generates a memory ker-\nnel because of the structure of the nonequilibrium time-\ndependent density matrix \u001aneq(t) obtained from TD-\nNEGF calculations. Here \u001aeqis the grand canonical equi-\nlibrium density matrix; \u001b= (ˆσx,ˆσy,ˆσz) is the vector of\nthe Pauli matrices; |i/angbracketrightelectron orbital centered on site i;\nand the operator |i/angbracketright/angbracketlefti|⊗\u001bacts in the composite Hilbert\nspaceH=Horb⊗Hspinof electronic orbital and spin\ndegrees of freedom. In this Section, we also discuss how\nin the limit of slow magnetization dynamics the memory\nkernel can be expanded in a Taylor series in order to ex-\ntract conventional Gilbert damping and magnetic inertia\nterms, but with time- and spatially-dependent parame-\ntersλD\ni(t) andID\ni(t). In Secs. III A–III C we compare the\ndynamics of local magnetic moments driven by an exter-\nnal magnetic field as computed by TDNEGF+LLG vs.\nconventional LLG simulations for three one-dimensional\n(1D) examples depicted in Fig. 1(a)–(c), respectively.\nSec. III C also compares pumped charge current due to\nthe DW motion as computed by TDNEGF+LLG vs. the\nwidely-used spin motive force (SMF) theory [12, 53] com-\nbined [54–56] with the conventional LLG equation. We\nconclude in Sec. IV.\nII. MODELS AND METHODS\nA. Coupled quantum and classical Hamiltonians\nThe conduction electron subsystem is modeled by a\nquantum Hamiltonian\nH(t) =−γ/summationdisplay\n/angbracketleftij/angbracketrightˆc†\niˆci−Jsd/summationdisplay\niˆc†\ni\u001b·Mi(t)ˆci, (4)\nwhich is (assumed to be 1D for simplicity) tight-binding\n(TB) model where electron interacts with magnetic mo-\nments localized at sites iand described by the classical\nvector Mi(t) of unit length. Here ˆ c†\ni= (ˆc†\ni↑,ˆc†\ni↓) is a row\nvector containing operators ˆ c†\niσwhich create an electron\nof spinσ=↑,↓at sitei; ˆciis a column vector that con-\ntains the corresponding annihilation operators; γ= 1 eV\nis the nearest neighbor hopping; and Jsdis thes-dex-\nchange coupling parameter between conduction electrons\nand local magnetic moments. The active region of de-\nvices depicted in Fig. 1(a)–(c) consists of 1, 11 and 21\nTB sites, respectively. These are attached to the left (L)\nand right (R) semi-infinite ideal leads modeled by the\nsame Hamiltonian in Eq. (4) but with Jsd= 0 eV. The\nleads are assumed to terminate into macroscopic reser-\nvoirs kept at the same chemical potential since we do not\napply any bias voltage to the devices in Fig. 1(a)–(c).The classical Hamiltonian describing the local mag-\nnetic moments is given by\nH=−J/summationdisplay\nijMi·Mj−µM/summationdisplay\niMi·Bi\next−K/summationdisplay\ni(Mx\ni)2\n−Jsd/summationdisplay\ni/angbracketleft^s/angbracketrighti·Mi,(5)\nwhereJis the Heisenberg exchange coupling parame-\nter;Bi\nextis the applied external magnetic field; Kis the\nmagnetic anisotropy (in the x-direction) and/angbracketleftˆs/angbracketrightiis the\nnonequilibrium electronic spin density computed from\nEq. (3).\nB. Time-retarded damping and magnetic inertia in\nthe LLG equation self-consistently coupled to\nTDNEGF\nThe quantum equation of motion for the nonequilib-\nrium density matrix of electrons [57, 58]\ni~∂\u001aneq(t)\n∂t= [H(t),\u001aneq(t)] +/summationdisplay\np=L,Ri[Πp(t) +Π†\np(t)].\n(6)\nis an example of a master equation for an open (i.e., con-\nnected to macroscopic reservoirs) quantum system [59]\ndue to the presence of the second term on the right hand\nside, in addition to standard terms of the von Neumann\nequation. This term and the density matrix itself can be\nexpressed using TDNEGF formalism [51, 52] as\n\u001aneq(t) =1\niG<(t,t/prime)|t=t/prime, (7)\nΠp(t/prime) =\u0002t/prime\n−∞dt1[G>(t/prime,t1)Σ<\np(t1,t/prime)−\nG<(t/prime,t1)Σ>\np(t1,t/prime)].(8)\nThe central quantities of the TDNEGF formalism are\nthe retarded Gr,σσ/prime\nii/prime(t,t/prime) =−iΘ(t−t/prime)/angbracketleft{ˆciσ(t),ˆci/primeσ/prime(t)}/angbracketright\nand the lesser G<,σσ/prime\nii/prime(t,t/prime) =i/angbracketleftˆc†\ni/primeσ/prime(t/prime)ˆciσ(t)/angbracketrightGreen\nfunctions (GFs) which describe the available density of\nstates and how electrons occupy those states, respec-\ntively. In addition, it is also useful to introduce the\ngreater GF, G>(t,t/prime) = [G<(t/prime,t)]†, and the advanced\nGF,Ga(t,t/prime) = [Gr(t,t/prime)]†. The current matrices Πp(t)\nmake it possible to compute directly [57, 58] charge cur-\nrent\nIp(t) =e\n~Tr[Πp(t)], (9)\nand spin current\nISα\np(t) =e\n~Tr[ˆσαΠp(t)], (10)4\nin the L and R semi-infinite leads. The equation of mo-\ntion for the lesser and greater GFs is given by\ni~∂G>,<(t,t1)\n∂t=H(t)G>,<(t,t1)+\n+∞\u0002\n−∞dt2/bracketleftbigg\nΣr\ntot(t,t2)G>,<(t2,t) +Σ>,<\ntot(t,t2)Ga(t2,t)/bracketrightbigg\n,\n(11)\nwhere Σr,>,<\ntot (t,t2) =/summationtext\np=L,RΣr,>,<\np (t,t2) and\nΣr,>,<\np (t,t2) are the lead self-energy matrices [52, 57, 58].\nThe classical equation of motion for the magnetic mo-\nment localized at site iis the Landau-Lifshitz equation\n∂Mi(t)\n∂t=−gMi(t)×Beff\ni(t), (12)\nwhere the effective magnetic field is\nBeff\ni(t) =−1\nµM∂H/∂MiandµMis the magnitude\nof the magnetic moment [5].\nThe full TDNEGF+LLG framework [50], which we\nalso denote as TDNEGF \u001cLLG, consists of self-\nconsistent combination of Eq.(6) and (12) where one first\nsolves for the nonequilibrium electronic spin density in\nEq.(3), which is then fed into Eq. (12) to propagate local\nmagnetic moments Mi(t) in the next time step. Evolving\n\u001aneq(t) via Eq. (6) requires time step δt= 0.1 fs for nu-\nmerical stability, and we use the same time step to evolve\nLLG or Landau-Lifshitz equations for Mi(t). These up-\ndated local magnetic moments are fed back into the quan-\ntum Hamiltonian of conduction electron subsystem in\nEq. (6). Thus obtained solutions for Mi(t),/angbracketleftˆs/angbracketrighti(t),Ip(t)\nandISαp(t) are numerically exact. For testing the im-\nportance of the self-consistent feedback loop, we also use\nTDNEGF←LLG where TDNEGF is utilized to obtain\nIp(t) andISαp(t) while the local magnetic moments are\nevolved solely by the conventional LLG Eq. (1), i.e., by\nusingJsd≡0 in Eq. (5) but Jsd/negationslash= 0 is used in Eq. (4).\nIn the weak-coupling limit [34, 60] (i.e., small Jsd)\nfor electron-spin/local-magnetic-moment interaction it is\npossible to extract explicitly the generalized LLG equa-\ntion with a memory kernel. For this purpose we use the\nfollowing expansions in the powers of small Jsd\n\u001aneq(t) =∞/summationdisplay\nn=0\u001an(t)Jn\nsd, (13)\nΠp(t/prime) =∞/summationdisplay\nn=0Π(n)\np(t/prime)Jn\nsd, (14)\nGr,a,>,<(t/prime,t1) =∞/summationdisplay\nn=0Gr,a,>,<\nn (t/prime,t1)Jn\nsd. (15)In Appendix A, we show how to combine Eqs.(6), (11),\n(13), (14) and (15) to obtain the perturbative equation\n∂Mi(t)\n∂t=−g/bracketleftbigg\nMi(t)×Beff,0\ni(t)+\nJ2\nsd\nµM/summationdisplay\np=L,RMi(t)×+∞\u0002\n−∞dt/prime/primeMi(t/prime/prime){Kp\ni(t/prime/prime,t)+Kp∗\ni(t/prime/prime,t)}/bracketrightbigg\n,\n(16)\nfor the dynamics of each local magnetic moment at site i,\nby retaining only the terms linear in Jsdin Eqs. (13)–(15).\nHere Beff,0\ni≡−1\nµM∂H0/∂Mi,H0is the classical Hamil-\ntonian in Eq. (5) with Jsd≡0 and Kp\ni(t/prime/prime,t) is defined in\nAppendix A. The physical origin [35] of time-retardation\neffects described by the second term in Eq. (16) is that,\neven though electron dynamics is much faster than the\ndynamics of local magnetic moments, the nonequilibrium\nspin density in Eq. (3) is always behind Mi(t) and, there-\nfore, never parallel to it which introduces spin torque\nterm into the Landau-Lifshitz Eq. (12). In other words it\ntakes finite amount of time for conduction electron spin\nto react to the motion of classical local magnetic mo-\nments, so that nonequilibrium electrons effectively me-\ndiate interaction of Mi(t) with the same local magnetic\nmoment at time t/prime< t. In the full TDNEGF+LLG,\nsuch retardation effects are mediated by the nonequilib-\nrium electrons starting at site iat timet/primeand returning\nback to the same site at time t > t/prime, while in the per-\nturbative limit the same effect is captured by the second\nterm in Eq. (16). The perturbative formula Eq. (16) is\nexpected [35] to breakdown after propagation over time\nt∼~/Jsd.\nFurther approximation to Eq. (16) can be made by\nconsidering sufficiently slow dynamics of local magnetic\nmoments so that higher order terms in the Taylor series\nMi(t/prime/prime)≈Mi(t)+∂Mi(t)\n∂t(t/prime/prime−t)+1\n2∂2Mi(t)\n∂t2(t/prime/prime−t)2+...,\n(17)\ncan be neglected. By defining the following quantities\nλD\np,i(t)≡+∞\u0002\n−∞dt/prime/prime(t/prime/prime−t)[Kp\ni(t/prime/prime,t) + K∗p\ni(t/prime/prime,t)],(18)\nand\nID\np,i(t)≡1\n2+∞\u0002\n−∞dt/prime/prime(t/prime/prime−t)2[Kp\ni(t/prime/prime,t) + K∗p\ni(t/prime/prime,t)],(19)\nand by retaining terms up to the second order in Eq. (17)5\nwe obtain the conventionally looking LLG equation\n∂Mi(t)\n∂t=−g/bracketleftbigg\nMi(t)×Beff,0\ni(t)+\nJ2\nsd\nµM/braceleftbigg/summationdisplay\np=L,RλD\np,n(t)/bracerightbigg\nMi(t)×∂Mi(t)\n∂t+\nJ2\nsd\nµM/braceleftbigg/summationdisplay\np=L,RID\np,i(t)/bracerightbigg\nMi(t)×∂2Mi(t)\n∂t2/bracketrightbigg\n.(20)\nHowever, the Gilbert damping term prefactor\nλD\ni(t) =J2\nsd\nµM/summationdisplay\np=L,RλD\np,i(t), (21)\nand the magnetic inertia term prefactor\nID\ni(t) =J2\nsd\nµM/summationdisplay\np=L,RID\np,i(t), (22)\nin Eq. (20) are now time- and position-dependent. This\nis in sharp contrast to conventional LLG Eq. (1) em-\nployed in classical micromagnetics where Gilbert damp-\ning and magnetic inertia prefactors are material specific\nconstants.\nIII. RESULTS AND DISCUSSION\nA. Single local magnetic moment in an external\nmagnetic field\nTo compare the dynamics of local magnetic moments\nin full TDNEGF+LLG quantum-classical simulations vs.\nconventional LLG classical simulations, we first consider\na well-known example [5] for which the conventional LLG\nequation can be analytically solved—a single local mag-\nnetic moment which at t= 0 points along the + x-\ndirection and then starts to precesses due to an external\nmagnetic field pointing in the + z-direction. Its trajectory\nis given by [5]\nMx(t) = sech/parenleftbigggλGB\n1 +λGt/parenrightbigg\ncos/parenleftbigggB\n1 +λ2\nGt/parenrightbigg\n,(23a)\nMy(t) = sech/parenleftbigggλGB\n1 +λGt/parenrightbigg\nsin/parenleftbigggB\n1 +λ2\nGt/parenrightbigg\n,(23b)\nMz(t) = tanh/parenleftbigggλGB\n1 +λGt/parenrightbigg\n, (23c)\nwhere B= (0,0,B) is the applied external mag-\nnetic field. Thus, if the conventional intrinsic Gilbert\ndamping parameter is set to zero, λG= 0, then\nthe local magnetic moment precesses steadily around\nthez-axis with Mz≡0. On the other hand,\nfor nonzero λG>0, the local magnetic moment\nFIG. 2. (a) Time dependence of tanh−1(Mz) for a sin-\ngle local magnetic moment in Fig. 1(a) obtained from TD-\nNEGF+LLG simulations. Colors red to blue indicate in-\ncreasings-dexchange coupling in steps of 0 .1 eV, ranging\nfromJsd= 0 eV toJsd= 1.9 eV. (b) The dynamical Gilbert\ndamping parameter in Eq. (21) extracted from panel (a) as\na function of Jsd. (c) Time dependence of Mzcomponent\nfor a single local magnetic moment in Fig. 1(a) at large\nJsd= 2.0 eV exhibits nutation as a signature of magnetic in-\nertia. To generate fast magnetization dynamics and reduce\nsimulation time, we use an unrealistically large external mag-\nnetic field of strength B= 1000 T. The conventional intrinsic\nGilbert damping parameter is set to zero, λG= 0, and the\nFermi energy is EF= 0 eV.\nwill relax towards the direction of magnetic field,\ni.e., lim\nt→∞(Mx(t),My(t),Mz(t)) = (0,0,1). Thus, such\ndamped dynamics is signified by a linear tanh−1(Mz)\nvs. time dependence. Figure 2 plots results of TD-\nNEGF+LLG simulations for the same problem. Even\nthough we set conventional intrinsic Gilbert damping\nto zero,λG= 0, Fig. 2(a) shows linear tanh−1(Mz) vs.\ntime, independently of the strength of s-dexchange cou-\npling as long as Jsd.2 eV. This means that the lo-\ncal magnetic moment is experiencing (time-independent)\ndynamical Gilbert damping λD∝J2\nsd, in accord with\nEq. (21) and as shown in Fig. 2(b), which is generated\nsolely by the TDNEGF part of the self-consistent loop\nwithin the full TDNEGF+LLG scheme.\nForJsd&2 eV, the dynamics of the local magnetic mo-\nment also exhibits nutation [35], as shown in Fig. 2(c),\nwhich is the signature of the magnetic inertia [19–24]\nterm∝Mi×∂2Mi/∂t2in Eq. (20). Thus, nutation be-\ncomes conspicuous when the dynamics of the local mag-\nnetic moments is sufficiently fast, so that ∂2Mi/∂t2is\nlarge, as well as when the interaction between the itiner-\nant and localized spins is sufficiently large.6\nFIG. 3. TDNEGF+LLG-computed trajectories\n(Mx(t),My(t),Mz(t)) on the Bloch sphere of local magnetic\nmoment in the setup of Fig. 1(b) at: (a) site 1; and (c) site\n6. The total number of local magnetic moments is N= 11,\nand they do not interact with each other via exchange\ncoupling [i.e., J= 0 eV in Eq. (5)]. Panels (b) and (d) show\nthe corresponding time dependence of Mzcomponent from\npanels (a) and (c), respectively. The external magnetic\nfield isB= 1000 T, and the s-dexchange coupling strength\nJsd= 0.1 eV is nonperturbative in this setup, therefore,\nnotallowing us to extract explicitly the dynamical Gilbert\ndamping parameter from Eq. (21). The conventional intrinsic\nGilbert damping parameter is set to zero, λG= 0, and the\nFermi energy is EF= 0 eV.\nB. Multiple exchange-uncoupled local magnetic\nmoments in an external magnetic field\nIn order to examine possible spatial dependence of the\ndynamical Gilbert damping parameter or emergence of\ndynamical exchange coupling [61, 62] between local mag-\nnetic moments, we consider a chain of N= 11 magnetic\nmoments which do not interact with each other ( J= 0)\nbut interact with conduction electron spin ( Jsd/negationslash= 0), as\nillustrated in Fig. 1(b). At t= 0, all magnetic moments\npoint in the + x-direction while the external magnetic\nfield is in the + zdirection, and the conventional intrin-\nsic Gilbert damping is set to zero, λG= 0.\nFigures 3(a) and 3(c) show the trajectory of selected\nlocal magnetic moments ( i= 1 and 6) on the Bloch\nsphere forJsd= 0.1 eV. In contrast to single local mag-\nnetic moment in Fig. 2(a), for which tanh−1(Mz) vs.\ntime is linear using Jsd= 0.1 eV, we find that in case of\nmultiple exchange-uncoupled magnetic moments this is\nno longer the case, as demonstrated by Figs. 3(b) and\n3(d). Hence, the trajectory followed by these local mag-\nnetic moments cannot be described by Eq. (23) so that\nFIG. 4. (a) TDNEGF+LLG-computed time dependence of\nMzcomponent of local magnetic moment on sites 1, 3 and 6 in\nthe setup of Fig. 1(b) with a total of N= 11 moments. (b) Po-\nsition dependence of the dynamical Gilbert damping param-\neter in Eq. (21). The external magnetic field is B= 1000 T,\nand thes-dexchange coupling strength Jsd= 0.01 eV is per-\nturbative in this setup, therefore, allowing us to extract the\ndynamical Gilbert damping explicitly from Eq. (21). The con-\nventional intrinsic Gilbert damping parameter is set to zero,\nλG= 0, and the Fermi energy is EF= 0 eV.\nthe conventional-like Gilbert damping parameter cannot\nbe extracted anymore. Thus, such a nonstandard damp-\ning of the dynamics of local magnetic moments originates\nfrom time-dependence of the dynamical damping param-\neterλD\niin Eq. (21).\nFigure 4(a) shows tanh−1(Mz) vs. time for selected\nlocal magnetic moments ( i= 1,3 and 6) and smaller\nJsd= 0.01 eV. Although all local magnetic moments fol-\nlow linear tanh−1(Mz) vs. time, as predicted by the\nsolution in Eq. (23c) of the conventional LLG equation,\nthe dynamical Gilbert damping extracted from Eq. (23)\nchanges from site to site as shown in Fig. 4(b). Further-\nmore, the linear tanh−1(Mz) vs. time relation breaks\ndown for times t&50 ps at specific sites, which then pre-\nvents extracting time-independent λD\niat those sites.\nC. Magnetic field-driven motion of a domain wall\ncomposed of multiple exchange-coupled local\nmagnetic moments\nIn order to examine difference in predicted dynam-\nics of exchange-coupled local magnetic moments by TD-\nNEGF+LLG framework vs. conventional LLG equa-\ntion, we consider the simplest example of 1D head-to-\nhead magnetic DW depicted in Fig. 1(c). Its motion is\ndriven by applying an external magnetic field in the + x-\ndirection. Some type of damping mechanism is crucial\nfor the DW to move, as demonstrated by solid lines in\nFig. 5(e)–(h), obtained by solving the conventional LLG\nequation with λG= 0, which show how local magnetic\nmoments precess around the magnetic field but without\nnet displacement of the center of the DW.\nOn the other hand, even though we set λG= 0 in TD-\nNEGF+LLG simulations in Fig. 5(a)–(d), the center of\nthe DW moves to the right due to dynamically generated7\nFIG. 5. (a)–(d) TDNEGF+LLG-computed snapshots of head-to-head DW in the setup of Fig. 1(c) driven by an external\nmagnetic field of strength B= 100 T pointing in the + x-direction, in the absence ( λG= 0) or presence ( λG= 0.01) of the\nconventional intrinsic Gilbert damping. Panels (e)–(h) show the corresponding snapshots computed solely by the conventional\nLLG Eq. (1) where in the absence ( λG= 0) of the conventional intrinsic Gilbert damping the DW does not move at all. The\nHeisenberg exchange coupling between local magnetic moments is J= 0.01 eV;s-dexchange coupling between electrons and\nlocal magnetic moments is Jsd= 0.1 eV; magnetic anisotropy (in the x-direction) is K= 0.01 eV; and the Fermi energy of\nelectrons is EF=−1.9 eV. The magnetic field is applied at t= 2 ps, while prior to that we evolve the conduction electron\nsubsystem with TDNEGF until it reaches the thermodynamic equilibrium where all transient spin and charge currents have\ndecayed to zero.\ntime-retarded damping encoded by the memory kernel\nin Eq. (16). Including the conventional intrinsic Gilbert\ndamping,λG= 0.01 as often used in micromagnetic sim-\nulations of DW along magnetic nanowires [43, 44, 63],\nchanges only slightly the result of TDNEGF+LLG sim-\nulations which demonstrates that the effective dynam-\nical Gilbert damping (which is also time-dependent) is\nabout an order of magnitude larger than λG. This is also\nreflected in the DW velocity being much larger in TD-\nNEGF+LLG simulations with λG= 0 in Fig. 5(a)–(d)\nthan in the conventional LLG equation simulations with\nλG= 0.01 in Fig. 5(e)–(h).\nIt has been predicted theoretically [12, 53, 64–68] and\nconfirmed experimentally [69] that a moving DW will\npump charge current even in the absence of any applied\nbias voltage. The corresponding open circuit pumping\nvoltage in the so-called spin motive force (SMF) the-\nory [12, 53] is given by\nVSMF=1\nG0\u0002\njxdx, (24a)\njα(r) =Pσ0~\n2e[∂tm(r,t)×∂αm(r,t)]·m(r,t),(24b)\nwherejxis the pumped local charge current along the\nx-axis. Here σ0=σ↑+σ↓is the total conductivity;\nP= (σ↑−σ↓)/(σ↑+σ↓) is the spin polarization of the\nferromagnet; and ∂t=∂/∂t. Equation (24) is typicallycombined [54–56] with classical micromagnetics which\nsupplies Mi(t) that is then plugged into the discretized\nversion [50]\njx(i)∝1\na[∂tMi(t)×(Mi+1(t)−Mi(t))]·Mi(t)\n∝1\na[∂Mi(t)×Mi+1(t)]·Mi(t). (25)\nof Eq. (24b). We denote this approach as SMF ←LLG,\nwhich is perturbative in nature [67, 70] since it considers\nonly the lowest temporal and spatial derivatives.\nOn the other hand, the same pumping voltage can be\ncomputed nonperturbatively\nVTDNEGF =Ip(t)\nG(t), (26)\nusing TDNEGF expression for charge current in lead p\nin Eq. 9, where TDNEGF calculations are coupled to\nLLG calculations either self-consistently (i.e., by using\nTDNEGF\u001cLLG) or non-self-consistently (i.e., by us-\ning TDNEGF←LLG). Here, G(t) is the conductance\ncomputed using the Landauer formula applied to two-\nterminal devices with a frozen at time ttexture of local\nmagnetic moments.\nFigures 6(a) and 6(b) plot the pumping voltage cal-\nculated by TDNEGF \u001cLLG for DW motion shown in\nFig. 5(a)–(d) in the absence or presence of conventional8\nFIG. 6. Time dependence of pumping voltage generated by\nthe DW motion depicted in Fig. 5(a)–(d) for: (a) λG= 0;\n(b)λG= 0.01. In panels (a) and (b) local magnetic mo-\nments evolve in time by the full TDNEGF+LLG framework\nwhere the arrows indicate how TDNEGF sends nonequilib-\nrium electronic spin density into the LLG equation which, in\nturn, sends trajectories of local magnetic moments into TD-\nNEGF. Time dependence of pumping voltage generated by\nDW motion depicted in Fig. 5(e)–(h) for: (c) λG= 0; (d)\nλG= 0.01. In panels (c) and (d) local magnetic moments\nevolve in time using the conventional LLG equation which\nsends their trajectories into either TDNEGF (green) or SMF\nformulas (blue) in Eq. (26) or Eq. (24), respectively, to obtain\nthe corresponding pumping voltage.\nGilbert damping, respectively. The two cases are virtu-\nally identical due to an order of magnitude larger dynam-\nical Gilbert damping that is automatically generated by\nTDNEGF\u001cLLG in both Figs. 6(a) and 6(b). The\nnonperturbative results in Figs. 6(a) and Fig. 6(b) are\nquite different from SMF ←LLG predictions in Figs. 6(c)\nand Fig. 6(d), respectively. This is due to both failure\nof Eqs. (24) and (25) to describe noncoplanar and non-\ncollinear magnetic textures with neighboring local mag-\nnetic moments tilted by more than 10◦[50] and lack\nof dynamical Gilbert damping in SMF ←LLG simula-\ntions [54–56]. The latter effect is also emphasized by the\ninability of TDNEGF ←LLG in Figs. 6(c) and Fig. 6(d)\nto reproduce the results of self-consistent TDNEGF \u001c\nLLG in Figs. 6(a) and Fig. 6(b), respectively.\nIV. CONCLUSIONS\nIn conclusion, we delineated a hierarchy of theoret-\nical descriptions of a nonequilibrium quantum many-\nbody system in which conduction electron spins inter-\nact with local magnetic moments within a ferromagneticlayer sandwiched between normal metal electrodes. On\nthe top of the hierarchy is a fully quantum approach,\nfor both electrons and local magnetic moments, whose\ncomputational complexity (using either original spin op-\nerators [71, 72] for local magnetic moments, or their\nmapping to bosonic operators in order to enable ap-\nplication of many-body perturbation theory within the\nNEGF formalism [73]) makes it impractical for systems\ncontaining large number of local magnetic moments.\nThe next approach in the hierarchy is computation-\nally much less expensive quantum-classical hybrid [74]\nbased on self-consistent coupling [50] of TDNEGF (which\ncan be implemented using algorithms that scale linearly\nwith both system size and simulation time [52, 58, 75])\nwith classical LLG equation for local magnetic moments.\nSuch TDNEGF+LLG approach is numerically exact and,\ntherefore, nonperturbative in the strength of electron-\nspin/local-magnetic-moment interaction, speed of local\nmagnetic moment dynamics and degree of noncollinearity\nbetween them. Even though electron dynamics is much\nfaster than localized spin dynamics, the most general sit-\nuation cannot be handled by integrating out [6, 34] the\nconduction electron degrees of freedom and by focusing\nonly on the LLG-type equation where a much larger time\nstep can be used to propagate spins only.\nNevertheless, in the limit [34, 60] of weak electron-\nspin/local-magnetic-moment interaction [i.e., small Jsd\nin Eqs. (4) and (5)] one can derive analytically a type\nof generalized LLG equation [34–37] for each local mag-\nnetic moment which is next approach in the hierarchy\nthat sheds light onto different effects included in the nu-\nmerically exact TDNEGF+LLG scheme. Instead of the\nconventional Gilbert damping term in Eq. (1), the gen-\neralized LLG equation we derive as Eq. (16) contains\na microscopically determined memory kernel which de-\nscribes time-retardation effects generated by the coupling\nto TDNEGF. Fundamentally, the memory kernel is due\nto the fact that electron spin can never follow instanta-\nneously change in the orientation of the local magnetic\nmoments [35]. In the limit of slow dynamics of local\nmagnetic moments, one can further expand the memory\nkernel into a Taylor series to obtain the final approach\nwithin the hierarchy whose LLG Eq. (20) is akin to the\nconventional one, but which contains both Gilbert damp-\ning (proportional to first time derivative of local mag-\nnetization) and magnetic inertia terms (proportional to\nsecond time derivative of local magnetization) with time-\ndependent parameters instead of usually assumed mate-\nrials specific constants.\nUsing three simple examples—single or multiple local\nmagnetic moments precessing in an external magnetic\nfield or magnetic-field-driven magnetic DW motion—\nwe demonstrate the importance of dynamically induced\ndamping which operates even if conventional static\nGilbert damping is set to zero. In the case of field-\ndriven magnetic DW motion, we can estimate that the\nstrength of dynamical damping is effectively an order of\nmagnitude larger than typically assumed [43, 44, 63] con-9\nventional static Gilbert damping λG/similarequal0.01 in classical\nmicromagnetic simulations of magnetic nanowires. In ad-\ndition, we show that charge pumping by the dynamics of\nnoncoplanar and noncollinear magnetic textures, which\nis outside of the scope of pure micromagnetic simulations\nbut it is often described by combining [54–56] them with\nthe SMF theory formula [12, 53], requires to take into ac-\ncount both the dynamical Gilbert damping and possiblylarge angle between neighboring local magnetic moments\nin order to reproduce numerically exact results of TD-\nNEGF+LLG scheme.\nACKNOWLEDGMENTS\nThis work was supported by NSF Grant No. ECCS\n150909.\nAppendix A: Derivation of Memory Kernel in LLG equation self-consistently coupled to TDNEGF\nIn this Appendix, we provide a detailed derivation of the memory kernel in Eq. (16). To obtain the perturbative\nequation of motion for local magnetic moments we start from Landau-Lifshitz Eq. (12) where the effective magnetic\nfield can be written as\nBeff\ni(t) =Beff,0\ni(t) +Jsd/angbracketleftˆs/angbracketrighti(t). (A.1)\nThe nonequilibrium spin density is expanded up to terms linear in Jsdusing Eq. (13)\n/angbracketleftˆs/angbracketrighti(t) =~\n2Tr[\u001aneq(t)|i/angbracketright/angbracketlefti|⊗\u001b]−/angbracketleftˆs/angbracketrighti\neq≈~\n2Tr/bracketleftbigg\n{\u001a0(t)+Jsd\u001a1(t)}|i/angbracketright/angbracketlefti|⊗\u001b/bracketrightbigg\n−/angbracketleftˆs/angbracketrighti\neq=Jsd~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b]−/angbracketleftˆs/angbracketrighti\neq.\n(A.2)\nHere/angbracketleftˆs/angbracketrighti\neqis the equilibrium electronic spin density i.e., /angbracketleftˆs/angbracketrighti\neq= (~/2) Tr [ \u001aeq|i/angbracketright/angbracketlefti|⊗\u001b]. Furthermore, the electronic\nspin density in the zeroth order must vanish, i.e., Tr [ \u001a0(t)|i/angbracketright/angbracketlefti|⊗\u001b] = 0 since for Jsd= 0 electrons are not spin-\npolarized. Hence, we can write Eq. (12) as\n∂Mi(t)\n∂t=−gMi(t)×/bracketleftbigg\nBeff,0\ni(t) +J2\nsd~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b]−Jsd/angbracketleftˆs/angbracketrighti\neq/bracketrightbigg\n. (A.3)\nTo obtain analytical results, we assume that the equilibrium spin density follows the direction of local magnetic\nmoments, so that Mi(t)×/angbracketleftˆs/angbracketrighti\neq= 0. By expanding Eq. (6) we obtain\ni~∂\u001a0(t)\n∂t= [H0(t),\u001a0(t)] +/summationdisplay\np=L,Ri[Π(0)\np(t) +Π(0)†\np(t)], (A.4)\nand\ni~∂\u001a1(t)\n∂t= [H1(t),\u001a0(t)] +i/summationdisplay\np=L,R[Π(1)\np(t) +Π(1)†\np(t)], (A.5)\nwhere H1(t) =−/summationtext\ni|i/angbracketright/angbracketlefti|⊗\u001b·Mi(t). One can formally integrate Eq. (A.5) which leads to\n~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b] =/summationdisplay\np=L,R1\n2t\u0002\n−∞dt/primeTr/bracketleftbigg\n{Π(1)\np(t/prime) +Π(1)†\np(t/prime)}|i/angbracketright/angbracketlefti|⊗\u001b/bracketrightbigg\n. (A.6)\nwhich requires to find an expression for Π(1)\np(t/prime). Using Eq. (8) and the fact that lead self-energy matrices do not\ndepend onJsdleads to\nΠ(1)\np(t/prime) =t/prime\u0002\n−∞dt1[G>\n1(t/prime,t1)Σ<\np(t1,t/prime)−G<\n1(t/prime,t1)Σ>\np(t1,t/prime)]. (A.7)\nEquations (11) and (15) can be formally integrated to yield lesser and greater GFs in Eq. (A.7)\nG>,<\n1(t/prime,t1) =1\ni~/parenleftbiggt/prime\u0002\n−∞dt/prime/primeH1(t/prime/prime)G>,<\n0(t/prime/prime,t1) +t/prime\u0002\n−∞dt/prime/prime+∞\u0002\n−∞dt2/bracketleftbigg\nΣr\ntot(t/prime/prime,t2)G>,<\n1(t2,t/prime/prime) +Σ>,<\ntot(t/prime/prime,t2)Ga\n1(t2,t/prime/prime)/bracketrightbigg/parenrightbigg\n.\n(A.8)10\nWe further assume that the active region in Fig. 1 is weakly coupled with semi-infinite leads and, therefore, macroscopic\nreservoirs into which they terminate. This means that after we substitute Eq. (A.8) into Eq. (A.7) we can keep only\nthose terms that are linear in the self-energy\nΠ(1)\np(t/prime) =1\n2it/prime\u0002\n−∞dt/prime/primeH1(t/prime/prime)t/prime\u0002\n−∞dt1/bracketleftbigg\nG>\n0(t/prime/prime,t1)Σ<\np(t1,t/prime)−G<\n0(t/prime/prime,t1)Σ>\np(t1,t/prime)/bracketrightbigg\n(A.9)\n=i\n2/summationdisplay\nit/prime\u0002\n−∞dt/prime/prime|i/angbracketright/angbracketlefti|⊗\u001b·Mi(t/prime/prime)t/prime\u0002\n−∞dt1/bracketleftbigg\nG>\n0(t/prime/prime,t1)Σ<\np(t1,t/prime)−G<\n0(t/prime/prime,t1)Σ>\np(t1,t/prime)/bracketrightbigg\n(A.10)\n=i/summationdisplay\nit/prime\u0002\n−∞dt/prime/prime|i/angbracketright/angbracketlefti|⊗\u001b·Mi(t/prime/prime)A0\np(t/prime/prime,t/prime), (A.11)\nwhere A0\np(t/prime/prime,t/prime) is an operator constructed out of the zeroth order terms in the expansion of GFs shown in Eq. (15)\nA0\np(t/prime/prime,t/prime)≡i\n2t/prime\u0002\n−∞dt1/bracketleftbigg\nG>\n0(t/prime/prime,t1)Σ<\np(t1,t/prime)−G<\n0(t/prime/prime,t1)Σ>\np(t1,t/prime)/bracketrightbigg\n. (A.12)\nBy plugging in Eqs. (A.11) and (A.12) into Eq. (A.6) we obtain\n~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗ˆσµ] =/summationdisplay\np=L,R/summationdisplay\nj/summationdisplay\nνt\u0002\n−∞dt/primet/prime\u0002\n−∞dt/prime/primeMν\nj(t/prime/prime) Tr/bracketleftbigg\n|j/angbracketright/angbracketleftj|⊗ˆσν{A0\np(t/prime/prime,t/prime)+A0†\np(t/prime/prime,t/prime)}|i/angbracketright/angbracketlefti|⊗σµ/bracketrightbigg\n.(A.13)\nSince A0\np(t/prime/prime,t/prime) is an operator constructed from the zeroth order GFs, it can be written in the followin form\nA0\np(t/prime/prime,t/prime) =1\n2/summationdisplay\nmnAp\nmn(t/prime/prime,t/prime)|m/angbracketright/angbracketleftn|⊗12, (A.14)\nwhere 12is a 2×2 identity matrix. 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B 93, 134506\n(2016)."    },    {        "title": "1811.00020v2.Anisotropic_and_controllable_Gilbert_Bloch_dissipation_in_spin_valves.pdf",        "content": "Anisotropic and controllable Gilbert-Bloch dissipation in spin valves\nAkashdeep Kamra,1,\u0003Dmytro M. Polishchuk,2Vladislav Korenivski,2and Arne Brataas1\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, Trondheim, Norway\n2Nanostructure Physics, Royal Institute of Technology, Stockholm, Sweden\nSpin valves form a key building block in a wide range of spintronic concepts and devices from\nmagnetoresistive read heads to spin-transfer-torque oscillators. We elucidate the dependence of the\nmagnetic damping in the free layer on the angle its equilibrium magnetization makes with that in\nthe \fxed layer. The spin pumping-mediated damping is anisotropic and tensorial, with Gilbert- and\nBloch-like terms. Our investigation reveals a mechanism for tuning the free layer damping in-situ\nfrom negligible to a large value via the orientation of \fxed layer magnetization, especially when the\nmagnets are electrically insulating. Furthermore, we expect the Bloch contribution that emerges\nfrom the longitudinal spin accumulation in the non-magnetic spacer to play an important role in a\nwide range of other phenomena in spin valves.\nIntroduction. { The phenomenon of magnetoresistance\nis at the heart of contemporary data storage technolo-\ngies [1, 2]. The dependence of the resistance of a multi-\nlayered heterostructure comprising two or more magnets\non the angles between their respective magnetizations has\nbeen exploited to read magnetic bits with a high spatial\nresolution [3]. Furthermore, spin valves comprised of two\nmagnetic layers separated by a non-magnetic conductor\nhave been exploited in magnetoresistive random access\nmemories [2, 4, 5]. Typically, in such structures, one\n`free layer' is much thinner than the other `\fxed layer'\nallowing for magnetization dynamics and switching in\nthe former. The latter serves to spin-polarize the charge\ncurrents \rowing across the device and thus exert spin-\ntorques on the former [6{9]. Such structures exhibit a\nwide range of phenomena from magnetic switching [5] to\noscillations [10, 11] driven by applied electrical currents.\nWith the rapid progress in taming pure spin cur-\nrents [12{20], magnetoresistive phenomena have found\na new platform in hybrids involving magnetic insulators\n(MIs). The electrical resistance of a non-magnetic metal\n(N) was found to depend upon the magnetic con\fgura-\ntion of an adjacent insulating magnet [21{24]. This phe-\nnomenon, dubbed spin Hall magnetoresistance (SMR),\nrelies on the pure spin current generated via spin Hall\ne\u000bect (SHE) in N [25, 26]. The SHE spin current accu-\nmulates spin at the MI/N interface, which is absorbed\nby the MI depending on the angle between its magne-\ntization and the accumulated spin polarization. The\nnet spin current absorbed by the MI manifests as ad-\nditional magnetization-dependent contribution to resis-\ntance in N via the inverse SHE. The same principle of\nmagnetization-dependent spin absorption by MI has also\nbeen exploited in demonstrating spin Nernst e\u000bect [27],\ni.e. thermally generated pure spin current, in platinum.\nWhile the ideas presented above have largely been ex-\nploited in sensing magnetic \felds and magnetizations,\ntunability of the system dissipation is a valuable, un-\nderexploited consequence of magnetoresistance. Such\nan electrically controllable resistance of a magnetic wire\nFIG. 1. Schematic depiction of the device under investigation.\nThe blue arrows denote the magnetizations. The \fxed layer\nF2magnetization remains static. The free layer F 1magneti-\nzation precesses about the z-axis with an average cone angle\n\u0002\u001c1. The two layers interact dynamically via spin pumping\nand back\row currents.\nhosting a domain wall [28] has been suggested as a ba-\nsic circuit element [29] in a neuromorphic computing [30]\narchitecture. In addition to the electrical resistance or\ndissipation, the spin valves should allow for controlling\nthe magnetic damping in the constituent magnets [31].\nSuch an in-situ control can be valuable in, for example,\narchitectures where a magnet is desired to have a large\ndamping to attain low switching times and a low dissipa-\ntion for spin dynamics and transport [13, 16]. Further-\nmore, a detailed understanding of magnetic damping in\nspin valves is crucial for their operation as spin-transfer-\ntorque oscillators [10] and memory cells [5].\nInspired by these new discoveries [21, 27] and previous\nrelated ideas [31{34], we suggest new ways of tuning the\nmagnetic damping of the free layer F 1in a spin valve\n(Fig. 1) via controllable absorption by the \fxed layer\nF2of the spin accumulated in the spacer N due to spin\npumping [31, 35]. The principle for this control over spin\nabsorption is akin to the SMR e\u000bect discussed above and\ncapitalizes on altering the F 2magnetization direction.\nWhen spin relaxation in N is negligible, the spin lost by\nF1is equal to the spin absorbed by F 2. This lost spin\nappears as tensorial Gilbert [36] and Bloch [37] damp-arXiv:1811.00020v2  [cond-mat.mes-hall]  10 Apr 20192\ning in F 1magnetization dynamics. In its isotropic form,\nthe Gilbert contribution arises due to spin pumping and\nis well established [31{33, 35, 38{40]. We reveal that\nthe Bloch term results from back\row due to a \fnite dc\nlongitudinal spin accumulation in N. Our results for the\nangular and tensorial dependence of the Gilbert damping\nare also, to best of our knowledge, new.\nWe show that the dissipation in F 1, expressed in terms\nof ferromagnetic resonance (FMR) linewidth, varies with\nthe angle\u0012between the two magnetizations (Fig. 3).\nThe maximum dissipation is achieved in collinear or or-\nthogonal con\fgurations depending on the relative size\nof the spin-mixing g0\nrand longitudinal spin glconduc-\ntances of the NjF2subsystem. For very low gl, which\ncan be achieved employing insulating magnets, the spin\npumping mediated contribution to the linewidth vanishes\nfor collinear con\fgurations and attains a \u0012-independent\nvalue for a small non-collinearity. This can be used to\nstrongly modulate the magnetic dissipation in F 1electri-\ncally via, for example, an F 2comprised by a magneto-\nelectric material [41].\nFMR linewidth. { Disregarding intrinsic damping for\nconvenience, the magnetization dynamics of F 1including\na dissipative spin transfer torque arising from the spin\ncurrent lost IIIs1may be expressed as:\n_^mmm=\u0000j\rj(^mmm\u0002\u00160HHHe\u000b) +j\rj\nMsVIIIs1: (1)\nHere, ^mmmis the unit vector along the F 1magnetization\nMMMtreated within the macrospin approximation, \r(<0)\nis the gyromagnetic ratio, Msis the saturation magneti-\nzation,Vis the volume of F 1, andHHHe\u000bis the e\u000bective\nmagnetic \feld. Under certain assumptions of linearity\nas will be detailed later, Eq, (1) reduces to the Landau-\nLifshitz equation with Gilbert-Bloch damping [36, 37]:\n_^mmm=\u0000j\rj(^mmm\u0002\u00160HHHe\u000b) + ( ^mmm\u0002GGG)\u0000BBB: (2)\nConsidering the equilibrium orientation ^mmmeq=^zzz, Eq. (2)\nis restricted to the small transverse dynamics described\nbymx;y\u001c1, while the z-component is fully determined\nby the constraint ^mmm\u0001^mmm= 1. Parameterized by a diagonal\ndimensionless tensor \u0014 \u000b, the Gilbert damping has been in-\ncorporated via GGG=\u000bxx_mx^xxx+\u000byy_my^yyyin Eq. (2). The\nBloch damping is parametrized via a diagonal frequency\ntensor \u0014\n asBBB= \n xxmx^xxx+ \nyymy^yyy. A more familiar,\nalthough insu\u000ecient for the present considerations, form\nof Bloch damping can be obtained by assuming isotropy\nin the transverse plane: BBB= \n 0(^mmm\u0000^mmmeq). This form,\nrestricted to transverse dynamics, makes its e\u000bect as a\nrelaxation mechanism with characteristic time 1 =\n0ev-\nident. The Bloch damping, in general, captures the so-\ncalled inhomogeneous broadening and other, frequency\nindependent contributions to the magnetic damping.\nConsidering uniaxial easy-axis and easy-plane\nanisotropies, parametrized respectively by Kzand\n0 30 60 9000.10.20.30.40.5FIG. 2. Normalized damping parameters for F 1magneti-\nzation dynamics vs. spin valve con\fguration angle \u0012(Fig.\n1). ~\u000bxx6= ~\u000byysigni\fes the tensorial nature of the Gilbert\ndamping. The Bloch parameters ~\nxx\u0019~\nyyare largest for\nthe collinear con\fguration. The curves are mirror symmetric\nabout\u0012= 90\u000e. ~g0\nr= 1, ~gl= 0:01, \u0002 = 0:1,!0= 10\u00022\u0019\nGHz, and!ax= 1\u00022\u0019GHz.\nKx[42], the magnetic free energy density Fmis ex-\npressed as: Fm=\u0000\u00160MMM\u0001HHHext\u0000KzM2\nz+KxM2\nx;with\nHHHext=H0^zzz+hhhrfas the applied static plus microwave\n\feld. Employing the e\u000bective \feld \u00160HHHe\u000b=\u0000@Fm=@MMM\nin Eq. (2) and switching to Fourier space [ \u0018exp(i!t)],\nwe obtain the resonance frequency !r=p\n!0(!0+!ax).\nHere,!0\u0011j\rj(\u00160H0+ 2KzMs) and!ax\u0011j\rj2KxMs.\nThe FMR linewidth is evaluated as:\nj\rj\u00160\u0001H=(\u000bxx+\u000byy)\n2!+t(\nxx+ \nyy)\n2\n+t!ax\n4(\u000byy\u0000\u000bxx); (3)\nwhere!is the frequency of the applied microwave \feld\nhhhrfand is approximately !rclose to resonance, and t\u0011\n!=p\n!2+!2ax=4\u00191 for a weak easy-plane anisotropy.\nThus, in addition to the anisotropic Gilbert contribu-\ntions, the Bloch damping provides a nearly frequency-\nindependent o\u000bset in the linewidth.\nSpin \row. { We now examine spin transport in the\ndevice with the aim of obtaining the damping parame-\nters that determine the linewidth [Eq. (3)]. The N layer\nis considered thick enough to eliminate static exchange\ninteraction between the two magnetic layers [31, 40]. Fur-\nthermore, we neglect the imaginary part of the spin-\nmixing conductance, which is small in metallic systems\nand does not a\u000bect dissipation in any case. Disregarding\nlongitudinal spin transport and relaxation in the thin free\nlayer, the net spin current IIIs1lost by F 1is the di\u000berence\nbetween the spin pumping and back\row currents [31]:\nIIIs1=gr\n4\u0019\u0010\n~^mmm\u0002_^mmm\u0000^mmm\u0002\u0016\u0016\u0016s\u0002^mmm\u0011\n; (4)\nwheregris the real part of the F 1jN interfacial spin-\nmixing conductance, and \u0016\u0016\u0016sis the spatially homogeneous3\nspin accumulation in the thin N layer. The spin current\nabsorbed by F 2may be expressed as [31]:\nIIIs2=g0\nr\n4\u0019^mmm2\u0002\u0016\u0016\u0016s\u0002^mmm2+gl\n4\u0019(^mmm2\u0001\u0016\u0016\u0016s)^mmm2;\n\u0011X\ni;j=fx;y;zggij\n4\u0019\u0016sj^iii; (5)\nwhereglandg0\nrare respectively the longitudinal spin\nconductance and the real part of the interfacial spin-\nmixing conductance of the N jF2subsystem, ^mmm2denotes\nthe unit vector along F 2magnetization, and gij=gji\nare the components of the resulting total spin conduc-\ntance tensor. glquanti\fes the absorption of the spin\ncurrent along the direction of ^mmm2, the so-called longi-\ntudinal spin current. For metallic magnets, it is domi-\nnated by the di\u000busive spin current carried by the itin-\nerant electrons, which is dissipated over the spin re-\nlaxation length [31]. On the other hand, for insulat-\ning magnets, the longitudinal spin absorption is domi-\nnated by magnons [43, 44] and is typically much smaller\nthan for the metallic case, especially at low tempera-\ntures. Considering ^mmm2= sin\u0012^yyy+ cos\u0012^zzz(Fig. 1),\nEq. (5) yields gxx=g0\nr,gyy=g0\nrcos2\u0012+glsin2\u0012,\ngzz=g0\nrsin2\u0012+glcos2\u0012,gxy=gyx=gxz=gzx= 0,\nandgyz=gzy= (gl\u0000g0\nr) sin\u0012cos\u0012.\nRelegating the consideration of a small but \fnite spin\nrelaxation in the thin N layer to the supplemental ma-\nterial [45], we assume here that the spin current lost by\nF1is absorbed by F 2, i.e.,IIIs1=IIIs2. Imposing this spin\ncurrent conservation condition, the spin accumulation in\nN along with the currents themselves can be determined.\nWe are primarily interested in the transverse (x and y)\ncomponents of the spin current since these fully deter-\nmine the magnetization dynamics ( ^mmm\u0001^mmm= 1):\nIs1x=1\n4\u0019grgxx\ngr+gxx(\u0000~_my+mx\u0016sz);\nIs1y=1\n4\u0019\u0014grgyy\ngr+gyy(~_mx+my\u0016sz) +gyz\u0016sz(1\u0000ly)\u0015\n;\n\u0016sz=~gr(lxmx_my\u0000lymy_mx\u0000p_mx)\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001;\n(6)\nwherelx;y\u0011gxx;yy=(gr+gxx;yy) andp\u0011gyz=(gr+gyy).\nThe spin lost by F 1appears as damping in the magneti-\nzation dynamics [Eqs. (1) and (2)] [31, 35].\nWe pause to comment on the behavior of \u0016szthus ob-\ntained [Eq. (6)]. Typically, \u0016szis considered to be \frst\nor second order in the cone angle, and thus negligibly\nsmall. However, as discussed below, an essential new\n\fnding is that it becomes independent of the cone an-\ngle and large under certain conditions. For a collinear\ncon\fguration and vanishing gl,gzz=gyz= 0 results\nin ~\u0016sz\u0011\u0016sz=~!!1 [38]. Its \fnite dc value con-\ntributes to the Bloch damping [Eq. (6)] [38]. For a\nnon-collinear con\fguration, \u0016sz\u0019\u0000~grp_mx=(gzz\u0000pgyz)\n0 45 90 135 18000.10.20.30.40.50.6FIG. 3. Normalized ferromagnetic resonance (FMR)\nlinewidth of F 1for di\u000berent values of the longitudinal spin\nconductance ~ gl\u0011gl=grof NjF2bilayer. The various parame-\nters employed are ~ g0\nr\u0011g0\nr=gr= 1, \u0002 = 0:1 rad,!0= 10\u00022\u0019\nGHz, and!ax= 1\u00022\u0019GHz.grandg0\nrare the spin-mixing\nconductances of F 1jN and NjF2interfaces respectively. Only\nthe spin pumping-mediated contribution to the linewidth has\nbeen considered and is normalized to its value for the case of\nspin pumping into a perfect spin sink [31].\nand contributes to Gilbert damping via Is1y[Eq. (6)].\nThus, in general, we may express the spin accumulation\nas\u0016sz=\u0016sz0+\u0016sz1[46], where \u0016sz0is the dc value\nand\u0016sz1/_mxis the linear oscillating component. \u0016sz0\nand\u0016sz1contribute, respectively, to Bloch and Gilbert\ndamping.\nGilbert-Bloch dissipation. { Equations (1) and (6) com-\npletely determine the magnetic damping in F 1. However,\nthese equations are non-linear and cannot be captured\nwithin our linearized framework [Eqs. (2) and (3)]. The\nleading order e\u000bects, however, are linear in all but a nar-\nrow range of parameters. Evaluating these leading or-\nder terms within reasonable approximations detailed in\nthe supplemental material [45], we are able to obtain the\nGilbert and Bloch damping tensors \u0014 \u000band\u0014\n. Obtaining\nthe general result numerically [45], we present the ana-\nlytic expressions for two cases covering a large range of\nthe parameter space below.\nFirst, we consider the collinear con\fgurations in the\nlimit of ~gl\u0011gl=gr!0. As discussed above, we obtain\n~\u0016sz0\u0011\u0016sz0=~!!1 and ~\u0016sz1\u0011\u0016sz1=~!!0 [Eq. (6)].\nThus the components of the damping tensors can be di-\nrectly read from Eq. (6) as ~ \u000bxx;yy\u0011\u000bxx;yy=\u000bss=ly;x=\ng0\nr=(gr+g0\nr) = ~g0\nr=(1+~g0\nr);and~\nxx;yy\u0011\nxx;yy=(\u000bss!) =\n\u0000lx;y\u0016sz0=(~!) =\u0000g0\nr=(gr+g0\nr) =\u0000~g0\nr=(1 + ~g0\nr). Here,\nwe de\fned ~ g0\nr\u0011g0\nr=grand\u000bss\u0011~grj\rj=(4\u0019MsV) is the\nGilbert constant for the case of spin-pumping into an\nideal spin sink [31, 35]. Substituting these values in Eq.\n(3), we \fnd that the linewidth, or equivalently damping,\nvanishes. This is understandable since the system we\nhave considered is not able to relax the z component of\nthe spin at all. There can, thus, be no net contribution to4\nFIG. 4. Normalized FMR linewidth of F 1for very small ~ gl.\nThe squares and circles denote the evaluated points while the\nlines are guides to the eye. The linewidth increases from being\nnegligible to its saturation value as \u0012becomes comparable to\nthe average cone angle \u0002. ~ g0\nr= 1,!0= 10\u00022\u0019GHz, and\n!ax= 1\u00022\u0019GHz.\nmagnetic damping. \u0016sz0accumulated in N opposes the\nGilbert relaxation via a negative Bloch contribution [38].\nThe latter may also be understood as an anti-damping\nspin transfer torque due to the accumulated spin [6].\nNext, we assume the system to be in a non-collinear\ncon\fguration such that ~ \u0016sz0!0 and may be disre-\ngarded, while ~ \u0016sz1simpli\fes to:\n~\u0016sz1=\u0000_mx\n!(~gl\u0000~g0\nr) sin\u0012cos\u0012\n~g0r~gl+ ~glcos2\u0012+ ~g0rsin2\u0012; (7)\nwhere ~gl\u0011gl=grand ~g0\nr\u0011g0\nr=gras above. This in turn\nyields the following Gilbert parameters via Eq. (6), with\nthe Bloch tensor vanishing on account of ~ \u0016sz0!0:\n~\u000bxx=~g0\nr~gl\n~g0r~gl+ ~glcos2\u0012+ ~g0rsin2\u0012;~\u000byy=~g0\nr\n1 + ~g0r;(8)\nwhere ~\u000bxx;yy\u0011\u000bxx;yy=\u000bssas above. Thus, ~ \u000byyis\u0012-\nindependent since ^mmm2lies in the y-z plane and the x-\ncomponent of spin, the absorption of which is captured\nby ~\u000byy, is always orthogonal to ^mmm2. ~\u000bxx, on the other\nhand, strongly varies with \u0012and is generally not equal\nto ~\u000byyhighlighting the tensorial nature of the Gilbert\ndamping.\nFigure 2 depicts the con\fgurational dependence of nor-\nmalized damping parameters. The Bloch parameters are\nappreciable only close to the collinear con\fgurations on\naccount of their proportionality to \u0016sz0. The\u0012range over\nwhich they decrease to zero is proportional to the cone\nangle \u0002 [Eq. (6)]. The Gilbert parameters are described\nsu\u000eciently accurately by Eq. (8). The linewidth [Eq.\n(3)] normalized to its value for the case of spin pump-\ning into a perfect spin sink has been plotted in Fig. 3.\nFor low ~gl, the Bloch contribution partially cancels the\nGilbert dissipation, which results in a smaller linewidthclose to the collinear con\fgurations [38]. As ~ glincreases,\nthe relevance of Bloch contribution and \u0016sz0diminishes,\nand the results approach the limiting condition described\nanalytically by Eq. (8). In this regime, the linewidth\ndependence exhibits a maximum for either collinear or\northogonal con\fguration depending on whether ~ gl=~g0\nris\nsmaller or larger than unity. Physically, this change in\nthe angle with maximum linewidth is understood to re-\n\rect whether transverse or longitudinal spin absorption\nis stronger.\nWe focus now on the case of very low ~ glwhich can\nbe realized in structures with electrically-insulating mag-\nnets. Figure 4 depicts the linewidth dependence close to\nthe collinear con\fgurations. The evaluated points are\nmarked with stars and squares while the lines smoothly\nconnect the calculated points. The gap in data for very\nsmall angles re\rects the limited validity of our linear\ntheory, as discussed in the supplemental material [45].\nAs per the limiting case ~ gl!0 discussed above, the\nlinewidth should vanish in perfectly collinear states. A\nmore precise statement for the validity of this limit is\nre\rected in Fig. 4 and Eq. (6) as ~ gl=\u00022!0. For su\u000e-\nciently low ~ gl, the linewidth changes sharply from a neg-\nligible value to a large value over a \u0012range approximately\nequal to the cone angle \u0002. This shows that systems com-\nprised of magnetic insulators bearing a very low ~ glare\nhighly tunable as regards magnetic/spin damping by rel-\natively small deviation from the collinear con\fguration.\nThe latter may be accomplished electrically by employ-\ning magnetoelectric material [41] for F 2or via current\ndriven spin transfer torques [6, 9, 47].\nDiscussion. { Our identi\fcation of damping contribu-\ntions as Gilbert-like and Bloch-like [Eq. (6)] treats \u0016sz\nas an independent variable that may result from SHE,\nfor example. When it is caused by spin pumping cur-\nrent and\u0016sz/!, this Gilbert-Bloch distinction is less\nclear and becomes a matter of preference. Our results\ndemonstrate the possibility of tuning the magnetic damp-\ning in an active magnet via the magnetization of a passive\nmagnetic layer, especially for insulating magnets. In ad-\ndition to controlling the dynamics of the uniform mode,\nthis magnetic `gate' concept [48] can further be employed\nfor modulating the magnon-mediated spin transport in a\nmagnetic insulator [43, 44]. The anisotropy in the result-\ning Gilbert damping may also o\u000ber a pathway towards\ndissipative squeezing [49] of magnetic modes, comple-\nmentary to the internal anisotropy-mediated `reactive'\nsqueezing [50, 51]. We also found the longitudinal accu-\nmulated spin, which is often disregarded, to signi\fcantly\na\u000bect the dynamics. This contribution is expected to\nplay an important role in a wide range of other phenom-\nena such as spin valve oscillators.\nSummary. { We have investigated the angular modu-\nlation of the magnetic damping in a free layer via control\nof the static magnetization in the \fxed layer of a spin\nvalve device. The damping can be engineered to become5\nlarger for either collinear or orthogonal con\fguration by\nchoosing the longitudinal spin conductance of the \fxed\nlayer smaller or larger than its spin-mixing conductance,\nrespectively. The control over damping is predicted to\nbe sharp for spin valves made from insulating magnets.\nOur results pave the way for exploiting magneto-damping\ne\u000bects in spin valves.\nAcknowledgments. { We acknowledge \fnancial support\nfrom the Research Council of Norway through its Centers\nof Excellence funding scheme, project 262633, \\QuSpin\",\nand from the Swedish Research Council, project 2018-\n03526, and Stiftelse Olle Engkvist Byggm astare.\n\u0003akashdeep.kamra@ntnu.no\n[1] Albert Fert, \\Nobel lecture: Origin, development, and\nfuture of spintronics,\" Rev. Mod. Phys. 80, 1517{1530\n(2008).\n[2] S. Parkin, Xin Jiang, C. Kaiser, A. Panchula, K. Roche,\nand M. 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Lett. 116, 146601 (2016).\n[51] Akashdeep Kamra, Utkarsh Agrawal, and Wolfgang\nBelzig, \\Noninteger-spin magnonic excitations in untex-\ntured magnets,\" Phys. Rev. B 96, 020411 (2017).\n[52] A.I. Akhiezer, V.G. Bar'iakhtar, and S.V. Peletminski,\nSpin waves (North-Holland Publishing Company, Ams-\nterdam, 1968).1\nSupplemental material with the manuscript Anisotropic and controllable\nGilbert-Bloch dissipation in spin valves by\nAkashdeep Kamra, Dmytro M. Polishchuk, Vladislav Korenivski and Arne Brataas\nCOLLINEAR CONFIGURATION WITHOUT LONGITUDINAL SPIN RELAXATION\nIn order to appreciate some of the subtleties, we \frst examine the collinear con\fguration in the limit of vanishing\nlongitudinal spin conductance. \u0012= 0;\u0019andgl= 0 imply the following values for the various parameters:\ngxx=gyy=g0\nr; g zz=gyz=p= 0; lx;y=g0\nr\ngr+g0r\u0011l; (S1)\nwhence we obtain:\n\u0016sz\n~=(mx_my\u0000my_mx)\nm2x+m2y; (S2)\n=!0+!ax\n1 +!ax\n2!0[1\u0000cos(2!t)]; (S3)\nwhere we have assumed magnetization dynamics as given by the Landau-Lifshitz equation without damping, and\nthe phase of mxis treated as the reference and set to zero. In order to obtain analytic expressions, we make the\nassumption !ax=!0\u001c1 such that we have:\n\u0016sz=\u0016sz0+\u0016sz2; with (S4)\n\u0016sz0=~\u0010\n!0+!ax\n2\u0011\n; (S5)\n\u0016sz2=~!ax\n4\u0000\ne\u0000i2!t+ei2!t\u0001\n: (S6)\nIn contrast with our assumptions in the main text, a term oscillating with 2 !appears. Furthermore, it yields\ncontributions to the Bloch damping via products such as my\u0016sz, which now have contributions oscillating at !due\nto the\u0016sz0as well as\u0016sz2. We obtain:\n~\u000bxx= ~\u000byy=l; (S7)\n~\nxx=\u0000l!0+3!ax\n4\n!0+!ax\n2and ~\nyy=\u0000l!0+!ax\n4\n!0+!ax\n2; (S8)\nsubstituting which into Eq. (3) from the main text yields a vanishing linewidth and damping. This is expected from\nthe general spin conservation argument that there can be no damping in the system if it is not able to dissipate the\nz-component of the spin. In fact, in the above considerations, \u0016sz2contributed with the opposite sign to ~\nxxand\n~\nyy, and thus dropped out of the linewidth altogether. This also justi\fes our ignoring this contribution in the main\ntext.\nFigure 1 depicts the dependence of the accumulated z-polarized spin and the normalized linewidth for small but\n\fniteglin the collinear con\fguration. The accumulated longitudinal (z-polarized) spin increases with the cone angle\nand the linewidth accordingly decreases to zero [38].\nNUMERICAL EVALUATION\nDespite the additional complexity in the previous section, we could treat the dynamics within our linearized frame-\nwork. However, in the general case, \u0016szhas contributions at all multiples of !and cannot be evaluated in a simple\nmanner. A general non-linear analysis must be employed which entails treating the magnetization dynamics numer-\nically altogether. Such an approach prevents us from any analytic description of the system, buries the underlying\nphysics, and is thus undesirable.\nFortunately, the e\u000bects of non-linear terms are small for all, but a narrow, range of parameters. Hence, we make\nsome simplifying assumptions here and continue treating our system within the linearized theory. We only show2\n10-310-210-100.10.20.30.40.5\nFIG. 1. Ferromagnetic resonance linewidth and the dc spin accumulation created in the spacer as a function of the average\ncone angle in the collinear con\fguration. Depending on ~ gl, there is a complementary transition of the two quantities between\nsmall and large values as the cone angle increases. ~ g0\nr= 1,!0= 10\u00022\u0019GHz, and!ax= 1\u00022\u0019GHz.\nresults in the parameter range where our linear analysis is adequate. Below, we describe the numerical routine for\nevaluating the various quantities. To be begin with the average cone angle \u0002 is de\fned as:\n\u00022=\nm2\nx+m2\ny\u000b\n; (S9)\nwhereh\u0001idenotes averaging over time. The spin accumulation is expressed as \u0016sz=\u0016sz0+\u0016sz1with:\n\u0016sz0=*\n~gr(lxmx_my\u0000lymy_mx\u0000p_mx)\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001+\n; (S10)\n\u0016sz1=\u0000*\ngrp\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001+\n~_mx: (S11)\nThe above expressions combined with the equations for the spin current \row (Eqs. (6) in the main text) directly yield\nthe Gilbert and Bloch damping tensors.\nVARIATION WITH ADDITIONAL PARAMETERS\nHere, we discuss the dependence of the FMR linewidth on the easy-plane anisotropy and the spin-mixing conduc-\ntanceg0\nrof the NjF2interface. The results are plotted in Fig. 2. A high easy-plane anisotropy is seen to diminish\nthe con\fguration dependence of the linewidth and is thus detrimental to the dissipation tunability. The easy-axis\nanisotropy, on the other hand, is absorbed in !0and does not need to be examined separately. We also see an increase\nin the con\fguration dependence of the damping with an increasing g0\nr. This is understood simply as an increased\ndamping when the spin is absorbed more e\u000eciently due to a larger g0\nr. The damping is expected to reach the case of\nspin pumping into a perfect spin sink in the limit of ~ g0\nr!1 and\u0012= 0;\u0019.\nEFFECT OF SPIN RELAXATION IN THE SPACER LAYER\nWe now address the role of the small but \fnite spin relaxation in the non-magnetic spacer layer. To this end, we\nconsider that a part of the spin current injected into N by F 1is lost as the \\spin-leakage current\" IIIsl, as depicted in\nFig. 3, such that IIIs1=IIIs2+IIIsl. In order to evaluate the leakage, we consider the spin di\u000busion equation in N which\nreads [31]:\nD@2\nx\u0016\u0016\u0016s=\u0016\u0016\u0016s\n\u001csf; (S12)3\n0 45 90 135 18000.10.20.30.40.50.6\n(a)\n0 45 90 135 18000.20.40.60.81 (b)\nFIG. 2. Normalized ferromagnetic resonance (FMR) linewidth of F 1. (a) Same as Fig. 3 in the main text with additional plots\nfor a large easy-plane anisotropy. (b) Linewidth dependence for di\u000berent spin-mixing conductances of N jF2interface. The\nparameters employed are the same as Fig. 2 in the main text.\nFIG. 3. Schematic depiction of the spin currents \rowing through the device, including the spin-leakage current IIIslthat is lost\non account of a \fnite spin relaxation in the spacer layer N.\nwhereDand\u001csfare di\u000busion constant and spin-\rip time, respectively. We now integrate the equation over the\nthickness of N:\nZ\nd(D@x\u0016\u0016\u0016s) =Zd\n0\u0016\u0016\u0016s\n\u001csfdx: (S13)\nSince the N-layer thickness dis typically much smaller than the spin di\u000busion length in N (e.g., a few nm versus a\nfew hundred nm for Cu), we treat \u0016\u0016\u0016son the right hand side as a constant. Furthermore, in simplifying the left hand\nside, we invoke the expression for the spin current [31]: IIIs= (\u0000~NSD=2)@x\u0016\u0016\u0016s, withNthe one-spin density of states\nper unit volume and Sthe interfacial area. Thus, we obtain\n2\n~NS(IIIs1\u0000IIIs2) =d\n\u001csf\u0016\u0016\u0016s; (S14)\nwhich simpli\fes to the desired relation IIIs1=IIIs2+IIIslwith\nIIIsl=~NVN\n2\u001csf\u0016\u0016\u0016s\u0011gsl\n4\u0019\u0016\u0016\u0016s; (S15)\nwhereVNis the volume of the spacer layer N.\nIt is easy to see that accounting for spin leakage, as derived in Eq. (S15), results in the following replacements to\nEqs. (6) of the main text:\ngxx!gxx+gsl; g yy!gyy+gsl; g zz!gzz+gsl: (S16)4\nSince all our speci\fc results are based on Eqs. (6) of the main text, this completes our assessment of the role played\nby spin relaxation in N. Physically, this new result means that the condition for no spin relaxation in the system,\nwhich was previously treated as gl!0, is now amended to gl+gsl!0. This, however, does not a\u000bect the generality\nand signi\fcance of the key results presented in the main text."    },    {        "title": "1811.04094v2.Switching_of_biaxial_synthetic_antiferromagnets__a_micromagentic_study.pdf",        "content": "Switching of biaxial synthetic antiferromagnets: a micromagentic study\nMichael S. Ackermann1, 2and Satoru Emori3,a)\n1)Academy of Integrated Science, Virginia Tech, Blacksburg, VA 24061, USA\n2)Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA\n3)Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n(Dated: November 21, 2018)\nWe simulate the switching behavior of nanoscale synthetic antiferromagnets (SAFs), inspired by recent\nexperimental progress in spin-orbit-torque switching of crystal antiferromagnets. The SAF consists of two\nferromagnetic thin \flms with in-plane biaxial anisotropy and interlayer exchange coupling. Staggered \feld-like\nRashba spin-orbit torques from the opposite surfaces of the SAF induce a canted net magnetization, which\ntriggers an orthogonal torque that drives 90\u000eswitching of the N\u0013 eel vector. Such dynamics driven by the\n\feld-like spin-orbit torque allows for faster switching with increased Gilbert damping, without a signi\fcant\ndetrimental increase of the threshold switching current density. Our results point to the potential of SAFs as\nmodel systems, based on simple ferromagnetic metals, to mimic antiferromagnetic device physics.\nI. INTRODUCTION\nAntiferromagnets are considered promising material\nplatforms for ultrafast spintronic information-technology\ndevices that are highly stable against external\nmagnetic \felds1{3. Recent experimental studies\nhave demonstrated switching of the antiferromagnetic\norder (N\u0013 eel vector) between two orthogonal states in\nepitaxial antiferromagnetic conductors of CuMnAs4\nand Mn 2Au5. This N\u0013 eel switching is driven by a\ncurrent-induced \\\feld-like\" spin-orbit torque (SOT)\nthat acts locally in opposite directions on the two\nmagnetic sublattices of the antiferromagnet6,7. The key\ningredient for this non-vanishing \feld-like N\u0013 eel SOT is\nthe inversion asymmetry around each magnetic atom\n(i.e., Mn) that is intrinsic to the speci\fc crystal structure\nof CuMnAs and Mn 2Au. However, the synthesis of\nepitaxial CuMnAs and Mn 2Au with the correct crystal\nstructure may not be straightforward, and so far no\nother conductive collinear antiferromagnets with the\ncompatible crystal structure for the \feld-like N\u0013 eel\nSOT have been realized4,5. It has also been shown\nthat epitaxial antiferromagnetic insulator NiO can\nbe switched by a SOT from an adjacent metal with\nstrong spin-orbit coupling (e.g., Pt)8,9. In this case, the\nlimitation may be the relatively small magnetoresistance\nsignal (i.e., spin-Hall magnetoresistance10) to read out\nthe N\u0013 eel vector state. Furthermore, it is generally\ndi\u000ecult to apply conventional laboratory-based\ncharacterization techniques (e.g., magnetometry,\nferromagnetic resonance, magnetic microscopy, etc.) to\nstudy the fundamental properties of antiferromagnets.\nThese points above may constitute a serious obstacle\nto studying and engineering viable materials for\nantiferromagnetic spintronics.\nHere, we study by micromagnetic simulations\nthe switching behavior of synthetic antiferromagnets\n(SAFs)11as a model system analogous to intrinsic\na)Electronic mail: semori@vt.educrystal antiferromagnets. The SAFs consist of two\nin-plane biaxial ferromagnetic metals (FMs) whose\nmagnetizations are locked antiparallel to each other\nby interlayer exchange coupling (e.g., through the\nRuderman-Kittel-Kasuya-Yosida mechanism across a\nnon-ferromagnetic metal such as Cr or Ru)12,13. Such\nbiaxial FMs can be readily synthesized by epitaxial\ngrowth on a cubic single-crystal substrate, e.g., body-\ncentered-cubic Fe on MgO (001) or GaAs (001)14{18.\nThis SAF structure has two orthogonal easy axes in\nthe \flm plane de\fned by cubic magnetocrystalline\nanisotropy. These two digital states, represented by\northogonal N\u0013 eel vector orientations, can be read through\nanisotropic magnetoresistance4,5; e.g., when the N\u0013 eel\nvector is oriented parallel (transverse) to the sense\ncurrent, the SAF exhibits a higher (lower) electrical\nresistance19,20. Switching is achieved when opposite local\n\felds are applied orthogonal to the magnetizations of\nthe two FM layers (Fig. 1), analogous to the opposite\nlocal \felds applied to the two sublattices in CuMnAu and\nMn2Au4,5. In the SAF, the required symmetry breaking\nfor such opposite local \felds occurs at the layer interfaces.\nWe simulate the e\u000bect of the interfacial Rashba spin-\norbit \felds (\feld-like SOTs)21{23arising from the top and\nbottom surfaces of the SAF interfaced with, e.g., an oxide\ncapping layer and substrate24,25. Our study therefore\nFM 1\nFM 2\nx ycurrentBso\nx y\nFigure 1. Schematic of the synthetic antiferromagnet (SAF)\nconsisting of two ferromagnetic metal (FM) layers. The\nSAF has two orthogonal easy axes along the x- andy-\naxes. The current-induced spin-orbit \feld Bsoswitches the\nantiferromagnetic order (N\u0013 eel vector) from the x- toy-axis.arXiv:1811.04094v2  [cond-mat.mes-hall]  22 Nov 20182\nsuggests a possible pathway for simple SAF spintronic\ndevices that inherit some of the switching behavior of\nantiferromagnets.\nAdvantages of SAFs have been reported previously\nfor engineering stable pinned and free layers in spin\nvalves26{28, rapid motion of domain walls29{32, and SOT-\ndriven switching of perpendicular magnetization33{35.\nIn contrast with these prior devices based on 180\u000e\nswitching, we emphasize that our proposed approach is\nbased on 90\u000eswitching. To the best of our knowledge,\nour study is the \frst to numerically examine such\northogonal switching in SAFs, speci\fcally driven by \feld-\nlike SOT. The orthogonal orientation between the initial\nmagnetization and the spin-orbit \feld in each FM layer\n(Fig. 1) maximizes the torque on the magnetization\nand enables rapid switching. This switching scheme\ndriven by the \feld-like torque also allows for faster\nswitching by increasing the Gilbert damping parameter\nwithout an adverse increase of the threshold switching\ncurrent density. Our simulations indicate that SAFs\nwith realistic material parameters are robust against\nup to\u00181 T of external magnetic \feld and can be\nswitched in\u00180.1 ns at a reasonable current density\nof<\u00181011A/m2. We also note that this proposed\ndevice scheme is operated by two orthogonal current\nlines, analogous to four-terminal toggle magnetic random\naccess memories (MRAMs)36. The use of the \feld-\nlike SOT as proposed here, instead of Oersted \felds in\ntoggle MRAMs, may enable alternative scalable memory\ndevices.\nII. MODEL PARAMETERS\nMagnetic switching was simulated using the Mumax3\nmicromagnetics package37. A series of square samples\nwith di\u000berent widths of 26 to 400 nm were studied\nwith a lateral cell size of 2 or 4 nm. Each FM layer\nhad the following \fxed properties: thickness tFM=\n1.5 nm, exchange constant Aex= 20 pJ/m, and the\ncubic anisotropy constant Kc= 30 kJ/m3with the easy\naxes parallel to the square edges. In most simulations,\nwe set the saturation magnetization Msat 1700 kA/m\n(typical value for Fe), the Gilbert damping parameter\n\u000bat 0.01 (typical value for nanometer-thick FMs), and\nthe interlayer exchange coupling energy density Jexat\n\u00000.2 or\u00001 mJ/m2(where the negative sign indicates\nantiferromagnetic interlayer coupling). We note that the\nvalues ofJexused here are similar to those experimentally\nachieved in SAFs consisting of FMs13,32,38,39.\nIII. RESULTS AND DISCUSSION\nA. Stability against global magnetic \feld\nWe \frst compare the stability of the magnetization\nstate against a global external \feld in the SAFs with\nsingle layerJex= -0.2 mJ/m2Jex= -1 mJ/m2(a)\n(b)\nJex= -1 mJ/m2single layer\nsize [nm]Bsw[T]\nBy[nm]myFigure 2. (a) External magnetic \feld Bswrequired to switch\nthe magnetization from the x-axis toy-axis for samples\nof di\u000berent lateral sizes. (b) Equilibrium magnetization\ncomponent myalong they-axis versus external magnetic \feld\nBy. Here the lateral sample size is 52 nm.\ntheir single-layer FM counterparts. With the initial\nmagnetizations set parallel to the x-axis (i.e.,mtop\nx= 1,\nmbot\nx=\u00001, andmtop\ny=mbot\ny= 0), an external\nmagnetic \feld Byalong the + y-direction was applied.\nThe critical switching \feld Bswis de\fned as Byrequired\nto pull the total magnetization, m=1\n2(mtop+mbot),\nto they-direction, i.e., my>0:99. Figure 2(a)\nshows that the single-layer FMs switch at low values\nofBsw<\u00180:01 T, indicating that these samples are\nvulnerable to spontaneous switching from external stray\n\felds. As evidenced by the substantial variation in Bsw{\nas much as an order of magnitude { with lateral size, the\nswitching behavior of the single-layer FMs is also heavily\nimpacted by the device geometry, e.g., due to dipolar\n\felds from the sample edges. A slight variation in the\nshape or edge defects of single-layer in-plane FM devices\ncan lead to a random distribution of switching thresholds.\nThe SAFs show about an order of magnitude greater\nBswthan the single-layer FMs. As shown in Fig. 2(a),\nBswis enhanced with increasing Jex. ForJex=\n\u00001 mJ/m2readily achievable in realistic SAFs13,32,38,39,\nan external \feld of nearly 1 T is required to orient\nthe magnetization along the y-direction. As shown in\nFig. 2(b), while the single-layer FM undergoes abrupt\nswitching at low By, the SAF undergoes a gradual\nmagnetization rotation until the magnetizations of the\ntwo layers are fully oriented along the \feld direction. We\nalso note that Bswonly varies by a factor of \u00192 with\nthe lateral sample dimensions of the SAFs (Fig. 2(a)),\nindicating that the dipolar \felds from the sample edges3\nplay relatively little role. The SAFs are therefore shown\nto be signi\fcantly more stable against disturbances from\nexternal magnetic \felds, and this stability is largely\nindependent of the sample geometry. We emphasize that\nthe stability at \felds of \u00180.1-1 T can be achieved in\nSAFs consisting of simple FMs (e.g., Fe), in contrast\nwith intrinsic crystal antiferromagnetic compounds4,5for\nwhich epitaxial growth is more challenging.\nB. Threshold spin-orbit \feld for switching\nHaving demonstrated the stability of the SAFs, we\ncompute how much spin-orbit \feld is required to switch\nthe antiferromagnetic order in the SAFs between the x-\nandy-axes (e.g., Fig. 1). For example, the magnetization\nof the top (bottom) FM layer, initially oriented along\nthe +x-direction (\u0000x-direction), sees an e\u000bective current-\ninduced \feld pointing along the - y-direction (+ y-\ndirection). When the magnitude of this e\u000bective \feld is\nsu\u000eciently large, the magnetization overcomes the cubic\nanisotropy energy barrier and switches from the x-axis\ntoy-axis. Unlike a global magnetic \feld (Sec. III A)\nthat cants the magnetizations toward the parallel state\nand hence results in a large interlayer antiferromagnetic\nexchange energy penalty, the local spin-orbit \feld rotates\nthe magnetization of each layer while maintaining the\nmostly antiparallel magnetization alignment across the\nlayers. We discuss the details of the switching process in\nSec. III C.\nWe de\fne the threshold Bth\nsoas the e\u000bective local \feld\nrequired to switch the N\u0013 eel vector, l=1\n2(mtop\u0000mbot),\nto they-axis, i.e.,jlyj>0:99. We simulated two cases\nwhere (1) only the top layer sees the spin-orbit \feld (and\nthe bottom layer magnetization is dragged by the top\nlayer magnetization), and (2) the top and bottom layers\nsee the spin-orbit \feld in opposite directions (Fig. 1).\nThese two con\fgurations of the spin-orbit \feld would\narise by enabling an interfacial Rashba \feld-like SOT at\n(1) only the top surface of the SAF and (2) both the top\nand bottom surfaces of the SAF.\nFigure 3 plots the computed Bth\nsoagainst the SAF\nlateral size. Bth\nsois somewhat dependent on the lateral\nsample size, increasing by nearly a factor of 2 when\nthe lateral sample size is decreased from 400 to 26 nm,\nas the mode of switching transitions from incoherent\nto coherent. More importantly, we \fnd a factor of 2\nreduction in Bth\nsowith the current-induced \feld active at\nboth the top and bottom surfaces compared to just one.\nThis \fnding con\frms that the spin-orbit \feld is additive\nand that engineering the Rashba e\u000bect at both surfaces\nwould lead to a more e\u000ecient SAF device. It should\nalso be noted that jJexjdoes not a\u000bect Bth\nso, suggesting\nthat biaxial SAFs can be switched e\u000eciently regardless of\nthe strength of interlayer exchange coupling. Here, since\nthe cubic magnetic anisotropy energy density Kc\u0018104\nJ/m3is signi\fcantly smaller than the interlayer exchange\nenergy densityjJexj=tFM\u0018105\u0000106J/m3, the energy\nBso: 2 layers, Jex= -1 mJ/m2\nBso: 2 layers, Jex= -0.2 mJ/m2Bso: 1 layer, Jex= -1 mJ/m2\nBso: 1 layer, Jex= -0.2 mJ/m2Bso[mT]th\nsize [nm]Figure 3. Threshold current-induced spin-orbit \feld Bth\nso\nrequired to switch the N\u0013 eel vector from the x-axis toy-axis\nfor SAF samples with di\u000berent lateral dimensions.\nbarrier for 90\u000eswitching of the N\u0013 eel vector is mostly\ndetermined by Kcrather thanjJexj. This \fnding is\nconsistent with a prior study of 180\u000eswitching in SAFs,\nwhere the energy barrier is governed by uniaxial magnetic\nanisotropy28. However, we show in the next subsection\n(Sec. III C) that the interlayer exchange coupling can\nin\ruence the switching speed by generating a torque on\nthe N\u0013 eel vector.\nProvided that the spin-orbit \feld arises entirely from\nthe interfacial Rashba-Edelstein e\u000bect, we can estimate\nthe critical threshold current density for switching Jth\nfromBth\nsowith40,41\nJth=\u0016BBth\nsoMs\n\u000bRP; (1)\nwhere\u0016Bis the Bohr magneton, \u000bRis the Rashba\nparameter, and Pis the e\u000bective spin polarization\n(proportional to the exchange interaction between\nthe Rashba-induced spin accumulation and the FM\nmagnetization). For Jthto be comparable to\n<\u00181011A/m2recently reported in antiferromagnetic\nmemory prototypes4,5,8,9, the product \u000bRPwould need\nto be>\u00180:1 eV\u0001\u0017A. This is reasonably achieved with\nRashba parameters similar to those reported in oxide\nsystems24,25,42{46. While the \feld-like SOT has not\nreceived as much attention (compared to the damping-\nlike SOT) for FM-based device applications, an enhanced\ninterfacial Rashba spin-orbit \feld would be a robust\ndriving force to e\u000eciently switch a biaxial SAF memory.\nC. Time-dependence of switching\nFinally, we discuss the mechanism and time\ndependence of SOT-driven switching in the SAFs. In\nthe following, switching is driven by opposite local spin-\nobit \felds acting on the top and bottom layers. The\ninitial orthogonal con\fguration between the spin-orbit\n\feld (e.g., Btop\nsojj\u0000^y,Bbot\nsojj+^y) and the magnetization\n(mtopjj+^x,mbotjj\u0000^x) in each layer maximizes the\ntorque that initiates the switching process. When this4\nmx,mymz(d)\n(e)lz-ly\nlxτdemag\nBdemagBdemagτdemag\nBsoSOT\nmbotτexch\nBexchBexchτexch\nBso(a) (b) (c)\nz\nx ySOT\nmtop\nlx, ly, lzmx, my, mz\ntime [ns]\nFigure 4. (a,b,c) Schematics of the torques due to the (a)\nspin-orbit \feld Bso, (b) demagnetizing \feld Bdemag , and\n(c) interlayer antiferromagnetic exchange \feld Bexch. (d,e)\nTime traces of the (d) N\u0013 eel vector ( lx,ly,lz) and (e) total\nmagnetization ( mx,my,mz) atBso= 6 mT in the 52-nm-\nwide SAF sample with Ms= 1700 kA/m, Jex= -1 mJ/m2,\u000b\n= 0.01.\nSOT (\u0000j\rjmtop\u0002Btop\nso,\u0000j\rjmbot\u0002Bbot\nso) is turned\non, the magnetization is tilted out of the \flm plane in the\nsame direction in both layers (Fig. 4(a)), thereby giving\nrise to a \fnite z-component in the total magnetization\nmz. This out-of-plane canting then yields two torques\nalong\u0000^yin the top layer (+ ^yin the bottom layer):\n(1) a torque due to the out-of-plane demagnetizing\n\feld (Fig. 4(b)) and (2) a torque due to the interlayer\nantiferromagnetic exchange penalty (Fig. 4(c)). These\ndemagnetizing and antiferromagnetic-exchange torques\nhave the same symmetry to drive 90\u000eswitching of the\nN\u0013 eel vector lfrom thex-axis to the y-axis.\nAn exemplary time evolution of the N\u0013 eel vector and\ntotal magnetization is shown in Fig. 4(d,e). The\ninitial rise in mzcon\frms the out-of-plane tilting of the\nmagnetization, while mx;y\u00190 indicates that the in-plane\nmagnetization components remain compensated between\nthe two layers. Moreover, the damped oscillation of l\n(Fig. 4(d)) exhibits a phase o\u000bset of \u0019=2 with respect\ntomz(Fig. 4(e)), i.e., the time rate of change of lis\nmaximized whenjmzjexhibits a maximum. This relationcon\frms that the torque on lis indeed related to the\nmagnetization canting jmzj.\nThe relative contributions of the torques can be tuned\nby varying the saturation magnetization Ms, since a\nsmaller value of Msshould decrease the demagnetizing\ntorque contribution. Figure 5 compares the time-\ndependence of switching for SAFs with Ms= 1700 kA/m\nand 170 kA/m, each with di\u000berent strengths of interlayer\nexchange coupling Jex. For each Ms, the magnitude of\nthe spin-orbit \feld Bsois chosen to be slightly above\nthe threshold for switching Bth\nso. In the case of Ms\n= 1700 kA/m, the switching speed changes only by\na factor of\u00192 whenJexis varied by a factor of 25\n(Fig. 5(a,b)). Evidently, for SAFs consisting of high-\nmoment FMs (e.g., Fe), the out-of-plane demagnetizing\ntorque dominates the switching process, whereas the\nantiferromagnetic-exchange torque plays a relatively\nminor role. We thus \fnd that although SAFs have\nzero net magnetization at equilibrium, their dynamics\ncan be driven predominantly by the demagnetizing \feld\nfrom nonequilibrium magnetization. By contrast, in\nthe case of Ms= 170 kA/m, increasing jJexjresults\nin an order of magnitude faster switching (Fig. 5(c,d)),\nindicating that the antiferromagnetic-exchange torque\nplays a relatively major role when the constituent FMs\nhave low magnetization.\nEnhancing the interlayer exchange coupling is not\nnecessarily an e\u000bective way to speed up switching\nin SAFs because (1) the exchange torque may not\nbe the dominant driving mechanism for switching\nif the constituent FMs have high Msand, (2)\neven if the exchange torque dominates, it would\nbe practically di\u000ecult to increase jJexjwell above\n\u00181 mJ/m2. We therefore explore an alternative method\nof enhancing the switching speed by increasing the\nGilbert damping parameter \u000b, which is experimentally\nmore straightforward (e.g., through alloying the FM\nwith a small concentration of rare-earth metal47).\nThe threshold current density for switching driven\nby the \feld-like torque is not signi\fcantly a\u000bected\nby damping48. This is in contrast with coherent\nswitching of a single-domain in-plane nanomagnet\ndriven by a damping-like torque, where the threshold\ncurrent density is inversely proportional to the damping\nparameter49; since lower damping would prolong\nthe magnetization oscillations before settling along\nthe equilibrium orientation, damping-like-torque-driven\nswitching leads to a trade-o\u000b between reducing the\npower consumption (threshold switching current density)\nand the switching time. Our proposed scheme of\nutilizing the \feld-like torque in the biaxial SAF allows\nfor speeding up switching by increasing the damping\nparameter, without adversely a\u000becting the threshold\nswitching current density.\nFigure 6 shows the in\ruence of the damping parameter\n\u000bon the time evolutions of landmin an SAF. The\noscillations around the y-axis are signi\fcantly suppressed\nat higher values of \u000bin Fig. 6. We note, however, that5\nJex= -5 mJ/m2\nJex= -1 mJ/m2\nJex= -0.2 mJ/m2Ms= 1700 kA/m\n(Bso= 6 mT)Ms= 170 kA/m\n(Bso= 74 mT)(a) (c)\n(d)Jex= -5 mJ/m2\nJex= -1 mJ/m2\nJex= -0.2 mJ/m2\n(b)\ntime [ns]mz -ly\n-ly\nmz\ntime [ns]\nFigure 5. Time traces of the (a,c) N\u0013 eel vector component lyand (b,d) total magnetization component mzfor samples with\n(a,b)Ms= 1700 kA/m and (c,d) Ms= 170 kA/m at di\u000berent strengths of interlayer exchange coupling Jex. The magnitude\nof the spin-orbit \feld Bsois chosen to be slightly above the threshold for switching. The sample width is 52 nm and \u000b= 0:01.\n(a)\n(b)= 0.05\n= 0.01\n= 0.002Ms= 1700 kA/m\n(Bso= 6 mT)\nmz -ly\ntime [ns]\nFigure 6. Time traces of the (a) N\u0013 eel vector component ly\nand (b) total magnetization component mzatBso= 6 mT in\nthe 52-nm-wide SAF sample for with Ms= 1700 kA/m and\nJex= -1 mJ/m2at di\u000berent Gilbert damping parameters \u000b.\nby increasing \u000bfurther to>\u00180:1, the switching process\nbecomes overdamped and is hence slowed down. These\nresults indicate that the switching time is minimized with\na moderately large value of \u000b. The time traces shown in\nFig. 6 are obtained at Bso= 6 mT, which is only slightly\nabove the threshold for switching for the simulated 52-\nnm-wide device (Fig. 3). The switching time can be\ndecreased further with a greater spin-orbit \feld (currentdensity). Our results thus suggest that 90\u000eswitching in\nan SAF device can be accomplished in <\u00180:1 ns at a\nreasonable current density of \u00181011A/m2, provided a\nsu\u000eciently strong interfacial Rashba-Edelstein e\u000bect (as\ndiscussed in Sec. III B). While \u00180.1-ns switching has been\ndemonstrated for SOT-driven perpendicular anisotropy\nmemories, the required current density exceeds 1012\nA/m250. Biaxial SAFs may therefore be an attractive\npower-e\u000ecient alternative to conventional spintronic\nmemory platforms.\nIV. SUMMARY\nWe have demonstrated by micromagnetic\nsimulations that biaxial SAFs { consisting of two\nantiferromagnetically-coupled FMs { are stable against\nlarge external magnetic \felds and can be switched\ne\u000eciently with a \feld-like SOT. Even though SAFs have\nzero net magnetization at equilibrium, the \feld-like SOT\nyields a \fnite nonequilibrium magnetization, which gives\nrise to switching driven mostly by the torque from the\ndemagnetizing \feld, particularly if the SAF consists of\nhigh-moment FMs (e.g., Fe). The 90\u000eswitching scheme\ncan enable fast dynamics, especially when combined with\nmoderately high Gilbert damping. Such SAFs can be\nreadily engineered from simple FMs and are attractive\nmodel systems that mimic some of the dynamics of\nintrinsic crystal antiferromagnets.\nThis work was supported in part by the Luther\nand Alice Hamlett Undergraduate Research Support\nProgram in the Academy of Integrated Science at\nVirginia Tech. 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For the case of the torus it is shown that for small nonlinear\ndamping\u0014>0 stationary spatially periodic solutions exist on branches that bifurcate from\nconstant solutions whereas all nonconstant solutions disappear when the damping parameter\n\u0014exceeds a critical value. These results apply both for normal ( d<0) and anomalous ( d>0)\ndispersion. For the case of the real line we show by the Implicit Function Theorem that for\nsmall nonlinear damping \u0014>0 and large detuning \u0010\u001d1 and large forcing f\u001d1 strongly\nlocalized, bright solitary stationary solutions exists in the case of anomalous dispersion d>0.\nThese results are achieved by using techniques from bifurcation and continuation theory and\nby proving a convergence result for solutions of the time-dependent Lugiato-Lefever equation.\n1.Introduction\nThe Lugiato-Lefever equation\n(1) i @ta=\u0000(i\u0000\u0010)a\u0000daxx\u0000jaj2a+ if\nwas proposed in 1987 by Lugiato and Lefever [14] as an approximative model for the electric\n\feld inside an optical cavity excited by a laser pump of strength f. Since then many authors\nhave derived (1) as a model, e.g., for the \feld a(x;t) =P\nk2Z^ak(t)eikxinside a continous\nwave(cw)-pumped ring resonator, cf. [1, 2, 10]. Here ^ ak(t) denotes the complex amplitude\nof thek-th excited mode in the ring resonator. The cw-laser frequency has a detuning\no\u000bset\u0010relative to the primarily excited 0-mode of the ring resonator, and the second-order\nlinear dispersion coe\u000ecient dof the ring resonator may be normal ( d < 0) or anomalous\n(d > 0). Nonlinear interaction of the strongly enhanced \feld due to the Kerr e\u000bect in\nthe microresonator eventually leads to modulation instability. Consequently, a cascaded\ntransfer of power from the primarily excited mode to a multitude of neighbouring modes\ntakes place. A resulting stable stationary pattern of spectrally equidistant excited modes is\ncalled a frequency comb. Spectrally broad octave spanning frequency combs have turned out\nto be extremely attractive sources for a variety of applications including time and frequency\nmetrology [5, 32], high-speed optical data communications [17, 26, 27], and ultrafast optical\nranging [30,31].\n2000 Mathematics Subject Classi\fcation. Primary: 34C23, 34B15; Secondary: 35Q55, 35Q60.\nKey words and phrases. Lugiato-Lefever equation, bifurcation, continuation, solitons, frequency combs,\nnonlinear damping, two photon absorption.\n1arXiv:1811.12200v3  [math.AP]  10 Dec 20182 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nRecently, semiconductors exhibiting two-photon-absorption (TPA) at telecommunication\nwavelengths such as silicon have been considered as waveguide materials for microresonators.\nTPA causes an electron from the valence band to be excited to the conduction band. There,\nfree-carrier absorption (FCA) of additional photons leads to a further excitement to other\nstates within the conduction band. While these nonlinear losses hinder the generation of\nfrequency combs in microresonators, at the same time comb formation bene\fts from a higher\nKerr nonlinearity that comes along with TPA. Furthermore, especially silicon is highly rel-\nevant from a practical point of view, since it is an established material used for photonic\nintegrated circuits.\nWe are not aware of mathematically rigorous studies on the Lugiato-Lefever equation with\nTPA or FCA. In this paper we want to start the analysis of the e\u000bect of TPA on the formation\nof frequency combs. For mathematical reasons the e\u000bect of FCA will be neglected in this\npaper, since the full model is currently out of reach for our analysis. TPA modi\fes the Kerr\ne\u000bect by adding an imaginary component i \u0014,\u0014>0 to the coe\u000ecient of the cubic nonlinear\nsusceptibility. Following [8,13] the model equation (1) is therefore modi\fed as follows\n(2) i @ta=\u0000(i\u0000\u0010)a\u0000daxx\u0000(1 + i\u0014)jaj2a+ if:\nSince FCA will not be considered we have set the free carried density to 0 so that the ODE\nfor the free carrier density, which is coupled to (2), cf. [8, 13], is not present. Stationary\nsolutions of (2) satisfy\n(3)\u0000da00\u0000(i\u0000\u0010)a\u0000(1 + i\u0014)jaj2a+ if= 0; a (\u0001) =a(\u0001+ 2\u0019)\nwhere the spatial period given by the circular nature of resonators is normalized to 2 \u0019. Due\nto the nonlinear damping e\u000bect of TPA in addition to the linear damping, TPA is unfavorable\nfor comb formation. However, in this paper we prove the converse: Kerr comb formation in\nsilicon based microresonators is still possible if the TPA coe\u000ecient \u0014is su\u000eciently small. For\nlarge\u0014above a certain threshold, for which we provide lower bounds, Kerr comb formation is\nprohibited. Our results apply both for normal and anomalous dispersion. Since soliton-like\nstationary solutions of (2) are of utmost importance in applications, we also consider the\nformation of bright solitary combs for anomalous dispersion in the presence of small \u0014.\nBefore describing our results for (2) and (3) in more detail, we \frst present the mathemat-\nical results which deal with the special case \u0014= 0 of purely linear damping. One important\nfact about (3) for \u0014= 0 and any \fxed f6= 0 is that there is a uniquely determined curve\nparameterized by \u0010consisting of constant solutions, see for instance Lemma 2.1 (a) [16] for\nan explicit parametrization. With \u0010as a bifurcation parameter bifurcation theory is a conve-\nnient tool for proving the existence of nonconstant solutions. A number of existence results\nfor (3) with \u0014= 0 were found using bifurcation results for dynamical systems via the spatial\ndynamics approach [4,6,7,23{25]. Here the requirement of 2 \u0019-periodicity is dropped and one\nis interested in nonconstant solutions of the four-dimensional (real) dynamical system that\ncorresponds to the second order ODE from (3) for the complex-valued function a. A detailed\nanalysis of the normal forms of this system around the constant equilibria reveals which types\nof solutions exist in a neighbourhood. In [6] (Theorem 2.1{2.6) periodic, quasiperiodic and\nhomoclinic orbits were proved to exist near the curve of constant solutions both in the caseTHE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 3\nof normal dispersion ( d<0) and anomalous dispersion ( d>0). Since solutions correspond-\ning to these orbits necessarily resemble constant functions on [0 ;2\u0019], soliton-like solutions\nwith a strong spatial pro\fle can not be analytically described by local bifurcation methods.\nTherefore, in order to see interesting spatial pro\fles, local bifurcations have to be continued,\ne.g., by numerical methods, cf. [16,23{25], far away from the curve of constant equilibria.\nProving local bifurcations of exactly 2 \u0019-periodic solutions requires a di\u000berent approach. A\n\frst local bifurcation bifurcation result from a speci\fc constant solution was proved in [19]\n(Theorem 3.1). This study was extended in [16] using local and global bifurcation results\ndue to Crandall-Rabinowitz and Krasnoselski-Rabinowitz. All (\fnitely many) bifurcation\npoints on the curve of constant solutions were identi\fed and the bifurcating solutions were\nshown to lie on bounded solution continua that return to another bifurcation point. Some\nof these continua even undergo period-doubling, period-tripling, etc. secondary bifurcations\nas was shown in Section 4 in [15]. The theoretical results from [15, 16] were accompanied\nby numerically computed bifurcation diagrams indicating that the most localized and thus\nsoliton-like solutions can be found at those turning points of the branches that are the farthest\naway from the curve of trivial solutions. We remark that a two-dimensional version of the\nLugiato-Lefever equation posed on the unit disk was recently discussed in [22].\nFinally, still in the case \u0014= 0 we mention some results about the time-dependent equa-\ntion (1). In [11] it was proved that the initial value problem is globally well-posed in\na2C(R+;H4(T))\\C1(R+;H2(T))\\C2(R+;L2(T)) for initial data in H4(T). Here, Tis the\none-dimensional torus, i.e., the interval [0 ;2\u0019] with both ends identi\fed, and R+= [0;1) is\nthe temporal half-line. Additionally, it was shown that all solutions of the initial value prob-\nlem remain bounded in L2while theH1-norm is proved to grow at most likep\ntast!1 .\nIn the corresponding model with an additional third order dispersion e\u000bect well-posedness\nresults and even the existence of a global attractor were proved in [21]. Convergence results\nfor the numerical Strang-splitting scheme can be found in [11]. Finally, the orbital asymp-\ntotic stability of 2 \u0019-periodic solutions was investigated in [29] (Theorem 1) with the aid of\nthe Gearhart-Pr uss-Theorem, see also [18,20]. Notice that the linearized operators (i.e. the\ngenerators of the semigroup) are not selfadjoint, which makes this result particularly interest-\ning. Using the center manifold approach, spectral stability and instability results as well as\nnonlinear stability with respect to co-periodic or subharmonic perturbations were obtained\nin [4].\nLet us now describe the results of our paper. We consider (3) with f6= 0,\u0014\u00150 andd6= 0\n\fxed. Our \frst theorem contains three results on the structure of solutions of (3). Notice\nthat for every \u00102R(3) has either one, two or three di\u000berent constant solutions a02C\nlying on a smooth curve. Theorem 1 addresses the question of bifurcation from the curve\nof trivial solutions. We show that for su\u000eciently small \u00142(0;1=p\n3) bifurcation from the\ncurve of trivial solutions happens, whereas for su\u000eciently large \u0014>\u0014\u0003the trivial curve has no\nbifurcation points at all. In case of small \u0014we give su\u000ecient conditions (4), (5) for bifurcation\nbased on the Crandall-Rabinowitz theorem on bifurcation from simple eigenvalues [3]. They\ncorrespond to simple kernels of the linearization around a given point of the trivial curve and\nto transversality, respectively.4 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nThe notion of bifurcation may depend on spaces and norms. In our context we use the\nfollowing set-up. Let Tbe the one-dimensional torus, i.e., the interval (0 ;2\u0019] with end-points\n0 and 2\u0019identi\fed. We consider solutions a= Rea+ i Ima2H2(T) of (3).\nTheorem 1. Forf6= 0;\u0014> 0the following holds:\n(i) All constant solutions of (3)form a smooth unbounded curve in H2(T)\u0002R.\n(ii) A point (\u0010;a0)on the curve of constant solutions is a bifurcation point provided exactly\none of the two numbers\n(4) k1;2:=s\n2ja0j2\u0000\u0010\u0006p\n(1\u00003\u00142)ja0j4\u00004\u0014ja0j2\u00001\nd\nis inNand\n(5) 2(3\u00142\u0000ja0j4)(ja0j2\u0000\u0010)\u00004\u0014ja0j2(3ja0j2\u0000\u0010)\n\u0006p\n(1\u00003\u00142)ja0j4\u00004\u0014ja0j2\u00001\u0010\n1 +\u00102\u0000ja0j4\u00004\u0014ja0j2+ 3\u00142\u0011\n6= 0\nwith \\ +\" ifk12Nand \\\u0000\" ifk22N.\n(iii) The curve of constant solutions does not contain bifurcation points provided \u0014 > \u0014\u0003\nwhere\n\u0014\u0003:= max(\n\u00142(0;1p\n3) :2\u0014+p\n1 +\u00142\n(1\u00003\u00142)3(1\u0000\u00142+\u0014p\n1 +\u00142)2\u0014f2)\niff2>1;\n\u0014\u0003:= 0 iff2\u00141:\nRemark 2. (i) Necessarily, we have \u0014 <p\n3in case (ii) since otherwise the values k1;2\nin(4)will not be real. Moreover, in case (ii) we may apply Rabinowitz' global bi-\nfurcation theorem from [28]. It says not only that (\u0010;a0)is a bifurcation point, but\nthat there is a global branch of non-trivial solutions that either returns to the trivial\nbranch at some other bifurcation point or becomes unbounded in the \u0010-direction or in\ntheH2(T)-direction.\n(ii) Notice that by strict monotonicity, the value \u0014\u0003is the uniquely determined solution of\n2\u0014+p\n1 +\u00142\n(1\u00003\u00142)3(1\u0000\u00142+\u0014p\n1 +\u00142)2=f2; (6)\ncf. Figure 1.\n(iii) Forjfj&1, we have\u0014\u0003!0. This is consistent with [16], where for \u0014= 0 it was\nshown that no bifurcations occur in the case jfj\u00141.\n(iv) By running pde2path for increasing values of \u0014 > 0we can determine numerically\nwhen bifurcations cease to exist. The values for \u0014?from Theorem 1 and these numer-\nically determined values from pde2path are very similar, cf Table1.\nTheorem 1 provides nontrivial solutions via bifurcation theory for \u00142(0;\u0014\u0003), i.e., the\nbifurcating branches described in [16] for \u0014= 0 persist for small \u0014>0. The natural question,\nwhat happens to the bifurcating branches when \u0014gets larger, is also answered in part (iii)THE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 5\n\u0014\n1=p\n3f2\n\u0014?\nFigure 1. Illustration of (6).f\u0014?\u0014num\n?\n1:10:045 0:042\n1:60:185 0:185\n20:248 0:245\n40:380 0:378\n100:474 0:473\n200:513 0:513\nTable 1. \u0014?from Theorem 1 and\nnumerical values from pde2path .\nof the theorem: bifurcation points disappear at latest when \u0014exceeds\u0014\u0003. In Figure 2 the\nvanishing of bifurcation points and nontrivial solutions for increasing \u0014is illustrated. Black\ncurves indicate the line of trivial solutions, colored curves show bifurcation branches. With\nincreasing nonlinear damping, more and more bifurcation branches vanish, until all have\ndisappeared when \u0014exceeds the value 0 :185.\nFigure 2. Bifurcation diagrams for d= 0:1,f= 1:6. Sub\fgure (a) corre-\nsponds to\u0014= 0 , (b) to \u0014= 0:05, (c) to\u0014= 0:1, (d) to\u0014= 0:15, (e) to\n\u0014= 0:185, (f) to \u0014= 0:186. Solutions at turning points A, B in (a), C, D in\n(b) and E in (c) are shown in Figure 3.\nIn Figure 3(a), the solutions corresponding to the turning points A,C in Figure 2 of the\ncurve of 1-solitons are shown. Additionally, the 1-soliton at the turning point of the corre-\nsponding branch for \u0014= 0:025 is depicted. In Figure 3(b) the turning points B, D, E of the6 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\ncurve of 2-solitions are shown for di\u000berent values of the nonlinear damping coe\u000ecient. It\nbecomes apparent that the solitons \ratten as \u0014increases.\nFigure 3. Sub\fgure (a) shows 1-solitons and sub\fgure (b) 2-solitons of (3)\nfor increasing values of \u0014.\nSince Theorem 1 only addresses the occurence and disappearance of bifurcations, it does\nnot answer the question what happens to the entire set of solutions when \u0014increases. This\nis answered in our next two results: all nontrivial solutions disappear for \u0014beyond a certain\npositive threshold. A \frst threshold for nonexistence of nontrivial solutions is given by the\nfollowing result.\nTheorem 3. Letd6= 0,\u0014>0,\u0010;f2Rand let\u0014?be given by\n\u0014?:= 6p\n6\u0000\n1 + 2\u00192f2jdj\u00001\u00013f2:\nThen all solutions of (3)are constant provided \u0014>\u0014?.\nA second threshold may be obtained by studying the time-dependent Lugiato-Lefever\nequation (2). Modifying slightly the proof by Jahnke, Mikl and Schnaubelt [11] for (1)\nwe \frst derive the global well-posedness of the initial value problem for (2) with initial data\na(0) =\u001e2H4(T). In [11] the corresponding well-posedness result for \u0014= 0 is based on\nthe observation that the \row remains bounded in L2(T) and that the H1(T)-norm grows at\nmost likep\ntast!1 . It is not known whether in\fnite time blow-up or convergence occurs\nin this case. We show that for su\u000eciently strong nonlinear damping \u0014\u00151p\n3the solutions\nconverge to a constant solution regardless of the initial datum.\nTheorem 4. Letd6= 0\u0010;f2Rand\u0014\u00151p\n3. Ifa(0) =\u001e2H4(T)then the solution of (2)is\ninC(R+;H4(T))and converges in H1(T)to a constant as t!1 . In particular, all solutions\nof(3)are constant.\nCombining Theorem 3 and Theorem 4 we obtain that for \u0014>minf\u0014\u0003;1p\n3gonly constant\nsolutions exist. Notice that all weak solutions of (3) are smooth and in particular lie inTHE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 7\nH4(T). Actually we can also prove convergence results for smaller \u0014assuming thatk\u001exk2is\nnot too big. We refer to Lemma 13 for details.\nFinally we discuss the e\u000bect of nonlinear damping to the Lugiato-Lefever equation on the\nreal line in the case of anomalous dispersion d>0. In this case the problem reads\n\u0000da00\u0000(i\u0000\u0010)a\u0000(1 + i\u0014jaj2)a+ if= 0 on R; a0(0) = 0 (7)\nand we are interested in even homoclinic solutions. More precisely, the solutions we will \fnd\nhave the form a= ~a+a1wherea12Cand ~a2H2(R). This is a valid approach, since\nhighly localized solutions of (7) serve as good approximations for solutions of (3), cf. [9].\nUsing a suitable singular rescaling of the problem as well as the Implicit Function Theorem,\nwe prove the existence of large solutions of (7) for large parameters \u0010andfand small\nnonlinear damping \u0014.\nTheorem 5. Letd;~\u0010 > 0and0<j~fj<2p\n3\n9~\u00103=2. Then for all \";\u0014 > 0su\u000eciently small\nthere are two even homoclinic solutions a\";\u0014of(7)with\u0010=~\u0010\"\u00001;f=~f\"\u00003=2satisfying\r\ra\";\u0014\u0000limjxj!1a\";\u0014(x)\r\r\nH2!1 as\"!0uniformly with respect to \u0014.\nRemark 6. The above theorem guarantees the existence of \u00140;\u000f0>0depending on d;~\u0010;~f\nwith the property that for 0< \u0014 < \u0014 0and0< \" < \" 0the parameter triple (\u0010;f;\u0014 )with\n\u0010=~\u0010\"\u00001andf=~f\"\u00003=2allows for a localized solution of (7). For \fxed \u00142(0;\u00140)let\nus take\"0=\"0(d;~\u0010;~f;\u0014)to be the largest value with the above property. Then we can\nconsider the curve (0;\"0)3\"7!(~\u0010\"\u00001;~f\"\u00003=2)in the (\u0010;f)-plane. By varying the parameters\n~\u0010and ~fthese curves cover regions in the (\u0010;f)-plane, such that above the lower envelope\n(~\u0010\"\u00001\n0;~f\"\u00003=2\n0)localized solutions of (7)exist.\nThe practical applicability of Theorem 5 is demonstrated in the following. We have used\nthe idea of the proof of the theorem as the basis for a numerical continuation method with\npde2path . This is done by replacing the real line with the interval [0 ;\u0019] and by considering\nthe rescaled version (50) of the Lugiato-Lefever equation on [0 ;\u0019] with Neumann boundary\nconditions at the endpoints. Then, for a given \fxed value of ~\u0010and ~f=\"=\u0014= 0 the\napproximate solution iq\n2~\u0010sech(xq\n~\u0010=d) is continued \frst in ~f, then in\"and \fnally in \u0014.\nRescalinga(x) =\"\u00001=2u(\"\u00001=2x) we obtain a function de\fned on [0 ;p\"\u0019] that we extend as a\nconstant to [p\"\u0019;\u0019 ]. The resulting function is mirrored on the vertical axis and shifted by \u0019\nso that an approximate 2 \u0019-periodic solution of (3) for parameter values ( \u0010;f) = ( ~\u0010\"\u00001;~f\"\u00003=2)\nis found. Re\fning this solution with a Newton step yields a periodic soliton solution asolving\n(3) on [0;2\u0019] for the parameters ( \u0010;f;\u0014 ). As an example, for \fxed d= 0:1;~\u0010= 5 we initially\nset~f=\"=\u0014= 0, and \frst continued the sech-type soliton with respect to ~f2[0;2:9]. For\n\fxed ~f= 2:9 the continuation is then done with respect to \"2[0;0:5]. Fixing both ~f= 2:9\nand\"= 0:5 the \fnal continuation is done in \u0014, and for three di\u000berent values of \u0014the resulting\nsolutions are shown in Figure 4. With \"= 0:5 the corresponding detuning and forcing values\nare\u0010=~\u0010\"\u00001= 10 andf=~f\"\u00003=2= 8:20.8 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nFigure 4. Solutions of (7) for d= 0:1,\u0010= 10,f= 8:20, and three di\u000berent\nvalues of\u0014.\nOne might ask if a similar result for heteroclinic solutions in the case of normal dispersion\nd < 0 could be achieved. In Section 5 we will point out that this cannot be done by our\ncontinuation method. The above result is of perturbative nature and therefore does not reveal\nwhether nontrivial solutions of (7) have to disappear for large nonlinear damping \u0014 >0 as\nit was shown in Theorem 3 and Theorem 4 for the case of 2 \u0019-periodic solutions of (3). The\nproofs of both theorems make use of the boundedness of [0 ;2\u0019] in an essential way. Since\nwe do not know how to adapt these results to solutions on Rwe have to leave this question\nopen.\n2.Proof of Theorem 1\nThis section is structured according to the results in Theorem 1.\n2.1.Proof of (i). Here we determine the curve of trivial solutions.\nLemma 7. Let\u001c2(0;1)be the unique value such that \u001c(1+\u0014f2\u001c)2= 1. Fort2(\u0000p\u001c;p\u001c)\nde\fne\nA(t) :=t\u00101 + 4\u0014f2\u001c+ 3\u00142f4\u001c2+t2(\u00003\u00142f4\u001c\u00002\u0014f2) +t4\u00142f4\n\u001c\u0000t2\u00111=2\n:\nThent7!(\u0010(t);a0(t))parametrizes the curve of trivial solutions with\n\u0010(t) :=f2(\u001c\u0000t2) +A(t);\na0(t) :=f(\u001c\u0000t2)\u0000\n1 +\u0014f2(\u001c\u0000t2)\u0000iA(t)\u0001\n:\nRemark 8. The curve (\u0010;a0) : (\u0000p\u001c;p\u001c)!R\u0002R2is smooth and unbounded in the \u0010-\ncomponent. The same is true if we consider (\u0010;a0)as a map from (\u0000p\u001c;p\u001c)intoR\u0002H2(T).\nThis is the claim of part (i) of Theorem 1.THE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 9\nProof. Constant solutions ( a0;\u0010) of (3) satisfy\n(8) ( \u0010\u0000i)a0\u0000(1 +i\u0014)ja0j2a0+ if= 0\nand in particular\n(9) ja0j2\u0000\n(\u0010\u0000ja0j2)2+ (1 +\u0014ja0j2)2\u0001\n=f2:\nLet us successively parametrize ja0j2,\u0010anda0. Since (\u0010\u0000ja0j2)2\u00150 we obtain from (9) that\n(10) 0 <ja0j2f\u00002\u0014\u001c;\nfor\u001c2(0;1) as in the statement of the lemma. Equation (9) suggests the following\nparametrization of ja0j2byt7!ja0j2(t) :=f2(\u001c\u0000t2) fort2(\u0000p\u001c;p\u001c). The sign of t\nis chosen according to sign( t) = sign(\u0010\u0000ja0j2). Due to (9), the value \u0010can be written as\nfollows:\n\u0010=f2(\u001c\u0000t2) +\u0010\u0000ja0j2\n=f2(\u001c\u0000t2) + sign(t)j\u0010\u0000ja0j2j\n(9)=f2(\u001c\u0000t2) + sign(t)p\nf2ja0j\u00002\u0000(1 +\u0014ja0j2)2:\nInserting the parametrization of ja0j2(t) yields the following parametrization of \u0010\n\u0010(t) =f2(\u001c\u0000t2) + sign(t)r\n1\n\u001c\u0000t2\u0000(1 +\u0014f2(\u001c\u0000t2))2=f2(\u001c\u0000t2) +A(t)\nNext we rearrange (8) to express a0in terms of f;\u0014;\u0010;ja0j2and use (9) to \fnd\na0=if\nja0j2\u0000\u0010+i(1 +\u0014ja0j2)\n=\u0000if(i(1 +\u0014ja0j2) +\u0010\u0000ja0j2)\n(1 +\u0014ja0j2)2+ (\u0010\u0000ja0j2)2\n(9)=ja0j2\nf\u0000\n1 +\u0014ja0j2+ i(ja0j2\u0000\u0010)\u0001\n:\nIf we insertja0j2(t) =f2(\u001c\u0000t2) and\u0010(t) =f2(\u001c\u0000t2) +A(t) into the previous expression we\n\fnally arrive at\na0(t) =f(\u001c\u0000t2)\u0000\n1 +\u0014f2(\u001c\u0000t2)\u0000iA(t)\u0001\n:\n\u0003\n2.2.Proof of (ii) { necessary and su\u000ecient conditions for bifurcation. In order to\nprove (ii) we need the following preliminary result, which is a generalization of Proposition 4.3\nin [16]. It provides the necessary condition for bifurcation.\nProposition 9. All bifurcation points (\u0010;a0)for(3)with respect to the curve of trivial\nsolutions satisfy\n(11) ( \u0010+dk2)2\u00004ja0j2(\u0010+dk2) + 3(1 +\u00142)ja0j4+ 4\u0014ja0j2+ 1 = 0\nfor somek2N. In other words, one of the two numbers k1;2from (4)needs to be in N.10 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nRemark 10. We exclude the case k1= 0 ork2= 0 in the bifurcation condition (11). It\nhappens exactly at the turning points of the curve of trivial solutions and corresponds to\nthe non-injectivity of \u0010(t). Since it creates only arti\fcial bifurcation points as explained in\nSection 4.2 in [16], we omit it.\nProof. By the implicit function theorem we know that a necessary condition for bifurcation\nis that the linearized operator\n(12) L=\u0000dd2\ndx2\u0000(i\u0000\u0010)\u0000Dg(a0) :H2(T)!L2(T)\nhas a nontrivial kernel. Here g(a) = (1 + i\u0014)jaj2a\u0000ifstands for the nonlinearity and\nDg(a)z:=d\ndtg(a+tz)jt=0= 2(1 + i\u0014)jaj2z+ (1 + i\u0014)a2\u0016zwitha;z2Cfor the derivative of g\nata. The derivative Dg(a) can also be written in the form\n(13)Dg(a)z=\u0012\nRe(a2) + 2jaj2\u0000\u0014Im(a2) Im(a2)\u00002\u0014jaj2+\u0014Re(a2)\nIm(a2) +\u0014Re(a2) + 2\u0014jaj22jaj2\u0000(Rea2) +\u0014Im(a2)\u0013\u0012\nRez\nImz\u0013\n:\nSinceLis a Fredholm operator, the space Ker Lis \fnite dimensional, and the adjoint operator\n(14) L\u0003=\u0000dd2\ndx2+ (i +\u0010)\u0000Dg(a0) :H2(T)!L2(T)\nhas a kernel with the same \fnite dimension as Ker L. Any element \u001e2KerLcan be expanded\nin the form \u001e(x) =P\nl2Z\u000bleilx. The condition that \u001e2KerLmeans that there is at least\none integer k2Zsuch thatL(\u000beikx) = (dk2\u0000i +\u0010\u0000Dg(a0))\u000beikx= 0 for some \u000b2Cnf0g.\nIn other words, dk2is an eigenvalue of the matrix\nN=Dg(a0) +\u0012\n\u0000\u0010\u00001\n1\u0000\u0010\u0013\nwithDg(a0) in matrix representation given by (13). Non-zero elements in Ker Lexist if\ndet(\u0000dk2Id +N) = 0 and computing this determinant yields (11). Solving for kleads to\nk1;2given by (4). Likewise, non-zero elements in Ker L\u0003exist if det(\u0000d~k2Id +NT) = 0 for\nsome integer ~k2N0. Solving det(\u0000d~k2Id +NT) = det(\u0000d~k2Id +N) = 0 leads to the same\nformula (4) as for k. Consequently, (4) is equivalent to both LandL\u0003having nontrivial\nkernels. If neither k1ork2are in Nthen KerL= KerL\u0003=f0g, and in this case the implicit\nfunction theorem, cf. [12][Theorem I.1.1], implies that solutions nearby the point ( \u0010;a0) are\nunique, i.e., trivial, and hence ( \u0010;a0) cannot be a bifurcation point. Therefore, k1ork2inN\nis a necessary condition for bifurcation. \u0003\n2.3.Proof of (ii) { simplicity of the kernel of the linearization. Notice that Ker L\nis either two-dimensional or four-dimensional, since \u000beikxbelonging to Ker Lalways implies\nthat\u000be\u0000ikxalso belongs to Ker L. The two-dimensional case happens if exactly one of the two\nnumbersk1;2from (4) is an integer and the four-dimensional case happens if both k1;k22N.\nIn order to achieve simple instead of multiple eigenvalues we need to change the setting\nfor (3) by additionally requiring a0(0) = 0, i.e., solutions need to be even around x= 0.\nTogether with 2 \u0019-periodicity this implies a0(\u0019) = 0, i.e., we consider (3) with vanishing\nNeumann boundary conditions at x= 0 andx=\u0019. If we de\fne H2\neven(T) andL2\neven(T) asTHE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 11\nthe subspaces of H2(T) andL2(T) with even symmetry around x= 0 thenL;L\u0003:H2\neven(T)!\nL2\neven(T) are again Fredholm operators, and Propositon 9 still holds. In this way we halve\nthe dimension of Ker Lfor everyksatisfying (11) since instead of both \u000beikxand\u000be\u0000ikxonly\n\u000bcos(kx) remains in the kernel of L. In particular, we get a one-dimensional kernel of Lif\nand only if exactly one the numbers k1;2from (4) belongs to N. The same is true for the\nkernel ofL\u0003.\n2.4.Proof of (ii) - computing the kernel of the linearization. Under the condition\nthat exactly one of the numbers k1;2from (4) belongs to Nlet us compute Ker Land KerL\u0003.\nTo describe the matrix N\u0000dk2Id let us introduce the real numbers \u000bj;~\u000bj;\u000b\u0003\nj;~\u000b\u0003\njforj= 1;2\nas follows\n(15)N\u0000dk2Id =\u0012\n\u0000\u000b2\u000b1\n~\u000b2\u0000~\u000b1\u0013\n=\u0012\n\u0000~\u000b\u0003\n2\u000b\u0003\n2\n~\u000b\u0003\n1\u0000\u000b\u0003\n1\u0013\n=\u0012\n\u0000\u0010\u0000dk2+ (Rea0)2+ 2ja0j2\u0000\u0014Im(a2\n0) Im(a2\n0)\u00001\u00002\u0014ja0j2+\u0014Re(a2\n0)\nIm(a2\n0) + 1 +\u0014Re(a2\n0) + 2\u0014ja0j2\u0000\u0010\u0000dk2+ 2ja0j2\u0000Re(a2\n0) +\u0014Im(a2\n0)\u0013\n:\nIn the matrix N\u0000dk2Id the o\u000b-diagonal elements have the property that\n\u000b1<Im(a2\n0)<~\u000b2\nand hence they cannot be zero simultaneously. Therefore, if Im( a2\n0)\u00140 we can de\fne\n(16) \u000b:= (\u000b1;\u000b2)T; \u000b\u0003:= (\u000b\u0003\n1;\u000b\u0003\n2)T\nand obtain eigenvectors of N\u0000dk2Id,NT\u0000k2Id, respectively, so that Ker L= spanf\u000beikxg,\nKerL\u0003= spanf\u000b\u0003eikxg. Likewise, if Im( a2\n0)\u00150 then\n(17) ~ \u000b:= (~\u000b1;~\u000b2)T;~\u000b\u0003:= (~\u000b\u0003\n1;~\u000b\u0003\n2)T\nare the eigenvectors of N\u0000dk2Id,NT\u0000k2Id leading to Ker L= spanf~\u000beikxg, KerL\u0003=\nspanf~\u000b\u0003eikxg.\n2.5.Proof of (ii) { tangent direction to the trivial branch of solutions. Let us\nassume that the curve of trivial solutions of (3) is parameterized by t7!(\u0010(t);a0(t)) as in\nLemma 7, and that ( \u0010;a0) = (\u0010(t0);a0(t0)) is a speci\fc bifurcation point. Let us compute the\ntangent ( _\u0010;_a0) =d\ndt(\u0010(t);a0(t)jt=t0. As explained in Remark 10 we can ignore turning points\nwhere _\u0010= 0. Di\u000berentiating the equation (i \u0000\u0010(t))a0(t) +g(a0(t)) = 0 with respect to tand\nevaluating the derivative at t0we get\n\u0000\nDg(a0) + i\u0000\u0010\u0001\n_a0=_\u0010a0:\nInsertingDg(a0)z= (1 + i\u0014)(2ja0j2z+a2\n0\u0016z) we \fnd\n(2(1 + i\u0014)ja0j2+ i\u0000\u0010)_a0+ (1 + i\u0014)a2\n0_a0=_\u0010a0\nand hence\n(18) _ a0=\u001c_\u0010a0with\u001c=(1\u00003i\u0014)ja0j2\u0000\u0010\u0000i\n3(1 +\u00142)ja0j4+ 4(\u0014\u0000\u0010)ja0j2+\u00102+ 1:12 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\n2.6.Proof of (ii) { su\u000ecient condition for bifurcation. According to the Crandall-\nRabinowitz theorem, see [3] or [12][Theorem I.5.1], two conditions are su\u000ecient for bifurca-\ntion. The \frst is that Ker Lis simple, i.e. one-dimensional. Above we proved this to hold\nprovidedk12N,k262Nor vice versa with k1;2from (4). In the following we write kfor the\none which is the integer. In view of the statement of (ii) it therefore remains to show that\nthe second condition, the so-called transversality condition, is satis\fed provided (5) holds.\nTo verify this we bring our problem into the form used in [3]. Nontrivial solutions of (3),\nwhich are even around x= 0 may be written as a(\u0001) =a0(t) +b(\u0001) withb0(0) =b0(\u0019) = 0.\nFrom (3) we derive the equation for the function bin the form\n(19) F(t;b) :=\u0000db00\u0000(i\u0000\u0010(t))(a0(t) +b)\u0000g(a0(t) +b) = 0\nwhereF:R\u0002H2\neven(T)!L2\neven(T). Notice that F(t;0) = 0 for all t, i.e., the curve of trivial\nsolutions (\u0010(t);a0(t)) for (3) has now become the line of zero solutions ( t;0) for (19). Let\nus writeD2\nb;tF(t0;0) for the mixed second derivative of Fwith respect to ( t;\u0010) at the point\n(t0;0). According to [3], the transversality condition is expressed by\nD2\nb;tF(t0;0)\u001e62RgDbF(t0;0);\nwith\u001esuch that Ker DbF(t0;0) = spanf\u001eg. In our case DbF(t0;0) =L, whereLis the lin-\nearized operator given in (12). By the Fredholm alternative, Rg L= (KerL\u0003)?= spanf\u001e\u0003g?,\nand\u001e(x) =\u000bcos(kx),\u001e\u0003(x) =\u000b\u0003cos(kx) if Im(a2\n0)\u00140, cf. (16), or \u001e(x) = ~\u000bcos(kx),\n\u001e\u0003(x) = ~\u000b\u0003cos(kx) with ~\u000b;~\u000b\u0003if Im(a2\n0)\u00150, cf. (17). The components of \u000b;\u000b\u0003and ~\u000b;~\u000b\u0003can\nbe read from (15). Since orthogonality of two functions u;vin the real Hilbert space L2\neven(T)\nmeans vanishing of the inner product hu;vi= ReR\u0019\n0u(x)\u0016v(x)dx, we \fnd that transversality\nis expressed as\n(20) hD2\nb;tF(t0;0)\u001e;\u001e\u0003i= ReZ\u0019\n0\u0000\nD2\nb;tF(t0;0)\u001e\u0001\n\u001e\u0003dx6= 0:\nUsingD2g(a0)(z;w) = 2(1 + i\u0014)(\u0016a0zw+a0z\u0016w+a0\u0016zw) we \fnd for the second derivative\n(21)D2\nb;tF(t0;0)\u001e=_\u0010\u001e\u0000D2g(a0)(\u001e;_a0)\n=_\u0010\u001e\u00002(1 + i\u0014)\u0000\n\u0016a0\u001e_a0+a0\u001e_a0+a0\u0016\u001e_a0\u0001\nwith _a0=\u001c_\u0010a0,\u001cfrom (18). As explained in Remark (10) we can ignore the turning points\nwhere _\u0010= 0. Hence, inserting (21) into the transversality condition (20) we get in case\nIm(a2\n0)\u00140\n(22) Re\u0010\n\u000b\u000b\u0003\u00002(1 + i\u0014)(2 Re\u001cja0j2\u000b\u000b\u0003+\u001ca2\n0\u0016\u000b\u000b\u0003\u0001\u0011\n6= 0\nand in case Im( a2\n0)\u00150 we replace \u000b;\u000b\u0003by ~\u000b;~\u000b\u0003. Let us \frst consider the case Im( a2\n0)\u00140.\nHere we obtain\n(23)\u000b\u000b\u0003=(\u000b1+ i\u000b2)(\u000b\u0003\n1\u0000i\u000b\u0003\n2|{z}\n=\u000b1) =\u000b1(\u000b\u0003\n1+\u000b2) + i(\u000b2\u000b\u0003\n1|{z}\n=\u000b1~\u000b\u0003\n1\u0000\u000b2\n1)\n=\u000b1\u0010\n2\u00100+ 2dk2\u00004ja0j2+ i(2 + 4\u0014ja0j2)\u0011\n;THE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 13\nLikewise, we use (15) and det( N\u0000dk2Id) = 0 to compute\n(24)\u0016\u000b\u000b\u0003=(\u000b1\u0000i\u000b2)(\u000b\u0003\n1\u0000i\u000b\u0003\n2|{z}\n=\u000b1) =\u000b1(\u000b\u0003\n1\u0000\u000b2)\u0000i(\u000b2\n1+\u000b2\u000b\u0003\n1|{z}\n=\u000b1~\u000b\u0003\n1)\n=\u000b12(1\u0000i\u0014)\u0016a2\n0:\nTaking the expressions for \u000b\u000b\u0003anda2\n0\u0016\u000b\u000b\u0003into the transversality condition (22) \fnally leads\nto\nRe\u0010\n\u000b\u000b\u0003(1\u00004(1 + i\u0014) Re\u001cja0j2)\u0011\n\u0000Re\u0010\n\u0016\u000b\u000b\u00032(1 + i\u0014)\u001ca2\n0\u0011\n= (1\u00004 Re\u001cja0j2) Re(\u000b\u000b\u0003) + 4\u0014Re\u001cja0j2Im(\u000b\u000b\u0003)\u00004\u000b1(1 +\u00142)ja0j4Re\u001c (25)\n=\u000b1\u0010\n2\u00100+ 2dk2\u00004ja0j2\u00004 Re\u001cja0j2(2\u00100+ 2dk2\u00003ja0j2(1 +\u00142)\u00002\u0014)\u0011\n6= 0:\nSince Im(a2\n0)\u00140 implies that \u000b1is non-zero, the non-vanishing of the expression in brackets\namounts to (after inserting Re \u001cfrom (18))\n(\u0010\u0000dk2)(ja0j4\u00003\u00142) + (\u00102+ 1)(\u0010+dk2\u00002ja0j2)\u00004\u0014ja0j2(ja0j2+dk2)6= 0:\nUsing (4) we obtain the transversality condition (5).\nChanges in case Im( a2\n0)\u00150 amount to replacing \u000b1in (23), (24) and (25) by ~ \u000b2, which is\nnon-zero in this case. Therefore, the \fnal transversality condition (5) is the same as before.\n2.7.Proof of (iii) { nonexistence of bifurcations. We assume that bifurcation for (3)\noccurs at some trivial solution ( \u0010;a0) so that the claim is proved once we show \u0014\u0014\u0014\u0003.\nBy Proposition 9 we know that the quadratic equation in \u0010+dk2from (11) holds for some\nk2N0. In particular, the discriminant is nonnegative and we obtain\n(26) 0\u0014(4ja0j2)2\u00004\u0001(3(1 +\u00142)ja0j4+ 4\u0014ja0j2+ 1) = 4\u0000\n(1\u00003\u00142)ja0j4\u00004\u0014ja0j2\u00001\u0001\n:\nFor\u0014\u00151p\n3this inequality is unsolvable, so we necessarily have \u00142[0;1p\n3) as well as\n(27) ja0j2\u00152\u0014+p\n1 +\u00142\n1\u00003\u00142:\nOn the other hand, the inequality (10) from the proof of (i) gives ja0j2\u0014f2\u001cwhere\u001cis the\nunique value such that \u001c(1 +kf2\u001c)2= 1. Therefore\n(28) ~ \u001c:=2\u0014+p\n1 +\u00142\n(1\u00003\u00142)f2\u0014ja0j2\nf2\u0014\u001c:\nSincez7!z(1 +\u0014f2z)2is increasing on [0 ;1), we deduce from the de\fnition of \u001cthe\ninequality\n~\u001c(1 +\u0014f2~\u001c)2\u0014\u001c(1 +\u0014f2\u001c)2= 1:\nInserting ~\u001cfrom (28) this is equivalent to\n2\u0014+p\n1 +\u00142\n(1\u00003\u00142)3(1\u0000\u00142+\u0014p\n1 +\u00142)2\u0014f2;\nwhich implies \u0014\u0014\u0014\u0003by de\fnition of \u0014\u0003. This \fnishes the proof of (iii). \u000314 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\n3.Proof of Theorem 3\nFollowing [16] we \frst provide some a priori bounds in L1(T) for solutions of (3).\nTheorem 11. Letd6= 0,\u0014>0and\u0010;f2R. Then every solution a2C2(T)of(3)satis\fes\nkak1\u0014\u0000\n1 + 2\u00192f2jdj\u00001\u0001\nmin(\njfj;\u0012jfj\n\u0014\u00131=3)\n: (29)\nRemark 12. One can obtain a more re\fned version of the bound (29) of the formkak1\u0014\u0000\n1 + 2\u00192f2jdj\u00001\u0001\nC\u0014where\n(30) C\u0014=3s\njfj\n2\u0014+r\nf2\n4\u00142+1\n27\u00143\u00003s\n\u0000jfj\n2\u0014+r\nf2\n4\u00142+1\n27\u00143:\nThis follows from Cardano's formula applied to (32). In this paper we do not make further\nuse of the re\fned value of C\u0014, since (29) already provides a meaningful a priori bound both\nfor small as well as for large values of \u0014. Indeed, as \u0014!0+theL1-bounds from (2) in [16]\n(valid for\u0014= 0) are partially recovered.\nProof. Leta2H2(T) be a solution of (3). Then we de\fne the 2 \u0019-periodic function g:=\n\u0000dIm(a0\u0016a)0. Using (3) we obtain\ng=\u0000dIm(a00\u0016a)\n= Im\u0010\n(i\u0000\u0010)jaj2+ (1 + i\u0014)jaj4\u0000if\u0016a\u0011\n(31)\n=jaj2+\u0014jaj4\u0000fRea:\nUsing the fact that gis 2\u0019-periodic together with H older's inequality we get from the previous\nidentity\n0 =Z2\u0019\n0gdx=Z2\u0019\n0(jaj2+\u0014jaj4\u0000fRea)dx\n\u0015\u0014kak4\n4+kak2\n2\u0000p\n2\u0019jfjkak2 (32)\n\u0015kak2\u0010\u0014\n2\u0019kak3\n2+kak2\u0000p\n2\u0019jfj\u0011\n:\nNeglecting once the kak3\n2and once thekak2term we obtain the L2-bound\nkak2\u0014p\n2\u0019~C\u0014with ~C\u0014= min(\njfj;\u0012jfj\n\u0014\u00131=3)\n: (33)THE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 15\nNext we derive a bound for ka0k2. First, the di\u000berential equation (3) yields the identity\nka0k2\n2= ReZ2\u0019\n0\u0000\n\u0000ida00\u0000i\u0010a+ (i\u0000\u0014)jaj2a+f\u00010\u0016a0dx\n= ReZ2\u0019\n0\u0000ida000\u0016a0+ i(jaj2a)0\u0016a0\u0000\u0014(jaj2a)0\u0016a0dx\n= ReZ2\u0019\n0i(jaj2)0a\u0016a0dx\u0000\u0014Z2\u0019\n0jaj2ja0j2dx\u0000\u0014ReZ2\u0019\n0(jaj2)0a\u0016a0dx (34)\n=\u0000ImZ2\u0019\n0(jaj2)0a\u0016a0dx\u0000\u0014Z2\u0019\n0jaj2ja0j2dx\u0000\u0014\n2Z2\u0019\n0(jaj2)0(jaj2)0dx\n\u0014\u0000ImZ2\u0019\n0(jaj2)0a\u0016a0dx:\nNext we set G:=\u0000dIm(a0\u0016a) =dIm(\u0016a0a) so thatG0=gas well asG(0) =G(2\u0019). Using the\nidentity (31) we get the pointwise estimate g\u0015\u0000f2\n4on [0;2\u0019] from which we deduce\n(35)G(x)\u0000G(0) =Zx\n0g(t)dt\u0015\u0000\u0019\n2f2(x2[0;2\u0019]) and\nG(x)\u0000G(2\u0019) =\u0000Z2\u0019\nxg(t)dt\u0014\u0019\n2f2(x2[0;2\u0019]):\nUsing the de\fnition of Gand (35) we deduce from (34)\njdjka0k2\n2\u0014\f\f\f\fdIm\u0012Z2\u0019\n0(jaj2)0\u0016a0adx\u0013\f\f\f\f=\f\f\f\fZ2\u0019\n0(jaj2)0Gdx\f\f\f\f\n\u0014Z2\u0019\n0(jaj2)0jG\u0000G(0)jdx\n\u0014\u0019f2\n2Z2\u0019\n0j(jaj2)0jdx=\u0019f2Z2\u0019\n0jajja0jdx\n\u0014\u0019f2kak2ka0k2\n\u0014p\n2\u00193=2f2~C\u0014ka0k2\nwith ~C\u0014from (33). So we \fnd\njdjka0k2\u0014p\n2\u00193=2f2~C\u0014: (36)\nFinally, we combine the previous estimates for kak2;ka0k2to deduce an L1-estimate.\nFrom (33) we obtain that there is an x12[0;2\u0019] satisfyingja(x1)j\u0014~C\u0014. Together with (36)16 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nthis implies\nkak1\u0014ja(x1)j+ka\u0000a(x1)k1\n\u0014~C\u0014+ka0k1\n\u0014~C\u0014+p\n2\u0019ka0k2\n\u0014\u0000\n1 + 2\u00192f2jdj\u00001\u0001~C\u0014:(37)\n\u0003\nWith these bounds the constancy of solutions for large \u0014is proved along the lines of the\nproof of Theorem 2 in [16]. However, from a technical point of view, several partial results\nfrom the proof presented in [16] break down and new di\u000eculties have to be overcome so that\nthe proof given next contains several new aspects.\nProof of Theorem 3. We equip the real Hilbert space H1(T) with the inner product generated\nby the norm\n(38) k\u001ek2\nH1:=\rk\u001e0k2\n2+k\u001ek2\n2 for\u001e2H1(T)\nwhere\r >0 will be suitably chosen later. We observe that a solution a: [0;2\u0019]!Cof (3)\nis constant if and only if the function A=a0is trivial. Since asolves (3) the function Ais a\n2\u0019-periodic solution of the di\u000berential equation\n(39) \u0000dA00= (i\u0000\u0010)A+ 2(1 + i\u0014)jaj2A+ (1 + i\u0014)a2\u0016A:\nWe introduce the di\u000berential operator L\u0014:H2(T)\u001aL2(T)!L2(T) by\n(40) L\u0014B:=\u0000dB00\u0000(i\u0000\u0010)B\u00002i\u0014jaj2B\u0000i\u0014a2\u0016B\nso that (39) may be rewritten as\n(41) L\u0014A= 2jaj2A+a2\u0016A:\nThe fact that L\u00001\n\u0014:L2(T)!H1(T) exists as a bounded linear operator will follow from the\ninjectivity of L\u0014, sinceL\u0014is a Fredholm operator of index 0. The injectivity is a consequence\nof the following estimate. For g2L2(T) letB2H2(T) satisfyL\u0014B=g. Testing with \u0016B\nyieldsZ2\u0019\n0\u0010\ndjB0j2\u0000(i\u0000\u0010)jBj2\u00002i\u0014jaj2jBj2\u0000i\u0014a2\u0016B2\u0011\ndx=Z2\u0019\n0g\u0016Bdx:\nTaking the real and imaginary part of this equation implies\ndkB0k2\n2+\u0010kBk2\n2+\u0014ImZ2\u0019\n0a2\u0016B2dx= ReZ2\u0019\n0g\u0016Bdx; (42)\nkBk2\n2+\u0014Z2\u0019\n0\u0010\n2jaj2jBj2+ Re(a2\u0016B2)|{z}\n\u0015jaj2jBj2\u0011\ndx=\u0000ImZ2\u0019\n0g\u0016Bdx: (43)THE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 17\nFrom (43) and \u0014\u00150 we getkBk2\u0014kgk2. Together with (42), (43) we obtain\njdjkB0k2\n2+ sign(d)\u0010kBk2\n2\u0000\u0014Z2\u0019\n0jaj2jBj2dx\u0014kgk2\n2;\nkBk2\n2+\u0014Z2\u0019\n0jaj2jBj2dx\u0014kgk2\n2:\nMultiplying the second equation with \u001b\u00151 and summing up both equations we \fnally get\njdjkB0k2\n2+ (\u001b+ sign(d)\u0010)kBk2\n2\u0014(\u001b+ 1)kgk2\n2:\nChoosing\u001bsu\u000eciently large and \rfrom (38) su\u000eciently small we obtain kBk2\nH1\u00144kgk2\n2.\nThis implies in particular the injectivity of L\u0014, consequently the boundedness of L\u00001\n\u0014:\nL2(T)!H1(T) and \fnally also the norm bound kL\u00001\n\u0014k\u00142 uniformly in \u0014>0.\nHaving proven this bound, we turn to the task to prove that solutions Aof (39) are trivial\nfor\u0014>\u0014\u0003. In view of (41) we de\fne the bounded linear operator\nKaB:=L\u00001\n\u0014\u0010\n2jaj2B+a2\u0016B\u0011\n:L2(T)!L2(T):\nIt remains to show that its operator norm is smaller than 1, because then Kais a contraction\nand therefore admits a unique \fxed point A, which must be the trivial one. Since\nk2jaj2B+a2\u0016Bk2\n2=Z2\u0019\n0\u0010\n5jaj4jBj2+ 2jaj2\u0016a2B2+ 2jaj2a2\u0016B2\u0011\ndx\u00149kak4\n1kBk2\n2\nwe \fnd that\nkKak\u00143kL\u00001\n\u0014kkak2\n1(37);(33)\n\u00146\u0000\n1 + 2\u00192f2jdj\u00001\u00012\u0010f2\n\u0014\u00112=3\n;\nwhich is smaller than 1 for \u0014>\u0014\u0003. This \fnishes the proof. \u0003\n4.Proof of Theorem 4\nLet us \frst recall a global existence and uniqueness results in the case \u0014= 0. It is\nshown in Theorem 2.1 in [11] that (1) with a(0) =\u001e2H4(T) has a unique solution a2\nC(R+;H4(T))\\C1(R+;H2(T))\\C2(R+;L2(T)). The proof of this result may be adapted to\nthe case\u0014>0 since the crucial estimate (6) in that paper is even better when \u0014>0 given\nthat the damping e\u000bect is stronger. The remaining parts of the proof need not be modi\fed\nso that we get the same estimates and gobal well-posedness result as in [11] also in the case\n\u0014 > 0. Since we will need the inequality ka(t)k2\u0014maxfp\n2\u0019jfj;ka(0)k2gin the proof of\nour convergence results, let us prove this \frst. For notational convenience we suppress the\nspatial variable in our notation.18 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nFor any given solution aof (2) the following estimate holds\nd\ndt\u0012ka(t)k2\n2\n2\u0013\n= Re\u0012Z2\u0019\n0at(t)a(t)dx\u0013\n(2)= Re\u0012Z2\u0019\n0\u0010\n(\u00001\u0000i\u0010+ (i\u0000\u0014)ja(t)j2)a(t) +f+ idaxx(t)\u0011\na(t)dx\u0013\n=\u0000ka(t)k2\n2\u0000\u0014ka(t)k4\n4+fZ2\u0019\n0Re(a(t))dx\n\u0014\u0000ka(t)k2\n2\u0000\u0014\n2\u0019ka(t)k4\n2+p\n2\u0019jfjka(t)k2:\nSoka(t)k2decreases provided the last term is negative. Since this is true is precisely for\nka(t)k2\u0015p\n2\u0019~C\u0014by (32),(33), we conclude\n(44)ka(t)k2\u0014maxfp\n2\u0019~C\u0014;ka(0)k2g(33)\n\u0014maxfp\n2\u0019jfj;ka(0)k2g for allt\u00150:\nFurthermore, using the equation for aand integration by parts we get\nd\ndt\u0012kax(t)k2\n2\n2\u0013\n= Re\u0012Z2\u0019\n0axt(t)ax(t)dx\u0013\n=\u0000Re\u0012Z2\u0019\n0at(t)axx(t)dx\u0013\n(2)=\u0000Re\u0012Z2\u0019\n0\u0010\n(\u00001\u0000i\u0010+ (i\u0000\u0014)ja(t)j2)a(t) +f+ idaxx(t)\u0011\naxx(t)dx\u0013\n=\u0000Z2\u0019\n0jax(t)j2dx\u0000\u0014Z2\u0019\n0ja(t)j2jax(t)j2dx\u00002\u0014Z2\u0019\n0Re\u0000\na(t)ax(t)\u00012dx\n\u00002Z2\u0019\n0Im\u0000\na(t)ax(t)\u0001\nRe\u0000\na(t)ax(t)\u0001\ndx:\nWritinga\u0016ax=s+ irand using the scalar inequality\n(45)\u0000\u0014(s2+r2)\u00002\u0014s2\u00002sr\u0014(\u00002\u0014+p\n1 +\u00142)|{z}\n=:\u000b\u0014(s2+r2) (s;r2R)\nwe get the estimate\nd\ndt\u0012kax(t)k2\n2\n2\u0013\n\u0014\u0000kax(t)k2\n2+\u000b\u0014Z2\u0019\n0ja(t)j2jax(t)j2dx for allt\u00150:\nSince we assumed \u0014\u00151p\n3, we have\u000b\u0014\u00140 so thatkax(t)k2\n2decays exponentially to 0.\nThe Poincar\u0013 e-Wirtinger inequality implies ka(t)\u00001\n2\u0019R2\u0019\n0a(t)dxk2decays exponentially as\nt!1 . TheL2-boundedness of a(t) derived in (44) now implies that the sequenceR2\u0019\n0a(t)dx\nis bounded, hence a(tm) converges in L2(T) for some sequence tm%1 to some constant\nsolutiona\u0003of (3). It remains to prove that this actually implies the convergence of the whole\nsequence.THE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 19\nBy the fundamental theorem of calculus we get\n(46)ka(t)\u0000a\u0003k1\u0014kax(t)k1+ min\n[0;2\u0019]ja(t)\u0000a\u0003j\u0014p\n2\u0019kax(t)k2+1p\n2\u0019ka(t)\u0000a\u0003k2:\nIn particular, the subsequence a(tm) converges uniformly to the constant a\u0003. So for any given\n\u000e2(0;1) we can \fnd an \">0 such that all h2Cwithjhj<\"satisfy the inequality\nRe\u0010\n(i\u0000\u0014)\u0010\nja\u0003+hj2(a\u0003+h)\u0000ja\u0003j2a\u0003\u0011\nh\u0011\n=\u0000\u0014ja\u0003j2jhj2\u00002\u0014\u0010\nRe\u0000\na\u0003h\u0001\u00112\n\u00002 Im\u0000\na\u0003h) Re(a\u0003h) +O(jhj3)\n(45)\n\u0014\u000b\u0014ja\u0003j2jhj2+O(jhj3)\n\u0014\u000ejhj2:(47)\nHere we used \u000b\u0014\u00140. Choosing tmlarge enough we can achieve\n(48)ka(tm)\u0000a\u0003k2\u0014p\n2\u0019\n4\" andkax(t)k2\u00141\n4p\n2\u0019\"for allt\u0015tm:\nSo the function h(t) :=a(t)\u0000a\u0003satis\fes for t\u0015tmthe following di\u000berential inequality\nprovidedkh(t)k1\u0014\"\nd\ndt\u0012kh(t)k2\n2\n2\u0013\n= Re\u0012Z2\u0019\n0@th(t)h(t)dx\u0013\n(2)=\u0000kh(t)k2\n2+ Re\u0012\n(i\u0000\u0014)Z2\u0019\n0\u0000\nja\u0003+h(t)j2(a\u0003+h(t))\u0000ja\u0003j2a\u0003\u0001\nh(t)dx\u0013\n(47)\n\u0014(\u00001 +\u000e)kh(t)k2\n2:\nGiven thatkh(tm)k1\u0014p\n2\u0019\n4\" < \" we infer thatkh(t)k2=ka(t)\u0000a\u0003k2decreases on some\nmaximal interval ( tm;tm+T) and we want to show T=1. From (48) we infer\nkh(t)k2\u0014p\n2\u0019\n4\";khx(t)k2\u00141\n4p\n2\u0019\" for allt2[tm;tm+T]\nso that (46) implies\nkh(t)k1\u0014p\n2\u0019\u00011\n4p\n2\u0019\"+1p\n2\u0019\u0001p\n2\u0019\n4\"\u0014\"\n2<\" for allt2[tm;tm+T]:\nAs shown above, this implies that kh(t)k2is decreasing on a right neighbourhood of tm+T.\nSo we conclude that there cannot be a \fnite maximal Twith the property mentioned above.\nAs a consequence, T=1,kh(t)k2is decreasing on [ tm;1) and we obtainka(t)\u0000a\u0003kH1(T)=\nkh(t)kH1(T)!0 as claimed. This \fnishes the proof. \u0003\nWe add an extension of this result that covers damping parameters \u0014 <1p\n3. In this case\nwe may obtain the convergence of the \row provided the initial condition \u001e=a(0) has the\nproperty thatk\u001exk2andk\u001ek2are not too large.20 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nLemma 13. Assumed6= 0;\u0010;f2Rand\u0014<1p\n3. Assume that the initial condition a(0) =\n\u001e2H4(T)satis\fes\n(49) 2 \u0019k\u001exk2+ maxfp\n2\u0019jfj;k\u001ek2g<r\n2\u0019\n\u000b\u0014:\nThen the uniquely determined solution a2C(R+;H4(T))of(2)converges in H1(T)to a\nconstant.\nProof. We argue as above. Using the same estimate as in the above proof we get now for\n\u000b\u0014>0\nd\ndt\u0012kax(t)k2\n2\n2\u0013\n\u0014\u0000kax(t)k2\n2+\u000b\u0014Z2\u0019\n0ja(t)j2jax(t)j2\n\u0014(\u00001 +\u000b\u0014ka(t)k2\n1)kax(t)k2\n2\n(46)\n\u0014\u0012\n\u00001 +\u000b\u0014(p\n2\u0019kax(t)k2+1p\n2\u0019ka(t)k2)2\u0013\nkax(t)k2\n2\n(44)\n\u0014\u0012\n\u00001 +\u000b\u0014(p\n2\u0019kax(t)k2+1p\n2\u0019maxfp\n2\u0019jfj;ka(0)k2g)2\u0013\nkax(t)k2\n2:\nSo the prefactor is negative for small t>0 by assumption (49). Hence, by monotonicity, it\nremains negative for all t>0 and we conclude as above. \u0003\nWe do not know whether the above convergence result is sharp in the sense that there are\ninitial data causing non-convergence or even blow-up in in\fnite time. As above we moreover\ninfer that all nonconstant stationary solutions afor\u0014<1p\n3satisfy\n2\u0019kaxk2+ maxfp\n2\u0019jfj;kak2g\u0015r\n2\u0019\n\u000b\u0014:\n5.Proof of Theorem 5\nIn this section we discuss (7) in the case of anomalous dispersion d>0, and we will prove\nthe existence of solitary-type localized solutions. At the end of this section we explain why\nour method fails in the case of normal dispersion d<0.\nLet us consider a rescaled version of (7) given by\n(50)\u0000du00+ (~\u0010\u0000\"i)u\u0000(1 + i\u0014)juj2u+ i~f= 0 on R; u0(0) = 0\nford;~\u0010 > 0 and\";\u0014\u00150. Notice that usolves (50) with ~\u0010;~fif and only if a(x) :=\n\"\u00001=2u(\"\u00001=2x) solves (3) with \u0010=~\u0010\"\u00001andf=~f\"\u00003=2onR.\nWe consider solutions of (50) of the form u= ~u+u1, where ~ubelongs to the space\nH2\neven(R;C) of even complex-valued H2-functions on the real line and u1:= limjxj!1u(x)\nsolves the algebraic equation\n(~\u0010\u0000\"i)u\u0000(1 + i\u0014)juj2u+ i~f= 0: (51)THE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 21\nThe strategy is to \fnd two purely imaginary solutions of (50) in the special case \"=\u0014= 0\nand to continue them into the situation \";\u0014 > 0 via the implicit function theorem. More\nprecisely, Theorem 5 is proved once we have shown Theorem 17 below.\nLet us begin with the case \u000f=\u0014= 0, where we consider solutions of\n\u0000du00+~\u0010u\u0000juj2u+ i~f= 0 on R; u0(0) = 0 (52)\nand whereu1= limjxj!1u(x) satis\fes\n~\u0010u\u0000juj2u+ i~f= 0: (53)\nWe will always work in the setting where (53) has three distinct solutions. Let us brie\ry\nexplain why this is ful\flled for 0 \u0014j~fj<2p\n3\n9~\u00103=2. Clearly, (53) only has purely imaginary\nsolutionsu1= iv1, wherev12Rsolves\n\u0000~\u0010v+v3=~f: (54)\nThe function v7!\u0000 ~\u0010v+v3has the local minimum \u00002p\n3\n9~\u00103=2and the local maximum2p\n3\n9~\u00103=2.\nTherefore, if 0\u0014j~fj<2p\n3\n9~\u00103=2then there are three distinct solutions v(j),j= 1;2;3 of (54)\nwithv(1)<\u0000p\u0010p\n3<v(2)<p\u0010p\n3<v(3).\nNext let us discuss the existence of two homoclinic solutions of (52). Their nondegeneracy\nwill be proved in Proposition 15.\nProposition 14. Letd;~\u0010 >0and0\u0014j~fj<2p\n3\n9~\u00103=2. There exist two purely imaginary and\neven solutions ui= ~ui+u1\niof(52) with ~ui2H2\neven(R;C)fori= 1;2andxIm(u0\n1)>0and\nxIm(u0\n2)<0onRnf0g.\nProof. Looking for purely imaginary even homoclinic solutions u= ivof (52) means that we\nneed to \fnd a real-valued even homoclinic solution vof\n\u0000dv00+~\u0010v\u0000v3+~f= 0 on R: (55)\nThe corresponding \frst integral is given by\nI(v0;v) :=\u0000dv02+~\u0010v2\u00001\n2v4+ 2~fv:\nAll trajectories of (55) are therefore bounded in the ( v;v0)-plane and symmetric with respect\nto thev-axis. Moreover, every trajectory crosses the v-axis.\nThe equilibria of (55) are given by the solutions of the algebraic equation (54). As we have\nseen, there are three distinct real-valued solutions v(j);j= 1;2;3 for 0\u0014j~fj<2p\n3\n9~\u00103=2. The\neigenvalues of the linearization in v(j)satisfy\n\u0015(j)\n1;2=\u0006p\n\u0000\u0001(j)=p\nd with \u0001(j):=\u0000~\u0010+ 3(v(j))2:\nThe linear stability analysis, which allows us to characterize the equilibria of the nonlinear\nsystem, reduces to the analysis of \u0001(j). Observe that we have \u0000~\u0010+3v2<0 on (\u0000p\u0010p\n3;p\u0010p\n3) and\n\u0000~\u0010+ 3v2>0 onRn[\u0000p\u0010p\n3;p\u0010p\n3]. Hence, for v(1)<v(2)<v(3), we have \u0001(j)>0 forj= 1;3 and22 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nvv0\nFigure 5. Homoclinic orbits for \"=\u0014= 0\n\u0001(j)<0 forj= 2. This means that for j= 2 we have two real eigenvalues of opposite sign,\nand the equilibrium is an unstable saddle. For j= 1;3, we have purely imaginary eigenvalues\nof opposite sign, and hence, these equilibria are stable centers surrounded by periodic orbits.\nSince the unstable manifold of the saddle is symmetric around the v-axis it connects to\nthe stable manifold and thus provides the two homoclinic orbits. \u0003\nFor the following nondegeneracy result let us recall from Section 2 the notation g(u) =\njuj2u\u0000if,Dg(u)z= 2juj2z+u2\u0016zforu;z2C.\nProposition 15. Letd;~\u0010 > 0with 0<j~fj<2p\n3\n9~\u00103=2. Ifu1;u2are the two homoclinic\nsolutions of (52) then\nkerH2(R;C)\u0010\n\u0000dd2\ndx2+~\u0010\u0000Dg(ui)\u0011\n= spanfu0\nig\nfori= 1;2.\nRemark 16. Here kerH2(R;C)(\u0000dd2\ndx2+~\u0010\u0000Dg(ui))refers to the kernel of the di\u000berential\noperator on the domain H2(R;C). As a consequence, if we set the domain of the di\u000berential\noperator as H2\neven(R;C)we get kerH2even(R;C)(\u0000dd2\ndx2+~\u0010\u0000Dg(ui)) =f0gfori= 1;2. This is\ntrue because H2\neven(R;C)only contains functions with u0(0) = 0 so thatu0\ni62H2\neven(R;C)for\ni= 1;2because (52) yields\ndu00\ni(0) = ~\u0010ui(0)\u0000jui(0)j2ui(0) + i ~f:\nThe latter expression is non-zero since ui(0)6= iv(j)forj= 1;2;3andiv(1);iv(2);iv(3)are the\nsolutions of (54). Note also that the proposition applies only for ~f6= 0 because for ~f= 0\nscaling with a complex phase factor produces another degeneracy so that spanfu0\ni;iuig \u001a\nkerH2(R;C)(\u0000dd2\ndx2+~\u0010\u0000Dg(ui))and this time iui2H2\neven(R;C).\nProof. We prove nondegeneracy only for i= 1. Since u0\n12H2(R;C), we may di\u000berentiate\n(52) to see that u0\n12kerH2(R;C)\u0010\n\u0000dd2\ndx2+~\u0010\u0000Dg(u1)\u0011\n.THE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 23\nFor the converse inclusion, let '='1+i'22kerH2(R;C)\u0010\n\u0000dd2\ndx2+~\u0010\u0000Dg(u1)\u0011\nfor real-valued\n'1;'2. Then we have\n\u0010\n\u0000dd2\ndx2+~\u0010\u0000v2\n1\u0011\n'1= 0 on R; (56)\n\u0010\n\u0000dd2\ndx2+~\u0010\u00003v2\n1\u0011\n'2= 0 on R; (57)\nand we need to show that '1= 0 and that '2is a real multiple of v0\n1. Due to (55) we also\nsee that\n(58)\u0000dv000\n1+ (~\u0010\u00003v2\n1)v0\n1= 0 on RandZ1\n0d(v00\n1)2\u0000(~\u0010\u00003v2\n1)(v0\n1)2dx= 0:\nWe split'1:='1;even+'1;oddinto even and odd part. Then we observe that '1;even2H2(R;R)\nsolves (56) with '0\n1;even(0) = 0 and that '1;odd2H2(R;R) solves (56) with '1;odd(0) = 0.\nLet us \frst show that either '1;even\u00110 or'1;even has no zero on R. Indeed, if '1;even\nhad a \frst positive zero x0>0 with'1;even(x0) = 0 then w.l.o.g. '1;even>0 on (0;x0).\nSince'1;even(x)!0 asx!1 there isx12(x0;1] such that '1;even<0 on (x0;x1) and\nlimx!x1'1;even(x) = 0. If we multiply the di\u000berential equation in (58) by '1;evenand subtract\n(56) multiplied by v0\n1then we \fnd\n0 =Zx1\nx0\u0000d(v000\n1'1;even\u0000'00\n1;evenv0\n1)dx\u0000Zx1\nx02v2\n1v0\n1|{z}\n>0'1;even|{z}\n<0dx\n\u0015\u0000Zx1\nx0dd\ndx(v00\n1'1;even\u0000'0\n1;evenv0\n1)dx\n=\u0000d0\nB@'0\n1;even(x0)|{z}\n<0v0\n1(x0)|{z}\n>0\u0000'0\n1;even(x1)|{z}\n\u00150v0\n1(x1)|{z}\n\u001501\nCA>0:\nThis is impossible and proves the assertion that '1;evenhas no zero on R. An almost identical\nargument applied to '1;oddprovides the alternative '1;odd\u00110 or'1;oddhas no zero on (0 ;1).\nNow suppose '1;odd6\u00110. Then'1;odd2H1\n0((0;1);R) is w.l.o.g a positive Dirichlet eigen-\nfunction to the eigenvalue 0 of L1:=\u0000dd2\ndx2+\u0010\u0000v2\n1on (0;1). Observe that v0\n12H1\n0((0;1);R)\nis a positive Dirichlet eigenfunction of L2:=\u0000dd2\ndx2+\u0010\u00003v2\n1on (0;1) corresponding to the\nsmallest eigenvalue 0. We also have the following inequality between the quadratic forms of\nL2andL1Z1\n0d(\u001e0)2+ (\u0010\u00003v1)2\u001e2dx<Z1\n0d(\u001e0)2+ (\u0010\u0000v1)2\u001e2dx\nfor all\u001e2H1\n0((0;1);R)nf0g. Therefore, by Poincar\u0013 e's min-max principle we obtain a strict\nordering between the Dirichlet eigenvalues of L1andL2, and hence the smallest Dirichlet\neigenvalue of L1is strictly positive, which yields a contradiction. This implies '1;odd\u00110.24 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nLet us now consider the even part '1;evenand suppose that '1;evenhas no zero on (0 ;1).\nTesting (55) with '1;even, integrating twice we obtain\n06=\u0000Z1\n0~f'1;evendx=Z1\n0\u0000dv00\n1'1;even+ (\u0010\u0000v2\n1)v1'1;evendx\n=Z1\n0\u0000dv1'00\n1;even+ (\u0010\u0000v2\n1)v1'1;evendx= 0:\nThis is a contraction, and hence, '1;even\u00110. Together with '1;odd\u00110 we \fnally see '1\u00110.\nNow we show that '2is a multiple of v0\n1. Multiplying the di\u000berential equation in (58) with\n'2, (57) with v0\n1and subtracting we obtain\n0 =\u0000d(v000\n1'2\u0000'00\n2v0\n1) =\u0000dd\ndx(v00\n1'2\u0000'0\n2v0\n1):\nAs bothv0\n1and'2together with their derivatives vanish at in\fnity we obtain\nv00\n1'2\u0000'0\n2v0\n1= 0;\nwhich means that '2is a multiple of v0\n1. This concludes the proof of the proposition. \u0003\nNow we will continue the purely imaginary nontrivial solutions u1;u2of (52) from Propo-\nsition 15 into the range where \";\u0014> 0. For the proof of the \fnal result, we rewrite (50) for\nu= ~u+u1\n\u000f;\u0014with ~u2Has follows\n\u0000d~u00+~\u0010~u\u0000\"i~u\u0000(1 + i\u0014)(g(~u+u1\n\";\u0014)\u0000g(u1\n\";\u0014)) = 0: (59)\nHereu1\n\";\u0014is given as the continuation of i v(2)into the range of \";\u0014 > 0. Note that three\ndistinct solutions of (51) persists for small \";\u0014> 0.\nTheorem 17. Letu1= ~u1+iv(2);u2= ~u2+iv(2)be the two solutions of (50) for(\";\u0014) = (0;0)\nfrom Proposition 14. Then there exist open neighborhoods Uiof~uiinH2\neven(R;C),Jiof(0;0)\ninR\u0002Rsuch that (59) is uniquely solvable for (~u;\u000f;\u0014 )inUi\u0002Ji,i= 1;2.\nProof. We de\fne F:H2\neven(R;C)\u0002R\u0002R!L2\neven(R;C) =f~u2L2(R;C) : ~u(\u0000x) =\n~u(x) for a.a.x2Rgby\nF(~u;\";\u0014 ) :=\u0000d~u00+~\u0010~u\u0000\"i~u\u0000(1 + i\u0014)(g(~u+u1\n\";\u0014)\u0000g(u1\n\";\u0014)):\nThen we have F(~ui;0;0) = 0 by de\fnition of ~ uiand@F\n@~u(~ui;0;0) =\u0000dd2\ndx2+~\u0010\u0000Dg(ui) :\nH2\neven(R;C)!L2\neven(R;C). Due to Remark 16 we know that ker H2even(R;C)(@F\n@~u(~ui;0;0)) =f0g\nfori= 1;2. Since@F\n@~u(~ui;0;0) is a Fredholm operator of index 0, it has a bounded inverse\nand thus the statement of the theorem follows from the implicit function theorem. \u0003\nRemark 18. Let us denote one of the two solution families of Theorem 17 by u\";\u0014. Taking\ninto account the rescaling a\";\u0014(x) =\"\u00001=2u\";\u0014(\"\u00001=2x)we have proved Theorem 5. Moreover,\n\r\r\ra\";\u0014\u0000lim\njxj!1a\";\u0014(x)\r\r\r\nH2\u0015\"\u00001=4\r\r\ru\";\u0014\u0000lim\njxj!1u\";\u0014(x)\r\r\r\nH2!1\nfor\"!0uniformly with respect to \u0014.THE LUGIATO-LEFEVER EQUATION WITH NONLINEAR DAMPING 25\nWe \fnish our discussion with a brief analysis of the case d<0 (normal dispersion). Here,\nwe also consider the rescaled equation (50) and write it in the form\n(60)\u0000jdju00+ (\"i\u0000~\u0010)u+ (1 + i\u0014)juj2u\u0000i~f= 0 on R; u0(0) = 0:\nStarting with \"=\u0014= 0 we consider purely imaginary solutions. The equilibria in the\nphase plane for (55) are the same as before, but due to d<0 their character changes. The\neigenvalues of the linearization are now given by\n\u0015(j)\n1;2=\u0006ip\n\u0000\u0001(j)=p\njdjwith \u0001(j):=\u0000~\u0010+ 3(v(j))2\nforj= 1;2;3. Now we have a center for j= 2 and two unstable saddles for j= 1;3. The\nunstable saddles are connected by two heteroclinic solutions. Going back to (60) we have\nfor\"= 0 two heteroclinic solutions u1;u2with Im(u0\n1)>0 and Im(u0\n2)<0 onR. Moreover\nlimx!1u1(x) = limx!\u00001u2(x) =u(3)= iv(3), limx!\u00001u1(x) = limx!1u2(x) =u(1)= iv(1).\nFor\";\u0014> 0 the unstable saddles persist and one might try to continue the heteroclinic solu-\ntionsu1;u2into the range \";\u0014> 0. Let us explain why the previous continuation argument\nfails in the case of u1(the argument for u2is the same). One could seek for heteroclinic\nsolutions of the form\nu= ~u+ \";\u0014with ~u2H2(R)\nand where \";\u0014is a smooth given function of x, continuous in \";\u0014with\n 0;0=u1and lim\nx!1 \";\u0014(x) =u(3)\n\u000f;\u0014;lim\nx!\u00001 \";\u0014(x) =u(1)\n\u000f;\u0014\nwhereu(j)\n\";\u0014are the continuations of the purely imaginary zeros u(j)of (51) into the range\n\";\u0014> 0. The implicit function continuation argument applied to\nF(\u000f;\u0014;~u) =\u0000jdj(~u+ \";\u0014)00+\u000fi(~u+ \";\u0014)\u0000g(~u+ \";\u0014)\nwould then provide ~ uas a function of \"and\u0014. Due to 0;0=u1we haveF(0;0;0) = 0\nand the linearized operator is given by@F\n@~u(0;0;0) =\u0000jdjd2\ndx2\u0000~\u0010+Dg(u1) :H2(R)!L2(R).\nNow there is the question of nondegeneracy of u1. Sinceu1is purely imaginary,@F\n@~u(0;0;0)\ndecouples into two real-valued, selfadjoint operators\nL1:=\u0010\n\u0000jdjd2\ndx2\u0000~\u0010+v2\n1(x)\u0011\n:H2(R)!L2(R); (61)\nL2:=\u0010\n\u0000jdjd2\ndx2\u0000~\u0010+ 3v2\n1(x)\u0011\n:H2(R)!L2(R): (62)\nSincev(j)solves (\u0000~\u0010+v2)v=f > 0 forj= 1;2;3 andv(1)<0< v(3)we see that\n\u0000~\u0010+ limx!\u00001v2\n1(x) =\u0000~\u0010+ (v(1))2<0 and\u0000~\u0010+ limx!\u00001v2\n1(x) =\u0000~\u0010+ (v(3))2>0. Hence\nwe get for the essential spectrum of L1the relation\n\u001bess(L1) = [\u0000~\u0010+ (v(1))2;1)\nand 02\u001bess(L1). 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Nature , 416(6877):233{237,\n2002.\nAcknowledgments\nWe gratefully acknowledge \fnancial support by the Deutsche Forschungsgemeinschaft\n(DFG) through CRC 1173.\nJ. Gartner\nKarlsruhe Institute of Technology\nInstitute for Analysis\nEnglerstra\u0019e 2\nD-76131 Karlsruhe, Germany\nE-mail address :Janina.Gaertner@kit.edu\nR. Mandel\nKarlsruhe Institute of Technology\nInstitute for Analysis\nEnglerstra\u0019e 2\nD-76131 Karlsruhe, Germany\nE-mail address :Rainer.Mandel@kit.edu28 JANINA G ARTNER, RAINER MANDEL, WOLFGANG REICHEL\nW. Reichel\nKarlsruhe Institute of Technology\nInstitute for Analysis\nEnglerstra\u0019e 2\nD-76131 Karlsruhe, Germany\nE-mail address :Wolfgang.Reichel@kit.edu"    },    {        "title": "1811.12626v2.Dynamical_precession_of_spin_in_the_two_dimensional_spin_orbit_coupled_systems.pdf",        "content": "arXiv:1811.12626v2  [cond-mat.mes-hall]  4 Jun 2019Dynamical precession of spin in the two-dimensional spin-o rbit coupled systems\nTsung-Wei Chen,1,∗Zhi-Yang Huang,1and Dah-Wei Chiou1,2,†\n1Department of Physics, National Sun Yat-sen University, Ka ohsiung 80424, Taiwan\n2Center for Condensed Matter Sciences, National Taiwan Univ ersity, Taipei 10617, Taiwan\n(Dated: June 5, 2019)\nWe investigate the spin dynamics in the two-dimensional spi n-orbit coupled system subject to\nan in-plane ( x-yplane) constant electric field, which is assumed to be turned on at the moment\nt= 0. The equation of spin precession in linear response to the switch-on of the electric field is\nderived in terms of Heisenberg’s equation by the perturbati on method up to the first order of the\nelectric field. The dissipative effect, which is responsible for bringing the dynamical response to an\nasymptotic result, is phenomenologically implemented ` a lathe Landau-Lifshitz-Gilbert equation by\nintroducing damping terms upon the equation of spin dynamic s. Mediated by the dissipative effect,\nthe resulting spin dynamics asymptotes to a stationary stat e, where the spin and the momentum-\ndependenteffectivemagnetic fieldarealigned again andhave nonzerocomponentsintheout-of-plane\n(z) direction. In the linear response regime, the asymptotic r esponse obtained by the dynamical\ntreatment is in full agreement with the stationary response as calculated in the Kubo formula, which\nis a time-independent approach treating the applied electr ic field as completely time-independent.\nOur method provides a new perspective on the connection betw een the dynamical and stationary\nresponses.\nPACS numbers: 71.70.Ej, 72.25.Dc, 73.43.Cd, 75.47.-m\nI. INTRODUCTION\nThe phenomenon of the spin-Hall effect is the appear-\nance of lateral bulk spin current in the spin-orbit coupled\nsystems driven solely by applying an electric field [1, 2].\nThe fact that the spin-Hall current, arising from the sep-\naration of opposite spin orientations without breaking\nthe time-reversal symmetry, is dissapationless, has enor-\nmous advantages in the development of spintronics [3].\nA lot of attention has been devoted to investigating the\ntheoretical foundations [4–6] of the spin-Hall effect and\nperforming the experiments [7] that test the validity of\ntheory [8] and advance the technology of spintronics.\nIntwo-dimensional(2D)spin-orbitcoupledsystems[9–\n12], thespin-Halleffectbecomesveryimportant, notonly\nfor its relation to the topological Berry phase [13–16]\nbut also for the development of the quantum spin-Hall\neffect [17–19] (2D topological insulators) and Chern in-\nsulators [20], where the definition of the bulk spin cur-\nrent [21–23] plays the key role in the bulk-edge corre-\nspondence [17]. Recently, it was shown that the spin-\nHall effect in the two-dimensionalWeyl fermion system is\ncaused by the spin torque current [24]. The phenomenon\nof the bulk spin current is usually understood as the sta-\ntionaryresponse of the system to the applied in-plane\nelectric field, which, as well known, can be directly cal-\nculated by the Kubo formula. However, the dynamical\norigin of this response remains rather mysterious. That\nis, if the applied electric field is switched on at the mo-\nmentt= 0, how does the (expectation value of) spin\ndynamically evolve from its original in-plane direction to\n∗twchen@mail.nsysu.edu.tw\n†dwchiou@gmail.comyield an out-of-plane component and eventually asymp-\ntote to the stationary value?\nThe connection between the dynamical evolution and\nthe stationary response is essential to understanding the\nunderlying mechanism of the spin-Hall effect. This con-\nnection has been addressed for the Berry curvature in-\nduced spin dynamics in the 3D p-type semiconductor [4],\nthe 2D k-linear Rashba system [5], the semiclassical\nDrude model [25], and the stationary response of the ki-\nnetic equation [26]. Recently, the spin dynamics in the\nhoneycomb lattice [27] has also been investigated using\nthe Landau-Lifshitz-Gilbert equation [28], by which the\nclassical and quantum correspondence appears at low-\nenergy spectra. Besides theoretical importance, under-\nstanding the dynamical evolution of spin is also crucial\nto the performance of spintronic devices, which largely\ndepends on their response time to the applied field. In\nthis work, we focus on the two-dimensional spin-orbit\ncoupled system subject to a constant in-plane electric\nfield switched on at t= 0.\nFor the two-dimensional spin-orbit coupled systems,\nthe Hamiltonian before the electric field is turned on is\nin general written as H0=ǫk+σxdx(k)+σydy(k), where\nǫkis the kinetic energy, dis referred to as the effective\nmagneticfield, and σiarePaulimatricesrepresentingthe\nreal electron spin in the system. It follows from the alge-\nbra[σi,σj] = 2iǫijkσkthat the equationofmotion ofspin\n(i.e., spin precession) is described by the Larmor preces-\nsion around the direction of d. As the quantum average\nof spin is parallel to the effective magnetic field, its com-\nponent in the out-of-plane ( z) direction remains zero if\nthe effective magnetic field is in the in-plane ( x-y) direc-\ntion. Therefore, in order to have a nonzero stationary\nresponse of zcomponent of spin, the effective magnetic\nfield has to be tilted from the in-plane direction by some2\nmechanism.\nIt turns outthat applyingan in-planeelectric field pro-\nvides such a mechanism. The quantum average of spin is\ngiven by the stationary response to the applied electric\nfield [16], which exhibits a nonzero component in the z\ndirection. That is, in the presence of an in-plane electric\nfield, the total Hamiltonian H=H0+eE·x, when evalu-\nated as expectation values with respect to the eigenstates\nofH0, can be rendered into the form σ·D(k), where\nD(k) represents the new effective magnetic field and has\na nonzero zcomponent. However, the dynamical origin\nof the spin- zcomponent — especially, the question how\nthe electric filed tilts the effective magnetic field from the\nin-plane direction — remains obscure.\nTo address the dynamical issues, we treat the Hamilto-\nnianastime-dependent byassumingthattheelectricfield\nis turned off for t <0 and turned on for t >0. Heisen-\nberg’s equation is then used to solve the dynamical evo-\nlution perturbatively up to the first order of the electric\nfield. In this dynamical picture, the effective magnetic\nfield becomes nonstatic due to the switch-on of the elec-\ntric field and exhibits a time-dependent component on\nthex-yplane. The time-varying effective magnetic field\nis no longer aligned with the spin for t >0 and therefore\ndrives the spin to precess around it.\nThe dynamical evolution is expected to asymptote to\na stationary result, where the spin and the effective mag-\nnetic field are aligned again. To obtain the asymptotic\nbehavior, we have to take into account the dissipative\nprocess that attenuates and eventually ceases the spin\nprecession. The fundamental mechanism for the dissipa-\ntion remains unclear and complicated [29], but it can\nbe phenomenologically implemented ` a lathe Landau-\nLifshitz-Gilbert equation [28] by introducing damping\nterms upon the equation of spin dynamics obtained from\nthe first-order perturbation. Via the dissipative process,\nthe precession of spin gives rise a backreaction that al-\nters the effective magnetic field and tilts it from the x-y\nplane. The resulting dynamical evolution asymptotically\napproaches a stationary state, where the spin and the\neffective magnetic field are aligned again and both have\nnonzero components in the zdirection.\nMeanwhile, we also directly compute the stationaryre-\nsponsein atime-independent approachwherethe electric\nfield is treated as always turned on. The linear term of\nthestationaryresponseisexactlyequaltotheasymptotic\nresult obtained in the dynamical analysis. Our dynami-\ncaltreatment notonly revealsthe dynamicaloriginofthe\nspin-zcomponent in terms of the dynamical response to\nthe switch-on of the electric field but also establishes its\nconnection to the stationary response. In particular, we\nuncover that the dissipative effect is crucial for connect-\ning the dynamical and stationary responses.\nThis paper is organized as follows. In Sec. II, the spin\ndynamics for spin-orbit coupled systems subject to an\nconstant electric field turned on at t= 0 is derived in\nterms of Heisenberg’s equation up to the first order of\nthe electric filed. In Sec. III, the equation of spin dynam-ics is explicitly solved for a two-dimensional system. By\nphenomenologically implementing the dissipative effect,\nthe spin dynamics is shown to approach an asymptotic\nresult. In Sec. IV, we use the time-independent method\nto directly calculate the stationary response of spin. The\nlinear response of the spin- zcomponent is exactly the\nsame as the asymptotic result obtained from the spin\ndynamics. The spin-Hall current in relation to spin- z\ncomponent is also discussed in this section. Finally, our\nconclusion is summarized and discussed in Sec. V.\nII. EQUATION OF SPIN DYNAMICS IN 3D\nSYSTEMS\nIn the presence of a constant and in-plane electric field\nE, the full Hamiltonian is given by\nH=H0+eE·x, (1)\nwhereH0isthe unperturbed Hamiltonian, andin general\ncan be written as the form\nH0=ǫk+σxdx(k)+σydy(k)+σzdz(k),(2)\nwhereσx,σyandσzare Pauli matrices and represents\nreal electron spin. The Hamiltonian H0is time-reversal\nsymmetric as it is invariant under the transformation of\nσi→ −σi,di→ −diandk→ −k.\nTo study the spin dynamics, we have to treat the full\nHamiltonian Hastime-dependentwithatime-dependent\nE, assuming that E(t)→0 ast→ −∞andE(t)→E0\nast→ ∞. Generally, it is challenging to solve the time-\ndependent Schr¨ odinger (or, equivalently, Heisenberg’s)\nequation. One strategy is to expanded E(t) in a Fourier\nseries in time (e.g., see [30]), but adding the results of all\nFourier modes together usually only yields a formal sum\nexpression. Another strategy is to model E(t) as an ex-\nponentially growing function given by E(t) =E0estwith\na regularizing parameter s >0 (e.g., see [31, 32]). To\nobtain the static response as the asymptotic result, the\nregularization is removed in the end by taking the s→0\nlimit. However, removing the regularization also erases\nthe dynamical evolution in response to the switch-on of\nE(t), which we aim to investigate.\nWe adopt a different approach, assuming that E(t) is\nswitched off for t <0 and abruptly switched on to a\nconstant value E0≡Efort >0. More precisely, Eq. (1)\nis modified as\nH=H0+eE·xθ(t), (3)\nwhereθ(t) is the step function. The advantage of this\nmodeling is that Hbecomes time-independent again for\nt >0 andthusthe evolutionoperator eiHt//planckover2pi1canbe easily\nexpanded order by order for t >0, enabling us to con-\nduct the perturbation method. One might raise doubt as\nto whether the model is legitimate, as the abrupt switch-\non gives discontinuity at t= 0. It turns out nothing3\nis illegitimate. Because the time evolution is governed\nby the Schr¨ odinger or, equivalently, Heisenberg’s equa-\ntion, which is a first-order differential equation in time,\ntheresultingsolutionremainscontinuous(moreprecisely,\nof classC0) with respect to t, even if the time-varying\npotential is given discontinuous (more precisely, of class\nC−1).1By exploiting the fact that the solution is of C0,\nwe focus on the dynamics for t >0 and take the value\natt= 0−as the initial condition for t≥0+.2The fact\nthat the resulting solution is continuous but not differ-\nentiable at t= 0 (i.e., of class C0but notC1) is just an\nartifact due to the idealized (but still legitimate) model-\ning of abruptness of the switch-on. In reality, the electric\nfield can in principle be switched on as abruptly as pos-\nsible, but it remain smooth if measured with arbitrarily\nhigh resolution in time.\nAccording to Eq. (3), the corresponding equation of\nmotion of momentum is given by\n/planckover2pi1dkt\ndt=−eEθ(t). (4)\nThe solution of Eq. (4) is kt=k−eEt//planckover2pi1fort≥0, where\nkis defined as the momentum at t= 0−. Because the\ndynamics of momentum kt=k−eEt//planckover2pi1shows that time\nand electric field couples in the form Et, this implies that\nthe linearresponseisvalid alsoforaveryshorttime. The\nHeisenberg picture of an observable Ois defined as\nOH(t) =eiHt//planckover2pi1Oe−iHt//planckover2pi1. (5)\nThe dynamics of spin can always be cast in the following\nHeisenberg’s equation\n∂\n∂tσH(t) =ΩH(t)×σH(t) (6)\nfor some function ΩH(t), which is referred to as the ef-\nfective magnetic filed.3Eq. (6) can be exactly solved if\nΩH(t) =Ωis time-independent. For the time-dependent\neffective magnetic field ΩH(t), as far as we know, Eq. (6)\nhas no exact solution in general due to the complication\nthat the unitary transformation for diagonalizing the ef-\nfective magnetic field is time dependent.\nWe can solve Eq. (6) perturbatively by expanding the\neffective magnetic field ΩH(t) in series of different orders\nof the applied electric field: ΩH(t) =Ω0+Ω1(t)+o(λ2),\n1For a first-order differential equation αx′(t) +βx(t) +f(t) = 0,\nif the driving term f(t) is of class Cn, the resulting solution x(t)\nis of class Cn+1. Also note that the step function θ(t) is of class\nC−1.\n2Accordingly, in the rest of this paper, the equations of moti on\nare derived only for t >0 (the point t= 0+is not included),\nunless stated otherwise.\n3Rigorously speaking, it is −ΩH, not +ΩH, that should be re-\nferred to as the effective magnetic field (in kspace). In this\npaper, we nevertheless call + ΩHthe effective magnetic field for\nconvenience.whereΩ0is ofo(λ0) andΩ1is ofo(λ). The dimension-\nless perturbative parameter λwith|λ|<1 is given as a\nconstat proportional to Eas\nλ=el\n/planckover2pi1Ω0E, (7)\nwhere Ω 0= 2|d|//planckover2pi1is the interband gap of the unper-\nturbed system H0andldenotes a length scale associ-\nated with k. The condition |λ|<1 is understood as that\nthe energy induced by Ehas to be smaller than the in-\nterband gap so that interaction between the upper and\nlower bands remains negligible.4The length scale l(k) is\nabout how far a wave packet centered at ktravels and\ntherefore is given by l(k) =|vg(k)|twithvg(k) =∇kǫk\nbeing the group velocity of the band at the point k. The\n|λ|<1 condition then implies that the result of the first-\norderperturbationis valid onlyif tis shortenough. More\nprecisely, the short-time condition is given by\nt </planckover2pi1Ω0\nevg|E|. (8)\nThe spin is also expanded in series accordingly:\nσH(t) =σH0(t) +σλ(t) +o(λ2). Up to the linear or-\nder ofλ, Eq. (6) can be written as\n∂\n∂tσλ(t) =Ω0×σλ(t)+Ω1(t)×σH0(t),(9)\nwhere the unperturbed equation\n∂\n∂tσH0(t) =Ω0×σH0(t) (10)\nwas used. The right hand side of Eq. (9) exhibits two\ndifferent kinds of torque. The first one Ω0×σλ(t) gives\nthe Larmor precession around the direction of the static\nmagnetic field Ω0. The second one Ω1(t)×σH0(t) gives\na non-Larmor precession.\nAsEis turned on at t= 0, the initial state at t= 0 is\ngiven by the eigenstate of the unperturbed Hamiltonian\nH0denoted as |nk/an}b∇acket∇i}ht, withnbeing the band index. We\nnow apply the expectation value /an}b∇acketle{tnk|OH(t)|nk/an}b∇acket∇i}ht, which\nis also simply written as /an}b∇acketle{tOH(t)/an}b∇acket∇i}ht. For the unperturbed\nsystem, we have /an}b∇acketle{tnk|OH0(t)|nk/an}b∇acket∇i}ht=/an}b∇acketle{tnk|O|nk/an}b∇acket∇i}ht. Conse-\nquently, Eq. (9) leads to\n∂\n∂t/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht=Ω0×/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht+Ω1(t)×/an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht.(11)\n4Note that Ω 0→0 ask→0, and the condition |λ|<1 cannot\nbe satisfied in the vicinity of the Gamma point k= 0. It is\na consequence of the fact that the perturbation method break s\ndown whenever the energy levels become degenerate. However ,\nin the end when we sum up contributions over all points in\nthe bands, only those inside the annulus circumscribed by th e\nFermi surface contribute. The region close to k= 0 is ex-\ncluded and thus causes no problem. More rigorously, when\nthe whole of the system is considered, Eq. (7) is given by\n|λ|= maxk−\nF≤|k|≤k+\nFel(k)|E|//planckover2pi1Ω0(k), where k±\nFare the upper\nand lower bounds of the Fermi momentum.4\nEq. (11) is the equation of motion of spin subject to the\nextra torque Ω1(t)× /an}b∇acketle{tσH0(t)/an}b∇acket∇i}htin addition to the Lar-\nmor torque. Furthermore, /an}b∇acketle{tσH0(t)/an}b∇acket∇i}htis parallel (or anti-\nparallel) to Ω0. This can be deduced from the fact that\nthe left hand side of Eq. (10) vanishes because /an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht\nis independent of time. Therefore, the second term of\nEq. (11) becomes Ω1× /an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht=±Ω1×Ω0/Ω0, i.e.,\nperpendicular to Ω0. This implies that the linear re-\nsponse/an}b∇acketle{tσλ(t)/an}b∇acket∇i}htis always perpendicular to Ω0,\nΩ0·/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht= 0. (12)\nEq. (12) can also be obtained as follows. Because the\nfirst term and the second term on the right hand side\nof Eq. (11) is always perpendicular to Ω0andΩ0is\ntime-independent, we have ∂(Ω0·/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht)/∂t= 0. This\nimpliesΩ0· /an}b∇acketle{tσλ(t)/an}b∇acket∇i}htis a constant for all time. On the\nother hand, as the perturbation term /an}b∇acketle{tσλ(t)/an}b∇acket∇i}htvanishes\natt= 0, the constant is zero and therefore we obtain\nEq. (12). Furthermore, Eq. (12) also leads to the result\nthat|/an}b∇acketle{tσH(t)/an}b∇acket∇i}ht|2=|/an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht|2+o(λ2). This means that\nthe magnitude of spin remains unchanged up to the first\norder. That is, even though the non-Larmor torque can\nalter the magnitude of spin, the change is of o(λ2).\nUp to the linear order of the applied electric field, the\nsecond derivative of Eq. (11) can be written as\n∂2\n∂t2/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht=Ω0×(Ω0×/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht)\n+Ω0×(Ω1×/an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht)\n+Ω1×(Ω0×/an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht)\n+∂Ω1\n∂t×/an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht.(13)\nFor the first term, we have\nΩ0×(Ω0×/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht) =Ω0·/an}b∇acketle{tσλ(t)/an}b∇acket∇i}htΩ0−Ω2\n0/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht\n=−Ω2\n0/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht,\n(14)\nwhereΩ0·/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht= 0 [see Eq. (12)] is used. The third\nterm vanishes, i.e., Ω1×(Ω0×/an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht) = 0 because Ω0\nis parallelto the unperturbed spin /an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht=/an}b∇acketle{tσ/an}b∇acket∇i}ht, which\nis independent of t. Therefore, the second derivative of\n/an}b∇acketle{tσλ(t)/an}b∇acket∇i}htis given by\n∂2\n∂t2/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht=−Ω2\n0/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht+Ω0×(Ω1×/an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht)\n+∂Ω1\n∂t×/an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht.\n(15)\nEq. (11), Eq. (12) and Eq. (15) play the key role in ob-\ntaining the spin dynamics, and can be further simplified\nin two-dimensional systems. In the next section, we will\nsolve the spin dynamics in the intrinsic (impurity-free)\nsystem.III. INTRINSIC 2D SYSTEMS\nIn two-dimensional systems, both Ω0andΩ1lie on\nthex-yplane; we have Ω0= (Ω0x,Ω0y,0) andΩ1=\n(Ω1x,Ω1y,0). This implies that Ω1×Ω0is always per-\npendicular to the plane. To the first order of the electric\nfield,/an}b∇acketle{tσλ(t)/an}b∇acket∇i}htprecesses around Ω0but also tends to be\ndragged out of plane at the same time. In this sense,\nwe separate the solution of /an}b∇acketle{tσλ(t)/an}b∇acket∇i}htinto a harmonic term\ndescribing the Larmor motion and an anharmonic term\ndescribing the non-Larmor motion for the drag effect.\nTherefore, the first order perturbation of spin /an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht\ncan be separated into two parts,\n/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht=ΣL(t)+ΣN(t). (16)\nΣLis referred to as the Larmor component of spin and\nis a harmonic function of time. ΣNis referred to as\nthe non-Larmor component of spin and is anharmonic in\ntime. Correspondingly, ΣLandΣNrespectively satisfy\n∂\n∂tΣL=Ω0×ΣL, (17)\nand\n∂\n∂tΣN=Ω0×ΣN+Ω1×/an}b∇acketle{tσ/an}b∇acket∇i}ht, (18)\nwhere\n/an}b∇acketle{tσH0(t)/an}b∇acket∇i}ht=/an}b∇acketle{tnk|σ|nk/an}b∇acket∇i}ht ≡ /an}b∇acketle{tσ/an}b∇acket∇i}ht (19)\nwas used.\nEq. (17) shows that ΣLprecesses around the static\nmagnetic field Ω0harmonically and as a consequence the\nmagnitude |ΣL|is independent of time. On the other\nhand, as Ω0andΩ1span the 2D plane, the third term\nat the right hand side of Eq. (15) is perpendicular to the\n2D plane (the unperturbed spin /an}b∇acketle{tσ/an}b∇acket∇i}htis parallel to Ω0).\nFurthermore, the second term at the right hand side of\nEq. (15) lies on the 2D plane. Therefore, the in-plane\nspin, denoted as /an}b∇acketle{tσλ\n/bardbl/an}b∇acket∇i}ht= (/an}b∇acketle{tσλ\nx(t)/an}b∇acket∇i}ht,/an}b∇acketle{tσλ\ny(t)/an}b∇acket∇i}ht). and the out-\nof-plane spin, denoted as /an}b∇acketle{tσλ\nz(t)/an}b∇acket∇i}ht, satisfy the following\nequations, respectively,\n∂2\n∂t2/an}b∇acketle{tσλ\n/bardbl(t)/an}b∇acket∇i}ht+Ω2\n0/an}b∇acketle{tσλ\n/bardbl(t)/an}b∇acket∇i}ht=G/bardbl(t),\n∂2\n∂t2/an}b∇acketle{tσλ\nz(t)/an}b∇acket∇i}ht+Ω2\n0/an}b∇acketle{tσλ\nz(t)/an}b∇acket∇i}ht=Gz(t),(20)\nwhere\nG/bardbl(t) = (Gx,Gy) =Ω0×[Ω1×/an}b∇acketle{tσ/an}b∇acket∇i}ht],\nGz(t)ˆez=∂Ω1\n∂t×/an}b∇acketle{tσ/an}b∇acket∇i}ht.(21)\nImportantly, Gzin Eq. (21) gives the dynamical origin\nof the nonzero spin- zcomponent. We note that Gzis\nrelated to time derivative of Ω1, which is nonzero in gen-\neral.5\nThe unperturbed Hamiltonian of the spin-orbit cou-\npledsystemsintwodimensionscanbewrittenas( dz= 0)\nH0=ǫk+σxdx(k)+σydy(k), (22)\nwhereǫkis the kinetic energy, and dxanddyare func-\ntions of momentum k= (kx,ky), which describe the\nspin-orbit coupling. The eigenenegry of Eq. (22) satis-\nfyingH0|nk/an}b∇acket∇i}ht=Enk|nk/an}b∇acket∇i}htis given by Enk=ǫk−nd,\nwhered=/radicalBig\nd2x+d2yandn=±represents the band in-\ndex. For generic k-linear systems, dican be written as\ndi=/summationtext\njβijkj[16]. For the k-cubic Rahsba system [9],\ndx=αhk3sin(3φ) anddy=−αhk3cos(3φ). For the k-\ncubicRashba-Dresselhaussystem[10], dx= [αhsin(3φ)+\nβhcos(φ)]k3, anddy= [−αhsin(3φ)+βhsin(φ)]k3. The\nprecession frequency is given by Ω 0= 2d//planckover2pi1. The ex-\npectation values of spin with respect to the unperturbed\neigenstates are given by /an}b∇acketle{tσ/an}b∇acket∇i}ht=/an}b∇acketle{tnk|σ|nk/an}b∇acket∇i}ht, and we have\n(see Appendix A)\n/an}b∇acketle{tnk|σx|nk/an}b∇acket∇i}ht=−ndx\nd=−nΩ0x\nΩ0,\n/an}b∇acketle{tnk|σy|nk/an}b∇acket∇i}ht=−ndy\nd=−nΩ0y\nΩ0,\n/an}b∇acketle{tnk|σz|nk/an}b∇acket∇i}ht= 0.(23)\nSinceΩ0= (Ω0x,Ω0y,0) lies on the plane, the quantum\naverageof spin- zcomponent is zero, i.e., /an}b∇acketle{tnk|σz|nk/an}b∇acket∇i}ht= 0,\nas expected. To simplify the following calculations, we\ndefinedx=dsinθ, dy=−dcosθ, and we have\n∂θ\n∂ka=∂\n∂katan−1/parenleftbiggdx\n−dy/parenrightbigg\n=1\nd2/parenleftbigg\ndx∂dy\n∂ka−dy∂dx\n∂ka/parenrightbigg\n.(24)\nThe equation of motion of spin for the total Hamiltonian\nH=H0+eE·xis then given by\n∂\n∂tσH(t) =2dH(t)\n/planckover2pi1×σH(t), (25)\nwheredH(t) = (dH\nx(t),dH\ny(t),0). The corresponding ef-\nfective magnetic field is given by ΩH(t) = 2dH(t)//planckover2pi1. Eq.\n(5) can be perturbatively expanded up to first order of\neEaxa[33], and the result is given by\nOH(t) =OH0(t)+eEa[Γa,OH0(t)]+o(λ2),(26)\nwhereOH0(t) = exp( iH0t//planckover2pi1)Oexp(−iH0t//planckover2pi1), and the\noperator Γ ais given by\nΓa=i\n/planckover2pi1/integraldisplayt\n0dt′eiH0t′//planckover2pi1xae−iH0t′//planckover2pi1.(27)\nBy using Eq. (26) and Eq. (27), we have\ndH\ni(t) =eiHt//planckover2pi1die−iHt//planckover2pi1\n=di−eEa\n/planckover2pi1∂di\n∂kat+o(λ2).(28)We notethat the resultEq.(28) is validin ashorttime as\ngivenin Eq. (8). The original(i.e., zeroth-order)effective\nmagnetic field is given by\nΩ0= (2dx\n/planckover2pi1,2dy\n/planckover2pi1,0), (29)\nand the first-order effective magnetic field is given by\nΩ1=2\n/planckover2pi1(−eEa∂dx\n/planckover2pi1∂kat,−eEa∂dy\n/planckover2pi1∂kat,0) (30)\nThemagnitude of Ω0is|Ω0|= 2d//planckover2pi1= Ω0. We firstfocus\non the solution of /an}b∇acketle{tσλ\nz(t)/an}b∇acket∇i}ht. The solution of the Larmor\ncomponent ΣL\nzin Eq. (20) is obtained by setting Gz= 0,\nand we must have ΣL\nz=Asin(Ω0t) +Bcos(Ω0t). On\nthe other hand, as the first-order magnetic field Ω1is\nlinear in time, its time derivative is a constant in time.\nThe solution of the non-Larmor component ΣN\nzmust be\nanharmonic, and thus, we have Ω2\n0ΣN\nz=Gz,\nΣN\nz=1\nΩ2\n0/parenleftbigg∂Ω1\n∂t×/an}b∇acketle{tnk|σ|nk/an}b∇acket∇i}ht/parenrightbigg\nz\n=1\nΩ2\n0/parenleftbigg∂Ω1x\n∂t×/an}b∇acketle{tnk|σy|nk/an}b∇acket∇i}ht−∂Ω1y\n∂t×/an}b∇acketle{tnk|σx|nk/an}b∇acket∇i}ht/parenrightbigg\n=−eEa2\n/planckover2pi12Ω2\n0/bracketleftbigg∂dx\n∂ka/parenleftbigg\n−ndy\nd/parenrightbigg\n−∂dy\n∂ka/parenleftbigg\n−ndx\nd/parenrightbigg/bracketrightbigg\n=−neEa\n/planckover2pi1Ω0∂θ\n∂ka.\n(31)\nSimply by requiring the ΣL\nz+ΣN\nz= 0 at the initial time\nt= 0, we have the coefficient B=−ΣN\nzand we conse-\nquently/an}b∇acketle{tσλ\nz(t)/an}b∇acket∇i}ht= ΣN\nz(1−cos(Ω0t)) +Asin(Ω0t). Since\nthez-component of Ω0is zero, i.e., Ω 0z= 0, the con-\nditionΩ0·/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht= 0 cannot determine the coefficient\nA. We have to consider the time derivative of /an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht\ntogether with the solution of the in-plane component of\nspin.\nThe solution of the Larmor precession in the in-plane\ndirection is given by ΣL\n/bardbl=C/bardblcos(Ω0t) +K/bardblsin(Ω0t).\nThe non-Larmor precession in Eq. (20) is given by\nΩ2\n0ΣN\n/bardbl=G/bardbl, and we have\nΣN\n/bardbl(t) = ΣN\nx(t)ˆex+ΣN\ny(t)ˆey\n=1\nΩ2\n0Ω0×[Ω1×/an}b∇acketle{tnk|σ|nk/an}b∇acket∇i}ht]\n=1\nΩ2\n0Ω0׈ez(Ω1x/an}b∇acketle{tnk|σy|nk/an}b∇acket∇i}ht−Ω1y/an}b∇acketle{tnk|σx|nk/an}b∇acket∇i}ht)\n= ΣN\nz(Ω0׈ez)t\n=/parenleftbigg\n−dyneEa\n/planckover2pi1d∂θ\n∂kat/parenrightbigg\nˆex+/parenleftbigg\n+dxneEa\n/planckover2pi1d∂θ\n∂kat/parenrightbigg\nˆey.\n(32)\nBy requiring that ΣL\n/bardbl+ΣN\n/bardbl= 0 at the initial time t= 0,\nwe haveC/bardbl= 0. For the non-Larmor precession, we find6\nthat Eq. (18) is satisfied by substitution of Eq. (31) and\nEq. (32) into Eq. (18). Up to this step, the remaining\ncoefficients are Kx,KyandAfor direct precession. By\nsubstituting ΣL\n/bardbland ΣL\nzinto Eq. (17), we can obtain\nA= 0,Kx=−Ω0yΣN\nz/Ω0, andKy= Ω0xΣN\nz/Ω0. Fur-\nthermore,byusingthecondition Ω0·/an}b∇acketle{tσλ(t)/an}b∇acket∇i}ht= 0,wehave\nΩ0xKx+Ω0yKy= 0, and it is easy to check that the re-\nsult is satisfied. Therefore, the solution of /an}b∇acketle{tnk|σH(t)|nk/an}b∇acket∇i}ht\ncan be written as\n/an}b∇acketle{tnk|σH(t)|nk/an}b∇acket∇i}ht=/an}b∇acketle{tσ/an}b∇acket∇i}ht+ΣN(t)+ΣL(t),(33)\nwhere the Larmor component of spin ΣL= (ΣL\nx,ΣL\ny,ΣL\nz)\nis given by [ dx=dsinθ,dy=−dcosθis used]\nΣL(t) =ΣN\nzcosθsin(Ω0t)ˆex+ΣN\nzsinθsin(Ω0t)ˆey\n−ΣN\nzcos(Ω0t)ˆez,(34)\nand the non-Larmor component of spin ΣN(t) =\n(ΣD\nx,ΣD\ny,ΣN\nz) is given by\nΣN=/parenleftbig\nΩ0×ΣN\nzˆez/parenrightbig\nt+ΣN\nzˆez. (35)\nFor the Larmor precession [see Eq. (34)], the magnitude\nofΣLis|ΣL|= ΣN\nz, andΣLprecesses about the in-\nplane axis Ω0with frequency Ω 0as can be verified from\nEq. (17). Forthe non-Larmorprecession, the spin- zcom-\nponentisnotnecessarytobezero. Thespin- zcomponent\nbehavesasΣN\nz(1−cos(Ω0t)), whichhasamaximumvalue\n2ΣN\nzand a minimum value 0.\nThe solution of the in-plane and out-of-plane compo-\nnents of spin obtained in Eq. (33), Eq. (34) and Eq. (35)\ncan be recast into the following form [see also Eq. (B17)]\n/an}b∇acketle{tnk|σH\n/bardbl(t)|nk/an}b∇acket∇i}ht=−nΩ0\nΩ0+ΣN\nz(Ω0׈ez)/bracketleftbigg\nt−sin(Ω0t)\nΩ0/bracketrightbigg\n,\n/an}b∇acketle{tnk|σH\nz(t)|nk/an}b∇acket∇i}htˆez= ΣN\nz[1−cos(Ω0t)]ˆez.\n(36)\nThe term linear in tfor/an}b∇acketle{tσH\n/bardbl/an}b∇acket∇i}htseems to grow arbitrarily\nlargewith t, but this is only anartifact due to the pertur-\nbation method, which is valid only for a short range of t\nas delimited by Eq. (8). In fact, the time-dependent part\nof/an}b∇acketle{tσH\n/bardbl/an}b∇acket∇i}htand the whole part of /an}b∇acketle{tσH\nz(t)/an}b∇acket∇i}htare both of o(λ),\nreassuring |/an}b∇acketle{tσH(t)/an}b∇acket∇i}ht|2= 1+o(λ2). From Eq. (36), we see\nthat, after Eis turned on at t= 0, the out-of-plane com-\nponent/an}b∇acketle{tσH\nz(t)/an}b∇acket∇i}htarises from zero and begins a harmonic\noscillation around ΣN\nzwith the frequency Ω 0. The typ-\nical value of the frequency Ω 0on the Fermi surface is\nabout 1010per second.5\nIn reality, after a while, the harmonic oscillation will\ndecay and /an}b∇acketle{tσH(t)/an}b∇acket∇i}htwill asymptote to a new value due to\nsome dissipative process, which has not been taken into\n5For pure Rashba system, Ω 0∼2αkF//planckover2pi1andkF∼10−2˚A−1and\nα∼10−3eV˚A[for GaAs [34]], and we have Ω 0∼1010.consideration. The switch-on of Ebridges the two time-\nindependent systems described by H0andH0+eE·x,\nrespectively. Measurement of the spin in the latter sys-\ntem does not exhibit any harmonic oscillation, because\na time-independent system gives a stationary solution\n(which will be the topic of Sec. IV). However, on the\nother hand, if the latter system is transited from the\nformer system, the transition will induce a harmonic os-\ncillation as we just calculated. Without any dissipative\nprocess, the harmonic oscillation will remain persistent,\nwhich is unphysical because after long enough the system\nshould become oblivious ofwhen the transition has taken\nplace.\nThe exact mechanismfor the dissipation is unclearand\ncomplicated, but it can be implemented phenomenologi-\ncally. In particular, the time-varying behavior of a mag-\nnetic dipole subject to a magnetic filed can be modeled\nby various forms of the Landau-Lifshitz-Gilbert equa-\ntion [28], which phenomenologically includes a damping\ntermtoaccountforthedissipation. TheLandau-Lifshitz-\nGilbert equation can be obtained more fundamentally\nin the context of irreversible statistical mechanics [29].\nUnfortunately, as the Landau-Lifshitz-Gilbert equation\nmodels the time-varying behavior only for the magnetic\ndipole but assumes the magnetic field as static, it does\nnot serve our purpose. Instead, we phenomenologically\nmodel the dissipative process by introducing damping\nterms directly upon the precessional equation Eq. (20):\n∂2\n∂t2/an}b∇acketle{tσλ\n/bardbl(t)/an}b∇acket∇i}ht+Ω2\n0/an}b∇acketle{tσλ\n/bardbl(t)/an}b∇acket∇i}ht+β/bardbl∂\n∂t/an}b∇acketle{tσλ\n/bardbl(t)/an}b∇acket∇i}ht=G/bardbl(t)e−αt,\n∂2\n∂t2/an}b∇acketle{tσλ\nz(t)/an}b∇acket∇i}ht+Ω2\n0/an}b∇acketle{tσλ\nz(t)/an}b∇acket∇i}ht+βz∂\n∂t/an}b∇acketle{tσH\nz(t)/an}b∇acket∇i}ht=Gz(t),\n(37)\nwhere the constants β/bardblandβzare two damping param-\neters responsible for damping the harmonic oscillation.\nFor generality, we keep β/bardblandβzas two different co-\nefficients. Additionally, we also include an exponential\ndecay factor e−αtfor the “source” term G/bardbl(t). This fac-\ntoris not onlyprescribedto subdue the pathologicaltrait\nof the linear growth in tforG/bardbl(t) but in fact is required\nto render the damping with β/bardblconsistent (as will be seen\nshortly). Provided that the dissipative effects are strong\nenough, i.e., phenomenologically characterized by\n1\nβ/bardbl,1\nβz,1\nα≪/planckover2pi1Ω0\nevd|E|, (38)\nthe spin precession will reach an asymptotic state within\nthe valid range of time given by Eq. (8).\nIf we require that the solution of Eq. (37) agrees with\nEq. (36) when β/bardbl,βz,α→0, we then have the new solu-7\ntion\n/an}b∇acketle{tnk|σH\n/bardbl(t)|nk/an}b∇acket∇i}ht=−nΩ0\nΩ0+ΣN\nzΩ2\n0(Ω0׈ez)\n/parenleftbig\nΩ2\n0−αβ/bardbl+α2/parenrightbig2\n×/bracketleftbigg\ne−αt(Ω2\n0−αβ/bardbl+α2)t\n+(2α−β/bardbl)/parenleftbigg\ne−αt−e−β/bardblt\n2+i˜Ω/bardblt/parenrightbigg\n−Ω0e−η/bardblt\n2sin(˜Ω/bardblt)/bracketrightbigg\n,\n/an}b∇acketle{tnk|σH\nz(t)|nk/an}b∇acket∇i}htˆez= ΣN\nz/bracketleftBig\n1−e−βzt\n2cos(˜Ωzt)/bracketrightBig\nˆez,(39)\nwhere the new oscillatory frequencies ˜Ω/bardbl,zare given by\n˜Ω2\n/bardbl,z= Ω2\n0−/parenleftbiggβ/bardbl,z\n2/parenrightbigg2\n. (40)\nPrescribing β/bardbl/ne}ationslash= 0 gives rise to a pathological trait that\n/an}b∇acketle{tσH\n/bardbl(t)/an}b∇acket∇i}htin general becomes complex, with an imaginary\npart ofo(λ). This problem can be avoided by choosing\n2α=β/bardbl. (41)\nTherefore, to implement dissipation consistently, inclu-\nsion of the damping term with β/bardblentails the exponential\ndecay term with α.\nBy defining\nF0(t) =Ω3\n0\nΩ2\n0−αβ/bardbl+α2e−αtt,\nF1(t) =Ω3\n0(2α−β/bardbl)\n/parenleftbig\nΩ2\n0−αβ/bardbl+α2/parenrightbig2/parenleftbigg\ne−αt−e−β/bardblt\n2+i˜Ω/bardblt/parenrightbigg\n,\nF2(t) =Ω4\n0/parenleftbig\nΩ2\n0−αβ/bardbl+α2/parenrightbig2e−β/bardblt\n2sin(˜Ω/bardblt),\nF3(t) =e−βzt\n2cos(˜Ωzt),(42)\nEq. (39) can be rewritten as\n/an}b∇acketle{tσH\n/bardbl(t)/an}b∇acket∇i}ht=−nΩ0\nΩ0+ΣN\nz/parenleftbiggΩ0\nΩ0׈ez/parenrightbigg\n×[F0(t)+F1(t)−F2(t)],\n/an}b∇acketle{tσH\nz(t)/an}b∇acket∇i}htˆez= ΣN\nz[1−F3(t)]ˆez,(43)\nwhich follows\n∂\n∂t/an}b∇acketle{tσH\n/bardbl(t)/an}b∇acket∇i}ht= ΣN\nz/parenleftbiggΩ0\nΩ0׈ez/parenrightbigg\n×[F′\n0(t)+F′\n1(t)−F′\n2(t)],\n∂\n∂t/an}b∇acketle{tσH\nz(t)/an}b∇acket∇i}htˆez= ΣN\nzF′\n3(t)ˆez.(44)\nEqs. (43) and (44) can be organized into the following\nform\n∂\n∂t/an}b∇acketle{tσH(t)/an}b∇acket∇i}ht=Ωλ(t)×/an}b∇acketle{tσH(t)/an}b∇acket∇i}ht+o(λ2),(45)with the effective magnetic field given by\nΩλ(t) =Ω0\n+nΣN\nz/parenleftbiggΩ0\nΩ0׈ez/parenrightbigg/bracketleftBig\n−Ω0F0(t)−Ω0F1(t)\n+/parenleftbig\nΩ0F2(t)−F′\n3(t)/parenrightbig/bracketrightBig\n+nΣN\nzˆez/bracketleftbigg\n−Ω0+F′\n0(t)+F′\n1(t)\n+/parenleftbig\nΩ0F3(t)−F′\n2(t)/parenrightbig/bracketrightbigg\n+o(λ2).\n(46)\nWithβ/bardbl,z= 0 and α= 0, we have F0(t) = Ω0t,F1(t) =\n0, Ω0F2(t)−F′\n3(t) = 0 and Ω 0F3(t)−F′\n2(t) = 0, and\nconsequently Eq. (46) leads to\nΩλ(t) =Ω0−nΣN\nz(Ω0׈ez)Ω0t+o(λ)\n≡Ω0+Ω1(t)+o(λ2).(47)\nThis affirms the consistency that the effective magnetic\nfieldΩλ(t) obtained from the equation of spin precession\nis identical to ΩH(t) in Eq. (6), where our calculation\nbegins.\nOn the other hand, with β/bardbl,βz,α >0, we have\nF0,1,2,3(t)→0 fort≫1/β/bardbl,1/βz,1/α. Therefore, when\ntis largeenough, the spin and the effective magnetic field\nasymptote to the fixed values:\n/an}b∇acketle{tσH(t)/an}b∇acket∇i}ht → −nΩ0\nΩ0+ΣN\nzˆez+o(λ2),(48)\nand\nΩλ(t)→Ω0−nΩ0ΣN\nzˆez+o(λ2) (49)\nInthe asymptoticlimit, both the spin /an}b∇acketle{tσH(t)/an}b∇acket∇i}htand theef-\nfective magneticfield Ωλ(t) areparallelto eachotherand\nhave constant nonzero components in the out-of-plane\ndirection. The asymptotic values of /an}b∇acketle{tσH(t)/an}b∇acket∇i}htandΩλ(t)\nobtained from the dynamical response to the switch-on\nof the applied electric field should be the same as those\nobtained as the stationary response to the electric field\nthat is viewed as never-changingin time. In the next sec-\ntion, we will perform the time-independent analysisupon\nthe time-independent Hamiltonian H0+eE·x. In terms\nof the matrix elements with respect to the eigenstates of\nH0, the full Hamiltonian H0+eE·xtakes the form\nˆH=ǫk+σ·D. (50)\nThedirectionoftheeffectivemagneticfield(i.e., D/D)as\nastationaryresponseisinfullagreementwiththeasymp-\ntotic value of the dynamical response given in Eq. (49).\nTherefore, we have arrived at a good understanding\nabout the dynamical origin of the out-of-plane spin com-\nponent. In the beginning, before the electric field Eis\nturned on, the spin is aligned with the original effective8\nmagnetic filed Ω0, lying on the x-yplane. When Eis\nturned on at t= 0, it deflects Ω0intoΩ0+Ω1+o(λ2),\nwhich remains on the x-yplane. The spin is no longer\naligned with the new effective magnetic field and starts\nto precess around it, thereby giving rise to the spin- z\ncomponent. The precession of spin alters the effective\nmagnetic field and tilts it from the x-yplane as a back-\nreaction via the dissipative process. Eventually, the pre-\ncession ofspin and the evolutionof the effective magnetic\nfieldasymptoticallyreachastationarybalance,wherethe\nspin is aligned again with the effective magnetic field and\nhas a nonzero spin- zcomponent. The dissipative effect\nplaysa crucialrolein establishingthe stationarybalance.\nWe close this section with a remark: The mathemat-\nical result obtained from solving Eq. (6) up to the first\norder of the electric field should be the same with that\nobtained directly from the Heisenberg picture. The later\ncalculation is presented in Appendix B.\nIV. TIME-INDEPENDENT ANALYSIS\nAs demonstrated in the previous section, an effective\nout-of-plane magnetic field is dynamically generated by\nan in-plane electric field and it asymptotes to an asymp-\ntotic result. In this section, complementary to the dy-\nnamical treatment, we conduct a time-independent anal-\nysis to directly derive the stationary response to the ap-\nplied electric field, which is now treated as always turned\non and completely time independent. It will shown that\nthe linear term ofthe stationaryresponseis exactlyequal\nto the asymptotic response obtained in the dynamical\ntreatment.\nThe unperturbed Hamiltonian Eq. (22) is represented\nin the spin space | ↑/an}b∇acket∇i}htand| ↓/an}b∇acket∇i}ht. In the basis |nk/an}b∇acket∇i}ht,H0\nis diagonalized, while the total Hamiltonian is not but\nreads as\nH=/parenleftbigg\n/an}b∇acketle{t+k|H|+k/an}b∇acket∇i}ht /an}b∇acketle{t+k|H|−k/an}b∇acket∇i}ht\n/an}b∇acketle{t−k|H|+k/an}b∇acket∇i}ht /an}b∇acketle{t−k|H|−k/an}b∇acket∇i}ht/parenrightbigg\n=/parenleftbigg\nǫk−d+VE/an}b∇acketle{t+k|xa|−k/an}b∇acket∇i}hteEa\n/an}b∇acketle{t−k|xa|+k/an}b∇acket∇i}hteEaǫk+d−VE/parenrightbigg\n,(51)\nwhere we have defined /an}b∇acketle{t+k|xa|+k/an}b∇acket∇i}hteEa≡VEand\n/an}b∇acketle{t−k|xa|−k/an}b∇acket∇i}hteEa≡ −VE, as in general the vector po-\ntential/an}b∇acketle{tnk|xa|nk/an}b∇acket∇i}htdepends on the band index n=\n±. In general, the quantity VE≡eEa(/an}b∇acketle{t+k|xa|+k/an}b∇acket∇i}ht −\n/an}b∇acketle{t−k|xa|−k/an}b∇acket∇i}ht)/2 is not invariant under the gauge transfor-\nmation|+k/an}b∇acket∇i}ht →eiφ+(k)|+k/an}b∇acket∇i}ht,|−k/an}b∇acket∇i}ht →eiφ−(k)|−k/an}b∇acket∇i}htunless\nwe choose φ+(k) =φ−(k). This raises an issue of how to\ndefine a gauge-independent spin current in the spin-Hall\neffect. Following Ref. [21], one has to apply an intricate\nprescription to render the spin current gauge indepen-\ndent. Nevertheless, up to the first order of the electric\nfield, the linear response is independent of VEand thus\nfree of this problem as will be seen shortly.\nFor the off-diagonal matrix elements of xa, we can\nwrite/an}b∇acketle{t±k|xa|∓k/an}b∇acket∇i}htin terms of matrix elements of σz,which is valid for all choice of wave functions, as proved\nin Appendix A. Using Eq. (A7), we have\n/an}b∇acketle{t+k|xa|−k/an}b∇acket∇i}ht=1\n2/an}b∇acketle{t+k|σz|−k/an}b∇acket∇i}ht∂θ\n∂ka,\n/an}b∇acketle{t−k|xa|+k/an}b∇acket∇i}ht=1\n2/an}b∇acketle{t−k|σz|+k/an}b∇acket∇i}ht∂θ\n∂ka.(52)\nSubstituting Eq. (52) into Eq. (51), Hcan be written as\nH=ǫk+axτx+ayτy+azτz, (53)\nwhere\nax=1\n2Re/an}b∇acketle{t−k|σz|+k/an}b∇acket∇i}ht∂θ\n∂kaeEa\nay=1\n2Im/an}b∇acketle{t−k|σz|+k/an}b∇acket∇i}ht∂θ\n∂kaeEa\naz=VE−d.(54)\nThe matrices τi, which are called pseudo-spin matrices,\nare mathematically Pauli matrices, but they are not real\nspin. This can also be seen as follows. The position op-\neratorxand the electric field Eare even under the time-\nreversal transformation. The real spin is odd σi→ −σi\nand the momentum is also odd under the time-reversal\noperation k→ −k. The effective magnetic field dmust\nbe odd under the time-reversal operation dx→ −dxand\ndy→ −dy. This implies that the Hamiltonian H0is in-\nvariantunderthetime-reversaltransformation. Sincethe\nHamiltonian H0isinvariantunder time-reversaltransfor-\nmation, the Hamiltonianin the basis |nk/an}b∇acket∇i}htmust notbreak\nthe time reversal symmetry. Therefore, we find that ax,\nayandazare even under time reversal operation, and\nthusτimust be even, which means that τiare not the\nreal spin. In order to transform Eq. (53) back to the real\nspin, we have to transform the spin in basis |nk/an}b∇acket∇i}htto the\nspin space | ↑/an}b∇acket∇i}htand| ↓/an}b∇acket∇i}ht.\nThe original spin σiin basis |nk/an}b∇acket∇i}htcan be written in\nterms of the new spin matrices denoted as /tildewideσi. For the\nspin-zcomponent, we have\n/tildewideσz≡/parenleftbigg\n/an}b∇acketle{t+k|σz|+k/an}b∇acket∇i}ht /an}b∇acketle{t+k|σz|−k/an}b∇acket∇i}ht\n/an}b∇acketle{t−k|σz|+k/an}b∇acket∇i}ht /an}b∇acketle{t−k|σz|−k/an}b∇acket∇i}ht/parenrightbigg\n=2\n∂θ\n∂kaeEa(axτx+ayτy).(55)\nFor spin- xnd spin-ycomponents, we have\n/tildewideσx≡/parenleftbigg\n/an}b∇acketle{t+k|σx|+k/an}b∇acket∇i}ht /an}b∇acketle{t+k|σx|−k/an}b∇acket∇i}ht\n/an}b∇acketle{t−k|σx|+k/an}b∇acket∇i}ht /an}b∇acketle{t−k|σx|−k/an}b∇acket∇i}ht/parenrightbigg\n=−dx\ndτz+dy\nd2\n∂θ\n∂kaeEa(ayτx−axτy).(56)\nand\n/tildewideσy≡/parenleftbigg\n/an}b∇acketle{t+k|σy|+k/an}b∇acket∇i}ht /an}b∇acketle{t+k|σy|−k/an}b∇acket∇i}ht\n/an}b∇acketle{t−k|σy|+k/an}b∇acket∇i}ht /an}b∇acketle{t−k|σy|−k/an}b∇acket∇i}ht/parenrightbigg\n=−dy\ndτz−dx\nd2\n∂θ\n∂kaeEa(ayτx−axτy).(57)9\nBy using Eq. (A5), it is easy to show that /tildewideσisatisfies\nthe algebra of the Pauli matrices, i.e., {/tildewideσi,/tildewideσj}= 2δij\nand [/tildewideσi,/tildewideσj] = 2iǫijk/tildewideσk. By using Eq. (55), Eq. (56) and\nEq. (57), the Hamiltonian Eq. (53) can be written in\nterms of /tildewideσiand the result is given by\nH=ǫk+Dx/tildewideσx+Dy/tildewideσy+Dz/tildewideσz,(58)\nwhere\nDx=−dx\ndaz=dx−dx\ndVE,\nDy=−dy\ndaz=dy−dy\ndVE,\nDz=1\n2∂θ\n∂kaeEa.(59)\nTherefore, the Hamiltonian in terms of the expectation\nvalues with respect to |±k/an}b∇acket∇i}htcan be again cast into the\nformH=ǫk+D·/tildewideσ. That is, the spin is aligned with\nan effective magnetic field given by Eq. (59). We also\nfind that D= (Dx,Dy,Dz) are odd under time-reversal\noperation. Therefore, the Hamiltonian in terms of the\nexpectation values with respect to |±k/an}b∇acket∇i}htcan be cast into\nthe form\nˆH=ǫk+Dxσx+Dyσy+Dzσz,(60)\nwhereσiare pauli matrices. Importantly, Eq. (60) shows\nthat the z-component of effective magnetic field is non-\nzero. The effective magnetic field is being tilted up in\nthe presence of an electric field. In the absence of electric\nfield, Eq. (60) goes back to the unperturbed Hamiltonian\nH0=ǫk+σxdx+σydy. It should be noted that Dx\nandDyin Eq. (59) contain unphysical gauge-dependent\npieces involving VE. Nevertheless, the gauge-dependent\nterms are of o(λ) and are exactly cancelled out when\nwe compute D/D(i.e., the direction of D) in the linear\nresponse regime. Noting that D2=d2−2VEd+o(λ2),\nwe have\nD\nD=dx\ndˆex+dy\ndˆey+1\n2d∂θ\n∂kaeEaˆez+o(λ2).(61)\nTherefore, up to o(λ2), the result of Eq. (61) is gauge\nindependent. We find that the direction of the effective\nmagnetic field given in Eq. (61) is exactly the same as\nthat of Eq. (49).\nThe eigenstates of ˆHare given by\n|Ψ−k/an}b∇acket∇i}ht=1/radicalBig\n2(1+ˆDz)/parenleftbigg\n1+ˆDz\nˆDx+iˆDy/parenrightbigg\n,\n|Ψ+k/an}b∇acket∇i}ht=1/radicalBig\n2(1+ˆDz)/parenleftbigg\n−ˆDx+iˆDy\n1+ˆDz/parenrightbigg\n,(62)\nwhereˆDi=Di/DandD=/radicalBig\nD2x+D2y+D2z. Thecorre-\nsponding eigenenergies are given by ˆH|Ψℓk/an}b∇acket∇i}ht=Eℓk|Ψℓk/an}b∇acket∇i}htwithEℓk=ǫk−ℓD. In the absence of electric field,\nthe eigenenergy is given by Eℓk=ǫk−ℓd, which is the\neigenenergy of the unperturbed Hamiltonian H0. As\nmentioned above, the spin is aligned in the direction\nD/D, and by using Eq. (62), we have /an}b∇acketle{tΨℓk|σ|Ψℓk/an}b∇acket∇i}ht=\n−ℓD/D. In the presence of the electric field, up to the\nfirst order of E, we have\n/an}b∇acketle{tΨℓk|σ|Ψℓk/an}b∇acket∇i}ht=−ℓD/D\n=−ℓdx\ndˆex−ℓdy\ndˆey,−ℓ\n2d∂θ\n∂kaeEaˆez+o(λ2).\n(63)\nOn the other hand, consider the projection of spin on the\nDdirection, which is given by\n/hatwideΣc=1\nD2D(σ·D). (64)\nBy noting that D2=d2−2VEd+o(λ2), the spin /hatwideΣc=\n(/hatwideΣcx,/hatwideΣcy,/hatwideΣcz) in the unperturbed basis |nk/an}b∇acket∇i}htis given by\n/an}b∇acketle{tnk|/hatwideΣcx|nk/an}b∇acket∇i}ht=−ndx\nd+o(λ2),\n/an}b∇acketle{tnk|/hatwideΣcy|nk/an}b∇acket∇i}ht=−ndy\nd+o(λ2),\n/an}b∇acketle{tnk|/hatwideΣcz|nk/an}b∇acket∇i}ht=−n1\n2d∂θ\n∂kaeEa+o(λ2).(65)\nIt can be shown that /an}b∇acketle{tΨnk|σ|Ψnk/an}b∇acket∇i}ht=/an}b∇acketle{tnk|/hatwideΣc|nk/an}b∇acket∇i}ht+o(λ2).\nImportantly, compare to Eq. (31), we have\n/an}b∇acketle{tnk|/hatwideΣcz|nk/an}b∇acket∇i}ht= ΣN\nz. (66)\nEq. (65) and Eq. (66) are in full agreement with the\nasymptotic response obtained in Eq. (49).\nWe close this section by showing the relation between\nthe spin current and ΣN\nzby using the Kubo formula [33].\nThe spin- zcomponent satisfies the following continuity\nequation [21, 35]\n∂\n∂tΨ†SzΨ+∇iRe/bracketleftbig\nΨ†Jz\niΨ/bracketrightbig\n= Re/bracketleftbigg\nΨ†dSz\ndtΨ/bracketrightbigg\n,(67)\nwhere the conventional spin current Jz\niis given by\nJz\ni=1\n2{J/planckover2pi1σz,vi}\n=J∂ǫk\n∂kiσz,(68)\nwherevi=∂H//planckover2pi1∂kiandJ= 1/2 for spin 1 /2 and so on.\nThe Kubo formula for spin current [4] is given by\nJz\ni=2q/planckover2pi1\nV/summationdisplay\nn(/negationslash=n′)fnkIm/an}b∇acketle{tnk|Jz\ni|n′k/an}b∇acket∇i}ht/an}b∇acketle{tn′k|vj|nk/an}b∇acket∇i}ht\n(Enk−En′k)2Ej,(69)\nwheren=±,q=−eforanelectronand fnkistheFermi-\nDiracdistribution. UsethematrixelementsEq.(A5)and\nEq. (A6) shown in Appendix A, we have\n/an}b∇acketle{tnk|σz|n′k/an}b∇acket∇i}ht/an}b∇acketle{tn′k|vj|nk/an}b∇acket∇i}ht=−2id2\n/planckover2pi1(Enk−En′k)∂θ\n∂kj,(70)10\nwhich is purely imaginary. Substituting Eq. (70) into\nEq. (69), we have\nJz\ni=J\nV/summationdisplay\nnkfnk∂ǫk\n∂ki/parenleftbigg−n\n2d/parenrightbigg∂θ\n∂kjeEj\n=J\nV/summationdisplay\nnkfnk∂ǫk\n∂kiΣN\nz\n=J\nV/summationdisplay\nnkfnk∂ǫk\n∂ki/an}b∇acketle{tnk|/hatwideΣcz|nk/an}b∇acket∇i}ht+o(λ2).(71)\nIn the second equality of Eq. (70), the definition of ΣN\nz\n[see Eq. (31)] was used. The third equality of Eq. (70)\ncan be directly obtained from Eq. (60) in the linear re-\nsponse regime, and the result is in agreement with the\nKubo formula. Furthermore, since the source term in\nEq. (67) is d/an}b∇acketle{tnk|σH\nz(t)|nk/an}b∇acket∇i}ht/dt→dΣN\nz/dt= 0, and thus\nwefind thatthe spin currentshowninEq.(71) isthe con-\nserved spin current. Another definition of spin current\nfrom the source term Jτ\ni=1\n2{xi,dσz/dt}(spin-torque\ncurrent) has been shown to be zero in the presence of\nconstant electric field [15, 16], which is in agreement with\nthe present result. The spin-torque current is non-zero\nonly when the electric field is nonhomogeneous in the\nspaceEexp(iqx), andJτ\niwill be the rate of change of\nthe torque spin density with respective to qin the limit\nq→0 [21]. In addition, Eq. (65) shows that the linear\nresponse of the in-plane spin is zero. This can be seen as\nfollows. By using Eq. (A2) and Eq. (A6), we have\n/an}b∇acketle{tnk|σx|n′k/an}b∇acket∇i}ht/an}b∇acketle{tn′k|vj|nk/an}b∇acket∇i}ht\n=2idy\nEn′k−Enk−2id2\n/planckover2pi1(Enk−En′k)|/an}b∇acketle{tnk|σz|n′k/an}b∇acket∇i}ht|2∂θ\n∂kj\n=−4dyd2\n(Enk−En′k)2|/an}b∇acketle{tnk|σz|n′k/an}b∇acket∇i}ht|2∂θ\n∂kj,(72)\nwhich is purely real, and therefore the imaginary part is\nzero. For σy, similar to the derivation shown in Eq. (72)\n(by using Eq. (A3) and Eq. (A6)), it can be shown that\n/an}b∇acketle{tnk|σy|n′k/an}b∇acket∇i}ht/an}b∇acketle{tn′k|vj|nk/an}b∇acket∇i}htis also purely real.\nV. CONCLUSION\nWe obtain the dynamical equation of spin in two-\ndimensional spin-orbit coupled systems by solving\nHeisengerg’s equation perturbatively up to the linear or-\nder of the applied electric field, which is assumed to be\nturned on at t= 0. As shown in Eqs. (29) and (30),\nthe switch-on of the electric field deflects the effective\nmagneticfieldfromitsoriginaldirectionbygivingatime-\ndependent component on the x-yplane. As the spin is no\nlonger aligned with the effective magnetic field, it starts\nto precess around the new direction.\nTaking into account the dissipative effect that atten-\nuates and eventually ceases the spin precession, we phe-\nnomenologically add damping terms upon the equationof spin dynamics as in Eq. (37). The solution of the re-\nsulting dynamics is given in Eq. (39) for the spin and\nEq. (46) for the effective magnetic field. When tis large\nenough, the dynamical solution asymptotes to an asymp-\ntotic state given by Eqs. (48) and (49), where the spin\nand the effective magnetic field are aligned again and ex-\nhibit nonzero components in the zdirection.\nOn the other hand, treatingthe applied electric field as\nalwaysturnedon,wealsodirectlycomputethestationary\nresponse in the time-independent approachby projecting\nthefull HamiltonianonthespinspaceasinEq.(60). The\nstationary response is obtained in Eqs. (65) [and (66)],\nwhich is exactly equal to the asymptotic result (48) ob-\ntained from the dynamical treatment. The direction of\neffective magnetic field [Eq. (61)] is also in agreement\nwith that of the asymptotic result [Eq. (49)]. Further-\nmore, the relation between the stationary response of the\neffective magnetic field and the spin current is derived,\nand the result is in agreement with the Kubo formula.\nOur dynamical treatment reveals the dynamical ori-\ngin of the spin- zcomponent and provides a method to\nstudy the connection between the dynamicaland station-\naryresponses. Thedissipativeeffectisfoundtobecrucial\nfor establishing the connection. However, our prescrip-\ntion of dissipative effect remains phenomenological and\nit should be derived more fundamentally by the methods\nof irreversible statistical mechanics following the lines of\nRef. [29]. Furthermore, it has been shown in Ref. [36]\nthat, in a many-body interacting system, the stationary\neffectivemagneticfieldiscreatedbythesumofthevector\ndin Eq. (2) that accounts for the spin-orbit coupling and\na mean-field contribution derived from the many-body\nproblem. Thisseemstosuggestthatthe dissipativeeffect\nmight be understood as the result of many-body interac-\ntion in a mean-field theory approach. In the dynamical\ntreatment, we study the evolution of the expectation val-\nues of the kind /an}b∇acketle{tnk|·|nk/an}b∇acket∇i}htbut disregard the off-diagonal\nterms of the kind /an}b∇acketle{t+k| · |−k/an}b∇acket∇i}htand/an}b∇acketle{t−k| · |+k/an}b∇acket∇i}ht, while in\nthe time-independent approach, both are included [see\nEq. (57)]. The fact that the dynamical treatment with\ndissipation asymptotically leads to the stationary result\nof the time-independent approach strongly suggests that\nthe dissipative effect in the dynamical picture is closely\nrelated to the equilibrium of the interband transition in\nthe stationarypicture. This relationshould becomemore\ntransparent if the dissipation can be more fundamentally\nderived.\nACKNOWLEDGMENTS\nT.-W. Chen would like to thank Wang-Chuang Kuo\nfor valuable discussions on the stationary response. This\nwork was supported in part by the Ministry of Science\nand Technology, Taiwan under the Grant MOST 106-\n2112-M-110-010.11\nAppendix A: Matrix Elements\nIn this appendix, we calculate unperturbed matrix el-\nements of the spin and velocity operators used in this\narticle without specifying any form of the wave functions\n|nk/an}b∇acket∇i}ht. By using {σx,H0}={σx,ǫk+σxdx+σydy}=\n2ǫkσx+2dx, and/an}b∇acketle{tnk|{σx,H0}|nk/an}b∇acket∇i}ht= 2/an}b∇acketle{tnk|σx|nk/an}b∇acket∇i}htEnk=\n2ǫk/an}b∇acketle{tnk|σx|nk/an}b∇acket∇i}ht −2nd/an}b∇acketle{tnk|σx|nk/an}b∇acket∇i}ht. We have the diagonal\nmatrix element of σxin the helicity basis,\n/an}b∇acketle{tnk|σx|nk/an}b∇acket∇i}ht=−ndx\nd. (A1)\nFor the off-diagonal matrix elements, we note that\n[σx,H0] = [σx,ǫk+σxdx+σydy] = 2iσzdy. It follows\n/an}b∇acketle{tnk|σx|mk/an}b∇acket∇i}ht=2idy\nEmk−Enk/an}b∇acketle{tnk|σz|mk/an}b∇acket∇i}ht.(A2)\nSimilar to the derivation, for the spin- ycomponent, we\nhave\n/an}b∇acketle{tnk|σy|nk/an}b∇acket∇i}ht=−ndy\nd,\n/an}b∇acketle{tnk|σy|mk/an}b∇acket∇i}ht=−2idx\nEmk−Enk/an}b∇acketle{tnk|σz|mk/an}b∇acket∇i}ht.(A3)\nFor the spin- zcomponent, we have {σz,H0}= 2ǫkσz,\nand this implies −2nd/an}b∇acketle{tnk|σz|nk/an}b∇acket∇i}ht= 0. If the splitting d\nis nonzero (the spin-orbit coupling does not vanish and\nk/ne}ationslash= 0), we have\n/an}b∇acketle{tnk|σz|nk/an}b∇acket∇i}ht= 0. (A4)\nThe off-diagonal matrix element of σzcannot be further\ndetermined, and in general /an}b∇acketle{tnk|σz|mk/an}b∇acket∇i}htdepends on the\nchoice of wave functions |nk/an}b∇acket∇i}ht. However, from σ2\nz= 1,\nwe have/an}b∇acketle{t+k|σzσz|+k/an}b∇acket∇i}ht= 1. By inserting/summationtext\nn|nk/an}b∇acket∇i}ht/an}b∇acketle{tnk|=\n1 into the result, we obtain /an}b∇acketle{t+k|σz|+k/an}b∇acket∇i}ht/an}b∇acketle{t+k|σz|+k/an}b∇acket∇i}ht+\n/an}b∇acketle{t+k|σz|−k/an}b∇acket∇i}ht/an}b∇acketle{t−k|σz|+k/an}b∇acket∇i}ht= 1. Because of /an}b∇acketle{t+k|σz|+k/an}b∇acket∇i}ht= 0\nas shown above, we have in general\n|/an}b∇acketle{tnk|σz|mk/an}b∇acket∇i}ht|2= 1, n/ne}ationslash=m. (A5)\nThe velocity operator is defined as vb=∂H0//planckover2pi1∂kb=\n∂ǫk//planckover2pi1∂kb+σx(∂dx//planckover2pi1∂kb) +σy(∂dy//planckover2pi1∂kb). By using\nEqs. (A2), (A3), the off-diagonal matrix element of ve-\nlocity operator is given by\n/an}b∇acketle{tnk|vb|mk/an}b∇acket∇i}ht=−2id2\n/planckover2pi1(Emk−Enk)/an}b∇acketle{tnk|σz|mk/an}b∇acket∇i}ht∂θ\n∂kb,(A6)\nwhere Eq. (24) was used. The off-diagonal matrix el-\nement of vbis related to the position operator xbby\nvb=dxb/dt= [xb,H0]/i/planckover2pi1. By using /an}b∇acketle{tnk|mk/an}b∇acket∇i}ht= 0 for\nn/ne}ationslash=m, we have\n/an}b∇acketle{tnk|xb|mk/an}b∇acket∇i}ht=i/planckover2pi1\nEmk−Enk/an}b∇acketle{tnk|vb|mk/an}b∇acket∇i}ht.(A7)\nFor the diagonal part of vb, by using Eqs. (A1) and (A3),\nwe can obtain /an}b∇acketle{tnk|vb|nk/an}b∇acket∇i}ht=∂Enk//planckover2pi1∂kb.Appendix B: Heisebgerg operator\nIn this appendix, we directly obtain the solutions of\nspin dynamics in terms of the time-evolving spin opera-\ntors in the Heisenburg picture. We will show that the re-\nsult obtained from the method of solving equation of mo-\ntion is the same with that obtained from the Heisenberg\ntime evolution method. The Hamiltonian under consid-\neration is given by\nH0=ǫk+K, (B1)\nwhereK≡σxdx+σydy, anddxanddydepends on the\nmomentum and the spin-orbit coupling. The band index\nnis defined as Enk=ǫk−nd, and the diagonalmatrixel-\nement of Kis/an}b∇acketle{tnk|K|nk/an}b∇acket∇i}ht=−nd. The Pauli matrices σx,\nσyandσzsatisfy{σi,σj}= 2δijand [σi,σj] = 2iǫijkσk.\nBy using K2=d2\nx+d2\ny≡d2, we have K2m+1=d2mK.\nThe time evolution operator of the unperturbed Hamil-\ntonian is given by\neiH0t//planckover2pi1=eiǫkt//planckover2pi1/bracketleftbigg\ncos/parenleftbiggd\n/planckover2pi1t/parenrightbigg\n+iK\ndsin/parenleftbiggd\n/planckover2pi1t/parenrightbigg/bracketrightbigg\n.(B2)\nThe time evolution of σx,σyandσzunder the unper-\nturbed Hamiltonian can be written as\nσH0(t) =eiH0t//planckover2pi1σe−iH0t//planckover2pi1\n=σcos2/parenleftbiggd\n/planckover2pi1t/parenrightbigg\n+i\n2dsin/parenleftbigg2d\n/planckover2pi1t/parenrightbigg\n[K,σ]\n+1\nd2sin2/parenleftbiggd\n/planckover2pi1t/parenrightbigg\n(KσK).(B3)\nThe operator [ K,σ] is given by\n[K,σx] =−2iσzdy,[K,σy] = 2iσzdx,\n[K,σz] =−2i(dxσy−dyσx).(B4)\nThe operator KσKis given by\nKσxK= 2dy(dxσy−dyσx)+σxd2,\nKσyK=−2dx(dxσy−dyσx)+σyd2,\nKσzK=−σzd2.(B5)\nTherefore, we have\nσH0\nx(t) =σx+dy\ndsin(Ω0t)σz\n−dy(d×σ)z\nd2[cos(Ω 0t)−1]\nσH0\ny(t) =σy−dx\ndsin(Ω0t)σz\n+dx(d×σ)z\nd2[cos(Ω 0t)−1]\nσH0\nz(t) =σzcos(Ω0t)+(d×σ)z\ndsin(Ω0t),(B6)12\nwhere Ω 0= 2d//planckover2pi1was used. In order to simplify the\ncalculations, we define dx=dsinθanddy=−dcosθ,\nand we have Eq. (24). For position operator xa, we have\nxH0\na(t) =eiH0t//planckover2pi1xae−iH0t//planckover2pi1\n=xa+eiH0t//planckover2pi1/parenleftbigg\ni∂\n∂kae−iH0t//planckover2pi1/parenrightbigg\n,(B7)\nBy substitution of Eq. (B2), we have\nxH0\na(t) =xa+/bracketleftbigg∂ǫk\n/planckover2pi1∂ka+K\n2d∂Ω0\n∂ka/bracketrightbigg\nt+1\n2σz∂θ\n∂ka[cos(Ω 0t)−1]\n+1\n2d∂θ\n∂ka(d×σ)zsin(Ω0t).\n(B8)\nIn the presence of applied electric field, the operator\nOcan be perturbatively expanded up to the first order\nof the electric field [33], and the result is given by\nOH(t) =OH0(t)+eEa[Γa,OH0(t)]+o(λ2),(B9)\nwhere the operator Γ ais given by\nΓa=i\n/planckover2pi1/integraldisplayt\n0dt′eiH0t′//planckover2pi1xae−iH0t′//planckover2pi1.(B10)\nBy substituting Eq. (B6) and Eq. (B8) into Eq. (B9) and\nEq. (B10), after straightforward calculations, we have\nσH(t) =σH0(t)+i\n/planckover2pi1[A1σx+A2σy\n+A3σz+BK+C(d×σ)z],(B11)\nwhereAx1,Ax2,Ax3,BxandCxare for the spin-\nxcomponent, the spin- ycomponent and so on. The\nseparation is convenient for obtaining the diagonal ma-\ntrix element /an}b∇acketle{tnk|(···)|nk/an}b∇acket∇i}htbecause /an}b∇acketle{tnk|σz|nk/an}b∇acket∇i}ht= 0 and\n/an}b∇acketle{tnk|(d×σ)z|nk/an}b∇acket∇i}ht= 0. The results of the spin- zcompo-\nnent are given by\nAx1=itdy\nd2[cos(Ω 0)−1]∂dy\n∂ka,\nAx2=−itdy\nd2[cos(Ω 0)−1]∂dx\n∂ka+i∂θ\n∂ka/bracketleftbigg1\nΩ0sin(Ω0t)−t/bracketrightbigg\n,\nAx3=it∂\n∂ka/parenleftbiggdy\ndsin(Ω0t)/parenrightbigg\n−it2\n2d∂Ω0\n∂kadycos(Ω0t)\n+i/planckover2pi1\n2d2∂θ\n∂ka[cos(Ω 0t)−1]dx,\nBx=i∂θ\n∂ka/bracketleftbigg1\nΩ0sin(Ω0t)−t/bracketrightbiggdy\nd2[cos(Ω 0t)−1]\n−i/planckover2pi1\n2d2∂θ\n∂ka[cos(Ω 0t)−1]dy\ndsin(Ω0t),\nCx=−it2\n2d∂Ω0\n∂kady\ndsin(Ω0t)−it∂\n∂ka/bracketleftbiggdy\nd2(cos(Ω 0t)−1)/bracketrightbigg\n.\n(B12)By using /an}b∇acketle{tnk|σH0x(t)|nk/an}b∇acket∇i}ht=/an}b∇acketle{tnk|σx|nk/an}b∇acket∇i}ht=−ndx/d,\n/an}b∇acketle{tnk|σy|nk/an}b∇acket∇i}ht=−ndy/d,/an}b∇acketle{tnk|K|nk/an}b∇acket∇i}ht=−ndand neglect-\ning the irrelevant terms CxandAx3, we obtain\n/an}b∇acketle{tnk|σH\nx(t)|nk/an}b∇acket∇i}ht\n=−ndx\nd+i\n/planckover2pi1/braceleftbiggdy\nd2[cos(Ω 0t)−1]int\nd/parenleftbigg∂d\n∂ka×d/parenrightbigg\nz\n−in∂θ\n∂ka/bracketleftbigg/planckover2pi1\n2dsin(Ω0t)−t/bracketrightbiggdy\ndcos(Ω0t)\n+in/planckover2pi1\n2d2∂θ\n∂ka[cos(Ω 0t)−1]dysin(Ω0t)/bracerightbigg\n=−ndx\nd+neEa\n/planckover2pi1dy\nd∂θ\n∂ka/bracketleftbigg1\nΩ0sin(Ω0t)−t/bracketrightbigg\n.\n(B13)\nSimilar to the derivation of /an}b∇acketle{tnk|σH\nx(t)|nk/an}b∇acket∇i}ht, the spin- y\ncomponent is given by\n/an}b∇acketle{tnk|σH\nx(t)|nk/an}b∇acket∇i}ht=−ndy\nd−neEa\n/planckover2pi1dx\nd∂θ\n∂ka/parenleftbigg1\nΩ0sin(Ω0t)−t/parenrightbigg\n.\n(B14)\nFor the spin- zcomponent, the coefficients are given by\nAz1=−itsin(Ω0t)1\nd∂dy\n∂ka,\nAz2=itsin(Ω0t)1\nd∂dx\n∂ka,\nAz3=−it2\n2∂Ω0\n∂kasin(Ω0t),\nBz=it\nd∂θ\n∂kasin(Ω0t)+i/planckover2pi1\n2d2∂θ\n∂ka[cos(Ω 0t)−1],\nCz=it2\n2d∂Ω0\n∂kacos(Ω0t)−itsin(Ω0t)1\nd2∂d\n∂ka.(B15)\nBy using /an}b∇acketle{tnk|σH0x(t)|nk/an}b∇acket∇i}ht=/an}b∇acketle{tnk|σx|nk/an}b∇acket∇i}ht=−ndx/d,\n/an}b∇acketle{tnk|σy|nk/an}b∇acket∇i}ht=−ndy/d,/an}b∇acketle{tnk|K|nk/an}b∇acket∇i}ht=−nd,/an}b∇acketle{tnk|σz|nk/an}b∇acket∇i}ht=\n0 and neglecting the irrelevant terms CxandAx3, we can\nobtain\n/an}b∇acketle{tnk|σH\nz(t)|nk/an}b∇acket∇i}ht=neEa\n/planckover2pi1Ω0∂θ\n∂ka[cos(Ω 0t)−1].(B16)\nBy using the definition of ΣN\nz, the components of spin\ncan be written as\n/an}b∇acketle{tnk|σH\n/bardbl(t)|nk/an}b∇acket∇i}ht=−nΩ0\nΩ0+ΣN\nz(Ω0׈ez)/bracketleftbigg\nt−sin(Ω0t)\nΩ0/bracketrightbigg\n,\n/an}b∇acketle{tnk|σH\nz(t)|nk/an}b∇acket∇i}htˆez= ΣN\nz[1−cos(Ω0t)]ˆez.\n(B17)\nand we can see that Eq. 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B 93, 239904 (2016)."    },    {        "title": "1812.00720v1.Microscopic_theory_of_magnon_drag_electron_flow_in_ferromagnetic_metals.pdf",        "content": "Microscopic theory of magnon-drag electron \row in ferromagnetic metals\nTerufumi Yamaguchi and Hiroshi Kohno\nDepartment of Physics, Nagoya University, Nagoya 464-8602, Japan\nRembert A. Duine\nInstitute for Theoretical Physics, Utrecht University,\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands and\nDepartment of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: December 4, 2018)\nA temperature gradient applied to a ferromagnetic metal induces not only independent \rows of\nelectrons and magnons but also drag currents because of their mutual interaction. In this paper, we\npresent a microscopic study of the electron \row induced by the drag due to magnons. The analysis\nis based on the s-dmodel, which describes conduction electrons and magnons coupled via the s-d\nexchange interaction. Magnetic impurities are introduced in the electron subsystem as a source\nof spin relaxation. The obtained magnon-drag electron current is proportional to the entropy of\nmagnons and to \u000b\u0000\f(more precisely, to 1 \u0000\f=\u000b), where\u000bis the Gilbert damping constant and\n\fis the dissipative spin-transfer torque parameter. This result almost coincides with the previous\nphenomenological result based on the magnonic spin-motive forces, and consists of spin-transfer\nand momentum-transfer contributions, but with a slight disagreement in the former. The result\nis interpreted in terms of the nonequilibrium spin chemical potential generated by nonequilibrium\nmagnons.\nI. INTRODUCTION\nTransport phenomena in ferromagnetic metals exhibit\nsurprisingly rich physics as unveiled by intensive stud-\nies in spintronics. This is largely because they involve\ntransport of not only charge and heat but also spin\nangular momentum. In the presence of magnetization\ntextures, applying an electric current induces magneti-\nzation dynamics because of spin-transfer torques that\nthe spin current of electrons exerts on the magnetiza-\ntion [1, 2]. In turn, a time-dependent magnetization\ninduces spin and charge currents of electrons via spin-\nmotive forces that are reciprocal to the spin-transfer\ntorques [3]. Even when the (equilibrium) magnetization\nis uniform, its thermal/quantum \ructuations, i.e., spin\nwaves or magnons, can interact with electrons. More-\nover, transport through an inhomogeneous region induces\nnonequilibrium spin accumulation, both in electrons and\nmagnons, which then induce di\u000busion spin currents. The\nconcept of \\spin chemical potential\" [4] and \\magnon\nchemical potential\" [5] have been introduced to describe\nsuch e\u000bects.\nOne of the important e\u000bects in the interplay of elec-\ntrons and magnons in transport phenomena are drag ef-\nfects. When subjected to a temperature gradient, elec-\ntrons and magnons start to \row, \frst independently, and\nthen by dragging with each other. Thermoelectric mea-\nsurements indicate the presence of magnon-drag contri-\nbutions in Fe [6], NiCu [7], NiFe [8], and in Fe, Co and\nNi [9]. Theoretical studies include both phenomenolog-\nical [7, 10, 11] and microscopic [12] ones. In particular,\nphenomenological studies based on the spin-motive force\npicture [10, 11] indicate the importance of the dissipative\n\fparameter, which stems from spin relaxation of elec-trons. Microscopic treatment of spin-relaxation e\u000bects\nrequires the consideration of so-called vertex corrections,\nbeyond the simple self-energy (damping or scattering-\ntime) e\u000bects, as noted in the study of current-induced\nspin torque [13, 14], but such studies are not available\nyet for the drag e\u000bects. In a related work, which stud-\nies spin torques due to magnons, a careful treatment of\nthe spin-relaxation e\u000bects revealed an additional contri-\nbution not obtained in a phenomenological analysis [15].\nTherefore, one may expect an analogous situation also in\nmagnon-drag transport phenomena.\nIn this paper, we present a microscopic analysis of\nmagnon-drag electric current (or electron \row) induced\nby a temperature gradient. Using as a microscopic model\nthes-dmodel that describes conduction electrons inter-\nacting with magnons, we calculate the electric current\ncaused by magnons that are driven by the temperature\ngradient. The temperature gradient is treated by its me-\nchanical equivalent, a \fctitious gravitational \feld, intro-\nduced by Luttinger [16]. The obtained result consists\nof two terms, which may be interpreted as due to the\nspin-transfer e\u000bect and the momentum-transfer e\u000bect,\nas in the phenomenological theory [11]. However, as to\nthe former (spin-transfer e\u000bect), there is a quantitative\ndi\u000berence, and our result is proportional to \u000b\u0000\f(or\n1\u0000\f=\u000b), where\u000bis the Gilbert damping constant. It\nvanishes, and changes sign, at \u000b=\f, which agrees with\nthe intuitive notion that the case \u000b=\fis very spe-\ncial. Although this is mostly of conceptual importance,\nit may acquire a practical one if one can determine the\nvalue of\f=\u000b from magnon-drag experiments. We inter-\npret the results in terms of the spin chemical potential\ninduced by magnons. In the course of our study, we give\nan argument that justi\fes the Luttinger's argument byarXiv:1812.00720v1  [cond-mat.mes-hall]  3 Dec 20182\nan explicit calculation.\nThe organization of the paper is as follows. In Sec. II,\nwe describe the microscopic model and some calcula-\ntional tools such as Green's functions. In Sec. III, we out-\nline the microscopic calculation of magnon-drag electron\n\row. The result is discussed in terms of spin-motive force\nand spin chemical potential. In Sec. IV, we revisit the\nphenomenological theory based on the spin-motive force,\nand compare the result with our microscopic result. In\nSec. V, we give an alternative analysis which \\derives\"\nthe spin chemical potential. Details of the microscopic\ncalculations are presented in Appendices A and B. In Ap-\npendix C, we reanalyze the phenomenological theory in\nanother way using the stochastic Landau-Lifshitz-Gilbert\nequation.\nII. MODEL\nA. Hamiltonian\nWe consider a system consisting of conduction elec-\ntrons and magnons in a ferromagnetic metal with uniform\nequilibrium magnetization. The Hamiltonian is given by\nH=H0\nel+Hmag+Hsd; (1)\nH0\nel=Z\ndr\u00141\n2m(@icy)(@ic) +cy(Vimp\u0000\u0016)c\u0015\n;(2)\nHmag=X\nq!qay\nqaq; (3)\nHsd=\u0000JsdZ\ndrcy(S\u0001\u001b)c; (4)\nwherec=t(c\";c#) andcy= (cy\n\";cy\n#) are annihilation and\ncreation operators of the electrons, aqanday\nqare those\nof magnons, mand\u0016are the mass and the chemical\npotential of the electrons, !q=Jq2+ \u0001 is the magnon\ndispersion with exchange sti\u000bness Jand energy gap \u0001,\nS=Sn(jnj= 1) is the localized spin with magnitude\nS,\u001b= (\u001bx;\u001by;\u001bz) are Pauli matrices, and Jsdis the\ns-dexchange coupling constant. We consider low enough\ntemperature and assume Sis constant. Hereafter we use\nM\u0011JsdSandninstead ofS. ForVimp, we consider\nboth nonmagnetic and magnetic impurities,\nVimp(r) =uiX\ni\u000e(r\u0000Ri) +usX\njSj\u0001\u001b\u000e(r\u0000R0\nj);\n(5)\nwhereSjis the impurity spin located at position R0\nj. We\naverage over the impurity positions, RiandR0\nj, as usual,\nand the impurity spin directions,\nS\u000b\niS\f\nj=\u000eij\u000e\u000b\f\u0002\u001a\nS2\n?(\u000b=\f=x;y)\nS2z(\u000b=\f=z): (6)The s-dexchange interaction describes the exchange-\nsplitting in the electron spectrum, and the electron-\nmagnon scattering,\nHsd=\u0000MZ\ndrcy\u001bzc+Hel\u0000mag; (7)\nHel\u0000mag=MZ\ndr\u00141\ns0aya^\u001bz\u0000r\n2\ns0\u0000\na^\u001b\u0000+ay^\u001b+\u0001\u0015\n;\n(8)\nwheres0=S=r3\n0is the spin density of the magnetization,\nr0the lattice constant, ^\u001b=cy\u001bc, and ^\u001b\u0006= (^\u001bx\u0006i^\u001by)=2.\nThe total Hamiltonian is given by\nH=Hel+Hmag+Hel\u0000mag; (9)\nHel=Z\ndr\u00141\n2m(@icy)(@ic) +cy(Vimp\u0000\u0016)c\u0000Mcy\u001bzc\u0015\n:\n(10)\nB. Green's function\nThe Green's functions of electrons Gk\u001b(i\"n) and\nmagnonsDq(i\u0017l) are given by\nGk\u001b(i\"n) =1\ni\"n+\u0016\u0000k2=2m+\u001bM\u0000\u0006\u001b(i\"n);(11)\nDq(i\u0017l) =1\ni\u0017l\u0000!q\u0000\u0005q(i\u0017l); (12)\nwith Matsubara frequencies, \"n= (2n+ 1)\u0019Tand\u0017l=\n2\u0019lT, and self-energies, \u0006 \u001b(i\"n) and \u0005 q(i\u0017l), for the elec-\ntrons and magnons, respectively.\nFIG. 1. (a) Self-energy of electrons, \u0006. (b) Self-energy of\nmagnons, \u0005. (c) Spin vertex \u0003\u000brenormalized by impurity-\nladder corrections. (d) Four-point vertex \u0000 \u001b\u001b0, which we\ncall the di\u000busion-type vertex correction, or simply, the dif-\nfusion propagator. The solid (wavy) lines represent electron\n(magnon) propagators, and the dashed line with a cross rep-\nresents impurity scattering.3\nWe assume the electron self-energy is dominated by im-\npurity scattering and treat it in the Born approximation\n[Fig. 1 (a)]. Thus, \u0006R\n\u001b(\") = \u0006\u001b(\"+i0) =\u0000i\r\u001b, with\n\r\u001b=\u0019(\u00001\u0017\u001b+ \u00002\u0017\u0016\u001b)\u00111\n2\u001c\u001b; (13)\nand\n\u00001=niu2\ni+nsu2\nsS2z; \u00002= 2nsu2\nsS2\n?: (14)\nHere,ni(ns) is the concentration of nonmagnetic (mag-\nnetic) impurities, and \u0017\u001bis the density of states of spin- \u001b\nelectrons.\nThe magnon self-energy comes from the electron-\nmagnon scattering [Fig. 1 (b)]. Expanding with respect\nto the wave vector qand the frequency \u0017of magnons, we\nwrite\n\u0005q(\u0017+i0) =\u0000\u0014\u000eS\nS+i\u000b\nz\u0015\n\u0017\u0000\u000eJq2+O(\u00172;q4):(15)\nHere,\u000eS,\u000eJandz\u0011S=(S+\u000eS) are the renormaliza-\ntion constants for spin, the exchange sti\u000bness, and wave\nfunction, respectively, of the localized spins. Also, \u000bis\nthe Gilbert damping constant calculated as [13]\n\u000b=\u0019nsu2\nsh\n2S2z\u0017\"\u0017#+S2\n?(\u00172\n\"+\u00172\n#)i\nz=s0: (16)\nHere and hereafter, we assume the s-dexchange coupling\nMis much larger than the spin-relaxation rate [15].\nAs seen fromHel\u0000mag[Eq. (8)], the natural expan-\nsion parameter in the electron-magnon problem is s\u00001\n0\n(orS\u00001). In this paper, we focus on the leading contri-\nbutions, which are O(s\u00001\n0). (As seen below, we need two\nelectron-magnon scattering vertices in the magnon-drag\nprocess, giving\u0018(s\u00001=2\n0)2=s\u00001\n0.) Since\u000eSand\u000eJare\nO(s\u00001\n0), we setz= 1 and\u000eJ= 0 in the magnon Green's\nfunction.\nIII. MICROSCOPIC CALCULATION\nA. Thermal linear-response theory\nTo treat the temperature gradient in the linear re-\nsponse theory, we introduce Luttinger's (\fctitious) gravi-\ntational potential  , which couples to the energy density\nh(r) of the system [16]. The coupling is described by the\nHamiltonian,\nH0=Z\ndrh(r) (r;t): (17)\nWe consider the case,  (r;t) = Q;!ei(Q\u0001r\u0000!t), whereQ\nand!are the wave vector and the frequency of  , and\nwrite the linear response of a physical quantity Ato as\nhAi \n!=\u0000hA;h(\u0000Q)i!+i0 Q;!; (18)\nFIG. 2. Feynman diagrams for Kij(i!\u0015) [Eq. (26)], which\ndescribe the magnon-drag processes. The solid (wavy) lines\nrepresent the electron (magnon) Green's functions. (a) Pro-\ncesses for Q=0. The gray triangles are de\fned in Fig. 1 (c).\n(b) Additional processes that contribute when Q6=0. The\ngray square represents the di\u000busion propagator \u0000 \u001b\u001b0de\fned\nin Fig. 1 (d). The diagrams in (b) vanish for Q=0, but\ncontribute for \fnite Qand lead to Eq. (57).\nwhereh(\u0000Q) is the Fourier component of h(r). The\nresponse function hA;Bi!+i0is obtained from\nhA;Bii!\u0015\u0011ZT\u00001\n0d\u001cei!\u0015\u001chT\u001cA(\u001c)Bieq; (19)\nby the analytic continuation, i!\u0015!!+i0, whereA\nandBare arbitrary operators. Here, Tis the tempera-\nture andh\u0001\u0001\u0001ieqrepresents the average in thermal equi-\nlibrium. Hereafter we use h\u0001\u0001\u0001i instead ofh\u0001\u0001\u0001ieqfor\nsimplicity. Using the continuity equation,\n@th(r) +@ijQ\ni= 0; (20)\nwhich de\fnes the heat-current density jQ\ni, we rewrite\nEq. (18) as a linear response to ( \u0000@i ) [17],\nhAi \n!=Ki(!+i0)\u0000Ki(0)\ni!\u0012\n\u0000@j \u0000@jT\nT\u0013\nQ;(21)\nKi(i!\u0015) =hA;jQ\ni(\u0000Q)ii!\u0015: (22)\nHere, we introduced the temperature gradient @iT\nthrough the combination, \u0000@i \u0000@iT=T. This is justi\fed\nfor operators Aof which the average vanishes naturally\nin the equilibrium state, where @iT=T +@i = 0 holds\n[16, 17]. Therefore, the response to ( \u0000@iT=T) is obtained\nas the response to ( \u0000@i ) [16].\nB. Magnon-drag process\nSpecializing to the present model, Eq. (9), we \fnd from\nEq. (20) that the heat-current density jQ\niconsists of two4\nparts,jQ\ni=jQ\nel;i+jQ\nmag, one for the electrons ( jQ\nel;i) and\none for magnons,\njQ\nmag;i=\u0000J\u0002\n_ay(@ia) + (@iay)_a\u0003\n; (23)\nwhere _a=@ta.\nIn this paper, we are interested in the magnon-drag\nprocess, which corresponds to taking the magnon heat-\ncurrent density jQ\nmag;iforjQ\niin Eq. (22). As for Ain\nEq. (22), we focus on the electron (number) current den-\nsity,\njel;i=~\n2mi\u0002\ncy(@ic)\u0000(@icy)c\u0003\n: (24)Therefore, we consider\nhjel;iidrag=Kij(!+i0)\u0000Kij(0)\ni!\u0012\n\u0000@j \u0000@jT\nT\u0013\n;\n(25)\nKij(i!\u0015) =hjel;i(Q) ;jQ\nmag;j(\u0000Q)ii!\u0015; (26)\ni.e., the correlation function between the electron (num-\nber) current and the magnon heat current. Here jel;i(Q)\nandjQ\nmag;j(\u0000Q) represent their respective Fourier com-\nponents. The combination \u0000@j \u0000@jT=T in Eq. (25) in-\ndicates that the current vanishes in the equilibrium state,\nin which\u0000@j \u0000@jT=T = 0 (Einstein-Luttinger relation)\nholds. We will verify this form by an explicit calculation\nin Secs. V-A and V-B.\nThe relevant magnon-drag processes are shown dia-\ngrammatically in Fig. 2 (a). These are the leading con-\ntribution with respect to 1 =s0, and expressed as\nKij(i!\u0015) =2M2\ns0TX\nl;quj\u001a\u0012\ni\u0017l+i!\u0015\n2\u0013\nDq(i\u0017l+i!\u0015)Dq(i\u0017l)\u00001\n2[Dq(i\u0017l+i!\u0015) +Dq(i\u0017l)]\u001b\nEi; (27)\nwhereui= 2Jqiis the magnon velocity, !\u0015is the Matsubara frequency of the external perturbation  , and we have\nsetQ=0for simplicity. The terms linear in Dqare \\corrections\" arising from the \u000e-function in the relation [17],\nhT\u001ca(\u001c) _ayi=\u0000hT\u001c_a(\u001c)ayi=d\nd\u001cD(\u001c) +\u000e(\u001c): (28)\nThese terms, combined with the \frst term ( \u0018DqDq) in the curly brackets, lead to f\u0001\u0001\u0001g =f!q+1\n2[\u0005q(i\u0017l+i!\u0015) +\n\u0005q(i\u0017l)]gDqDq. This amounts to making a replacement, i\u0017l+i!\u0015=2!!q, in the \frst term if the self-energies are\nneglected.\nThe last factorEiin Eq. (27) is the electron part coming from the electron triangles in Fig. 2 (a),\nEi=TX\nn;kvi\u0002\nGk#(i\"n+i!\u0015) \u0003\u0000\n#\"Gk\u0000q;\"(i\"n\u0000i\u0017l) \u0003+\n\"#Gk#(i\"n)\n+Gk\"(i\"n+i!\u0015) \u0003+\n\"#Gk+q;#(i\"n+i\u0017l+i!\u0015) \u0003\u0000\n#\"Gk\"(i\"n)\u0003\n; (29)\nwherevi=ki=mis the electron velocity and \u0003\u0006\n\u001b\u001b0's are\nthe renormalized spin ( \u001b\u0006) vertices; see Appendix A for\nthe de\fnition. After the analytic continuations, i\u0017l!\u0017\nandi!\u0015!!+i0, an expansion is made with respect to!and/or\u0017. From Eq. (25), we are primarily interested\nin the!-linear terms. The factor !comes either from the\nmagnon part or from the electron part. Hence we write\nKij(!+i0)\u0000Kij(0)\n'2M2\ns0i!\n2\u0019(Z\nd\u0017\u0012\n\u0000@n\n@\u0017\u0013\n\u0017X\nqujDR\nq(\u0017)DA\nq(\u0017)E(2)\ni\u00002Z\nd\u0017n(\u0017)\u0017X\nqujIm\"\nDR\nq(\u0017)DR\nq(\u0017)E(1)\ni\n\u0000i!#\n\u00001\n2Z\nd\u0017\u0012\n\u0000@n\n@\u0017\u0013X\nquj\u0002\nDR\nq(\u0017) +DA\nq(\u0017)\u0003\nE(2)\ni+ 2Z\nd\u0017n(\u0017)X\nqujIm\"\nDR\nq(\u0017)E(1)\ni\n\u0000i!#)\n;(30)\nwheren(\u0017) = (e\u0017=kBT\u00001)\u00001is the Bose-Einstein dis- tribution function. The terms in the second line are the5\ncorrections mentioned above. E(1)\niis obtained fromEiby\nthe analytic continuation, i(\u0017l+!\u0015)!\u0017+!+i0 and\ni\u0017l!\u0017+i0, andE(2)\nibyi(\u0017l+!\u0015)!\u0017+!+i0 and\ni\u0017l!\u0017\u0000i0. In the term with E(2)\ni,!is picked up from\nthe magnon part, whereas in the term with E(1)\ni,!is\nobtained from the electron part.\nAt this point, it is worth noting that not only DRDA\nbut alsoDRDRappears in Eq. (30) for the pair of\nmagnon propagators. This is not surprising in diagram-\nmatic calculations as being done here, but seems incom-\npatible with the spin-motive force picture, in which there\nshould be a causal relationship between the magnetiza-\ntion dynamics and the resulting current (see Sec. IV).\nWe retain low-order terms with respect to \u0017, which is\njusti\fed because the magnon energy \u0017is typically small\ncompared to the electron Fermi energy. Deferring the de-\ntails to Appendix B, the electron part has been calculated\nas\nE(2)\ni=1\n(2M)2\u001b\"\u0000\u001b#\ne2f2\fel\u0017\u0000i!(1 +i\fel)gqi;(31)\nE(1)\ni'1\n(2M)2\u001b\"\u0000\u001b#\ne2(\u0000i!qi); (32)\nwhere\u001b\u001b=e2(v2\nF\u001b=3)\u0017\u001b\u001c\u001bis the conductivity of elec-trons with spin \u001b,\n\fel=\u0019nsu2\ns\nMh\n(S2\n?+S2z)\u0017++P\u00001(S2\n?\u0000S2z)\u0017\u0000i\n;(33)\nis the so-called \fparameter that parametrizes the dissi-\npative corrections to the spin-transfer torque [13, 14] and\nto the Berry-phase spin-motive force [18, 19]. We de\fne\n\u0017\u0006=\u0017\"\u0006\u0017#andP= (\u001b\"\u0000\u001b#)=(\u001b\"+\u001b#). For the present\npurpose, we can discard the !-linear term inE(2)\ni. It will\nbe used in Sec. IV when we discuss the spin-motive force.\nThe magnon part is calculated by using\n1\n2\u0019Z\nd\u0017\u0012\n\u0000@n\n@\u0017\u0013\n\u00172X\nquiqjDR\nqDA\nq=1\n2\u000bTSmag\u000eij;\n(34)\n1\n\u0019Z\nd\u0017n(\u0017)\u0017X\nquiqjImh\u0000\nDR\nq\u00012i\n=Emag\u000eij;(35)\n1\n\u0019Z\nd\u0017n(\u0017)X\nquiqjIm\u0002\nDR\nq\u0003\n= \n mag\u000eij;(36)\nwhereEmag=P\nq!qn(!q) is the energy density, \n mag=\nkBTP\nqln(1\u0000e\u0000~!q=kBT) is the thermodynamic poten-\ntial density, andSmag=\u0000@\nmag=@T is the entropy den-\nsity of magnons. Thus the magnon-drag electron (num-\nber) current is obtained as\nhjel;iidrag=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0014\nEmag\u0000\fel\n\u000bTSmag\u0000\nmag\u0015\n(\u0000@i )\n=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag(\u0000@i ); (37)\nwhere we used \n mag=Emag\u0000TSmag. Note that \n mag,\nwhich arises as \\corrections\" here, turned the energy\nEmaginto the entropy TSmag, and the result depends on\nmagnons only through their entropy. This is the main\nresult of this paper.\nC. Result\nA physical result is obtained by replacing @i by\n@iT=T,\nhjelidrag=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag\u0012\n\u0000rT\nT\u0013\n(38)\n=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nr\nmag: (39)\nIn the second line, we noted Smag=\u0000@\nmag=@T and as-\nsumed that \n magis position (r) dependent only throughthe local temperature, T=T(r).\nThe obtained magnon-drag current, Eq. (39), is pro-\nportional to \u001b\"\u0000\u001b#[20]. This indicates that the magnons\nexert on the electrons a spin-dependent force,\nF\u001b=\u0000\u001b\n2s0\u0012\n1\u0000\fel\n\u000b\u0013\nr\nmag; (40)\nwhere\u001b= 1 or\u00001 depending on the electron spin projec-\ntion,\u001b=\"or#. Some discussion will be given in Sec. IV\nin relation to the spin-motive force.\nEquation (40) has the form of total gradient, suggest-\ning that it is of di\u000busive nature and is induced by a spin-\ndependent, nonequilibrium chemical potential,\n\u000e\u0016\u001b=\u001b\n2s0\u0012\n1\u0000\fel\n\u000b\u0013\n\u000e\nmag; (41)\nwhere\u000e\nmagis the deviation of \n magfrom its thermal-\nequilibrium value. In Sec. V-B, we will give a further\nanalysis that supports this picture of the spin chemical\npotential.6\nIV. PHENOMENOLOGY BASED ON\nSPIN-MOTIVE FORCE\nIn this section, we revisit the phenomenology based\non the spin-motive force along the lines of Refs. [10, 11],\nand compare the result with the microscopic result. The\nphysical pictures that emerge from the microscopic study\nare also discussed.\nWhen the magnetization vector nvaries in space and\ntime, it exerts a spin-dependent force, \u0006Fi, on electrons,\nwhere\nFi=~\n2[n\u0001(_n\u0002@in)\u0000\f_n\u0001@in]: (42)\nThis is called the spin-motive force. The \frst term is\nthe \\Berry phase term\" and the second term with a\ndimensionless coe\u000ecient \fis the dissipative correction,\nwhich we call the \f-term [18, 19, 21]. ( \fis equal to\n\fel[Eq. (33)], but we continue to use these notations;\n\felfor the microscopically-calculated one, and \ffor the\nphenomenologically-introduced one.) These e\u000bects are\nreciprocal to the current-induced spin torques; the former\nis reciprocal to the spin-transfer torque, and the latter to\nits dissipative correction.\nSpin waves, or magnons, can also be the origin of the\nspin-motive force. Although they are \ructuations, they\nwill induce a net electron current\nhjel;iismf=\u001b\"\u0000\u001b#\ne2hFii; (43)\nif the average survives, hFii6= 0. This will contribute\nto the magnon-drag electron current. Here we assume\na uniformly magnetized state at equilibrium, njeq=\n^z, and consider small \ructuations \u000enaround it, such\nthatn= ^z+\u000en. With magnon operators, fa;ayg=\n(s0=2)1=2(\u000enx\u0006i\u000eny), we rewrite Fias\nFi=i\n2s0\u0002\n\u0000_ay@ia+ (@iay)_a\u0003\n\u0000\f\n2s0\u0002\n_ay@ia+ (@iay)_a\u0003\n:\n(44)\nAs noted previously [10, 11], the second term is essen-\ntially the magnon heat current jQ\nmag;i[Eq. (23)]. Here\nwe note that the \frst term is expressed by the magnon\nenergy,hmag=i(ay_a\u0000_aya)=2, and the magnon current\njmag;i=\u0000iJ[ay@ia\u0000(@iay)a]. Thus,\nFi=1\n2s0\u001a\n@ihmag+1\n2J@\n@tjmag;i+\f\nJjQ\nmag;i\u001b\n:(45)\nLet us evaluate each term in Eq. (45) for a steady\nstate with a temperature gradient. Since the \frst term\nhas a spatial derivative @i, we evaluate it in the local\nequilibrium state as hhi=Emag(T), which depends on r\nthrough the local temperature T=T(r). This leads to\n@ihhmagi= (@Emag=@T)(@iT). The second term vanishes\nin the steady state because of the overall time deriva-\ntive. The third term is evaluated as hjQ\nmag;ii=\u0000\u0014@iTwith the magnon heat conductivity \u0014. This is calculated\nusing, e.g., the Kubo-Luttinger formula as [22]\n\u0014=1\nTZd\u0017\n2\u0019\u0012\n\u0000@n\n@\u0017\u0013\n\u00172X\nqu2\nxDR\nqDA\nq=J\n\u000bSmag;(46)\nwhere we used Eq. (34). This expression for \u0014in terms\nof magnon entropy also follows from an intuitive argu-\nment. Following Drude, one may express the magnon\nheat-current density at position xas [23]\njQ\nx(x) =1\n2X\nqux!q[n(x\u0000ux\u001c)\u0000n(x+ux\u001c)];(47)\nwheren(x) is the Bose distribution function de\fned with\na local temperature T(x),\u001c= (2\u000b!q)\u00001is the lifetime\nof magnons, and the temperature gradient is assumed in\nthexdirection. The \frst term represents the energy \row\nfrom the left region, and the second term from the right,\nwhich are due to magnons that experienced their last\ncollision at x\u0006ux\u001c; the factor 1/2 is there because half of\nmagnons at x\u0006ux\u001c(namely, those with qx>0 orqx<0)\npropagate to x. Expanding as n(x\u0000ux\u001c)\u0000n(x+ux\u001c)'\n\u00002ux\u001c(@n=@T )(@T=@x ) and using Eq. (B16), one has\n\u0014=1\n2\u000b@\n@TX\nqu2\nxn(!q) =J\n\u000bSmag; (48)\nin agreement with Eq. (46).\nTaken together, we obtain\nhFii=1\n2s0\u001a\n\u0000@Emag\n@T+\f\n\u000bSmag\u001b\n(\u0000@iT): (49)\nThe same result has been obtained by other methods; see\nAppendix C. Therefore, we may conclude that any (phe-\nnomenological) theories starting from the spin-motive\nforce lead to Eq. (49). The \frst term is somewhat di\u000ber-\nent from the one obtained in Ref. [11], and gives a slight\nrevision to it (see Appendix C-3).\nWe now compare Eq. (49) with the microscopic result,\nEq. (38). One readily sees a disagreement in the \frst\nterm, namely, the entropy Smagappears in the micro-\nscopic result instead of @Emag=@T in the phenomenolog-\nical result.\nTo identify the the origin of the di\u000berence, let us look\nat the Feynman diagram. To calculate the spin-motive\nforce, one calculates the electric current induced by mag-\nnetization dynamics [18]. This can be done by consider-\ning small \ructuations \u000enaround the uniform magnetiza-\ntion, and look at the second-order (nonlinear) response\nto\u000en[24]. This is expressed diagrammatically in Fig. 3\n(a), and the response function is given by E(2)in Eq. (31).\nTherefore, the induced current is calculated as\nhjel;ii=2M2\ns0E(2)\niaq;\u0017+!a\u0003\n\u0000q;\u0000\u0017; (50)\nwherefa;a\u0003gis a classical (c-number) counterpart of\nfa;aygde\fned just above Eq. (44), and the subscripts7\nFIG. 3. Feynman diagrams for the electric current in-\nduced by magnetization dynamics. Arrows in the electron\nlines (solid lines) are suppressed for simplicity. (a) Nonlin-\near response to the (classical) magnetization dynamics. The\nwavy lines represent the perturbations due to (classical) mag-\nnetization. Because of causality (retarded response), the in-\ncoming Matsubara frequencies should satisfy the conditions,\n\u0017l+!\u0015>0 and\u0000\u0017l>0 [25]. (b) Part of the diagram of\nthe present magnon-drag process (Fig. 2). The wavy lines\nrepresent (quantum) magnon propagators. Note that the\n\row of the Matsubara frequency in the lower magnon line\nis reversed compared to (a). The same causality relation as\n(a) leads to the analytic continuation, D(i\u0017l+i!\u0015)D(i\u0017l)!\nDR(\u0017+!)DA(\u0017), for the pair of magnon propagators, and\nthis is associated with E(2)\nigiven by Eq. (31).\nindicate their wave vector and frequency. This leads to\nEq. (44), hence to Eq. (42). Therefore, the spin-motive\nforce is described by the \u0017- and!-linear terms inE(2)\ni.\nThe appearance of E(2)\ni(originally from the magnon-drag\ncalculation) in the nonlinear response here is due to the\nmatching of the causality relationship; see Fig. 3 (b) and\nthe caption thereof.\nOn the other hand, in the present magnon-drag pro-\ncess, the \frst term comes from the !-linear term inE(1)\ni,\nnot from the !-linear term inE(2)\ni; the latter is irrelevant\nfor the magnon-drag DC electron current. Since E(1)\niis\naccompanied by DRDR(notDRDA), the physical in-\nterpretation of this term (in the magnon-drag current)\ndoes not necessarily rely on the causal relationship to\nthe magnetization dynamics. In fact, the spin-transfer\nprocess may be understood to occur in the quasi- or\nlocal-equilibrium situation, as will be discussed in the\nparagraph containing Eq. (64).\nV. SPIN CHEMICAL POTENTIAL\nIn this section, we give an alternative argument that\nintroduces a spin chemical potential. This is intended to\ncomplement the heuristic discussion in Sec. III-C.\nOur strategy here is as follows. From the viewpoint\nof microscopic theory, statistical quantities such as the\nchemical potential and temperature, which characterize\nthe distribution function, cannot be easily handled. In-\nstead, we can disturb the system by \\mechanical\" per-\nturbations (which are described by the Hamiltonian and\nthus controllable theoretically) and then observe the re-\nsult. By examining how the distribution function is de-formed, we may read o\u000b the change of statistical param-\neters such as chemical potential and temperature. For\nexample, an inhomogeneous potential (or electric \feld)\ninduces a density modulation. This e\u000bect is described\nby an inhomogeneous change of chemical potential, and\nappears in the current as a di\u000busion current [21].\nIn the following, we examine the possibility that the\nmagnon-drag e\u000bects are described in a similar manner.\nWe \frst illustrate the procedure using a simple model\n(Sec. V-A), and then consider the present problem of\nmagnon-drag process (Sec. V-B). In both cases, we take\nthe \feld as a mechanical perturbation.\nA. Electron-only process: E\u000bective temperature\nWe begin by reviewing the relation between the grav-\nitational \feld @i and temperature gradient, @iT. For\nsimplicity, we consider a (spin-unpolarized) free elec-\ntron system subject to nonmagnetic impurities, forget-\nting about magnons and even the magnetization (ex-\nchange splitting). We calculate the electron density \u000enel\nand current density hjel;ii \n!induced by the disturbance\n having \fnite Q(and!). In this case, it is essential\nto consider the di\u000busion-type vertex corrections [Fig. 1\n(d)], hence the diagrams shown in Fig. 4. The results are\ngiven by\n\u000enel=Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"\u0011(\"); (51)\nhjel;ii \n!=Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"\u001b(\")(\u0000@i )Q\n\u0000Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"D(\") [@i\u0011(\")]Q; (52)\nwhere\n\u0011(\") =\u0000\u0017(\")D(\")Q2\nD(\")Q2\u0000i! Q; (53)\ndescribes \\di\u000busive corrections\" which arise since Qis\n\fnite. We de\fned e2\u001b(\") =e2(v2\nF=3)\u0017(\")\u001c(\") andD(\") =\n(v2\nF=3)\u001c(\"), which are the Boltzmann conductivity and\nthe di\u000busion constant, respectively, evaluated at energy\n\"(measured from the chemical potential \u0016).\nIf we consider a local modi\fcation of temperature, T!\nT+\u000eT(r), the electron density changes by\n\u000enel=Z\nd\"\u0017(\")[f(\";T+\u000eT)\u0000f(\";T) ]\n'\u000eT\nTZ\nd\"\u0017(\")\"\u0012\n\u0000@f\n@\"\u0013\n: (54)\nIn the `slow' limit !!0, Eq. (51) may be compared with\nEq. (54), and we may identify the e\u000bective temperature\nchange\u000eTQby\n\u0000lim\n!!0D(\")Q2\nD(\")Q2\u0000i! Q=\u000eTQ\nT: (55)8\nFIG. 4. Feynman diagrams for the electron density \u000enel\n[Eq. (51)] and the current density jel;i[Eq. (52)] induced by\nrT.helandjQ\nel;iare the Hamiltonian density and heat cur-\nrent density, respectively, of the conduction electrons. The\nshaded rectangle represents the di\u000busion-type ladder vertex\ncorrection due to impurities [Fig. 1(d)], which describes dif-\nfusive motion of the electrons.\nThis is nothing but the Einstein-Luttinger relation,  Q+\n\u000eTQ=T= 0, that holds in the equilibrium state (under\na static potential,  Q). Using this \u000eTQ, we may rewrite\nEq. (52) as\nhjel;ii \n!=Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"\u001b(\")(\u0000@i )Q\n\u0000Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"D(\")\u0017(\")\u0012\n\u0000@iT\nT\u0013\nQ\n=Z\nd\"\u0012\n\u0000@f\n@\"\u0013\n\"\u001b(\")\u0012\n\u0000@i \u0000@iT\nT\u0013\nQ:(56)\nThis shows the \\equivalence\" of the mechanical force @i \nand the statistical force @iT=T, and forms a basis of Lut-\ntinger's thermal linear-response theory.\nB. Magnon-drag process: Spin chemical potential\nLet us apply a similar procedure to the magnon-drag\nprocess. For this purpose, we calculate the magnon-drag\nelectron current in response to a spatially-modulated po-\ntential, / Qei(Q\u0001r\u0000!t), with \fnite wave vector Q. As\nin the preceding subsection, we consider the di\u000busion-\ntype vertex corrections and the diagrams in Fig. 2 (b).\nThe result is obtained as\nhjel;iidrag=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag(\u0000@i )\n\u0000@i\u0010\nD\"\u000en\"\nel+D#\u000en#\nel\u0011\n; (57)\nwhere\n\u000en\u001b\nel=\u001b\n2s0\u001b\u001b\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmagQ2\nD\u001bQ2\u0000i! Q;(58)\nis the change of the electron density (of spin \u001b) caused by\nthe perturbation  Q, andD\u001b= (v2\nF\u001b=3)\u001c\u001bis the di\u000busionconstant. From the form of Eq. (58), it is natural to\nregard the density change \u000en\u001b\nelas caused by the change of\nthe electrons' chemical potential, instead of temperature\nas in Eq. (54). Namely, Eq. (58) in the `slow' limit, !!\n0, may be compared with\n\u000en\u001b\nel=Z\nd\"\u0017\u001b(\")[f(\";T\u0016+\u000e\u0016\u001b)\u0000f(\";T;\u0016) ]\n'Z\nd\"\u0017\u001b(\")\u0012\n\u0000@f\n@\"\u0013\n\u000e\u0016\u001b\n'\u0017\u001b\u000e\u0016\u001b; (59)\nwhere\u000e\u0016\u001bis the change in (spin-dependent) chemical\npotential. From the comparison, we may identify [26]\n\u000e\u0016\u001b=\u001b\n2s0\u0012\n1\u0000\fs\n\u000b\u0013\nTSmag Q; (60)\n=\u001b\n2s0\u0012\n1\u0000\fs\n\u000b\u0013\nTSmag\u0012\n\u0000\u000eTQ\nT\u0013\n: (61)\nIn the second line, we used the Einstein-Luttinger rela-\ntion, Q=\u0000\u000eTQ=T. Note that Eq. (61) is consistent\nwith Eq. (41). Using Eq. (61) for \u000e\u0016\u001bin Eq. (59), we\nrewrite Eq. (57) as\nhjel;iidrag=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag(\u0000@i )\n+1\ne2\u001b\"(\u0000@i\u000e\u0016\") +1\ne2\u001b#(\u0000@i\u000e\u0016#) (62)\n=\u00001\n2s0\u001b\"\u0000\u001b#\ne2\u0012\n1\u0000\fel\n\u000b\u0013\nTSmag\n\u0002\u0012\n\u0000@i \u0000@iT\nT\u0013\n: (63)\nThis reproduces the form of Eq. (25).\nThe nonequilibrium chemical potential \u000e\u0016\u001bis spin de-\npendent,\u000e\u0016\"=\u0000\u000e\u0016#(because of the overall factor\n\u001b=\u00061). Thus the electrons feel the e\u000bects of the\nnonequilibrium magnons as a \\spin chemical potential\",\nor spin accumulation, \u0016s=\u0016\"\u0000\u0016#. This is quite natural\nsince the local change \u000eTin temperature modulates the\nmagnon density, and the balance of the \\reaction\"\nm +e\"\u001de#; (64)\nshifts in the left or the right direction. Here, m, e\"and\ne#represent a magnon, an electron with spin up, and\nan electron with spin down, respectively. If we focus on\nthe electrons ( e\"ande#), this is precisely controlled by\nthe chemical-potential di\u000berence, \u0016\"\u0000\u0016#. This process\ncorresponds to the \frst term (the spin-transfer term).\nThe absence of the causality relationship, as discussed at\nthe end of Sec. IV, may be due to the local equilibrium\nnature of this process.\nThe second term (proportional to \fel=\u000b) acts in the\nopposite way; it increases the density of up-spin (down-\nspin) electrons in the hotter (colder) region. Let us inter-\npret this e\u000bect in terms of momentum transfer process.9\nFor this, we consider the e\u000bects of magnon \row. The\nmagnons \row from the hotter to the colder region, and\nwill scatter electrons into the colder region. If a magnon\nis absorbed by an electron, the scattered electron has\ndown spin and \rows downstream. This means that the\ndown-spin electrons \row to colder regions and this e\u000bect\nwill increase the density of down-spin electrons in the\ncolder region. There is also a reverse process: if a down-\nspin electron emits a magnon and \rips its spin, and if the\nmagnon \rows downstream, the up electron will \row up-\nstream. This process will increase the density of up-spin\nelectrons in the hotter region.\nVI. SUMMARY\nIn this paper, we studied magnon-drag electron \row in-\nduced by a temperature gradient. The analysis is based\non a microscopic model that contains spin relaxation,\nand on the linear response theory due to Luttinger that\nexploits a gravitational potential  . The obtained re-\nsult is physically interpreted in terms of the spin-transfer\nprocess and the momentum-transfer process from the\nmagnons to the electrons. It is found that the e\u000bect\nof nonequilibrium magnons yields a nonzero spin chem-\nical potential of the electrons. In the process, we gave\na microscopic procedure that leads to the Luttinger's\nform of the response, namely, a combination of the form,\n\u0000@i \u0000@iT=T. We supplemented the analysis with a phe-\nnomenological one that is based on the spin-motive force,\nand found that the agreement with the microscopic result\nis good for the dissipative \f-term, but di\u000bers slightly for\nthe Berry-phase (spin-transfer) term.\nACKNOWLEDGEMENT\nWe are grateful to Y. Imai for fruitful discussions.\nValuable comments by G. E. W. Bauer and J. P. Here-\nmans are also appreciated. This work is supported by\nJSPS KAKENHI Grant Numbers 25400339, 15H05702\nand 17H02929. TY is supported by a Program for Lead-\ning Graduate Schools \\Integrative Graduate Education\nand Research in Green Natural Sciences\". RD is a mem-\nber of the D-ITP consortium, a program of the Nether-\nlands Organisation for Scienti\fc Research (NWO) that\nis funded by the Dutch Ministry of Education, Culture\nand Science (OCW). This work is in part funded by\nthe Stichting voor Fundamenteel Onderzoek der Materie\n(FOM) and the European Research Council (ERC).\nAppendix A: Vertex corrections\nIn this Appendix, we calculate the vertex corrections\nto the electron spin \u001b\u0006due to impurity potentials in\nthe ladder approximation. The renormalized vertex \u0003\u0006\n\u001b\u0016\u001bsatis\fes\n(\u0003\u0006\n\u001b\u0016\u001b)ab= (\u001b\u0006)\u001b\u0016\u001b+ \u00000Yab\n\u001b\u0016\u001b(\u0003\u0006\n\u001b\u0016\u001b)ab; (A1)\nwhere (\u001b+)\"#= (\u001b\u0000)#\"= 1 (other elements vanish),\n\u00000=niu2\ni\u0000nsu2\nsS2z; (A2)\nandYab\n\u001b\u0016\u001b=P\nkGa\nk\u001bGb\nk\u0016\u001bwith \u0016\u001b=\u0000\u001b. We write the\nGreen's function as Ga\nk\u001b= (i\"a+\u001bM\u0000~2k2=2m\u0000\u0006a\n\u001b)\u00001,\nwherea,bspecify retarded (R) or advanced (A), namely,\na= R for\"a>0, anda= A for\"a<0. Writing the\nself-energy as\n\u0006a\n\u001b= \u00001ga\n\u001b+ \u00002gb\n\u0016\u001b; (A3)\nwith \u0000 1=niu2\ni+nsu2\nsS2z, \u00002= 2nsu2\nsS2\n?[Eq. (14)], and\nga\n\u001b=P\nkGa\nk\u001b, we evaluate Yab\n\u001b\u0016\u001bas\nYab\n\u001b\u0016\u001b=gab\n\u001b\u0016\u001b\ni\"ba\u00002\u001bM+ \u0006ab\n\u001b\u0016\u001b; (A4)\nwhere\"ba=\"b\u0000\"a,gab\n\u001b\u001b0=ga\n\u001b\u0000gb\n\u001b0and \u0006ab\n\u001b\u001b0= \u0006a\n\u001b\u0000\u0006b\n\u001b0.\nThen, from Eq. (A1), we obtain\n(\u0003\u0006\n\u001b\u0016\u001b)ab=(\u001b\u0006)\u001b\u0016\u001b\n1\u0000\u00000Yab\n\u001b\u0016\u001b=i\"ba\u00002\u001bM+ \u0006ab\n\u001b\u0016\u001b\ni\"ba\u00002\u001bM+ \u0001ab\n\u001b\u0016\u001b(\u001b\u0006)\u001b\u0016\u001b;\n(A5)\nwhere \u0001ab\n\u001b\u001b0= \u0001a\n\u001b\u0000\u0001b\n\u001b0with\n\u0001a\n\u001b= (\u0000 1\u0000\u00000)ga\n\u001b+ \u00002ga\n\u0016\u001b\n= 2nsu2\ns(S2zga\n\u001b+S2\n?ga\n\u0016\u001b): (A6)\nExplicitly, \u0003+and \u0003\u0000are given by\n(\u0003+\n\"#)ab=i\"ba\u00002M+ \u0006ab\n\"#\ni\"ba\u00002M+ \u0001ab\n\"#; (A7)\n(\u0003\u0000\n#\")ab=i\"ba+ 2M+ \u0006ab\n#\"\ni\"ba+ 2M+ \u0001ab\n#\": (A8)\n(Other elements vanish, \u0003+\n#\"= \u0003\u0000\n\"#= 0, etc.) Therefore,\nGa\nk\"(\u0003+\n\"#)abGb\nk#=Ga\nk\"\u0000Gb\nk#\ni\"ba\u00002M+ \u0001ab\n\"#; (A9)\nGa\nk#(\u0003\u0000\n#\")abGb\nk\"=Ga\nk#\u0000Gb\nk\"\ni\"ba+ 2M+ \u0001ab\n#\": (A10)\nFor example,\nGR\nk\"(\u0003+\n\"#)RAGA\nk#=GR\nk\"\u0000GA\nk#\n\u00002M+ \u0001RA\n\"#\n'\u00001\n2M \n1 +\u0001RA\n\"#\n2M!\n\u0000\nGR\nk\"\u0000GA\nk#\u0001\n;\n(A11)\nGR\nk#(\u0003\u0000\n#\")RAGA\nk\"=GR\nk#\u0000GA\nk\"\n2M+ \u0001RA\n#\"\n'1\n2M \n1\u0000\u0001RA\n#\"\n2M!\n\u0000\nGR\nk#\u0000GA\nk\"\u0001\n:\n(A12)10\nIn the second lines, we assumed that \u0001ab's, which are on\nthe order of spin relaxation rate, are much smaller than\nthe exchange splitting M.\nAppendix B: Details of microscopic calculation\nIn this Appendix, we present the details of the calcu-\nlation of the magnon-drag electron current. It is divided\ninto the electron part and the magnon part.\n1. Electron part\nAs described in the text, the electron part, given\nby Eq. (29), contributes in two di\u000berent ways, E(2)\ni\u0011Ei(q;\u0017\u0000i\u0011;!+ 2i\u0011) andE(1)\ni\u0011 Ei(q;\u0017+i\u0011;!+i\u0011),\nwhere\u0011is a positive in\fnitesimal. For the magnon-drag\ncontribution, the former is calculated by setting != 0\nand retaining the \u0017-linear terms, and the latter by set-\nting\u0017= 0 and retaining the !-linear terms. They are\ngiven, respectively, by\nE(2)\ni=\u0017\n2\u0019q`(E0\ni`+E00\ni`); (B1)\nwith\nE0\ni`=\u0000iX\nkviv`h\nGR\n#n\n(\u0003\u0000\n#\")RR\u0000\nGR\n\"\u00012(\u0003+\n\"#)RA\u0000(\u0003\u0000\n#\")RA\u0000\nGA\n\"\u00012(\u0003+\n\"#)AAo\nGA\n#\n+GR\n\"n\n(\u0003+\n\"#)RR\u0000\nGR\n#\u00012(\u0003\u0000\n#\")RA\u0000(\u0003+\n#\")RA\u0000\nGA\n#\u00012(\u0003\u0000\n#\")AAo\nGA\n\"i\n; (B2)\nE00\ni`=\u0000X\nkviv`Imh\nGR\n#(\u0003\u0000\n#\")RR\u0000\nGR\n\"\u00012(\u0003+\n\"#)RRGR\n#+GR\n\"(\u0003+\n\"#)RR\u0000\nGR\n#\u00012(\u0003\u0000\n#\")RRGR\n\"i\n; (B3)\nand\nE(1)\ni'1\n2\u0019q`X\nkviv`h\n\u0000GR\n#(\u0003\u0000\n#\")RA\u0000\nGA\n\"\u00012(\u0003+\n\"#)AAGA\n#+GR\n\"(\u0003+\n\"#)RR\u0000\nGR\n#\u00012(\u0003\u0000\n#\")RAGA\n\"i\n; (B4)\nwhereGR(A)\n\u001b =GR(A)\nk\u001b(0).\nTo calculateE0\ni`, we use Eqs. (A9)-(A10) and the approximations as in Eqs. (A11)-(A12) valid for weak spin\nrelaxation (compared to M). With short notations, \u0001ab= \u0001ab\n\"#and ~\u0001ab= \u0001ab\n#\", we write\nE0\ni`'i\n(2M)2X\nkviv`\" \n1 +\u0001RA\u0000~\u0001RR\n2M!\n(GR\n#\u0000GR\n\")(GR\n\"\u0000GA\n#)\u0000 \n1 +\u0001AA\u0000~\u0001RA\n2M!\n(GR\n#\u0000GA\n\")(GA\n\"\u0000GA\n#)\n\u0000 \n1 +\u0001RA\u0000~\u0001AA\n2M!\n(GR\n\"\u0000GA\n#)(GA\n#\u0000GA\n\") + \n1 +\u0001RR\u0000~\u0001RA\n2M!\n(GR\n\"\u0000GR\n#)(GR\n#\u0000GA\n\")#\n'i\n(2M)2X\nkviv`\u0014\u00011\n2MGR\n\"GA\n\"\u0000\u00012\n2MGR\n#GA\n#\u00002iIm\b\n(GR\n\"\u0000GR\n#)2\t\u0015\n; (B5)\nwhere \u0001 1\u0011\u0001RA\u0000\u0001RR+~\u0001RA\u0000~\u0001AAand \u0001 2\u0011\u0001RA\u0000\n\u0001AA+~\u0001RA\u0000~\u0001RR, and we retained the leading terms\nwith respect to the electron damping. On the other hand,\nE00\ni`is calculated as\nE00\ni`'\u00002\n(2M)2X\nkviv`Im\u0002\n(GR\n\"\u0000GR\n#)2\u0003\n: (B6)Therefore, we have\nE0\ni`+E00\ni`=i\n(2M)2X\nkviv`\u0014\u00011\n2MGR\n\"GA\n\"\u0000\u00012\n2MGR\n#GA\n#\u0015\n=i\u000ei`\n(2M)22\u0019\ne2\u0014\u00011\n2M\u001b\"\u0000\u00012\n2M\u001b#\u0015\n: (B7)\nHere we noted\nX\nkviv`GR\n\u001bGA\n\u001b=\u000ei`v2\nF\u001b\n3\u0019\u0017\u001b\n\r\u001b=\u000ei`2\u0019\ne2\u001b\u001b; (B8)11\nwith\u001b\u001b=e2(v2\nF\u001b=3)\u001c\u001bbeing the conductivity of spin-\n\u001belectrons. From Eq. (A6), we have \u0001 1= 2\u0001RA\n##and\n\u00012= 2\u0001RA\n\"\"with \u0001RA\n\u001b\u001b=\u00004\u0019insu2\ns(S2z\u0017\u001b+S2\n?\u0017\u0016\u001b), and\nthus\n\u00011\n2M\u001b\"\u0000\u00012\n2M\u001b#=\u00002i\fel(\u001b\"\u0000\u001b#); (B9)\nwhere\felis given by Eq. (33). Using these relations in\nEq. (B1), we obtain\nE(2)\ni=\fel\u0017\n2M2qi\u001b\"\u0000\u001b#\ne2: (B10)\nThe!-linear terms inE(2)\ni[as given in Eq. (31)], which\ncontributes to the spin-motive forces, can be obtained in\na similar way.\nSimilarly, we obtain\nE(1)\ni'1\n2\u00191\n(2M)2q`X\nkviv`\u0002\nGR\n\"GA\n\"\u0000GR\n#GA\n#\u0003\n'1\n(2M)2qi\u001b\"\u0000\u001b#\ne2: (B11)\n2. Magnon part\nFor the magnon part, we encounter the following inte-\ngrals,\nI1=1\n2\u0019Z\nd\u0017\u0012\n\u0000@n\n@\u0017\u0013\n\u00172X\nquiqjDR\nq(\u0017)DA\nq(\u0017);(B12)\nI2=1\n\u0019Z\nd\u0017n(\u0017)\u0017X\nquiqjImh\u0000\nDR\nq(\u0017)\u00012i\n; (B13)\nI3=1\n\u0019Z\nd\u0017n(\u0017)X\nquiqjIm\u0002\nDR\nq\u0003\n; (B14)\nTo calculate I1, we useDR\nq(\u0017)DA\nq(\u0017)'(\u0019=\u000b\u0017 )\u000e(\u0017\u0000!q).\nThen,\nI1'1\n2\u000bX\nq!quiqj\u0012\n\u0000@n\n@\u0017\u0013\n\u0017=!q\n=1\n2\u000bT@\n@TX\nqn(!q)uiqj: (B15)\nBy noting ( @=@qi)kBTln\u0000\n1\u0000e\u0000~!q=kBT\u0001\n=n(!q)ui, we\nsee\nX\nqn(!q)uiqj=X\nqqj@\n@qikBTln\u0010\n1\u0000e\u0000~!q=kBT\u0011\n=\u0000\u000eij\nmag; (B16)\nwhere\n\nmag=kBTX\nqln\u0010\n1\u0000e\u0000~!q=kBT\u0011\n; (B17)is the thermodynamic potential of magnons. Therefore,\nI1=\u00001\n2\u000b\u000eijT@\n@T\nmag=1\n2\u000b\u000eijTSmag; (B18)\nwhereSmag=\u0000@\nmag=@T is the entropy (density) of\nmagnons.\nForI2, we useui(DR)2=@DR=@qiand ImDR\nq(\u0017)'\n\u0000\u0019\u000e(\u0017\u0000!q), and calculate as\nI2=\u0000\u000eij\n\u0019Z\nd\u0017n(\u0017)\u0017X\nqIm\u0002\nDR\nq\u0003\n'\u000eijX\nq!qn(!q)\n=\u000eijEmag: (B19)\nSimilarly,I3is calculated as\nI3'\u0000X\nquiqjn(!q) =\u000eij\nmag: (B20)\nAppendix C: Semi-classical analysis based on\nspin-motive force\nIn this Appendix, we calculate\nhFii=1\ns0\b\nImh_ay@iai\u0000\fReh_ay@iai\t\n; (C1)\nsemi-classically using the stochastic Landau-Lifshitz-\nGilbert (LLG) equation. This method has been used in\nthe calculation of magnonic spin torques [27, 28].\n1. Formulation\nThe stochastic LLG equation is given by\n_n=\u0000Jn\u0002@2\nin+n\u0002h\u0000\u000bn\u0002_n; (C2)\nwherenis the magnetization unit vector, and his\nthe Langevin noise \feld that satis\fes the \ructuation-\ndissipation theorem,\nhhi(r;t)hj(r0;t0)i= 2\u000bs0T\u000eij\u000e(r\u0000r0)\u000e(t\u0000t0);(C3)\nwhereTis the temperature. We consider the case that\nthe temperature is nonuniform and assume Tin Eq. (C3)\nis position-dependent, T=T(r), and calculatehFiithat\nis proportional to @iT.\nIn the complex notation, a= (s0=2)1=2(\u000enx+i\u000eny)\nandh=hx+ihy, Eq. (C2) becomes\ni_a= (\u0000J@2\ni+ \u0001)a+\u000b_a\u00001p2s0h(r;t); (C4)\nwhere \u0001 is the magnon energy gap, and hsatis\fes\nhh(r;t)h\u0003(r0;t0)i= 4\u000bs0T(r)\u000e(r\u0000r0)\u000e(t\u0000t0):(C5)12\nUsing the retarded Green's function DRthat satis\fes\n\u0002\ni@t+J@2\ni\u0000\u0001\u0000\u000b@t\u0003\nDR=\u000e(r\u0000r0)\u000e(t\u0000t0);(C6)\nEq. (C4) is solved as\na(r;t) =\u00001p2s0Z\ndt0Z\ndr0DR(r\u0000r0;t\u0000t0)h(r0;t0):\n(C7)\nIn the Fourier representation, DR\nq(\u0017) = (\u0017\u0000!q+i\u000b\u0017)\u00001,\nwhere!q=Jq2+ \u0001, it reads\naq(\u0017) =\u00001p2s0DR\nq(\u0017)h(q;\u0017); (C8)anda\u0003is given by the complex conjugate of Eq. (C7).\nFor a quantum system (in the present case, magnons),\nwe consider the Fourier transform of Eq. (C3) with re-\nspect to time, wherein the temperature is replaced as\nT!\u0017\n2coth\u0017\n2T=\u0017[n(\u0017) +1\n2] for the Fourier component\nof frequency \u0017. Its gradient is thus replaced as\n@iT!\u0017\u0012@n\n@T\u0013\n@iT: (C9)\n2. Calculation of hFii\nTo obtainhFii, it is su\u000ecient to calculate\n_ay@ia\u000b\n.\nWith Eq. (C8), this proceeds as follows,\n\n_ay(r;t)@ia(r;t)\u000b\n=\u001c\n@t\u0012\n\u00001p2s0Z\ndt1Z\ndr1DR(r\u0000r1;t\u0000t1)h(r1;t1)\u0013\u0003\n\u0002@i\u0012\n\u00001p2s0Z\ndt2Z\ndr2DR(r\u0000r2;t\u0000t2)h(r2;t2)\u0013\u001d\n=1\n2s0ZZ\ndt1dt2ZZ\ndr1dr2\u0000\n@tDR(r\u0000r1;t\u0000t1)\u0001\u0003\u0000\n@iDR(r\u0000r2;t\u0000t2)\u0001\nhh\u0003(r1;t1)h(r2;t2)i\n=4\u000bs0\n2s0Z\ndt1Z\ndr1\u0000\n@tDR(r\u0000r1;t\u0000t1)\u0001\u0003\u0000\n@iDR(r\u0000r1;t\u0000t1)\u0001\nT(r1)\n= 2\u000bZ\ndt1Z\ndr1X\nq;q0;q1ZZd\u0017d\u00170\n(2\u0019)2i\u00170iqiDA\nq0(\u00170)DR\nq(\u0017)Tq1ei(q0\u0000q+q1)\u0001r1e\u0000i(\u0017\u0000\u00170)(t\u0000t1)ei(q\u0000q0)\u0001r\n= 2\u000bX\nq;q1Zd\u0017\n2\u0019i\u0017\u0001i\u0010\nqi+q1;i\n2\u0011\nDR\nq+q1=2(\u0017)DA\nq\u0000q1=2(\u0017)Tq1eiq1\u0001r; (C10)\nwhereDA\nq(\u0017)\u0011\u0000\nDR\nq(\u0017)\u0001\u0003= (\u0017\u0000!q\u0000i\u000b\u0017)\u00001. We are interested in the term linear in q1, which, combined with Tq1,\ngives the temperature gradient. Thus,\n\n_ay(r;t)@ia(r;t)\u000b\n'\u000bX\nq;q1Zd\u0017\n2\u0019i\u0017DRDAiq1;iTq1eiq1\u0001r+ 2\u000bX\nq;q1Zd\u0017\n2\u0019i\u0017iqiuj2iImh\u0000\nDR\u00012DAiq1;j\n2Tq1eiq1\u0001r;(C11)\nwhereDR=DR\nq(\u0017) andDA=DA\nq(\u0017). With the replacement (C9), we obtain\n\n_ay(r;t)@ia(r;t)\u000b\n'\u000b(@iT)@\n@T(\niX\nqZd\u0017\n2\u0019\u00172n(\u0017)DRDA\u00002X\nqZd\u0017\n2\u0019qiuj\u00172n(\u0017) Imh\u0000\nDR\u00012DAi)\n: (C12)\nUsing the relations,\nX\nqZd\u0017\n2\u0019\u00172n(\u0017)DRDA'X\nqZd\u0017\n2\u0019\u00172n(\u0017)\u0001\u0019\n\u000b\u0017\u000e(\u0017\u0000!q)\n=1\n2\u000bEmag; (C13)\nand\nX\nqZd\u0017\n2\u0019\u00172n(\u0017)qiujImh\u0000\nDR\u00012DAi=X\nqZd\u0017\n2\u0019\u00172n(\u0017)qiujIm\u0014\nDR1\n2i\u000b\u0017\u0000\nDA\u0000DR\u0001\u0015\n'X\nqZd\u0017\n2\u0019\u00172n(\u0017)qiujIm\u00141\n2i\u000b\u0017DRDA\u0015\n'\u00001\n4\u000b2X\nqqiujn(!q)\n=1\n4\u000b2\nmag\u000eij; (C14)13\nwhere \n magis given by Eq. (B17), we obtain\n\n_ay@ia\u000b\n=1\n2\u0012@iT\nT\u0013\nT@\n@T\u0014\niEmag\u00001\n\u000b\nmag\u0015\n=1\n2\u0012@iT\nT\u0013\u0014\niT@\n@TEmag+1\n\u000bTSmag\u0015\n:(C15)\nFrom Eq. (C1), this leads to\nhFii=1\n2s0\u001a\n\u0000@Emag\n@T+\f\n\u000bSmag\u001b\n(\u0000@iT): (C16)\n3. Comparison with the previous study\nTo compare the phenomenological result (C16) ob-\ntained here with the one obtained previously [11], letus consider the case, T\u001d\u0001, where every quantity\nshows power-law dependence on temperature T. In\nthis case,T(@Emag=@T)'(1 +d=2)EmagandTSmag'\n(1 + 2=d)Emag, and Eq. (C16) becomes\nhFii=\u00001\n2s0\u0012\n1 +d\n2\u0013\u0012\n1\u00002\nd\f\n\u000b\u0013\nEmag\u0012\n\u0000@iT\nT\u0013\n:\n(C17)\nCompared with the result of Ref. [11], the coe\u000ecient of\n\f=\u000b is di\u000berent by a factor of 2.\n[1] D. C. Ralph and M. D. Stiles, J. Mag. Mag. Mat. 320,\n1190 (2008).\n[2] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[3] V. Korenman, J. L. Murray, and R. E. Prange, Phys.\nRev. B 16, 4032 (1977); G. E. Volovik, J. Phys. C 20,\nL83 (1987); A. Stern, Phys. Rev. Lett. 68, 1022 (1992); S.\nE. Barnes and S. 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Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n[14] R. A. Duine, A. S. N\u0013 u~ nez, J. Sinova and A. H. MacDon-\nald, Phys. Rev. B 75, 214420 (2007).\n[15] T. Yamaguchi, and H. Kohno, J. Phys. Soc. Jpn. 86,\n063706 (2017).\n[16] J. M. Luttinger, Phys. Rev. 135, A1505 (1964).\n[17] H. Kohno, Y. Hiraoka, M. Hatami, and G.E.W. Bauer,\nPhys. Rev. B 94, 104417 (2016).\n[18] R. A. Duine, Phys. Rev. B 77, 014409 (2008).[19] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n[20] It is possible to argue this di\u000berently if we note\n(\u001b\"\u0000\u001b#)\fel= (\u001b\"\u0000\u001b#)\fs+ (\u001b\"+\u001b#)\fc;\nwhere\n\fs=\u0019nsu2\ns\nM(S2\n?+S2z)\u0017+;\n\fc=\u0019nsu2\ns\nM(S2\n?\u0000S2z)\u0017\u0000:\nThus Eq. (39) consists of a term proportional to \u001b\"\u0000\u001b#\nand a term proportional to \u001b\"+\u001b#. This suggests the\nspin-dependent force that magnons exert on electrons,\nF\u001b=\u001bFs+Fc, contains a spin-independent part Fcas\nwell, where\nFs=\u00001\n2s0\u0012\n1\u0000\fs\n\u000b\u0013\nr\nmag;\nFc=\u00001\n2s0\u0012\n\u0000\fc\n\u000b\u0013\nr\nmag:\nSimilarly, the spin-dependent nonequilibrium chemical\npotential,\u000e\u0016\u001b=\u001b\u000e\u0016 s+\u000e\u0016c, contains a spin-independent\npart\u000e\u0016c, where\n\u000e\u0016s=1\n2s0\u0012\n1\u0000\fs\n\u000b\u0013\n\u000e\nmag;\n\u000e\u0016c=1\n2s0\u0012\n\u0000\fc\n\u000b\u0013\n\u000e\nmag:\nThe presence of the charge current (the term proportional\nto\fc) seems interesting from the viewpoint of the mo-\nmentum transfer e\u000bect. However, the sign of \fchere is\nnot de\fnite and it seems di\u000ecult to give a clear physical\nmeaning. It vanishes when the magnetic impurities are\nisotropic,S2\n?=S2z, anyway. Hence we will not pursue\nthis aspect in this paper.\n[21] For example, see: J. Shibata and H. Kohno, Phys. Rev.\nB84, 184408 (2011).14\n[22] Y. Imai and H. Kohno, J. Phys. Soc. Jpn. 87, 073709\n(2018).\n[23] N. W. Aschcroft and N. D. Mermin: Solid State Physics\n(Saunders College, Philadelphia, 1976).\n[24] H. Kohno et al. , inProceedings of ISQM-Tokyo '08 , pp.\n111-117, Eds. S. Ishioka and K. Fujikawa (World Scien-\nti\fc, 2009) (arXiv:0912.1676).[25] H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710\n(2007).\n[26] If a comparison is made with Eq. (54), we see that \u000eT=\n0, namely, the electron temperature is not changed by\nthe coupling to nonequilibrium magnons.\n[27] A. A. Kovalev, Phys. Rev. B 89, 241101(R) (2014).\n[28] S. K. Kim and Y. Tserkovnyak, Phys. Rev. B 92,\n020410(R) (2015)."    },    {        "title": "1812.07244v2.Thermal_gradient_driven_domain_wall_dynamics.pdf",        "content": "arXiv:1812.07244v2  [cond-mat.mes-hall]  26 May 2019Thermal gradient driven domain wall dynamics\nM. T. Islam,1,2X. S. Wang,3,4and X. R. Wang1,5,∗\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2Physics Discipline, Khulna University, Khulna, Banglades h\n3School of Electronic Science and Engineering and State Key L aboratory of Electronic Thin Film and Integrated Devices,\nUniversity of Electronic Science and Technology of China, C hengdu 610054, China\n4Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Tr ondheim, Norway\n5HKUST Shenzhen Research Institute, Shenzhen 518057, China\nThe issue of whether a thermal gradient acts like a magnetic fi eld or an electric current in the\ndomain wall (DW) dynamics is investigated. Broadly speakin g, magnetization control knobs can\nbe classified as energy-driving or angular-momentum drivin g forces. DW propagation driven by a\nstatic magnetic field is the best known example of the former i n which the DW speed is proportional\nto the energy dissipation rate, and the current-driven DW mo tion is an example of the latter. Here\nwe show that DW propagation speed driven by a thermal gradien t can be fully explained as the\nangular momentum transfer between thermally generated spi n current and DW. We found DW-\nplane rotation speed increases as DW width decreases. Both D W propagation speed along the wire\nand DW-plane rotation speed around the wire decrease with th e Gilbert damping. These facts\nare consistent with the angular momentum transfer mechanis m, but are distinct from the energy\ndissipation mechanism. We further show that magnonic spin- transfer torque (STT) generated by a\nthermal gradient has both damping-like and field-like compo nents. By analyzing DW propagation\nspeed and DW-plane rotational speed, the coefficient ( β) of the field-like STT arising from the non-\nadiabatic process, is obtained. It is found that βdoes not depend on the thermal gradient; increases\nwith uniaxial anisotropy K/bardbl(thinner DW); and decreases with the damping, in agreement w ith the\nphysical picture that a larger damping or a thicker DW leads t o a better alignment between the\nspin-current polarization and the local magnetization, or a better adiabaticity.\nI. INTRODUCTION\nManipulating domain walls (DW) in magnetic nanos-\ntructures has attracted much attention because of its po-\ntential applications in data storage technology [ 1] and\nlogic gates [ 2]. The traditional DW control knobs,\nnamely magnetic fields and spin-polarized currents, have\ncertain drawbacks in applications. In the magnetic-field-\ndriven DW motion, energy dissipation is the main cause\nofDWpropagationwhosespeedisproportionaltotheen-\nergy dissipation rate [ 3,4], and the magnetic field tends\ntodestroyunfavorabledomainsandDWs, insteadofdriv-\ning a series of DWs synchronously [ 5–7]. An electrical\ncurrent drives a DW to move mainly through the angu-\nlar momentum transfer so that it pushes multiple DWs\n[8–11] in the same direction. To achieve a useful DW\nspeed, it requires high electrical current densities that\nresult in a Joule heating problem [ 12–14]. To avoid these\nproblems, spin-wave spin current has been proposed as a\nmoreenergy-efficientcontrolparameter[ 15–18]. Thermal\ngradient, a way to generate spin-wave spin current, is an\nalternative control knob of the DW motion. The inves-\ntigation on thermal-gradient-driven domain wall motion\nis meaningful not only for conventional applications, but\nalso for the understanding of spin wave and domain wall\ndynamics [ 16,17,20–23], as well as for possible recycling\nof waste heat [ 19,24].\n∗[Corresponding author:]phxwan@ust.hkTo understand the mechanism behind thermal-\ngradient-drivenDWdynamics, therearemicroscopicthe-\nories [15–17,25,26] and macroscopic thermodynamic\ntheories [ 21,22]. Briefly speaking, the microscopic theo-\nries suggest that magnons populated in the hotter region\ndiffuses to the colder region to form a magnon spin cur-\nrent. The magnon spin currentpassesthrough a DWand\nexerts a torque on the DW by transferring spin angular\nmomentum to the DW. Thus, magnons drive the DW\npropagating toward the hotter region of the nanowire,\nopposite to the magnon current direction [ 15,16,18].\nThe thermodynamic theories anticipate that a thermal\ngradient generates an entropy force which always drives\nthe DW towards the hotter region in order to minimize\nthe system free energy. The macroscopic theories do not\nprovide any microscopic picture about DW dynamics al-\nthough a thermal gradient is often considered as an effec-\ntive magnetic field to estimate DW speed [ 21,22] from\nfield-driven DW theories. Thus, one interesting issue is\nwhether a thermal gradient in DW dynamics acts like a\nmagnetic field or an electric current. DW propagation\nspeed should be sensitive to both DW width and types\nof a DW (transverse DW) under an energy-driving force\nwhile the speed should be insensitive to the DW and DW\nstructure in the angular-momentum-driving force. This\nis the focus of the current work.\nIn this paper, we investigate DW motion along a uni-\naxial wire with the easy axis along the wire direction\nunder a thermal gradient. We found that the DW al-\nways propagates to the hotter region with an accom-\npanied DW-plane rotation. DW propagation speed and2\nz\nxy\nFIG. 1. Schematic diagram of a uniaxial magnetic nanowire\nwith a head-to-head DW at the center under a thermal gra-\ndient∇T. Black (white) color represents colder (hotter) end\nof the sample.\nDW-plane rotation speed increases as the magnetic easy-\naxis anisotropy and damping decreases. We show that\nDW motion can be attributed to the angular momen-\ntum transfer between magnonic spin current and the\nDW. Thus, we conclude that a thermal gradient in-\nteracts with DW through angular-momentum transfer\nrather than through energy dissipation. Similar to an\nelectric current [ 27], a thermal gradient can generate\nboth damping-like (or adiabatic) STT and field-like (or\nnon-adiabatic) STT. From the damping-dependence and\nanisotropy-dependence of the average DW velocity and\nDW-plane rotation angular velocity, we extract field-like\nSTT coefficient ( β). It is found that βis independent\nof thermal gradient; is bigger for a thinner DW; and de-\ncreases with the damping coefficient. We also show that\nin the presence of a weak hard-axis anisotropy perpen-\ndicular to the wire, the DW still undergoes a rotating\nmotion. The DW propagation speed increases slightly\nwhile the DW-plane rotation speed decreases with the\nstrength of the hard-axis anisotropy.\nII. MODEL AND METHOD\nWe consider a uniaxial nanowire of length Lxand\ncross-section Ly×Lzalong the x-axis (easy axis) with\na head-to-head DW at the center, as shown in Fig. 1.\nLy,Lzis much smaller than the DW width ∆, and ∆\nis much smaller than Lx. A thermal gradient is applied\nalong the wire. The highest temperature is far below\nthe Curie temperature Tc. The magnetization dynam-\nics is governed by the stochastic Landau-Lifshitz-Gilbert\n(LLG) equation [ 28,29],\ndm\ndt=−γm×(Heff+hth)+αm×∂m\n∂t,(1)\nwherem=M/MsandMsare respectively the magne-\ntization direction and the saturation magnetization. α\nis the Gilbert damping constant and γis the gyromag-netic ratio. Heff=2A\nµ0Ms/summationtext\nσ∂2m\n∂x2σ+2K/bardbl\nµ0Msmxˆx+hdipoleis\nthe effective field, where Ais the exchange constant, xσ\n(σ= 1,2,3) denote Cartesian coordinates x,y,z,K/bardblis\nthe easy-axis anisotropy, and hdipoleis the dipolar field.\nhthis the stochastic thermal field.\nThe stochastic LLG equation is solved numerically by\nMUMAX3 package [ 30] in which we use adaptive Heun\nsolver. To balance stability and efficiency, we choose the\ntime step 10−14s with the cell size (2 ×2×2) nm3. Mag-\nnetic charges at the two ends of the wire are removed to\navoid their attraction to the DW. The saturation mag-\nnetization Ms= 8×105A/m and exchange constant\nA= 13×10−12J/m are used to mimic permalloy in\nour simulations. The thermal field follows the Gaussian\nprocess characterized by following statistics [ 31]\n/angb∇acketlefthth,ip(t)/angb∇acket∇ight= 0,\n/angb∇acketlefthth,ip(t)hth,jq(t+∆t)/angb∇acket∇ight=2kBTiαi\nγµ0Msa3δijδpqδ(∆t),(2)\nwhereiandjdenote the micromagnetic cells, and p,q\nrepresent the Cartesian components of the thermal field.\nTiandαiare respectively temperature and the Gilbert\ndamping at cell i, andais the cell size. kBis the Boltz-\nmann constant [ 28]. The numerical results presented in\nthis study are averaged over 15 random configurations\n(for DW velocity) and 4000-5000 random configurations\n(for spin current).\nUnderthethermalgradient ∇xT,magnetizationatdif-\nferent positions deviate from their equilibrium directions\ndifferently and small transverse components myandmz\nare generated. The transverse components vary spatial-\ntemporally and depend on the local temperature. This\nvariation generates a magnonic spin current [ 16]. This\nmagnonic spin current can interact with spin textures\nsuch as DWs. In the absence of damping (the thermal\nfield also vanishes), the spin currentalong the xdirection\ncan be defined from the spin continuity equation derived\nfrom Eq. ( 1) as follows [ 15],\n∂m\n∂t=−1\n1+α2m׈xmxK/bardbl−∂J\n∂x,(3)\nwhere\nJ(x) =2γA\nµ0Msm×∂m\n∂x, (4)\nis the spin current density along x-direction due to the\nexchangeinteraction. J(x) can be numerically calculated\n[15,23]. In the presence of damping as well as the ther-\nmal field, the contribution of the damping term and the\nthermal term is proportional to α, which is relatively\nsmall. More importantly, according to the fluctuation-\ndissipation theorem [ 28], the damping term and the ther-\nmal term should cancel each other after average over a\nlong time. Since the time scale of DW dynamics is much\nlonger than the thermal fluctuation, the combined con-\ntribution of damping and thermal terms should be very\nsmall.3\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s52/s56/s49/s50/s49/s54/s50/s48\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s50/s52/s54/s56/s49/s48/s49/s50/s45/s56/s48/s48 /s45/s52/s48/s48 /s48 /s52/s48/s48 /s56/s48/s48/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50\n/s32/s32\n/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41/s32/s32/s32\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41\n/s32/s32\n/s48/s46/s48/s55/s32 /s75/s47/s110/s109\n/s48/s46/s49 /s75/s47/s110/s109\n/s48/s46/s49/s53/s32 /s75/s47/s110/s109\n/s48/s46/s50/s32 /s75/s47/s110/s109\n/s48/s46/s50/s53/s32 /s75/s47/s110/s109\n/s48/s46/s51/s32 /s75/s47/s110/s109/s74\n/s116/s111/s116/s40/s120/s41/s40\n/s115/s41/s41\n/s120 /s32/s40/s110/s109/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s40/s100/s41/s40/s98/s41/s40/s97/s41\n/s32\n/s75 /s32/s40/s49/s48/s52\n/s32/s74/s47/s109/s51\n/s41/s118 /s32/s40/s109/s47/s115/s41/s40/s99/s41/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41\n/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41\n/s51/s54/s51/s57/s52/s50\n/s32\nFIG. 2. (a) The spatial dependence of spin current densities\nJtot(x) for various ∇xT. The DW center is chosen as x=\n0. (b) Thermal gradient dependence of DW velocity vsimu\nfrom micromagnetic simulations (open squares) and vcurrent\ncomputedfrom total spin current(solid squares). (c)Therm al\ngradient dependence of DW-plane rotation angular velocity\n(squares). In (a)(b)(c) model parameters are Lx= 2048 nm,\nLy=Lz= 4 nm, α= 0.004 and K/bardbl= 5×105J/m3. (d)\nvsimu(solid squares) and dφ/dt(open squares) as a function\nofK⊥forLx= 1024 nm and ∇xT= 0.5 K/nm.\nIntegrating the x−component of Eq. ( 3) over a space\nenclosed the DW in the center and noticing the absence\nof the first term on the right, we have\nvcurrent=1\n2/integraldisplayLx/2\n−Lx/2∂mx\n∂tdx\n=−2γA\nµ0Ms/bracketleftbig1\n2(Jx|left−Jx|right)].(5)\nwhere we have assumed the fluctuations in the domains\nare small and the DW is not far from a symmetric one.\nJx|left,Jx|rightmean the x-components of the total spin\ncurrent on the left and right sides of the DW. The equa-\ntion clearly shows that the DW propagates opposite to\nthe spin current. This is the theoretical DW velocity un-\nder the assumption of angular momentum conservation,\nand it will be compared with the directly simulated DW\nvelocity below.\nIII. RESULTS\nA. Average spin current and DW velocity\nTosubstantiateourassertionthatDWpropagationun-\nder a thermal gradient is through angular-momentum ef-\nfect instead of energy effect, we would like to compare\nthe DW velocity obtained from micromagnetic simula-\ntions and that obtained from total spin current based on/s49 /s50 /s51 /s52 /s53 /s54 /s55/s56/s49/s50/s49/s54/s50/s48\n/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116\n/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41/s32/s118 /s32/s40/s109/s47/s115/s41\n/s32/s76\n/s120/s61/s50/s48/s52/s56/s32/s110/s109\nFIG. 3. Damping αdependence of the DW dynamics: vsimu\n(Open squares); vcurrent(solid squares ); and dφ/dt(solid\ncircles). Model parameters are ∇xT= 0.2 K/nm, K/bardbl= 5×\n105J/m3,Lx= 2048 nm and Ly=Lz= 4 nm.\nEq. (5). Eq. (4) is used to calculate Jx(x). Fig.2(a) is\nspatial distribution of the ensemble averaged Jx(x) with\nDW atx= 0 for various thermal gradients. The sud-\nden sign change of Jx(x) at the DW center is a clear\nevidence of strong angular-momentum transfer from spin\ncurrent to the DW. Technically, magnetizationof the two\ndomains separated by the DW point to the opposite di-\nrections, thus the spin current polarization changes its\nsign. In calculating DW velocity vcurrentfrom Eq. ( 5),\nthe spin currents before entering DW and after passing\nDW are the averages of Jx(x) overx∈[−2∆,−∆] and\nx∈[∆,2∆], where ∆ is the DW width which is 16 nm\nin the current case. The thermal gradient dependence\nofvcurrentis shown in Fig. 2(b) (solid squares). vcurrent\ncompares well with the velocity vsimu(open squares) ob-\ntained directly from simulations by extracting the speed\nof the DW center along x-direction. The DW veloc-\nity is linearly proportional to the temperature gradient\nv=C∇xT, with the thermal mobility C= 6.66×10−8\nm2s−1K−1forvsimuorC= 6.59×10−8m2s−1K−1for\nvcurrent. It is noted that vcurrentalmost coincides with\nvsimuexcept a small discrepancy at very high thermal\ngradient when the nonlinear effects is strong. The small\ndiscrepancy may be attributed to the large fluctuations\nas well as the contribution from the damping, the dipo-\nlar and stochastic fields. These observations are consis-\ntent with magnonic STT [ 15,16,25,26]. It is observed\nthat the DW-plane rotates around the x-axis counter-\nclockwise for head-to-head DW and clockwise for tail-to-\ntail DW during DW propagation. DW rotation speed\ndφ/dt(squares) is shown in Fig. 2(c)) as a function of\n∇xT.4\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54\n/s52/s54/s56/s49/s48/s49/s50/s49/s52/s32/s32\n/s75\n/s124/s124/s32/s40/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41\n/s100 /s47/s100/s116/s40/s100/s101/s103/s47/s110/s115/s41/s32\nFIG. 4. Anisotropy K/bardbldependence of the DW dynamics:\nvsimu(open squares); vcurrent(solid squares); and dφ/dt(solid\ncircles). Model parameters are Lx= 2048 nm, Ly=Lz=4 nm,\nα= 0.004 and ∇xT= 0.2 K/nm.\nB. Damping and anisotropy dependence of DW\ndynamics\nAn energy-effect and angular-momentum-effect\nhave different damping-dependence and anisotropy-\ndependence of DW dynamics. To distinguish the roles of\nenergy and the angular-momentum in thermal-gradient\ndriven DW dynamics, it would be useful to probe how\nthe DW dynamics depends on αandK/bardbl. Damping have\ntwo effects on the spin currents: one is the decay of\nspin current during its propagation so that the amount\nof spin angular momentum deposited on a DW should\ndecrease with the increase of the damping coefficient.\nAs a result, the DW propagation speed and DW-plane\nrotation speed should also be smaller for a larger α.\nIndeed, this is what we observed in our simulations\nas shown in Fig. 3(a) for DW speed and DW-plane\nrotation speed (open squares for vsimu, solid circles for\nvcurrent, and stars for dφ/dt). The model parameters are\nLx= 2048, Ly=Lz= 4 nm, ∇xT= 0.2 K/nm and\nK/bardbl= 5×105J/m3. The second damping effect is that\nthe larger αhelps the spin current polarization to align\nwith the local spin. This second effect enhances the\nadiabatic process that is important for non-adiabatic\nSTT or field-like torque discussed in the next subsection.\nTherefore, α−dependence of DW dynamics supports\nthe origin of thermal driven DW dynamics to be the\nangular-momentum effect, not the energy effect that\nwould lead to a larger vsimuanddφ/dtfor a larger α\n[3,4,33–35] instead of a decrease observed here.\nHere we would like to see how the DW dynamics de-\npendsonuniaxialanisotropy K/bardbl. Fig.4showsboth vsimu\n(open squares), vcurrent(filled squares) and dφ/dt(cir-\ncles) for Lx= 2048 nm, α= 0.004 and ∇xT= 0.2. The\nDW propagation speed, vsimudecreases with K/bardblwhileDW-plane rotational speed increases with K/bardbl. These re-\nsults seem follow partially the behavior of magnetic-field\ninduced DW motion, in which DW propagation speed\nis proportional to DW width (∆ ∼/radicalBigg\nA\nK/bardbl) or decrease\nwithK/bardbl, and partially electric current driven DW mo-\ntion, in which DW-plane rotational speed increases with\nK/bardbl. Thus, one may tend to conclude that a thermal gra-\ndient behaves more like a magnetic field rather than an\nelectric current from the DW width dependence of DW\npropagation speed, opposite to our claim of the angular-\nmomentum effects of the thermal gradient. It turns out,\nthis is not true. The reason is that magnon spectrum,\nωk=2γ\nµ0Ms/parenleftbig\nAk2+K/bardbl/parenrightbig\n, has a gap in a system with mag-\nnetic anisotropy. The larger K/bardblis, the bigger the energy\ngap will be. Thus, it becomes harder to thermally excite\nmagnon. As a result, the spin current decreasesas K/bardblin-\ncreases. To see whether the thermal-gradient driven DW\nmotion is due to the angular-momentum transfer or not,\none should compare whether vsimuandvcurrentmaintain\na good agreement with each other as K/bardblvaries. Indeed,\na good agreement between vsimuandvcurrentis shown in\nFig.4. This conclusion is also consistent with existing\nmagnonic STT theories [ 33–35].\nC. Separation of adiabatic and non-adiabatic\ntorques\nWe have already demonstrated that a thermal gradi-\nent interacts with DW through magnonic STT rather\nthan through energy dissipation. It is then interesting to\nknow what kind of STTs a thermal gradient can gen-\nerate. Specifically, whether a magnonic spin current\ngenerates damping-like (adiabatic), or field-like (Non-\nadiabatic) torques, or both just like an electric current\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s73/s32/s40/s49/s48/s49/s48\n/s32/s65/s47/s109/s50\n/s41\n/s32\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41\n/s32/s32\nFIG. 5. Model parameters are K/bardbl= 5×105J/m3,α=\n0.004,Lx= 1024 and Ly=Lz=4 nm. Effective electric current\ndensityI(open squares) and β(solid squares) are plotted as\nfunctions of ∇xT.5\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s46/s49/s52/s48/s46/s50/s49/s48/s46/s50/s56\n/s40/s49/s48/s45/s51\n/s41\n/s32/s32\n/s75\n/s124/s124/s32/s40/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41/s40/s97/s41\n/s40/s98/s41/s32/s32\nFIG. 6. Model parameters are ∇xT=0.5 K/nm, Lx= 1024\nnm and Ly=Lz=4 nm. (a) α-dependence of βforK/bardbl= 106\nand J/m3. (b)K/bardbl-dependence of βforα= 0.004.\n[27] does. To extract the STT generated from a thermal\ngradient, we approximate DW dynamics by the motion\nof its collective modes of DW center Xand the titled\nangleφof DW-plane. Subject to both damping-like and\nfield-like torques, using the travelling-wave ansatz [ 33–\n35], tan(θ/2) = exp[( x−X)/∆] where ∆ ∼/radicalbig\nA/K/bardbl, one\ncan derive the equations for X and φ,\nα\n∆dX\ndt+dφ\ndt=β\nαu,1\n∆dX\ndt−αdφ\ndt=u\nα.(6)\nFrom the above two equations, one can straightfor-\nwardly find DW propagating speed and DW-plane ro-\ntation speed,\nv=(1+αβ)\n(1+α2)u,˙φ=(β−α)\n(1+α2)u. (7)\nOne can extract βand equivalent electric current den-\nsityI= (2eMsu)/gµBPfromvanddφ/dtobtained in\nsimulations. For α= 0.004,K/bardbl= 106J/m3, theIandβ\nare obtained and plotted in Fig. 5as a function of ∇xT.\nIt is evident that Ilinearly increases with ∇xTandβ\nis independent of ∇xTas it should be. We then fixed\n∇xT= 0.5 K/nm, and repeat simulations and analysis\nmentioned above for various αandK/bardbl. Fig.6(a) and\n(b) shows βas a function of αandK/bardbl. From the figure,\nit is evident that βdecreases with α. This is because\nthe larger damping favors the alignment of spin current\npolarization with the local spin so that the non-adiabatic\neffect,β, becomes smaller. βincreases with K/bardblfor the\nsimilar reason: Larger K/bardblmeans a thinner DW so that\nit is much harder for the spin current polarization to re-\nverse its direction after passing through the thinner DW,\ni.e. a stronger non-adiabatic effect.\nIn some experiments, the temperature gradient is gen-\nerated by a laser spot[ 36]. The laser spot will induce a\nGaussian distribution of the temperature over the space\n[36,37]. In Fig. 7, weshowtheDWmotionin aGaussian\ntemperature profile T(x) =T0exp/parenleftBig\n−(x−xL)2\n2σ2/parenrightBig\nby plot-\nting the DW position against the time. Here we use the/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s32/s32/s68/s87/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s40/s110/s109/s41\n/s116/s105/s109/s101/s32/s40/s110/s115/s41\nFIG. 7. Domain wall position versus time in a Gaussian tem-\nperature profile. The gray lines are raw data for different\nrandom seeds and the red line is the averaged result. The\ngreen dashed line is theoretical result using the thermal mo -\nbilityC= 6.66×10−8m2s−1K−1obtained from Fig. 2(b).\nsame parameters as those in Fig. 2(b), except a longer\nwireLx= 2048 nm, and T0= 400 K, σ= 200 nm,\nxL= 200 nm. Theoretically, if the instantaneous DW\nspeed under a Gaussian temperature is the same as that\nin the constant thermal-gradient case, we should expect\ndx\ndt=CdT\ndx, where the thermal mobility Cis the same as\nthat in Fig. 2(b). Using C= 6.66×10−8m2s−1K−1,\nthe above differential equation for x(t) can be numeri-\ncally solved with initial condition x(0) = 0. The result\nis plotted in Fig. 7in green dashed line. The simu-\nlated speed is smaller than this theoretical result. This\nis probably because, for the constant thermal-gradient,\nwe focus on the steady-state DW motion speed. In a\nGaussian temperature, the DW cannot immediately fol-\nlow the local temperature gradient. Before the DW can\nreach the steady-state speed corresponding to the local\ntemperature, it already moves to a position of smaller\ntemperature gradient. More details about DW motion in\nGaussian temperature profile may be an issue of future\nstudies.\nIV. DISCUSSION AND SUMMARY\nWe have studied the thermal gradient-driven DW dy-\nnamicsinanuniaxialnanowire. Inreality, thereisalways\ncertain hard anisotropy in a wire whose cross-section is\nnot a perfect ellipse. Thus, it is interesting to see how\nthe above results will change in a weak biaxial nanowire\nwith a small hard anisotropy K⊥= 1/2µ0M2\ns(Nz−Ny),\nsay along y-direction. Our simulations show that a DW\nstill propagatestowardsthe higher temperature region in\na similar way as that in a uniaxial wire. Interestingly, as\nshown in Fig. 2(d) for the K⊥-dependence of vsimu(solid\nsquares) and dφ/dt(open squares), DW speed increases6\nslightly with K⊥. This may be due to the increase of\ntorque along θ-direction [ 33] since Γ θis proportional to\n(Nz−Ny). This is also consistent with the early results\nforthe uniaxialwire that vsimu(which includes stochastic\nthermal field and demagnetisation fields) is always larger\nthanvcurrent(where the transverse fields are neglected).\nAt the meanwhile, dφ/dtdecreases with K⊥.\nThe main purpose of this paper is to study the\nmagnonic effects in thermal-gradient-driven domain wall\ndynamics. We consider the spin waves explicitly and\nall the material parameters (exchange constant A, crys-\ntalline anisotropy K, saturation magnetization Ms, and\nGilbert damping α) are assumed to be constant. Indeed,\nthe atomistic magnetic moments are independent of tem-\nperature. At the atomistic level, the exchange constant\nAoriginating from the Pauli exclusion principle and the\ncrystalline anisotropy Koriginating from the spin-orbit\ncoupling onlyweaklydepend on the temperature because\nof the vibration of atoms [ 39]. In micromagnetic models,\nbecause finite volumes that contains many magnetic mo-\nments are considered as unit cells, the parameters A,K,\nandMsdepend on the temperature. This is because the\nthermally excited spin waves with wavelengths shorter\nthan the length scale of the unit cells are included in\nthe effective A,K, andMsby doing an average [ 16,38].\nSince we use small mesh size 2 ×2×2 nm3, only spin\nwaves of very short wavelength affect the parameters A,\nK, andMsin our model. Those short-wavelength spin\nwavespossess high energyaswell as low density ofstates,\nso their contributions to the effective A,K, andMsare\nnot significant. The Gilbert damping αdepends on the\ntemperature non-monotonically [ 40–43]. The underlying\nmechanism is still under debate, but for many cases the\ndependence is not significant in a wide range of temper-\nature.\nIn summary, our results show that the uniform ther-\nmal gradient always drives a DW propagating towards\nthe hotter region and the DW-plane rotates around the\neasy axis. The DW velocity and DW-plane rotational\nspeed decrease with the damping coefficient. The DW\nvelocity obtained from simulation agrees with the veloc-\nity obtained from angular momentum conservation when\nthe magnon current density ( J(x)) from the simulation is\nusedtoestimatetheamountofangularmomentumtrans-\nferred from magnon current to the DW. All the above\nfindings lead to the conclusion that the thermal gradient\ninteracts with DW through angular-momentum transfer\nrather than through energy dissipation. Furthermore,\nwe demonstrated that the magnonic STT generated by\na thermal gradient has both damping-like and field-like\ncomponents. The field-like STT coefficient βis deter-\nmined from DW speed and DW-plane rotation speed. β\ndoes not depend on the thermal gradient as expected,\nbut increases with a decrease of DW width. This behav-\nior can be understood from the expected strongmisalign-\nment of magnon spin polarization and the local spin so\nthat non-adiabatic torque (also called field-like torque)\nis larger. For the same reason, a larger Gilbert dampingresults in a better alignment between spin current polar-\nization and the local spin, thus βshould decrease with\nα. The thermal gradientcan be a veryinteresting control\nknob for nano spintronics devices, especially those made\nfrom magnetic insulators.\nThis work was supported by the National Natural Sci-\nence Foundation of China (Grant No. 11774296) as well\nas Hong Kong RGC Grants Nos. 16300117, 16301518\nand 16301816. X.S.W acknowledges support from NSFC\n(GrantNo. 11804045),ChinaPostdoctoralScienceFoun-\ndation (Grant No. 2017M612932and 2018T110957),and\nthe Research Council of Norway through its Centres of\nExcellence funding scheme, Project No. 262633, “QuS-\npin.” M. T. I acknowledges the Hong Kong PhD fellow-\nship.7\n[1] Parkin S S P, Hayashi M and Thomas L 2008 Science\n320 190\n[2] Allwood D A, Xiong G, Faulkner C C, Atkinson D, Petit\nD and Cowburn R P 2005 Science309 1688\n[3] Wang X R, P Yan, Lu J and He C 2009 Ann. Phys. (N.\nY.)324 1815\n[4] Wang X R, Yan P and Lu J 2009 Europhys. Lett. 86\n67001\n[5] Atkinson D, Allwood D A, Xiong G, Cooke M D,\nFaulkner C C, and Cowburn R P 2003 Nat. Mater. 2\n85\n[6] Beach G S D, Nistor C, Knutson C, Tsoi M, and Erskine\nJ L 2005 Nat. Mater. 4 741\n[7] Hayashi M, Thomas L, Bazaliy Ya B , Rettner C, Moriya\nR, Jiang X, and Parkin S S P 2006 Phys. Rev. Lett. 96\n197207\n[8] Berger L 1996 Phys. Rev. B 54 9353\n[9] Slonczewski J 1996 J. Magn. Magn. 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B84 014412\n[43] Maier-Flaig H, Klingler S , Dubs C, Surzhenko O, Gross\nR, Weiler M, Huebl H, and Goennenwein S T B 2017\nPhys. Rev. B 95 214423"    },    {        "title": "1812.08404v1.Laser_Controlled_Spin_Dynamics_of_Ferromagnetic_Thin_Film_from_Femtosecond_to_Nanosecond_Timescale.pdf",        "content": "1\n \n \nLaser Controlled Spin Dynamics \nof Ferromagnetic Thin Film \nfrom \nFemtosecond to Nanosecond Timescale\n \nSucheta Mondal and Anjan Barman\n*\n \nDepartment of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for \nBasic Sciences, Block JD, Secto\nr III, Salt Lake, Kolkata 700 106, India.\n \n*\nabarman@bose.res.in\n \nKey words: (\nThin Film Heterostructures, Ultraf\na\nst Demagnetization, Gilbert Damping, Time\n-\nresolved\n \nMagneto\n-\noptical Kerr Effect\n)\n \n \n \nLaser induced modulatio\nn of the \nmagnetization dynamics \noccurring over various time\n-\nscales \nhave been unified\n \nhere \nfor\n \na \nNi\n80\nFe\n20\n \nthin \nfilm excited \nby\n \namplified \nfemtosecond laser pulses. \nThe weak correlation between demagnetization time and pump fluence with substantial \nenhancemen\nt in remagnetization time is demo\nn\nstrated using three\n-\ntemperature model \nconsidering the temperatures of electron, spin\n \nand \nlattice.\n \nThe \npicosecond\n \nmagnetization \ndynamics is modeled using \nthe L\nandau\n-\nLifshitz\n-\nGilbert equation. \nW\nith increasing pump fluence \nth\ne Gilbert damping parameter shows significant enhancement\n \nfrom its intrinsic value \ndue to\n \nincrement in the \nratio\n \nof electronic temperature to Curie temperature within very short time \nscale. The precessional frequency experiences \nnoticeable \nred shift with i\nncreasing pump fluence. \nThe changes in the local magnetic properties due to accumulation and dissipation of thermal \nenergy within the probed volume are described by the \nevolution of \ntemporal chirp parameter in a \ncomprehensive manner.\n \nA unification of ultra\nfast magnetic processes and its control over broad \ntimescale would enable the integration of various magnetic processes in a single device and use \none effect to control another.\n \n \n \n \n \n \n 2\n \n \nI. INTRODUCTION\n \nRecent development in \nmagnetic \nstorage \n[1] \nand \nmemory\n \n[2\n] \ndevice\ns\n \nheavily relies\n \nup\non \nincreasing\n \nswitching \nspeed and \ncoherent \nswitching \nof \nmagnetic states \nin \nferromagnetic thin films\n \nand patterned structures\n.\n \nO\nperating speeds of information storage devices have progressed into \nthe \nsub\n-\ngig\nahertz \nregime\n \nand controlled switching in \nindividual \nlayers of magnetic \nmultilayers \nand hetero\nstructures \nhas become \nimportant\n. \nThe relaxation processes \ninvolved in magnetization \ndynamics set \nnatural limit\ns\n \nfor \nthese \nswitching times and data transfer rates.\n \nIn the context of \nprecessional \nmagnetization dynamics \nthe \nnatural \nrelaxation \nrate \nagainst the small perturbation is \nexpressed as Gilbert damping\n \n(\nα\n)\n \naccording to the Landau\n-\nLiftshiz\n-\nGilbert (LLG) equation\n \n[3, \n4]\n.\n \nThis \nis analogous to viscous damping of the mechanical frictional torque\n \nand\n \nleads to the \ndirect dissipation of energy from the uniform precessional mode to thermal bath in case of zero \nwav\ne\n-\nvector excitation.\n \nGilbert damping \noriginates from spin\n-\norbit coupling and depends on the \ncoupling strength and \nd\n-\nband width of the \n3\nd\n \nferromagnet\n \n[5]\n. Th\ne damping\n \ncan be \nvaried \nby\n \nvarious \nintrinsic and extrinsic \nmechanisms including\n \nphonon drag\n \n[6]\n, Edd\ny current\n \n[7],\n \ndoping\n \n[8]\n \nor\n \ncapping\n \n[9]\n \nwith other material\n, injection of spin current\n \n[10]\n, magnon\n-\nmagnon scattering\n \n[11]\n \nand\n \ncontrolling \ntemperature of\n \nthe system\n \n[12]\n.\n \nIntri\nn\nsic and extrinsic \nnature of Gilbert \ndamping\n \nwere primarily studied by using fe\nrromagnetic resonance (FMR) technique. When the \nmagnetization is aligned with either in\n-\nplane or \nout\n-\nof\n-\nplane\n \napplied magnetic field, the \nlinewidth is proportional to the frequency with a slope determined by\n \ndamping co\nef\nf\nicient. This \nis the homogeneous or \nintrinsic contribution to the FMR linewidth. However, experiments show \nan additional frequency\n-\nindependent contribution to the linewidth\n \ncorresponds to \ninhomogeneous line broadening\n \n[13, 14]\n.\n \n \n \nHowever\n,\n \nstate\n-\nof\n-\nthe\n-\nart technique based on \npump\n-\nprobe geomet\nry\n \nhas been developed and rigorously exploited for measuring ultrafast \nmagnetization dynamics of ferromagnetic thin films during last few decades\n \n[15, 16]\n.\n \nUsing \ntime\n-\nresolved magneto\n-\noptical Kerr effect (TR\n-\nMOKE) \ntechnique \none can directly address the \npro\ncesses which are responsible for the excitation and relaxation of a magnetic system on their \ncharacteristic time scales\n \n[17\n-\n19]\n.\n \nGenerally \nduring\n \nthe pump\n-\nprobe measurements pump fluence \nis kept low to avoid nonlinear effects and sample surface degradation\n. Some recent experiments \nreveal that nonlinear spin waves play a \ncrucial \nrole in high power thin film precession\nal\n \ndynamics by introducing spin\n-\nwave instability\n \n[20]\n \nsimilar to \nFMR\n \nexperiments \nby \nappl\nication \nof\n \nhigh rf power\n \n[21]\n. \nThe coercivity and aniso\ntropy of the ferromagnetic thin films \ncan \nalso\n \nbe\n \nlowered by pump fluence\n,\n \nwhich \nmay\n \nhave potential application in heat assisted \nmagnetic \nrecording\n \n(HAMR)\n \n[22]\n. \nRecent\n \nreport\ns\n \nreveal \nthat damping \ncoefficient\n \ncan be increased or \ndecreased noticeably in the \nhigher excitation regime due to opening of further energy dissipation\n \nchannels\n \nbeyond a threshold\n \npump power\n \n[23\n-\n25]\n. \nNot only relaxation parameters but also \nfrequency \nshift\n \ndue to enhancement in pump power\n \nhas been \nobserved\n \n[20]\n. \nHowever,\n \nthe \nexperimental\n \nevidence for \nlarge \nmodulation of Gilbert damping along with frequency \nshift \nand \ntemporal chirping of the uniform precessio\nn\nal \nmotion \nis absent \nin the literature\n. \nThis \ninvestigation demands suitable choice of material, \nand here we have chosen \nPermalloy\n \n(Ni\n80\nFe\n20\n 3\n \n \nor Py here\n \non)\n \nbecause \nof its\n \nhigh permeability, \nnegligible magneto\n-\ncrystalline anisotropy, \nvery \nlow coercivity, large anisotropic magnetoresistance with reasonably low damping. Also, \ndue to \nits\n \nnegligible\n \nmagnetostriction P\ny\n \nis less sensitive to st\nrain and stress exerted during the thermal \ntreatment in \nHAMR\n \n[22]\n. \n \nIn this \narticle\n,\n \nwe have \nused \nfemto\n-\nsecond amplified laser\n \npulses\n \nfor excitation and detection of \nultrafast magnetization\n \ndynamics in \na \nP\ny\n \nthin \nfilm. Pump fluence dependent ultrafast \ndemag\nnetization is \ninvestigated\n \nalong with fast \nand slow \nremagnetization\n. \nOur comprehensive \nstudy \nof \nthe \npicosecond\n \ndynamics \nreveals transient nature of enhanced Gilbert damping in \npresence of high pump fluence\n. Also\n,\n \nthe time\n-\nvarying precession is subjected to\n \ntemporal \nchirping \nwhich occurs due to enhancement of temperature of the probed volume within a very \nshort time scale \nbeing followed by\n \nsuccessive\n \nheat dissipation. \nThis fluence dependent \nmodulation of magnetization dynamics will undoubtedly found suitable\n \napplication\n \nin spintronic \nand magnonic devices\n.\n \nII. SAMPLE PREPARATION AND CHARACTERIZATION\n \n20\n-\nnm\n-\nthick Permalloy (Ni\n80\nFe\n20\n, Py hereafter) film was deposited by using electron\n-\nbeam \nevaporation technique (SVT Associates, model: Smart Nano Tool AVD\n-\nE01) (ba\nse pressure = 3 \n× 10\n−8\n \nTorr, deposition rate = 0.2 Å/S) on 8 × 8 mm\n2\n \nsilicon (001) wafer coated with 300\n-\nnm\n-\nthick SiO\n2\n. Subsequently, 5\n-\nnm\n-\nthick SiO\n2\n \nis deposited over the Ni\n80\nFe\n20\n \nusing rf sputter\n-\ndeposition technique (base pressure = 4.5 × 10\n−7\n \nTorr, Ar pressure = 0.5 mTorr\n, deposition r\nate = \n0.2 Å/S, rf power = 60 W). \nThis capping layer protects the surface from environmental \ndegradation, oxidation and laser ablation during the pump\n-\nprobe experiment using femtosecond \nlaser pulses. \nFrom the vibrating sample magnetometry (VSM\n) we have obtained the saturation \nmagnetization (M\ns\n) and Curie temperature (T\nc\n) to be 850 emu/cc and 86\n3\n \nK\n,\n \nrespectively\n \n[26]\n.\n \n \nTo study the ultrafast magnetization dynamics of this sample, we have used a custom\n-\nbuilt time \nresolved magneto optical Kerr eff\nect (TRMOKE) magnetometer based on optical pump\n-\nprobe \ntechnique as shown in Fig. 1 (a). Here, the second harmonic (λ = 400 nm, repetition rate = 1 kHz, \npulse width > 40 fs) of amplified femtosecond laser pulse generated from a regenerative \namplifier system\n \n(Libra, Coherent) is used to excite the dynamics while the fundamental laser \npulse (λ = 800 nm, repetition rate = 1 kHz, pulse width ≈ 40 fs) is used as probe to detect the \ntime\n-\nresolved polar Kerr signal from the sample. The temporal resolution of the me\nasurement is \nlimited by the cross\n-\ncorrelation between the pump and probe pulses (\n≈120 fs). The probe beam \nhaving diameter of about 100 µm is normally incident on the sample whereas the pump beam is \nkept slightly defocused (spot size is about 300 µm) and is\n \nobliquely (\n≈ 30\n◦\n \nwith normal to the \nsample plane) incident on the sample maintaining an excellent spatial overlap with the probe \nspot. Time\n-\nresolved Kerr signal is collected from the uniformly excited part of the sample and \nslight misalignment during the \ncourse of the experiment does not affect the pump\n-\nprobe signal \nsignificantly. A large magnetic field of 3.5 kOe is first applied at a small angle of about 10° to 4\n \n \nthe sample plane to saturate its magnetization. This is followed by reduction of the magnetic \nfield to the bias field value (\nH \n= in\n-\nplane component of the bias field), which ensures that the \nmagnetization remains saturated along the bias field direction. The tilt of magnetization from the \nsample plane ensures a finite demagnetizing field along the \ndirection of the pump pulse, which is \nfurther modified by the pump pulse to induce a precessional dynamics within the sample\n \n[17]\n. In \nour experiment a 2\n-\nns time window has been used, which gave a damped uniform precession of \nmagnetization. The pump beam is\n \nchopped at 373 Hz frequency and the dynamic signal in the \nprobe pulse is detected by using a lock\n-\nin amplifier in a phase sensitive manner. Simultaneous \ntime\n-\nresolved reflectivity and Kerr rotation data were measured \nand no significant breakthrough \nof one\n \ninto another has been found\n \n[26]\n.\n \nThe probe fluence is kept constant at 2 mJ\n/\ncm\n2\n \nduring \nthe measurement to avoid additional contribution to the modulation of spin dynamics via laser \nheating. Pump fluence (\nF\n) was varied from 10 to 55 mJ\n/\ncm\n2\n \nto study the fl\nuence dependent \nmodulation in magnetization dynamics. All the experiments were performed under ambient \ncondition and room temperature. \n \nIII. RESULTS AND DISCUSSIONS\n \nA.\n \nLaser\n \ninduced ultrafast demagnetization\n \n \nWhen a femtosecond laser pulse \ninteracts\n \nwith\n \na\n \nferromagnetic \nthin film in its saturation \ncondition\n, \nthe magnetization of the system is partially or fully lost within hundreds of \nfemtosecond as measured by the time\n-\nresolved Kerr rotation or ellipticity.\n \nThis is known as \nultrafast \ndemagnetization of the\n \nferromagnet\n \nand was first observed by Beaurepire et al. in 1996 \n[\n27\n]\n. \nThis is generally followed by a fast recovery of the magnetization within sub\n-\npicosecond to \nfew picosecond\ns\n \nand a slower recovery within tens to hundreds of picoseconds, known as the fa\nst \nand slow remagnetization\n.\n \nIn many cases the slower recovery is accompanied by a coherent \nmagnetization precession and damping [\n17\n]. \nIn\n \nour\n \npump\n-\nprobe experiment, \nthe sample \nmagnetization is maintained in \nthe saturated state by application of a magnetic \nfield \nH\n \n= \n2.4 kOe\n \nbefore zero delay\n. \nRight after the zero\n-\ndelay\n \nand the \ninteraction\n \nof the pump pulse\n \nwith the \nelectrons in the \nferromagnetic \nmetal, ultrafast demagnetization takes place\n.\n \nThe local \nmagnetization is immediately quenched within first few hun\ndreds of fs \nfollowed by a subsequent \nfast \nremagnetization \nin next\n \nfew ps\n \n[27]\n. \nFigure \n1\n(b) shows ultrafast demagnetization \nobtained \nfor\n \ndifferent pump fluences. \nSeveral models have been proposed over \ntwo decades to explain the \nultrafast demagnetization\n \n[16\n, 28\n-\n31]\n.\n \nOut of those \na phenomenological thermodynamic model\n, \ncalled three temperature m\nodel\n \n[\n27, \n32, 33]\n \nhas been most widely used\n,\n \nwhere the dynamics of \nthese spin fluctuations can be describes as:\n \n)\n(\n)\n)(\n(\n0\n)\n(\n2\n1\n)\n(\n1\n2\n1\nt\nM\ne\nA\nA\ne\nA\nA\nA\nt\nM\nlat\nel\nsp\nel\nt\nlat\nel\nsp\nel\nlat\nel\nt\nsp\nel\nlat\nel\nsp\nel\nlat\nel\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n    \n \n(1)\n.\n \nThis is an approximated\n \nform based on the assumption that the electron temperature rises \ninstantaneously upon laser excitation\n \nand can be applied to fit time\n-\nresolved \nKerr rotation \ndata \ntaken within few picoseconds timescale\n.\n \nT\nhe whole system i\ns divided into three subsystems: 5\n \n \nel\nectron, spin and lattice system. On laser excitation the hot electrons are created above Fermi \nlevel. Then during energy rebalancing between the subsystems\n,\n \nquenched magnetization relaxes\n \nback to the initial state.\n \nThe two exponential functions \nin the abov\ne equation \nmirror the \ndemagnetization given by demagnetization time (τ\nel\n-\nsp\n) for energy transfer between electron\n-\nspin \nand the decay of electron temperature \n(\nτ\nel\n-\nlat\n) \nowing to the tr\nansfer of energy to the lattice. In \naddition to these characteristics time\n \nconstants, the spin\n-\nlattice relaxation time also can be \nextracted \nby including another exponential term in the above equation\n \nif the spin specific heat is \ntaken into account [\n34\n]\n.\n \nθ\n \nis the Heaviside step function and \nΓ(t) \nstands for \nthe Gaussian function \nto be convoluted\n \nwith the laser pulse envelope determining the temporal resolution\n \n(showing the \ncross c\norrelation between the probe an\nd pump pulse)\n.\n \nThe constant, \nA\n1\n \nindicates the ratio \nbetween amount of magnetization after equilibrium between electrons, s\npins, and lattice is \nrestored and the initial magnetization. A\n2 \nis proportional to the initial electronic temperature rise.\n \nWe have plotted A1 and A2, normalized with their values at the highest fluence, as a function of \npump fluence in Fig. \n3S\n \nof the supp\nlemental material\n \nwhich shows that magnitude of both \nparameters increases with \nlaser \nfluence\n \n[26]\n.\n \nWe have \nobserved\n \nthat with increasing fluence the \ndemagnetization time has been \nnegligibly varied within a range of 250\n±40\n \nfs.\n \nThe weak or no \ncorrelation bet\nween the pump fluence and the demagnetization rate describes the intrinsic nature \nof the spin scattering\n, governed by various mechanisms including Elliott\n-\nYafet mechanism\n \n[\n35\n]\n. \nAnother \nimportant\n \nobservation here is that \nthe delay of demagnetization process\nes \nwhich is the \ntime delay between pump pulse (full width at half maxima, FWHM \n≈ 130±20 fs) and starting \npoint of the ultrafast demagnetization,\n \nbecomes shorter due to increase in pump fluences. A \nplausible explanation for this is the dependence of \ndelay o\nf \ndemagnetization on the electron\n-\nthermaliz\nation\n \ntime which is eventually proportional to electron density or pump fluences\n \n[\n3\n6\n]\n. \nOn the other hand\n,\n \nfast \nremagnetization \ntime has been found to be increased noticeably from \n0.40\n \n±\n \n0.05 ps to 0.8\n0 \n±\n \n0.05 ps w\nithin the experimental fluence range\n \nof 10\n-\n55 mJ/cm\n2\n. The \nlarger is the pump fluence, the higher is the electron temperature or further the spin temperature. \nTherefore, it is reasonable that magnetization recovery time increases with the pump fluence.\n \n \nB. \nPump fluence dependent modulation in Gilbert damping\n \n \nF\nig\nure\n \n1 \n(c) shows the representat\nive Kerr rotation data for \nF\n \n= 25\n \nmJ/cm\n2\n \nconsisting of three \ntemporal\n \nregions\n,\n \ni.e.\n \nultrafast \ndemagnetization\n, fast remagnetization and slow\n \nremagnetization \nsuperposed \nwith \ndamped \nprecession within\n \nthe time window\n \nof 2 ns\n. We \nprocess\n \nthe \nmagnetization \nprecession part\n \nafter subtracting \na\n \nbi\n-\nexponential background \nto estimate the \ndamping and its modulation\n. \nThe slower remagnetization is \nmainly \ndue to heat diffusion from th\ne \nlattice to the substrate and surrounding. Within our experimental fluence range the slow \nremagnetization time has increased from \n≈0.4 ns to ≈1.0 ns. \nThe precessiona\nl dynamics is \ndescribed\n \nby phenomenological Landau\n-\nLifshitz\n-\nGilbert \n(LLG) \nequation, \n \ndt\nM\nd\nM\nM\nH\nM\ndt\nM\nd\ns\neff\n\n\n\n\n\n\n\n\n\n\n\n\n                                                                                  \n        \n    \n(2)\n 6\n \n \nwhere \nγ\n \nis the gyromagnetic ratio, \nM\n \nis magnetization\n, \nα\n \nis Gilbert damping constant\n \nand \nH\neff\n \nis \nthe effective magnetic field consisting of \nseveral field components. \nThe \nvariation \nof precessional \nfrequency with the angle between sample plane and bias \nmagnetic \nfield direction is plotted in \nF\nig. \n1 \n(d\n), which\n \nreveals that there is no uniaxial anisotropy present in this sample.\n \n \nThe energy deposit\ned by the pump pulse, in terms of heat within the probed volume, plays a very \ncrucial role in modification of local magnetic properties\n,\n \ni.e. magnetic moment, anisotropy, \ncoercivity, magnetic susceptibility\n,\n \netc. With increasing fluence the precessional fr\nequency \nexperienced a red shift\n \n[20, 25]. Thus, at the onset of the precessional dynamics (about 10 ps \nfrom zero delay), for relatively high fluence, the initial frequency (\nf\ni\n) will be smaller than its \nintrinsic value (in absence of any significant heat di\nssipation). As time progresses and the sample \nmagnetization gradually attains its equilibrium value, the precessional frequency continuously \nchanges, causing a temporal chirping of the damped oscillatory Kerr signal.\n \nThe frequency shift \ncan be \nestimated\n \nfr\nom the amount of temporal chirping\n \n[\n3\n7\n].\n \nFigure \n2 \n(a) shows the background \nsubtracted \ntime\n-\nresolved Kerr rotation data (\nprecessional \npart)\n \nfor different pump fluences fitted \nwith \na \ndamped sinusoidal function with added temporal chirping\n,\n \n \n)\n)\n(\n2\nsin(\n/\n\n\n\n\n\nt\nbt\nf\nAe\ni\nt\nk\n\n\n\n                                                                                         \n \nwhere \nA\n, \nτ\n, \nf\ni\n, b\n \nand \nΦ \nare the amplitude of the magnetization precession, the relaxation time, the \ninitial precessional frequency, \nchirp parameter \nand \ninitial \nphase, respectively. \n \nAt this point, we \nare unsure of the exact nature of the damping, \ni.e.\n \nit may consis\nt of both intrinsic and extrinsic \nmechanisms and hence we term it as\n \neffective damping parameter \n(\nα\neff\n)\n \nwhich can be\n \nextracted \nusing the following formula\n \n[3\n8\n]\n, \n \n)\n2\n4\n(\n1\neff\neff\nM\nH\n\n\n\n\n\n                                                                        \n                                        \n(\n3\n)\n \nγ\n \n= 1.\n83\n \n×10\n7\n \nHz/\nOe\n \nfor Py \nan\nd \nM\neff\n \nis the effecti\nve magnetization including pump\n-\ninduced \nchanges\n \nat \nH\n \n= 2.4 kOe\n. \nThis formula is exploited to extract effective damping parameter \nprecisely\n \nin the moderate bias fi\neld regime. \nThe variation of relaxation time and effective \ndamping \nare\n \nplotted with pump fluence in \nF\nig. \n2 \n(b) and (c). \nHere, \nτ \ndecreases with fluence while \ndamping increases \nsignificantly with respect to\n \nits \nintrinsic value within this fluence range. We \nh\nave repeated the experiment for two different field values (2.4 and 1.8 kOe). The slope of \nfluence dependent damping remains unaltered\n \nfor both the field values\n. \nWe have also observed \nincrease in relative amplitude\ns\n \nof precession with pump fluence as shown\n \nin the inset of Fig. 2 \n(c). \nTo verify the transient nature of damping we have performed another set of experiment \nwhere the probed area is exposed to different pump fluences \n(\nF\ni\n) \nfor several minutes. After the \nirradiation, the precessioanl dynamics is mea\nsured from that area\n \nwith fixed probe and pump \nfluences 2 and 10 mJ/cm\n2\n, respectively. We found that damping remains almost constant for all \nthe measurements\n \n(as shown in Fig.\n \n2 (d))\n. These results demonstrate that the enhancement of 7\n \n \ndamping is transient a\nnd only exists in the presence of high pump fluence but dropped to its \noriginal value when the pump laser was set to initial fluence.\n \n \nThe bias field dependence of precessional dynamics at four different pump fluence\ns\n \nis studied to \ngain more insight about \nthe origin of fluence dependent damping.\n \nFirst, we plotted the average \nfrequency \n(\nf\nFFT\n)\n \nwith bias field which is obtained from the fast Fourier transformation (FFT) of \nthe precessional data in \nF\nig. \n3\n \n(a). The experimental data points are fitted with the Ki\nttel formula, \n \n)\n4\n(\n2\neff\nFFT\nM\nH\nH\nf\n\n\n\n\n\n      \n                                                                                             \n      \n \n(\n4\n)\n \nM\neff\n \nis the effective magne\ntization of the sample. Figure \n3 \n(b) shows that effective magnetization \ndoes not v\nary much within the applied fluence range. So\n,\n \nwe infer that \nwith increasing fluence \nthere is no \ninduced \nanisotropy \ndeveloped in the system\n \nwhich can modify the effective damping\n \nup to this extent\n \n[23]\n. The variation of relaxation time with bias field for \nfour different pump \nfluences are plotted in \nF\nig. \n3 \n(c). Relaxation\n \ntime is increased with decreasing\n \nfield for each case \nbut for the higher fluence regime, those value\ns seem\n \nto be fluctuating. \nThis depend\nence of τ on \nfield\n \nwas fitted with eq\nuation\n \n3 to extract damping coefficient at different fluence values. We \nhave further plotted the damping coefficient as a function of precession frequency (\nf\nFFT\n)\n \n[see \nsupplementa\nl\n \nmaterial\n, \nF\nig. \n4\nS\n]\n \n[26]\n, which shows a\nn invariance of \nα\neff\n \nwith \nf\nFFT\n. From that we \ncan infer that the damping coefficient in our sample within the experimental field and fluence \nregime are intrinsic in nature and hence, we may\n \nnow\n \nterm it as the intrinsic damping coefficient \nα\n0\n.\n \nThe extrinsic \ncontributions to damping mainly come from magnetic anisotropy field, two\n-\nmagnon scattering, multimodal dephasing for excitation of several spin\n-\nwave modes, etc, which \nare negligible in our present case. \n \nF\nigure\n \n3\n(d)\n \nshows the variation of \nα\n0\n \nwith pump flue\nnce, which shows that even the intrinsic \ndamping is significantly increasing with pump fluence \n[20, \n3\n9\n]\n.\n \nFor generation of perpendicular \nstanding spin\n-\nwave modes the film needs to be thick enough. Though the film thickness is 20 nm \nhere, but within the app\nlied bias field range we have not found any other magnetic mode \nappearing with the uniform Kittel mode within the frequency window of our interest\n \n(as shown \nin \nF\nig. \n5\nS of suppleme\nn\ntal material\n)\n \n[26]\n.\n \nAlso\n,\n \nfor 20\n-\nnm\n-\nthick \nPy \nfilm,\n \nthe effect of eddy \ncurre\nnt will be negligible\n \n[\n40\n]\n.\n \nThe overlap between spatial profile of focused probe and pump \nlaser spot may lead to the generation of magnons that propagate away from the region that is \nbeing probed. \nGenerally\n,\n \nenhancement of \nnonlocal damping by spin\n-\nwave emi\nssion becomes \nsignificant\n \nwhen the excitation area is less than 1 \nµm\n. Recently \nJ. \nWu \net al.\n \nshowed that \npropagation of magnetostatic spin waves could be significant even for probed regions of tens of \nmicrons in size\n \n[\n4\n1\n]\n. \nAlso\n,\n \nby generating spin\n-\nwave trap\n \nin the pump\n-\nprobe experiment \nmodification of precessional frequency in ferromagnetic thin film due to accumulation and \ndissipation of thermal energy within the probed volume has been reported\n \n[\n4\n2\n]\n. D\nuring \nour\n \nexperiment\n \nthe \noverlap between \nprobe \nand pump \nspot \nis \nmaintai\nned carefully\n \nand\n \nKerr signal is 8\n \n \ncollected from the uniformly excited part of the sample so that slight misalignment during the \ncourse of experiment does not \nintroduce \nany \nnonlocal effects\n. \nWe will now substantiate our \nresults with some theo\nretical arguments which involve the calculation of electronic temperature \nrise in the system due to \napplication of higher \npump fluence. The electronic temperature (\nT\ne\n) is \nrelated to absorbed laser energy per unit volume (\nE\na\n) according to the following \nequa\ntion\n \n[\n4\n3\n]\n, \n \n2\n/\n)\n(\n2\n0\n2\nT\nT\nE\ne\na\n\n\n\n                                                                                                           \n \n(\n5\n)\n \nwhere, \nξ\n \nis the electronic specific heat of the system and \nT\n0\n \nis \nthe\n \ninitial electronic temperature \n(room temperature here). \nFirst, \nwe have estimated \nE\na\n \naccording to the optical parameters of the \nsample\n \nby using the following equation,\n \n \n]\n/\n)\n1\n(\n)\n1\n[(\nd\nR\nF\ne\nE\nd\na\n\n\n\n\n\n    \n                                                                                                 \n(\n6\n)\n \nwhere\n, \nd\n \nis sample thickness, \nΨ\n \nis optical penetration depth (\n~\n17 nm for \n400\n-\nnm pump\n \nlaser \nin \n20\n-\nnm\n-\nthick Py\n \nfilm\n), \nR\n \nis the reflectivity of the sample \n(0.\n5 \nmeasured \nfor \nthe \nPy\n \nfilm\n) \nand \nF\n \nis \napplied pump fluence. \nBy solving equations (\n5\n) and (\n6\n) \nwe have \nobserved \nthat \nT\ne\n \nincreases from \n≈\n1800\n \nto\n \n4\n5\n00 K within our experimental fluence range\n \nof \n10\n \nto \n55 mJ/cm\n2\n. \nDecay time of the \nelectron temperature and other r\nelevant parameters (i.e. \nE\na\n, T\ne\n \nat various flue\nnce\ns\n) are described \nin the \nsupplementa\nl\n \nmaterial\n \n[26]\n. \nThe sample remains in its magnetized state even if the \nelectronic temperature exceeds the \nCurie temperature \nT\nc\n \n. \nImportantly, ratio of the system \ntemperat\nure\n,\n \nT\n \n(as decay of electronic temperature is strongly correlated with rise of lattice \ntemperature)\n \nto \nT\nc\n \nis affecting the magnetization relaxation time\n \nwhich\n \nfundamentally depends \non susceptibility\n. Accordingly damping \nshould\n \nbe proportional to susceptibi\nlity\n \nwhich \nis\n \nstrongly \ntemperature dependent\n \n[\n40\n]\n.\n \nVarious procedures for exciting precessional dynamics in \nferromagnets show the different mechanisms to be responsible for exploration of different energy \ndissipation channels. The spin\n-\nphonon interaction m\nechanism, which historically has been \nthought to be the main contribution to magnetization damping, is important for picosecond\n-\nnanosecond applications at high temperatures such as spin caloritronics. But for laser\n-\ninduced \nmagnetization dynamics, where spi\nn\n-\nflips occur mainly due to electron scattering, quantum \nLandau\n-\nLifshitz\n-\nBloch equation is sometimes exploited to explain the temperature dependence \nof damping by considering a simple spin\n-\nelectron interaction as a source for magnetic relaxation\n \n[4\n4\n]\n. This\n \napproach suggests that increasing ratio between \nsystem \ntemperature and Curie \ntemperature\n \ninduces electron\n-\nimpurity \nled \nspin\n-\ndependent scattering. Even slightly below \nT\nc\n \na \npure change in the magnetization magnitude oc\nc\nurs\n \nwhich causes \nthe\n \nenhancement of \nda\nmping\n. \nAlso our experimental results revea\nl that the precession amplitude and\n \ndamping have been \nsubjected to a sudden change for F > 30 mJ/cm\n2\n. Energy density deposited in the probed volume \nis proportional to pump fluence. For higher fluence, the temperatu\nre dependence of the electronic \nspecific heat plays major role\n.\nThe \nincrease in the electronic \nspecific heat\n \nvalue\n \nwith temperature 9\n \n \nmay lead\n \nto\n \nlonger thermal\n-\nrelaxation time\n. We infer that relative balance between the energy \ndepo\nsited\n \ninto the lattice and electron system is \nalso \ndifferent for higher fluence regime \ncompared to that in the lower fluence regime. Thus\n,\n \nthe system temperature remains well above \nCurie temperature for F > 30 mJ/cm\n2\n, during the onset of precession for t \n≥\n \n10 ps. This may open \nup additional energy dissipation channel for the magnetization relaxation process over \nnanoseconds time scale.\n \nSometimes within very short time scale the spin temperature can go \nbeyond the Curie temperature leading towards formation o\nf paramagnetic state but that is a \nhighly non\n-\nequilibrium \ncase\n \n[\n45\n]\n. \nHowever\n \nwe believe that \nin our experiment, \neven for the \nhigh \nfluence limit and \nin \nlocal thermal \nequilibrium\n \nthe ferro\nmagnetic\n \nto paramagnetic tra\nn\nsition is not \nobserved\n.\n \nR\nepetitive measur\nements established \nthe \nreversibility of the damping parameter\n \nand \nbias\n-\nmagnetic\n-\nfield dependence of precessional frequency confirms ferromagnetic \nnature\n \nof the \nsample\n.\n \nC. \nFrequency modulation and temporal chirping\n \nPump fluence also eventua\nlly modulates the precessional frequency by introducing temporal \nchirping in the uniform precession. After immediate arrival of pump pulse, due to enhancement \nof the surface temperature, the net magnetization is reduced in \npicosecond \ntime scale which \nresul\nts in chirping of the precessional \noscillation\n. The initial frequency\n \n(\nf\ni\n)\n \nis reduced with \nrespect to its intrinsic value at a constant field. But when the probed volume cools with time, the \nspins try to retain their original precessional frequency. Thus\n,\n \nwithin a fixed time window, the \naverage frequency (\nf\nFFT\n) \nalso undergoes slight modification. In the high fluence regime, \nsignificant red shift is observed in both \nf\nFFT\n \nand \nf\ni\n. For \nH\n \n= 2.4 and 1.8 kOe, modulation of \nfrequency is found to be 0.020 GHz.cm\n2\n/mJ\n \nfor \nf\nFFT\n \nand 0.028 GHz.cm\n2\n/mJ for \nf\ni\n, from the slope \nof linear fit (as shown in \nF\nig. 4(a)). The \nf\nFFT\n \nis redu\nced by 7.2\n% of the extrapolated value at zero \npump fluence for both the fields.\n \nOn the other hand, \nf\ni\n \nis decreased \nby\n \n8.7% of its zero pump value\n \nf\nor the highest pump fluence\n. \nThe temporal chirp parameter\n, \nb \nshows giant enhancement within the experimental fluence range \n(\nF\nig. \n4\n(b)). \nFor \nH\n \n= 2.4 kOe, \nb\n \nhas increased up\n \nto \nten \ntimes (from 0.03 GHz/ns to 0.33 GHz/ns) \nin this fluence limit which implies a\nn increase in frequency of 0.66 GHz. Within our \nexperimental \nscan\n \nwindow (2 ns)\n, the maximum frequency \nshift\n \nis found to be 4.5% for \nF\n \n= 55 \nmJ/cm\n2\n.\n \nFor another bias field (\nH\n \n= 1.8 kOe), the enhancement of chirp parameter follows the \nsimilar\n \ntrend. \nThis \nult\nrafast \nmodulation is attributed to the \nthermal effect\n \non the\n \nlocal magnetic \nproperties within \nthe \nprobe\nd\n \nvolume\n \nand \nis inferred to be reversible\n \n[\n3\n7\n]\n.\n \nWe \nhave also \nplotted \nthe variation of \nb\n \nwith applied bias field for four different pump \nfluencies\n. \nI\nt see\nms to be almost \nconstant for all the \nfield values\n \nin moderate fluence regime \n(as shown in \nF\nig. \n4\n \n(c))\n. \nBut for \nF\n \n= \n40 mJ/cm\n2\n, data points are relatively scattered and large errors have been considered to take care \nof those fluctuations. \n \n 10\n \n \nIV. CONCLUSION\n \nIn\n \nessence, \nfluence dependent study \nof ultrafast magnetization dynamics \nin \nNi\n80\nFe\n20\n \nthin film\n \nreveals very weak correlation between ultrafast demagnetization time and Gilbert damping \nwithin our experimental fluence range. W\ne have reported \nlarge\n \nenhancement o\nf damping with \npump fluence. \nF\nrom the bias field \nas well as pump fluence \ndependence of \nexperimentally \nobtained \ndynamic\nal\n \nparameters we have excluded all the possible extrinsic contributions \nand \nobserved a pump\n-\ninduced modulation of intrinsic Gilbert dampin\ng. Also\n,\n \nfrom repetitive \nmeasurements with different pump irradiat\nion we have shown that the pump\n-\ninduced changes are \nreversible in nature. \nEnha\nn\ncement of the system temperature to Curie temperature ratio is \nbelieved to be responsible for increment in \nrema\ngnetization\n \ntimes and damping. \nThe temporal \nchirp parameter has been found to be increased \nby \nup to \nten \ntimes within the experimental \nfluence range\n,\n \nwhile the frequency experiences a significant red\n \nshift. \nF\nrom application point of \nview, \nas increasing dema\nnd for faster and efficient magnetic memory devices, has led the \nscientific community in the extensive research field of ultrafast magnetization dynamics, our \nresults will further enlighten the understanding of modulation of magnetization dynamics in \nferro\nmagnetic systems in presence of higher pump fluence.\n \nUsually l\now damping materials are \npreferred because it is easier to switch their magnetization in expense of smaller energy\n, lower \nwrite current in STT\n-\nMRAM devices and longer propagation length of spin \nwaves im magnonic \ndevices\n. \nOn the other hand,\n \nhigher damping is also required to \nstop \nthe post switching ringing of \nthe signal. \nThe results also have important implications on the emergent field of\n \nall\n-\noptical\n \nhelicity dependent\n \nswitching [4\n6\n-\n4\n8\n]. \nIn thi\ns context, the \ntransient \nmodulation of Gilbert \ndamping \nand other dynamical parameters \nin ferromagnetic materials is of fundamental interest \nfor characterizing and controlling ultrafast responses in magnetic structures. \n \n \nAcknowledgements:\n \nWe gratefully ack\nnowledge the financial support from S. N. Bose National \nCentre for Basic Sciences (grant no.: \nSNB/AB/12\n-\n13/96\n \nand SNB/AB/18\n-\n19/211\n) and \nDepartment of Science and Technology (DST), Government of India (grant no.: \nSR/NM/NS\n-\n09/2011\n).  We also gratefully ackno\nwledge the technical assistance of Dr. Jaivardhan Sinha and \nMr. Samiran Choudhury for preparation of the sample. SM acknowledges DST for INSPIRE \nfellowship.\n \n \nReferences\n:\n \n[\n1\n]\n \nO. Hellwig, A. Berger, T. Thomson, E. Dobisz, Z. Z Bandic, H. Yang, D. S. Kercher \nand E. \nE. Fullerton, \nSeparating dipolar broadening from the intrinsic switching field distribution in \nperpendicular patterned media\n,\n \nAppl. Phys. Lett.\n \n90\n,\n \n162516\n \n(2007)\n.\n 11\n \n \n[\n2\n]\n \nS. Tehrani, E. Chen, M. Durlam, M. DeHerrera, J. M. Slaughter, J. Shi and G. Kersz\nykowski,\n \nHigh density submicron magnetoresistive random access memory\n,\n \nJ. Appl. Phys. \n85\n,\n \n5822\n \n(1999)\n.\n \n[\n3\n] \nL. Landau and E. 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Phys.\n \nMaterials \nand devices for all\n-\noptical helicity dependent switching,\n \n50\n, 133002 (2017).\n \n 15\n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1\n: (a) Schematic of experimental geometry\n.\n \nIn the inset,\n \nφ is\n \nshown as\n \nin\n-\nplane rotational angle of\n \nH, \n(b) \npump fluence dependence of ultrafast demagnetization; \nSolid lines are fit\nting line\ns. P\nump fluences (\nF\n) having unit of \nmJ/cm\n2 \nare mentioned in numerical figure. The Gaussian envelope of laser pulse is presented to describe the \nconvolution. (c) Repre\nsentative time resolved Kerr rotation data with three distinguished temporal regions for \nF\n \n= \n25 mJ/cm\n2\n. (d) Angular variation of precessional frequency at \nH\n \n= 1.1 kOe for 20\n-\nnm\n-\nthick Py film. \nφ is presented \nin degree.\n \n \n \n 16\n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2\n: (a) Background subtracted time\n-\nresolved Kerr rotation data for different pump fluences at \nH\n \n= 2.4 kOe. \nF\n \nhaving unit of mJ/cm\n2 \nis mentioned in numerical figure. \nSolid lines are fit\nting lines\n. \nPump f\nluence\n \ndependen\nce of\n \n(b) \nrelaxation time (\nτ\n) and (c) effective damping (\nα\neff\n). \nBlack \nand \nblue \nsymbols represent the variation of these \nparameters at two different field values, \nH\n \n= 2.4 and 1.8 kOe, respectively.\n \nA\nmplitude of precession is also plotted \nwith pump \nfluence for \nH\n \n= 2.4 kOe\n,\n \n(d) Variation of effective damping with irradiation fluence \n(\nF\ni\n) \nat \nH\n \n= 2.4 kOe. \nIn order to check the possible damage in the sample as high fluence values the pump fluence was taken up to the \ntargeted value of \nF\ni\n \nfor several minut\nes followed by reduction of the pump fluence to a constant value of 10 mJ/cm\n2\n \nand the pump\n-\nprobe measurement was performed. The damping coefficient is found to be unaffected by the \nirradiation fluence as shown in (d). \n \n \n \n \n \n \n \n \n \n17\n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3\n: (a) Bias\n \nfield dependence of precessional frequency for \nF\n \n= 10 mJ/cm\n2\n. The red solid line indicates the \nKittel fit. (b) Pump fluence dependence of effective magnetization (M\neff\n) of the probed volume. (c) Bias field \ndependence of relaxation time (\nτ\n) for four differ\nent fluences. \nF\n \nhaving unit of mJ/cm\n2 \nis mentioned in numerical \nfigures. Solid lines are the fitted data. (d) Variation of intrinsic Gilbert damping (\nα\n0\n) with pump fluence.\n \n \n \n \n \n \n \n \n \n \n \n \n \n 18\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4\n: (a) Pump\n-\nfluence dependence of precessional fre\nquencies for \nH\n \n= 2.4 and 1.8 kOe. Red and black symbols \nrepresent the variation of average frequency (\nf\nFFT\n) and initial frequency (\nf\ni\n) respectively. (b) Variation of temporal \nchirp parameter ‘\nb’\n \nwith pump fluence for two different magnetic field values. (c\n) Variation of temporal chirp \nparameter with bias field for four different pump fluences. \nF\n \nhaving unit of mJ/cm\n2 \nis mentioned in numerical figure.\n \nDotted lines are guide to eye.\n \n \n \n "    },    {        "title": "1812.09596v1.Spin_dynamics_of__3d__and__4d__impurities_embedded_in_prototypical_topological_insulators.pdf",        "content": "Spin dynamics of 3dand 4dimpurities embedded in prototypical\ntopological insulators\nJuba Bouaziz,\u0003Manuel dos Santos Dias, Filipe Souza Mendes Guimar~ aes, and Samir Lounis\nPeter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich and JARA, 52425 J ulich, Germany\n(Dated: September 16, 2021)\nAbstract\nTopological insulators are insulating bulk materials hosting conducting surface states. Their\nmagnetic doping breaks time-reversal symmetry and generates numerous interesting e\u000bects such\nas dissipationless transport. Nonetheless, their dynamical properties are still poorly understood.\nHere, we perform a systematic investigation of transverse spin excitations of 3 dand 4dsingle\nimpurities embedded in two prototypical topological insulators (Bi 2Te3and Bi 2Se3). The impurity-\ninduced states within the bulk gap of the topological insulators are found to have a drastic impact\non the spin excitation spectra, resulting in very high lifetimes reaching up to microseconds . An\nintuitive picture of the spin dynamics is obtained by mapping onto a generalized Landau-Lifshitz-\nGilbert phenomenological model. The \frst quantity extracted from this mapping procedure is the\nmagnetic anisotropy energy, which is then compared to the one provided by the magnetic force\ntheorem. This uncovers some di\u000eculties encountered with the latter, which can provide erroneous\nresults for impurities with a high density of states at the Fermi energy. Moreover, the Gilbert\ndamping and nutation tensors are obtained. The nutation e\u000bects can lead to a non-negligible shift\nin the spin excitation resonance in the high-frequency regime. Finally, we study the impact of\nthe surface state on the spin dynamics, which may be severely altered due to the repositioning of\nthe impurity-induced state in comparison to the bulk case. Our systematic investigation of this\nseries of magnetic impurities sheds light on their spin dynamics within topological insulators, with\nimplications for available and future experimental studies as, for instance, on the viability of using\nsuch impurities for solid-state qubits.\n1arXiv:1812.09596v1  [cond-mat.mes-hall]  22 Dec 2018I. INTRODUCTION\nThe ever-increasing need for higher storage density oriented research towards the minia-\nturization of magnetic memories, constricted by the super-paramagnetic limit1. The real-\nization of smaller magnetic bits requires materials with a high magnetic anisotropy energy\n(MAE), originating from the relativistic spin-orbit interaction. The extreme limit for high-\ndensity magnetic storage consists of a single atomic bit2, for which quantum e\u000bects can be\npredominant. Therefore, a deep fundamental understanding underlying the stability mech-\nanisms is crucial for future technological applications. Moreover, the manipulation of these\nmagnetic units relies on external time-dependent \felds, with their dynamical properties\nbeing of prime relevance as well.\nThe standard tool for probing the dynamical magnetic properties ( i.e.spin excitations)\nof single atoms is the inelastic scanning tunneling spectroscopy (ISTS). It was employed to\ninvestigate magnetic adatoms on non-magnetic surfaces3{12. The spin excitations signature\nin the di\u000berential conductance (dI\ndV, withIbeing the tunneling current and Vthe applied\nvoltage) consists of step-like features at the excitation frequencies. They are determined\nby the applied external magnetic \feld and the MAE, which can also be accessed via other\nexperimental methods such as X-ray magnetic circular dichroism (XMCD)13,14. The nature\nof both the substrate and the adsorbate play a major role in the determination of the\nresonance frequency and lifetime of the excitation.\nSeveral theoretical investigations of spin excitations of magnetic atoms deposited on non-\nmagnetic surfaces have been performed. In the limit of weak coupling ( i.e.low hybridization)\nbetween the adsorbate and the substrate, the ISTS spectra can be interpreted employing\na Heisenberg model with localized atomic moments possessing an integer (or half integer)\nspin. Such a scenario occurs when the substrate is of insulating or semi-conducting na-\nture6,15,16. When the coupling to the substrate is strong, the hybridization e\u000bects must\nbe taken into account and a more accurate description of the electronic structure is re-\nquired. This was achieved using real-space \frst-principles calculations in the framework of\nthe Korringa-Kohn-Rostoker Green function (KKR-GF) method, which was extended to the\ndynamical regime17{20relying on time-dependent density functional theory (TD-DFT) in its\nlinear response formulation21.\nTopological insulators are intermediate between metallic and insulating substrates, con-\n2sisting of bulk insulators hosting conducting topologically protected surface states22{24. The\nmagnetic doping of topological insulators breaks time-reversal symmetry and generates ex-\notic phenomena such as the quantum anomalous Hall e\u000bect25,26. In this case, one also expects\na rather low but \fnite hybridization (with the surface state) in the region of the bulk gap,\nleading to unconventional dynamical behaviour. For instance, the magnetization dynam-\nics of a ferromagnet coupled to the surface state of a three-dimensional (3D) topological\ninsulator has already been investigated, and an anomalous behaviour in the ferromagnetic\nresonance was predicted27. Other studies with a similar focus were done in Refs. 28{31.\nFurthermore, arrays of magnetic adatoms interacting with a topological surface state were\nconsidered in Ref. 32, with the surface magnons following a linear dispersion, very unusual\nfor a ferromagnetic ground state. Moreover, the electron spin resonance of single Gd ions\nembedded in Bi 2Se3was examined in Ref. 33. The temperature dependence of the g-factor\nwas investigated and the coexistence of a metallic and an insulating phase (dual character)\nwas reported.\nIn this paper, we systematically investigate the spin dynamics of 3 dand 4dsingle im-\npurities embedded in prototypical 3D topological insulators, namely Bi 2Te3and Bi 2Se3.\nThin \flm (with a topological surface state) and inversion symmetric bulk (insulating) ge-\nometries are considered. For an accurate description of the dynamical electronic properties\nof these impurities, we employ linear response TD-DFT as implemented in the KKR-GF\nmethod17,18,20. We compute the dynamical transverse magnetic susceptibility, which rep-\nresents the magnetic response of the system to frequency-dependent transverse magnetic\n\felds. It incorporates the density of spin excitations and can be connected to ISTS mea-\nsurements34. The spin excitation spectra we obtain reveals astonishing results, with lifetimes\nspanning six orders of magnitude: from picoseconds tomicroseconds for Fe and Mn impu-\nrities embedded in Bi 2Se3, respectively. These contrasting values of the lifetimes correlate\nwith the presence (or absence) of in-gap states in the impurity local density of states (LDOS)\nnear the Fermi energy35. Next we gain further insight on the magnetization dynamics by\nmapping the transverse dynamical magnetic susceptibility to the phenomenological Landau-\nLifshitz-Gilbert (LLG) equation36. A generalized formulation of the LLG equation including\ntensorial Gilbert damping Gand nutationIis employed37. The static limit of the response\nfunction via the LLG formulation was used to extract the MAE. The latter is then compared\nto the values obtained with conventional ground state methods relying on the magnetic force\n3theorem: band energy di\u000berences38{40and torque method41. A connection between the MAE\nobtained within the linear response theory and the torque method using small deviations\nis established. Moreover, for elements with high resonance frequencies, the signature of the\nnutation is observed as a resonance shift, proving that inertial e\u000bects are relevant at such\nhigh precession rates37,42,43. Finally, we compare the LLG parameters obtained when the 3 d\nand 4dimpurities are embedded in the bulk and at the surface of Bi 2Te3. Our results show\nthat the modi\fcation of the in-gap state due to the presence of the surface state may play\na major role in the dynamics depending on the nature of the impurity.\nThis paper is structured as follows. Sec. II is dedicated to the description of the linear\nresponse TD-DFT approach employed to compute the spin excitation spectra. It also in-\ncludes the mapping of the transverse dynamical magnetic susceptibility into the generalized\nphenomenological LLG model and the di\u000berent methods used to compute the MAE. Sec. III\nis devoted to the analysis of the electronic structure and the ground state properties of 3 d\nand 4dtransition metal impurities embedded in Bi 2Te3and Bi 2Se3. In Sec. IV, we present\nthe MAE for the considered magnetic impurities and explain the discrepancies between the\ndi\u000berent methods. Sec. V contains a detailed discussion of the spin excitation spectra of\n3dand 4dimpurities embedded at the surface of both Bi 2Te3and Bi 2Se3. The \ftted LLG\nparameters are given as well, which are interpreted in terms of the impurity LDOS. Finally,\nin Sec. VI, the dynamical properties of the 3 dimpurities in the bulk and at the surface\nare compared. The contribution of the topological surface state for each impurity is then\nanalyzed.\nII. THEORETICAL DESCRIPTION\nThe description of the spin excitations of the investigated systems relies on linear re-\nsponse TD-DFT17,20,21,44. The central quantity in our approach is the dynamical magnetic\nsusceptibility, which displays poles at the excitation energies of the system. The calcula-\ntions are performed in two steps: First we determine the ground state of the system using\nconventional DFT calculations; then, we compute the dynamical response of the system\nto an external perturbing time-dependent magnetic \feld. To gain further physical insights\ninto the results, we also describe how to map the results of TD-DFT calculations onto an\nextended phenomenological LLG model. Lastly, we compare the MAE obtained from the\n4dynamical calculations with the ones computed from DFT calculations in di\u000berent ways.\nA. Density functional theory\nThe ground state DFT simulations are done using the KKR-GF method45,46in the atomic\nsphere approximation (ASA) including the full charge density, and the exchange-correlation\npotential is taken in the local spin density approximation (LSDA)47. The spin-orbit inter-\naction is included in a self-consistent fashion within the scalar relativistic approximation.\nSince we investigate impurities embedded in periodic crystals, we perform two types of cal-\nculations. The ground state of the clean host is determined \frst. Then, the impurities are\nself-consistently embedded in its crystalline structure. The host crystals investigated in this\nwork consist of Bi 2Te3and Bi 2Se3. The bulk unit cell contains \fve atoms (one quintuple\nlayer) in a rhombohedral structure (space group R \u00163m)48. The corresponding self-consistent\ncalculations employ a 30 \u000230\u000230k-mesh. The surface is simulated using a slab containing\nsix quintuple layers and 60 \u000260k-points, as in our previous work35.\nB. Time-dependent density functional theory\nThe dynamical magnetic susceptibility encodes the spin excitation spectra. It describes\nthe linear change in the spin magnetization density \u000e~M(~ r;!) upon the application of a\nfrequency-dependent external magnetic \feld \u000e~B(~ r;!) as\n\u000eM\u000b(~ r;!) =X\n\rZ\nd~ r0\u001f\u000b\r(~ r;~ r0;!)\u000eB\r(~ r0;!); (1)\nwhere\u000b;\r2fx;y;zg. For a speci\fc direction of ~M(~ r), the susceptibility tensor can be\ndivided into longitudinal and transversal blocks. In presence of the spin-orbit interaction\nor magnetic non-collinearity, the two blocks are coupled. However, for the systems that\nwe analyze in this paper, the coupling is negligible and we focus only on the transversal\nmagnetic response of systems (the xyblock when the magnetic moment is along the z-\ndirection). Within TD-DFT, the magnetic susceptibility \u001f\u000b\f(~ r;~ r0;!) is determined starting\nfrom the non-interacting magnetic susceptibility of the Kohn-Sham system, \u001fKS\n\u000b\f(~ r;~ r0;!),\n5using a Dyson-like equation17,20,21:\n\u001f\u000b\f(~ r;~ r0;!) =\u001fKS\n\u000b\f(~ r;~ r0;!) +\nX\n\r\u0016=x;yZ\nd~ r1d~ r2\u001fKS\n\u000b\r(~ r;~ r1;!)Kxc\n\r\u0016(~ r1;~ r2;!)\u001f\u0016\f(~ r2;~ r0;!);(2)\nwhere\u000b;\f;\r;\u00162fx;ygandKxc\n\r\u0016(~ r;~ r0;!) is the transverse part of the exchange-correlation\nkernel, with Kxc\n\r\u0016(~ r;~ r0;!) =\u000e\r\u0016Kxc\n?(~ r;~ r0;!). In the framework of the adiabatic LDA21,49,\nKxc\n?(~ r;~ r0;!) =\u000e(~ r\u0000~ r0) 2Bxc(~ r)=M(~ r) is frequency-independent and local in space. The\ndynamical Kohn-Sham susceptibility is evaluated from the single particle Green function\nG(~ r;~ r0;\") (de\fned in Eq. (B1)) as:\n\u001fKS\n\u000b\f(~ r;~ r0;!) =\u00001\n\u0019Z\"F\n\u00001d\"Trf\u001b\u000bG(~ r;~ r0;\"+!+ i0)\u001b\fImG(~ r0;~ r;\")\n+\u001b\u000bImG(~ r;~ r0;\")\u001b\fG(~ r0;~ r;\"\u0000!\u0000i0)g:(3)\nSince the frequency range of interest is relatively low20,44, the frequency dependence of the\nKohn-Sham susceptibility is incorporated via a Taylor expansion as\n\u001fKS\n\u000b\f(~ r;~ r0;!)\u0019\u001fKS\n\u000b\f(~ r;~ r0;0) +!d\u001fKS\n\u000b\f(~ r;~ r0;!)\nd!\f\f\f\f\f\n!=0+!2\n2d2\u001fKS\n\u000b\f(~ r;~ r0;!)\nd!2\f\f\f\f\f\n!=0:(4)\n\u001fKS\n\u000b\f(~ r;~ r0;0) being the static Kohn-Sham susceptibility. Moreover, for a system with uni-\naxial symmetry, the transversal excitations can be summarized in the spin-\rip magnetic\nsusceptibility20\n\u001f+\u0000(~ r;~ r0;!) =1\n4[\u001fxx(~ r;~ r0;!) + i\u001fxy(~ r;~ r0;!)\u0000i\u001fyx(~ r;~ r0;!) +\u001fyy(~ r;~ r0;!)]:(5)\nFurther details on the computation of the Kohn-Sham susceptibility and exchange-correlation\nkernel can be found in Refs. 17, 20, and 44. Finally, we can obtain an intuitive picture of\nthe spin excitations via the spatial average of \u001f+\u0000(~ r;~ r0;!) over a suitably-de\fned volume\nenclosing the magnetic impurity,\n\u001f+\u0000(!) =Z\nVd~ rZ\nVd~ r0\u001f+\u0000(~ r;~ r0;!); (6)\nwhich corresponds to its net response to a uniform external magnetic \feld20.\nC. Generalized Landau-Lifshitz-Gilbert equation\nIn order to develop a more intuitive picture of the magnetization dynamics, we make a\nconnection with a phenomenological model for the magnetization dynamics. We consider a\n6generalized formulation of the Landau-Lifshitz-Gilbert (LLG) equation36including a tenso-\nrial Gilbert damping G, as well as a nutation tensor Iaccounting for inertial e\u000bects37,50{52.\nThe latter can be important at relatively high frequencies37,42,43. The equation of motion of\nthe magnetic moment ~M(t) =R\nVd~ r~M(~ r;t) then reads\nd~M\ndt=\u0000\r~M\u0002 \n~Be\u000b+G\u0001d~M\ndt+I\u0001d2~M\ndt2!\n: (7)\nHere\ris the gyromagnetic ratio ( \r= 2 in atomic units) and ~Be\u000bis the e\u000bective magnetic\n\feld acting on the magnetic moment. ~Be\u000bcan be split into two contributions: ~Be\u000b=\n~Bext+~Ba, with~Bextbeing the external magnetic \feld, and ~Bais an intrinsic anisotropy\n\feld which arises due to the spin-orbit interaction20. The relation between ~Baand the\nmagnetocrystalline anisotropy energy (MAE) Kis detailed in Appendix A.\nTo establish a connection between the LLG equation and the transverse magnetic sus-\nceptibility computed using Eq. (2), we \frst consider that the local equilibrium direction is\nalong thez-axis and apply a small time-dependent transverse magnetic \feld:\n~Bext(t) =\u000eBx(t)~ ex+\u000eBy(t)~ ey; with\u000eBx(t);\u000eBy(t)\u001cj~Baj: (8)\nThen, we linearize Eq. (7) with respect to transverse components of ~Bext(t) and~M(t), which\nbecomes, in the frequency domain,\nX\n\f=x;y\u0012Ba\nz\nM\u000e\u000b\f+i!\n\rM\u000f\u000b\f+ i!G\u000b\f+!2I\u000b\f\u0013\n\u000eM\f(!) =\u000eB\u000b(!); (9)\nwith\u000f\u000b\fbeing the 2-dimensional Levi-Civita symbol ( \u000fxy= +1) and \u000eM\f(!) the\fcompo-\nnent of the frequency dependent magnetization ~M(!). The preceding equation combined\nwith Eq. (1) provides a direct connection between \u001f\u000b\f(!) obtained within TD-DFT and the\nphenomenological LLG parameters:\n8\n><\n>:(\u001fxx(!))\u00001=\u00002KSusc\nM2\u0000i!\n\rMGs\nk\u0000!2\n\rMIs\nk;\n(\u001fxy(!))\u00001=i!\n\rM(1 +Ga\nk) +!2\n\rMIa\nk;(10)\nwhereKSuscis the MAE, and the subscript indicates that this quantity is extracted from\nthe static magnetic susceptibility obtained from the TD-DFT calculations. Gs\nk(Is\nk) andGa\nk\n(Ia\nk) are the symmetric and anti-symmetric components of the Gilbert damping (nutation)\ntensor, respectively. A more detailed description of the Gilbert damping and nutation tensors\n7for the uniaxial symmetry that applies to the systems under consideration is provided in\nAppendix A. The previous equation shows in a clear fashion that the static limit of \u001fxx(!)\nis inversely proportional to the anisotropy. In the limit of small nutation, the MAE is\nconnected to the resonance frequency !LLG\nresvia (see Appendix A)\n!LLG\nres=\u0000\rq\n1 +\u0000\nGs\nk\u00012+ 2Ga\nk+\u0000\nGa\nk\u000122KSusc\nMs: (11)\nThis is the resonance frequency for precessional motion about the z-axis. Note that !LLG\nres\nis renormalized by Gs\nkandGa\nk, accounting for the damping of the precession and the renor-\nmalization of \r, respectively (see Eq. (A7)).\nD. Magnetocrystalline anisotropy\nIn absence of external magnetic \felds, the gap opening in the spin excitation spectrum is\nuniquely due to the MAE ( i.e.anisotropy \feld) breaking the SU(2) rotational symmetry20.\nThe expression of !LLG\nresin the LLG model provided in Eq. (11) shows that the resonance\nfrequency is proportional to K, which can also be computed from ground state DFT calcula-\ntions. Here, we discuss two di\u000berent ground state methods to compute this quantity relying\non the magnetic force theorem38{40,53and establish a connection with the MAE obtained\nusing linear response theory, Ksusc.\nFor uniaxial systems, the energy depends on the direction of the magnetic moment in\na simple way:E(\u0012)\u0018 K cos2\u0012, where\u0012is the angle that the magnetic moment makes\nwith thez-axis, i.e.~M=j~Mj= ^n(\u0012;') = (cos'sin\u0012;sin'sin\u0012;cos\u0012). To lowest order in\nthe phenomenological expansion, the axial symmetry renders the energy independent of the\nazimuthal angle '. It follows that the magnitude of the MAE, K, can be obtained from\ntotal energy di\u000berences for two di\u000berent orientations of the magnetization (out-of-plane and\nin-plane). However, as Kis at most a few meV's, this approach requires very accurate total\nenergies, which is computationally demanding.\nAlternatively, one can use the magnetic force theorem , which states that, if the changes\nin the charge and magnetization densities accompanying the rotation of the spin moment\nare small, the total energy di\u000berence can be replaced by the band energy di\u000berence38{40:\nKBand=EBand(0\u000e)\u0000E Band(90\u000e); (12)\n8whereEBand(\u0012) is the band energy (sum of Kohn-Sham energy eigenvalues) of the system\nwhen the spin moment makes an angle \u0012with thez-axis:\nEBand(\u0012) =Z\"F\n\u00001d\"(\"\u0000\"F)\u001a(\";\u0012): (13)\nIt contains the e\u000bect of the orientation of the magnetic moment through how the density of\nstates\u001a(\";\u0012) is modi\fed upon its rotation. This quantity is evaluated with a single non-self-\nconsistent calculation, by orienting the exchange-correlation magnetic \feld in the desired\ndirection,~Bxc(~ r) =Bxc(~ r) ^n(\u0012;') (rigid spin approximation54).\nThe MAE can also be evaluated from the magnetic torque, which corresponds to the \frst\nderivative ofEBand(\u0012) with respect to the magnetic moment direction. Using the Hellman-\nFeynman theorem, the torque reads41,55,56:\nT\u0012=@EBand\n@\u0012;\n=Z\nd~ rB xc(~ r)@^n(\u0012;')\n@\u0012\u0001~M(~ r;\u0012):(14)\nAs for the band energy calculations, the torque is also obtained from a single non-self-\nconsistent calculation, under the same approximations. It is non-vanishing if the output spin\nmagnetization density ~M(~ r;\u0012) is not collinear with the input magnetic moment direction.\nConsidering the expected form of the MAE for uniaxial symmetry, we should \fnd\nT\u0012=\u0000K Torque sin(2\u0012): (15)\nIn practice, the torque can be evaluated at di\u000berent angles \u0012. In this work, two deviation\nangles have been considered: a large deviation angle with \u0012= 45\u000e, as done in Ref. 41, and a\nsmall one near self-consistency, \u0012= 5\u000e. For such small deviations, one can connect KTorque\nto the value of the MAE obtained from the magnetic susceptibility, KSusc. It is shown in\nAppendix B that when considering a small rotation angle \u0012and a constant magnitude of\nthe exchange-correlation spin-splitting (frozen potential approximation),\nKSusc=KTorque\n1\u00004\u001fKS\n+\u0000(0)KSusc\nM2z;\n\u0018KTorque\n1 +Ba\nBxc:(16)\nThe previous expression shows that Ksusccorresponds to the KTorque (evaluated for a small\ndeviation angle) renormalized by a prefactor (1 +Ba\nBxc)\u00001. In fact, this result is similar to the\n9\u000010\u000050510LDOS (States/eV)CrMnFeCo\n\u00006\u00004\u000020246LDOS (States/eV)NbMoTc\n\u00005\u00004\u00003\u00002\u00001012\"\u0000\"F(eV)\u00006\u00004\u000020246LDOS (States/eV)NbMoTcRuPd\n\u00005\u00004\u00003\u00002\u00001012\"\u0000\"F(eV)\u000010\u000050510LDOS (States/eV)CrMnFeCo(a)(b)\n(c)(d)Bi2Te3surface\nBi2Se3surfaceBi2Te3surface\nBi2Se3surface3d4dFIG. 1. Spin-resolved LDOS for 3 dimpurities (Cr, Mn, Fe and Co) and 4 dimpurities (Nb, Mo,\nTc, Ru, Pd) embedded in a Bi 2Te3(Bi2Se3) surface. (a) 3 din Bi 2Te3, (b) 4din Bi 2Te3, (c) 3din\nBi2Se3and (d) 4din Bi 2Se3. The full lines represent the majority-spin states, with dashed lines\nfor the minority-spin ones. The energies are given with respect to the Fermi energy \"Fand the\nenergy window associated with the bulk band gap is highlighted with light blue color.\nrenormalization observed for magnetic interactions computed from the magnetic suscepti-\nbility57,58. For the systems of interest (3 dand 4dtransition metals impurities), Bais in the\nmeV range while Bxcis in the order of eV. Therefore, one expects small corrections due to\nthis renormalization, and the two quantities should be in good agreement.\n10III. ELECTRONIC STRUCTURE OF 3dAND 4dIMPURITIES IN Bi 2Te3AND\nBi2Se3\nIn this section, we brie\ry recap the discussion of the electronic structure and ground\nstate properties of 3 dimpurities embedded in the Bi 2Te3(Bi2Se3) surface already addressed\nin Ref. 35. Furthermore, we also consider 4 dimpurities which have a stronger hybridization\nwith the host electrons compared to the 3 dones. This information will be employed for\nthe analysis of their dynamical properties, such as the Gilbert damping. The LDOS of 3 d\nand 4dmagnetic impurities embedded into Bi 2Te3and Bi 2Se3(111) surfaces are shown in\nFig. 1. The bulk band gap (\u0001 gap) is depicted in light blue | with \u0001 gap\u00190:25 eV for Bi 2Te3\nand \u0001 gap\u00190:35 eV for Bi 2Se335. We consider that the impurity spin moment is oriented\nperpendicularly to the surface ( i.e.along the [111] direction). The full lines represent the\nmajority spin channel ( \"), while the dashed lines account for the the minority spin channel\n(#). All the 3 dand 4dimpurities donate electrons to the host atoms (see Table I). It can\nalso be seen in Fig. 1 that the spin splitting of the 4 dimpurities is weaker compared to the\n3dones, resulting in smaller spin moments, as listed in Table I. This is attributed to the\nStoner parameter being larger for 3 dthan for 4delements59.\nAll 3delements except Cr display a completely \flled majority-spin d-resonance. Mn and\nCr have a nearly-empty minority-spin d-resonance, resulting in a large spin moment and\nCr Mn Fe Co Nb Mo Tc Ru Pd\nQBi2Te35.154 6.160 7.282 8.448 3.488 4.717 5.892 7.147 9.421\nBi2Se34.841 5.863 6.963 8.136 3.077 4.316 5.474 6.734 9.041\nMsBi2Te33.843 4.412 3.395 2.108 1.097 2.678 2.493 0.000 0.000\nBi2Se33.671 4.421 3.482 2.231 0.906 2.574 2.534 0.564 0.578\nMlBi2Te30.065 0.050 0.260 0.883 -0.143 -0.004 0.202 0.000 0.000\nBi2Se30.008 0.024 0.144 0.942 -0.048 -0.093 0.079 0.378 0.135\nTABLE I. Ground state properties of 3 dand 4dimpurities embedded in the Bi 2Te3and Bi 2Se3\nsurfaces including: the valence charge on the impurity Q, spin moment Msand orbital moment\nMl. The spin and orbital moments are given in units of \u0016B.\n11a small orbital moment ( Ml). Fe and Co have a partially-\flled minority-spin d-resonance,\nleading to higher values for Ml, as shown in Table I. The LDOS also reveals impurity-induced\nin-gap states near the Fermi energy, which arise from the hybridization with the bulk sp\nstates of Bi 2Te3(Bi2Se3)35. When replacing the Bi 2Te3host by Bi 2Se3, the valence charge\nand the spin moment are mildly a\u000bected, in contrast to the orbital moments which are\nconsiderably altered35.\nFor 4dimpurities, both minority- and majority-spin d-resonances are partially occupied\ndue to a weak spin-splitting. The LDOS is broader and \ratter in comparison with the\n3dones, indicating a stronger hybridization with the host material, as the 4 d-orbitals are\nspatially more extended than the 3 dones, and so overlap more with the orbitals of the\nhost. In the Bi 2Te3host, Nb, Mo and Tc are found to be magnetic, while Ru, Rh and\nPd impurities were found to be nonmagnetic. The analysis of the paramagnetic LDOS (not\nshown here) reveals that, when moving in the periodic table from Tc towards Pd ( i.e.adding\nelectrons), the 4 dpeak is shifted to lower energies. This leads to a drastic decrease of the\nLDOS at\"Fand makes the Stoner criterion unful\flled. Nb has a less than half-\flled d-shell,\ninducing an orbital moment anti-parallel to its spin moment, as shown in Table I. For Mo\nand Tc, a half \flled d-shell results in the highest values for Msbetween the 4 delements.\nThese observations are in qualitative agreement with Hund's rules60. In-gap states are also\nobserved near \"F, as for the 3 dimpurities. Interestingly, in the Bi 2Se3host, Ru and Pd\nacquire a magnetic moment, while Rh remains nonmagnetic. Higher values of the LDOS at\n\"Fcompared to the Bi 2Te3host now satisfy the Stoner criterion for these elements. Pd is a\nrather peculiar case, since the increase of the LDOS at \"Fis related to the presence of an\nin-gap state in the minority-spin LDOS, as shown in Fig. 1d.\nThe electronic structure, especially in the vicinity of the Fermi energy, governs the be-\nhaviour of the MAE and spin excitations of the system. In particular, the presence of\nd-resonances near \"Fmay result in inaccuracies in the computation of the MAE. Together\nwith in-gap states, it can also induce high values of the Gilbert damping, as discussed in the\nnext sections.\n12IV. MAGNETOCRYSTALLINE ANISOTROPY OF 3dAND 4dIMPURITIES IN\nBi2Te3AND Bi 2Se3\nWe now investigate the MAE employing the di\u000berent methods discussed in Sec. II D. In\nour convention, a positive (negative) MAE stands for an in-plane (out-of-plane) easy-axis.\nIn Fig. 2a, we show the evolution of the MAE for 3 dimpurities embedded in Bi 2Te3and\nBi2Se3, respectively. For every impurity, all the methods predict the same easy-axis. In the\nBi2Te3host, Cr and Fe present an in-plane magnetic anisotropy, while Mn and Co favor an\nout-of-plane orientation. The trend is mostly accounted for by Bruno's formula61, where the\nMAE is given by the anisotropy of the orbital moment ( Ml):K/\u00102(Mx\nl\u0000Mz\nl), with\u0010\nbeing the spin-orbit interaction strength. Mn displays a small MAE, as it has a small orbital\nmoment, while the large anisotropy energies obtained for Fe and Co stem both from their\nlarge orbital moments and their substantial dependence on the spin orientation. However,\nthe results obtained for the MAE of Cr do not agree with the predictions of Bruno's formula,\nsince the MAE reaches \u00181 meV, despite a rather small anisotropy in the orbital moment of\nthe adatom (see Table. II). For the Bi 2Se3host, the anisotropy follows very similar trends in\ncomparison with the Bi 2Te3case. Nonetheless, the easy axis of Cr switches from in-plane to\nout-of-plane, while the MAE of Fe and Co present a noticeable increase, as shown in Fig. 2a.\nThese changes in the MAE are attributed to the modi\fcation of the ground state properties,\nparticularly the orbital moments (as listed in Table II), according to Bruno's formula.\nIn Fig. 2b, we show the MAE of 4 dimpurities embedded in Bi 2Te3and Bi 2Se3computed\nwith the di\u000berent approaches outlined in Section II D. For the Bi 2Te3case, all the impurities\n(Nb, Mo and Tc) display an in-plane easy-axis. Nb displays a large MAE, while Mo and\nTc have a rather small one (with the exception of KTorque (45\u000e) andKBand). For Mo, the\nsmall MAE correlates with its small orbital moment. In the Bi 2Se3host, Nb, Mo, and Tc\nare characterized by an in-plane easy-axis as well. Note that, due to a strong hybridization\nwith the host (broad LDOS in Fig. 1b and d), the MAE of Tc is drastically a\u000bected by the\nsurrounding environment. Ru and Pd acquire a magnetic moment in Bi 2Se3displaying an\nout-of-plane easy-axis. Particularly, Ru displays a very large MAE in comparison with the\nrest of the 4 delements.\nWe now focus on the reasons why di\u000berent methods may provide contrasting values for\nthe MAE (see Fig. 2). The origin of these divergences can be traced back to the features of\n13the electronic structure at the impurity site. Fig. 2a shows that the obtained MAE energies\nof Fe and Co can be separated in two groups, according to the method used to compute them:\nOne for large angle methods, including the band energy di\u000berences ( KBand [Eq. (12)]) and\nthe torque method at 45\u000e(KTorque (45\u000e) [Eq. (14)]); and the other for small perturbations,\nencompassing the torque method at 5\u000e(KTorque (5\u000e)[Eq. (14)]) and linear response theory\n(KSusc[Eq. (10)]). The results from the two methods in each group are in good agreement\nwith each other, but the results from one group do not agree with those from the other.\nThis can be understood via Table II, which lists the change in the ground state properties\nof the impurity upon 90\u000erotation of the spin moment ( z!xaxis), in a frozen potential\ncalculation. There is a large variation in the valence charge and in the spin moment of Fe\nand Co in comparison to Cr and Mn, owing to the change in the position of the 3 dpeak\nin the minority spin channel in the vicinity of \"F(see Fig. 1a and 1c). This violates the\nassumptions justifying the magnetic force theorem (in the frozen potential approximation),\nas previously observed in Ref. 62 for Co adatoms deposited on a Cu(111) surface. The\ndisagreement between the di\u000berent methods for Tc and Ru observed in Fig. 2b is attributed\nto a high occupation at \"Fas well (see Fig. 1b and 1d). An exception occurs for Nb, where\ngood agreement between the di\u000berent methods is observed. In this case, the high LDOS at\nCr Mn Fe Co Nb Mo Tc Ru Pd\n\u0001QzxBi2Te3-0.016 0.001 -0.224 -0.484 0.018 0.002 -0.287 0.000 0.000\nBi2Se3-0.001 0.000 -0.320 -0.583 -0.004 0.001 -0.319 -0.347 0.000\n\u0001Mzx\nsBi2Te3-0.016 -0.001 0.224 0.483 0.0147 -0.000 0.288 0.000 0.000\nBi2Se3-0.001 -0.000 0.320 0.582 -0.009 0.001 0.286 0.320 -0.003\n\u0001Mzx\nlBi2Te30.019 0.003 -0.323 0.484 -0.081 -0.002 -0.188 0.000 0.000\nBi2Se30.003 0.002 -0.493 0.487 -0.261 0.003 -0.284 0.285 0.008\nTABLE II. Change in the valence charge of the impurity \u0001 Qzx, spin moment \u0001 Mzx\nsand orbital\nmoment \u0001Mzx\nlfor 3dand 4dimpurities embedded in a Bi 2Te3and a Bi 2Se3surface, using the\nfrozen potential approximation. For Fe and Co, \u0001 Qzxand \u0001Mzx\nsare relatively large, invalidating\nthe use of the magnetic force theorem to compute the MAE.\n14 j.bouaziz@fz-juelich.de                                  1RuNbMoTcNbMoTcRuPd\u000010\u000050510MAE (meV)Bi2Te3Bi2Se3\nKBandKSuscKTorque(45\u0000)KTorque(5\u0000)CrMnFeCoCrMnFeCo\u000010\u000050510MAE (meV)Bi2Te3Bi2Se3\nKBandKSuscKTorque(45\u0000)KTorque(5\u0000)\n(a)(b)FIG. 2. Comparison of the MAE for (a) 3 dimpurities and (b) 4 dimpurities, embedded in a\nBi2Te3and a Bi 2Se3surface. The black curve is obtained using the band energy di\u000berences ( KBand\n[Eq. (12)]) (with a 90\u000erotation of the spin moment). The red curve shows the MAE computed\nfrom the static part of the magnetic susceptibility ( KSusc[Eq. (10)]). The green and blue curves\nare obtained using the torque method at 45\u000eand 5\u000e(KTorque (\u0012)[Eq. (14)]), respectively. Most of\nthe impurities display an in-plane magnetic anisotropy ( K>0).\n\"Fis due to the majority spin states, which are weakly a\u000bected by the spin rotation.\nThe previous analysis indicates that, if a high density of electronic states is present at\n\"F(Fe, Co, Tc and Ru), a large rotation angle may lead to large changes in the charge\ndensity and invalidate the use of the magnetic force theorem in combination with the frozen\npotential approximation. Therefore, a small deviation angle, for which the system remains\nnear self-consistency, should be considered. This can be achieved through the torque method\nor the magnetic susceptibility. The MAE obtained in these cases ( KTorque (5\u000e) andKSusc)\nshould be comparable with the one extracted for inelastic scanning tunneling spectroscopy\nmeasurements, since in such experiments the deviation of the magnetic moment from the\neasy-axis are rather small.\nV. SPIN EXCITATIONS OF 3dAND 4dIMPURITIES IN Bi 2Te3AND Bi 2Se3\nIn Sec. III, we addressed the ground state properties of 3 dand 4dimpurities embedded in\nBi2Te3and Bi 2Se3. Here, we investigate their spin dynamics, relate it to the MAE obtained\nin Sec. IV, and study the possibility of exciting and manipulating these impurities with\n150510152025!(meV)0.00.20.40.60.81.0\u00001⇡\u0000+\u0000(!) (states/meV)NbMoTc\n020406080100!(meV)0.000.010.020.030.040.05\u00001⇡\u0000+\u0000(!) (states/meV)TcRuPd05101520!(meV)0.00.51.01.52.02.53.0\u00001⇡\u0000+\u0000(!) (states/meV)CrMnFeCo\n05101520!(meV)0246810\u00001⇡\u0000+\u0000(!) (states/meV)CrMnFeCo(a)(b)\n(c)(d)0.100.150.20!(meV)0123\n012!(meV)012345\n0.00.20.4!(meV)0246810024!(meV)0.00.20.40.60.81.03d4dFIG. 3. Density of states of transverse spin excitations for magnetic impurities. The panels\nshow the results for (a) 3 dand (b) 4dimpurities embedded in Bi 2Te3, and (c) 3 dand (d) 4d\nimpurities embedded in Bi 2Se3. They present an almost-Lorentzian, with resonances located at\nthe excitation energies of the system. The dashed lines mark the resonance frequency without\ndynamical corrections, !0\nres=\u00002\rKSusc\nMs. For Mn, Co, Ru and Pd, \u001f\u0000+(!) is plotted instead, to\naccount for their easy-plane MAE.\ntime-dependent external magnetic \felds. We focus on the transverse spin excitations en-\ncoded in the dynamical magnetic susceptibility, which have been observed experimentally for\nmagnetic impurities on nonmagnetic surfaces by means of ISTS measurements3,8,11,63{65. In\nthese experiments, the spin excitations yield a step in the di\u000berential tunneling conductance\nat well-de\fned energies.\nWe show in Fig. 3 the imaginary part of \u001f+\u0000(!) (i.e.the density of states of the magnetic\nexcitations) as function of the frequency of the external \feld for both 3 dand 4dimpurities\n16embedded in Bi 2Te3and Bi 2Se3. Only the response of the magnetic impurities is considered,\nsince the induced moments in the surrounding (host) atoms are rather small. Nonetheless,\ntheir contribution is accounted for when computing the transverse exchange-correlation ker-\nnelKxc\n?at the impurity site via the spin-splitting sum rule17,20. The LLG parameters ob-\ntained by \ftting the data to Eq. (10) are given in Table III. As depicted in Fig. 3, Im \u001f+\u0000(!)\nhas a Lorentzian-like shape, and the resonance frequency ( !res) is \fnite even in absence of\nan external magnetic \feld. This resonance arises from the MAE, which breaks the SU(2)\nrotational symmetry ( i.e.no Goldstone mode), as explained previously in Sec. II D. The\nhighest resonance frequencies are obtained for Nb and Ru due to their strong anisotropy\ncombined with a small magnetic moment complying with Eq. (11), while the smallest value\nof!resis obtained for Mn impurities in Bi 2Se3. The dashed lines in Fig. 3 represent the\nresonance position obtained neglecting dynamical corrections in Eq. (11), leading to the es-\ntimate!0\nres=\u00002\rKSusc\nMs(with\r= 2 andG= 0)20. There is a qualitative agreement between\n!0\nresand the resonance position extracted from the spin excitation spectra, !res, including\ndamping and nutation. Nonetheless, their values are quantitatively di\u000berent, illustrating\nthat dynamical corrections can be of crucial importance for an accurate determination of\nthe resonance frequency.\nAnother quantity which is strongly dependent on the nature of the impurity and the\nhost is the full width at half maximum (FWHM) \u0000. This quantity is proportional to the\nsymmetric part of the Gilbert damping tensor ( Gs\nk) and provides information about the\nlifetime of the excitations66as\u001c=2\n\u0000. This lifetime ranges from picoseconds (comparable\nto lifetimes obtained at metallic surfaces20,66) to very high values reaching microseconds for\nMn in Bi 2Se3as shown in Fig. 4. Furthermore, the values of Gs\nk, shown in Table III, can be\ninterpreted in terms of the LDOS at \"F, sinceGs\nk/n#(\"F)n\"(\"F) (wheren#(\") andn\"(\")\nrepresents the LDOS of the minority and majority spin channels, respectively)44. The highest\nvalues ofGs\nkare obtained for Ru, which coincide the lowest excitation lifetime as displayed\nin Fig. 4. The anti-symmetric part of the Gilbert damping tensor Ga\nkis also displayed in\nTable III. It accounts for the renormalization of the gyromagnetic ratio, \re\u000b=\r\n1+Ga\nk(see\nAppendix A). This renormalization is attributed to the presence of a \fnite LDOS at \"Fas\nwell44.Ga\nkis negative for Cr, Nb and Ru indicating an enhancement of the gyromagnetic\nratio ( i.e.\re\u000b>2), while\re\u000b<2 for the remaining impurities. Note that the spin excitation\nspectra of Nb and Mo impurities in Bi 2Se3is not shown in Fig. 3, since for these elements\n17the Taylor expansion shown in Eq. (4) fails due to contributions from higher order terms in\nfrequency becoming too large.\nCr Mn Fe Co Nb Mo Tc Ru Pd\nMsBi2Te33.844 4.412 3.395 2.109 1.097 2.678 2.493 | |\nBi2Se33.671 4.421 3.482 2.231 0.906 2.574 2.534 0.564 0.578\nGs\nkBi2Te30.019 0.000 0.143 0.164 0.053 0.000 0.172 | |\nBi2Se30.037 0.000 0.112 0.012 0.003 0.000 0.512 0.852 0.094\nGa\nkBi2Te3-0.245 0.109 0.286 0.274 -0.087 0.096 0.099 | |\nBi2Se3-0.153 0.101 0.125 0.196 -0.021 0.134 0.081 -0.396 1.824\n!cBi2Te377.68 3439 135.7 277.4 21.91 224.5 31.64 | |\nBi2Se3283.2 1340 100.4 73.37 2.784 403.5 4.481 10.11 437.0\n\u0011cBi2Te37.154 298.3 65.66 38.39 30.36 752.2 234.4 | |\nBi2Se330.97 17820 76.31 40.19 8.703 171.5 84.93 341.8 502.5\nKSuscBi2Te30.959 -0.201 4.302 -6.725 4.091 0.417 0.353 | |\nBi2Se30.090 0.005 6.019 -5.894 5.453 0.102 3.845 -8.178 -0.431\n!LLG\nresBi2Te31.322 0.164 3.917 9.926 16.31 0.568 0.509 | |\nBi2Se30.115 0.004 6.113 8.833 24.08 0.158 5.073 55.49 1.055\n!LLG\nres\n!cBi2Te30.017 0.000 0.029 0.036 0.744 0.003 0.016 | |\nBi2Se30.000 0.000 0.063 0.125 8.836 0.000 1.132 5.487 0.002\nTABLE III. LLG parameters for 3 dand 4dimpurities embedded in the surface of Bi 2Te3(Bi2Se3),\nobtained by \ftting the TDDFT dynamical susceptibility data to Eq. (10). Msis the spin moment\nof the impurity.Gs\nkis the symmetric part and Ga\nkis the antisymmetric part of the damping tensor,\nboth unitless.KSuscis the MAE obtained from the magnetic susceptibility, in meV. !LLG\nresis the\nresonance frequency without including nutation, in meV, as de\fned in Eq. (11). A large ratio\nbetween!LLG\nresand!c=Ga\nk\nIs\nkindicates that the nutation makes a substantial contribution to !res,\nwhile\u0011c=Gs\nk\nIa\nkprovides information on the contribution of the nutation to the damping of the spin\nexcitation. Ru and Pd in Bi 2Te3were found to be nonmagnetic, so the corresponding entries are\nmarked with a dash.\n18Cr Mn Fe Co Nb Mo Tc Ru Pd10\u00001210\u0000910\u00006⌧(s)3d in Bi 2Te3\n4d in Bi 2Te3\n3d in Bi 2Se3\n4d in Bi 2Se3FIG. 4. Excitation lifetime of 3 dand 4dmagnetic impurities embedded in Bi 2Te3and Bi 2Se3. Note\nthat the lifetime axis is on a logarithmic scale. The highest excitation lifetime is obtained for Mn\nin Bi 2Se3and reaches microseconds, while the lowest one is obtained for Ru. Elements without\ndata were found to be nonmagnetic in the respective hosts.\nThe importance of the nutation can be estimated from the real part of the denominator\nof Eq. (A5). Both damping and nutation terms, Ga\nk!andIs\nk!2, contribute to the resonance.\nWhen it occurs at frequencies higher than !c=Ga\nk\nIs\nk,!rescan be substantially a\u000bected by the\nnutation. The ratio between !LLG\nresobtained using Eq. (11) (without including nutation) and\n!c(shown in Table III) is employed to evaluate the importance of this contribution. The\nsymmetric parts of the Gilbert damping and nutation tensors can be also related via43,67\nIs\nk/Gs\nk,i.e.the damping and nutation coe\u000ecients are proportional. The ratio !cis fairly\nsmall for the majority of the elements, indicating that nutation has no signi\fcant impact\non the resonant spin precession. However, for some elements such as Nb and Tc (in Bi 2Se3)\nthe nutation leads to a shift of \u00181:3 and 0:4 meV in the resonance frequency, respectively.\nFinally, the most striking element is once again Ru, with a shift of the resonance frequency\nfrom!LLG\nres= 55:49 to!res= 25:52 due to the nutation.\nVI. SURFACE AND BULK SPIN DYNAMICS\nWe now compare di\u000berent cases of 3 dand 4dmagnetic impurities embedded in a surface\nand in a bulk inversion symmetric Bi 2Te3(i.e.insulating phase with no topological surface\nstate). This enables us to disentangle the surface and bulk contributions to the spin dy-\n19namics. The analysis of the ground state properties of the 3 dimpurities embedded in bulk\nBi2Te3is given in Ref. 35. The impurity-induced electronic in-gap states are also present\nin 4dimpurities embedded in bulk Bi 2Te3. The LLG parameters obtained in the bulk (de-\nnoted with a subscript \\b\") and at the surface (denoted with a subscript \\s\") are displayed\nin Table IV. With the exception of Mn, the MAE obtained from the susceptibility di\u000bers\nconsiderably between the bulk and surface cases | Cr even has its easy-axis switched. The\noverall change in the MAE is a decrease from the surface to the bulk cases. The immediate\nenvironment of the embedded impurities is the same in bulk and at surface. However, for\nthe bulk case, the missing contribution of the surface state leads to modi\fcations in the\nelectronic structure, altering the virtual bound and the in-gap states35. This results in a\nreduction of the MAE. The spectral weight at the Fermi level is also a\u000bected leading to a\nmodi\fcation of the damping parameter44. For Cr, Fe and Tc, Gs\nkdecreases, while for Co,\nNb and Mo, it increases. Ga\nkfollows similar trends as in the surface case. Co and Nb are\nthe exception since Ga\nkswitches sign, resulting in a change of \re\u000b. The nutation is negligible\nfor most of elements, except for Nb and Co | for the latter, it leads to a noticeable shift\nof the resonance frequency from !LLG\nres= 4:24 meV to!res= 4:68 meV. In summary, Co and\nNb impurities are very sensitive to the the presence of the surface state, where the impurity\nstates display rather di\u000berent behaviours in the bulk and at the surface leading to a di\u000berent\nspin excitational nature. In contrast, Mn impurities have a similar behavior in the bulk and\nat the surface, showing that the topological surface state plays a negligible role for their spin\ndynamics.\nVII. CONCLUSIONS\nIn this paper, we employed a \frst-principles approach for the investigation of the spin\nexcitation spectra of 3 dand 4dimpurities embedded in two prototypical topological insu-\nlators, namely Bi 2Te3and Bi 2Se3. The simulations were carried out using linear response\nTD-DFT in the framework of the KKR-GF method, suitable for computing the properties\nof spin excitations at the nanoscale. A mapping onto a generalized LLG model allowed to\nextract from \frst-principles the MAE and transversal components of the Gilbert damping\nand nutation tensor. The obtained values of the MAE were then compared systematically\nto the ones obtained using the torque method and band energy di\u000berences, that rely on the\n20MsGs\nkGa\nk!c\u0011cKSusc!LLG\nres!LLG\nres\n!c\nCrs 3.844 0.018 -0.245 77.68 7.154 0.959 1.322 0.017\nCrb 3.823 0.004 -0.215 332.6 47.48 -0.824 1.090 0.003\nMns 4.412 0.000 0.109 3439 298.4 -0.201 0.164 0.000\nMnb 4.335 0.000 0.118 860.7 590.4 -0.216 0.178 0.000\nFes 3.395 0.143 0.286 135.7 65.66 4.302 3.917 0.029\nFeb 3.294 0.045 0.234 58.98 20.87 3.055 3.004 0.053\nCos 2.109 0.164 0.274 277.4 38.39 -6.725 9.926 0.037\nCob 1.977 0.307 -0.011 1.015 56.09 -2.168 4.237 4.174\nNbs 1.097 0.053 -0.087 21.91 30.36 4.091 16.31 0.769\nNbb 0.740 0.314 0.049 10.59 488.5 1.028 5.074 0.479\nMos 2.678 0.000 0.096 224.5 752.2 0.417 0.568 0.003\nMob 2.527 0.012 0.151 323.9 1083 0.454 0.624 0.002\nTcs 2.493 0.172 0.099 31.64 234.4 0.353 0.509 0.016\nTcb 2.057 0.059 0.072 12.67 29.32 0.755 1.368 0.111\nTABLE IV. LLG parameters for 3 dand 4dimpurities embedded in the surface (subscript s) and\nin the bulk (subscript b) of Bi 2Te3, obtained by \ftting the TDDFT dynamical susceptibility data\nto Eq. (10). Msis the spin moment of the impurity. Gs\nkis the symmetric part and Ga\nkis the\nantisymmetric part of the damping tensor, both unitless. KSuscis the MAE obtained from the\nmagnetic susceptibility, in meV. !LLG\nresis the resonance frequency without including nutation, in\nmeV, as de\fned in Eq. (11). A large ratio between !LLG\nresand!c=Ga\nk\nIs\nkindicates that the nutation\nmakes a substantial contribution to !res, while\u0011c=Gs\nk\nIa\nkprovides information on the contribution\nof the nutation to the damping of the spin excitation. The MAE and the Gilbert damping are\nconsiderably a\u000bected when going from surface to bulk. The largest changes occur in the case of\nthe Co impurity.\nmagnetic force theorem and the frozen potential approximation.\nAll the considered 3 dimpurities acquire a \fnite magnetic moment in both hosts, while\nthe strong hybridization of the 4 dimpurities with the host states makes them more sensitive\n21to the surrounding environment. For instance, Ru and Pd were found to be nonmagnetic\nin Bi 2Te3but became magnetic in Bi 2Se3. Furthermore, and independently from nature of\nthe orbitals (3 dor 4d), large rotation angles result in signi\fcant changes in the electronic\nproperties when a high electronic density of states is found at the Fermi energy, invalidating\nthe assumptions made to invoke the magnetic force theorem. The MAE must be then\ncomputed employing perturbative methods such as linear response theory or the torque\nmethod with small deviation angles. The MAE obtained using linear response theory is found\nto coincide with the one computed from the torque method di\u000bering only by a negligible\nrenormalization factor.\nThe spin excitation spectra of the impurities displays diverse trends. When the impurity\nvirtual bound states or in-gap states are located away from the Fermi energy, the Gilbert\ndamping is rather low and the lifetime of the excitation reaches high values compared to\nthe ones observed in metallic hosts20,66. The most striking example is a Mn impurity in\nBi2Se3, where the lifetime reaches microseconds . A contrasting situation is observed for Ru,\nwhich displays a \rat excitation resonance in conjunction with a low lifetime. Moreover, we\nfound that nutation e\u000bects can be important and lead to important shifts of the resonance\nfrequency for some elements such as Nb, Tc and Ru. Moreover, we examined the contribution\nof the surface state to the spin dynamics by comparing the LLG parameters of the impurities\nembedded in the surface with those of impurities embedded in the bulk. For Co and Nb\nimpurities, it was found that the topological surface state has a drastic impact on the\ndynamics via the spectral shift of the impurity-induced electronic in-gap states, while it\nplays a minor role for Mn impurities.\nWe provided a systematic investigation of the spin dynamics of 3 dand 4dimpurities\nembedded in topologically insulating hosts. The results obtained for excitation lifetimes of\nsome speci\fc impurities (Mn) provide insights on the dual (metal and insulator) nature of\nthese materials. In addition to that, the MAE computed employing perturbative methods\nsuch as the linear response can be compared to the one extracted from ISTS measure-\nments. Finally, several aspects remain to be uncovered from \frst principles: the zero-point\nspin \ructuations60of these impurities, which can be accessed via the dynamical magnetic\nsusceptibility, as well the spin dynamics of magnetic nanoclusters or full magnetic layers\ndeposited on topological insulators.\nAcknowledgements We thank Dr. Julen Iba~ nez-Azpiroz for fruitful discussions. This\n22work was supported by the European Research Council (ERC) under the European Union's\nHorizon 2020 research and innovation programme (ERC-consolidator grant 681405 DYNA-\nSORE). We gratefully acknowledge the computing time granted by the JARA-HPC Ver-\ngabegremium and VSR commission on the supercomputer JURECA at Forschungszentrum\nJ ulich.\nAppendix A: Phenomenological parameters from the generalized Landau-Lifshitz-\nGilbert equation\nIn this Appendix, we provide the explicit forms of the phenomenological quantities\n(anisotropy \feld, damping and nutation tensors) discussed in section II C. First, we es-\ntablish a connection between the anisotropy \feld ~Baand the magnetocrystalline anisotropy\nusing the phenomenological form of the band energy EBand. For ease of connection with\nthe LLG, we present the derivation using a vector formalism. For systems with uniaxial\nsymmetry, the expansion of the band energy in terms of the magnetization up to second\norder reads41\nEBand=E0(j~Mj) +K\nM2(~M\u0001~ en)2+::: : (A1)\nE0(j~Mj) contains the isotropic energy contributions and ~ enrepresents the direction of the\neasy-axis. The anisotropy \feld is then given by the \frst order derivative of EBandwith respect\nto~M(the longitudinal component does not a\u000bect the dynamics within the LLG):\n~Ba=\u0000@EBand\n@~M;\n=\u00002K\nM2(~M\u0001~ en)~ en:(A2)\nSecond, the Gilbert damping ( G) and nutation (I) tensors shown in section II C are rank-\n2 tensors, which can be split into a symmetric part (labeled with the superscript s) and\nan anti-symmetric part (labeled with the superscript a). Moreover, due to the uniaxial\nsymmetry, the Gilbert damping tensor has the following structure:\nG=\u00001\n\rM0\nBBB@Gs\nk\u0000Ga\nkGa\n?\nGa\nkGs\nk\u0000Ga\n?\n\u0000Ga\n?Ga\n?Gs\n?1\nCCCA: (A3)\nThe symbolkdenotes the spin dynamics parameters describing the transverse components\nof the precessional motion when the spin moment is along the [111] direction in its ground\n23state. As the system has uniaxial symmetry, the spin dynamics can be anisotropic, and we\nintroduce the symbol ?to account for this possibility. The nutation tensor has the same\nstructure:\nI=\u00001\n\rM0\nBBB@Is\nk\u0000Ia\nkIa\n?\nIa\nkIs\nk\u0000Ia\n?\n\u0000Ia\n?Ia\n?Is\n?1\nCCCA: (A4)\nThe previous decomposition of Gilbert damping and nutation tensors is identical to the\none performed on magnetic exchange interactions68,69. The trace of the the damping tensor\ncoincides with the conventional Gilbert damping constant for a cubic system36, while the\no\u000b-diagonal components account for the renormalization of \r, which controls the precession\nrate. Considering the previous forms for the Gilbert damping and nutation combined with\nEqs. (9) and (5), the spin-\rip dynamical magnetic susceptibility obtained from the LLG\nequation reads then:\n\u001fLLG\n+\u0000(!) =1\n2M\r\n\u00002K\r\nM\u0000(1 +Ga\nk+iGs\nk)!+ (\u0000Is\nk+iIa\nk)!2: (A5)\nThe resonance frequency is de\fned as@Im\u001fLLG\n+\u0000(!)\n@!\f\f\n!LLGres= 0. In absence of nutation, it can be\ncomputed analytically and is given by:\n!LLG\nres=\u0000\rq\n1 +\u0000\nGs\nk\u00012+ 2Ga\nk+\u0000\nGa\nk\u000122Ksusc\nMs: (A6)\nThe latter can be written in terms of the e\u000bective gyromagnetic ratio as:\n!LLG\nres=\u0000\re\u000br\n1 +\u0010Gs\nk\n1+Ga\nk\u001122Ksusc\nMs;with\re\u000b=\r\n1 +Ga\nk: (A7)\nAppendix B: Torque method and linear response theory\nIn this appendix, we consider small deviations of the spin moment from the equilibrium\ndirection and connect the MAE obtained within the torque method and linear response. This\nwill be done employing the retarded single-particle Green function (GF), which is de\fned\nas the resolvent of the single-particle Hamiltonian H(~ r),\n\u0000\n\"+ i0\u0000H(~ r)\u0001\nG(~ r;~ r0;\"+ i0) =\u000e(~ r\u0000~ r0): (B1)\n24To keep the notation as light as possible, we do not introduce the partition of space into\ncells around each atom, as is customary in the KKR-GF approach. The expressions can\neasily be generalized to take that aspect into account. We shall require the following two\nbasic properties (note that the GF is a spin matrix):\n@\n@\"G(~ r;~ r;\"+ i0) =\u0000Z\nd~ r0G(~ r;~ r0;\"+ i0)G(~ r0;~ r;\"+ i0); (B2)\n@\n@XG(~ r;~ r;\"+ i0) =Z\nd~ r0G(~ r;~ r0;\"+ i0)@H(~ r0)\n@XG(~ r0;~ r;\"+ i0); (B3)\nwhereXis some parameter or quantity upon which the Hamiltonian depends. Both relations\nfollow trivially from the de\fning equation of the GF (Eq. (B1)). The electronic density of\nstates is given by\n\u001a(\") =\u00001\n\u0019Im Tr\u001bZ\nd~ rG(~ r;~ r;\"+ i0); (B4)\nfrom which the connection between the GF and the band energy of the main text Ebandis\nestablished. The spin magnetization density is given by\n~M(~ r) =\u00001\n\u0019Im Tr\u001bZ\"F\n\u00001d\"~\u001bG(~ r;~ r;\"+ i0); (B5)\nand we make the assumption that the Hamiltonian depends on the direction of the spin\nmagnetization density in a coarse-grained way\nH(~ r) =H0(~ r) +Bxc(~ r) ^n(\u0012;')\u0001~\u001b: (B6)\n^n(\u0012;') being the direction of the exchange-correlation magnetic \feld. Assuming that the\neasy axis is along the z-direction, a small rotation angle \u0012in thexz-plane of ^nresults in\na torqueT\u0012given in Eq. (14). Using the de\fnition of the band energy and the density of\nstates (Eqs. (13) and (B4)), T\u0012can be expressed in terms of the GF as\nT\u0012=\u00001\n\u0019Im Tr\u001bZ\nd\"Z\nd~ r(\"\u0000\"F)@G(~ r;~ r;\"+ i0)\n@\u0012; (B7)\nRelying on Eq. (B3), the \frst order derivative of the GF with respect to \u0012can expressed in\nterm of the derivative of H(~ r) which reads:\n@H(~ r)\n@\u0012=Bxc(~ r)@^n(\u0012)\n@\u0012\u0001~\u001b:\n=Bxc(~ r) [cos\u0012\u001bx\u0000sin\u0012\u001bz](B8)\n25The combination of the previous equation with Eq. (B3) and Eq. (B7) leads to the following\nexpression for the torque:\nT\u0012=\u00001\n\u0019Im Tr\u001bZ\"F\n\u00001d\"Z\nd~ rBxc(~ r) [cos\u0012G(~ r;~ r;\" )\u001bx\u0000sin\u0012G(~ r;~ r;\" )\u001bz]: (B9)\nThe previous expression was obtained after performing a partial energy integration. Fur-\nthermore, considering a small rotation angle, then G(~ r;~ r;\" ),i.e.the Green function for the\nrotated~Bxcis related to the unperturbed Green function G0(~ r;~ r;\" ) (with~Bxc(~ r)kz-axis)\nvia a Dyson equation:\nG(~ r;~ r;\" )\u0019G0(~ r;~ r;\" ) +Z\nd~ r0G0(~ r;~ r0;\") \u0001~Bxc(~ r0)\u0001~\u001bG 0(~ r0;~ r;\"): (B10)\n\u0001~Bxc(~ r) being the change in the exchange-correlation spin-splitting given by:\n\u0001~Bxc(~ r) =Bxc(~ r) (sin\u0012;0;cos\u0012\u00001);\n\u0019Bxc(~ r)\u0012\n\u0012;0;\u0000\u00122\n2\u0013\n:(B11)\nThen, the expression of G(~ r;~ r;\" ) from Eq. (B10) is plugged back into Eq. (B9) and cos \u0012\nand sin\u0012are expanded for small \u0012as well (retaining linear terms), resulting in the following\nfrom for the torque:\nT\u0012=\u00001\n\u0019Im Tr\u001bZ\"F\n\u00001d\"Z\nd~ rBxc(~ r)Z\nd~ r0[\u001bxG0(~ r;~ r0;\")Bxc(~ r0)\u001bxG0(~ r0;~ r;\")]\u0012\n+1\n\u0019Im Tr\u001bZ\"F\n\u00001d\"Z\nd~ rBxc(~ r)\u001bzG0(~ r;~ r;\" )\u0012 :\n=Z\nd~ rBxc(~ r)\u0014Z\nd~ r0\u001fKS\nxx(~ r;~ r0;0)Bxc(~ r0)\u0000M(~ r)\u0015\n\u0012 :(B12)\n\u001fKS\nxx(~ r;~ r0;0) is the static Kohn-Sham magnetic susceptibility and M(~ r) is the magnetization\ndensity. Using the de\fnition of the spin-\rip Kohn-Sham magnetic susceptibility given in\nEq. (5) in the static limit ( i.e.\u001fKS\nxy(~ r;~ r0;0) =\u001fKS\nyx(~ r;~ r0;0) = 0) and xandy-directions are\nequivalent due to uniaxial symmetry), the torque reads:\nT\u0012=Z\nd~ rBxc(~ r)\u0002\n2\u001fKS\n+\u0000(~ r;~ r0;0)Bxc(~ r0)\u0000M(~ r)\u0003\n\u0012 : (B13)\nThe spin-splitting and the transversal exchange-correlation kernel Kxc\n?(~ r) are related via17,20:\nBxc(~ r) =Kxc\n?(~ r)M(~ r)\n2: (B14)\n26To obtain a simple result, we coarse-grain the exact equations by integrating out the spatial\ndependence and work with e\u000bective scalar quantities. This allows us to write the transversal\nexchange-correlation kernel as:\nKxc\n?=\u0000\n\u001fKS\n+\u0000(0)\u0001\u00001\u0000\u001f\u00001\n+\u0000(0): (B15)\nPlugging the two previous expressions into the coarse-grained form of Eq. (B13), T\u0012can be\nwritten in terms of the static spin-\rip magnetic susceptibilities (Kohn-Sham and enhanced)\nas:\nT\u0012=\u0000M2\n2\u0002\n\u001f\u00001\n+\u0000(0)\u0000\u001fKS\n+\u0000(0)\u001f\u00002\n+\u0000(0)\u0003\n\u0012 : (B16)\nOn one hand, considering that \u001f+\u0000(0) (static limit) obtained from TD-DFT relates to KSusc\nvia\u001f+\u0000(0) =M2\n4KSusc, Eq. (B16) can be recast into:\nT\u0012=\u0000\u0012\n2KSusc\u00008\u001fKS\n+\u0000(0)K2\nSusc\nM2\u0013\n\u0012 : (B17)\nOn the other hand, the torque T\u0012is also given by the \frst order derivative of the phenomeno-\nlogical form of the band energy as:\nT\u0012=@EBand\n@\u0012;\n=\u0000K Torque sin 2\u0012 :(B18)\nAfter expanding for a small angle, T\u0012reads:\nT\u0012=\u00002KTorque\u0012 : (B19)\nThe connection between KTorque andKSuscshown in Eq. (16) of the main text can be estab-\nlished when comparing Eq. (B17) and Eq. (B19).\n\u0003j.bouaziz@fz-juelich.de\n1Y. Shiroishi, K. Fukuda, I. Tagawa, H. 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Heinonen,2Yizheng Wu,3, 7,\u0003Axel Ho\u000bmann,2,yand Wei Zhang1, 2,z\n1Department of Physics, Oakland University, Rochester, MI 48309, USA\n2Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA\n3State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, China\n4Computational Sciences Division, Argonne National Laboratory, Argonne, IL 60439, USA\n5Department of Physics, Illinois Institute of Technology, Chicago IL 60616, USA\n6Department of Physics, Bogazici University, Bebek 34342, Istanbul, Turkey\n7Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China\n(Dated: January 8, 2019)\nTailoring Gilbert damping of metallic ferromagnetic thin \flms is one of the central interests in\nspintronics applications. Here we report a giant Gilbert damping anisotropy in epitaxial Co 50Fe50\nthin \flm with a maximum-minimum damping ratio of 400 %, determined by broadband spin-torque\nas well as inductive ferromagnetic resonance. We conclude that the origin of this damping anisotropy\nis the variation of the spin orbit coupling for di\u000berent magnetization orientations in the cubic lattice,\nwhich is further corroborate from the magnitude of the anisotropic magnetoresistance in Co 50Fe50.\nIn magnetization dynamics the energy relaxation rate\nis quanti\fed by the phenomenological Gilbert damping\nin the Landau-Lifshits-Gilbert equation [1], which is a\nkey parameter for emerging spintronics applications [2{\n6]. Being able to design and control the Gilbert damp-\ning on demand is crucial for versatile spintronic device\nengineering and optimization. For example, lower damp-\ning enables more energy-e\u000ecient excitations, while larger\ndamping allows faster relaxation to equilibrium and more\nfavorable latency. Nevertheless, despite abundant ap-\nproaches including interfacial damping enhancement [7{\n9], size e\u000bect [10, 11] and materials engineering [12{14],\nthere hasn't been much progress on how to manipulate\ndamping within the same magnetic device. The only\nwell-studied damping manipulation is by spin torque [15{\n18], which can even fully compensate the intrinsic damp-\ning [19, 20]. However the requirement of large current\ndensity narrows its applied potential.\nAn alternative approach is to explore the intrinsic\nGilbert damping anisotropy associated with the crys-\ntalline symmetry, where the damping can be continu-\nously tuned via rotating the magnetization orientation.\nAlthough there are many theoretical predictions [21{25],\nmost early studies of damping anisotropy are disguised\nby two-magnon scattering and linewidth broadening due\nto \feld-magnetization misalignment [26{29]. In addition,\nthose reported e\u000bects are usually too weak to be consid-\nered in practical applications [30, 31].\nIn this work, we show that a metallic ferromagnet can\nexhibit a giant Gilbert damping variation by a factor\nof four along with low minimum damping. We inves-\ntigated epitaxial cobalt-iron alloys, which have demon-\nstrated new potentials in spintronics due to their ultralow\ndampings [32, 33]. Using spin-torque-driven and induc-\ntive ferromagnetic resonance (FMR), we obtain a four-\nfold (cubic) damping anisotropy of 400% in Co 50Fe50thin\n\flms between their easy and hard axes. For each angle,the full-range frequency dependence of FMR linewidths\ncan be well reproduced by a single damping parame-\nter\u000b. Furthermore, from \frst-principle calculations and\ntemperature-dependent measurements, we argue that\nthis giant damping anisotropy in Co 50Fe50is due to the\nvariation of the spin-orbit coupling (SOC) in the cu-\nbic lattice, which di\u000bers from the anisotropic density of\nstate found in ultrathin Fe \flm [30]. We support our\nconclusion by comparing the Gilbert damping with the\nanisotropic magnetoresistance (AMR) signals. Our re-\nsults reveal the key mechanism to engineer the Gilbert\ndamping and may open a new pathway to develop novel\nfunctionality in spintronic devices.\nCo50Fe50(CoFe) \flms were deposited on MgO(100)\nsubstrates by molecular beam epitaxy at room temper-\nature, under a base pressure of 2 \u000210\u000010Torr [34]. For\nspin-torque FMR measurements, i) CoFe(10 nm)/Pt(6\nnm) and ii) CoFe(10 nm) samples were prepared. They\nwere fabricated into 10 \u0016m\u000240\u0016m bars by photolithog-\nraphy and ion milling. Coplanar waveguides with 100-\nnm thick Au were subsequently fabricated [18, 35]. For\neach layer structure, 14 devices with di\u000berent orienta-\ntions were fabricated, as shown in Fig. 1(a). The geom-\netry de\fnes the orientation of the microwave current, \u0012I,\nand the orientation of the biasing \feld, \u0012H, with respect\nto the MgO [100] axis (CoFe [1 10]).\u0012Iranges from 0\u000e\nto 180\u000ewith a step of 15\u000e(D1 to D14, with D7 and D8\npointing to the same direction). For each device we \fx\n\u0012H=\u0012I+ 45\u000efor maximal recti\fcation signals. In addi-\ntion, we also prepared iii) CoFe(20 nm) 40 \u0016m\u0002200\u0016m\nbars along di\u000berent orientations with transmission copla-\nnar waveguides fabricated on top for vector network an-\nalyzer (VNA) measurements. See the Supplemental Ma-\nterials for details [36].\nFig. 1(b) shows the angular-dependent spin-torque\nFMR lineshapes of CoFe(10 nm)/Pt devices from dif-\nferent samples (D1 to D4, hard axis to easy axis) atarXiv:1901.01941v1  [cond-mat.mtrl-sci]  7 Jan 20192\nFIG. 1. (a) Upper: crystalline structure, axes of bcc Co 50Fe50\n\flm on MgO(100) substrate and de\fnition of \u0012Hand\u0012I.\nLower: device orientation with respect to the CoFe crystal\naxis. (b) Spin-torque FMR lineshapes of i) CoFe(10 nm)/Pt\ndevices D1 to D4 measured. (c) Resonances of D1 and D4\nfrom (b) for \u00160Hres<0. (d) Resonances of iii) CoFe(20\nnm) for\u0012H= 45\u000eand 90\u000emeasured by VNA FMR. In (b-d)\n!=2\u0019= 20 GHz and o\u000bset applies.\n!=2\u0019= 20 GHz. A strong magnetocrystalline anisotropy\nas well as a variation of resonance signals are observed.\nMoreover, the linewidth increases signi\fcantly from easy\naxis to hard axis, which is shown in Fig. 1(c). We have\nalso conducted rotating-\feld measurements on a sec-\nond CoFe(10 nm)/Pt device from a di\u000berent deposition\nand the observations can be reproduced. This linewidth\nanisotropy is even more pronounced for the CoFe(20 nm)\ndevices without Pt, measured by VNA FMR (Fig. 1d).\nFor the CoFe(10 nm) devices, due to the absence of the\nPt spin injector the spin-torque FMR signals are much\nweaker than CoFe/Pt and completely vanish when the\nmicrowave current is along the easy axes.\nFigs. 2(a-b) show the angular and frequency de-\npendence of the resonance \feld Hres. In Fig. 2(a), the\nHresfor all four sample series match with each other,\nwhich demonstrates that the magnetocrystalline proper-\nties of CoFe(10 nm) samples are reproducible. A slightly\nsmallerHresfor CoFe(20 nm) is caused by a greater e\u000bec-\ntive magnetization when the thickness increases. A clear\nfourfold symmetry is observed, which is indicative of the\ncubic lattice due to the body-center-cubic (bcc) texture\nof Co 50Fe50on MgO. We note that the directions of the\nhard axes has switched from [100] and [010] in iron-rich\nalloys [33] to [110] and [1 10] in Co 50Fe50, which is con-\nω/2πμ0Hres  (T) μ0Hres  (T) [110]\n[110][100][010](a) (b) CoFe(10 nm)/Pt \nω/2π=2045o90 o135o\n135o180o 225oCoFe(10 nm)/Pt \nCoFe(10 nm) CoFe(20 nm) θH:\n[100]\n[110][010]FIG. 2. (a) Resonance \feld \u00160Hresas a function of \u0012Hat\n!=2\u0019= 20 GHz for di\u000berent samples. Diamonds denote the\nrotating-\feld measurement from the second CoFe(10 nm)/Pt\ndevice. The black curve denotes the theoretical prediction.\n(b)\u00160Hresas a function of frequency for the CoFe(10 nm)/Pt\ndevices. Solid curves denote the \fts to the Kittel equation.\nsistent with previous reports [37, 38].\nThe magnetocrystalline anisotropy can be quanti-\n\fed from the frequency dependence of \u00160Hres. Fig.\n2(b) shows the results of CoFe(10 nm)/Pt when HB\nis aligned to the easy and hard axes. A small uniax-\nial anisotropy is found between [1 10] (0\u000eand 180\u000e) and\n[110] (90\u000e) axes. By \ftting the data to the Kittel equa-\ntion!2=\r2=\u00162\n0(Hres\u0000Hk)(Hres\u0000Hk+Ms), where\n\r= 2\u0019(geff=2)\u000128 GHz/T, we obtain geff= 2:16,\n\u00160Ms= 2:47 T,\u00160H[100]\nk= 40 mT,\u00160H[010]\nk= 65 mT\nand\u00160H[110]\nk=\u00160H[110]\nk=\u000043 mT. Taking the disper-\nsion functions from cubic magnetocrystalline anisotropy\n[39, 40], we obtain an in-plane cubic anisotropy \feld\n\u00160H4jj= 48 mT and a uniaxial anisotropy \feld \u00160H2jj=\n12 mT. Fig. 2(a) shows the theoretical predictions from\nH4jjandH2jjin black curve, which aligns well with all\n10-nm CoFe samples.\nWith good magnetocrystalline properties, we now turn\nto the energy relaxation rate. Fig. 3(a) shows the full-\nwidth-half-maximum linewidths \u00160\u0001H1=2of the spin-\ntorque FMR signals at !=2\u0019= 20 GHz. Again, a fourfold\nsymmetry is observed for CoFe(10 nm)/Pt and CoFe(10\nnm), with the minimal (maximal) linewidth measured\nwhen the \feld lies along the easy (hard) axes. For\nCoFe(10 nm) devices, we did not measure any spin-torque\nFMR signal for HBalong the hard axes ( \u0012H= 45\u000e, 135\u000e\nand 225\u000e). This is due to the absence of the Pt spin\ninjector as well as the near-zero AMR ratio when the rf\ncurrent \rows along the easy axes, which will be discussed\nlater. For all other measurements, the linewidths of CoFe\ndevices are smaller than for CoFe/Pt by the same con-\nstant, independent of orientation (upper diagram of Fig.\n3a). This constant linewidth di\u000berence is due to the spin\npumping contribution to damping from the additional Pt\nlayer [41, 42]. Thus we can deduce the intrinsic damp-\ning anisotropy from CoFe(10 nm)/Pt devices, with the3\nω/2π 105, 195 deg 75, 165 deg 120, 210 deg 135, 225 deg(HA) 45, 135 deg (HA) \n60, 150 deg \n90, 180 deg(EA) θHCoFe(10 nm)/Pt \n(b) = -\n=-\n[100] [110] [110] [010](a) ω/2π=20 \nω/2π θH\n0, 90 deg 15, 75 deg 22.5, 67.5 deg 30, 60 deg 42.5, 50 deg \n40, 52.5 deg \n37.5, 55 deg CoFe(20 nm) (VNA) \n(c)CoFe(10 nm)/Pt CoFe(10 nm) \n90 deg (EA) \nfor CoFe \nFIG. 3. (a) \u00160\u0001H1=2as a function of \u0012Hat!=2\u0019= 20 GHz\nfor the CoFe(10 nm) series in Fig. 2(a). Top: Addtional\nlinewidth due to spin pumping of Pt. The green region de-\nnotes the additional linewidth as 4 :5\u00060:7 mT. (b-c) \u00160\u0001H1=2\nas a function of frequency for (b) CoFe(10 nm)/Pt and (c)\nCoFe(20 nm) samples. Solid lines and curves are the \fts to\nthe data.\ndamping shifted from CoFe(10 nm) devices by a constant\nand is much easier to measure.\nIn Fig. 3(b-c) we show the frequency dependence of\n\u00160\u0001H1=2of CoFe(10 nm)/Pt devices from spin-torque\nFMR and CoFe(20 nm) devices from VNA FMR. For\nboth the easy and hard axes, linear relations are ob-\ntained, and the Gilbert damping \u000bcan be extracted\nfrom\u00160\u0001H1=2=\u00160\u0001H0+ 2\u000b!=\r with the \fts shown\nas solid lines. Here \u00160\u0001H0is the inhomogeneous broad-\nening due to the disorders in lattice structures. In Fig.\n3(b) we also show the linewidths of the CoFe(10 nm)\ndevice along the easy axis ( \u0012H= 90\u000e), which has a\nsigni\fcant lower linewidth slope than the easy axis of\nCoFe(10 nm)/Pt. Their di\u000berences yield a spin pump-\ning damping contribution of \u0001 \u000bsp= 0:0024. By using\n\u0001\u000bsp=\r\u0016hg\"#=(4\u0019MstM), we obtain a spin mixing con-\nductance of g\"#(CoFe/Pt) = 25 nm\u00002, which is compa-\nrable to similar interfaces such as NiFe/Pt [43, 44]. For\n\u0012Hbetween the easy and hard axes, the low-frequency\nlinewidth broadenings are caused by the deviation of\nmagnetization from the biasing \feld direction, whereas\nat high frequencies the \feld is su\u000ecient to saturate the\nmagnetization. In order to \fnd the damping anisotropy,\nwe \ft the linewidths to the angular model developed bySuhl [45, 46], using a single \ft parameter of \u000band the\nextractedH2jjandH4jjfrom Fig. 2. The solid \ftting\ncurves in Fig. 3(b) nicely reproduce the experimental\npoints.\nThe obtained damping anisotropy for all the samples\nare summarized in Fig. 4, which is the main result of\nthe paper. For CoFe(10 nm)/Pt samples, \u000bvaries from\n0.0056 along the easy axis to 0.0146 along the hard axis.\nBy subtracting the spin pumping \u0001 \u000bspfrom both values,\nwe derive a damping anisotropy of 380%. For CoFe(20\nnm) samples measured by VNA FMR, \u000bvaries from\n0.0054 to 0.0240, which yields an anisotropy of 440% and\nreproduces the large anisotropy from spin-torque FMR.\nThis giant damping anisotropy implies, technologically,\nnearly four times smaller critical current to switch the\nmagnetization in a spin-torque magnetic random access\nmemory, or to excite auto-oscillation in a spin-torque os-\ncillator, by simply changing the magnetization orienta-\ntion from the hard axis to the easy axis within the same\ndevice. In addition, we emphasize that our reported\ndamping anisotropy is not subject to a dominant two-\nmagnon scattering contribution, which would be mani-\nfested as a nonlinear linewidth softening at high frequen-\ncies [28, 31]. For this purpose we have extended the fre-\nquency of spin-torque FMR on CoFe(10 nm)/Pt up to 39\nGHz, see the Supplemental Materials for details [36]. We\nchoose CoFe(10 nm)/Pt samples because they provide\nthe best signals at high frequencies and the additional Pt\nlayer signi\fcantly helps to excite the dynamics. Linear\nfrequency dependence of linewidth persists throughout\nthe frequency range and \u0001 H0is unchanged for the two\naxes, with which we can exclude extrinsic e\u000bects to the\nlinewidths. We also note that our result is substantially\ndi\u000berent from the recent report on damping anisotropy\nin Fe/GaAs [30], which is due to the interfacial SOC and\ndisappears quickly as Fe becomes thicker. In compari-\nson, the Gilbert damping anisotropy in Co 50Fe50is the\nintrinsic property of the material, is bonded to its bulk\ncrystalline structure, and thus holds for di\u000berent thick-\nnesses in our experiments.\nIn order to investigate the dominant mechanism for\nsuch a large Gilbert damping anisotropy, we perform\ntemperature-dependent measurements of \u000band the re-\nsistivity\u001a. Fig. 5(a) plots \u000bas a function of 1 =\u001afor\nthe CoFe(10 nm)/Pt and CoFe(20 nm) samples and for\nHBalong the easy and hard axes. The dominant lin-\near dependence reveals a major role of conductivitylike\ndamping behavior. This is described by the breathing\nFermi surface model for transition-metal ferromagnets,\nin which\u000bcan be expressed as [23, 24, 47{49]:\n\u000b\u0018N(EF)j\u0000\u0000j2\u001c (1)\nwhereN(EF) is the density of state at the Fermi level, \u001c\nis the electron relaxation time and \u0000\u0000=h[\u001b\u0000;Hso]iE=EF\nis the matrix for spin-\rip scatterings induced by the SOC\nHamiltonian Hsonear the Fermi surface [48, 49]. Here \u001c4\n(b) CoFe(10 nm) CoFe(20 nm) CoFe(20 nm) CoFe(10 nm)/Pt - ∆α sp \nCoFe(10 nm ) 400 %\n100 %\nFIG. 4. Renormalized damping and its anisotropy for\nCoFe(10 nm) and CoFe(20 nm), measured from spin-torque\nFMR and VNA FMR, respectively. For CoFe(20 nm)/Pt sam-\nples, \u0001\u000bsphas been subtracted from the measured damping.\nis proportional to the conductivity (1 =\u001a) from the Drude\nmodel, with which Eq. (1) gives rise to the behaviors\nshown in Fig. 5(a).\nFor the origin of damping anisotropy, we \frst check\nthe role of N(EF) by ab-initio calculations for di\u000berent\nordered cubic supercells, which is shown in the Supple-\nmental Materials [36]. However, a negligible anisotropy\ninN(EF) is found for di\u000berent magnetization orienta-\ntions. This is consistent with the calculated anisotropy\nin Ref. [30], where less than 0.4% change of N(EF) was\nobtained in ultrathin Fe \flms. The role of \u001ccan also be\nexcluded from the fact that the resistivity di\u000berence be-\ntween the easy and hard axes is less than 2% [36]. Thus\nwe deduce that the giant damping anisotropy of 400% is\ndue to the change of j\u0000\u0000j2, or the SOC, at di\u000berent crys-\ntalline directions. In particular, unlike the single element\nFe, disordered bcc Fe-Co alloy can possess atomic short-\nrange order, which gives rise to local tetragonal crystal\ndistortions due to the di\u000berent lattice constants of Fe and\nCo [2{4]. Such local tetragonal distortions will preserve\nglobal cubic symmetry but can have large e\u000bects on the\nSOC. We emphasize that our CoFe samples, which did\nnot experience annealing, preserve the random disorder.\nOur \frst principle calculations also con\frm the role of lo-\ncal tetragonal distortions and its enhancement on SOC,\nsee the Supplemental Materials for details [36].\nThe anisotropy of the SOC in Co 50Fe50can be re\rected\nby its AMR variation along di\u000berent crystalline orienta-\ntions. The AMR ratio can be de\fned as:\nAMR(\u0012I) =\u001ak(\u0012I)\n\u001a?(\u0012I)\u00001 (2)\nwhere\u001ak(\u0012I) and\u001a?(\u0012I) are measured for the biasing\n\feld parallel and perpendicular to the current direction,\nrespectively. The main contribution of AMR is the asym-\nmetrics-delectron scatterings where the s-orbitals are\nmixed with magnetization-containing d-orbitals due toSOC [53, 54]. Since both the damping and AMR origi-\nnate from SOC and, more precisely, are proportional to\nthe second order of SOC, a large damping anisotropy is\nexpected to be accompanied by a large AMR anisotropy\nand vice versa. Furthermore, due to the fourfold sym-\nmetry, the AMR should be invariant when the current\ndirection is rotated by 90 degrees, as the AMR is a func-\ntion of\u0012Ias de\fned in Eq. (1). Thus the damping and\nAMR should exhibit similar angular dependence on \u0012H\nand\u0012I, respectively.\nIn Fig. 5(b) we compare renormalized \u000b(\u0012H) with\nCoFe(20 nm) CoFe(10 nm)/Pt : (a)\n300 K 8 K F(θI)/F max (b) \n,10 nm \n20 nm 10 nm \n20 nm \nFIG. 5. (a) \u000b(T) as a function of 1 =\u001a(T).T= 8 K, 30 K, 70\nK, 150 K and 300 K for CoFe(10 nm)/Pt and T= 8 K and\n300 K for CoFe(20 nm). Dashed and dotted lines are guides\nto eyes. (b) Renormalized \u000b(\u0012H) and AMR( \u0012I) andF(\u0012I) for\nCoFe(10 nm)/Pt and CoFe(20 nm). Circles, crosses and +\ndenote\u000b, AMR and F, respectively.\nAMR(\u0012I) for 10-nm and 20-nm CoFe samples, where the\nAMR values are measured from Hall bars with di\u000berent\n\u0012I. The AMR ratio is maximized along h100iaxes and\nminimized alongh110iaxes, with a large anisotropy by a\nfactor of 10. This anisotropy is also shown by the inte-\ngrated spin-torque FMR intensity for CoFe(10 nm)/Pt,\nde\fned asF(\u0012I) = \u0001H1=2Vmax\ndc [17, 18] and plotted in\nFig. 5(b). The large AMR anisotropy and its symme-\ntry clearly coincide with the damping anisotropy mea-\nsured in the same samples, which con\frms our hypoth-\nesis of strong SOC anisotropy in CoFe. Thus we con-\nclude that the damping anisotropy is dominated by the\nvariation of SOC term in Eq. (1). In parallel, we also\ncompare\u000b(\u0012H) and AMR( \u0012I) for epitaxial Fe(10 nm)\nsamples grown on GaAs substrates [36]. Experimentally\nwe \fnd the anisotropy less is than 30% for both damping\nand AMR, which helps to explain the presence of weak\ndamping anisotropy in epitaxial Fe [30].5\nWe compare our results with prior theoretical works on\ndamping anisotropy [23, 24]. First, despite their propor-\ntional relationship in Fig. 5(a), the giant anisotropy in\n\u000bis not re\rected in 1 =\u001a. This is because the s-dscatter-\ning, which dominates in the anisotropic AMR, only con-\ntributes a small portion to the total resistivity. Second,\nneither the anisotropy of damping nor AMR are sensitive\nto temperature. This is likely because the thermal excita-\ntions at room temperature ( \u00180:025 eV) are much smaller\nthan the spin-orbit coupling ( \u00180:1 eV [47]). Third, the\ndamping tensor has been expressed as a function of M\nanddM=dt[24]. However in a fourfold-symmetry lat-\ntice and considering the large precession ellipticity, these\ntwo vectors are mostly perpendicular to each other, point\ntowards equivalent crystalline directions, and contribute\nequivalently to the symmetry of damping anisotropy.\nIn summary, we have experimentally demonstrated\nvery large Gilbert damping anisotropy up to 400% in\nepitaxial Co 50Fe50thin \flms which is due to their bulk,\ncubic crystalline anisotropy. We show that the damping\nanisotropy can be explained by the change of spin-orbit\ncoupling within the breathing Fermi surface model, which\ncan be probed by the corresponding AMR change. Our\nresults provide new insights to the damping mechanism\nin metallic ferromagnets, which are important for opti-\nmizing dynamic properties of future magnetic devices.\nWe are grateful for fruitful discussions with Bret Hein-\nrich. W.Z. acknowledges supports from the U.S. Na-\ntional Science Foundation under Grants DMR-1808892,\nMichigan Space Grant Consortium and DOE Visit-\ning Faculty Program. Work at Argonne, including\ntransport measurements and theoretical modeling, was\nsupported by the U.S. Department of Energy, Of-\n\fce of Science, Materials Science and Engineering Di-\nvision. Work at Fudan, including thin \flm growth\nand fabrication, was supported by the Nat'l Key Ba-\nsic Research Program (2015CB921401), Nat'l Key Re-\nsearch and Development Program (2016YFA0300703),\nNSFC (11734006,11474066,11434003), and the Program\nof Shanghai Academic Research Leader (17XD1400400)\nof China. O.O. and V.K. acknowledge supports\nfrom Bogazici University Research Fund (17B03D3),\nTUBITAK 2214/A and U.S. Department of State Ful-\nbright Visiting Scholar Program.\n\u0003wuyizheng@fudan.edu.cn\nyho\u000bmann@anl.gov\nzweizhang@oakland.edu\n[1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).\n[2] S. 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B 10, 4626 (1974).7\nSupplemental Materials:Giant anisotropy of Gilbert damping in epi-\ntaxial CoFe \flms\nbyYi Li, Fanlong Zeng, Steven S.-L. Zhang, Hyeondeok Shin, Hilal Saglam, Vedat Karakas, Ozhan Ozatay, John E.\nPearson, Olle G. Heinonen, Yizheng Wu, Axel Ho\u000bmann and Wei Zhang\nCrystallographic quality of Co 50Fe50\flms\nFIG. S-1. Crystallographic characterization results of CoFe \flms. (a) RHEED pattern of the CoFe(10 nm) \flm. (b) XRD\nof the CoFe(10 nm) and (20 nm) \flms. (c) X-ray re\rectometry measured for the CoFe(20 nm) \flm. (d) AFM scans of the\nCoFe(20 nm) \flm. (e) Rocking curves of the CoFe(20 nm) \flm for [100] and [110] rotating axes.\nFig. S-1 shows the crystallographic characterization for the epitaxial CoFe samples. Re\rection high-energy electron\ndi\u000braction (RHEED) shows very clear and sharp patterns which shows high quality of the epitaxal \flms. X-ray\ndi\u000braction (XRD) yields clear CoFe(002) peaks at 2 \u0012= 66:5\u000e. X-ray re\rectometry scan of the CoFe (20 nm) \flm\nshows a good periodic pattern and the \ft gives a total thickness of 19.84 nm. Atomic-force microscopy (AFM) scans\nfor 10\u0016m\u000210\u0016m and 100 nm\u0002100 nm scales show smooth surface with a roughness of 0.1 nm. Lastly XRD rocking\ncurves for [100] and [110] rotating axes show a consistent linewidth of 1.45\u000e, which indicates isotropic mosaicity of\nthe CoFe \flms.\nAs a result of the crystallographic characterizations, we believe our MBE-grown CoFe samples are epitaxial, have\nsmooth surfaces and exhibit excellent crystalline quality. Moreover, we can exclude the source of inhomogeneity\nfrom misorientation of crystallities (mosaicity) due to isotropic rocking curves. This means the inhomogeneous FMR\nlinewidth broadening is isotropic, as is consistent with the experiments.\nDevice geometries for Spin-torque FMR and VNA FMR measurements.\nFig. S-2 shows the device geometry for Spin-torque FMR and VNA FMR measurements. For spin-torque FMR,\nwe have prepared CoFe(10 nm)/Pt, CoFe(10 nm) and Fe(10 nm) devices. A second CoFe(10 nm)/Pt sample is also\nprepared for rotating-\feld measurements. For VNA FMR, we have prepared CoFe(20 nm) samples. All the CoFe\n\flms are grown on MgO(100) substrates; the Fe \flm is grown on a GaAs(100) substrate. Au (100 nm) coplanar\nwaveguides are subsequently fabricated on top of all devices. For VNA FMR samples, an additional SiO 2(100 nm) is8\nFIG. S-2. (a) Spin-torque FMR devices of CoFe(10 nm)/Pt samples. (b) Illustration of the Spin-torque FMR circuit. (c) Front\nand (d) back view of the VNA FMR devices for CoFe(20 nm) samples.\ndeposited between CoFe and Au for electric isolation. The CoFe(20 nm) bars is only visible from the back view in\nFig. S-2(d).\nSpin-torque FMR lineshapes\nFigure S-3 shows the full lineshapes of (a) CoFe(10 nm)/Pt(6 nm), (b) CoFe(10 nm) and (c) Fe(10 nm) devices\nmeasured at !=2\u0019= 20 GHz. The Fe \flms were deposited on GaAs substrates by MBE growth. (a) and (b) are used\nto extract the resonance \felds and linewidths in Figs. 2(a) and 3(a) of the main text. (c) is used to examine the\ncorrelation between damping anisotropy and AMR anisotropy.\nSpin-torque FMR linewidths as a function of frequency for CoFe(10 nm) devices.\nFigure S-4(a) shows the spin-torque FMR linewidths for CoFe(10 nm) devices. Because there is no spin torque\ninjection from Pt layer, the FMR signals are much weaker than CoFe(10 nm)/Pt and the extracted linewidths are\nmore noisy. The excitation of the dynamics is due to the magnon charge pumping e\u000bect [1] or inhomogeneities of the\nOersted \felds. No signal is measured for the rf current \rowing along the easy axis (magnetic \feld along the hard\naxis, see Fig. S-3b), because of the negligible AMR ratio.\nFigure S-4(b) shows the angular dependence of the extracted Gilbert damping for CoFe(10 nm)/Pt and CoFe(10\nnm). The former is extracted from Fig. 3(b) of the main text. The latter is extracted from Fig. S-4(a). The blue\ndata points for CoFe(10 nm)/Pt are obtained from the resonances at negative biasing \felds. Those data are used in\nFig. 4 of the main text.9\nFIG. S-3. Spin-torque FMR lineshapes of (a) CoFe(10 nm)/Pt, (b) CoFe(10 nm) and (c) Fe(10 nm) devices measured at\n!=2\u0019= 20 GHz. \u0012H\u0000\u0012Iis \fxed to 45\u000e.\nFIG. S-4. (a) \u00160\u0001H1=2as a function of frequency for CoFe(10 nm) devices. Solid lines and curves are the \fts to the experiments.\n\u0012H\u0000\u0012Iis \fxed to 45\u000e. (b)\u000bas a function of \u0012Hfor CoFe(10 nm)/Pt and CoFe(10 nm) devices.\nSpin-torque FMR for CoFe(10 nm)/Pt up to 39 GHz.\nFig. S-5 shows the spin-torque FMR lineshapes and linewidths up to 39 GHz for CoFe(10 nm)/Pt devices along\nthe easy and hard axes ( \u0012H= 90\u000eand45\u000e). At!=2\u0019= 32:1 GHz (Fig. S-5a), the spin-torque FMR amplitude is\n0.1\u0016V for the easy axis and 0.02 \u0016V for the hard axis. 10 seconds of time constant is used to obtained the signals.\nThroughout the frequency range, linewidths demonstrate good linear dependence on frequency as shown Fig. S-5(b).\nFor the hard axis the signal has reached the noise bottom limit at 32.1 GHz. For the easy axis the noise bottom limit\nis reached at 39 GHz. The two linear \fts yield \u000b= 0:0063 and\u00160\u0001H0= 1:8 mT for the easy axis and \u000b= 0:00153\nand\u00160\u0001H0= 1:5 mT for the hard axis. The two damping parameters are close to the values obtained below 20 GHz\nin the main text. Also the inhomogeneous linewidth \u00160\u0001H0nicely match between easy and hard axes.10\nFIG. S-5. High-frequency ST-FMR measurement of i) CoFe(10 nm)/Pt for the biasing \feld along the easy axis ( \u0012H= 90\u000e) and\nhard axis (\u0012H= 45\u000e). Left: lineshapes of ST-FMR at !=2\u0019= 32:1 GHz. Right: linewidth as a function of frequency. Lines\nare linear \fts to the data by setting both \u000band \u0001H0as free parameters.\nLow-temperature FMR linewidths and dampings for CoFe(10 nm)/Pt and CoFe(20 nm).\nFIG. S-6. (a-b) \u00160\u0001H1=2as a function of frequency for CoFe(10 nm)/Pt devices at di\u000berent temperatures. (c) Extracted\ndamping at di\u000berent temperatures, same as in Fig. 4 of the main text.\nFigure S-6 shows the frequency dependence of linewidths for extracting temperature-dependent Gilbert damping\nin Fig. 5(a) of the main text.\nFor CoFe(10 nm)/Pt samples, we plot both \u000band resistivity \u001ameasured at di\u000berent temperatures in Fig. S-6(c).\nThe measurements of \u001awere conducted with a biasing magnetic \feld of 1 Tesla parallel to the current direction, so\nthat the AMR in\ruence is excluded. Also the resistivity variation between the easy and hard axes is very small, about\n1%, which is much smaller than the damping anisotropy.\nWe have also conducted the low-temperature VNA FMR of the new CoFe(20 nm) samples at 8 K, in addition to\nthe room-temperature measurements. The linewidths data are shown in Fig. S-6(d) for both easy and hard axes.\nThe extracted damping are: \u000b= 0:0054 (EA, 300 K), 0.0061 (EA, 8 K), 0.0240 (HA, 300 K) and 0.0329 (HA, 8 K).\nThose values are used in Fig. 4(b) and Fig. 5(a) of the main text.\nFor CoFe(10 nm) the damping anisotropy decreases from 380 % at 300 K to 273 % at 30 K by taking out the spin\npumping damping enhancement (an unexpected reduction of alpha happens at 8 K for the hard axis). For CoFe(2011\nnm) the damping anisotropy increases from 440 % at 300 K to 540 % at 8 K. Thus a clear variation trend of damping\nanisotropy in CoFe \flms remains to be explored.\nFirst-principle calculation of N(EF)anisotropy for Co 50Fe50\nFIG. S-7. Density of states as a function of energy. EFis the Fermi level.\nFirst-principle calculations were done using QUANTUM ESPRESSO for a cubic lattice of Co 50Fe50of CsCl, Zintl\nand random alloy structures. Supercells consisting of 4 \u00024\u00024 unit cells were considered with a total of 128 atoms (64\ncobalt and 64 iron atoms). The calculations were done using plane-wave basis set with a 180 Ry kinetic energy cut-o\u000b\nand 1440 Ry density cut-o\u000b. For both Co and Fe atoms, fully relativistic PAW pseudopotentials were used. Figure\nS-7 shows the density of states (DOS) of the CsCl form for di\u000berent magnetization orientations \u0012in thexy-plane.\nClearly, DOS exhibits no anisotropy ( <0:1% variation at E=EF). No anisotropy was found in the Zintl form, either.\nThus, we conclude that the Gilbert damping anisotropy in Co 50Fe50cannot be caused by a variation of N(EF) with\nrespect to magnetization direction in ideal ordered structures.\nSOC induced by atomic short-range order (ASRO)\nIn our experiment, because the Co 50Fe50\flms were grown by MBE at low temperatures, they do not form the\nordered bcc B2 structure but instead exhibit compositional disorder. Transition metal alloys such as CoPt, NiFe, and\nCoFe tend to exhibit ASRO [2{4]. The ASRO in CoFe is likely to give rise to local tetragonal distortions because of the\ndi\u000berent lattice constants of bcc Fe and (metastable) bcc Co at 2.856 \u0017A and 2.82 \u0017A, respectively. Such local tetragonal\ndistortions will preserve global cubic (or four-fold in-plane) symmetry, but can have large e\u000bects on the SOC, with\nconcomitant e\u000bect on spin-orbit induced magnetization damping. For example, \frst-principle calculations using the\ncoherent-potential approximation for the substitutionally disordered system shows that a tetragonal distortion of 10%\nin the ratio of the tetragonal axes aandcgives rise to an magnetocrystalline anisotropy energy (MAE) density [2, 3]\nof about 1 MJ/m3. These results are consistent with our observed MAE in Co 50Fe50.\nTo con\frm this mechanism, we performed DFT-LDA calculations on 50:50 CoFe supercells consisting of a total\nof 16 atoms for CsCl, zintl, and random alloy structures; in the random alloy supercell, Co or Fe atoms randomly\noccupied the atomic positions in the supercell. Note that all CoFe geometries are fully relaxed, including supercell\nlattice vectors.\n1. Structural relaxation including spin-orbit coupling (SOC) shows local tetragonal distortions for random alloy\nsupercell. Among the three di\u000berent CoFe phases, tetragonal c/a ratio for the supercell in optimized geometry\nis largest (1.003) in the random alloy supercell with SOC, which means local tetragonal distortions are more12\nFIG. S-8. Density of states (DOS) for (a) CsCl, (b) Zintl, and (c) alloy form of CoFe with SOC (black solid) and without SOC\n(red solid).\nTABLE I. Relaxed atomic positions (including SOC) of the alloy structure. In the ideal CsCl or Zintl structures, the atomic\npositions are all multiples of 0.25 in units of the lattice vector components.\nAtom x-position y-position z-position\nCo 0.003783083 0.000000000 0.000000000\nFe -0.001339230 0.000000000 0.500000000\nFe -0.002327721 0.500000000 0.000000000\nFe 0.002079922 0.500000000 0.500000000\nFe 0.502327721 0.000000000 0.000000000\nFe 0.497920078 0.000000000 0.500000000\nCo 0.496216917 0.500000000 0.000000000\nFe 0.501339230 0.500000000 0.500000000\nCo 0.250000000 0.250000000 0.254117992\nFe 0.250000000 0.250000000 0.752628048\nFe 0.250000000 0.750000000 0.247371952\nCo 0.250000000 0.750000000 0.745882008\nCo 0.750000000 0.250000000 0.250415490\nCo 0.750000000 0.250000000 0.746688258\nCo 0.750000000 0.750000000 0.253311742\nCo 0.750000000 0.750000000 0.749584510\ndominant in random alloy compared to CsCl and Zintl structures. [c/a values : CsCl (0.999), Zintl (0.999),\nAlloy (1.003)]. In addition, the alloy system exhibited local distortions of Co and Fe position relative to their\nideal positions. In contrast, in CsCl and Zintl structures the Co and Fe atoms exhibited almost imperceptible\ndistortions. Table shows the relaxed atomic positions in the alloys structure in units of the lattice vectors. In\nthe ideal (unrelaxed) system, the positions are all at multiples of 0.25; the relaxed CsCl and Zintl structures no\ndeviations from these positions larger than 1 part in 106\n2. SOC changes the density of states (DOS) at the Fermi energy, notably for the random alloy but notfor the CsCl\nand Zintl structures. Figure S-8 shows DOS for (a) CsCl, (b) Zintl, and (c) random alloy structure with SOC\n(black lines) and without it red lines). We can see signi\fcant DOS di\u000berence for the random alloy supercell\nwith SOC where tetragonal distortions occurred, while almost no changes are observed in the CsCl and Zintl\nstructures.\n3. The local distortions in the alloy structure furthermore gave rise to an energy anisotropy with respect to the\nmagnetization direction. The energy (including SOC) of the relaxed alloy structure for di\u000berent directions of\nthe magnetization is shown in Fig. S-9. While the supercell was rather small, because of the computational\nexpense in relaxing the structure with SOC, so that no self-averaging can be inferred, the \fgure demonstrates\nan induced magnetic anisotropy that arises from the SOC and local distortions. No magnetic anisotropy was\ndiscernible in the CsCl and Zintl structures.\nAs a result from the DFT calculation, we attribute the large SOC e\u000bect in damping anisotropy of Co 50Fe50to local\ntetragonal distortions in disordered Co and Fe alloys. These distortions give rise to SOC-induced changes of DOS at\nthe Fermi level, as well as magnetic anisotropy energy with respect to the crystallographic axes.13\nFIG. S-9. Change in total energy (per supercell) of the alloy structure as function of the magnetization direction.\n\u0003wuyizheng@fudan.edu.cn\nyho\u000bmann@anl.gov\nzweizhang@oakland.edu\n[1] C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak, Y. Tserkovnyak, H. Kurebayashi, A. Brataas and A. Ferguson, Nature\nNano. 10, 50 (2015)\n[2] S. Razee, J. Staunton, B. Ginatempo, E. Bruno, and F. Pinski, Phys. Rev. B 64, 014411 (2001).\n[3] Y. Kota and A. Sakuma, Appl. Phys. Express 5, 113002 (2012).\n[4] I. Turek, J. Kudrnovsk\u0013 y, and K. Carva, Phys. Rev. B 86, 174430 (2012)."    },    {        "title": "1901.05753v1.Spin_transport_parameters_of_NbN_thin_films_characterised_by_spin_pumping_experiments.pdf",        "content": "1 \n  \nSpin transport parameters of NbN thin films  characterised by spin \npumping experiments  \nK. Rogdakis1,*, A. Sud1, M. Amado2, C. M. Lee2, L. McKenzie -Sell2, K.R. Jeon2, \nM. Cubukcu1, M. G. Blamire2, J. W. A. Robinson2, L. F. Cohen3, and H. Kurebayashi1,† \n 1London Centre for Nanotechnology, University College London, London WC1H 0AH, United \nKingdom  \n2Department of Materials Science & Metallurgy, University of Cambridge, Cambridge CB3 0FS, \nUnited Kingdom  \n3The Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom  \n \nAbstract  \nWe present  measurements of ferro magnetic -resonance - driven  spin pumping and  inverse spin -\nHall effect in NbN/ Y3Fe5O12 (YIG) bilayers . A clear enhancement  of the (effective) Gilbert \ndamping constant of the thin -film YIG was observed due to  the presence of the NbN  spin sink . \nBy varying the NbN thickness and em ploying spin-diffusion theory , we have estimated the  \nroom temperature  values of the spin diffusion length and the spin Hall angle in NbN  to be  14 \nnm and -1.1×10-2, respectively.  Furthermore, we have determined  the spin-mixing conductance \nof the NbN/YIG  interface to be 10 nm-2. The experimental quantification of these spin transport  \nparameters is an important step towards  the development of  superconducti ng spintronic devices \ninvolving NbN  thin films . \n \nIntroduction  \nThe extraction of key functional materials parameters associated with electron transport \nis important for the development of new solid -state device schemes as well as testing \nprototypes. In the field of spintronics, the spin Hall angle ( θSH) represents the strength of spin -\nHall effect  (SHE)  [1] that converts charge currents into spin currents via the relativistic spin -\norbit interaction. T he spin diffusion le ngth (𝑙𝑆𝐷) [2] is a parameter that describes  the distance \nover which non-equilibrium spin currents  can diffuse  before dissipation and is crucial in \ndetermining the useful device dimensions  of future spintronic applications. Moreover, t he spin \nangular momentum  transfer  across  a ferromagnetic (FM) and non -magnetic (NM) interfac e can \nbe parameterised by the spin mixing conductance  (𝑔𝑟↑↓) which governs the spin current \ngeneration efficiency in spin pumping  process es [3]. These s pin transport parameters can be 2 \n determined by employing  different  measurement techniques. For example, it is possible to use \nlateral spin -valves to quantify 𝑙𝑆𝐷  and θSH in non -magnetic materials [ 4, 5, 6, 7]. Spin pumping  \n[3, 8, 9] is another established method to investigate spin transport parameters in a variety of \nmaterials, such as metals  [10], inorganic  [11, 12] and organic semiconductors  [13, 14], graphene  \n[15] and topological insulators  [16]. It should be noted that spin pumping  relies on the transfer \nof angular momentum from a ferromagnet with precessing moments into an adjacent non -\nmagnetic layer , and do es not suffer from the conductance mismatch problem which causes \ndifficulties in electric al spin injection through ohmic contacts  [11].  Using  a FM conductor as \nspin injector in a spin pumping experiment can potentially give rise to microwave  (MW) -\ninduced photo -voltages  [17] due to time -varying resistance changes produced by the magnetic \nprecession coupled with a time -varying current, as  well as the ISHE in the FM layers  [18, 19]. \nThe use of FM insulators such as Y3Fe5O12 (YIG) to conduct spin pumping experiments has the \nadvantage because these effects are negate d. In addition , YIG has a low bulk Gilbert damping \nconstant (α ≃ 6.7 × 10−5) and a high Curie temperature ( TC = 560 K) , enabling efficient spin \npumping at  room temperature ( RT) [20].  \nIn this paper, we report spin pumping in thin -film YIG/NbN bilayers with the aim of \nextracting multiple spin transport parameters of NbN thin films  in the normal state . NbN  is a \nkey material for superconducting (SC) spintronics  [21] with a  bulk TC of approximately 16.5 \nK, a SC energy gap of 2.5  meV, and a SC coherence length of 5  nm [22]. NbN is increasingly \nused in the field of SC spintronics , for example  in spin -filter Josephson junctions  [23, 24, 25] \nand to demonstrate spectroscopic evidence for odd frequency (spin -triplet) superconductivity \nat the interface with GdN [26]. Recently , Wakamura et al.  observed an unprecedented \nenhancement of the SHE  at 2K, interpreted in terms of  quasiparticle mediated transport [ 27]. \nQuasiparticle spin transport has also been investigated by spin pumping and by monitoring the \nspin Seebeck effect [ 28, 29]. To the best of our knowledge, s pin trans port parameters in NbN  \nsuch as 𝑙𝑆𝐷 and θSH have only been extracted  by Wakamura  et al.  [27] by the spin absorption \nmethod in lateral spin -valves , and it is vitally important to extract  these  parameters also by other \ncharacterisation techniques  and with NbN  grown by different growth method s. This can, for \nexample, help to understand whether spin transport parameters in NbN have a significant \ndependence on  the growth conditions . In our study, b y using high-quality epitaxial thin-film \nYIG it is possible to obs erve a modulation of the Gilbert damping constant  (α) with NbN \nthickness and therefore  extract 𝑙𝑆𝐷 of NbN (14 nm) and 𝑔𝑟↑↓ of the YIG/NbN interface (1 0 nm-\n2). Furthermore, we have investigated the NbN -thickness -dependence of the ISHE  voltage \n(VISHE) and have determined θSH of NbN ( -1.1 ×10-2) by the spin pumping technique . We 3 \n compare 𝑙𝑠𝑑 extracted by three independent methods , namely the thickness dependence of α and \nVISHE as well as Hanle spin precession , and we find  good agreement between them . Determining \nthe normal -state spin -transport parameters in NbN from spin -pumpi ng-induced ISHE is \nimportant, which enables the comparison between  parameters extract ed using various  \ntechniques from  different research groups [e.g. 27-29]. By accumulation of a body  of results , \nwe will then be able to understand the fundamental nature of SHE  and spin transport in  NbN  \nwhich can be useful and transferable to future spintronics research using SC NbN [ 21, 30].  \n \nMaterial growth   \n          Epitaxial  YIG thin films are grown on (111) -oriented GGG single crystal substrate s by \npulse laser depositio n (PLD) in a n ultra-high vacuum chamber (UHV) with a base pressure  \nbetter than 5×10-7 mbar. Prior to film growth, the GGG substrate are ultrasonically cleaned by \nacetone and isopropyl alcohol and  annealed ex-situ at 1000 oC in a constant O2 flow \nenvironment for 8 hours. The YIG is deposited from a stoichiometric ( polycrystalline ) target \nusing a KrF excimer laser (248 nm wavelength) , with a nominal energy of 450 mJ  and fluence \nof 2.2 W cm-2 in 0.12 mbar of O 2 at 680  oC, and pulse frequency of 4 Hz for 60 minutes. The \nYIG is p ost-anneal ed at 750  oC for 1.5 hours in 0.5 mb ar partial pressure of static O 2 and \nsubsequently cooled to RT at a rate of -10 K/min. Atomic force microscopy (AFM ) reveal s that \na root -mean -squared roughness of the YIG films is  less than 0.16 nm over 10 ×10 µm scan size  \n[Fig. 1(a) ]. The YIG films were characterised by a SC quantum interference device (SQUID) \nmagnetometer and have a saturation magnetisation ( MS) of 140  3 emu cm-3 [Fig. 1(b) ], which  \nmatches the bulk value  [31]. In Fig. 1(c) we have plotted a high -angle X -ray diffraction trace \nof the same film where Laue fringes indicate layer -by-layer growth of YIG  and good lattice -\nmatching with the substrate. Figure 1(d ) shows  low-angle X -ray reflectivity from a  YIG film  \nand from the  decay and angle separation of the Kiessig fringes , we determined a  nominal \nthickness tYIG = 60  2 nm.  Following the growth  of YIG,  films were  directly  transferred in air \nto a UHV  sputter deposition system with a base pressure  of 1×10-9 mbar. NbN is grown by \nreactive sputtering in a gas mixture of argon (72%) and nitrogen (28%) with the deposition rate \nof 85 nm  min-1. The growth temperature is RT,  giving  polycrystalline NbN  layers . We grew \nNbN with different thicknesses (tNbN) from  5 to 50  nm.  \n \nFerromagnetic resonance ( FMR ) setup and spin pumping measurements  4 \n FMR is performed using a broadband coplanar waveguide  (CPW)  and ac-field \nmodulation  technique as illustrated i n Fig. 2(a). The sample s are placed face down on top of \nthe CPW s where an insulator tape is used for electrical insulation. We generate dc ( H) and ac \n(hac) magnetic fields by electromagnets  and the absorbed power  at the modulation frequency  is \nmeasured by a MW power detector and a lock -in amplifier while  H is swept . An input MW \npower  (PMW) of 100  mW is used unless otherwise is stated . We kept t he modulation field  \namplitude  (hac) smaller than the  measured FMR  linewidth s of all samples tested , in order to \navoid artefacts by strong  modulation . The magnetic field  is applied along different in -plane and \nout-of-plane directions  related to the samples as shown  in Fig. 2(a ). The  FMR absorption ( VP) \nwas measured using a MW power detector for different frequencies  typically ranging from 2 -\n12GHz  as depicted  in Fig. 2(b) (for a sample  with tNbN = 10 nm ). For each scan, the resonance \nfield ( Hres) and the  half-width -at-half-maximum  linewidth ( ΔH) of the FMR signal are \ndetermined  by a fit using differential forms of symmetric and anti -symmetric Lorentzian \nfunctions  (Appendix A) . Figure 2(c) shows the frequency dependence of the extracted Hres for \ndifferent NbN thicknesses. The curves of the frequency dependen ce for all samples, including \ntNbN  = 0 nm, overlap suggesting no significant modification of the YIG magnetic anisotrop y due \nto the presence of NbN. We note here that the effective magnetisation (Meff) extracted from the \nfits for each sample shows larger values than the Ms value measured in the SQUID. This \nenhanced Meff has often been observed in other thin -film studies [ 32, 33] and a detailed \nunderstanding of this lies outside of the scope of the present  work . For spin transport analysis \ndiscussed later, we use the values extracted by SQUID measurements since it is a more direct \nmeasurement of magnetisati on, while we confirmed that the  discrepancy between Ms and Meff \ndoes not alter the calculated spin tran sport parameters significantly. Although the magnetic \nanisotropies of the YIG films are unchanged with or without the presence of NbN, the \nmagnetization  relaxation of YIG represented by ΔH shows a clear dependence on tNbN as shown \nin Fig. 3( a). With a linear fit to  the data for each thickness using  ΔH =ΔH0 + (4πα/γ)f where \nΔH0 and γ describe  the inhomogeneous broadening and  the gyromagnetic ratio respectively, we \nhave quantif ied α for each sample  as shown  in Fig. 3(b) . α = (5.4 ± 0.2) × 10−4 was obtained for \nbare YIG, which  compare s well to  previously reported values  [34, 35]. A gradual increase of  α \nis observed  with increasing  NbN thickness , in agreement with spin pumping through the \nYIG/NbN interface  where  the α dependence with tNbN is given by  [36]:  \n𝛼(𝑡𝑁𝑏𝑁)=𝛼0+(𝑔𝐿𝜇𝐵𝑔𝑟↑↓\n4𝜋𝑀𝑠𝑡𝑌𝐼𝐺)∙[1+𝑔𝑟↑↓𝜌𝑁𝑏𝑁𝑙𝑠𝑑𝑒2\n2πℏ tanh(𝑡NbN\n𝑙𝑠𝑑)]−1\n (1). 5 \n Here,  α0  is the Gilbert damping constant for tNbN= 0 nm and the second term represents the \ndamping enhancement by spin pumping  into NbN ; 𝑔𝐿 is the free electron Land é factor  which is \nassumed equal to  2,  𝑔𝑟↑↓ is the effective real -part spin -mixing conduct ance across the NbN/YIG \ninterface;  𝜌𝑁𝑏𝑁 is the resistivity of NbN  which was  measured for each sample  [see inset of Fig . \n3(b)], and 𝑒 is the electron charge. A best f it of the data in Fig. 3(b) using  Eq. (1) yield s 𝑔𝑟↑↓ = \n10 ±2 nm-2 and 𝑙𝑠𝑑 =14 ± 3 nm. The extracted 𝑙𝑠𝑑 can be well compared with the value  (7 nm)  \nby Wakamura et al.  [27] using the spin -absorption method in lateral spin -valve devices . We \nalso found that the spin couplin g of NbN/YIG  is as good as  heavy-metals/YIG interfaces since  \n𝑔𝑟↑↓ is comparable  to those of  YIG/Pt, YIG/Ta and YIG/W  [35]. We note from analytic \ncalculations (Appendix B)  that the additional damping expected from eddy currents cannot \nexplain t he observed NbN  thickness  dependence of α. \nWe now discuss the  ISHE  voltage (VISHE) measurements . In Figs. 4(a) and 4(b) we show \ntypical data set s for VISHE (for direct comparison we present also corresponding Vp data) for tNbN \n= 20 nm and f = 3 GHz . Note that , since we use d the lock-in ac field-modulation  method for \nboth detections , the curves represent the derivative of the signals  without the ac field -\nmodulation : for both VP and VISHE a symmetric Lorentzian lineshape is expected without the ac \nfield modulation. As expected  from spin pumping and ISHE , we observe a clear VISHE peak at \nthe YIG precession  frequency . By changing the sign of H [observe the sign of magnetic field \naxis for Figs. 4 (a) and 4 (b)], we observe  a sign change of VISHE  in agreement with the symmetry \nof spin pumping  [37]. Corresponding measurements for tNbN = 5 nm are depicted in Figs. 4(c) \nand 4(d). By using the known ac field modulation amplitude as well as differential forms of \nsymmetric and anti -symmetric Lorentzian fu nctions (Appendix A), we quantify the peak \namplitude of ISHE voltage defined as 𝑉𝐼𝑆𝐻𝐸∗. The PMW-dependence of 𝑉𝐼𝑆𝐻𝐸∗ shown in Figs. 5 (a) \nand 5 (b) suggests that 𝑉𝐼𝑆𝐻𝐸∗ is proportional to  PMW, consistent with standard spin -pumping \ntheory  [36].  \nWe have  also performed H - angular dependen t measurements of V*ISHE along in -plane \nand out-of-plane directions of the NbN/YIG  films . The in -plane angular dependence  of the  spin \npumping experiment  follows  the expres sion  𝑉𝐼𝑆𝐻𝐸∗∝ 𝝐𝒙∙(𝑱𝐬×𝝈)∙|𝝈×𝒉𝒓𝒇| where the first \npart is due to  the ISHE  symmetry , 𝐸𝐼𝑆𝐻𝐸∝(𝑱𝐬×𝝈) , multiplied by the amplitude of magnetic \ntorque generated by MW-induced magnetic field  |𝝈×𝒉𝒓𝒇|; here, 𝝐𝒙 is the unit vector along x \ndirection in the measurement ’s framework shown in Fig. 2(a). The first component gives  a cos \ndependence  whereas the second produces a |cos| dependence,  which combined nicely matches \nour expe rimental results shown in Fig. 6 (a).  The rationale to plot 𝑉𝐼𝑆𝐻𝐸∗𝑡𝑁𝑏𝑁/𝜌𝑁𝑏𝑁 against 𝑡𝑁𝑏𝑁 6 \n is to include the thickness dependence of 𝜌𝑁𝑏𝑁 allowing to fit the data points based on bare  \nNbN as well as those of the YIG/NbN bi-layers . In addition, this analysis can display the \nasymptotic behaviour of the data/fit -curves towards  the long thickness limit. The in -plane \nsymmetry re -confirm s that spin rectification effects are not a dominant mechanism in our \nmeasurements since in this case  a higher order sin  2θ component is expected in the voltage \nsymmetry  [17]. We also  measure d the out -of-plane angular dependence of 𝑉𝐼𝑆𝐻𝐸∗ as shown in \nFig. 6 (b) and moreover we applied the Hanle precession model  [38] to fit our data . In this case \nthe out -of-plane 𝑉𝐼𝑆𝐻𝐸∗  is given by:  \n𝑉𝐼𝑆𝐻𝐸∗(𝜙)∝{cos(𝜙)∙cos(𝜙−𝜙𝑀)+sin𝜙∙sin(𝜙−𝜙𝑀)∙[1\n1+(𝜔𝐿∙𝜏𝑠)2]}    (2) \n𝜔𝐿=𝑔𝐿𝜇𝐵∙(𝜇0𝐻)/ℏ is the Larmor frequency  and 𝜏𝑠 is the  spin relaxation time  in NbN ;  𝜙 and \n𝜙𝑀 represent the angle of between  the z -axis and  H and the equilibrium magnetic moment \ndirection, respectively. By minimizing the total magnetic energy of the FM layer consisting of \nthe Zeeman and demagnetization energy, the following equation is derived to determine the \nvalue of 𝜙𝑀 with respect to 𝜙: 𝜙𝑀=\n𝜙−arctan\n[    \nsgn(𝜙).√(cos(2𝜙)+(𝜇0𝐻𝑟𝑒𝑠\n𝜇0𝑀𝑒𝑓𝑓)\nsin(2𝜙))2\n+1−(cos(2𝜙)+(𝜇0𝐻𝑟𝑒𝑠\n𝜇0𝑀𝑒𝑓𝑓)\nsin(2𝜙))\n]    \n [39]. After spin currents are \ninjected  inside NbN,  they start precessing due to  the external ly applied  H. This is described by \nthe well -known  Hanle  precession model which is the basis of Eq. (2). The equilibrium spin \norientation depends on the precession rate ( 𝜔𝐿) and the spin relaxation rate (1/ 𝜏𝑠), both of which \ncontribute  in the equation. When 𝜏𝑠 is much shorter than 1/ 𝜔𝐿, the injected spins do not precess \nand instead generate 𝑉𝐼𝑆𝐻𝐸 with spin orientation along M (𝜙M). This is the case for the red curve \nin Fig. 6(b). In the opposite extreme condition ( depicted as blue curve in Fig. 6(b)), spins \nprecess many times and dephase along the H orientation (𝜙), resulting in an approximately  \ncos(𝜙) angle dependence . Fitting the data in Fig. 6(b) using  Eq. (2) allows us to  estimate 𝜏𝑠. In \nparticular, the  best fit of the measured  𝑉𝐼𝑆𝐻𝐸∗(𝜙) was obtained  giving  an extracted  𝜏𝑠 = 11 ± 2 \nps. This value quantified by the Hanle model can be compared with 𝜏𝑠 independently calculated \nfrom the spin diffusion model as already discussed above, i.e. 𝜏𝑠 =(𝑙𝑠𝑑)2/𝐷 where  D is the \nEinstein diffusion coefficient (its  value equal to  0.4−0.56 cm2/s was taken from Ref. [ 40]). \nFollowing this approach and by using 𝑙𝑠𝑑=14 nm as extracted from the thickness dependence \nof damping modulation , we calculated 𝜏𝑠 = 3.6-5.9 ps which is  a fair agreement between the \ntwo different  𝜏𝑠 extraction methods .  7 \n In the following section,  the 𝜃SH of NbN is determined from the thickness dependence \nof 𝑉𝐼𝑆𝐻𝐸∗ as shown in Fig. 7. Using the spin transport parameters discussed above  and Eq. (3) , \nwe estimate the  spin current emitted at the NbN/ YIG interface,  js, as well as  the value of   𝜃SH \nextracted by fitting the thickness dependence  of 𝑉𝐼𝑆𝐻𝐸∗ [39]: \n𝑉𝐼𝑆𝐻𝐸∗=(𝑤𝑦𝜌𝑁𝑏𝑁\n𝑡𝑁𝑏𝑁)∙𝜃SH𝑙𝑠𝑑∙tanh(𝑡𝑁𝑏𝑁\n2𝑙𝑠𝑑)∙𝑗𝑠                                          (3) \nwhere 𝑗𝑠=(𝐺𝑟↑↓ℏ\n8𝜋)∙(𝜇0ℎ𝑟𝑓𝛾\nα)2\n∙[𝜇0𝑀𝑠𝛾 + √(𝜇0𝑀𝑠𝛾)2 + 16(𝜋𝑓)2 \n(𝜇0𝑀𝑠𝛾)2 + 16(𝜋𝑓)2]∙(2𝑒\nℏ) \nwith   𝐺𝑟↑↓≡𝑔𝑟↑↓∙[1+𝑔𝑟↑↓𝜌𝑁𝑏𝑁𝑙𝑠𝑑𝑒2\n2πℏ tanh(𝑡𝑁𝑏𝑁\n𝑙𝑠𝑑)]−1\n. \nHere w e assume that  YIG is a perfect insulator ; 𝜇0ℎ𝑟𝑓 is the amplitude of MW magnetic field \n(56 µT for 100 mW);  𝑤𝑦 is defined by the width of MW transmission line . For the data fitting \nprocedure  we use 𝜃SH and 𝑙𝑠𝑑 as free parameters , where the best fitting was achieved for 1.1 \n×10-2 and 14 nm, respectively. We also confirm ed the sign of 𝜃SH to be negative by comparing \nYIG/NbN data with a YIG/Pt control sample where Pt is known to have a positive 𝜃SH [1]. We \nemphasise that  the value of  𝑙𝑠𝑑 extracted by the thickness dependence of 𝑉𝐼𝑆𝐻𝐸∗ agrees very well \nwith the one extracted from the thickness dependence  of damping . The former approach \nincludes spin -orbit and spin -transport properties of NbN, whereas the latter is purely related \nwith magnetic propert ies of YIG . We found that t he value we extract by our spin pumping \nexperiments is similar to  θSH quantified by Wakamura et al.  using lateral spin -valve samples \n(θSH ~-1× 10-2) [27] for the temperature region between 20 to 200 K.  Although there is \ndifference in temperature between experiments by Wakamura et al.  and ours, an agreement of \nthe same sign and magnitude in  θSH quantified by different techniques ( i.e. spin pumping and \nspin-absorption) has been observed. T he value of θSH of the same material but grown and \nmeasured by different research groups can vary rather significantly , for example as  in the case s \nof Pt  [41] and some topological insul ators [ 42, 43, 44]. Such differences might result from \nvariation in  sample qualit y where the density of scattering impurities can particularly  influence  \nθSH via the extrinsic spin-Hall mechanisms  [1]. We note that the resistivity of NbN used in  the \nWakamura  et al.  study measured at 20 K  (220 μΩcm ) is roughly three times greater than our \nNbN  films at the same temperature (65  μΩcm ). This highlights  that the resistivity and mobility \nof NbN might be  highly growth -dependent, possibly due to the stoichiometry of Nb and N  as \nwell as the nitrogen vacancy concentration . The NbN spin-Hall resistivity of Wakamura  et al.  \nis 2.2 μΩ∙cm at 20 K  [27], whereas  our spin-Hall resistivity  at RT is calculated to be  0.5 μΩ∙cm \nwhich is smaller owing to the resistivity difference . For the relevance of SC spintronics, w e also 8 \n compare our θSH value with those of Nb thin films  reported in previous works . Morota et al.  \nmeasured θSH of several 4d and 5d transition metals by the spin absorption method in the lateral \nspin valve structures  [6] including Nb. They quantified θSH of Nb to be  -8.7 ×10-3 at 10K, which \nis close  to our  θSH in NbN at RT. There is recent work by Jeon et al.   who measured  θSH = -\n1×10-3 in Nb at RT  [39]. Direct comparison between θSH of Nb and NbN is not possible but they \nare within the same order, suggesting that there are similar atomistic spin-orbit contributions \nfrom Nb atoms both for Nb and NbN. Details of this will be further clarified when more realistic \ntheoretical studies  of SHE  in NbN become available .   \nAs a final remark, we also performed FMR measurements as a fun ction of temperature  to \ndetermining the low -temperature spin -pumping properties of NbN through the SC Tc. However, \na significant increase of magnetic damping was observed as the temperature was lowered  (this \nbehaviour is summarised in Appendix  C). This enhanced damping complica ted the \ninvestigation of VISHE across the SC Tc. \n \nConclusions  \nWe determined the spin transport parameters  of polycrystalline NbN thin -films  by the spin \npumping technique  using epitaxial YIG thin-films  at RT . We observe a modification of  the YIG \nGilbert damping  param eter as a function of the variation of the  NbN film thickness, confirming \nspin current injection  in the NbN  layer . By applying a spin-diffusion model , we have  estimate d \n𝑙𝑠𝑑 =14 ± 3 nm in NbN and 𝑔𝑟↑↓ = 10 ±2 nm-2 at the NbN/YIG interface . From the  NbN  thickness \ndependence of the ISHE  voltages , we determine  θSH to be equal to  -1.1 ×10-2. We also compare \n𝑙𝑠𝑑 of NbN extracted by  three different techniques (thickness dependence of both α and 𝑉ISHE \nas well as the Hanle measurements) and found good agreement between them . The measured \nparameters are a good reference to understand the NbN spin -orbit and spin transport properties  \nand to aid the design of  feasible spintronic experiments/ devices in the normal and  SC state.  \nAcknowledgment  This work was supported by the Engineering and Physical Sciences \nResearch Council  through the  Programme Grant “Superspin” ( Grant No. EP/N017242/ 1) and \nInternational Network Grant ( Grant No. EP/P026311/1 ). \n \nAppendix A: Derivation of FMR fit curves  \nIn normal dc FMR analysis, the measured dc voltage can be decomposed into symmetric and \nanti-symmetric Lorentzian functions with respect to μ0Hres, with weights of Asym and Aasy \nrespectively , where combined lead to the following general power absorption expression  \n[which is applicable both for  FMR absorption (V p) and ISHE voltage (V ISHE)]:  9 \n 𝑃𝑑𝑐(𝐻)=𝐴𝑠𝑦𝑚(𝐻)+𝐴𝑎𝑠𝑦(𝐻)+𝑉0=𝐴𝑠𝑦𝑚∆𝐻2\n(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2+𝐴𝑎𝑠𝑦∆𝐻(𝐻−𝐻𝑟𝑒𝑠)\n(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2+𝑉0,       (4)                                  \nwhere V0 is a background voltage. The first term gives  the symmetric lineshape and the second \nterm produces the anti -symmetric one. For FMR mea surements based on a c magnetic -field \nmodulation, where an additional pair of coils on electromagnets provide small ac magnetic field, \nPac has the following relationship with Pdc. \n𝑃𝑎𝑐=𝑑𝑃𝑑𝑐\n𝑑𝐻ℎ𝑎𝑐                                                          (5) \nwhere, hac is the amplitude o f ac magnetic field modulation. With these two equations, we can \ncalculate 𝑃𝑎𝑐 as: \n𝑃𝑎𝑐(𝐻)=−𝐴𝑠𝑦𝑚ℎ𝑎𝑐2(𝐻−𝐻𝑟𝑒𝑠)∆𝐻2\n{(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2}2−𝐴𝑎𝑠𝑦ℎ𝑎𝑐∆𝐻{(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2}\n{(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2}2                  (6) \nThis equation was used to fit the ac field modulated signals, bot h Vp and VISHE, in our study. \nThe first term gives now the anti -symmetric lineshape and the second term pr oduces the \ndistorted symmetric one . Figure 8  (a) and (b) display typical FMR  data together with best fit \ncurves using Eq. ( 4) and (6 ), respectively, with corresponding extracted parameters presented \nin Fig 8 as legends . We also checked that there was no experimental artifact by doing our ac \nexperiments, by directly confirming that  ac (Fig. 8a) and dc (Fig. 8b) measurements for the \nsame experimental conditions generate the same fit parameters.  \n \nAppendix B: A simpl ified model for the eddy -current damping  \nWe consider a slab of magnet containing a  chain of distributed magnetic moments m as shown \nin Fig 9 (a).  In order to model the eddy -current damping in NbN, we first calculate the magnetic \nflux at point P where the distance between the point and the slab is x (Fig 9a). We can estimate \nthe magnetic field at point P generated by a moment at (0, y) using the Biot -Savart law, as: \n𝐵=𝜇0\n4𝜋𝑚\n(𝑥2+𝑦2)3/2                                                                  (7) \nwhere 𝜇0 is the permeability of free space. We assume that  the length of the chain is infinitely \nlong, which is a valid assumption by taking in consideration that the film thickness is much \nshorter than the sample lateral dimensions. By integrating the contribution of the individual \nmoments, we calculate the tota l magnetic field  𝐵𝑡𝑜𝑡𝑎𝑙 as: 10 \n 𝐵𝑡𝑜𝑡𝑎𝑙= 2∫𝜇0\n4𝜋𝑚\n(𝑥2+𝑦2)3/2𝑑𝑦∞\n0=𝜇0\n2𝜋𝑚\n𝑥2                                            (8) \nUsing  this 𝐵𝑡𝑜𝑡𝑎𝑙 expression within this quasi -2D picture, we can calculate the magnetic flux Φ \nat point P. By definition, Φ  = ∬𝐵𝑡𝑜𝑡𝑎𝑙𝑑𝑠 , where the integration surface is defined by the  \nthickness 𝑡𝑁𝑏𝑁 and the width w of the NbN film. This reads : \nΦ = ∬𝐵𝑡𝑜𝑡𝑎𝑙𝑑𝑠=𝑤×∫𝜇0\n2𝜋𝑚\n𝑥2𝑡𝑌𝐼𝐺\n2+𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺/2𝑑𝑥=𝜇0𝑤𝑚\n𝜋𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺(𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁)                         (9) \nFor the definition of the integration region,  we assume that the chain of the magnetic moments \nis locate d at the centre of the YIG film . \nAfter estimating the magnetic flux, we can calculate t he radiative dissipation power P as: \n𝑃=𝜔\n2𝑍𝑁𝑏𝑁 Φ2=𝜔\n2𝑍𝑁𝑏𝑁( 𝜇0𝑤𝑚\n𝜋𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺(𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁))2\n                                   (10) \nHere 𝑍𝑁𝑏𝑁 is the impedance of the NbN film and for simplification  we assume that the  real part  \n(resistance)  dominates, meaning that 𝑍𝑁𝑏𝑁≈𝑅𝑁𝑏𝑁=𝜌𝑁𝑏𝑁(𝑑/𝑤𝑡𝑁𝑏𝑁). Using the total non -\nequilibrium magnon energy generated during the experiments as ħ𝜔𝑁𝑉  (here, 𝑁 is the number \nof the non -equilibrium magnons and V is the volume of YIG), we can express t he rate of energy \ndissipation being:  \n1\n𝜏=𝑃\n𝐸=𝜔 𝑤𝑡𝑁𝑏𝑁\n2𝜌𝑁𝑏𝑁𝑑ħ𝜔𝑁( 𝜇0𝑤𝑚\n𝜋𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺(𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁))2\n                                       (11) \nFinally, the damping component caused by eddy currents generated by the time -dependent flux \nchange can be given by:  \n    𝛼𝑒𝑑𝑑𝑦=1\n2𝜔(1/𝜏)=𝑤𝑡𝑁𝑏𝑁\n4𝜌𝑁𝑏𝑁𝑑ħ𝜔𝑁𝑉( 𝜇0𝑤𝑚\n𝜋𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺(𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁))2\n                           (12) \nAs this model is a simplified one, we only discuss 𝛼𝑒𝑑𝑑𝑦 qualitatively. In particular, we c an \nextract the NbN thickness dependence of 𝛼𝑒𝑑𝑑𝑦 by using this expression and find that it is \nproportional to ( 𝑡𝑁𝑏𝑁3/2\n𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁)2\n. We plot the dependence in F ig. 9 (b) which indicates  that the \ndamping based on this mechanism should monotonically increase with thickness. However,  this \ntrend is different from what we experimentally observed, where 𝛼 becomes constant for the \nlarger thickness limit. This suggest that the damping mechanism through the eddy current in \nthe NbN layers is no t significant and can be neglected for the examined NbN thicknesses . \nMoreover, in the  work by Flovik et al.  [45] they discuss the  eddy current effect on the lineshape \nof the FMR spectrum. They show ed that when  eddy currents exist in an FM/ NM bi-layer, the 11 \n FMR lineshape can be significantly affected, varying from a pure symmetric shape to a mix ture \nof symmetric and anti -symmetric ones. Experimentally, we have not observed strong 𝐴𝑎𝑠𝑦 \ncomponent, suggesting that the eddy current in our NbN film does not pla y a significant role in \nour measurements. In addition, similar eddy current and radiative damping mechanisms has \nalso been discussed by Schoen et al.  [46]. They demonstrated that when their sample is placed \n100 μm away from the waveguide, radiative damping with the waveguide is largely supressed. \nSince we also inserted an insulating tape be tween our samples and the waveguide , we believe \nthat the radiative damping is minor  in our experiments. Furthermore, Qaid et al.  [47] reported \nthat although eddy -current da mping can be observed in a weak spin -orbit material ( in their case \na conducting polymer) , this is not the case for a high spin orbit metal (Pt). For instance, they \nshowed that the damping enhancement  in a YIG/Pt structure can still be dominated by the spin-\npumping effect in Pt. Since our NbN is a suffici ently high spin -orbit material, w e believe that \nthe eddy -current component is much smaller (an order of magnitude at least) than that of spin -\npumping into NbN.  \n \nAppendix  C: Low temperature measurements of spin pumping in NbN/YIG samples  \nIt is widely reported that  YIG thin -films tend to show signif icant temperature dependent \nmagnetic damping  [32, 33 , 48, 49], where the superb damping character at RT is  lost when the \nfilms are cooled to lower  temperatures. The origin of this remains  under debate but enhanced \nlow temperature two -magnon scattering (due to interfac ial defects  in ultrathin films) [32] in \ncombination with  rare-earth or Fe2+ impurity scattering [ 50, 51] are likely mechanisms . Jermain \net al.  [33] discuss  that, if the FMR linewidth has a peaked temperature -dependence that \ndominates over the  proportionality expected with MS(T) increase , impurity scattering is the \nmore  likely  mechanism. Although  the nature of the impurities remains ambiguous, other reports \nof the high frequency characterisation of PLD -grown and sputtered YIG thin film s have pointed \nout the likely significan ce of Gd3+ diffusion from the GGG substrate [ 52, 53, 54]. \nOur own extensive FMR measurements of bare YIG on GGG ( of comparable \nthicknesses)  [55] show that, when Gd3+ impurities are concentrate d in a thin (1 -5 nm) layer near \nthe substrate  interface , they form a  ferromagnetic sublattice that, as its moment increases at low \ntemperatures, opposes the net YIG magnetisation  [50, 56], and also introduces magnetic \ndisorder and additional damping channels that dominate the film’s FMR response . \nHere  we describe the  low-temperature characterisatio ns of our YIG/NbN samples . \nFigure 10 summarises both FMR absorption spectra and ISHE voltages as a function of 12 \n temperature for the sample with NbN thickness of 10 nm. With decreasing temperature, there \nis a clear increase of  ΔH, leading to a corresponding reduction of the FMR absorption signal , \nas shown in Fig. 10(a). The FMR spectrum at 3K can be extracted by taking multiple scans to \nimprove the signal to noise ratio through data averaging.   Figure 10(b) shows  that ΔH increases \nby a factor of 5 between 300K and 3K, with a steep enhancement below 100 K.  For direct \ncomparison we present data in Fig. 10(b) both of  YIG/NbN (black points) and bare YIG samples \n(red points). It is clear that linewidth enhancement at low tempe ratures is due to YIG. In \ncomparison with the previous low temperature FMR studies on YIG, we can detect an FMR \nsignal down to 3K in the  MW transmission geometry, possibly owing to a relativ ely thick film . \nUnfortunately, the  ΔH enhancement significantly hindered our ISHE detection plotted in Fig s. \n10(c) and (d). The voltage peak is comparable or below the noise level at 50 K and it was not \npossible to investigate the evolution of VISHE across the Tc of NbN  which is 11 K for the 10 nm \nfilm, measured by the four point dc resistance R4p shown in the inset of Fig. 10(d). To study the \nspin transport properties in NbN across Tc, by spin -pumping technique , would require an  \nimprovement of YIG thin -film quality to overcome the observed  ΔH enhancement. We note \nthat a very recent work by Umeda et al.  exploited the spin -Seebeck effe ct as a spin -injection \nmethod, demonstrating an interesting coherent peak in spin-Seebeck coefficient related to  the \nquasi -particle spin transport [ 29].   \n \n*rogdakis7@gmail.com  \n†h.kurebayashi@ucl.ac.uk  \n \nReferences  \n[1] J. Sinova , S. O. Valenzuela, J. Wunderlich, C.H. Back, and T. Jungwirth, Rev. Mod. Phys. \n87, 1213 (2015).  \n[2] J. Bass and W.  P. Pratt Jr, J. Phys.: Condens. Matter 19 183201 (2007).  \n[3] Y. Tserkovnyak, A. 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Gomez -Perez, et al., arXiv:1803.05545 . \n[55] L. McKenzie -Sell, M. Amado, G. Kimbell, G. Divitini, C. Ciccarelli, and J. W. A. \nRobinson, to be submitted (n.d.).  \n[56] T. Yamagishi, J. Awaka, Y. Kawas hima, M. Uemura, S. Ebisu, S. Chikazawa, and S. \nNagata, Philos. Mag. 85, 1819 (2005).  \n \nFigure captions  \nFIG.1: Structural and magnetic properties of a bare (111) -oriented YIG film (nominally 60 -nm-\nthick) used in this work and deposited onto GGG. (a) 10×10 µm2 AFM topography scan \nshowing a root -mean -square roughness of less than 0.16  nm. (b) Magnetization hysteresis loops \ncharacterised by a superconducting quantum interference device magnetometer show ing a \nsaturation volume magnetization of 140   3 emu  cm-3. (c) High angle X -ray diffraction data \ndemonstrating (111) orientation with visible Laue fringes on the (444) and (888) diffraction \npeaks characteristic of layer -by-layer growth. (d) Low angle X -ray reflectometry data (black) \nwith a best -fit (red) curve fr om which we estimate a nominal thickness of 60   2 nm.  \n \n \nFIG.2: (a) A schematic of the spin -pumping setup. The lateral area of all samples is 5x5mm2. \nMW magnetic fields ( hrf) were generated by the transmission line to generate magnetic \ndynamics in the YIG  film. Spin currents ( js) were emitted at the YIG/NbN interface, which can \ninduce ISHE voltages detected through the two electrodes attached to the edges of the sample. \nWe simultaneously measured the FMR absorption signal as a voltage in a microwave power \nmeter ( VP) connected to the microwave line and the ISHE signal ( VISHE) using two lock -in \namplifiers. (b) FMR absorption measurements for different MW frequencies. (b) FMR \nabsorption measurements for different MW frequencies. Voltages in our MW power detect or \nwere measured while magnetic fields were swept. Dots in red, green, blue, cyan, pink, yellow \nand black represents measurement results for 3, 4, 5, 6, 8, 10 and 12 GHz respectively. (c) A \nplot of frequency versus FMR field ( Hres) for samples with differe nt NbN thicknesses. Dots \nrepresent experimental results and curves are produced by fitting using the Kittel formula.  \n \n \nFIG.3: (a) Frequency dependence of FMR linewidth of YIG/NbN samples with different NbN \nthicknesses. Experimental data (filled points ) is fitted by a linear line ΔH =ΔH 0 + (4πα/γ) f, 15 \n                                                                                                                                                                                               \nwhere ΔH 0 and γ describe the inhomogeneous broadening and the gyromagnetic ratio \nrespectively, from which the Gilbert damping coefficient , α, is extracted . (b) Plots of α for \ndifferent YIG/NbN  samples. Equation (1) was used to fit to the thickness dependence with the \nspin-diffusion length and the real part of mixing conductance as fitting parameters. The inset \ndepicts  the resistivity as a function of NbN thickness.  \n \n \nFIG.4: ISHE  measurements. Simultaneous measurements of FMR absorption and ISHE \nvoltages for positive (a) and negative (b) magnetic field values for a tNBN = 20 nm sample. \nCorresponding data for tNBN = 5 nm are depicted in (c) and (d), respectively. Both VP and VISHE \npeaks appear at the same magnetic field, confirming that the voltages were generated when YIG \nmagnetic moments were preccessing. The sign change in voltage peaks observed between the \npositive and negative magnetic field regions is consistent with the spin -pumping/ISHE picture.  \n \nFIG.5: Microwave power dependent measurements. (a) ISHE voltage measurements with \ndifferent insertion powers ( PMW). (b) A plot of ISHE voltage peak to peak amplitude ( V*ISHE) \nas a function of PMW. VISHE scales with PMW as expected from the spin pumping theory in the \nlinear regime.  \n  \nFIG.6: In -plane (a) and out -of-plane (b) angular dependences of VISHE signal peak to peak \namplitude ( V*ISHE). Fit curves in both angular dependences are discussed in the main text. We \nshow f it curves with four different spin -relaxation time ( 𝜏𝑠) in (b) to illustrate how the model \ncurve changes with 𝜏𝑠.  The best fit curve was produced with 𝜏𝑠 = 11 ± 2 ps. We define three \nangles ( ϕ, ϕM, θ) as depicted in Figure’s insets.  \n  \nFIG.7: 𝑉ISHE𝑡NbN/𝜌NbN as a function of NbN thickness. We plot 𝑉ISHE𝑡NbN/𝜌NbN to normalise  \n𝑉ISHE  with NbN thickness and resistivity. By using Eq. (3) in the main text, we extract the spin -\nHall angle ( θSH) and spin -diffusion length ( 𝑙𝑠𝑑) of NbN to be  1.1 ×10-2 and 14 nm . The best fit \ncurve is shown along with the experimental data.  \n \nFIG.8: Comparison of (a) ac and (b) dc  VP measurements. The extracted parameters using \nEquations in Appendix A for each measurement method are depicted in the legends of the \nfigur es. We can confirm that the extracted values are almost the same for both measurements.  \nFIG.9: Eddy -current damping contribution. (a) A schematic of our model for the eddy -current \ndamping. A chain of Magnetic moments (red arrow) lines up along the y direct ion and we \nconsider the magnetic field at Point P (x, 0) . (b) A plot of calculated eddy -current damping as \na function of the NbN thickness. The unit of the eddy -current damping is arbitrary in order to \ndiscuss them qualitatively. The thickness dependence i s clearly different from our experimental \nresults in Fig. 3(b), suggesting that this damping mechanism is not significant in our \nexperiments.  \n \nFIG.10: FMR absorption spectra and ISHE voltages as a function of temperature for tNbN=10 \nnm sample. (a) FMR abso rption spectra measured at 3 GHz, with temperature ranging from \n260K to 3K. (b) Linewidth evolution with temperature for the 3 GHz measurements. Black data \ncorresponds to an YIG/NbN sample and red data to a bare YIG sample. (c) ISHE voltages \nmeasured at 3 GHz for the temperature region of 50K -300K. We confirm that the peak height \nis below the signal -to-noise ratio around 50 K. (d) The normalised ISHE voltage amplitude as \na function of temperature. The inset represents our four point probe measurements of Nb N \nresistivity (for tNbN = 10 nm).  16 \n                                                                                                                                                                                               \n(a) (b) (c) (d)Figure 1:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 17 \n                                                                                                                                                                                               \nFigure 2:\n(a)\n-400 -200 0 200 400-2-1012\n \n  \nH (mT)VP (mV)\nf=2GHz\nf=12GHz\nISHE(b)\n(c)\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 18 \n                                                                                                                                                                                               \n(a)Figure 3:\n0 5 10 150.00.51.01.5\n 0nm \n 5nm \n 10nm\n 15nm  \n 20nm  \n 30nm  \n 50nm  H (mT)\nf (GHz)\n(b)\n0 10 20 30 40 500.00040.00060.00080.00100.00120.0014\na\ntNbN(nm) Data\n gr = 10.44 nm-2 , lsd = 14.46 nm \n gr = 8 nm-2 , lsd = 14 nm\n gr = 13 nm-2, lsd = 18 nm\n gr = 22 nm-2 , lsd = 20 nm\n gr = 9 nm-2 , lsd = 16 nm\n0 10 20 30 40 5030405060 ( - cm)\ntNbN (nm)\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 19 \n                                                                                                                                                                                               \n(a)\n-0.40.00.4\n35 40 45 50 550.450.600.75 VP (mV)20nm\nT=300K\nf = 3GHz VISHE (V)\nH (mT) (b)Figure 4:\n-101\n35 40 45 50 5501T=300K\nf=3GHzVP (mV)5nmVISHE (V)\n 0H (mT)\n-101\n-55 -50 -45 -4001VP (mV)5nmVISHE  (V)\n0H (mT)T=300K\nf=3GHz\n-0.40.00.4\n-55 -50 -45 -40 -350.450.600.75T = 300K VP (mV)20nm\nf = 3GHzVlSHE  (V)\nH (mT)\n(c) (d)\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 20 \n                                                                                                                                                                                               \n(a)\n40 45 500.40.60.8 VISHE (V)\nH (mT) 8mW\n 18mW\n 32mW\n 56mW\n 100mW\n 178mW 300K \n3GHz(b)\n0 50 100 150 2000.00.10.20.30.40.5V*\nISHE (V)\nP\nMW (mW) 300K\n3GHzFigure 5:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 21 \n                                                                                                                                                                                               \n(b)Figure 6:\n(a)\n0 90 180 270 360-1.0-0.50.00.51.0\n  \n 10nm \n  Fit\n 20nm \n  FitV*\nISHE (V)\n (deg)\n90 75 60 45 30 15 00.00.51.0V*\nISHE()/V*=\nISHE (-)\n (deg) VISHE/V=\nISHE\n s = 11 ps\n s = 53 ps ( s = 1)\n s << 1/ \n s >> 1/ \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 22 \n                                                                                                                                                                                               \n0 10 20 30 40 500123\ntNbN (nm) V*\nISHEtNbN/NbN (A)\n data\n fitFigure 7:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 23 \n                                                                                                                                                                                               \nFigure 8\n(a)(b)\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 24 \n                                                                                                                                                                                               \nFigure 9:\n(b)\n (a)\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 25 \n                                                                                                                                                                                               \nFigure 10:\n(a)\n(c)(b)\n(d)\n50 100 150 200 250 3000.00.20.40.60.81.01.2V*\nISHE/ V*300 K\nISHE\nT (K)\n20 30 40-3-2-10123\n  \n30K\n3K210K\n170KVP (mV)\n H (mT)260K\n100K\n(x10)\n20 30 40-3.0-1.50.01.53.0\n  \n100K170K210K260K\n50KVISHE(V)\nH (mT)75K\n10 1000510152025R4p (ohm)\nT (K)\n0 50 100 150 200 250 3000.51.01.52.02.53.0\n0H (mT)\nT (K)\n "    },    {        "title": "1901.08358v4.Generalization_of_Stokes_Einstein_relation_to_coordinate_dependent_damping_and_diffusivity__An_apparent_conflict.pdf",        "content": "arXiv:1901.08358v4  [cond-mat.stat-mech]  18 Sep 2019Generalization of Stokes-Einstein relation to coordinate dependent damping and\ndiffusivity: An apparent conflict\nA. Bhattacharyay1,∗\n1Indian Institute of Science Education and Research, Pune, I ndia\nBrownian motion with coordinate dependent damping and diffu sivity is ubiquitous. Understand-\ning equilibrium of a Brownian particle with coordinate depe ndent diffusion and damping is a con-\ntentious area. In this paper, we present an alternative appr oach based on already established\nmethods to this problem. We solve for the equilibrium distri bution of the over-damped dynamics\nusing Kramers-Moyal expansion. We compare this with the ove r-damped limit of the generalized\nMaxwell-Boltzmann distribution. We show that the equipart ition of energy helps recover the Stokes-\nEinstein relation at constant diffusivity and damping of the homogeneous space. However, we also\nshow that, there exists no homogeneous limit of coordinate d ependent diffusivity and damping with\nrespect tothe applicability of Stokes-Einstein relation w hen it does not hold locally. In the other sce-\nnario where the Stokes-Einstein relation holds locally, on e needs to impose a restriction on the local\nmaximum velocity of the Brownian particle to make the modifie d Maxwell-Boltzmann distribution\ncoincide with the modified Boltzmann distribution in the ove r-damped limit.\nPACS numbers: 05.40.Jc, 05.10,Gg, 05.70.-a\nINTRODUCTION\nDiffusion shows a lot of variety. Normal Fickian dif-\nfusion is characterized by a mean square displacement\n(MSD) which scales linearly with time and the corre-\nsponding probabilitydistribution of position is Gaussian.\nDeviation of the MSD from this linear scaling with time\nis termed as anomalous diffusion which falls into super-\nor sub-diffusive category based on MSD scaling as tα\nwith 1< α <2 and 0 < α <1 respectively. There\nare observed variant of diffusion which are normal ac-\ncording to the linear scaling of the MSD with time, but,\nare characterized by non-Gaussian distributions at inter-\nmediate times which crosses over to Gaussian at larger\ntimes [1, 2]. An averaging over a distributed diffusivity\nof particles of the system has been seen to result in the\nrequired Laplace form of the density distribution [1, 3].\nThis method is similar in spirit to the concept of super-\nstatistics which employs a superposition of Boltzmann\nstatistics at smaller scales to get a non-Boltzmann distri-\nbution with a variable intensive quantity at larger scales\n[4]. ChubinskyandSlatercameupwiththeideaofdiffus-\ning diffusion (time dependent diffusivity) and by employ-\ning that they got a short time probability distribution of\nLaplace form which crosses over to a Gaussian form at\nlarge times [5]. Chechkin et al., developed a minimal\nmodel using the concept of diffusing diffusion embeded\nin a two component Langevin dynamics to capture this\ncrossoverbetween Laplace to Gaussian regime as well [6].\nCoordinate dependence of diffusivity and damping [7]\nof a Brownian particle (BP) is observed (or invoked) in\nmany experiments [8–11] where the BP resides near a\nwall or a boundary. Position dependent diffusion is sup-\nposed to be playing major role in protein folding [12–14].\nThe same is also invoked in hydrodynamic (large wave\nlength) models of some optical systems [15, 16] and openquantum systems [17]. A covariant formulation of state\ndependent diffusion and related issues with equilibration\nin such systems has been reported by Polettini [18]. The\ndifficulty of experimentally determining position depen-\ndent diffusivity in protein folding is highlighted in a re-\ncent paper by Foster et al., [19].\nIt is generally believed that the hydrodynamic effects\nnearawall areat the originofthe coordinatedependence\nof diffusivity and damping of a BP [8] and there may be\nother reasons as well. Imagine a BP diffusing in a finite\nspace filled with some network of static obstacles. The\nBP will be subjected to a coordinate dependent diffusiv-\nity and damping almost everywhere in such a stationary\ncrowded space. Another example could be the Brown-\nian motion of a polymer or a protein in its state space\nwhich is finite. Depending upon relative proximity of the\nmonomers or residues in various configurations the dif-\nfusivity and damping could become a function of state\nspace. Due to the finite extent of the space and static in-\nhomogeneity, when kept at a constant temperature, such\naBrownianmotionshouldequilibrateatlargetimes. The\npurpose of this paper is to ask if the equilibrium distri-\nbution and other features of such a system is different\nfrom that when diffusivity and damping are constant.\nInthepresentkindofaproblem,whilelookingforequi-\nlibrium, we are basically dealing with quenched coordi-\nnatedependent diffusivity and dampingwhich depend on\nbath degrees of freedom and also things other than bath\ndegrees of freedom. Had this not been the case, i.e. if\nthe finite space of Brownian motion is homogeneous (be-\ning characterizedby a constant diffusivity and damping),\nwe would be in a regime of equilibrium governed by the\nStokes-Einstein relation (fluctuation-dissipation relation\n(FDR)). The theory of equilibrium Brownian motion is\nwell established for such homogeneous spaces. One may\nreasonably ask - what happens in the general case? does2\nthe Stock’s-Einstein relation get generalized or it gets\nmodified?\nAn over-damped BP under confinement equilibrates\nwith heat-bath at large times. The Boltzmann distribu-\ntion (BD) characterizes position distribution of the BP\nin equilibrium when the damping and the diffusivity are\nconstant. This is a well known and tested result. What\nhappens when the diffusivity and the damping are func-\ntions of space (i.e., coordinate dependent) is a question\npeople have pondered over a long time and there exists\ncontroversy [20, 21]. The main theme of the approach to\nthis problem has so far been to demand the BD as an\nirrevocable condition for equilibrium [22–27]. This ne-\ncessitates replacing constant diffusivity D and constant\ndampingΓwithcoordinatedependentD( x) andΓ(x) (for\nexample, in 1D) to generalize the BD of such systems to\nP(x) = Ne−V(x)\nD(x)Γ(x)= Ne−V(x)\nkTwhere N is a normalization\nconstant, V( x) is the potential that confines the particle,\nk is the Boltzmann constant and T is the temperature of\nthe system. This indicates a local generalization of the\nStokes-Einstein relation DΓ = kT to D( x)Γ(x) = kT.\nOne of the central issues, here, would be whether or\nnot to take D( x)Γ(x) = kT as the local generalization of\ntheStokes-Einsteinrelationandinthispaperwewilllook\nat two cases where D( x)Γ(x)/negationslash= kT and D( x)Γ(x) = kT\nwithin the framework of Kramers-Moyalexpansion with-\nouta priori imposition of BD as an equilibrium con-\ndition. There are other issues when one imposes BD\nfor equilibrium in such systems, like: (a) In the deriva-\ntion of the probability distribution using Smoluchowski\nequation one needs to keep the Fick’s law in its con-\nstant diffusivity form giving the diffusion current density\njdiff=−D(x)∂P(x)\n∂x[22]. (b) The BD does not include\ncoordinate dependent diffusivity D( x) or damping Γ( x)\nand, thus, does not reflect the inhomogeneity of space\nwhich cannot be accommodated in a potential. We will\nsee in what follows that, the clue to have consistent so-\nlution to this problem lies with taking the correct form\nof Fick’s law over inhomogeneous space where the diffu-\nsivity is a function of coordinates.\nWe show in the first part of our results where\nD(x)Γ(x)/negationslash= kT that, all the above mentioned issues get\nnaturallyresolvedifonegoesbythe analysisbasedonthe\nKramers-Moyal expansion (in the over-damped regime)\nand the resolution happens in an unexpected way. By\nderiving the Smoluchowski equation for such a BP us-\ning Kramers-Moyal expansion, one gets the modification\nof the Fick’s law to jdiff=−∂\n∂xD(x)P(x) instead of\na generalization to jdiff=−D(x)∂\n∂xP(x). The Stokes-\nEinstein relation in this inhomogeneous case results from\nthe equipartition of kinetic energy as /angbracketleftD(x)Γ(x)/angbracketright= kT\nindicating that DΓ = kT is only strictly valid for the\nconstant diffusivity and damping.\nThere is no controversy in deriving the Fokker-Planck\nequation of a generalized Langevin dynamics that in-cludes inertial term. This is so because one can easily\nconvert the problem to a stochastic dynamics with ad-\nditive noise. The distribution one gets here is a direct\ngeneralization of the Maxwell-Boltzmann (M-B) form.\nMoreover, correspondence between this generalized M-\nB distribution at the over-damped limit to the modified\nBD as obtained from the Smoluchowski dynamics pro-\nduces D( x) =CΓ(x) where C=/angbracketleftD(x)2/angbracketright\nkT=kT\n/angbracketleftΓ(x)2/angbracketrightis a\nconstant in equilibrium.\nIt can be easily inferred that in the presence of the\nproportional relationship D( x) =CΓ(x) between the lo-\ncal diffusivity and damping the product D( x)Γ(x) cannot\nbe locally equivalent to any quantity of an independent\nphysical origin because that will make diffusivity locally\ninversely proportional to damping in direct conflict with\nD(x) =CΓ(x). This subtle constraint will set the im-\npossibility of having local temperatures in the form of\n(D(x)Γ(x))/k in equilibrium of such systems. As a result\nD(x)Γ(x) comes out to be the local energy scale that sets\nthe width of distributions and this is only equal to the\nthermal energy scale set by the bath on an average over\nthe inhomogeneity.\nIn this part, we will see that the general theory in the\npresence of equipartition of energy gives Stocks-Einstein\nrelation in homogeneous space (constant D and Γ), but,\nthe homogeneous limit of the relation in weakly inhomo-\ngeneous space does not exist. This observation will turn\nout to be crucial to rule out the extension of the the-\nory for homogeneous space to the case of even weakly\nnon-homogeneous cases when D( x)Γ(x)/negationslash= kT. The non-\nexistence of this homogeneous limit indicates a severe\nconstraint on the Stokes-Einstein relation and its use\neven in weakly inhomogeneous space.\nInthesecondpart,wegobylocalvalidityoftheStokes-\nEinstein relation within the realm of an analysis strictly\nbasedon Kramers-Moyalexpansion. We showthat a cru-\ncialconsiderationisneededtomakethetwoover-damped\nlimits - one taken on the dynamics and the other taken\non the modified M-B distribution - coincide. The con-\nsideration is that the local normalization of the velocity\ndistribution cannot be done on limits from −∞to +∞.\nThe velocity limits on the integral has to be set between\nthequantities −D(x)/Land+D( x)/L whereListhe only\nlength scale available that does not involve diffusivity\nand this length scale is the system size. Note that the\nStokes-Einstein relation holds locally, D( x) = kT/Γ(x).\nThis means, the local maximal velocity limit is inversely\nproportional to Γ( x) and is proportional to thermal en-\nergy kT given a system size L which makes sense. This\nwe identify as an important requirement for the Stokes-\nEinstein relation to hold locally when diffusivity is coor-\ndinate dependent and the over-dampedlimit on modified\nM-B distribution resulting in the required modified BD\nthat comes from the Smoluchowski equation.\nThe Kramers-Moyal expansion is a formal procedure3\nperfectly suited for a coordinate dependent diffusivity.\nTheequilibriumdistributionthatresultsfromtheSmolu-\nchowski equation as obtained from the Kramers-Moyal\nexpansion is a modified Boltzmann distribution with the\ndiffusivity D( x) dependent amplitude [28]. This is an ex-\npected equilibrium distribution as compared to the BD\nbecause the BD does not manifest the broken spatial ho-\nmogeneity. Thus, beyond the so far used methods of\nessentially extending the Brownian motion theory of a\nhomogeneous space to inhomogeneous conditions, if one\nfollows already existing method (Kramers-Moyal expan-\nsion) for inhomogeneous space, one gets a set of consis-\ntent results and possibly the clue as to why the notions\nbelonging to the homogeneous space theory cannot be\nextendedtoinhomogeneoussituations. Thepresentanal-\nysis based on Kramers-Moyal expansion indicates either\nthe Stokes-Einstein relation is not locally applicable or\nwhen it holds even locally there exists a maximum limit\nof locally accessible velocity of the BP at each point in\nspace.\nThe plan of the paper is as in the following. We first\nconsider the over-damped Brownian dynamics and em-\nploytheKramers-MoyalexpansiontofindouttheSmolu-\nchowskiequationanditsequilibriumsolutionasthemod-\nified BD. We then derive the Fokker-Planckdynamics for\nthe generalized Langevin equation of the system which\nincludes the inertial term. Following that we show our\nresults in two subsections. In one subsection we employ\nequipartition to recover Stokes-Einstein relation in ho-\nmogeneous space. We then take the over-damped limit\nof the generalized M-B distribution to compare this with\nthemodified BDtogettherelationbetweentheD( x) and\nΓ(x). In the next subsection we consider the local valid-\nity of the Stokes-Einstein relation D( x)Γ(x) = kT and\nshow how one has to modify the integration limits of the\nlocal velocity normalization to get to the over-damped\nlimit that follows from the Smoluchowski dynamics. We\nconclude the paper with a discussion of main results.\nOVER-DAMPED DYNAMICS\nLet us consider a 1D model of Brownian motion (for\nthe sake of simplicity) as\n˙x=v\nm˙v=−mζ(x)v+F(x)+mζ(x)/radicalbig\n2D(x)η(t),(1)\nwherexis the position of the BP and vis its velocity.\nWe have kept the mass mof the BP explicitly present\nfor the ease of taking the over-damped limit, mζ(x) =\nΓ(x) is the damping coefficient and F( x) is an external\nforce resulting from some potential F( x) =−dV(x)\ndx. The\nGaussian white noise of unit strength is represented by\nη(t).At the over-damped limit of this dynamics we get the\nvery standard form of the equation\n˙x=F(x)\nΓ(x)+/radicalbig\n2D(x)η(t). (2)\nIn what follows, we will never impose any a priorire-\nlationship between the D( x) and Γ(x). The relations will\nfollow from the over-damped limit of the generalized M-\nB distribution and the equipartition of kinetic energy.\nLet us have a look at a few well known but important\ndetails of the over-damped Langevin dynamics eqn.(2).\nIn the absence of the force F( x) it represents free diffu-\nsion. The diffusivity D( x) gets defined by the dynam-\nics in the presence of the Gaussian noise η(t). Thus,\nD(x) =/angbracketleft(x(t+δt)−x(t))2/angbracketright\n2δtwherex≡x(t) [28].\nThe diffusion time scale δtwould depend on the local-\nity of the D( x) and the average is over noise. Inclusion\nof the force term F( x) makes the damping Γ( x) explic-\nitly appear and fix the local drift current. We, therefore,\nare effectively considering normal diffusion here, the only\nmodificationisinthelocalcharacterofthediffusivityand\nthe damping. A very important property of normal dif-\nfusion is the isotropy of the process and in the present\ncase although the diffusivity is inhomogeneous in space,\nit is isotropic, i.e., the same in both directions at every\npoint in one dimensional space.\nLet us first have a look at the Smoluchowski equation\nfor the over-damped dynamics (eqn.(2)) using Kramers-\nMoyal expansion [28]. The Kramers-Moyal expansion\ngives dynamics of probability density P( x,t) as\n∂P(x,t)\n∂t=∞/summationdisplay\nn=1/parenleftbigg\n−∂\n∂x/parenrightbiggn\nD(n)(x,t)P(x,t),(3)\nwhere the expansion coefficients are\nD(n)(x,t) =1\nn!lim\nτ→01\nτ/angbracketleft[ξ(t+τ)−x]n/angbracketright(4)\nwithξ(t) =xand the angular brackets indicate average\nover noise [29]. Consistent with Pawula’s theorem, there\nwillbe twotermsonthe r.h.s., oftheSmoluchowskiequa-\ntion for the BP whose dynamics is given by the Langevin\nequation (eqn.(2)) as\n∂P(x,t)\n∂t=∂\n∂x/bracketleftbigg\n−F(x)P(x,t)\nΓ(x)+∂D(x)P(x,t)\n∂x/bracketrightbigg\n.(5)\nAt this stage, a discussion on the so-called spuri-\nous current is in order. The drift current density to\nbejdrift=F(x)\nΓ(x)P(x) is determined by the drift veloc-\nityvd(x) =F(x)\nΓ(x)which results from a balance between\nthe damping term and the external force. In the over-\ndamped limit this force balance is always there and does\nnot depend on the coordinate dependence of damping.\nThis is so because the over-damped limit is taken by set-\ntingm→0 andζ(x)→ ∞such that Γ( x) is finite. This4\nlimit practically sets the relaxation time of the system\nτ(x) =1\nζ(x)→0 everywhere and thus the relaxation\ntime scale becomes negligible compared to the diffusion\ntime scale δt.\nDue to non-validity of mean value theorem on stochas-\ntic integrals with multiplicative noise, a convention is\nneededtoevaluatethe driftcoefficient D(1)(x,t) andhere\ncomes the Itˆ o vs Stratonovich dilemma [30, 31]. Where\nItˆ oconventioncorrectlygivesthedriftcurrentinitsform,\nthe Stratonovich convention produces spurious drift cur-\nrent on top of it which has to be neglected in a straight\nforward manner if one wants strictly to be in the over-\ndamped limit. The drift velocity is completely defined\nin the over-damped limit everywhere because of the rea-\nson that diffusion being isotropic at all points in space\neven when diffusivity is coordinate dependent the dif-\nfusion gradient cannot result in a drift current. It is\nessential to break isotropy of space to get a drift cur-\nrent, however, the coordinate dependent diffusivity does\nnot do that symmetry breaking. The inhomogeneity of\nspacedue toD( x) showsanapparentbreakingofisotropy\nby the presence of gradients, but, diffusive transport re-\nmaining isotropic there cannot be a drift proportional to\nthese gradients. If such a drift appears that appears as\nan artifact of the convention followed. Thus, it is not at\nall difficult to identify the spurious conventiondependent\ncomponent of drift current here.\nMoreover, the diffusion current does not involve any\nspurious contribution in any convention and, therefore,\ncannot be altered. This is exactly where, in the existing\nliterature, manipulations are made. One not only cancels\nthespuriousdriftcurrentbutalsothrowsawaythepartof\nthe diffusion current appearing in the form −P(x)dD(x)\ndx\nto ensure Boltzmann distribution. For example, in the\npaper by Lau and Lubensky [22], which explains the ex-\nisting practice in this regard in a general way, one can\nidentify the omission of the above mentioned part of the\ndiffusion current in the considered definition of the diffu-\nsion current density as J(x,t) =−D(x)∂P(x,t)\n∂x. But, how\ncould this be done even when there is no spurious con-\ntribution in diffusion current? As is clearly mentioned\nby Lau and Lubensky [22], this is done to get the BD as\nthe equilibrium solution of the resulting Smoluchowski\nequation.\nThe equilibrium distribution that results from the\nSmoluchowski dynamics as given by eqn.(5) is\nP(x) =N\nD(x)exp/integraldisplayx\n−∞F(x′)\nD(x′)Γ(x′)dx′.(6)\nThisisamodifiedBoltzmanndistributionin thepresence\nof coordinate dependent diffusivity and damping where\nN =/bracketleftBig/integraltext∞\n−∞dx\nD(x)exp/integraltextx\n−∞F(x′)\nD(x′)Γ(x′)dx′/bracketrightBig−1\nis a normaliza-\ntion constant. Note that, the temperature of the bath\ndoes not show up in this expression since we have not\nyet considered any relation between the diffusivity anddamping as the one results from the Stokes-Einstein re-\nlation in homogeneous space. Obviously, we do not want\nto impose the Stokes-Einstein relation. The way to bring\nin the temperature is to employ the equipartition of ki-\nnetic energy of the BP and for that we will be needing\nto find out the equilibrium distribution for the model\ninvolving inertial term given by eqs.(1).\nThe modified Boltzmann distribution as shown in\neqn.(6) can always be given a Boltzmann form by expo-\nnentiatingtheD( x)dependentamplitude. Thiswouldre-\nsult in an effective potential involving the D(x) and Γ( x)\nas shown in [28]. Making use of this effective potential,\none can write a Langevin dynamics with additive noise\nto simulate equilibrium fluctuations of a system. After\nall, it is the equilibrium fluctuations of the Langevin dy-\nnamics which are ofany practicaluse. If the temperature\nis identified properly, then this alternative procedure can\npossiblyworkfineforawholeclassofstochasticproblems\nin inhomogeneous space.\nBefore going to the next section to capture the tem-\nperature in this formalism let us have critical look at\nthe issue as to why the Boltzmann distribution can-\nnot be an acceptable equilibrium distribution for the\nover-damped dynamics as given by eqn.(2) whereas the\nmodified Boltzmann distribution is a perfectly accept-\nable equilibrium distribution. It is important to no-\ntice that, if the eqn.(2) is characterized by a Boltzmann\ndistribution in equilibrium the average mean velocity\n/angbracketleft˙x/angbracketright=/angbracketleftF(x)\nΓ(x)/angbracketrightis not identically equal to zero whereas\n/angbracketleftF(x)\nΓ(x)/angbracketright=/integraltext∞\n∞dxF(x)\nD(x)Γ(x)P(x) =/integraltext0\n0dP(x)≡0 when\nP(x) is the modified Boltzmann distribution without the\n1/D(x) normalization factor as is given by eqn.(6). This\nisacrucialcheck. Theequilibriumdistributionisstation-\nary by construction as the BP equilibrates at the min-\nimum of a potential. Existence of this average current\ndue to Boltzmann distribution will produce entropy in\ncontradiction with the thermodynamic demand of equi-\nlibrium to be the highest entropy state under given con-\nditions. Had the manipulations normally done on the\nSmoluchowski dynamics to get the Boltzmann distribu-\ntion for eqn.(2) been correct this inconsistency would\nhavenotresulted. However,theappearanceofthisincon-\nsistency clearly indicates that the modified Boltzmann\ndistribution as obtained from the methods following the\nKramers-Moyal expansion is just perfectly consistent to\nbe the equilibrium distribution.\nGENERALIZED LANGEVIN DYNAMICS\nConsidering the change of variable u=v\nχ(x)where\nχ(x) =ζ(x)/radicalbig\n2D(x), eqs.(1) take the form5\n˙x=χ(x)u\n˙u=−ζ(x)u−χ′(x)u2+F(x)\nmχ(x)+η(t).(7)In eqs.(7), the χ′(x) =∂χ(x)\n∂xand, these equations be-\ning in additive noise form, its Fokker-Planck dynamics\ncanbederivedinastraightforwardmanner. TheFokker-\nPlanck dynamics for eqn.(7) is\n∂P(x,u,t)\n∂t=−∂\n∂xχ(x)uP(x,u,t)−∂\n∂u/bracketleftbigg\n−ζ(x)u−χ′(x)u2+F(x)\nmχ(x)/bracketrightbigg\nP(x,u,t)+1\n2∂2\n∂u2P(x,u,t).(8)\nTo obtain the stationary equilibrium distribution with\nthe detailed balance maintained, one sets the part of the\nequation involving the operators symmetric in uto zero\nto obtain the velocity distribution. This requirement ofdetailed balance in equilibrium (see for reference chap. 6\nof [32]) requires the r.h.s., of eqn.(8) be separated in the\nfollowing manner for a stationary solution.\n−∂\n∂xχ(x)uP(x,u)+∂\n∂uχ′(x)u2P(x,u)−∂\n∂uF(x)\nmχ(x)P(x,u) =−∂\n∂u/bracketleftbigg\nζ(x)uP(x,u)+1\n2∂\n∂uP(x,u)/bracketrightbigg\n.(9)\nSetting the current density within the square bracket on\nthe r.h.s., of the above equation to zero one gets the\nMaxwellian distribution of the velocity and the station-\nary probability density now assumes the shape\nP(x,u) = P(x)M(x)e−ζ(x)u2= P(x)M(x)e−ζ(x)v2\nχ(x)2.(10)Intheabovementionedexpressionforprobabilitydensity\nthe local normalization factor M( x) =1\nχ(x)/radicalBig\nζ(x)\nπfor the\nvelocity ( v) distribution is explicitly considered. With\nthese, eqn.(9) now takes the form\n−u∂\n∂xχ(x)P(x)M(x)e−ζ(x)u2+ 2uχ′(x)P(x)M(x)e−ζ(x)u2−2u3χ′(x)ζ(x)P(x)M(x)e−ζ(x)u2\n+2uζ(x)F(x)\nmχ(x)P(x)M(x)e−ζ(x)u2= 0. (11)\nRemoving the common factor of ufrom all the terms\nand then integrating out vwhile keeping in mind that\nthe average /angbracketleftv2/angbracketrightlocal=χ(x)2\n2ζ(x)we get\n−P(x)χ′(x)−χ(x)∂\n∂xP(x)+2P(x)χ′(x)−P(x)χ′(x)\n+2ζ(x)F(x)\nmχ(x)P(x) = 0. (12)\nEqn.(12) results in a distribution over position space\nP(x) =e/integraltextx\n−∞dx′2ζ(x′)F(x′)\nmχ(x′)2. (13)\nIncluding all the terms, therefore, the generalized M-Bdistribution is\nP(x,v) =N/radicalbiggm\n2πΓ(x)D(x)e/integraltextx\n−∞dx′F(x′)\nΓ(x′)D(x′)e−mv2\n2Γ(x)D(x)\n(14)\nwhereN=/bracketleftBig/integraltext∞\n−∞dxP(x)/bracketrightBig−1\nis an overall normalization\nconstant. Note that, this M-B distribution is an exact\ngeneralization of the M-B distribution over homogeneous\nspace [33]. If one replaces D( x) by D and Γ( x) by Γ, one\nwould get the standard M-B distribution of a BP over\nhomogeneous space.6\nStokes-Einstein relation does not hold locally\nSo far, the temperature has not been introduced in\nthe expressions we have got and that now can easily be\nobtained from the equipartition of kinetic energy. The\nequipartition of energy is a general feature of equilibrium\nand the kinetic energy being quadratic in momentum its\naverage value iskT\n2. The average must be done over the\nwhole phase space. Using the general M-B distribution\nthe equipartition results in\n/angbracketleftmv2/angbracketright=N/integraldisplay∞\n−∞dxP(x)mχ(x)2\n2ζ(x)=/angbracketleftΓ(x)D(x)/angbracketright= kT\n(15)\nwhere the angular brackets indicate a space average over\ntheboundedregioninwhichtheBPhasequilibratedwith\nthe bath. Obviously, when diffusivity and damping are\nconstant, we recoverthe Stokes-Einstein relation D =kT\nΓ\nfrom the equipartition and this relation does not hold in\ngeneral for a coordinate dependent damping and diffu-\nsion.\nImportant to note that, arrivingat the Stokes-Einstein\nrelationat the homogeneouscasejustifies a posteriori the\nuse of equipartition relation eqn.(15). Equipartition of\nkinetic energy giving1\n2kT is a consequence of the M-B\ndistribution, however, the distribution we have arrivedat\nis a generalized form without the temperature being ex-\nplicitly present. The recovery of Stokes-Einstein relation\nfor constant D and Γ now raises the question - does the\nhomogeneous limit exist where the Stokes-Einstein rela-\ntion can be used at least for weak inhomogeneity? We\nwilltrytofindananswertothisquestioninthefollowing.\nThe relation between the coordinate dependent damp-\ning and diffusion can be arrived at by taking the over-\ndamped limit m→0 andζ(x)→ ∞keeping Γ( x) fi-\nnite on the generalized M-B distribution (eqn.(14)) and\ncomparing that with the modified BD as already ob-\ntained in eqn.(6). This limit sets the factor e−mv2\n2Γ(x)D(x)\nto unity and the resulting limit of the normalization fac-\ntor/radicalBig\nm\n2πΓ(x)D(x)→0 is a consequence of the flatness\nof the velocity distribution however, the normalization\nfactor must be kept explicitly present in the expression\nof the distribution. At this limit, correspondence be-\ntween the generalized M-B distribution and the gener-\nalized BD needs D( x) =CΓ(x) where the proportion-\nality constant comes out from the equipartition to be\nC=/angbracketleftD(x)2/angbracketright\nkT=kT\n/angbracketleftΓ(x)2/angbracketright.\nThese two relations (1) D( x) =CΓ(x) and (2) D =kT\nΓ\nare completely consistent so long one takes into account\nthe fact that the latter is valid strictly for constant diffu-\nsivity and damping. It is obvious that even at weak inho-\nmogeneity limit the Stokes-Einstein relation cannot ap-\nproximate for the relation D( x) =CΓ(x). This fact can\neasilybecheckedbyTaylorexpandingtwoexpressionsfor\nD(x) namely (a) D( x) =kT\nΓ(x)and (b) D( x) =kT\n/angbracketleftΓ(x)2/angbracketrightΓ(x)for smalldΓ(x)\ndxwhich takes into account the weak spa-\ntial variation of Γ( x) over its average value Γ = /angbracketleftΓ(x)/angbracketright.\nConsider the average of the diffusivity to be D = /angbracketleftD(x)/angbracketright.\nTakingthe relation(a) intoaccountand truncatingthe\nTaylor expansion about /angbracketleftD(x)/angbracketright= D and /angbracketleftΓ(x)/angbracketright= Γ at\nthe second term we get\nD+/parenleftbiggdD(x)\ndΓ(x)dΓ(x)\ndx/parenrightbigg\nD,Γδx=kT\nΓ−kT\nΓ2/parenleftbiggdΓ(x)\ndx/parenrightbigg\nD,Γδx,\n(16)\nwhere the expansion is truncated at the second term due\nto smallness ofdΓ(x)\ndxin the weakly inhomogeneous space.\nThis gives\n/parenleftbiggdD(x)\ndΓ(x)/parenrightbigg\nD,Γ=−kT\nΓ2. (17)\nGoing by the relation (b) and the same procedure\n/parenleftbiggdD(x)\ndΓ(x)/parenrightbigg\nD,Γ=C=kT\nΓ2. (18)\nSo, the mismatch of the sign cannot be cured unless\nT→0 or equivalently Γ → ∞which is essentially a\nnon-stochastic limit. Therefore, the relations (a) and (b)\ncannot be limiting cases of each other at even small in-\nhomogeneity. This indicates the Stokes-Einstein relation\ncannot be generalized to situations where the diffusivity\nand damping are even weakly coordinate dependent.\nLet ushavea closerlookatthe implicationsofthe non-\nexistence of this limit. The equipartition gives a gener-\nalization of the Stokes-Einstein relation as /angbracketleftΓ(x)D(x)/angbracketright=\nkT and the correspondence gives D( x) =CΓ(x). While\nthe equipartition clearly indicates that the temperature\nis defined globally, the relation D( x) =CΓ(x), which is\nat conflict with the local generalization of the Stokes-\nEinstein relation, relates the local fluctuation and dis-\nsipation. Moreover, the latter indicates, the local tem-\nperature is proportional toD(x)\nΓ(x)and not to D( x)Γ(x) as\nwould be the demand of a generalized Stokes-Einstein\nrelation.\nTheenergyscaleW( x) = D(x)Γ(x) maybe interpreted\nin analogy as kT( x) over an inhomogeneous space, how-\never, this analogy cannot bring in a local temperature\nkT(x) of an independent physical origin (i.e. a prop-\nerty of the bath) than what the product D( x)Γ(x) it-\nself is. This is so because, existence of any other in-\ndependent physics (for example thermal) giving rise to\nsuch a quantity will impose an inverse relation between\nthe local diffusivity and damping in direct conflict with\nD(x) =CΓ(x). Therefore, it is clear that, although an\nanalogy apparently exists, but, it is of no physical con-\nsequence to actually create thermal gradients in equi-\nlibrium as captured by the modified Boltzmann distribu-\ntion. Thus, the appearance of the relation D( x) =CΓ(x)\npreserves the basic tenet of existence of no temperature\ngradients in equilibrium. In other words, the failure of7\nthe local generalization of the Stokes-Einstein relation\nrids us ofthe problem of appearanceof local temperature\nin the equilibrium scenario of such a spatially inhomoge-\nneous space.\nStokes-Einstein relation holds locally\nThe knowledge gained in the previous subsection in-\ndicates that if Stokes-Einstein relation holds locally,\nthe over-damped limit on the generalized Maxwell-\nBoltzmann distribution cannot correspond to the mod-\nified Boltzmann distribution that we have got from the\nSmoluchowski dynamics because Γ( x)D(x) is a constant\nkT. There is a simple way out of this problem. Although\nwe are used to integrating over all velocities while nor-\nmalizing the velocity distribution, however, the situation\nat hands indicates that we cannot do that when D( x) is\ncoordinate dependent.\nThe natural local velocity cut off for such a system can\nbe taken as D( x)/L where L is the system size i.e. the\nlength scale of the space in which the BP equilibrates.\nThere is no other length scale available in this system\nthan this which does not depend on D( x) when Stokes-\nEinstein equation is locally valid. When employed, this\ngives a local normalization factor for the velocity distri-\nbution in the following way.\n/integraldisplayD(x)\nL\n−D(x)\nLdve−mv2\n2Γ(x)D(x)=/radicalbigg\n2kT\nm/integraldisplayD(x)\nL√m\n2kT\n−D(x)\nL√m\n2kTdze−z2,\n(19)\nwherez=/radicalbigm\n2kTv.\nEqn.(19) readily gives the value of the integral to be\n/radicalbigg\n2πkT\nmerf(D(x)\nL/radicalbiggm\n2kT),\nwhich at the overdamped limit can be written simply\nas√πD(x)/L and that gives the normalization constant\nproportional to 1 /D(x).\nThe modified Maxwell-Boltzmann distribution in this\ncase becomes\nP(x,v) =N/radicalBig\n2πkT\nmerf(D(x)\nL/radicalbigm\n2kT)e/integraltextx\n−∞dx′F(x′)\nΓ(x′)D(x′)e−mv2\n2Γ(x)D(x).\n(20)\nNow, taking the over-damped limit i.e. m→0 on this\nwe get,\nP(x,v) =N\nD(x)exp/integraldisplayx\n−∞dx′F(x′)\nΓ(x′)D(x′),(21)\nwhere in the above relations Nstands for normaliza-\ntion constant. Note that, eqn.(21) is identical to eqn.(6)and we get to the same expressionfor the modified Boltz-\nmanndistributionattheover-dampedlimit bytakingthe\nlimit both ways - on the dynamics and on the modified\nM-B distribution. When Stokes-Einstein relation holds\nlocally, it needs the local maximum velocity be restricted\nto D(x)/L is the physics which goes very much contrary\nto the common sense that at m→0 all velocities should\nbe allowed. However, this length scale remains hidden in\nthe normalization constant and instead of this, any other\nconstant emergent length scale would result in the same.\nHere experiments can possibly look for the existence of\nan emergent length scale which fixes the local maximum\nvelocities.\nDISCUSSION\nIn this paper we have lookedat the problem of a Brow-\nnian particle moving in a finite space where its diffusiv-\nity and damping are stationary and are coordinate de-\npendent. We have been investigating the equilibrium of\nsuch a finite system. Coordinate dependent diffusivity\nand damping makes the space inhomogeneous even in\nthe absence of a force, however, the isotropy of the space\nremains intact at every point over space in this diffusive\nprocess. A global force may break the isotropy of the\nsystem and result in drift current but the diffusion does\nnot do that.\nIn the over-damped limit our approach has been\nto consider the Smoluchowski equation as obtained\nfrom Kramers-Moyal expansion and solve it for equilib-\nrium distribution without imposing any condition. On\nthe other hand, we derived the generalized Maxwell-\nBoltzmanndistribution forequilibrium ofthe systemand\nthen took the over-damped limit on it. On compari-\nson of results obtained from the over-damped dynam-\nics and over-damped limit of the generalized M-B distri-\nbution we see that there exists a proportional relation\nbetween coordinate dependent diffusivity and damping\nwhen the Stokes-Einstein relation does not hold locally.\nThe equipartition of energy results in recovery of Stokes-\nEinstein relation for constant diffusivity and damping.\nHowever, in terms of validity of the Stokes-Einstein re-\nlation the limit of the inhomogeneous space going to the\nhomogeneous space does not exist.\nOn the other hand, when we have taken into consid-\neration the local validity of the Stokes-Einstein relation,\nwe see that, we have to impose a local maximal velocity\nlimit to the velocity distribution to recover the modi-\nfied Boltzmann distribution of over-damped limit from\nthe generalized M-B distribution. The generalized M-B\ndistribution in this case is not a straight forward general-\nization of the M-B distribution with constant diffusivity\nand damping and involves an error function in the nor-\nmalization factor. This is an interesting situation where\nthe local maximal velocity a BP can take is proportional8\nto kT and inversely proportional to Γ( x). The equiparti-\ntion will hold in this case locally unlike where the Stokes-\nEinstein relation is not locally valid.\nThese modified equilibrium distributions and rela-\ntions between the local diffusivity and damping could\nbe checked within present experimental access. To the\nknowledge of the author, experiments so far have not\nparticularly looked for such an inversion of the Stokes-\nEinetein relation or local maximum velocity of a BP.\nOn the contrary, Stokes-Einstein relation has been ex-\ntensively employed to get diffusivity from damping and\nvice versa even when the diffusivity and damping are\nspace dependent. These new results, which are based\non already established formal methods, if experimentally\nverified,canhavefarreachingconsequenceonourpresent\nunderstanding of equilibrium of such systems.\nLetus tryto understandwhysuchanequilibriumanal-\nysis of the Brownian motion in inhomogeneous space is\nimportant. Consider the biophysical environment of a\ncell. This is a very crowded and confined environment\nand of course the processes are not happening strictly\nin equilibrium in the true thermodynamic sense. How-\never, many of the processes are weakly non-equilibrium\nstochastic processes whose statistics to be mostly gov-\nerned by equilibrium fluctuations. In other words, many\nprocesses fall in the linear response regime where the\nequilibrium distribution dictates the physics. This is\nexactly the reason we care about an otherwise ideal-\nized equilibrium conditions because the same physics ap-\nplies to a plethora of phenomena in the weakly non-\nequilibrium regime. The importance of the present re-\nsults lie in this wide area of applicability.\nACKNOWLEDGEMENT\nI would like to acknowledge discussions with J. K.\nBhattacharjee.\n∗Electronic address: a.bhattacharyay@iiserpune.ac.in\n[1] Wang B, Anthony S M, Bae S C and Granick S 2009\nPNAS 106 15160\n[2] WangB, Kuo J, Bae S C and Granick S 2012 Nat. Mater.\n11481[3] Hapca S, Crawford J W and Young I M 2009 J. R. Soc.\nInterface 6111\n[4] Beck C and Cohen E D G 2003 Physica A 322267-275\n[5] Chubynsky M V and Slater G W 2014 Phys. Rev. Lett.\n113098302\n[6] Chechkin A V, Seno F, Metzler R and Sokolov I M 2017\nPhys. Rev. X 7021002\n[7] Berezhkovskii A and Szabo A 2011 J. Chem. Phys. 135\n074108\n[8] Faucheux L P and Libchaber A J 1994 Phys. Rev. E 49\n5158-5163\n[9] Lancon P, Batrouni G, Lobry L and Ostrowsky N 2001\nEurophys. Lett. 5428-34\n[10] Volpe G, Helden L, Brettschneider T, Wehr J and\nBechinger C 2010 Phys. Rev. Lett. 104170602\n[11] Wolfson W, Liepold C, LinB andRice SA 2018 J. Chem.\nPhys.148194901\n[12] Best R B and Hummer G 2010 PNAS1071088-1093\n[13] Hummer G 2005 New J. Phys. 734\n[14] Oliveira R J, Whitford P C, Chahine J, Leite V B P and\nWang J 2010 Methods 5291-98\n[15] Yamilov A G, Sarma R, Redding B, Payne B, Noh H and\nCao H 2014 Phys. Rev. Lett. 112023904\n[16] Neupane P and Yamilov A G 2015 Phys. Rev. B 92\n014207\n[17] Sargsyan V V, Palchikov Yu V, Kanokov Z, Adamian G\nG and Antonenko N V 2007 Phys. Rev. A 75062115\n[18] Polettini M 2013 J. Stat. Mech. P07005\n[19] Foster D A N, Petrosyan R, Pyo A G T, Hoffmann A,\nWang F and Woodside M T 2018 Biophys. J 1141657-\n1666\n[20] Sokolov I M 2010 Chem. Phys. 375359-363\n[21] Tupper P F and Yang X 2012 Proc. R. Soc. A 4683864-\n3881\n[22] Lau A W C and Lubensky T C 2007 Phys. Rev. E 76\n011123\n[23] Sancho J M, San Miguel M and D¨ urr D 1982 J. Stat.\nPhys.28291-305\n[24] Sancho J M 2011 Phys. Rev. E 84062102\n[25] Schnitzer M J 1993 Phys. Rev. E 482553-2568\n[26] Farago O and Grønbech-Jensen N 2014 Phys. Rev. E 89\n013301\n[27] Farago O and Grønbech-Jensen N 2014 J Stat. Phys. 156\n1093-1110\n[28] Bhattacharyay A 2019 Physica A 515665-670\n[29] Risken H 1984 The Fokker-Planck Equation: Methods of\nSolution and Applications (Springer-Verlag).\n[30] Itˆ o K 1944 Proc. Imp. Acad. Tokyo 20519-524\n[31] Stratonovich R L 1966 SIAM J. Control 4362-371\n[32] Gardiner C 2009 Stochastic Methods: A handbook for\nnatural and social sciences (Springer)\n[33] Schwabl F 2002 Statistical Mechanics (Springer)"    },    {        "title": "1902.04605v1.Ultra_low_damping_in_lift_off_structured_yttrium_iron_garnet_thin_films.pdf",        "content": "1 \n This article may be downloaded for personal use only. Any other use requires prior permission of the \nauthor and AIP Publishing. This article appeared in Applied Physics Letters 111 (19), 192404 (2017) \nand may be found at https://aip.scitation.org/doi/abs/10.1063/1.5002004  \n \n \nUltra -low damping in lift-off structured y ttrium iron garnet  thin films  \nA. Krysztofik ,1 L. E. Coy,2 P. Kuświk ,1,3 K. Załęski,2 H. Głowiński ,1  \nand J. Dubowik1 \n1Institute of Molecular Physics, Polish Academy of Sciences, PL -60-179 Poznań, Poland  \n2NanoBioMedical Centre, Adam Mickiewicz University, PL -61-614 Poznań, Poland  \n3Centre for Advanced Technology, Adam Mickiewicz University, PL -61-614 Poznań, Poland  \nElectronic mail: adam.krysztofik@ifmpan.poznan.pl , hubert .glowinski @ifmpan.poznan.pl  \n \nWe show that using maskless photolithography and the lift-off technique patterned \nyttrium iron garnet thin films possessing ultra -low Gilbert damping can be  \naccomplished . The films of the 70 nm thickness we re grown on (001)-oriented \ngadolinium gallium garne t by means of pulsed laser deposition and exhibit high \ncrystalline quality, low surface roughness and effective  magnetization of 127 \nemu/cm3. The Gilbert damping parameter is as low as 5×10−4. The obtained \nstructures have well-defined sharp edges  which  along with good structural and \nmagnetic film properties, pave a path in the fabrication of high -quality magnonic \ncircuits as well as  oxide -based  spintronic devices.  \n \n \nYttrium iron garnet (Y 3Fe5O12, YIG) has become an intensively studied material in recent years due \nto exceptionally low damping of magnetization precession and electrical insulation enabling its \napplication in research on spin -wave propagation1–3, spin-wave based logic devices4–6, spin pumping7, \nand thermally -driven spin caloritronics8. These applications inevitably entail film structurization in \norder to construct complex integrated devices . However, the fabrication of high -quality thin YIG films \nrequires  deposition temperatures over 500 C6,9–18 leading to top -down lithographical approach that is \nion-beam etching  of a previously  deposited plain film where as patterned resist layer serves as a mask.  \nConsequently, this metho d introdu ces crystallographic  defects , imperfections to surface structure  and, \nin the case of YIG films, causes significant  increase of the damping parameter .19–21 Moreover, it does \nnot ensure well-defined  structure edges  for insulators , which play a crucial role in devices utilizing 2 \n edge spin waves22, Goos -Hänchen spin wave shifts23,24 or standing spin waves modes25. On the \ncontrary, t he bottom -up structurization  deals with th ese issues  since it allows  for the film grow th in the \nselect ed, patterned  areas followed by  a removal of the resist layer along with redundant film during \nlift-off process.  Additionally, it reduces the patterning  procedure by one step , that is ion etching , and \nimposes room -temperature deposition  which both are particularly important  whenever  low fabrication  \nbudget is required.  \nIn this letter  we report on ultra -low damping in the bottom -up structured YIG film by means of \ndirect writing photolithography technique.  In our case, t he method allows for structure patterning \nwith 0.6 µm resolution across full writing area . In order to not preclude the lift -off process, the pulsed \nlaser deposition (PLD) was conducted at room temperature and since such as -deposited films are \namorphous19,27 the ex-situ annealing was performed for recrystallization.  Note that post -deposition \nannealing of YIG films  is commonly carried out regardless the substrate temperature  during film \ndeposition6,12,13,28,29. As a reference we  investigated  a plain film which was grown in the same \ndeposition process and underwent the same fabrication procedure except for patterning. Henceforth, \nwe will refer to the structured and the plain film as Sample 1 an d Sample 2, respectively . We \nanticipate that such a procedure may be of potential for fabrication of other magnetic oxide structures \nuseful in spintronics.  \nStructural characteriza tion of both samples was performed by means  of X-Ray Diffraction (XRD). \nAtomic force microscopy (AFM) was applied to investigate  surface morphology and the quality of \nstructure edges.  SQUID magnetometry  provided information on the saturation magnetization  and \nmagnetocrystalline anisotropy field . Using a coplanar waveguide connected to a vector network \nanalyzer , broadband ferromagnetic resonance (VNA -FMR) was performed to determine Gilbert \ndamping parameter  and anisotropy fields . All the experiments were co nducted at the room \ntemperature.  \nThe procedure of samples preparation was as follows. The (001) -oriented gadolinium gallium \ngarnet substrates were ultrasonicated in acetone, trichloroethylene and isopropanol to remove surface \nimpurities. After a 1 minute o f hot plate baking for water evaporation, a positive photoresist was spin -\ncoated onto the substrate (Sample 1). Using maskless photolithography an array of 500  μm x 500  μm \nsquares separated over 500 μm was patterned and the exposed areas were developed. Detailed \nparameters of photolithography process can be found in Ref.26. We chose rather large size of the \nsquares to provide a high signal -to-noise ratio in the latter measurements. Thereafter, plasma etching \nwas performed to remove a residual resist. We would like to emphasize the importance of this step in \nthe fabrication p rocedure as the resist residues may locally affect crystalline structure of a YIG film \ncausing an undesirable increase of overall magnetization damping. Both substrates were then placed in \na high vacuum chamber of 9×10-8 mbar base pressure and a film was d eposited from a stoichiometric \nceramic YIG target under 2×10-4 mbar partial pressure of oxygen. We used a Nd:YAG laser (λ  = 355 \nnm) for the ablation with pulse rate of 2 Hz which yielded 1 nm/min growth rate. The target -to-3 \n substrate distance was approximat ely 50 mm. After the deposition the l ift-off process for the Sample 1  \nwas performed using sonication in acetone  to obtain the expected structures. Subsequently, both \nsamples were annealed in a tube furnace under oxygen atmosphere (p ≈ 1 bar) for 30 minutes  at \n850°C. The heating and cooling rates were about 50  C/min and 10  C/min, respectively.  \n \n \n \nFIG. 1.  (a) XRD θ−2θ plot near the (004) reflection of structured ( Sample 1 ) and plain ( Sample 2 ) YIG film. Blue arrows \nshow clear Laue reflections of the plain film. Insets show schematic illustration of the structured and plain film used in this \nstudy. (b) Height  profile (z(x)) taken from the structured sample (left axis), right shows the differential of the p rofile, clearly \nshowing the slope change.  Inset shows 3D map of the structure’s edge.  \n \n \nThe structure of YIG films was determined by the X -ray diffraction.  Although the as-deposited \nfilms were amorphous, with the annealing treatment they inherited the lattice orientation of the GGG \nsubstrate and recrystallized along [ 001] direction. Figure 1 (a) presents diffraction curves  taken in the \nvicinity of ( 004) Bragg reflection.  The ( 004) reflection position of structured YIG well coincide s with \nthe reflection of the plain film. The 2 θ=28.70 9 corresponds to the cubic lattice constant of 12.428  Å. \nA comparison of this value with lattice parameter of a bulk YIG (12.376 Å) suggest distortion of unit \n4 \n cells due to slight nonstoichiometry.16,30 Both samples exhibit distinct Laue oscillation s depicted by \nthe blue arrows,  indicating  film uniformity and high crystalline order , although the structured film \nshowed lower intensity due to the lower mass of the film . From the oscillation period we estimated \nfilm thickness of 73 nm in agreement with the nominal thickness and the value determined using AFM  \nfor Sample 1 ( Fig. 1 (b)). By measuring the diffraction  in the expanded angle range w e also confirmed \nthat no additional phases like Y 2O3 or Fe 2O3 appeared.  \nThe surface morphology of the structured film was investigated by means of AFM. In Fig. 1 (b) \nprofile of a square’s edge is shown. It should be highlighted that no edge irregularities has formed \nduring lift -off process. The horizontal distance between GGG substrate and the surface of YIG film is \nequal to 170 nm as marked in Fig. 1 (b) by the shaded area. A fitting with Gaussian function to the \nderivative of height profile yields the full width at half maximum of 61 nm. This points to the well -\ndefined struct ure edges achieved with bottom -up structurization. Both samples have smooth and \nuniform surface s. The comparable values of root mean square (RMS) roughness (0.306 nm for Sample \n1 and 0.310 nm for Sample 2) indicate that bottom -up structurization process  did not leave any resist \nresidues. Note that a roughness of a bare GGG substrate before deposition was 0.281 nm, therefore, \nthe surface roughness of YIG is increased merely by 10%.  \n \n \nFIG. 2. Hysteresis loops of structured (Sample 1) and plain (Sample 2) YIG films measured by SQUID \nmagnetometry  along [100] direction at the room temperature . \n \nFigure 2 shows magnetization reversal curves measured along [ 100] direction. For each hysteresis \nloop a paramagnetic contribution arising for the GGG substrates was subtracted. The saturation \nmagnetization 𝑀𝑠 was equal to 117 emu/cm3 and 118.5 emu/cm3 for Sample 1 and 2, respectively . \nBoth hysteresis loops demonstrate in -plane anisotropy. For the (001) -oriented YIG the [ 100] direction \nis a “hard” in -plane axis and the magnetization saturates at 𝐻𝑎 = 65 Oe. This value we identify as \n-100 -75 -50 -25 0 25 50 75 100-1.0-0.50.00.51.0  Sample 1\n Sample 2M / MS\nMagnetic Field (Oe)5 \n magnetocrystalline anisotropy field. The VNA -FMR measurements shown in Fig. 3 (a) confirm these \nresults. Using Kitte l dispersion relation, i.e. frequency 𝑓 dependence of resonance magnetic field 𝐻: \n 𝑓=𝛾\n2𝜋√(𝐻+𝐻𝑎cos 4𝜑)(𝐻+1\n4𝐻𝑎(3+cos 4𝜑)+4𝜋𝑀𝑒𝑓𝑓), (1) \n 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑢, (2) \nwe derived 𝐻𝑎 and the effective magnetization 𝑀𝑒𝑓𝑓, both comparable to the values determined using \nSQUID and close to the values of a bulk YIG (see Table I .). Here,  the azimuthal angle 𝜑 defines the \nin-plane orientation of the magnetization direction with respect to the [100] axis of YIG  and  𝛾 is the \ngyromagnetic ratio ( 1.77×107𝐺−1𝑠−1). To better compare the values of 𝐻𝑎 between samples and to \ndetermine if the results are influenced by additional anisotropic contribution arising  from  the squares’ \nshape in the  structured film we performed angular resolved resonance measurements  (inset in Fig. \n3(a)) . The fitting according to Eq. (1) gives |𝐻𝑎| equal to 69.5±0.6 for Sample 1 and 69.74±0.28 for \nSample 2 in agreement with the values derived from 𝑓(𝐻) dependence and better accuracy.  Hence, we \nconclude  that the structurization did not affect the in -plane anisotropy. The deviations of  the derived  \n𝑀𝑠 and 𝐻𝑎 from bulk values can be explained in the framework of Fe vacancy model developed for \nYIG films as a result of nonstoichiometry.13,30 For the experimentally determined 𝑀𝑠 and 𝐻𝑎 the \nmodel yields the chemical unit Y 3Fe4.6O11.4 which closely approximates to  the composition of a \nstoichiometric YIG Y 3Fe5O12. \n \n \nTABLE I.  Key parameters reported for PLD and LPE YIG films.  \n  AFM  SQUID  VNA -FMR  \n Film \nthickness  RMS rough -\nness (nm)  Ms  \n(emu/cm3) Ha  \n(Oe) Field  \norientation  Meff \n(emu/cm3) |Ha|  \n(Oe) Hu \n(Oe) α  \n(× 10-4) ΔH 0  \n(Oe) \nSample 1  70 nm  0.306  117±1  65±5  (100):  \n(110):  \n(001):  125±1  \n126±1  \n129±2  64±1  \n63±1  \n− -101±18  \n-113±18  \n-151±28  5.53±0.13  \n5.24±0.12  \n5.19±0.64  1.45±0.09  \n2.86±0.09  \n2.61±0.34  \nSample 2  70 nm  0.310  118.5±2  65±5  (100):  \n(110):  \n(001):  124±1  \n127±1  \n131±2  62±1  \n65±1  \n− -69±28  \n-107±28  \n-157±36  5.05±0.07  \n5.09±0.09  \n5.02±0.18  0.97±0.05  \n1.28±0.06  \n1.48±0.09  \nLPE-YIG31 106 nm  0.3 143 − (112):  − − − 1.2 0.75 \nLPE-YIG30 120 μm  − 139±2  − (111):  133±2  85±6  76±1  0.3 − \n \n \nAlthough the saturation magnetization  of the films  is decreased by 15% with respect to the bulk \nvalue we can expect similar spin wave dynamics since magnon  propagation does not solely depend on \n𝑀𝑠 but on the effective magnetization or equivalently, on the uniaxial anisotropy field  𝐻𝑢.12 \nSubstitution of 𝑀𝑠 into Eq. (2) gives  average values of  𝐻𝑢 equal to -122 Oe and -111 Oe for Sample 1 \nand 2, respectively  (to determine 𝐻𝑢 from the out -of-plane FMR measurements when H || [001] we 6 \n used the 𝑓=𝛾\n2𝜋(𝐻+𝐻𝑎−4𝜋𝑀𝑒𝑓𝑓) dependence13 to fit the data  and assumed  the value of 𝐻𝑎 from \nangular measurements ). As 𝑀𝑒𝑓𝑓𝑆𝑎𝑚𝑝𝑙𝑒  1,2≈𝑀𝑒𝑓𝑓𝑏𝑢𝑙𝑘, it follows  that the low value of 𝑀𝑠 in room -\ntemperature deposited thin films is “compensated ” by uniaxial anisotropy field.  Note that for bulk YIG \nsaturation magnetization is diminished  by 𝐻𝑢/4𝜋 giving  a lower value of 𝑀𝑒𝑓𝑓 while for Sample 1 \nand 2,  𝑀𝑠 is augmented  by 𝐻𝑢/4𝜋 giving a higher value of 𝑀𝑒𝑓𝑓 (Table I .). The negative sign of \nuniaxial anisotropy field  is typical for PLD -grown YIG films  and originates from preferential \ndistribution of Fe vacancies between different si tes of YIG’s octahedral  sublattice.30 This point s to the \ngrowth -induced anisotropy mechanism while the stress -induced contribution is of ≈10 Oe29 and, as it \ncan be estimated  according to Ref.32, the transition layer at the substrate -film interface due to Gd, Ga, \nY ions diffusion is ca. 1.5 nm thick for the 30 min of annealing treatment.  We argue that the growth -\ninduced anisotropy due to ordering of the magnetic ions is related to the growth condition which in our \nstudy is specific. Namely, it is crystallization of an amorphous material.  \nGilbert damping parameter 𝛼 was obtained by fitting dependence of linewidth 𝛥𝐻 (full width at \nhalf maximum ) on frequency 𝑓 as shown in Fig. 3 (b): \n 𝛥𝐻 =4𝜋𝛼\n𝛾𝑓+𝛥𝐻0, (3) \nwhere  𝛥𝐻0 is a zero -frequency linewidth broadening . The 𝛼 parameter of both samples is nearly the \nsame , 5.32×10−4 for Sample 1 and 5.05×10−4 for Sample 2  on average (see Table I.) . It proves \nthat bottom -up patterning does not compromise magnetization damping. The value of 𝛥𝐻0 \ncontribution is around 1.5 Oe although small variations of 𝛥𝐻0 on 𝜑 can be noticed. Additional \ncomments on  angular dependencies of  𝛥𝐻 can be found in the supplementary material. The derived \nvalues of 𝛼 remain one order of magnitude smaller than for soft ferromagnets like Ni 80Fe2033, CoFeB34 \nor Finemet35, and are comparable to values reported for YIG film s deposited at hi gh temperatures \n(from 1×10−4 up to 9×10−4).6,9,11,14,15,17,18 It should be also highlighted  that 𝛼 constant is \nsignificantly increased in comparison to the bulk YIG made by means of Liquid Phase Epitaxy (LPE) . \nHowever, recently reported LPE-YIG films of  nanometer thickness , suffer from  the increased damping \nas well (Table I.) due to impurity elements  present in the high -temperature solutions used in LPE \ntechnique31. As PLD method allow s for a good  contamination control , we attribute the increase as a \nresult of slight nonstoichiometry determined above with Fe vacancy model .30 Optimization of growth \nconditions , which further improve the film composition may resolve this issue and allow to cross the \n𝛼=1×10−4 limit. We also report that additional annealing of the samples (for 2h) did not influence \ndamping nor it improved the value of 𝐻𝑎 or 𝑀𝑒𝑓𝑓 (within 5% accuracy).  7 \n  \nFIG. 3. (a) Kittel dispersion relation s of the structured (Sample 1) and plain (Sample 2) YIG film.  The i nset \nshows angular dependence of resonance field  revealing perfect fourfold anisotropy for both samples . (b) \nLinewidth dependence on frequency fitted with Eq. (3). The inset shows resonance absorptions peaks with very \nsimilar width (5.3 Oe for Sample 1 and 4.7 Oe for Sample 2  at 10 GHz ). Small differences of the resonance field  \noriginate from different values of 4𝜋𝑀𝑒𝑓𝑓. \n \nIn conclusion , the lift-off patterned YIG films possessing low damping have  been presented. \nAlthough the structurization procedure required deposition at room temperature , the 𝛼 parameter does \nnot diverge from those reported for YIG thin films grown at temperatures above 500 C. Using the \nplain, reference film fabricated along with the structured one, we have shown  that structurization does \nnot significantly affect structural nor magnetic properties  of the films,  i.e. out-of-plane lattice constant, \nsurface roughness, saturation magnetization, anisotropy fields and damping. The structures obtain ed \nwith bottom -up structurization indeed possess sharp , well-defined edges . In particular, o ur findings \nwill help in the development of magnonic and spintronic devices utilizing film boundary effects  and \nlow damping of magnetization precession . \n8 \n  \nSupplementary Material  \nSee supplementary material for the angular dependence of resonance linewidth . \n \nThe research received funding from the European Union Horizon 2020 research and innovation \nprogra mme under the Marie Skłodowska -Curie grant agreement No 644348 (MagIC).  We would like \nto thank Andrzej Musiał for the assistance during film annealing.  \n \n1 H. 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Matter 29, 435803 (2017).  \n35 H. Głowiński, I. Gościańska, A. Krysztofik, J. Barnaś, M. Cecot, P. Kuświk, and J. Dubowik, in 21st Int. Conf. Microwave, \nRadar Wirel. Commun. MIKON 2016  (2016).  \n \n \n "    },    {        "title": "1902.04608v1.Characterization_of_spin_wave_propagation_in__111__YIG_thin_films_with_large_anisotropy.pdf",        "content": "1 \n Characterization of spin wave propagation in (111) YIG thin films with \nlarge anisotropy  \nA. Krysztofik,1,b) H. Głowiński,1,a) P. Kuświk,1,2 S. Ziętek,3 L. E. Coy,4 J. N. Rychły ,5 \nS. Jurga,4 T. W. Stobiecki,3 J. Dubowik1 \n1Institute of Molecular Physics, Poli sh Academy of Sciences, M. Smoluchowskiego 17, PL -60-179 Poznań, Poland  \n2Centre of Advanced Technology, Adam  Mickiewicz University, Umultowska 89c, PL -61-614 Poznań, Poland  \n3Department of Electronics, AGH University of Science and Technology, Al. Mickiewic za 40, PL -30-059 Kraków, Poland  \n4NanoBioMedical Centre, Adam Mickiewicz University, Umultowska 85, PL -61-614 Poznań, Poland  \n5Faculty of Physics, Adam  Mickiewicz University, Umultowska 85, PL -61-614 Poznań, Poland  \n6Faculty of Physics and Applied Computer Sc ience, AGH University of Science and Technology, Al. Mickiewicza 30, PL -30-059 Kraków, \nPoland  \n \na)E-mail: hubert.glowinski@ifmpan.poznan.pl  \nb)E-mail: adam.krysztofik@ ifmpan.poznan.pl  \n \n \n \nAbstract  \nWe report on  long-range spin wave  (SW)  propagation in nanomete r-thick Yttrium \nIron Garnet (YIG) film with  an ultralow Gilbert damping. The knowledge of a wavenumber \nvalue  |𝑘⃗ | is essential  for design ing SW devices. Although determining  the wavenumber |𝑘⃗ | \nin experiments like Brillo uin light scattering spect roscopy is straightforward , quantifying \nthe wavenumber in all -electrical experiments has not been widely commented  so far. \nWe analyze magnetost atic spin wave (SW) propagation in  YIG films  in order to determine \nSW wavenumber |𝑘⃗ | excited by the coplanar waveguide . We show  that it is crucial  to \nconsider influence of magnetic anisotropy fields present in YI G thin films  for precise \ndetermination of SW wavenumber . With the proposed  methods  we find that experimentally \nderived v alues of |𝑘⃗ | are in perfect agreement with  that obtained from  electromagnetic \nsimulation only if anisotropy fields are included.  \n \n \n \n \n \n \n 2 \n  \nSpin wave (SW) propagation in magnetic thin film structures  has become intensively \ninvestigated topic in recent year s due to promising applications in modern electronics  [ 1, 2, \n3, 4 ]. The wavenumber (or equivalently – the wavelength 𝜆=2𝜋/|𝑘⃗ |) is an important \nparameter to  account for propagation characteristics.  For example, it is essential to choose \nSW wavenumber and correlate  it to certain device dimension in order to ensure observation \nof expected phenomena in SW devices e.g. i n magnonic crystals  [ 5, 6 ] or devices based on  \nwave interference such as SW transistor [  2 ], SW logic gates  [ 2 ], Mach -Zender  type \ninterferometer s [ 7 ]. The knowledge  of SW wavenumber is also very important  in the \nassessment of the  effective magnitude of Dzaloshinskii -Moriya interaction using collective \nspin-wave dynamics  [ 8 ]. \nIn propagating SW spectroscopy experiments two s horted coplanar waveguides  \n(CPW s) are commonly used as a transmitter and a receiver  [ 9 ]. Each CPW , integrated \nwithin the film , consists of a signal line and two ground lines conn ected at one end. When \na rf-current flows through the transmitter it induces  an oscillating magnetic field around the \nlines that exerts a torque and causes spin precession in the magnetic material beneath. The \ninverse  effect is then used for SW detection by the receiver . Since  the generated magnetic \nfield is not homogenous with reference to the film plane and solely depends on CPW \ngeometry , it determines the distribution of SW wavenumber that can be excited.  \nIt is assumed that the transmitter excites a broa d spectrum of SW wavevectors \nof wavenumber 𝑘 exten ding to 𝑘𝑚𝑎𝑥≈𝜋/𝑊 (𝑊 is a width of  CPW line) with a maximum \nof excitation amplitude approximately around 𝑘𝑚𝑎𝑥𝐴𝑚𝑝≈𝜋/2𝑊 [ 10 ]. The question now is : \nwhat is the actual wavenumber of the SW with the la rgest amplitude  detected by the receiver \nsituated at a certain distance f rom the transmitter. It appears that while in Brillo uin light \nscattering spect roscopy 𝑘 is easily accessible,  in all electrical spin wave spectroscopic \nexperiment s the determination of SW wavenumber  is rather challenging  [ 11 ]. \nWe aim to ans wer this question by analyzing our experimental results of  SW \npropagation  in yttrium iron garnet (Y 3Fe5O12, YIG)  thin film s. YIG films are known \nas possessing the lowe st Gilbert damping parameter enabl ing the SW transmission over  the \ndistances of several hundred micrometers  [ 2, 12 ]. However, YIG films synthesized by \npulsed laser deposition (PLD) exhibit substan tially disparate  values of  anisotropy fields and \nsaturation magnetization , depending on the growth process parameters and , consequently , \nstoichiometr y of the obtained film [ 13, 14, 15 ]. It has already been theoretically predicted 3 \n that anisotropy may significantly affect SW propagat ion and the transmission characteristics  \n[ 16, 17 ]. Therefore , for such  YIG films , SW spectra analysis requires careful consideration \nof anisotropic properties of a given film.  \nHere, we compare two methods of experimental determination of the SW \nwavenumber  which include anisotropy fields. The experimental results are then compared \nwith electromagnetic simulations.  \n \n \nFig. 1 . A θ-2θ XRD  scan of epitaxial YIG film on GGG (111) substrate near the GGG (444) reflection . \n \n \nYIG film was grown on a monocrystalline, 111-oriented Gadolinium Gallium \nGarnet substrate (Gd 3Ga5O12, GGG) by means of  PLD  technique . Substrate temperature was \nset to 650℃  and under  the 1.2×10−4 𝑚𝑏𝑎𝑟  oxygen pressure ( 8×10−8 𝑚𝑏𝑎𝑟  base \npressure)  thin film was deposited at the  0.8 𝑛𝑚/𝑚𝑖𝑛 growth rate using third harmonic \nof Nd:YAG Laser ( 𝜆=355 𝑛𝑚). After the growth, the sample was additionally ann ealed \nex situ at 800℃  for 5 𝑚𝑖𝑛. X-ray diffraction  and reflection  measurements showed that \nthe YIG film was single -phase, epitaxial with the GGG substrate  with the thickness of 82 𝑛𝑚 \nand RMS  roughness of 0.8 𝑛𝑚. XRD θ-2θ scan, presented in Fig. 1, c learly shows the high \ncrystallinity of the YIG film, displaying well defined Laue oscillations , typical for highly \nepitaxial films, which clearly point to the high quality and well textured YIG  (111) film \n[ 18 ]. Subsequently , a system of two CPW s made of 100 𝑛𝑚 thick alumin um was integrated  \nonto YIG film  (Fig. 2) using a maskless photolit hography techniqu e. The width 𝑊 of signal \nand ground lines was equal to  9.8 𝜇𝑚 and the gaps between them were 4 𝜇𝑚 wide. \nThe distance between the centers  of signal l ines was 150 𝜇𝑚. \n49.5 50.0 50.5 51.0 51.5 52.0 52.5101102103104105106Intensity [a.u.]\n2 [deg]YIG (444)\nGGG (444)4 \n  \nFig. 2. SEM  image of the integrated CPW s on the YIG film.  The distance 𝑑 between the transmitter and the \nreceiver  is equal to  150 𝜇𝑚. The depicted  Cartesian and crystallographic coordinate  system is used throughout \nthis paper. The width of signal and ground lines  is marked  with 𝑊. 𝐺 denotes the gap width between the lines.  \n \n \nTo investigate SW propagation we follow ed approach presented in Ref. [ 9 ] and \n[ 12 ]. Using a Vector  Network Analyzer  transmission signal S21 was measured for Damon -\nEshbach surface modes with  wavevector 𝑘⃗  perpendicular to the magnetization  for magnetic \nfields ranging from −310 𝑂𝑒 to +310 𝑂𝑒 (Fig.  3(a)). Exemplary S 21 signal s (imaginary \npart) , whic h are shown in Figs  3(b) and (c) , reveal  a series of oscillations as a function \nof frequency with a Gaussian -like envelope  corresponding to the excited SW wave number  \ndistribution.  Figure  3(c) shows  that frequency separation ∆𝑓 between two oscillation maxi ma \ndiffers noticeably in value  depending on the magnetic field . The decrease in signal amplitude \nis also observed  since SW decay length is inversely proportional to the frequency , so that the \nlow-frequency SWs propagate further away  [ 12, 19 ]. \n \n \n \n5 \n  \nFig. 3. (a) Color -coded SW propagation data S 21 measured at different magnetic fields.  \nWith a red line 𝑓(𝐻) dependence of the uniform excitation ( 𝑘=0) is depicted. The red line corresponds to \nthe maximum in S 11 signal  in (b). The blue dashed line represents a  dispersion relation with 𝐻𝑎=𝐻𝑢=0.     \n(b) Reflection (S 11, 𝑘=0) and transmission (S 21, 𝑘≠0) signals. The plot illustrates a magnified cross -section \nof (a) at 𝐻=−67.5 𝑂𝑒. (c) SW spectra measured  at different magnetic fields.  Color -coding in (b) and (c) \ncorresponds to the one defined in (a).  \n \n \nFor the frequencies of the highest signal am plitude, the wavenumber 𝑘𝑚𝑎𝑥𝐴𝑚𝑝  can \nbe determined according to the dispersion relation derived for (111) crystalline orientation of \nthe YIG film [ 16, 17 ]: \n6 \n  𝑓=𝜇𝐵\n2𝜋ℏ𝑔√(𝐻+2𝜋𝑀𝑠𝑡𝑘)(𝐻−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠−2𝜋𝑀𝑠𝑡𝑘)−1\n2(𝐻𝑎sin (3𝜙))2, (1) \nwhere 𝑓 is the microwave frequency, 𝜇𝐵 – the Bohr magneton constant, ℏ – the reduced \nPlanck constant, 𝑔 – the spectroscopic splitting factor, 𝐻 – the ex ternal magnetic field, 𝑀𝑠 – \nthe saturation magnetization, 𝑡 – the film thickness, 𝑘 – the wavenumber, 𝐻𝑎 – the cubic \nanisotropy field and 𝐻𝑢 – the out -of-plane uniaxial anisotropy field. 𝐻𝑎=2𝐾𝑎\n𝑀𝑠 and 𝐻𝑢=2𝐾𝑢\n𝑀𝑠, \nwhere 𝐾𝑎 and 𝐾𝑢 are anisotropy constants. It should be highlighted that when 𝐻𝑎=𝐻𝑢=0, \nEq. 1 becomes equivalent to the one originally obtained by Damon and Eshbach  [ 20 ]. The \nazimuthal angle 𝜙 define s the in -plane orientation of magnetization direction with respect to \nthe (112̅) axis of YIG film. In our study the term −1\n2(𝐻𝑎sin (3𝜙))2 in Eq. 1 vanishes since \nmagnetic field 𝐻 is parallel to  (112̅) axis and 𝜙=0°. \nAs can be seen from Eq.  1, in order to determine wavenumber  𝑘 one need s \nto evaluate many material constants, namely 𝑔, 𝑀𝑠, 𝑡, 𝐻𝑎, 𝐻𝑢 in the first instance. \nThis problem can be partially  solved with a broadband ferromagnetic resonance measurement  \nof the film. Fo r 𝑘=0 Eq.1  simplifies to the form ula, which  allows for  the determination of \nthe spectroscopic factor 𝑔 and the effective magnetization 4𝜋𝑀𝑒𝑓𝑓∗=−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠: \n 𝑓𝑘=0=𝜇𝐵\n2𝜋ℏ𝑔√𝐻(𝐻+4𝜋𝑀𝑒𝑓𝑓∗). (2) \nTherefore , within this approach , the film thickness and the saturation magnetization should be \ndetermined using  other experimental methods.  \nTo investigate ferromagnetic resonance of the YIG film , the reflection signal S 11 was \nmeasured. In order to avoid extrinsic  contribution to the resonance linewidth caused by non -\nmonochromatic excitation of the CPW (2𝜋∆𝑓𝑒𝑥𝑡𝑟=𝑣𝑔∆𝑘) [ 21 ] and, consequently , possible \nambiguities in  the interpret ation of  resonance peak position , it is recommended to perform \nthis measurement with the use of a wide CPW . Note that the full width a t half maximum of a \nCPW excitation spectra ∆𝑘≈𝑘𝑚𝑎𝑥𝐴𝑚𝑝  [ 21 ]. In our study  we used a CPW with signal and \nground lines  of the width equal to  450 𝜇𝑚 and with the  20 𝜇𝑚 wide gaps between  them. For \nsuch a CPW,  the simulated value of  𝑘𝑚𝑎𝑥𝐴𝑚𝑝  is equal to 49 𝑐𝑚−1 and, therefore, yields \nnegligible broadening that is of the order of a few MHz.  \nThe measured S 11 signal (imaginary part) is depicted in Fig.  3(a) with the red line. It \nappears to lie just below th e S 21 signal. Fitting to the experimental data with Eq.  2 gave  \nfollowing value  of the spectroscopic facto r  𝑔=2.010±0.001 and the effective 7 \n magnetization 𝑀𝑒𝑓𝑓∗=169±7 𝑒𝑚𝑢/𝑐𝑚3. A comparison of 4𝜋𝑀𝑒𝑓𝑓∗ with 4𝜋𝑀𝑠 (𝑀𝑠=\n120±19 𝑒𝑚𝑢/𝑐𝑚3 was measured  using  Vibrating Sample Magnetometry) gives  −1\n2𝐻𝑎−\n𝐻𝑢 of 616 𝑂𝑒, showing the substantial difference  between  obtained  values  of 𝑀𝑒𝑓𝑓∗ and 𝑀𝑠. \nThe determined value of −1\n2𝐻𝑎−𝐻𝑢 remain s in th e midst of the  range reported for  PLD -\ngrown YIG thin films , from 229 𝑂𝑒 up to 999 𝑂𝑒 [ 14, 22 ]. It is worth to mention that for \nfully stoichiometric , micrometer -thick  YIG films made by means of  liquid phase epitaxy  \n(LPE)  technique −1\n2𝐻𝑎−𝐻𝑢=101 𝑂𝑒 [ 14 ]. From  the analysis of resonance linewidth vs . \nfrequency  [ 23 ] we additionally  extracted Gilbert damping parameter of the YIG film , which \nequals to  𝛼=(5.5±0.6)×10−4 and impli es low damping of magnetization precession . \nSubstitution  of the 𝑔, 𝑀𝑒𝑓𝑓∗, 𝑀𝑠 and 𝑡 values  into Eq. 1 enabled the determination of \nwavenumber 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1980±102 𝑐𝑚−1. It sho uld be noted  that if anisotropy fields were \nneglected in the Eq.1 ( 𝐻𝑎=𝐻𝑢=0), yet only saturation magnetization was taken into \naccount , a fitting to the experimental data would  not converge . The calculated dispersion \nrelation with the derived valu e of 𝑘𝑚𝑎𝑥𝐴𝑚𝑝 , assuming 𝐻𝑎=𝐻𝑢=0 is depicted with blue \ndashed line in Fig. 3 (a). Omission of anisotropy fields in magnetization dynamic \nmeasurements  may therefore lead to the significant misinterpretation of  experimental results  \nfor YIG thin films.  \nTypical values of cubic magnetocrystalline anisotropy field 𝐻𝑎 range from  −18 𝑂𝑒 \nto −64 𝑂𝑒 for PLD grown YIG films  [ 14, 15, 22 ], what indicates that resonance \nmeasurements as well as spin wave propagation  are govern ed by the out -of-plane uniaxial \nanisotropy.  For the film employed in our study , the 𝐻𝑢 value is of about  −600 𝑂𝑒 in \nagreement with previous reports  [ 14, 15, 22 ]. For any more complex architecture  of \nmagnonic  waveguides and circuits  it is likewise  imperative t o investigate the  in-plane \nanisotropy properties  [ 24 ]. As  can be seen from Eq.  1 one would  expect  a six-fold \nanisotropy in the plane of (111) -oriented single crystals , that is common  among  rare-earth \nsubstituted YIG garnet s and LPE -YIG films  [ 18, 25, 26, 27 ]. To examine this issue , we \nperformed VSM and angular resolved ferromagnetic resonance measurements. Hysteresis \nloops for all measured in -plane directions exhibit no substantial differences regarding  \ncoercive field ( ≈1.2 𝑂𝑒), saturation field  and saturation magnetization (Fig.  4(a)). The \nangular resolved resonance measurement s confirm  this result  and show  that the (111) YIG \nfilm is isotropic in the film plane  (Fig. 4(b)). The main reason for this behavior is the low  \nvalue of cubic anisotropy field which cause s the resonance frequency modulation  by a value 8 \n of the fraction of  MHz. Such  small differences do not surpass  the experimental error, nor \nwould they  significantly affect the coherent SW propagation.  It is expecte d that t he SW \npropagation characteristics, measured for any other crystallographic orientation, would \ntherefore remain unaltered.  \n \n \n \n \nFig. 4. (a) VSM hysteresis loops measured in the film plane for three different  crystallographic directions. \nThe magneti zation is normalized to the saturation magnetization 𝑀𝑠=120±19 𝑒𝑚𝑢/𝑐𝑚3. A paramagnetic \ncontribution of the GGG substrate was subtracted for each loop.  (b) Resonance frequency as a function \nof azimuthal angle 𝜙 taken at 𝐻=150 𝑂𝑒. The red li ne depicts the calculated values of resonance frequency \naccording to Eq.1 for 𝑘=0, 𝐻𝑎=−30 𝑂𝑒 and 𝐻𝑢=−600 𝑂𝑒. \n \n \n \n-50 -40 -30 -20 -10 0 10 20 30 40 50-1.0-0.50.00.51.0M / MS\nMagnetic Field [Oe] \n \n (a)\n1.41.61.8\n0306090\n120\n150\n180\n210\n240\n2703003301.4\n1.6\n1.8\n(110)(101)(211)\no\n(112)o\n(011)o\n(121)o\noo (211)o (101)oo (110)\no (121)\no (011) Resonance frequency [GHz]\nH = 150 Oeo (112)(b)9 \n Another me thod of extracting SW wavenumber  involves the analysis of the SW \ngroup velocity  𝑣𝑔. Following Ref. [ 21 ], 𝑣𝑔 can be determined from frequency difference ∆𝑓 \nbetween two oscillation maxima in S 21 signal according to the relation:  \n 𝑣𝑔=𝑑∆𝑓, (3) \nwhere 𝑑 is the distance between two CPW s. To determine ∆𝑓 we chose two neighboring \noscillation maxima of the highes t S21 signal  amplitude  as it is shown in Fig. 3 (b) and (c) . \nIn Fig. 5 the derived values of group velocity are shown as a function of magnetic \nfield. It is found that 𝑣𝑔 reaches the  value of 7.6 𝑘𝑚/𝑠 for the field of 1.3 𝑂𝑒 (preferable in  \nmagnonic  information processing devices  of high efficiency ) and 1.4 𝑘𝑚/𝑠 for the field  of \n285 𝑂𝑒. It should be highlighted that such big difference s in 𝑣𝑔 values can be further utilized \nto design  tunable , impulse -response delay  lines  as 𝑣𝑔 changes up to five times  with the \nmagnetic field.  At a distance of 150 𝜇𝑚 between CPWs  it would allow to achieve 20 to \n110 𝑛𝑠 delay times of an impulse.  \n \n \nFig. 5. Spin wave group velocity  as a function of the external magnetic f ield. The red line represents a fit \naccording to Eq.  4. \n \n \nWith the red line in Fig. 5 a fitting is depicted according to:  \n 𝑣𝑔=2𝜋𝜕𝑓\n𝜕𝑘=𝜇𝐵\nℏ𝑔2𝜋𝑀𝑠𝑡(−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠−4𝜋𝑀𝑠𝑡𝑘)\n2√(𝐻+2𝜋𝑀𝑠𝑡𝑘)(𝐻−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠−2𝜋𝑀𝑠𝑡𝑘). (4) \nThe main advantage of extracting SW wavenumber  from 𝑣𝑔(𝐻) dependence is  that it does \nnot require additional measurement of 𝑀𝑠 which is often notably influenced by an error in the \nestimated  film volu me. Since  the saturation magnetization 𝑀𝑠 can be treated as a fitting \n-300 -200 -100 0 100 200 30012345678vg [km/s]\nH [Oe]10 \n parameter in Eq.  4, the derivation  of SW wavenumber  involves only S 11, S21 and thickness \nmeasurement s. The determined values of 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1690±53 𝑐𝑚−1 and 𝑀𝑠=116±\n2 𝑒𝑚𝑢/𝑐𝑚3 remain  in a good agreement with that obtained above  - directly derived from \ndispersion relation  (𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1980±102 𝑐𝑚−1, 𝑀𝑠=120±19 𝑒𝑚𝑢/𝑐𝑚3). \nAs can be seen from Fig ure 5, SW group velocity attains the maximum va lue as the \nmagnetic field approaches 𝐻=0. The maximum value of 𝑣𝑔 is given by:   \n 𝑣𝑔(𝐻=0)≅𝜇𝐵\nℏ𝑔√𝜋𝑀𝑠𝑡\n2𝑘(−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠[1−𝑡𝑘]). (5) \nThe zero -field region may therefore become the subject of interest  for magnonic applications. \nMoreover, Eq. 5 shows that the maximum value of 𝑣𝑔 depends on the anisotropy fields.  PLD -\ngrown YIG films possessing  a high anisotropy would allow faster information processing in \nSW circuits than LPE films for which  the value of  −1\n2𝐻𝑎−𝐻𝑢 is smaller (as it was pointed \nout above).  \nTo confront our experimental results with the expected, theoretical value of  \n𝑘𝑚𝑎𝑥𝐴𝑚𝑝 , we performed electromagnetic simulations in Comsol Multiphysics . Here,  CPW \nwas modeled accordin g to the geometry  of the performed CPW  (Fig.  2), assuming lossless \nconductor metallization, relative permittivity of the substrate 𝜀𝑟=12 and 50 𝛺 port \nimpedance. From the simulated in-plane  distribution of the dynamic magnetic field ℎ𝑥 (inset \nof Fig. 6) an excitation spectra of CPW was obtained  using d iscrete Fourier transformation  of \nℎ𝑥(𝑥). The highest excitation strength is observed for 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1838 𝑐𝑚−1, which \ncorrespond s well to the experimentally  obtained values  within 7% accuracy . The second \nobserved maxima is at 𝑘2=6770 𝑐𝑚−1. However,  as its amplitude is 2 0 times lower with \nrespect to  the amplitude of 𝑘𝑚𝑎𝑥𝐴𝑚𝑝  it is not observed in  the measured  S21 signal . \n \n 11 \n  \nFig. 6. Excitation spectrum  of the CPW with 9.8 𝜇𝑚 wide signal line s and 4 𝜇𝑚 gaps. The inset shows  in-plane \ncomponent of the dynamic  magnetic field excited by the CPW.  \n \n \nTo extend  our study , we performed a series of further simulations for the CPW \ndimensions , which are  achievable with electron - and photolithography.  We assumed equal \nwidths of signal and ground lines (𝑊) as well as equal widths of gaps between them  (𝐺). The \nresults are presented in Fig.  7. It is found  that for the width s 𝑊 ranging from 300 𝑛𝑚 to \n40 𝜇𝑚, the wavenumber 𝑘𝑚𝑎𝑥𝐴𝑚𝑝  vary between 70000 𝑐𝑚−1 and 250 𝑐𝑚−1, respectively , \nrevealing the CPW wavenumber probing limits. We also note that the gap width significantly \naffects  𝑘𝑚𝑎𝑥𝐴𝑚𝑝 . In order to accurately extrapolate its contribution to 𝑘𝑚𝑎𝑥𝐴𝑚𝑝 , we \ndeveloped empirical formula which incorporates width 𝐺: \n 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=2.27\n𝑊+0.6 𝐺. (6) \nThe fittings , according to  the Eq. 6, are depicted in Fig.  7 with solid lines. We f ound that Eq. \n6 is valid  for gap  width 0.1 𝑊<𝐺<2 𝑊. For 𝐺=0.74 𝑊 this formula is equivalent to the \none previously proposed in Ref.  [ 10 ] (𝑘𝑚𝑎𝑥𝐴𝑚𝑝≈𝜋/2𝑊). \n \n0 4000 8000 120000.00.20.40.60.81.0\n-50 -25 0 25 50-2502550hx [Oe]\nx [m]\n-50 -25 0 25 50-2502550hx [Oe]\nx [m]\nk2= 6770 cm-1Amplitude [a.u.]\nk [cm-1]\n-50 -25 0 25 50-2502550hx [Oe]\nx [m]kmaxAmp= 1838 cm-112 \n  \nFig. 7. Wavenumber of the highest amplitude as a function of CPW signal line width. The solid lines represent \na fit according to Eq.  6. \n \nTo conclude, we  report ed on long-range spin wave propagation in the 82 𝑛𝑚 thick \nYIG film  over the distance as large as 150 𝜇𝑚. In order to precisely determine excited \nwavenumber by the coplanar antenna, it is essential to take in to account anisotropy fields \npresent in YIG films. We show ed that anisotropy significantly affect s SW propagation \ncharacteristics, namely it causes an increase in SW frequency as well as in SW group \nvelocity. The main contribution comes from  the out -of-plane uniaxial anisotropy field. T he \ncubic anisotropy field is neglig ibly small  in the YIG (111) film  and it does not affect \nmagnetization dynamics  in the film plane.  We explain ed that the wavenumber determination \nfrom group velocity vs . magnetic field depend ence requires only two types of measurement , \nthat is  broadband SW spectroscopy and the measurement of film thickness.  \n \n \nAcknowledgements  \nThis work was carried out within the Project NANOSPIN  PSPB -045/2010 supported by a \ngrant from Switzerland through the S wiss Contribution to the enlarged European Union . J. \nRychły and J. Dubowik would like to acknowledge support from the European Union’s \nHorizon 2020 MSCA -RISE -2014: Marie Skłodowska -Curie Research and Innovation Staff  \nExchange (RISE) Grant Agreement No. 644 348 (MagIC).  The authors would like to thank  \nProfessor Maciej Krawczyk  for thoughtful suggestions . We also acknowledge valuable \ncomments from Dr. Piotr Graczyk and Paweł Gruszecki .  \n1 10100010000100000\n G = W / 10\n G = W / 4\n G = W / 2\n G = W\n G = 2 WkmaxAmp [1/cm]\nW [m]13 \n References  \n \n[1] Jamali M , Kwon J H, Seo S -M, Lee K -J and Yang H 2016 Spin wave nonreciprocity \nfor logic device applications. Sci. Rep . 3, 3160  \n[2] Chumak A V, Serga A A and Hillebrands B 2014 Magnon transistor for all -magnon \ndata processing. Nat. Comun . 5, 4700  \n[3] Vogt K, Fradin F Y, Pearson J E, Sebastian T, Bader S D, Hille brands B, Hoffmann A \nSchultheiss H 2014 Realization of a spin -wave multiplexer. Nat. Comun . 5, 3727  \n[4] Gertz F, Kozhevnikov A V, Filimonov Y A, Nikonov D E and Khitun A 2015 \nMagnonic Holographic Memory: From Proposal to Device. IEEE J. Explor. Solid -State \nComputat. Devices Circuits  1, 67-75 \n[5] Serga A A, Chumak A V and Hillebrands B 2010 YIG magnonics J. Phys. 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B  77, 054425  "    },    {        "title": "1902.07563v1.CoFeB_MgO_CoFeB_structures_with_orthogonal_easy_axes__perpendicular_anisotropy_and_damping.pdf",        "content": "CoFeB/MgO/CoFeB structures with orthogonal easy\naxes: perpendicular anisotropy and damping\nH. G lowi\u0013 nskia, A. _Zywczakb, J. Wronac, A. Kryszto\fka, I. Go\u0013 scia\u0013 nskaa,\nT. Stobieckid,e, J. Dubowika,\u0003\naInstitute of Molecular Physics, Polish Academy of Sciences, ul. Smoluchowskiego 17,\n60-179 Poznan, Poland\nbAGH University of Science and Technology, Academic Centre of Materials and\nNanotechnology, Al. Mickiewicza 30, 30-059 Krakow, Poland\ncSingulus Technologies AG, Hanauer Landstrasse 103, 63796 Kahl am Main, Germany\ndAGH University of Science and Technology, Department of Electronics, Al. Mickiewicza\n30, 30-059 Krakow, Poland\neAGH University of Science and Technology, Faculty of Physics and Applied Computer\nScience, Al. Mickiewicza 30, 30-059, Krakw Poland\nAbstract\nWe report on the Gilbert damping parameter \u000b, the e\u000bective magnetization\n4\u0019Meff, and the asymmetry of the g-factor in bottom-CoFeB(0.93 nm)/MgO(0.90{\n1.25 nm)/CoFeB(1.31 nm)-top as-deposited systems. Magnetization of CoFeB\nlayers exhibits a speci\fc noncollinear con\fguration with orthogonal easy axes\nand with 4\u0019Meffvalues of +2 :2 kG and\u00002:3 kG for the bottom and top\nlayers, respectively. We show that 4 \u0019Meffdepends on the asymmetry g?\u0000gk\nof theg-factor measured in the perpendicular and the in-plane directions re-\nvealing a highly nonlinear relationship. In contrast, the Gilbert damping is\npractically the same for both layers. Annealing of the \flms results in collinear\neasy axes perpendicular to the plane for both layers. However, the linewidth\n\u0003Corresponding author\nEmail address: dubowik@ifmpan.poznan.pl (J. Dubowik)\nPreprint submitted to Journal of Physics: Condensed Matter February 21, 2019arXiv:1902.07563v1  [cond-mat.mes-hall]  20 Feb 2019is strongly increased due to enhanced inhomogeneous broadening.\nKeywords: ferromagnetic resonance, perpendicular magnetic anisotropy,\nmagnetization precession damping\nPACS: 75.30.Gw, 75.70.Tj, 75.78.-n, 76.50.+g\n1. Introduction\nCoFeB/MgO/CoFeB systems are extensively employed in magnetic tun-\nnel junctions (MTJs), which are important for modern spintronic devices\nsuch as read-heads and magnetic random-access memory [1]. In these ap-\nplications the two key features are the perpendicular magnetic anisotropy\n(PMA) with PMA constant K?and magnetization damping with inhomoge-\nneous (extrinsic) and Gilbert (intrinsic) contributions to the ferromagnetic\nresonance (FMR) linewidth.\nThe FMR linewidth is usually enhanced in Ta/CoFeB/MgO stacks for\nwhich the values of PMA and the Gilbert damping parameter \u000bare scattered\n[2, 3, 4]. Recent experimental results [4, 5] indicate that there is no correlation\nbetweenK?and\u000bin these systems. Speci\fcally, \u000bis approximately constant\nwhile the PMA tends to improve on annealing. However, systems with a high\nPMA have often an increased linewidth due to an inhomogeneous broadening\n[6, 7] so that an extrinsic contribution to the linewidth may be as high as\n400{500 Oe [8] despite \u000bis of 0.01 { 0.02 in these systems. An increase in\nlinewidth is attributed to an angular dispersion of the easy PMA axis, which\nresults in a high inhomogeneous broadening attributed to the zero-frequency\nlinewidth \u0001 H0[6].\nIt has been shown that PMA in CoFe/Ni multilayers is linearly propor-\n2tional to the orbital-moment asymmetry [7, 9] in accordance with the Bruno's\nmodel [see Ref. [7] for discussion]. On the other hand, substantial PMA in\nTa/CoFeB/MgO systems [2] has been considered as related to an inhomoge-\nneous concentration of the anisotropy at the interface [10] so that the Bruno's\nmodel may be not valid in this case. Based on our experimental results, we\naim to shed some light on possible correlation between asymmetry of the\ng-factor and the e\u000bective magnetization 4 \u0019Meff, which are the magnetic\nparameters measured directly in a broadband FMR experiment. According\nto well known Kittel's formula, a departure from the free electron g-factor\nis proportional to \u0016L=\u0016S[11] so that we can discuss the asymmetry of the\ng-factor as well as on the asymmetry of the orbital moment on equal footing.\nHere, we prefer to use asymmetry in g-factor for evaluating the relationship\nbetween orbital moment and PMA.\nAs far as we know, FMR has not yet been thoroughly investigated in\n\"full\" Ta/CoFeB/MgO/CoFeB/Ta MTJ structures. In particular, a depen-\ndence of PMA on the asymmetry in the g-factor has not yet been proved\nin CoFeB/MgO/CoFeB systems. In this paper, we aim to independently\ncharacterize each CoFeB layer separated by a MgO tunnel barrier in terms\nof the\u000bparameter and 4 \u0019Meff. By analyzing FMR measurements in the\nin-plane and out-of-plane con\fgurations, we \fnd that PMA correlates with\ntheg-factor asymmetry in a highly nonlinear relationship.\n2. Experimental methods\nThe samples were sputtered in an Ar atmosphere using a Singulus Timaris\nPVD Cluster Tool. The CoFeB magnetic \flms were deposited by dc-sputtering\n3from a single Co 40Fe40B20target, whereas the MgO barriers were deposited\nby rf-sputtering directly from a sintered MgO target. The samples were de-\nposited on an oxidized silicon wafer with 5 Ta/ 20 Ru /Ta 3 bu\u000ber layers\nand capped with 5 Ta/ 5 Ru (numbers indicate the nominal thickness in\nnanometres). The studied structures consist of two ferromagnetic CoFeB\n(0.93 nm { bottom and 1.31 nm { top) \flms separated by a MgO barrier of\ndi\u000berent thicknesses (0.90, 1.1, and 1.25 nm). It is important to note that we\ninvestigated the as-deposited samples so that the CoFeB layers were amor-\nphous [3, 12]. The e\u000bect of annealing treatment (330oC for 1 hr) on magnetic\nproperties of the system will be discussed at the end of the paper.\nHysteresis loops of the samples were measured by vibrating sample mag-\nnetometer (VSM) with the perpendicular and in-plane magnetic \felds. The\nsaturation magnetization Msof 1200 G in the as-deposited state was deter-\nmined from magnetic moment per unit area vs. CoFeB thickness dependen-\ncies [13]. To investigate anisotropy and damping in studied samples, vector\nnetwork analyzer ferromagnetic resonance (VNA-FMR) spectra of the S21\nparameter were analyzed [14]. VNA-FMR was performed at a constant fre-\nquency (up to 40 GHz) by sweeping an external magnetic \feld, which was\napplied either in-plane or perpendicular to the sample plane. These two con-\n\fgurations will be referred to as the in-plane and out-of-plane con\fgurations.\nExperimental data were \ftted using the Kittel formula\n!\n\rk=q\n(Hr+Ha) (Hr+Ha+ 4\u0019Meff) (1)\nfor the in-plane con\fguration and\n!\n\r?= (Hr\u00004\u0019Meff) (2)\n4for the out-of-plane con\fguration, where != 2\u0019fis the angular microwave\nfrequency,Hrthe resonance \feld, \rk;?=gk;?\u0016B=~the gyromagnetic ratio,\ngkandg?are the spectroscopic g-factors for the in-plane and out-of-plane\ncon\fgurations, respectively, ~the reduced Planck constant, \u0016Bthe Bohr\nmagneton, and Hathe in-plane uniaxial anisotropy \feld. 4 \u0019Meff= 4\u0019Ms\u0000\nH?is the e\u000bective magnetization , where Msis the saturation magnetization,\nandH?= 2K?=Msis the perpendicular anisotropy \feld and K?is the\nperpendicular anisotropy constant. For the in-plane easy axis 4 \u0019Meff>0\nwhereas for the perpendicular to the plane easy axis 4 \u0019Meff<0. According\nto Eqs. (1) and (2), 4 \u0019Meff=\u00002Keff=Ms, whereKeffis the e\u000bective\nanisotropy constant de\fned as K?\u00002\u0019M2\ns[15].\n3. Results and discussion\nFigure 1 (e) presents hysteresis loops of the sample with a 1.25 nm thick\nMgO barrier measured in the out-of-plane (red line) and in-plane con\fgu-\nration (black line). The shape of the loops in both directions is nearly the\nsame for each con\fguration as the saturation \felds (of Hs\u00192 kOe) for both\nlayers have nearly the same magnitude with the opposite signs in 4 \u0019Meff.\nEach hysteresis loop is a sum of the loops typical for the easy and hard axis\nand, as explained below, we can infer from magnetization reversals which\nlayer possesses PMA.\nLet us assume that the bottom CoFeB layer (B) has an in-plane easy axis\nand the top layer (T) has a perpendicular to the plane easy axis so that their\nmagnetization directions are orthogonal at remanence. Three con\fgurations\nof a magnetic \feld Happlied for the magnetization measurements are shown\n5-10 -5 0 5 10-101\n  Normalized moment\nH (kOe)HTT\nBH\ne.a.e.a.\nB\nHe.a.e.a. T\nBe.a.\ne.a.a) b) c)\nB\nT\nB+T\nd)\n 10 51\n-1\n-10 -5 0e)\nH (kOe)Figure 1: (a)-(c) Con\fgurations used for the magnetic measurements with a magnetic\n\feld applied perpendicular or parallel to the \flm plane. (d) Example of schematic pictures\nof the magnetization reversals of a CoFeB/MgO/CoFeB structure for con\fguration (a).\n(e) Hysteresis loops of a CoFeB/MgO/CoFeB structure measured in con\fgurations (a)\n- black line and (b) - red line. The inset shows schematically the model reversals for\ncon\fgurations (a)-black and (b)-red\n.\nin Figs. 1 (a) - (c). These con\fgurations enable magnetization reversals to be\nobserved with Horiented parallel- (a) (perpendicular- (b)) to the easy axis of\nB (T) layer, respectively, or perpendicular to both easy axes (c). Further, we\nwill refer to these con\fgurations as (a), (b), and (c) con\fgurations. As it is\nschematically shown in Fig. 1 (d), an apparent magnetization reversal of B+T\nfor the con\fguration (a) is a sum of independent magnetization reversals of\nB and T. For the perfectly asymmetric structure with 4 \u0019MB\neff=\u00004\u0019MT\neff\nwith the same thickness (i.e. with the same magnetic moments MSVT;B) the\n6apparent magnetization reversals taken in con\fgurations (a) and (b) would\noverlay. However, as it is seen in Fig. 1 (e) they do not completely overlay\nso that the curve taken in the con\fguration (b) lies a bit higher than that\ntaken in (a). As it is shown in the inset of (e), a simple model explains that\nthe T layer (i.e. the with nominal thickness tof 1.3 nm) possesses an easy\naxis perpendicular to the plane, while the B layer with t= 0:93 nm has an\nin-plane easy axis.\nIn the model, the magnetization reversals in each layer can be approxi-\nmated with a normalized relation [16] M(H;S) = arctan[H=H s\u0002tan(\u0019S=2)]=\narctan[H=H max\u0002tan(\u0019S=2)], where Hsof 2 kOe is a saturation \feld for\nthe hard direction and Sis de\fned as a ratio of remanence to the satura-\ntion moment. For Hkparallel to the easy axis, S= 1 (B layer in Fig. 1\n(d)) and for H?perpendicular to the easy axis (T layer in Fig. 1 (d)),\nS= 0:66 as well as Hmax= 10 kOe are arbitrary chosen for the sake of\nsimplicity. The apparent magnetization curve for con\fguration (a) is a sum\n[tB\u0002M(H;S = 1) +tT\u0002M(H;S = 0:66)]=(tB+tT). For the con\fguration\n(b),tTandtBare reversed in the sum. In order to satisfy the experimental\ndata shown in (e), a ratio tB=tT= 0:79. It is easily seen that if the B layer\nhad an in-plane easy axis and the T layer had an easy axis perpendicular to\nthe plane, a curve taken in con\fguration (b) would lie lower than that taken\nin con\fguration (a). Hence, the thin B layer is that with the in-plane easy\naxis.\nFigures 2 (a) and (b) show typical VNA-FMR spectra of the CoFeB/MgO(1.25\nnm)/CoFeB system measured (see Figs. 1) in con\fguration (a) and (b) , re-\nspectively. Two FMR peaks associated with the bottom and top CoFeB lay-\n76 8ImS21(a.u.)\nH(kOe)topbottom(a) 20GHz\nin-planeconfiguration\n4 6 8 10(b)ImS21(a.u.)\nH(kOe)top\nbottom20GHz\nout-of-planeconfiguration\n( )\n()Figure 2: Typical VNA-FMR spectrum of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB\nstructure with resonance peaks from bottom (B) and top (T) layers measured in the in-\nplane (a) and out-of-plane (b) con\fgurations. Solid red lines represent the Lorentzian\n\fts to the experimental data. (c) Dependence of the FMR \feld on the polar angle \u0002\nof applied \feld in X band (9.1 GHz). The easy axis of magnetization of the B is in the\nin-plane orientation. For the T layer, the out-of-plane direction becomes the easy axis.\ners are clearly visible. To determine the resonance \feld Hrand the linewidth\n\u0001Hat constant frequency with a high precision, the spectra were \ftted with\nLorentzians (marked by solid lines in Fig. 2 (a) and (b)). Figure 2 (c) shows\ndependencies of the X-band (9.1 GHz) resonance \felds of the B and T layers\non the polar angle between the \flm normal and the direction of an applied\n\feld. It is clearly seen that the T layer has 4 \u0019Meff<0 (i.e., a perpendicular\neasy axis) and the B layer with 4 \u0019Meff>0 has an in-plane easy axis. From\nFigs. 2 (a) and (b), we can clearly see that the intensity (area under the FMR\n8peak) of the T layer is higher than that of the B layer. This additionally\ncon\frms that the bottom layer has the lower magnetic moment than that of\nthe top layer.\nA typicalHrvs.fdependence, observed for the CoFeB/MgO(1.25 nm)/CoFeB\nsystem is shown in Fig. 3 (a) and (b) for the in-plane (a) and out-of-plane\n(b) con\fguration, respectively. The observed data points are \ftted using\nEqs. (1) and (2). The values of 4 \u0019Meff, obtained from the \ftting are found\nto be of +2 :2 kG and\u00002:3 kG for the bottom and top layers, respectively.\nThefversusHrdata for the B layer were \ftted assuming Haof 30 Oe as\ncon\frmed by VSM measurements (not shown) in the con\fguration presented\nin Fig. 1(c). The values of gkof the top and bottom layers are equal to 2.04\nand 2.08, respectively, in contrast, the values of g?for these layers are 2.06\nand 2.22. One can notice the di\u000berences in values of g?resulting from clear\ndi\u000berences in the slopes of the f(Hr) dependencies (see, Fig. 3 (b)) for the\nbottom (\r?= 2:88 MHz/Oe) and top ( \r?= 3:11 MHz/Oe) layer, respec-\ntively.\nTo sum up, VSM and FMR measurements con\frmed the presence of or-\nthogonal easy axes in our CoFeB/MgO/CoFeB systems and showed that the\nthickness ratio tB=tT= 0:79 is slightly higher than the ratio of nominal thick-\nness (tB\nnom=tT\nnom= 0:71). The thinner B layer has an in-plane easy axis while\nthe T layer has a perpendicular easy axis. However, keeping in mind our for-\nmer studies of a dead magnetic layer (DML) in the Ta/CoFeB/MgO (B) and\nMgO/CoFeB/Ta (T) structures [13] deposited in the same Timaris system,\nwe estimated DMLB'0:23 nm and DMLT'0:4. With such asymmetric\nDMLs the e\u000bective thickness tB\neff'0:7 nm andtT\neff'0:9 nm which satis\fes\n90 5 1005101520\n0 5 10010203040\nf (GHz)\nHr(kOe)\n(b)(a)\nin-plane configuration\nout-of-plane configuration\nf (GHz)\nHr(kOe)1.25 nm MgO\nbottom\ntop0 2 46 8 10048121620\ntop 1.25 nm MgO \n 1.25 nm MgO \n 0.96 nm MgO\n 0.96 nm MgO\n 0.85 nm MgO\n 0.85 nm MgOf (GHz)\nH (kOe)bottomFigure 3: FMR dispersion relations of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB\nstructure measured in the in-plane con\fguration (a) and out-of-plane con\fguration (b).\nThe solid lines show the \fts given in accordance with Eqs. (1) and (2). Inset in (a) shows\nthat the \ftting parameter practically do not depend on the MgO thickness.\ntB=tT= 0:78. VNA-FMR measurements, which o\u000ber a greater precision than\nVSM measurements, give 4 \u0019Meff=\u00002:3 kG (K?= 10:4\u0002106erg/cm3) and\n4\u0019Meff= +2:2 kG (K?= 7:7\u0002106erg/cm3) for the T and B layers, respec-\ntively. All \ftting parameters for a CoFeB/MgO(1.25 nm)/CoFeB structure\nare juxtaposed in Table 1. As it is shown in the inset of Fig. 3 (a), the thick-\nness of MgO spacer within a range of 0.9 { 1.25 nm had almost no in\ruence\non the \ftting parameters, therefore, the values of \ftting parameters 4 \u0019Meff,\ng,\u000b, and \u0001H0are typical for all samples with various MgO thickness.\n10Table 1: Parameters determined from VNA-FMR spectra for the as-deposited\nCoFeB(0.93 nm)/MgO (1.25 nm)/CoFeB(1.31 nm) for the in-plane and out-of-plane con-\n\fgurations: the in-plane anisotropy \feld ( Ha), the e\u000bective magnetization (4 \u0019M eff), spec-\ntroscopicg-factors for in-plane and out-of-plane con\fguration, Gilbert damping ( \u000b), the\nfrequency-independent FMR linewidth (\u0001 H0). The values of the \ftting parameters do\nnot depend on the MgO thickness. The values of g?are marked by asterisks.\nIn-plane con\fguration\nHa(Oe) 4\u0019Meff(kG)gk,g? \u000b \u0001H0(Oe)\ntop 0 -2.29\u00060.05 2.04\u00060.02 0.018\u00060.002 102\u000622\nbottom 30 2.22\u00060.15 2.08\u00060.03 0.017\u00060.002 69\u000623\nOut-of-plane con\fguration\ntop { -2.3\u00060.01 2.22\u00060.01?0.018\u00060.001 95\u000613\nbottom { 2.19\u00060.04 2.06\u00060.02?0.017\u00060.003 160\u000630\nAlthough it is counter-intuitive that the thinner B layer possesses an in-\nplane easy axis, the same feature has been reported for other Ta/CoFeB(1\nnm)/MgO systems deposited in the same Timaris equipment [17]. Similar ef-\nfect has been recently observed in a substrate/MgO/CoFeB/Ta/CoFeB/MgO\nstructure, where the thicker CoFeB layer exhibits a strong PMA in con-\ntrast to the relatively weak PMA in the thinner CoFeB layer [18, 19]. It is\npossible that the growth mode of the MgO layer in contact with an amor-\nphous CoFeB layer might be responsible. The perpendicular anisotropy in\nthese systems originates from the CoFe/MgO interface [20]. The structure\nof the unannealed CoFeB layers is amorphous regardless of underlying lay-\ners, whereas the MgO barrier deposited on the amorphous CoFeB has an\namorphous structure of up to four monolayers (that is about 0.9 nm) [21].\n11Hence, there are subtle di\u000berences between the CoFeB/MgO (bottom) and\nMgO/CoFeB (top) interfaces; the interface of the bottom CoFeB layer is\nmainly amorphous whereas the interface of the top layer is crystalline, be-\ncause the barrier thickness of the investigated samples is above the transition\nfrom amorphous to crystalline phase. Therefore, di\u000berent structures for the\nCoFeB/MgO interfaces may result in di\u000berent values of anisotropy constant.\nAnother explanation is that the measured dependence Keff\u0002teffvs.teff\nin \flms with PMA is often strongly nonlinear due to either intermixing at\ninterfaces [22] or magnetoelastic e\u000bects [15], with Keff\u0002teffexhibiting a\nmaximum as a function of decreasing teffand with the PMA eventually\nbeing lost for small teffof, for example, 0.7 nm.\nThe values of gfactor yield the ratio of the orbital \u0016Land spin\u0016Smag-\nnetic moments in accordance with equation [9, 11]\n\u0016L\n\u0016S=g\u00002\n2; (3)\nwhere\u0016S=\u0016B. Hence, the di\u000berence between orbital moments \u0001 \u0016Lalong\nthe easy and hard direction in the in-plane [Fig. 1 (a)] and out-of-plane [Fig. 1\n(b)] con\fgurations is proportional to ( g?\u0000gk) and reads \u0001 \u0016L=\u0016B(g?\u0000\ngk)=2. \u0001\u0016Lis of 0.09\u0016Band\u00000:01\u0016Bfor the T and B layer, respectively.\nIn CoFe/Ni multilayers [7], the PMA has been shown to be proportional\nto the orbital moment anisotropy in accordance to Bruno model [23]. How-\never, in the case of the CoFeB/MgO systems this direct relationship between\nthe orbital moment asymmetry and the perpendicular anisotropy is not ful-\n\flled. As can be seen in Table 1, ( g?\u0000gk)\u00190 for the B layer corresponds to\n4\u0019Meff= 2:2 kG. Hence, while ( g?\u0000gk) is negligible, a decrease in 4 \u0019Meff\ndue to PMA from 4 \u0019MS= 15 kG to 2.2 kG is substantial. In contrast,\n12(g?\u0000gk)\u00190:18 is exceptionally large for the T layer, while 4 \u0019Meffmerely\ndecreases to - 2.3 kG. In accordance with the earlier report [24], this con\frms\nthat any relationship between the orbital moment asymmetry and the per-\npendicular anisotropy in CoFeB/MgO systems is highly nonlinear. Of course,\nother factors controlled by annealing such as disorder at interfaces and over-\nor underoxidized interfaces would also play a signi\fcant role in PMA [20].\nFuture work con\frming such a nonlinear relationship for a broad range of\ntCoFeB might resolve this issue.\nAt present, there is no doubt that PMA in MgO/CoFeB structures is\nan interface e\u000bect and it is correlated with the presence of oxygen atoms\nat the interface despite the weak spin-orbit coupling [20, 25]. The origin\nof PMA is attributed to hybridization of the O-p with Co(Fe)-d orbitals at\nthe interface [20] and/or to a signi\fcant contribution of thickness dependent\nmagnetoelastic coupling [15]. A deviation of the g-factor from the 2.0 value\nis expressed by g'2\u00004\u0015=\u0001 , where\u0015 < 0 is the spin-orbit constant for\nFe(Co) and \u0001 is the energy levels splitting in the ligand \feld [11]. While\nthe deviation of the g-factor is inversely proportional to \u0001, PMA (and hence\n4\u0019Meff) is proportional to the enhanced spin-orbit-induced splitting around\nthe Fermi level [20]. This may result in a complex relationship between PMA\nandg-factor anisotropy.\nThe Gilbert damping parameter \u000bis evaluated from the dependence of\nthe linewidth \u0001 Hon the resonance frequency as shown in Fig. 4 for the\nin-plane (a) and the out-of-plane (b) con\fgurations. The lines are linear \fts\nto\n\u0001H=\u000b4\u0019f\n\rk;?+ \u0001H0; (4)\n13/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s50/s48/s48/s52/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s40/s98/s41/s40/s97/s41\n/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56/s32/s32/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110\n/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56\n/s32/s32/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41Figure 4: Linewidth as a function of frequency measured in the in-plane con\fguration (a)\nand out-of-plane con\fguration (b). The \u000bdamping parameter is obtained using Eq. (4).\nThe thickness of MgO was 1.25 nm.\nwhere \u0001H0is the inhomogeneous broadening related to CoFeB layer quality.\nThe values of \u000band \u0001H0are shown in Table 1. The top and the bottom layers\nshow almost the same \u000bof 0.017 - 0.018. This suggests that the damping has\nno relation to PMA. While \u0001 H0for the top layer is almost the same for both\ncon\fgurations, \u0001 H0for the bottom layer at the (b) con\fguration is nearly\ntwice as large as that for the (a) con\fguration. Such a behavior suggests\nthat the layer B is rather inhomogeneous with a large angular dispersion of\nmagnetization across the layer [26, 27].\nSpin pumping to Ta layers (which are a part of the bu\u000ber and cap-\n14ping layers, as shown in Fig. 1 (e)) may also in\ruence the damping in\nCoFeB/MgO/CoFeB systems since magnetization precession induces a spin\ncurrent to the adjacent nonmagnetic Ta layers that result in an enhanced\ndamping [8]. This is an interface e\u000bect and hence scales inversely propor-\ntional to the CoFeB layer thickness. Because the bottom layer with an in-\nplane easy axis is thinner than the top layer with a perpendicular easy axis,\nthe spin pumping e\u000bect a\u000bects it more. To estimate spin pumping e\u000bect the\nstandard equation [28] without back\row is used\n\u0001\u000b=g\u0016Bg#\"\n4\u0019Msteff; (5)\nwhereteffis the e\u000bective thickness of CoFeB and g#\"is the mixing con-\nductance. The measured damping of both layers is of 0.017 - 0.018, while\ndamping of a bulk CoFeB is around 0.004 [12]. Therefore, an increase of \u0001 \u000b\ndue to spin pumping is of 0.014 which gives the mixing conductance g#\"= 0:8\nand 1\u00021015cm\u00002for the e\u000bective thickness 0.7 nm and 0.9 nm of B and\nT layer, respectively. The value of mixing conductance g#\"for Ta/CoFeB\ninterface found in the literature lies in a broad range from 1 :67\u00021014to\n1:4\u00021015cm\u00002[29, 30, 31, 32]. Taking into account our simpli\fcation (the\nlack of back\row), this estimation gives the maximal values of mixing conduc-\ntance. Hence, we can conclude that spin pumping substantially in\ruences\nthe damping in our structures. It is worth mentioning that the measured \u000b\nof 0.017 - 0.018 for CoFeB/MgO/CoFeB systems agrees with \u000b= 0:015 for\nthe Ta/CoFeB(1)/MgO structure reported in [3].\nFinally, we would like to make a further comment on postdeposition an-\nnealing of our CoFeB/MgO/CoFeB systems. We found that annealing at\n330oC for 1 hr, beside increasing Msto 1500 G, enhances also PMA so that\n15both layers possess easy axes perpendicular to the plane. 4 \u0019Meffattains\n-1 kG and -4 kG for the B and T layers, respectively. We found that an\nincrease in K?of 7:7\u0002106erg/cm3equally contributes to both layers and,\nfor example, K?= 17\u0002106erg/cm3for the T layer. On the other hand, the\nlinewidth \u0001 Hstrongly broadens to \u0018400 Oe and\u0018700 Oe for the B layer\nand the T layer, respectively. These values are in agreement with recently\nreported values for a similar systems [17]. Moreover, as it is shown in Fig. 5,\n\u0001Hdoes not follow the linear dependence described by Eq. (4). Therefore,\nit is impossible to determine \u000bprecisely for the annealed systems. Such a\nbehavior of \u0001 Hand the decreased remanence with respect to the saturation\nmagnetization (see, [17]) both con\frm a strong angular dispersion of the easy\nPMA axis in both layers. It has been observed that with increasing PMA\nthe dispersion of anisotropy also increases [6, 7, 27]. As a result, dispersion\nin PMA leads to a large two magnon scattering contribution to the linewidth\nfor in-plane magnetization and to an enhanced Gilbert damping [6]. While\nthe magnetic parameters practically do not depend on the MgO thickness in\nas-deposited structures, the annealed structures show a substantial spread in\n4\u0019Meffas it is shown in Fig. 6, which may imply some di\u000berent CoFeB/MgO\ninterfaces due to, for example, boron di\u000busion [30, 33].\n4. Conclusion\nWe investigated the CoFeB/MgO/CoFeB as-deposited systems with the\nin-plane and out-of-plane orthogonal easy axes due to the substantial dif-\nference in PMA for the bottom (B) and the top (T) CoFeB layers, respec-\ntively. The T and the B layer had comparable Gilbert damping \u000bsuggesting\n16/s53 /s49/s48 /s49/s53 /s50/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s32/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66\n/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 5: Linewidth as a function of frequency measured in the in-plane con\fguration for\nthe annealed structure. The thickness of MgO was 1.25 nm.\nthat there is no correlation between the Gilbert damping and PMA. We\nalso showed that 4 \u0019Meffcorrelates with the asymmetry in the g-factor (and\nhence with \u0001 \u0016L) and this correlation is highly nonlinear. Annealing enhances\nPMA in both layers but it has detrimental e\u000bect on the linewidth, however.\nTherefore, despite the Gilbert parameter shows no correlation with PMA, it\nseems that there is some correlation between the linewidth (see Eq. 4) and\nPMA in the annealed systems through a combined e\u000bect between dispersion\nof local anisotropy easy axes in crystallites with a high PMA.\n17/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s32\n/s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s107/s79/s101/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 6: FMR dispersion relations of CoFeB/MgO(0.9 { 1.25 nm)/CoFeB annealed struc-\nture measured in the in-plane con\fguration.\nAcknowledgments\nWe acknowledge support from the the project \\Marie Sk lodowska-Curie\nResearch and Innovation Sta\u000b Exchange (RISE)\" Contract No. 644348 with\nthe European Commission, as part of the Horizon2020 Programme, and\npartially by the project NANOSPIN PSPB-045/2010 under a grant from\nSwitzerland through the Swiss Contribution to the enlarged European Union.\n18References\n[1] B. 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Kurinec,\n\\Study of boron di\u000busion in MgO in CoFeB/MgO \flm stacks using par-\nallel electron energy loss spectroscopy,\" Applied Physics Letters , vol. 94,\nno. 8, p. 082110, 2009.\n24"    },    {        "title": "1902.08700v1.Strongly_Enhanced_Gilbert_Damping_in_3d_Transition_Metal_Ferromagnet_Monolayers_in_Contact_with_Topological_Insulator_Bi2Se3.pdf",        "content": "1 \n Strongly Enhanced Gilbert Damping in 3 d Transition Metal \nFerromagnet Monolayers in Contact with Topological Insulator Bi 2Se3 \nY. S. Hou1, and R. Q. Wu1 \n1 Department of Physics and Astronomy, University of California, Irvine, California \n92697 -4575, USA  \n \nAbstract  \nEngineering Gilbert damping of ferromagnetic metal films is of great importance to \nexploit and design spintronic devices that are operated with an ultrahigh speed. Based on \nscattering theory of Gilbert damping, we extend the torque method originally  used in \nstudies of magnetocrystalline anisotropy to theoretically determine Gilbert dampings of \nferromagnetic metals. This method is utilized to investigate Gilbert dampings of 3 d \ntransition metal ferromagnet iron, cobalt and nickel monolayers that are co ntacted by the \nprototypical topological insulator Bi 2Se3. Amazingly, we find that their Gilbert dampings \nare strongly enhanced by about one order in magnitude, compared with dampings of their \nbulks and free -standing monolayers, owing to the strong spin -orbit coupling of Bi 2Se3. \nOur work provides an attractive route to tailoring Gilbert damping of ferromagnetic \nmetallic films by putting them in contact with topological insulators.  \n  \n \n \n \n \nEmail: wur@uci.edu   \n \n \n \n \n \n 2 \n I. INTRODUCTION  \nIn ferromagnets, the time -evolution of their magnetization M can be described by the \nLandau -Lifshitz -Gilbert (LLG) equation [1-3]  \n1Meff\nSdd\ndt dt   MMM H M\n, \nwhere \n0B g  \n  is the gyromagnetic ratio, and \nMSM  is the saturation \nmagnetization. The first term describes the precession motion of magnetization M about \nthe effective magnetic field, Heff, which  includes contributions from external field, \nmagnetic anisotropy, exchange, dipole -dipole and Dzyaloshinskii -Moriya  interactions [3]. \nThe second term represents the decay of magnetization prece ssion with a dimensionless \nparameter \n , known as the Gilbert damping [4-8]. Gilbert  damping is known to be \nimportant for the performance of various spintronic devices such as hard drives, magnetic \nrandom access memories, spin filters, and magnetic sensors [3, 9, 10]. For example, \nGilbert damping in the free layer of reader head in a magnetic hard drive determines its \nresponse speed and signal -to-noise ratio [11, 12]. The bandwidth, insertion loss , and \nresponse time of a magnetic thin film microwave device also critically depend on the \nvalue of \n  in the film [13]. \n \nThe rapid developm ent of spintronic technologies calls for the ability of tuning Gilbert  \ndamping in a wide range. Several approaches have been proposed for the engineering of \nGilbert damping in ferromagnetic (FM) thin films, by using non -magnetic or rare earth \ndopants, addi ng differ ent seed layers for growth, or adjusting composition ratios in the \ncase of alloy films [9, 14-16]. In par ticular, tuning \n  via contact with other materials \nsuch as heavy metals, topological insulators (TIs), van der Waals monolayers or magnetic \ninsulators is promising as the selection of material combinations is essentially unlimited.  \nSome of these materials may have fundamentally different damping mechanism and offer \nopportunity for studies of new phenomena such as spin -orbit torque, spin -charge \nconversion, and thermal -spin-behavior [17, 18].  \n \nIn this work, we systematically investigate the effect of Bi 2Se3 (BS), a prototypical TI, on \nthe Gilbert damping of 3d transitio n metal (TM) Fe, Co and Ni monolayers (MLs) as they 3 \n are in contacted with each other. We find that the Gilbert dampings in the TM/TI \ncombinations are enhanced by about an order of magnitude than their counterparts in \nbulk Fe, Co and Ni as well as in the fr ee-standing TM MLs.  This drastic enhancement \ncan be attributed to the strong spin -orbit coupling (SOC) of the TI substrate and might \nalso be related to its topological nature . Our work introduces an appealing way to \nengineer Gilbert dampings of FM metal fi lms by using the peculiar physical properties of \nTIs.  \n \nII. COMPUTATIONAL DETAILS  \nOur density functional theory (DFT) calculations are carried out using the Vienna Ab-\ninitio  Simulation Package (VASP) at the level of the generalized gradien t approximation \n[19-22]. We treat Bi -6s6p, Se -4s4p, Fe -3d4s, Co -3d4s and Ni -3d4s as valence electrons \nand employ the projector -augmented wave pseudopotentials to d escribe core -valence \ninteractions [23, 24].  The energy cutoff of plane -wave expansion is 450 eV [22]. The BS \nsubstrate is simulated by five quintuple layers ( QLs), with an in -plane lattice constant of \naBS = 4.164 Å and a vacuum space of 13 Å between slabs along the normal axis. For the \ncomputational convenience, we put Fe, Co and Ni MLs on both sides of the BS slab. For \nthe structural optimization of the BS/TM slabs, a 6× 6× 1 Gamma -centered k -point grid is \nused, and the positions of all atoms except those of the three central BS QLs are fully \nrelaxed with a criterion that the force on each atom is less than 0.01 eV/Å. The van der \nWaals (vdW) correction in the form of the nonlocal vdW functional (optB86b -vdW) [25, \n26] is included in all calculations.  \n \nThe Gilbert dampings are determined by extending the torque method that we developed \nfor the study of magnetocrystalline anisotropy [27, 28]. To ensure the numerical \nconvergence, we use very dense Gamma -centered k -point grids and, furthermore, large \nnumbers of unoccupied bands. For example, the first Bri llouin zone of BS/Fe is sampled \nby a 37× 37× 1 Gamma -centered k -point grid, and the number of bands for the second -\nvariation step is set to 396, twice of the number (188)  of the total valence electrons. More \ncomputational details are given in Appendix A. Mag netocrystalline anisotropy energies \nare determined by computing total energies with different magnetic orientations [29].  4 \n  \nIII. TORQUE METHOD OF DETERMINING GILBERT DAMPING  \nAccording to the scattering theory of Gilbert damping [30, 31], the energy dissipation \nrate of the electroni c system with a Hamiltonian, H(t), is determined by  \n dis 2i j j i F i F j\nijE E E E Euu\n         \nHHuu\n. \nHere, EF is the Fermi level and \nu  is the deviation of  normalized magnetic moment away \nfrom its equilibrium, i.e., \n0m m u  with \n00 MsM m . On the other hand, the time \nderivative of the magnetic energy in the LLG equation is [32] \n mag 3S\neffM dEdt\n MH\n mm\n. \nBy taking \ndis magEE\n  , one obtains the Gilbert damping as:  \n4i j j i F i F j\nij SE E E EM u u\n        \nHH\n. \nNote that, to obtain Eq. (4), we use \n mu  since the eq uilibrium normalized \nmagnetization m0 is a constant. In practical numerical calculations, \nFEE  is \ntypically substituted by the Lorentzian function \n 22\n0 0.5 0.5 L          . \nThe half maximum parameter, \n1 , is adjusted to reflect different scattering rates of \nelectron -hole pairs created by the precession of magnetization M [10]. This procedure  \nhas been already used in several ab initio  calculations for Gilbert dampings of metallic \nsystems [8, 9, 32-35], where the electronic responses play the major role for energy \ndissipation .  \n \nIn this work, we focus on the primary Gilbert damping in FM metals that arises from \nSOC [10, 36-38]. There are two important effects in a uniform precession of \nmagnetization M, when SOC is taken into consideration. The first is the F ermi surface \nbreathing as M rotates, i.e., some occupied states shift to above the Fermi level and some \nunoccupied states shift to below the Fermi level. The second is the transition between \ndifferent states across the Fermi level as the precession can be viewed as a perturbation to 5 \n the system. These two effects generate electron -hole pairs near the Fermi level and their \nrelaxation through lattice scattering leads to the Gilbert damping.  \n \nNow we demonstrate how to obtain the Gilbert damping due to SOC thro ugh extending \nour previous torque method [27]. The general Hamiltonian in Eq. (4) can be replaced by \n SOC j j jr\n H l s\n [4, 27] where  the index j refers to atoms, and \njji lr and s \nare orbital and spin operators, respectively. This is in the same spirit for the determination \nof the magnetocrystal line anisotropy [27], for which  our torque meth od is recognized as a \npowerful tool in the framework of spin -density theory [27]. When m points at the \ndirection of \n , , ,x y zm m m n , the term \nls  in HSOC is written as follows:   \n22\n2211cos sin sin22\n1sin sin cos 52 2 2\n1sin cos sin2 2 2ii\nz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n\n\n\n\n  \n\n     \n   \n  \nn ls\n \nTo obtain the derivatives of H in Eq. (4), we assume that the magnitude of M is constant \nas its direction changes [36]. The processes of getting angular derivatives of H are \nstraightforward and the results are given by Eq. (A1) -(A5) in Appendix B.  \n \nIV. RESULTS AND DISCUSSION  \nIn this section, we first show that our approach of determining  Gilbert damping works \nwell for  FM metals such as 3d TM Fe, Co and Ni bulks. Following that, we demonstrate \nthe strongly enhanced Gilbert dampings of Fe, Co and Ni MLs due to the contact with BS \nand then discuss the underlying physical mechanism of these enhancements.  \n \nA. Gilbe rt dampings of 3d TM Fe, Co and Ni bulks  \nGilbert dampings of 3d TM bcc Fe, hcp Co and fcc Ni bulks calculated by means of our \nextended torque method are consistent with previous theoretical results [10]. As shown in \nFig. 1, the intraband contributions decrease whereas the interband contributions increase \nas the scattering rate \n  increases. The minimum values of  \n have the same magnitude 6 \n as those in Ref. [10] for all three metals, showing the applicability of our approach for the \ndetermination of Gilbert dampings of FM metals.  \n \n \nFigure 1 (color online) Gilbert dampings of (a) bcc Fe, (b) hcp Co and (c) fcc Ni bulks. Black \ncurves give the total Gilbert damping. Red and blue curves give the intraband and interband \ncontributi ons to the total Gilbert damping, respectively.  \n \nB. Strongly enhanced Gilbert dampings of Fe, Co and Ni MLs in contact with BS  \nWe now investigate the magnetic properties of heterostructures of BS and Fe, Co and Ni \nMLs. BS/Fe is taken as an example and its atom arrangement is shown in Fig. 2a. From \nthe spatial distribution of charge density difference \nBS+Fe-ML BS Fe-ML        in Fig. \n2b, we see that there is fairly obvious charge transfer between Fe and the topmost Se \natoms. By taking  the average of \n  in the xy plane, we find that charge transfer mainly \ntakes place near the interface (Fig.2c). Furthermore, the charge transfer induces non -\nnegligible magnetization  in the topmost QL of BS (Fig. 2b). Similar charge  transfers and \ninduced magnetization are also found in BS/Co and BS/Ni (Fig. A1 and Fig. A2 in \n7 \n Appendix C). These suggest that interfacial interactions between BS and 3 d TMs are very \nstrong.  Note that BS/Fe and BS/Co have in -plane easy axes whereas the BS/ Ni has an \nout-of-plane one.     \n \n \nFigure 2 (color online) (a) Top view of atom arrangement  in BS/Fe. (b) Charge density difference \n\n near the interface in BS/Fe. Numbers give the induced magnetic moments (in units of \nB ) in \nthe top most  QL BS. Color bar indicates the weight of negative (blue) and positive (red) charge \ndensity differences. (c) Planer -averaged charge density difference \n in BS/Fe. In (a), (b), (c), \ndark green, light gra y and red balls represent Fe, Se and Bi atoms, respectively.  \n \nFig. 3a and 3b show the \n  dependent Gilbert dampings of BS/Fe, BS/Co and BS/Ni. It is \nstriking that Gilbert dampings of BS/Fe, BS/Co and BS/Ni are enhanced by about one  or \ntwo order s in magnitude  from the counterparts of Fe, Co and Ni bulks as well as their \nfree-standing MLs, depending on the choice of scattering rate in the range from 0. 001 to \n1.0 eV.  Similar to Fe, Co and Ni bulks, the intraband contributions monotonically  \ndecrease while the interband contributions increase as the scattering rate \n  gets larger \n(Fig. A3 in Appendix D).  Note that our calculations indicate that there is no obvious \ndifference between the Gilbert dampings of BS/Fe when f ive- and six -QL BS slabs are \nused (Fig. A4 in Appendix E). This is consistent with the experimental observation that \nthe interaction  between the top and bottom topological surface states is negligible  in BS \nthicker than five QLs [39].  \n \n8 \n  \nFigure 3 (color online) Scattering rate \n  dependent Gilbert dampings of (a) Fe ML, bcc Fe bulk, \nBS/Fe and PbSe/Fe, (b) Co ML, hcp Co bulk, BS/Co, Ni ML, fcc Ni bulk and BS/Ni. (c) \nDependence of the Gilbert dampin g of BS/Fe on the scaled SOC \nBS  of BS in the range from \nzero (\nBS0 ) to full strength (\nBS1 ). Solid lines show the fitting of Gilbert damping \nBS/Fe  \nto Eq. (6). The inse t shows Gilbert damping comparisons between BS/Fe at \nBS0 , bcc Fe bulk \nand Fe ML.  \n \nAs is well -known, TIs are characterized by their strong SOC and topologically nontrivial \nsurface states. An important issue is how they affect the Gilbert damping s in BS/TM \nsystems. Using BS/Fe as an example, we artificially tune the SOC parameter \nBS of BS \nfrom zero to full strength and fit the Gilbert damping \nBS/Fe  in powers of \nBS  as \n2\nBS/Fe 2 BS BS/Fe BS 0 (6)       \n. \nAs shown in Fig. 3c, we obtain two interesting results: (I) when \nBS  is zero, the \ncalculated residual Gilbert damping \nBS/Fe BS 0    is comparable to Gilbert dampings of \nbcc Fe bulk and Fe free -standin g ML (see the inset in Fig. 3c) ; (II) Gilbert damping \nBS/Fe\n increases almost linearly with \n2\nBS  , simi lar to previous results [36]. These reveal \nthat the strong SOC of BS is crucial for the enhancement of Gilbert damping.    \n  \nTo gain insight i nto how the strong SOC of BS affects the damping of BS/Fe, we explore \nthe k-dependent contributions to Gilbert damping, \nBS/Fe . As shown in Fig. 4a , many \nbands near the Fermi  level show strong intermixing between Fe and BS orbitals (mar ked \nby black arrows ). Accordingly, these k-points have large contributions to the Gilbert \n9 \n damping  (marked by red arrows in Fig. 4b). However, if the hybridized states are far \naway from the Fermi level, they make almost zero contribution to the Gilbert damp ing. \nTherefore, we conclude that only hybridizations at or close to Fermi level have dominant \ninfluence on the Gilbert damping. This is understandable, since energy differences EF-Ei \nand EF-Ej are important in the Lorentzian functions in Eq. (4).   \n \n \nFigu re 4 (color online) (a) DFT+SOC calculated band structure of BS/Fe. Color bar indicates \nthe weights of BS (red) and Fe ML (blue). Black dashed line indicates the Fermi level. (b) k -\ndependent contributions to Gilbert damping \nBS/Fe at sc attering rate \n26meV . Inset shows \nthe first Brillouin zone and high -symmetry k -points \n , \nK and \n .  \n \nIt appears that there is no direct link between the topologic al nature of BS and the strong \nenhancement of Gilbert damping. The main contributions to Gilbert damping are not \nfrom the vicinity around the \n -point,  where the topological nature of BS manifests.  \nBesides,  BS should undergo a topol ogical phase transition from trivial to topological as \nits SOC \nBS  increases [40]. If the topological nature of BS dictates the e nhancement of \nGilbert damping, one should expect a kink in the \nBS  curve at this phase transition \npoint but this is obviously absent i n Fig. 3c.  \n \n10 \n To dig deeper into this interesting issue, we replace the topologically nontrivial BS with a \ntopologically trivial insulator PbSe, because the latter has a nearly the same SOC as the \nformer. As shown in Fig. 3a, the Gilbert damping of PbSe/Fe is noticeably smaller than \nthat of BS/Fe, although both are significantly enhanced from the values of \n  of Fe bulk \nand Fe free -standing ML. Taking the similar SOC and surface geometry between BS and \nPbSe  (Fig. A5 in Appendix F) , the large difference between the Gilbert dampings of \nBS/Fe and PbSe/Fe suggests that the topological nature of BS still has an influence on \nGilbert damping. One possibility is that the BS surface is metallic with the presence of \nthe time -reversal protected t opological surface states and hence the interfacial \nhybridization is stronger.      \n \n \nFigure 5 (color online) Comparisons between Gilbert damping \n  of BS/Fe at \n26meV  and \n(a) total DOS, (b) Fe projected DOS and (c) BS projected DOS. Red arrows and light cyan \nrectangles highlight the energy windows where Gilbert damping \n  and the total DOS and Fe \nPDOS have a strong correlation . In (a), (b) and (c), all DOS are in units of state per eV and \nFermi level EF indicated by the vertical green lines is set to be zero.  \n \nA previous study of Fe, Co and Ni bulks suggested a strong correlation between Gilbert \ndamping and total density of states (DOS) around the Fermi level [36]. To attest if this is \n11 \n applicable here, we show the total DOS and Gilbert damping \nBS/Fe  of BS/Fe as a \nfunction of the Fermi level based on the rigid band a pproximation. As shown in Fig. 5a, \nGilbert damping \nBS/Fe and the total DOS behave rather differently in most energy \nregions. From the Fe projected DOS (PDOS) and BS projected PDOS (Fig. 5b and 5c), \nwe see a better  correlation between G ilbert damping \nBS/Fe and Fe -projected DOS, \nespecially in regions highlighted by the cyan rectangles . We perceive that although the \n\n-DOS correlation might work for simple systems, it doesn’t hold when hybridiza tion and \nSOC are complicated as the effective SOC strength may vary from band to band.  \n \nV. SUMMARY  \nIn summary, we extend our previous torque method from determining magnetocrystalline \nanisotropy energies [27, 28] to calculating Gilbert da mping of FM metals and apply this  \nnew approach to Fe, Co and Ni MLs in contact with TI  BS. Remarkably, the presence of \nthe TI BS substrate causes order of magnitude enhancements in their Gilbert dampings. \nOur studies demonstrate such strong enhancement is mainly due to the strong SOC of TI \nBS substrate . The topological nature of BS may also play a role by facilitati ng the \ninterfacial hybridiz ation and leaving more states around the Fermi level . Although \nalloying with heavy elements also enhances Gilbert dampings  [32], the use of  TIs pushes \nthe enhancement into a much wide r range. Our work thus establishes an attractive way \nfor tuning the Gilbert damping of FM metallic films, especially in the ultrathin reg ime.  \n \nACKNOWLEDGMENTS  \nWe thank Prof. A. H. MacDonald and Q. Niu at University Texas, Austin, for insightful \ndiscussions. We also thank Prof. M. Z. Wu at Colorado State University and Prof. J. Shi \nat University of California, Riverside for sharing their ex perimental data before \npublication. Work was supported by DOE -BES (Grant No. DE -FG02 -05ER46237). \nDensity functional theory calculations were performed on parallel computers at NERSC \nsupercomputer centers.   \n \n 12 \n Appendix A: Details of Gilbert damping calcula tions  \nTo compare Gilbert dampings of Fe, Co and Ni free -standing MLs with BS/Fe, BS/Co, \nand BS/Ni, we use \n33  supercells containing three atoms and set their lattice \nconstants to 4.164 Å, same as that of the BS substrate. This means that the lattice \nconstant of their primitive unit cell (containing one atom) is fixed at 2.40 Å. The relaxed \nlattice constants of Fe (2.42 Å), Co (2.35 Å) and Ni (2.36 Å) free -standing MLs are close \nto this value.   \nSystems  a (Å)  b (Å) c (Å) k-point grid            \nFe bulk  2.931  2.931  2.931  35× 35× 35  16 36 2.25 \n Co bulk  2.491  2.491  4.044  37× 37× 23  18 40 2.22 \nNi bulk  3.520  3.520  3.520  31× 31× 31  40 80 2.00 \nFe ML  4.164  4.164  -- 38× 38× 1  24 56 2.33 \nCo ML  4.164  4.164  -- 37× 37× 1  27 64 2.37    \nNi ML  4.164  4.164  -- 39× 39× 1  30 72 2.40 \nBS/Fe  4.164  4.164  -- 37× 37× 1  188 396 2.11 \nBS/Co  4.164  4.164  -- 37× 37× 1  194 408 2.10 \nBS/Ni  4.164  4.164  -- 37× 37× 1  200 432 2.16 \nPbSe/Fe  4.265  4.265  -- 37× 37× 1  174 376 2.16 \n \nTable A1. Here are details of Gilbert damping calculations of all systems that are studied \nin this work.     is abbreviated for the number of  valence electrons and     stands for the \nnumber of total bands.   is the ratio between     and    , namely,          . Note that \nfive QLs of BS are used in calculations for BS/Fe, B S/Co and BS/Ni.   \n \n \n \n \n \nAppendix B: Derivatives of SOC Hamiltonian HSOC with respect to the \nsmall deviation \nu  of magnetic moments  \nBased on the SOC Hamiltonian HSOC in Eq. (5) in the main text, derivatives of the term \nls\n against the polar angle \n  and azimuth angle \n  are    13 \n \n11sin cos cos22\n1 1 1cos sin sin A1 ,2 2 2\n1 1 1cos sin sin2 2 2ii\nnz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n  \n  \n\n\n  \n\n       \n   \n  \nls \nand \n \n  \n 22\n22110 sin sin22\n10 sin cos A2 .2 2 2\n10 cos sin2 2 2ii\nn\nii\niis i l e i l e\ns i l e i l e\ns i l e i l e\n\n\n\n\n\n\n  \n\n         \n     \n    \nls\n \nNote that magnetization M is assumed to have a constant magnitude when it precesses , so \nwe have \n0SOC SOC    H M H m . When the normalized magnetization  m points \nalong the direction of \n , , ,x y zm m m n , we have: \nsin cosxm , \nsin sinym  \nand \ncoszm . Taking \n0m m u  and the chain rule together, we obtain derivatives of \nSOC Hamiltonian HSOC with respect to the small deviation  of magnetic moments as \nfollows:   \nsincos cos A3 ,sinSOC SOC SOC SOC SOC\nx x x x x\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \ncoscos sin A4 ,sinSOC SOC SOC SOC SOC\ny y y y y\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \nand  \nu14 \n \n sin A5 .SOC SOC SOC SOC SOC\nz z z z z\nSOCu m m m m\n\n                 \nH H H H H m\nm\nH \nCombining Eq. (5) and Eq. (A1 -A6), we can easily obtain the final formulas of \nderivatives of SOC Hamiltonian HSOC of magnetization m. \n \n \n \n \nAppendix C: Charge transfers and induced magnetic moments in BS/Fe, \nBS/Co and BS/Ni  \n \nFigure A1 (color online) Planar -averaged char ge difference \nBS TM ML BS TM-ML         \n(TM = Fe, Co and Ni) of  (a) BS/Fe, (b) BS/Co and (c) BS/Ni . The atoms positions are given along \nthe z axis.  \n \n15 \n  \nFigure A2 (Color online) Charge density difference \nBS TM ML BS TM ML          (TM = Fe, \nCo and Ni) nea r the interface betwee n the TM monolayer and the top most  QL BS of (a) BS/Fe, (b) \nBS/Co and (c) BS/Ni. The color bar shows the weights of the negative (blue) and positive (red) \ncharge density differences. Numbers give the induced magnetic moments (in units  of \nB ) in the \ntopmost  QL BS. Bi and Se atoms are shown by the purple and light green balls, respectively.  \n \n \nAppendix D: Contributions of intraband and interband to the Gilbert \ndampings of BS/Fe, BS/Co and BS/Ni  \n \nFigure A3 (color  online) Calculated Gilbert dampings of (a) BS/Fe, (b) BS/Co and (c) BS/Ni. \nBlack curves give the total damping. Red and blue curves give the intraband and interband \ncontributions, respectively.  \n16 \n Appendix E: Gilbert dampings of BS/Fe with five - and six -QLs of BS slabs  \n8  \nFigure A4 (color online). Gilbert dampings of BS/Fe with five (red) and six (black) QLs of BS \nslabs. In the calculations of the Gilbert damping of BS/Fe with six QLs of BS, we use a 39 ×39×1 \nGamma -centered k -point grid, and the number of the  total bands is 448 which is twice \nlarger than the number of the total valence electrons (216).    \n \nAppendix F: Structural c omparisons between BS/Fe and PbSe/Fe  \n \n17 \n Figure A5 (color online) (a) Top view and (c) side view of atom arrangement  in BS/Fe. (b) Top  \nview and (d) side view of atom arrangement  in PbSe/Fe. In (a) and (c), the xyz -coordinates are \nshown by the red arrows. In (b) and (d), the rectangles with blue dashed lines highlight the most \ntop QL BS in BS/Fe which is similar to the Pb and Se atom laye rs in PbSe/Fe. 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Svedlindh1 \n1Department of Engineering Sciences , Uppsala University, Box 534, SE -751 21 Uppsala, Sweden  \n2Seagate Technology, BT48  0BF, Londonderry, United Kingdom  \n3Department of Physics and Astronomy, Uppsala University, Box 516, SE -751 20 Uppsala,  \nSweden  \n \nThe effect s of rhenium doping  in the range 0 – 10 at% on the static and dynamic magnetic \nproperties of Fe65Co35 thin films have been studied experimentally as well as with first principles \nelectronic structure calculations  focussing on the change of the saturation magnetization (𝑀𝑠) and \nthe Gilbert damping parameter ( 𝛼) Both experiment al and theoretical results show that 𝑀𝑠 \ndecreases with increasing Re doping level, while at the same time 𝛼 increases. The experimental \nlow temperature saturation magnetic induction exhibits a 2 9% decrease,  from 2.3 1T to 1. 64T, in \nthe investigated doping concentration range , which is more than predicted by the theoretical \ncalculations. The room temperature value of the damping parameter obtained from ferromagnetic \nresonance measurements , correcting for extrinsic contributions  to the damping,  is for the undoped \nsample 2.7×10−3, which is close to the theoretically calculated Gilbert damping parameter . With \n10 at% Re doping , the damping parameter increases to 9.0×10−3, which is in good agreement \nwith the theoretical value of 7.3×10−3. The increase in damping parameter with Re doping is \nexplained by the increase in density of states at Fermi level, mostly contributed by the s pin-up \nchannel of Re. Moreover, both experimental and theoretical values  for the da mping parameter are \nobserved to be weakly decreas ing with decreasing temperature .  \n  1. INTRODUCTION  \nDuring the last decades , thin films of soft magnetic alloys such as NiFe and FeCo  have been in \nfocus  due to possible use in applications such as spin valves ,1,2 magnetic tunneling junctions ,3,4,5 \nspin injectors ,6 magnetic storage technologies  and in particular in magnetic recording write heads .7 \nBeside s spintronic and magnetic memory devices , such materials are useful for shielding \napplications that are necessary in order to reduce the effect of electromagnetic fields created  by \nelectronic devices. The magnetic damping parameter  of the material play s a critical role for the \nperformance of such spintronic and memory devices as well as for shielding applications.  On the \none hand, a low damping parameter is desired in order to get low critical switching current in \nspintronic devices .8,9,10 On the other hand , a high damping parameter is necessary in order to \nreduce the magetization switching time in magnetic memory devices and to be able to operate \ndevices at high speeds .11 FeCo alloys are promising materials for  high frequency spintronic \napplications and magnetic recording devices due to their high saturation magnetization  (𝑀𝑠), high \npermeability, thermal stability and comparably high resistivity .12,13,14 One possible drawback is \nthat FeCo alloy s exhibit  high coercivity  (𝐻𝑐), which is not favorable for such applications , however \nthis problem can be solved by thin film growth  on suitable buffer layer s.15,16,12 Except coercivity \nproblems, the damping parameter of these materials should be increased to make them com patible \nfor high speed devices . \nDynamic properties of  magnetic materials are highly dependent on the damping parameter. This \nparameter is composed of both intrinsic and extrinsic contributions. The intrinsic contribution is \ncalled  the Gilbert damping and  depends  primarily  on the spin-orbit coupling .17 Intrinsic damping \nis explained as scattering of electrons by phonons and magnons .18,19 Beside s electron scattering , \ndue to the close relation between magnetocrystalline  anisotropy and spin-orbit coupling , it can be \nassumed that the intrinsic damping is also related to the magnetocrystalline anisotropy constant .20 \nRegarding extrinsic damping , there can be a number of different contributions. The most common \ncontribution originates  from two magnon scattering  (TMS) .21 However , this contribution vanishes \nwhen ferromagnetic resonance  (FMR) measurements are performed by applying the static \nmagnetic field  along  the film normal  in inplane anisotropic thin films .22 Beside s TMS , there are \nsome other extrinsic contributions to the damping that are not possible to get rid of  by changing \nthe measurement configuration . One of these contributions  is radiative damping , which arises from \ninductive coupling between  the precessing magnetization and the waveguide  used for FMR \nmeasurem ents.23 Another contribution  for metallic ferromagnetic films is the eddy current \ndamping related to microwave magnetic field induced eddy currents in the thin film s during  \nmeasurement s.23,24 \nIn order to make a soft magnetic thin film  suitible for  a specific  applica tion, taking into account  \nrequirements set by the device application , its damping paramete r should be tailored. As mentioned \nabove , an increased damping parameter is necesssary for devices requiring high switching speed . \nSeveral efforts have been made  for enhanching the damping parameter of soft magnetic materials. \nNiFe alloys constitu te one of the most studied systems in this respect . The most common way to enhance  the intrinsic damping of an all oy is to dope it with differ ent elements . Rare earth elements  \nwith large spin-orbit coupling  have revealed  promising results  as dopant s in terms of increas ed \ndampin g parameter .25,26,27 3d, 4d and 5d transition metals  dopants have also been studied \nexperimentally , revealing an increase of the damping parameter .28,29 Beside s experimental results , \ntheoretical calculations support the idea that transition metals and especially 5d elements can \nenhance the damping parameter of NiF e alloys due to   scattering in presence of chemical disorde r \n, as well as due to the effect of spin -orbit coupling .30 \nAlthough NiFe alloys have been the focus in several extensive studies, FeCo alloys have so far not \nbeen studied to the same extent . Attempts have been made  to  dope  FeCo with Yb,20 Dy,31 Gd,32 \nand Si ,33 where in all cases an increase of the  damping parameter  was observed . Apart from  doping \nof alloys , the addition of  adjacent layers to NiFe and CoFe has also been studied . In particular , \nadding layers consisting of rare earth elem ents with large orbital moment s  gave positive results in \nterms of increased damping parameter .34 \nFe65Co35 alloy s are attractive material s because of high 𝑀𝑠 and reduced 𝐻𝑐 values. However , not \nmuch is known about the magnetic damping mechanism s for this composition . Since it is of \ninterest for high data rate magnetic memory devices, the damping parameter should be increased \nin order to make the magnetic switching faster. To the best of our knowledge , systematic doping \nof Fe 65Co35 with 5d elements has not been studied so far  experimentally . Some of us have found \nfrom ab initio  calculations  that 5d transition metal dopants can increase the damping parameter \nand Re is one of the potential candidates.35 Re is particularly interesting as it has a nice compromise \nof having not so much reduced saturation magnetization and a quite enh anced damping parameter. \nIn this work, we have perfomed a systematic ab initio study of Fe65Co35 doped with increasing Re \nconcentration to find an increasing  damping parameter . The theoretical prediction s are confirmed  \nby results obtained from temperature  dependent FMR measurements performed on Re doped  \nFe65Co35 films.  \n \n2. EXPERIMENTAL  AND THEORETIC AL METHOD S \nRhenium  doped Fe 65Co35 samples were prepared by varying the Re concentration from 0 to 10.23  \nat%. All samples were deposited using DC magnetron sputtering on Si/SiO 2 substrate s. First a 3 \nnm thick  Ru seed layer was deposited on the Si/SiO 2 substrate followed by room temperature \ndeposition of  20 nm  and 40 nm  thick Re -doped  Fe65Co35 films by co -sputtering between Fe 65Co35 \nand  Re target s. Finally, a 3 nm thick Ru layer was deposited as a capping layer over  the Re -doped  \nFe65Co35 film. The nominal Re concentration was derived from the calibrated deposition rate used \nin the deposition system. The nominal Re doping concentration s of the Fe65Co35 samples are as \nfollows ; 0, 2.62, 5.45 and  10.23  at%.  \nThe crystalline structure  of the fims were  investigated by utilizing grazing incident X -Ray \ndiffraction  (GIXRD). The i ncidence angle was fixed at  1o during  GIXRD measurements  and a CuKα source was used.  Accurate values for film thickness and interface roughness were  \ndetermined by X -ray reflectivity  (XRR) measurements.   \nBeside XRD , composition and areal density of the films were deduced by Rutherford \nbackscattering spectrometry36 (RBS)  with ion beams of 2 MeV 4He+ and 10 MeV 12C+. The beams \nwere provided by a 5 MV 15SDH -2 tandem accelerator at the Tandem Laboratory at Uppsala \nUniversity. The experiments were performed with the incident beam at 5° with respect to the \nsurface normal and scattering angles of 170° and 120° . The experimental data was evaluated with \nthe SIMNRA program .37 \nIn-plane magnetic hysteresis measurments were performed using a Magnetic Property \nMeasurement System (MPMS, Quantum Design) .  \nFerromagnetic  resonance  measurements were  performed using  two different techniques. First in-\nplane X -band (9.8 GHz) cavity FMR measurements were performed . The setup  is equipped with a \ngoniometer making it possible to rotate the sample with respect  to the applied magnetic field; in \nthis way the in -plane anisotropy fields of the different samples have been determine d. Beside s \ncavity FMR  studies , a setup for broadband out-of-plane FMR measurements have been utilized . \nFor out -of-plane measurements a vector network analyzer (VNA) was used. Two ports of the VNA \nwere connected to a coplanar waveguide (CPW) mounted on a Ph ysical Property Measurement \nSystem (PPMS, Quantum Design) multi -function probe . The PPMS is equipped with a 9 T \nsuperconducting magnet, which is needed to saturate Fe65Co35 films out -plane and to detect the \nFMR signal. The broadband FMR  measurements were carried out a t a fixed microwave frequency \nusing the field -swept mode, repeating the measurement for different f requencies in the range 15 – \n30GHz.  \nThe theoretical calculations are based on spin -polarized relativistic m ultiple scattering theory using \nthe Korringa -Kohn -Rostoker (KKR) formalism implemented in the  spin polarized relativistic \nKKR code  (SPR-KKR) . The Perdew -Burke -Ernzerhof (PBE) exchange -correlation functional \nwithin generalized gradient approximation was used. The equilibrium lattice parameter s were \nobtained by energy minimization for each composition. Substitutional disorder was treated within \nthe Coherent Potential Approximation (CPA). The damping parameters were calcu lated by the \nmethod proposed by Mankovsky et al.,38 based on the ab initio Green's function technique and  \nlinear res ponse formalism where one takes into consi deration scattering processes as well as spin -\norbit coupling built in Dirac's relativistic formulation. The calculations of Gilbert damping \nparameters at finite temperatures were  done using an alloy -analo gy model of atomic displacements \ncorresponding to the thermal average of the root mean square displacement at a given temperature.  \n3. RESULTS AND  DISCUSSION  \nRe concentrations and layer thickness (areal densities) of the 20 nm  doped films were obtained by \nRBS experiments. RBS employing a beam of 2 MeV He primary ions was used to deduce the areal \nconcentration of each layer. Additional measurements with 10 MeV C probing particles permit to resolve the atomic fractions of Fe,  Co and Re. The spectra for the  samples with different Re \nconcentration are shown in Fig. A1 . The measured Re concentrations are 3.0±0.1  at%, 6.6±0.3  at% \nand 12.6±0.5  at%. Moreover, the results for Fe and Co atomic fractions show that there is no \npreferential replacement by Re , implying that the two elements are replaced according to their \nrespective concentration . \nFigure 1 (a) shows GIXRD spectra in the 2𝜃-range from 20o to 120o for the Fe65Co35 films with \ndifferent Re concentration. Diffraction peaks corresponding to the body centered cubic Fe 65Co35 \nstructure have been indexed in the figure; no other diffraction peaks appear in the different spectra.  \nDepending on the Re -dopant concentration shi fts in the peak positions are observed, the diffraction \npeaks are suppressed to lower 2𝜃-values with increasing dopant concentration . The shift for the \n(110) peak for the different dopant concentrations is given as an inset in Fig. 1 (a). Similar shifts \nare observed for the other diffraction peaks. This trend in peak shift is an experimental evidence \nof an increasing amount of Re dopant within the deposited thin films. Since the peaks are shifted \ntowards lower 2𝜃-values  with increasing amount of Re dopant , the lattice parameter increases with \nincreasing Re concentration.39 Figure 1 (b) shows the experimental as well as theoretically \ncalculated lattice parameter versus Re concentration.  The qualitative agreement between theory \nand experiment is obtained. However, t he rate of lattice parameter increase with increasing Re \nconcentration is larger for the theoretically calculated lattice parameter. This is not unexpected as \nthe generalized gradie nt approximation for the exchange -correlation potential has a tendency to \noverestimate the lattice parameter. Another possible explanation for the difference in lattice \nparameter is that the increase of the lattice parameter for the Re -doped Fe 65Co35 films  is held back \nby the compressive strain due to lattice mismatch with Si/SiO 2/Ru. XRR measurements revealed \nthat the surface roughness of the Fe 65Co35 films is less than 1 nm , which cannot affect static and \nmagnetic properties drastically. Results from XRR measurements are given in table 1.  \nRoom temperature normalized magnetization curves for the Re-doped Fe 65Co35 films  are shown  \nin Fig. 2 (a) . The coercivity for  all films is in the range of 2 mT and all films, except for the 1 2.6 \nat% Re doped film that show a slightly rounded hysteresis loop, exhibit rectangular hysteresis \nloops. The low value for the coercivity is expected for seed layer grown films .15 The \nexperimentally determined  low temperature saturation magnetization  together with the \ntheoretically calculated magnetization versus Re concentra tion are shown in Fig. 2 (b). As \nexpected, both experimental and theoretical r esults show that the saturation magnetization \ndecreases with increasing Re concentration . A linear decrease in magnetization is observed in the \ntheoretical calculations whereas a non -linear behavior is seen in the experimental data.  \nAngle resolved cavity FMR measurements were used to study the in -plane magnetic anisotropy . \nThe angular -dependent resonant field ( 𝐻𝑟) data was analyzed using the following equation ,40 \n 𝑓=µ0𝛾\n2𝜋[{𝐻𝑟cos(𝜙𝐻−𝜙𝑀)+𝐻𝑐\n2cos4(𝜙𝑀−𝜙𝐶)+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}{𝐻𝑟cos(𝜙𝐻−\n𝜙𝑀)+𝑀𝑒𝑓𝑓+𝐻𝑐\n8(3+cos4(𝜙𝑀−𝜙𝐶))+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}]12⁄\n,  (1) where  𝑓 is the cavity resonance frequency  and 𝛾 is the gyromagnetic ratio . 𝜙𝐻, 𝜙𝑀, 𝜙𝑢 and 𝜙𝐶 \nare the in -plane directions for the magnetic field,  magnetization, uniaxial anisotropy and cubic \nanisotropy, respectively, with respect to  the [100 ] direction of the Si substrate.   𝐻𝑢=2𝐾𝑢\nµ0𝑀𝑠 and \n𝐻𝑐=4𝐾𝑐\nµ0𝑀𝑠  are the uniaxial and cubic anisotropy fields, where 𝐾𝑢 and 𝐾𝑐 are the uniaxial and cubic \nmagnetic anisotropy constants , and  𝑀𝑒𝑓𝑓 is the effective magnetization.  Fitting parameters were \nlimited to 𝑀𝑒𝑓𝑓,  𝛾 and 𝐻𝑢, since the Hr versus ϕH curves did not give any indication of a cubic \nanisotropy.  \nFigure 3 shows 𝐻𝑟 versus 𝜙𝐻 extracted from the angular -dependent FMR measurements together \nwith fits according to Eq. (1), clearly revealing dominant twofold uniaxial in -plane magnetic \nanisotropy. Extracted anisotropy field and effective magnetization values are given in  Table 2 . The \nresults show that 𝐻𝑢 is within the accuracy of the experiment independent of the Re concentration . \nTemperature dependent o ut-of-plane FMR measurements  were  performed in the temperature range \n50 K to 300 K  recording the complex transmission coefficient 𝑆21. Typical field -swept results for \nthe r eal and imaginary components of  𝑆21 for the undoped and 1 2.6 at% Re-doped samples are \nshown  in Fig. 4. The field -dependent 𝑆21 data was fitted to the following set of equations,41  \n𝑆21(𝐻,𝑡)=𝑆210+𝐷𝑡+𝜒(𝐻)\n𝜒̃0    \n𝜒(𝐻)=𝑀𝑒𝑓𝑓(𝐻−𝑀𝑒𝑓𝑓)\n(𝐻−𝑀𝑒𝑓𝑓)2−𝐻𝑒𝑓𝑓2−𝑖𝛥𝐻 (𝐻−𝑀𝑒𝑓𝑓) .  (2) \nIn these equations 𝑆210 corresponds to  the non-magnetic contribution to the complex transmission \nsignal , 𝜒̃0 is an imaginary function of the microwave frequency and film thickness  and 𝜒(𝐻) is the \ncomplex susceptibility  of the magnetic film. The term 𝐷𝑡 accounts for a linear drift of the recorded \n𝑆21 signal.  𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠, where 𝐻𝑘⫠ is the perpendicular anisotropy field  and 𝐻𝑒𝑓𝑓=2𝑓\n𝛾µ0. The \n𝑆21 spectra were  fitted to Eq. (2 ) in order to extract the linewidth 𝛥𝐻 and 𝐻𝑟 values. Fits t o Eq. (2) \nare shown as solid lines in Fig. 4.  \nThe experimentally measured total d amping parameter ( 𝛼𝑡𝑜𝑡𝑎𝑙 ), including both the intrinsic \ncontribution  (Gilbert damping) and extrinsic contributions , was extracted by fitting 𝛥𝐻 versus \nfrequency to the following expression, 41 \nµ0𝛥𝐻=4𝛼𝑡𝑜𝑡𝑎𝑙 𝑓\n𝛾+µ0𝛥𝐻0 ,   (3) \nwhere 𝛥𝐻0 is the frequency independent linewidth broadening  due to sample inhomogeneity . \nBeside s 𝛼𝑡𝑜𝑡𝑎𝑙 , 𝑀𝑒𝑓𝑓 can also be  extracted by fitting the 𝐻𝑟 versus frequency results to the \nexpression  µ0𝐻𝑟=2𝜋𝑓\n𝛾+µ0𝑀𝑒𝑓𝑓 .   (4) \nTypical temperature dependent results for 𝑓  versus 𝐻𝑟  and 𝛥𝐻 versus 𝑓  are shown in Fig. 5 for \nthe 1 2.6 at% Re -doped Fe65Co35 film. Extracted values of 𝑀𝑒𝑓𝑓  at different temperatures are given \nin Table 3 for all samples . As expected, the results show that 𝑀𝑒𝑓𝑓 decreas es with increasing \ndopant concentration.  Since 𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠  and the film thickness is large enough to make a \npossible contribution from out -of-plane anisotropy negligible one can make the justified \nassumption that 𝑀𝑒𝑓𝑓≈𝑀𝑠. The analysis using Eqs. (2) – (4) also give values for the Land é 𝑔-\nfactor ( 𝛾=𝑔µ𝐵\nħ), yielding 2.064 and 2.075 for the undoped and 12.6 at% doped samples, \nrespectively (similar values are obtained at all temperatures).  \nAs indicated above, the d amping parameters extracte d from FMR measurements ( 𝛼𝑡𝑜𝑡𝑎𝑙) include  \nboth intrinsic and extrinsic contributions. One of the most common extrinsic contribution s is TMS , \nwhich is avoided in this study by measuring FMR with the magnetic field applied out of the film \nplane. Except TMS , extrinsic contributions  such as eddy curr ent damping  and radiative damping \nare expected to contribute the measured damping . In a metallic  ferromagnet, which is placed on \ntop of a CPW , precession  of spin waves induces AC currents in the ferromagnet ic film, thereby \ndissipating energy . The radiative  damping has similar origin as the eddy current damping, but here \nthe precession of the magnetization induces microwave -frequency currents in the CPW where \nenergy is dissipated. Thus,  there are two extrinsic  contributions to the measured damping ; one that \nis caused by eddy currents in the ferromagnet ic film (𝛼𝑒𝑑𝑑𝑦) and  another one caused by  eddy \ncurrents in the CPW ( 𝛼𝑟𝑎𝑑).23 In order to  obtain the reduced  damping of the films  (𝛼𝑟𝑒𝑑), which \nwe expect to be close to the intrinsic damping of the films, the extrinsic contributions should be \nsubtracted from  𝛼𝑡𝑜𝑡. We have neglected any contribution to the measured damping originating \nfrom spin -pumping into seed and capping layers. However, since spin -pumping in low spin -orbit \ncoupling materials like Ru with thickness quite less than the spin -diffusion length is quit e small, \nthe assumption  of negligible contribution from spin -pumping  is justified.  The t otal damping can \nthus be given as  𝛼𝑡𝑜𝑡=𝛼𝑟𝑒𝑑+𝛼𝑟𝑎𝑑+𝛼𝑒𝑑𝑑𝑦 .  \nWhen the precession of the magnetization is assumed to be uniform in the sample , the expression \nfor  radiative damping can be given as23 \n𝛼𝑟𝑎𝑑=𝜂𝛾µ02𝑀𝑠𝛿𝑙\n2𝑍0𝑤 ,  (5) \nwhere 𝑍0 =50 Ω is the waveguide impedance, 𝑤=240 µm is the width  of the CPW center \nconductor , 𝜂 is a dimensionless parameter that accounts for FMR mode profile, δ is the thickness \nand 𝑙 is the length of the sample. The l ength of all samples were  4mm and the thickness  20nm for  \nthe undoped and 1 2.6 at% Re-doped films and 40nm for  the 3.0 at% and 6.6 at% Re-doped films. \nTemperature dependent radiative damping contributions for all Fe 65Co35 films are given  in Table \n4. Beside s 𝛼𝑟𝑎𝑑, the 𝛼𝑒𝑑𝑑𝑦 contribution should also be calculated and extracted from 𝛼𝑡𝑜𝑡𝑎𝑙 to extract \nthe reduced  damping parameter. 𝛼𝑒𝑑𝑑𝑦 can be estimated by the expression23 \n𝛼𝑒𝑑𝑑𝑦 =𝐶𝛾µ02𝑀𝑠𝛿2\n16𝜌 ,  (6) \nwhere  𝐶 is a parameter describing the  distribution of eddy current s within the films  and its value \nis 0.5 in our studied samples and 𝜌 is the resistivity of the films. Resistivity is measured for all \nfilms with different dopant concentrations at different temperatures. It is in the range of 8.2×10-8 \nto 5.6 ×10-8 𝛺𝑚 for undoped, 5.7 ×10-7 to 5.3 ×10-7 𝛺𝑚 for 3.0  at% doped , 6.9 ×10-7 to 6.1 ×10-\n7 𝛺𝑚 for 6.6  at% doped and 3.9×10-7 to 3.6 ×10-7 𝛺𝑚 for 12.6 at% doped films. Temperature \ndependent eddy current damping contributions , which are negligible,  for all Fe 65Co35 films are \ngiven in Table  5. \n𝛼𝑡𝑜𝑡 (filled symbols) and 𝛼𝑟𝑒𝑑  (open symbols) versus temperature for the differently Re -doped \nFe65Co35 films are shown in Fig. 6 .  Both  damping parameter s slowly decrease  with decreasing \ntempera ture. Moreover, the damping parameter increases with increasing Re concentration; the \ndamping parameter is 4 times as large for  the 12.6 at% Re -doped sample compared to the undoped \nsample . Since the damping parameter depends both on disorder induced scattering and spin-orbit \ncoupling, the observed enhanc ement of the damping parameter can emerge from the electronic \nstructure of the  alloy and  large spin -orbit coupling of Re.  \nA c omparison between temperature dependent experimental 𝛼𝑡𝑜𝑡 and 𝛼𝑟𝑒𝑑 values and \ntheoretically calculated intrinsic damping parameters is shown in Fig. 7 for the undoped and 12.6 \nat% Re -doped Fe 65Co35 films.  In agreement with the experimental results, the theoretically \ncalculated damping parameters decrease in magnitude with decreasing temperature . It has been \nargued  by Schoen et al., 42 that the contribution to the intrinsic Gilbert damping parameter comes \nprimarily from the strong electron -phonon coupling at high temperatures  due to interband \ntransition whereas at a low temperature, density of states at Fermi level (𝑛(𝐸𝐹)) and spin -orbit \ncoupling  give rise to intraband transition.  In Fig. 8, we show the correspondence between the \ncalculated damping parameter at 10 K with the density of states (spin up +spin down) at Fermi \nlevel for varying Re concentration. The increasing trend in both  properties is obviously seen. The \nincrease in DOS mainly comes from increasing DOS at Re sites in the spin -up channel. In the \ninset, the calculated spin -polarization as a function of Re concentration is shown. Spin polarization \nis defined as   𝜁=𝑛(𝐸𝐹)↑−𝑛(𝐸𝐹)↓\n𝑛(𝐸𝐹)↑+𝑛(𝐸𝐹)↓ where the contribution from both spin channels are seen. It is \nclearly observed that Re doping decreases the spin polarization.  \n \nOne should note that a quantitative comparison between theory and experiment requires more \nrigoro us theoretical considerations. The difference  between experimental and theoretical results \nfor the damping parameter may be  explained by the incompleteness of the model used to calculate \nthe Gilbert damping parameter  by neglecting several complex scatterin g processes. Firstly, the effect of spin fluctuations was neglected, which in principle could be considered in the present \nmethodology if the temperature dependent magnetization and hence information about the \nfluctuations of atomic moments were available from Monte -Carlo simulations. Other effects such \nas non-local damping and more sophisticated treatment of atomic displac ements in terms of \nphonon self -energies40  that may contribute to the relaxation of the magnetization in magnetic thin \nfilm materials  have been neglected . Nevertheless, a qualitative agreement has been achieved where  \nboth experimental  and theoretical results show that there is a significant increase of the damping \nparameter with increasing concentration of Re.  \n \n4. CONCLUSION  \nStatic and dynamic magnetic properties of rhenium doped Fe 65Co35 thin films have been \ninvestigated and clarified in a combined experimental and theoretical study. Results from first \nprinciples theoretical calculations show that the saturation magnetization gradually decreases with \nincreasing Re concentration, from 2.3T  for the undoped sample to 1.95T for the 10% Re -doped \nsample. The experimental results for the dependence of the saturation magnetization on the Re -\ndoping are in agreement with the theoretical results, although indicating a more pronounced \ndecrease of the saturation magnetization for the largest doping concentrations. The theoretical \ncalculations show that the intrinsic Gilbert damping increases with increasing Re concentration; at \nroom temperature the damping parameter  is 2.8×10−3, which increases to 7.3×10−3 for the 10  \nat% Re -doped sample. Moreover, temperature dependent calculations of the Gilbert damping \nparameter reveal a weak decrease of the value with decreasing temperature . At a low temperature, \nour theoretical analysis showed the prominence of intra band contribution arising from an increase \nin the density of states at Fermi level. The experimental results for the damping parameter were \ncorrected for radiative and eddy current contributions to the measured damping parameter and \nreveal similar trends as observed in the theoretical results; the damping parameter increases with \nincreasing Re concentration and the damping parameter value decreases with decreasing \ntemperature. The room temperature value for the reduced damping paramet er was  2.7×10−3 for \nthe undoped sample, which increased to 9.0×10−3 for the 1 2.6 at% Re -doped film. The \npossibility to e nhanc e the damping parameter for Fe65Co35 thin films is a  promising result since \nthese materials are used in magnetic memory applications and higher data rates are achievable if \nthe damping parameter of the material is increased.   \n \nACKNOWLEDGEMENT  \nThis work is supported by the Knut and Alice Wallenberg (KAW) Fou ndation, Grant No. KAW \n2012.0031 and by the Marie Curie Action “Industry -Academia Partnership and Pathways” (ref. \n612170, FP7 -PEOPLE -2013 -IAPP). The authors acknowledge financial support from Swedish \nResearch Council (grant no. 2016 -05366) and Carl Tryggers Stiftelse (grant no. CTS 12:419 and \n13:413). The simulations were performed on resources provided by the Swedish National Infrastructure (SNIC) at National Supercomputer Centre at Link öping University (NSC). M. 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Nembach, O. Eriksson, O. \nKaris, and J. M. Shaw, Nature Physics, 12, 839 –842 (2016).  \n \n     \nFigure 1 (a) GIXRD plot for Fe 65Co35 films with dif ferent Re concentrations. S hift of (110) peak \ndiffraction peak with Re concentration is given as insert . (b) Lattice parameter versus Re \nconcentration.  Circles are lattice parameters extracted from XRD measurements and squares are \ncalculated th eoretical values. Line s are guide to the eye. \n  \nFigure 2 (a) Normalized room temperature magnetization versus magnetic field for Fe 65Co35 \nfilms with different Re concentration . (b) Low temperature saturation magnetization versus Re \nconcentration. Circles  are experimental data and squares  corresponding calculated results.  \nExperimental 𝝁𝟎𝑴𝒔 values were extracted from temperature dependent FMR results. Lines are \nguide s to the eye .  \n  \n \n \nFigure 3 𝝁𝟎𝑯𝒓 versus in -plane angle of magnetic field 𝝓𝑯 for different dopant concentrations  of \nRe. Black squares are experimental  data and red line s are fits to Eq. (1).  \n  \n \n \nFigure 4 Room temperature  real (a and c) and imaginary  (b and d)  𝑺𝟐𝟏 components versus out -\nof-plane magnetic field  for Fe65Co35 thin  films with 0% and 12.6 at% Re recorded at 20GHz . \nBlack squares are data points and red lines are fit s to Eq. (2). \n \n  \n \n \nFigure 5 (a) Frequency versus  𝝁𝟎𝑯𝒓 values at different temperatures for  the Fe65Co35 thin film \nwith 12.6 at% Re. Coloured lines correspond to fits to Eq. ( 4). (b) Linewidth 𝝁𝟎∆𝑯 versus \nfrequency at different temperatures for the same Re doping concentration. Coloured  lines \ncorrespond to  fits to Eq. ( 3). Symbols represent  experimental  data. \n  \n \n \nFigur e 6  𝜶𝒕𝒐𝒕 versus  temperature for Fe 65Co35 thin films with different concentration of Re. \nBesides showing  𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of radiative \ndamping and eddy current damping contributions from  𝜶𝒕𝒐𝒕. Error bars are  given for measured \n𝜶𝒕𝒐𝒕 (same size as symbol size ). \n  \n \nFigur e 7 𝜶𝒕𝒐𝒕 versus  temperature for Fe 65Co35 thin films with 0  at% and 1 2.6 at% concentration \nof Re. Beside s showing  𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of \nradiative damping and eddy current damping contribution s from  𝜶𝒕𝒐𝒕. In addition to \nexperimental results theoretically calculated intrinsic damping parameters  are given for the \nsimilar  concentrations  of Re . Error bars are given for  measured 𝜶𝒕𝒐𝒕 (same size as symbol size) . \n  \n \n \nFigure 8  Calculated density of states at Fermi level (left axis) and damping parameter (right \naxis) are shown as a function of Re concentration. In the inset, spin -polarization is shown  as a \nfunction of Re concentration.  \n  \n0 0.03 0.06 0.09 0.12\nRe concentration0.90.951DOS at EF (States/eV)\n0 0.03 0.06 0.09 0.12\nRe concentration0.350.40.450.50.55Spin polarization\n0123456\nDamping parameter (x 10-3)Re \n(at%)  𝑡𝑅𝑢,𝑐𝑎𝑝 \n(nm)  𝜎  \n(nm)  𝑡𝐹𝑒𝐶𝑜 \n(nm)  𝜎  \n(nm)  𝑡𝑅𝑢,𝑠𝑒𝑒𝑑 \n(nm)  \n  \n(nm)  \n0 2.46 1.89 39.71  0.67 2.74 0.66 \n3.0 2.47 1.80 37.47 0.59 2.45 1.03 \n6.6 1.85 0.50 37.47 0.51 2.13 0.90 \n12.6 2.15 1.49 37.38 0.64 1.89 1.03 \nTable 1 Thickness and roughness (𝝈) values for different layers in films  extracted  from XRR \ndata. Error margin is 0.02nm for all thickness and roughness values.   \n \nRe (at%) 𝜇0𝐻𝑢 (mT) 𝜇0𝑀𝑒𝑓𝑓 (T) \n0 2.20 2.31 \n3.0 2.10 2.12 \n6.6 2.30 1.95 \n12.6 2.20 1.64 \nTable 2 Room temperature  𝝁𝟎𝑴𝒆𝒇𝒇 and  𝝁𝟎𝑯𝒖 values  for Fe 65Co35 films with different \nconcentration  of Re extracted by fitting the angle dependent cavity FMR data to Eq. (1). \n \n \nTemperature  (K) 0% Re  3.0 at% Re 6.6 at% Re 12.6 at% Re \n𝜇0𝑀𝑒𝑓𝑓  (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) \n300 2.29 2.16 1.99 1.61 \n200 2.31 2.16 2.04 1.67 \n150 2.33 2.24 2.06 1.70 \n100 2.36 2.25 2.07 1.72 \n50 2.36 2.27 2.08 1.74 \nTable 3 Temperature dependent  𝝁𝟎𝑴𝒆𝒇𝒇 values for Fe65Co35 films with  different concentrati on \nof Re extracted by fitting broadband out -of-plane FMR data to Eq. (4). Error margin is about  10 \nmT. \n \n \nTemperature(K)  𝛼𝑟𝑎𝑑 (×10-3) \n0% Re  3.0 at% Re 6.6 at% Re 12.6 at% Re \n300 0.218  0.482  0.438  0.154  \n200 0.222  0.494  0.450  0.160  \n150 0.216  0.499  0.454  0.162  \n100 0.225  0.502  0.456  0.219  \n50 0.221  0.505  0.458  0.166  \nTable 4 Temperature dependent r adiative damping contribution to total damping parameter for \nFe65Co35 films with  different concentration of Re  calculated  using Eq.  (5). \n  \nTemperature(K)  𝛼𝑒𝑑𝑑𝑦  (×10-3) \n0% Re  3.3 at% Re 6.6 at% Re 12.6 at% Re \n300 0.038  0.077  0.064  0.006  \n200 0.047  0.081  0.067  0.006  \n150 0.050  0.084  0.070  0.006  \n100 0.055  0.084  0.073  0.007  \n50 0.058  0.086  0.075  0.007  \nTable 5 Temperature dependent eddy current damping contribution to total damping parameter \nfor Fe 65Co35 films with  different concentration of Re calculated using Eq. ( 6).     \n \nFigure A1  RBS spectra for the Re -doped Fe 65Co35 films.  \n"    },    {        "title": "1903.00704v4.Complex_Stiffness_Model_of_Physical_Human_Robot_Interaction__Implications_for_Control_of_Performance_Augmentation_Exoskeletons.pdf",        "content": "Complex Stiffness Model of Physical Human-Robot Interaction:\nImplications for Control of Performance Augmentation Exoskeletons\nBinghan He1, Huang Huang, Gray C. Thomas and Luis Sentis\nAccepted for publication in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) ©2019 IEEE. Personal use of this material is permitted. Permission\nfrom IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating\nnew collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. DOI: 10.1109/IROS40897.2019.8968005Abstract — Human joint dynamic stiffness plays an important\nrole in the stability of performance augmentation exoskeletons.\nIn this paper, we consider a new frequency domain model of the\nhuman joint dynamics which features a complex value stiffness.\nThis complex stiffness consists of a real stiffness and a hysteretic\ndamping. We use it to explain the dynamic behaviors of the\nhuman connected to the exoskeleton, in particular the observed\nnon-zero low frequency phase shift and the near constant\ndamping ratio of the resonance as stiffness and inertia vary.\nWe validate this concept with an elbow-joint exoskeleton testbed\n(attached to a subject) by experimentally varying joint stiffness\nbehavior, exoskeleton inertia, and the strength augmentation\ngain. We compare three different models of elbow-joint dynamic\nstiffness: a model with real stiffness, viscous damping and\ninertia; a model with complex stiffness and inertia; and a\nmodel combining the previous two models. Our results show\nthat the hysteretic damping term improves modeling accuracy\n(via a statistical F-test). Moreover, this term contributes more\nto model accuracy than the viscous damping term. In addition,\nwe experimentally observe a linear relationship between the\nhysteretic damping and the real part of the stiffness which\nallows us to simplify the complex stiffness model down to a\n1-parameter system. Ultimately, we design a fractional order\ncontroller to demonstrate how human hysteretic damping\nbehavior can be exploited to improve strength amplification\nperformance while maintaining stability.\nI. I NTRODUCTION\nWhile the concept of a personal augmentation device or\nexoskeleton is an old idea [1], [2], [3], a system which\ndelivers on the dream of transparent interaction, of “feeling\nlike the system is not there,” through augmentation of sensed\nhuman interaction forces is still an ambitious goal of force\ncontrol technology today [4], [5], [6], [7]. Unlike assistive\nexoskeletons which help complete predictable behaviors [8],\n[9] or rehabilitation exoskeletons [10], [11] which simulate\nrehabilitation therapy, human augmentation exoskeletons [4],\n[12] use non-passive feedback control to amplify the user’s\nstrength. But this type of feedback control brings the system\ncloser to instability. And since the exoskeleton is in a feed-\nback interconnection with the human, a model of the human’s\ndynamic behavior plays a critical role in determining the\nstability of an augmentation exoskeleton [13], [14].\nAmong all different kinds of dynamic model of an individ-\nual human joint, perhaps the most popular one is the mass-\nspring-damper model—with the additional non-linearity that\nThis work was supported by the U.S. Government and NASA Space\nTechnology Research Fellowship NNX15AQ33H. We would also like to\nthank the members of the Human Centered Robotics Lab, University of\nTexas at Austin and Apptronik Systems Inc for their supports. Authors are\nwith The Departments of Mechanical Engineering (B.H., H.H., G.C.T.) and\nAerospace Engineering (L.S.), University of Texas at Austin, Austin, TX.\nSend correspondence to1binghan at utexas dot edu .the spring stiffness of the human joint can be modified\nby both voluntary muscle contractions or external torques\nexerted on the joint [15]. Several studies demonstrated a\nlinear relationship between the stiffness of the human (found\nby fitting a linear mass-spring-damper model for a single\njoint) and an external torque [16], [17], [18]. For modeling\nthe human joint damping, some other studies explored the\nfact that not only the stiffness but also the damping increases\nwith muscle contractions [19] and external torques [20]. A\nlinear relationship between the damping and the external\ntorque has also been identified for the human ankle joint,\nbut it is statistically weaker than the strong linear relationship\nbetween the stiffness of the ankle and the external torques\n[16], [18]. However, it is not clear from the literature that a\nlinear relationship between the damping and the stiffness of\na human joint can be expected in more general cases.\nAnother way to model the damping in the linear mass-\nspring-damper model is through the empirical observation\nthat a relatively consistent damping ratio is maintained by\nthe human elbow across different joint stiffnesses [14].\nFrequency domain identification of the ankle joint impedance\n[16], [21] also showed a consistent damping ratio within the\nrange from 0.22 to 0.49. This damping ratio consistency on\nthe ankle is also supported by the fact that the ankle damping\nratio does not have significant change with large variations\nof mean external torques exerted on the subjects [20]. For\nupper limbs, a multi-joint impedance study on human arms\n[22] showed that the damping ratio of the minimally damped\nmode for the 2-D endpoint impedance in the transverse plane\nis distributed with a mean of 0.26 and a standard deviation\nof 0.08. Although this could be explained as the effect of\nhumans adapting their damping to stabilize movement [23],\na more detailed explanation of how humans achieve this\nconsistency remains unclear.\nHysteretic damping models have seen success in biome-\nchanical modelling before. In [16], experimental results\nshowed a hysteretic relationship between the applied torque\nand the ankle angle at very low frequencies. Hysteretic\ndamping is shown indirectly in [17] (see Fig. 6 of that\npaper), where the human elbow stiffness has a phase shift\naround 25 degrees in a wide range of low frequencies—\ncontradicting the viscous damping hypothesis. This type\nof phase behavior is explained (in the field of structural\nmechanics) by defining a hysteretic damping whose damping\ncoefficient is proportional to the inverse of frequency [24].\nModels with hysteretic damping have also been adapted to\ndescribe the dynamic properties of the whole body of a seated\nhuman [25] as well as cockroach legs [26].arXiv:1903.00704v4  [cs.RO]  30 Apr 2020In this paper we study the human stiffness and damp-\ning behavior when coupled to an exoskeleton inertia, and\ntest the effectiveness of a hysteretic damping term in the\nsystem model. More specifically we compare three models\n1) a linear mass, spring, and viscous damper model, 2)\na nonlinear complex-stiffness-spring and mass model (that\nis, a spring, mass, and hysteretic damper model), and 3)\na combination model with mass, spring, and both viscous\nand hysteretic damping. Our results show that there is a\nstatistically significant benefit of the hysteretic damping term\n(comparing model 1 to model 3 with an F-test), and a less\nsignificant benefit for the viscous damping term (comparing\nmodel 2 to model 3). This hysteretic damping explains the\nconsistent damping-ratio of the human–exoskeleton resonant\npeak even as the stiffness and exoskeleton inertia change—\nwhich is not well explained by the linear model. And it\nalso explains the low frequency phase lag in human stiffness\n(previously observed in [17]). Our elbow joint experiments\nvary parameters which would result in a differing damping\nratio if the linear model were true: we change the inertia of\nthe exoskeleton, and (indirectly, using an adjustable exercise\nhand grip and a bias torque) the stiffness of the human joint.\nWe also test different exoskeleton strength amplification\nfactors, and it does not appear to elicit a different human\nbehavior than when the inertia is simply reduced. One further\ncontribution of the paper is the theorizing of an amplification\ncontroller which uses fractional order filtering to exploit\nthe hysteretic damping of the human, offering improved\nperformance over previous strategies.\nII. M ETHODS\nA. Apparatus\nFor this study we employed the P0 series elastic elbow-\njoint exoskeleton from Apptronik Systems, as shown in\nFig. 1. This exoskeleton has a moment of inertia of 0.1\nkg\u0001m2with no load on it, but allows for attaching additional\nweights to it. A load, attached 0.45 m from the exoskeleton\njoint, is pictured in Fig. 1.b. The contact force fcbetween\nthe human and the exoskeleton is measured by a six-axis\nforce/torque sensor situated below the white 3D printed\n“cuff” (which includes the adjustable strap which clamps\nthe forearm). This force torque signal is cast as a torque\n(tc) using the motion Jacobian Jof the sensor frame ( tc=\nJTfc). Rubber pads are adhered to the inside surfaces of\nthe cuff and the cuff strap to improve user comfort. Joint\nposition qeis directly measured by a dedicated encoder at\nthe exoskeleton joint. The series elastic actuator (SEA) has a\nspring force control bandwidth of 10 Hz and provides high\nfidelity actuator torque tstracking using the force control\ndisturbance observer of [27].\nIn parallel with an excitation chirp command (which\nessentially performs system identification of the human sub-\nject), a gravity compensation controller, a human augmen-\ntation controller, and a bias torque comprise the desired\nactuator torque signal. The gravity compensation controller\ntakes the measurement of qeto calculate and compensate\nthe gravity torque tgacting on the exoskeleton system. The\n(a)\nSEA6-Axis Force Sensor\nEncoder(b)\nLoad\nHand Grip\nFig. 1: Experimental apparatus: a series elastic P0 exoskele-\nton from Apptronik Systems, featuring an ATI Mini40 force\nsensitive cuff and a P170 Orion air cooled series elastic\nactuator module acting through a simple 3 bar linkage.\nSets\n1\nqe\nSe\ntgIMe\n0Setc\n1 RBhIMh\nC\nKh+Chja\u00001\nGSEAtd Excitation\n+Augmentation\n\u0000\nGravity Compensation\u0000Bias\n+\nFig. 2: Block diagram consisting of augmentation, gravity\ncompensation and experiment perturbation. Dynamics of\nhuman with exoskeleton are expressed as a bond graph with\neffort source of ts,tcandtg.\nhuman augmentation controller takes the measurement of tc\nand multiplies tcby negative a\u00001. With the assistance of\nactuator torques produced from the augmentation command,\nthe human’s interaction forces with the exoskeleton are\namplified by a factor of a. This exoskeleton augmentation\nstrategy differs from the one we applied in [14] in the\ndirectness of the augmentation feedback.\nB. Experimental Protocol\nThe experimental protocol was approved by the Institu-\ntional Review Board (IRB) at the University of Texas at\nAustin. It consists of fifteen perturbation experiments with a\n28-year old male subject. The experiments are separated into\nthree groups (Exp. I-III) of five experiments. The first three\nexperiments in each group are conducted with loads of 0.6\nkg, 2.3 kg and 4.5 kg and an avalue of 1 (corresponding to a\nnon-augmentation controller) while the last two experiment\nin each group are conducted with a load of 4.5 kg and a\nvalues of 2 and 4. The mass of the loads and the mass\nof the exoskeleton have their gravitational bias torque fully\ncompensated through gravity compensation control, while\ntheir inertia is attenuated by a factor of adue to the cuff\ntorque feedback.\nThe stiffness of the human elbow is influenced by muscle\nco-contraction as well as by contraction to resist the bias\ntorque. In order to obtain different values of elbow stiffness\nfor the three experiment groups, both the bias torque com-\nponent of the controller and the co-contraction are varied.\nThe three groups have, respectively, 0 Nm, 4 aNm, and 8 a\nNm of bias torque. Co-contraction is controlled by having\nthe subject squeeze an adjustable force hand grip. The three\ngroups have a 10-kg, a 14-kg, and a 27-kg gripping forcerespectively. The amplitude of the perturbation chirp signal\nis set to be 2 aNm.\nTo avoid fatigue of the subject, the duration of each\nperturbation experiment is set to be 100 seconds. The\nperturbation is set to be an exponential chirp signal, and\nthe results are typically analyzed in the frequency domain.\nTo sufficiently capture the natural frequency for damping\nfeature identification, we set different ranges of frequency\nfor the chirp signal according to the stiffness values the\nsubject achieved from the bias torque and the gripping force.\nFrequency ranges of 2-20 rad/s, 3-30 rad/s and 4-40 rad/s are\nset for the chirp signals for the three experiment groups.\nAfter the chirp perturbation experiments, we transfer the\ntime domain data into the frequency domain and identify the\ndynamic stiffness model of the subject by linear regression.\nThe parameters of the three experiment groups are summa-\nrized in Tab. I.\nC. Models\nIn our models, we define Khas the human elbow-joint real\nstiffness, Chas the human elbow-joint hysteretic damping,\nBhas the human elbow-joint viscous damping, Mhas the\nmoment of inertia of the human and Meas the moment of\ninertia of the exoskeleton. See list of symbols in Tab. II.\nA passive linear model of human dynamic stiffness with\nviscous damping can be expressed as\nSh=tc=qe=Mhs2+Bhs+Kh: (1)\nIf we consider a human model with hysteretic damping\n(complex stiffness) we have a nonlinear model\nSh=tc=qe=Mhs2+Chj+Kh: (2)\nAnd to generalize the two, we also consider a nonlinear\nmodel with both viscous and hysteretic damping\nSh=tc=qe=Mhs2+Bhs+Chj+Kh: (3)\nHowever, these models are difficult to identify from the\nexperimental tcandqevalues because the natural frequency\nof the human dynamic stiffness can easily go beyond the\nrange of the frequency for the experiments. With the augmen-\ntation controller, the operator feels an attenuated inertia from\nthe exoskeleton. Therefore, we added a nominal attenuated\ninertia of Me=ato the frequency domain data of tc=qe\nfor the model identification. In essence, we desensitize our\nidentification to errors far above the natural frequency of the\nhuman spring and the exoskeleton inertia. Combining this\nadditional term with (1), (2) and (3), the three models of\nhuman-exoskeleton interaction can be expressed as\nSh-e/a=Mh-e/as2+Bhs+Kh; (M1)\nSh-e/a=Mh-e/as2+Chj+Kh; (M2)\nSh-e/a=Mh-e/as2+Bhs+Chj+Kh; (M3)\nwhere Mh-e/a=Mh+Me=ais the perceived inertia at the\nhuman joint.\nWe also calculate the damping ratio zh-e/aofSh-e/a, as a\nmeasure of the degree of oscillation at the resonant zero-\npair. Because M2 and M3 have the Chjterm which providesTABLE I: Experiment Parameters\nExp a Load\n(kg)Grip\n(kg)Bias\n(Nm)Amplitude\n(Nm)Frequency\n(rad/s)\nI.1 1 0 :6\n10 0 2 a 2\u000020I.2 1 2 :3\nI.3 1 4 :5\nI.4 2 4 :5\nI.5 4 4 :5\nII.1 1 0 :6\n14 4 a 2a 3\u000030II.2 1 2 :3\nII.3 1 4 :5\nII.4 2 4 :5\nII.5 4 4 :5\nIII.1 1 0 :6\n27 8 a 2a 4\u000040III.2 1 2 :3\nIII.3 1 4 :5\nIII.4 2 4 :5\nIII.5 4 4 :5\nTABLE II: List of Symbols\nSymbol Meaning\ntd Actuator desired torque\nts Actuator actual torque\nGSEA Transfer function from tdtots\ntc Human-exoskeleton interaction torque\ntg Exoskeleton gravity torque\nqe Joint angular position\nKh Human joint real stiffness parameter\nCh Human joint hysteretic damping parameter\nBh Human joint viscous damping parameter\nMh Moment of inertia of human forearm\nMe Moment of inertia of exoskeleton\nMh-e Moment of inertia of human with exoskeleton\nMh-e/a Moment of inertia of human with attenuated exoskeleton\nSh Joint dynamic stiffness of human\nSh-e Joint dynamic stiffness of human with exoskeleton\nSh-e/a Joint dynamic stiffness of human with attenuated exoskeleton\nPa;Ca Plant and controller transfer functions of augmentation\nwSEA Natural frequency of GSEA\nwh;zh Natural frequency and damping ratio of Sh\nwh-e;zh-e Natural frequency and damping ratio of Sh-e\nwh-e/a;zh-e/aNatural frequency and damping ratio of Sh-e/a\na damping effect in addition to Bhs, we define the damping\nratio of each model using the imaginary part of the transfer\nfunction evaluated at the resonance:\nzh-e/a=Bh\n2pKhMh-e/afor M1, (4)\nzh-e/a=Chw\u00001\nh-e/a\n2pKhMh-e/a=Ch\n2Khfor M2, and (5)\nzh-e/a=Bh\n2pKhMh-e/a+Ch\n2Khfor M3, (6)\nwhere wh-e/a=p\nKh=Mh-e/ais the natural frequency of Sh-e/a.\nD. Statistical Analysis\nIn order to compare the significance of BhsandChjin\nthe human-exoskeleton interaction model, we calculate the\nresidual square sum (RSS) for all three models, denotedMagnitude (dB) Magnitude (dB) Magnitude (dB)Magnitude (dB) Magnitude (dB)\n(b) (d) (f)(a) (e) (c) Magnitude (dB)\n10 2 3 4 6 20\nPhase (deg)10 4 6 20 30 40\nPhase (deg)10 3 4 6 20 30\nPhase (deg)w(rad/s) w(rad/s) w(rad/s)\n10 2 3 4 6 20 10 4 6 20 30 40 10 3 4 6 20 30\nw(rad/s) w(rad/s) w(rad/s)\n10 2 3 4 6 20\nPhase (deg)10 3 4 6 20 30\nPhase (deg)10 4 6 20 30 40\nPhase (deg)w(rad/s) w(rad/s) w(rad/s)\n10 2 3 4 6 20 10 3 4 6 20 30 10 4 6 20 30 40\nw(rad/s) w(rad/s) w(rad/s)2550\n04590135180\n2040\n045901351802550\n04590135180\n3050\n045901351804060\n04590135180\n3050\n04590135180Exp. I.1\nExp. I.2\nExp. I.3\nExp. I.4\nExp. I.5Exp. II.1\nExp. II.2\nExp. II.3\nExp. II.4\nExp. II.5Exp. III.1\nExp. III.2\nExp. III.3\nExp. III.4\nExp. III.5\nFig. 3: Bode plots of frequency domain data of Sh-e/awith Exp. I.1-5 on (a) and (b), Exp. II.1-5 on (c) and (d), and Exp.\nIII.1-5 on (e) and (f). The dash lines on each plot show the fitted curves from M3.\nRSS M1,RSS M2andRSS M3respectively. For each experiment,\nwe conduct F-tests for each of the two three-parameter\nmodels (M1 and M2) against the generalizing four-parameter\nmodel (M3). Our F-statistic accounts for complex value data,\nFMi-M3=RSS Mi\u0000RSS M3\nRSS M3(2n\u00004);fori=1;2 (7)\nwhere nis the number of complex value samples at the\nfrequency domain and the real and imaginary parts of each\nsample are statistically independent. The significance of Bhs\nandChjthen will be evaluated by comparing this F statistic\nagainst a critical F statistic threshold based on a 0.05 false-\nrejection probability.\nWe split the 100 seconds of time domain data for each\nexperiment into 10 sequences. For each of the 10 second\nsequences, only the data from the first 5.78 seconds is used\nfor calculating the frequency domain sample. The remainder\nperiod of 4.22 seconds is greater than the 2% settling time\nfor all the 2nd order dynamics of Sh-e/aidentified in the\nexperiments. By this method we can safely assume statistical\nindependence between the 10 single-frequency data points\ncomprising our estimate of the frequency response function\nfor the purposes of statistical testing.\nIII. R ESULTS\nA. Phase Shift\nIn the frequency domain results of Sh-e/a(Fig. 3), the\nphase starts (at low frequencies) from a value between 25°\nto 45° instead of zero and changes very little across all the\nfrequencies before it reaches the second order zero at wh-e/a\nfor each experiment. This type of phase shift is very differentfrom the phase shift usually experienced by a linear system\nwith a constant time delay or a constant damping property in\nwhich the phase shift approaches zero in the limit as w!0.\nAs shown in Fig. 4, this phase shift is clearly visible even\nin time domain comparisons of tcandqe. The data show\nthat the human joint motion qeis not perfectly sinusoidal—\nit stops following the trend of the torque after they both\nreach their peak values and “waits” before following the\ntorque tcin its descent. At low frequencies, these peaks seem\nespecially flat.\nB. Model Comparisons\nThe results of the identified parameters (Tab. III) show\nthat the three models give the same values of Kh,Mh-e/a\nand consequently wh-e/ato two decimal places for each\nexperiment. This is because the difference between the three\nmodels is restricted to the imaginary part of Sh-e/awhile\nKhandMh-e/aare the coefficients of the real part of Sh-e/a.\nAlthough the identified values of BhandChare quite different\nbetween the three models, the values of zh-e/aare still very\nclose for each experiment. This means that the three models\ngive very similar values for the slope of the phase at the\nresonant frequency wh-e/a.\nFrom M1 to M3, the values of Bhhave been reduced\nconsiderably. This means that M3 uses the Chjterm to\nreplace part of the Bhterm in M1 while maintaining a similar\nphase behavior at the frequency wh-e/a. From M2 to M3, the\nvalues of Chhave been reduced except for Exp. I.3, III.3\nand III.5 in which M3 gives a negative value for Bh. These\nnegative value of Bhis because there is no lower boundconstraint on the value of Bhduring the frequency domain\nregression for M3. Although a negative value of Bhbrings\nnon-passivity to a linear mass-spring-damper system in the\ncommon sense, the Chjterm in M3 enforces the dynamics\nofSh-e/ato remain passive across the range of frequencies\nin our experiments.\nThe results from the F-tests (Fig. 5) relate to the signif-\nicance of BhsandChjin M3. Based on the 20 statistically\nindependent data values for each experiment, a critical F-\nstatistic value of 4.49 is calculated for 0.05 false-rejection\nprobability. The results show that values of FM1-M3for\nall the experiments are much higher than the critical F-\nstatistic value, with the values of FM1-M3in Exp. II.3 and\nII.5 exceeding 100 (c.f. the critical value of 4.49). This\nproves that the existence of the Chjterm in M3 significantly\nimproves modeling accuracy of Sh-e/a. The values of FM2-M3\nare mostly below the critical F-statistic value except for Exp.\nI.5, II.1, III.1 and III.2. The other observation is that the value\nofFM2-M3is always much lower than the value of FM2-M3for\nall experiments. Although the effect of the Bhsterm cannot\nbe completely ignored based on the results of these F-tests,\nwe can claim that the Chjterm is still much more significant\nthan the Bhsterm in M3.\nC. Linear Regression between C hand K h\nBecause the Chjterm is created to describe the phase shift\neffect from the complex human stiffness in M2 and M3, we\nsuspect that the identified value of Chhas a linear relation\nwith the value of Kh. Therefore, we apply linear regression\nbetween the values of ChandKhidentified from M2 and M3\n(Fig. 6). Compared with M3, the linear regression result with\nM2 shows a stronger linear relationship with a much higher\ncoefficient of determination ( R2). The regression equation\nidentified from the M2 parameters also has a smaller value\nof bias from the origin of the Ch-Khplane compared with\nthe regression equation identified from the M3 parameters.\nIntuition leads us to expect low bias in the regression\nequation, since a nonzero value of Chwhen the value of Kh\nis zero could not be explained as hysteretic spring behavior.\nBased on linear regression equations, we can express the\nphase shift (with respect to 0°) at the low frequencies as\nPhase Shift =tan\u00001(Ch\nKh) =tan\u00001(ch+dh\nKh)for M2, and (8)\nPhase Shift =tan\u00001(ch+dh+Bhw\nKh) for M3 ; (9)\nwhere Ch=chKh+dhis the regression equation identified\nfrom the values of ChandKhin M2 and M3 with chand\ndhbeing the slope and the bias of the regression equation.\nBy substituting Ch=chKh+dhinto (5) and (6), the value of\nzh-e/afor M2 and M3 can be expressed as\nzh-e/a=Ch\n2Kh=ch\n2+dh\n2Khfor M2 ;and (10)\nzh-e/a=ch\n2+dh\n2Kh+Bh\n2pKhMh-e/afor M3 : (11)\nBecause the values of dhof the regression equations for M2\nand M3 and the values of Bhfor M3 are relatively small,TABLE III: Subject Dynamic Stiffness Parameters\nExp Model Kh(Nm\nrad)Ch(Nm\nrad)Bh(Nms\nrad)Mh-e/a(kgm2)wh-e/a(rad\ns)zh-e/a\nI.1M1 10 :05 -- 1 :03 0 :28 5 :95 0 :31\nM2 10 :05 5 :89 -- 0 :28 5 :95 0 :29\nM3 10 :05 4 :97 0 :18 0 :28 5 :95 0 :30\nI.2M1 11 :80 -- 1 :51 0 :60 4 :44 0 :28\nM2 11 :80 6 :68 -- 0 :60 4 :44 0 :28\nM3 11 :80 5 :44 0 :31 0 :60 4 :44 0 :29\nI.3M1 15 :74 -- 2 :09 1 :18 3 :65 0 :24\nM2 15 :74 8 :33 -- 1 :18 3 :65 0 :26\nM3 15 :74 10 :44\u00000:60 1 :18 3 :65 0 :26\nI.4M1 13 :82 -- 1 :46 0 :60 4 :78 0 :25\nM2 13 :82 6 :87 -- 0 :60 4 :78 0 :25\nM3 13 :82 6 :01 0 :21 0 :60 4 :78 0 :25\nI.5M1 12 :09 -- 1 :22 0 :28 6 :59 0 :33\nM2 12 :09 6 :84 -- 0 :28 6 :59 0 :28\nM3 12 :09 4 :26 0 :52 0 :28 6 :59 0 :32\nII.1M1 12 :73 -- 1 :41 0 :20 7 :94 0 :44\nM2 12 :73 10 :18 -- 0 :20 7 :94 0 :40\nM3 12 :73 5 :86 0 :66 0 :20 7 :94 0 :44\nII.2M1 18 :79 -- 1 :91 0 :57 5 :72 0 :29\nM2 18 :79 11 :77 -- 0 :57 5 :72 0 :31\nM3 18 :79 11 :54 0 :04 0 :57 5 :72 0 :31\nII.3M1 25 :95 -- 3 :08 1 :03 5 :02 0 :30\nM2 25 :95 16 :75 -- 1 :03 5 :02 0 :32\nM3 25 :95 15 :48 0 :26 1 :03 5 :02 0 :32\nII.4M1 25 :77 -- 2 :83 0 :52 7 :02 0 :39\nM2 25 :77 20 :49 -- 0 :52 7 :02 0 :40\nM3 25 :77 16 :60 0 :60 0 :52 7 :02 0 :40\nII.5M1 19 :07 -- 1 :88 0 :28 8 :32 0 :41\nM2 19 :07 16 :27 -- 0 :28 8 :32 0 :43\nM3 19 :07 15 :72 0 :08 0 :28 8 :32 0 :43\nIII.1M1 48 :15 -- 1 :97 0 :23 14 :4 0 :29\nM2 48 :15 25 :45 -- 0 :23 14 :4 0 :26\nM3 48 :15 16 :66 0 :76 0 :23 14 :4 0 :29\nIII.2M1 48 :60 -- 2 :85 0 :58 9 :13 0 :27\nM2 48 :60 25 :61 -- 0 :58 9 :13 0 :26\nM3 48 :60 15 :19 1 :23 0 :58 9 :13 0 :27\nIII.3M1 42 :23 -- 3 :19 1 :01 6 :47 0 :24\nM2 42 :23 23 :60 -- 1 :01 6 :47 0 :28\nM3 42 :23 24 :08\u00000:07 1 :01 6 :47 0 :28\nIII.4M1 32 :22 -- 2 :82 0 :46 8 :35 0 :37\nM2 32 :22 25 :36 -- 0 :46 8 :35 0 :39\nM3 32 :22 20 :83 0 :55 0 :46 8 :35 0 :39\nIII.5M1 42 :33 -- 2 :08 0 :27 12 :43 0 :31\nM2 42 :33 26 :50 -- 0 :27 12 :43 0 :31\nM3 42 :33 27 :66\u00000:11 0 :27 12 :43 0 :31\n0:02:5\n\u00002:50:0\n0:02:5\n\u00002:50:0tc(Nm)\ntc(Nm)\ntc(Nm)\ntc(Nm)0:000:05\n\u00000:050:00\n0:000:05\n\u00000:050:00qe(rad)\nqe(rad)\nqe(rad)\nqe(rad)Phase Shift between tcandqetc(Nm)\nqe(rad)\n0 1 2 3 4 5 t (s)0 1 2 3 4 5 t (s)0 1 2 3 4 5 t (s)0 1 2 3 4 5 t (s)\nFig. 4: Four pieces of time data of tcandqefrom Exp.III.1\nused for identifying the frequency data of Sh-e/aat the\nfrequencies of 4.0, 5.0, 6.3 and 8.0 rad/s show the phase\nshift in the time domain.the phase shift at the low frequencies is dominated by the\nvalue of tan\u00001(ch)and the value of zh-e/ais dominated by\nthe constant ch=2 term. This explains the fact that the phase\nshift is non-zero at low frequencies and the fact that the value\nofzh-e/achanges very little compared to the changes of Kh\nandMh-e/aacross all our experiments.\nIV. I MPLICATIONS FOR CONTROL OF PERFORMANCE\nAUGMENTATION EXOSKELETONS\nA. 1-Parameter Complex Stiffness Model\nOne of the challenges of augmentation control is to design\nan augmentation controller to stabilize the exoskeleton with\nall possible human impedances. This requires a robust human\nimpedance model with bounded parameter uncertainties for\nthe augmentation controller design.\nSimilar to [13], a robust model version of M1 can be\ndefined with bounded uncertainties for Khand Bh, which\ncould be obtained from multiple measurements in advance.\n(We assume Mhdoes not change for the elbow-joint.)\nBecause both Khand Bhvary in large ranges, the 2-D\nuncertain parameter space of Kh-Bhbecomes very huge,\nand the augmentation controller can easily end up as an\nextremely low-bandwidth conservative controller. Since such\nan uncertain model includes all combinations of possible Kh\nand Bh, the damping ratio can be a very limiting design\nconstraint. This is not realistic, given that zh-e/ais relatively\nconsistent in our experiment results. Therefore, we propose\na 1-parameter model simplification to reduce the uncertain\nparameter space for augmentation controller design.\nOne strategy is to model Bhas a linear function of Kh\nwhich allows us to create a robust model of M1 with bounded\nuncertainty for only Kh. Based on a linear relationship Bh=\nahKhbetween BhandKh, M1 can be expressed as\nSh-e/a=Mh-e/as2+Kh(1+ahs): (12)\nBy substituting Bh=ahKhto (4), zh-e/acan be expressed as\nzh-e/a=ahKh\n2pKhMh-e/a=ah\n2\u0001wh-e/a; (13)\nwhich is proportional to wh-e/a. However, we do not observe\nthis proportional relationship between zh-e/aandwh-e/afrom\nour experimental results for M1 in Tab. III. On the other\nhand, because (12) is a simplification from M1, it also fails\nto explain the non-zero phase shift at low frequencies.\nIf we assume dh\u00190, a simplified complex stiffness model\nof M2 can be expressed as\nSh-e/a=Mh-e/as2+Kh(1+chj): (14)\nBased on (8) and (10), (14) is able to explain both the non-\nzero phase shift at low frequencies and the near constant\nvalue of zh-e/aacross all the experiments. This, in turn,\nsupports the use of (14) as a 1-parameter model of Sh-e/a\nfor augmentation controller design.\nAdopting this 1-parameter model allows simplifying (2),\nSh=tc=qe=Mhs2+Kh(1+chj); (15)\nand the dynamic stiffness of the human coupled with the\n10\u00002100102\n10\u00002100102\n10\u00002100102F-test ResultsFM1-M3\nFM2-M3\n20.6261.71133.83\n36.75134.55\n13.30\n0.031.012.32\n0.2048.0726.60 33.73 43.7127.82\n1.93 1.58 1.531.0312.39\n32.68 30.2856.1824.4850.30\n10.81 15.78\n0.021.29\n0.10\nIII.1 III.2 III.3 III.4 III.5II.3 II.1 II.2 II.4 II.5I.1 I.2 I.3 I.3 I.5\nFig. 5: F-statistics on log scale for all experiments show the\nsignificant improvement on modeling accuracy from M1 to\nM3 and a partial improvement from M2 to M3. The dashed\nline appears on a bar if the F-statistic value is over the critical\nF-statistic value of 4.49 (false-rejection probability of 0.05).\n(a)M3 (b)M2Linear Regression Between ChandKhExp. I\nExp. II\nExp. III\n0102030\nCh=0:58Kh\u00000:85;R2=0:55 Ch=0:65Kh\u00000:26;R2=0:85\n0 20 40 0 20 40102030\n0 Ch(Nm)\nKh(Nm) Kh(Nm)Ch(Nm)\nFig. 6: Linear regressions between ChandKhfor M3 (a)\nand M2 (b) show that the parameters of M2 have a stronger\nlinear relationship (that is, a higher R2value).\nexoskeleton Sh-e,\nSh-e=ts=qe=Mh-es2+Kh(1+chj); (16)\nwhere Mh-e=Mh+Meis the combined inertia between the\nhuman and the exoskeleton. We consider wh-e=p\nKh=Mh-e\nto be the natural frequency of Sh-e, despite the chterm.\nB. Fractional-Order Augmentation Controller\nAs in [14], the augmentation control we discuss here is\ndesigned to eliminate the augmentation error signal ta=\n(a\u00001)tc+tsby feeding it back to the actuator command\ntdwith an augmentation controller. Different from the direct\naugmentation feedback shown in Fig. 2 in which the aug-\nmentation command is \u0000tcmultiplied by a\u00001, this strategy\nallows us to design an augmentation controller completely\nseparated from the augmentation factor a.By substituting (15) and (16), the transfer function from\ntstotacan be expressed as\nta\nts=(a\u00001)\u0001Sh+Sh-e\nSh-e=a\u0001Sh-e/a\nSh-e: (17)\nBased on (17), the augmentation plant transfer function Pa\nfrom tdtotacan then be expressed as\nPa(s) =ta\ntd=a\u0001Sh-e/a\nSh-e\u0001GSEA(s); (18)\nwhere the SEA transfer function GSEA(s) =ts=tdacts as a\n2nd order low-pass filter. Because of the high bandwidth of\nthe SEA force controller, the natural frequency wSEAofGSEA\nis much greater than both wh-eandwh-e/a.\nLooking at the bode plot from low to high frequencies,\nPa(s)has a pair of conjugate poles at wh-e, then a pair of\nconjugate zeros at wh-e/aand then another pair of conjugate\npoles at wSEA (Fig. 7). Before wh-e, both Sh-eand Sh-e/a\nare dominated by the complex stiffness. Therefore, Pa(s)\nhas magnitude aand 0\u000ephase. Between wh-eandwh-e/a,\nSh-eis dominated by its inertia effect and the magnitude\nofPa(s)decreases while the phase leaves 0\u000e. On the other\nhand, Sh-e/ais still dominated by the complex stiffness and\nprevents the phase moving below tan\u00001(ch)\u0000180°. At the\nfrequency between wh-e/aandwSEA, the inertia effects in Sh-e\nandSh-e/acompletely dominate their frequency behaviors.\nThe magnitude of Pa(s)stays at (aMh+Me)=(Mh+Me)\nwhich is in the range from 1 to a.\nHowever, the gain crossover of Pafalls beyond wSEA\nwithout an augmentation controller. The phase margin with\nsuch crossover is very close to zero because of the 2nd order\nSEA dynamics. Also, the closed loop behavior amplifies\nthe high frequency sensor noise from the actual signal of\ntc. (tcis usually de-noised by a low-pass filter beyond the\nfrequency of wSEAwhich makes the closed loop even more\nunstable.) Therefore, the augmentation controller must lower\nthe crossover frequency in order to achieve a minimum phase\nmargin.\nSimilar to [14], the new crossover cannot be placed at the\nfrequency between wh-e/aandwSEAbecause multiple other\ncrossovers can be easily triggered. Instead, a new crossover\ncan be safely placed at the frequency between wh-eandwh-e/a\nwith a fractional-order augmentation controller\nCa(s) =kf=sf; (19)\nwhere fis the fractional order (that is, a non-integer power\nof s) of Ca(s)and kfis a gain which allows tuning the\nmagnitude of Ca(s)in the frequency domain. The fractional-\norder controller in (19) has its magnitude decreasing \u000020\u0001f\ndB per decade and its phase staying at \u000090\u0001fdegrees at\nall frequencies. Because of the non zero phase shift from\nthe complex stiffness, a positive phase margin fcan be\nguaranteed if\n0<90f<tan\u00001(ch)\u0000f;\n0<f<tan\u00001(ch)=90\u0000f=90; (20)\nwhere fis chosen in the range of (0;tan\u00001(ch)). The value of\nkfcan then be tuned to achieve a crossover in the frequency\nPa(s) Ca(s) Pa(s)\u0001Ca(s)wh-e/a wSEANoise CrossoverMulti-Magnitude (dB)\nPotential Bandwidthwh-e\nPhase (deg)\nPotential Bandwidthwh-e/a wSEAMulti-Crossover Noisewh-e020\u0001log(a) -180 -90 0\nFig. 7: Conceptual bode plots show the augmentation plant\nPa(s)with its poles (crosses) and zeros (circles). Regions\nare color-coded: the model is trustworthy in the green\nregion, the blue region reflects the multi-crossover behavior\nwhich makes an augmentation controller design unreliable,\nand the yellow region is dominated by sensor noise from\ntc. A fractional-order filter Ca(s)brings Pa(s)to a lower\ncrossover and increases the phase margin. The stars indicate\nthe crossovers of Pa(s)andPa(s)with Ca(s).\nrange from wh-etowh-e/a.\nAs a fractional-order controller, (19) cannot be imple-\nmented directly into the control system. However, we can\napproximate it as the product of many 1st order lag filters,\nCa(s) =kf\npf\n1\u0001n\nÕ\ni=11+s=zi\n1+s=pi; (21)\nzi=pi=rzp;fori=1;2;\u0001\u0001\u0001;n (22)\npi=pi\u00001=rpp;fori=2;3;\u0001\u0001\u0001;n; (23)\nwhere nis the number of lag filters and the pole and the zero\nfor each lag filter are \u0000piand\u0000zi. We define rzpsuch that\nall the lag filters have an equal distance between the pole\nand the zero, and we define rppsuch that there is a constant\ndistance between adjacent lag filters (in log frequency space).\nThe augmentation controller in (21) functions as a fractional-\norder filter in the frequency range of [p1;zn]rad/s. The\nfractional order fcan be approximated as log (rzp)=log(rpp).\nV. D ISCUSSION\nAllowing negative (non-passive) linear damping parame-\nters in M3 makes the model non-passive, but a better fitting\nF-test comparison model.\nIn Fig. 4 the flat peaks of qecould be explained by a\nhysteretic coulomb-friction-like non-linearity of the human.\nIf this was the case, then our frequency domain measure-\nments would be measurements of the describing function\nof the non-linearity. It is not yet clear how this hypothesis\nwould hold up to testing at different amplitudes of the\nforce input, since we did not include such tests in our\nexperimental plan. It would be a complex task to measure\nthe relationship between the amplitude and the hystereticdamping, since increasing the amplitude would potentially\nincrease the stiffness as well. Regardless of the cause, such\nhysteresis can not be modeled by M1, and we find that\nthe additional Chjterm in M2 and M3 helps to model the\nhysteresis by creating a non zero phase shift for the human\nstiffness. Because a similar type of phase shift has also been\nidentified with passive cockroach legs [26], we suspect that\nhuman joint hysteresis is a passive material property resulting\nfrom the contracted muscles around the joint.\nAs for the proposed fractional-order controller, if our\nfrequency domain model was due to hysteretic coulomb-\nfriction-like non-linearity of the human, then we would\nexpect that the Chvalue would be a function of the interaction\nforce signal amplitude. Fitting the model to tests performed\nat some maximal force amplitude, our controller would also\nbe stable for lower amplitudes. The result would be that our\ncontroller would perform well below a force threshold, at\nwhich point we would need to switch to another controller—\nperhaps saturating the desired force or employing a backup\nsafety controller [28]—to avoid having a non-robust con-\ntroller in situations with high force amplitude.\nThis study employed a single subject, so we cannot claim\nthat all humans have damping behavior that fits this model,\nbut some do. More subjects would allow us to be more clear\nabout the observed linear relationship between KhandCh,\nand to learn the population variance of our model parameters.\nFractional-order controllers can be hand tuned for different\nusers without parameter variance estimates.\nVI. C ONCLUSION\nAugmentation exoskeletons rely on a human model to\ndetermine stability. While ideal force feedback maps passive\nenvironments to passive human experiences, force feedback\nwith finite bandwidth will add energy due to the inevitable\nphase lag. 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Crago, “Multijoint dynamics and\npostural stability of the human arm,” Experimental Brain Research ,\nvol. 157, no. 4, pp. 507–517, 2004.\n[23] T. E. Milner and C. Cloutier, “Compensation for mechanically unstable\nloading in voluntary wrist movement,” Experimental Brain Research ,\nvol. 94, no. 3, pp. 522–532, 1993.\n[24] R. E. D. Bishop and D. C. Johnson, The Mechanics of Vibration .\nCambridge University Press, 1960.\n[25] S. Kitazaki and M. J. Griffin, “Resonance behaviour of the seated\nhuman body and effects of posture,” Journal of Biomechanics , vol. 31,\nno. 2, pp. 143 – 149, 1997.\n[26] D. M. Dudek and R. J. Full, “Passive mechanical properties of legs\nfrom running insects,” Journal of Experimental Biology , vol. 209,\nno. 8, pp. 1502–1515, 2006.\n[27] N. Paine, S. Oh, and L. Sentis, “Design and control considerations for\nhigh-performance series elastic actuators,” IEEE/ASME Transactions\non Mechatronics , vol. 19, no. 3, pp. 1080–1091, 2014.\n[28] G. C. Thomas, B. He, and L. Sentis, “Safety control synthesis with\ninput limits: a hybrid approach,” in 2018 American Control Conference\n(ACC) . AACC, June 2018, pp. 792–797."    },    {        "title": "1903.02812v1.Current_induced_motion_of_twisted_skyrmions.pdf",        "content": "1 \n Current -induced  motion of twisted skyrmions  \nChendong Jin1, Chunlei Zhang1, Chengkun Song1, Jinshuai Wang1, Haiyan Xia1, Yunxu \nMa1, Jianing Wang1, Yurui Wei1, Jianbo Wang1,2 and Q ingfang  Liu1,* \n \n1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, \nPeople’ s Republic of China . \n2Key Laboratory for Special Function Materials and Structural Design of the Ministry of the Education , Lanzhou University, \nLanzhou 730000, People ’s Republic of China.  \n \nAbstract  \nTwisted skyrmions, whose helicity angles are different from that of Bloch skyrmion s and Né el skyrmion s, have \nalready been demonstrated  in experiments recently. In this work, we first contrast  the magnetic structure and origin of the \ntwisted skyrmion with other three types of skyrmion  including Bloch skyrmion , Né el skyrmion  and antiskyrmion.  \nFollowing, we investigate the dynamics of twisted skyrmions driven by the spin transfer toque (STT) and the spin Hall \neffect (SHE) by using micromagnetic simulations. It is found that the spin Hall angle of the twisted skyrmion is related to \nthe dissipative force tensor and the Gilbert damping both for the motions induced by the  STT and the SHE, especially for \nthe SHE induced motion, the skyrmion Hall angle depends s ubstantially  on the skyrmion helicity. At last, we \ndemonstrate  that the trajectory  of the twisted skyrmion can be controlled in a t wo dimensional plane with  a Gilbert  \ndamping gradient . Our results provide the understanding of current -induced motion of twisted skyrmions, which may \ncontribute to the applications of skyrmion -based racetrack memories.  \n \nKeywords: Twisted skyrmion, spin transfer torque, spin Hall effect  \n \n \n \n \n \n_____________________________  \n*Corresponding author:  Qingfang Liu , liuqf @lzu.edu.cn  2 \n Introduction  \nIt has been recognized that the spin -polarized current -induced the motion and reversal of magnetic structures arises \nas a result of the spin transfer torque (STT) effect [1-4], which has attracted large interests due to the fundamental physics \nand potential applications in spintronic devices, such as magnetic random access memorie s (MRAM s)[5, 6] , racetrack \nmemori es[7, 8] , nano -oscillator s[9-11] and logic device s[12-14]. Recently, it has been reported that the spin Hall effect \n(SHE )[15, 16] , generated by the pure spin currents flowed from the heavy metal  substrate due to t he strong spin -orbit \ncoupling at the interface of ferromagnet/heavy -metal , is an  alternative  efficient method to manipulate the magnetization \ndynamics in magnetic materials [17-20]. Compared with the STT, the SHE  does not require currents flow through the \nmagnetic layer, and then reducing the Joule h eat and electromigration, i.e., avoiding the restricted effect of large current \ndensity in traditional STT devices [21].  \nMagne tic skyrmions are chiral spin magnetization structures with topological properties  and can be divided into the \nfollowing types according  to different types of Dzyaloshinskii -Moriya interaction (DMI) [22-26]: (i) Bloch skyrmmions  \nare first discovered in bulk non -centrosymmetric B20 -type lattice structures such as MnSi [27], FeCoS i[28-30], and \nFeGe [31, 32]  due to the presence of bulk DMI; (ii) Né el skyrmions are observed in multilayered ultrathin films lacking \ninversion symmetry with str ong spin -orbit coupling like Ir (111)/F e[33], Ta/CoFeB [34]  and Pt/Co [35] due to the presence \nof interfacial DMI ; (iii)  Antiskyrmions are reported in Heu sler compound s such as MnPtSn [36] due to the presence of \nanisotropic  DMI [37, 38] . Recently, at the interface of chiral bulk  Cu2OSeO 3 below a certain  thickness, the so -called \ntwisted skyrmions are demonstrated  directly  by the circularly polarized resonant e lastic x -ray scattering, due to the \nbreaking of translational symmetry at the surface of bulk ferromagnet [39, 40] . Up to now, the dynamics of twisted \nskyrmions driven by current have not been reported. Therefore, in this paper,  on the basis of comparing the magnetic \nstructure , origin and topological properties of the above four types of skyrmion,  we focus on the dynamics of twisted \nskyrmions driven by the STT and the SHE and also analysis the simulation results by using Thiele ’s equations [41].   3 \n Micromagnetic simulation details  \nOur magnetic simulation results are performed by using t he Object Oriented MicroMagnetic Framework (OOMMF) \npublic code [42], which includ es the additional modules for bulk DMI, interfacial DMI, anisotropic  DMI  and twisted DMI. \nThe magnetization  dynamics is described  by numerical ly solving the Landau -Lifshitz -Gilbert (LLG) equation  containing \nterms  of the STT and the SHE [17, 20] , as follow : \neff STT SHEd+ + ,m dmm H mdt dt      \n                                                    (1) \nwhere \nm  is the unit vector of the local magnet ization, is the gyromagnetic ratio,   is the Gilbert damping,  \neffH\n  is \nthe effective field including the exchange field, anisotropy field, demagnetization field and DMI  effective field . The STT \nterm is expressed  as \nSTT s s ( ) ( ),mmv m m v mxx    \n                                                        (2) \nwhere is the non -adiabatic factor , and vs is the velocity of the conduction electrons with the form \ns\n0s2PvJeM\n\n , \nwhere J is the current density, e is the electron char ge, P is the spin polarizatio n, \n is the reduced Planck c onstant , 0 is \nthe permeability of  free space , and Ms is the saturation magnetization . The electrons flowing toward + x direction when  vs > \n0. The SHE term is given by  \nSHE SH HM\n0s( ),2m m z jeM L   \n                                                     (3) \nwhere L is the thickness of the magnetic  layer with the value of 1 nm,  SH is the  spin-Hall angle  of Pt substrate with the \nvalue of 0.07 , \nz\n is the unit vectors o f the surface normal direction,  and \nHMj\n  is the current  density  injected into the \nheavy metal . \nIn order to eliminate the influence of the  boundary  effect on the size and dynamics of skyrmions, t he 2D plane  is \nassumed to 500 × 500 ×  1 nm3 (length × width × thickness) with the mesh size  of 1 ×  1 ×  1 nm3, and the initial position \nof the skyrmion is set in the center of the 2D plane. The material parameters  are chosen  similar to  Ref. [8]: saturation \nmagnetization Ms = 580  × 103 A/m, exchange constant A = 1.5  × 10-11 J/m, perpendicular magnetic anisotropy constant 4 \n Ku = 8 × 105 J/m3, and DMI strength DDMI = 2.5 ~  3.5 × 10-3 J/m2.  \nFour types of skyrmions  \nAccording  to the different helicity of skyrmions, there are four types of skyrmions: Bloch skyrmion, Né el skyrmion , \nantiskyrmion and twisted skyrmion as shown in Figs. 1(a)–(d), respectively. Figure s 1(e)–(h) display the corresponding \nspatial profiles of the  local magnetization  across the skyrmions. It can  be seen that the mz of the four types of skyrmions \nare consistent, while the mx and my of the four types of skyrmions are different, which again proves the different \ndistribution of the in-plane  magnetic moments of the four types of skyrmions. We emphasize that the distribut ion of the \nin-plane magnetic moments in the skyrmion structure is determined by the direction of the DMI vector , that is to say, the \nexistence  of the twisted skyrmion in this work is achieved by changing the DMI vector, which is much different as the \nreaso n that observed in the experiments.  Figure s 1(i)–(l) show the four types of DMIs: bulk DMI, interfacial DMI, \nanisotropic  DMI and twisted DMI that promise the existence of the Bloch skyrmion, Né el skyrmion , antiskyrmion and \ntwisted skyrmion, respectively. The four types of DMI considered in C4 symmetry can be written  as: \nˆ ˆ ˆ ˆ BulkDMI\nˆ ˆ ˆ ˆ InterDMI\nˆ ˆ ˆ ˆ AnisoDMI\nTwisDMIˆ ˆ ˆ ˆ ( ),2\nˆ ˆ ˆ ˆ ( ),2\nˆ ˆ ˆ ˆ ( ),2\n(2i i x i x i y i y\ni\ni i x i x i y i y\ni\ni i x i x i y i y\ni\niiDE S S x S x S y S y\nDE S S y S y S x S x\nDE S S y S y S x S x\nDE S S   \n   \n   \n        \n         \n        \n\n\n\nˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆˆ ˆˆ ( ) ( ) ( ) ( ),x i x i y i y\nix y S x y S x y S x y              \n              (4) \nwhere D is the DMI constant representing the DMI strength , \niS\n is the  atomic moment unit vector, \nˆx  and \nˆy  are \nthe unit vectors in the model.  \nTopological properties  of four types of skyrmions  \nA. Helicity, winding number and topological number  \nIn order to better understand the helicity  and winding number of skyrmions , we use the  two-dimensional polar \ncoordinates to describe a general magnetic skyrmion structure, as shown in Fig . 2 which display s a Bloch skyrmion, as 5 \n example, in the polar coordinates with azimuthal angle ( ) and radial coordinate ( ). Therefore , the unit vector of  the \nlocal magnetization  mx, my and mz in the C artesian coordinates  can be written as [26, 43, 44] : \nsin ( )cos ( ),\nsin ( )sin ( ),\ncos ( ),x\ny\nzm\nm\nm   \n   \n\n\n\n                                                                     (5) \nwhere () is the radial profile of the perpendicular component of the magnetization, and fro m the center  to the boundary, \nits value chan ges from 0.5 to 0.5; () is the angle between the magnetic moment and the radial coordinate . The \nvorticity of skyrmions is obtained by calculat ing the full turns of the transverse magnetic moments on the perimeter and \nis defined by the winding num ber[45] \n2\n01d ( )2W\n . Therefore, the winding number W = 1 for twisted \nskyrmion , Bloch skyrmion  and Né el skyrmion  as shown in  Fig. 3(a), and W = 1 for antiskyrmion  as shown in  Fig. 3(b). \nThe helicity of a skyrmion is given by \n( ) ( 0) W           with the value ranging  form  to, that is, for \nthe Bloch skyrmion, 0.5; for Né el skyrmion , 0 or ; for twisted skyrmion , 0.5, 0 and , and the helicity \n of the twi sted skyrm ion shown in  Fig. 1 (d) equals to 0.25; for antiskyrmion  as shown in Fig.  1(c), . The \ntopological  number Q relates to the winding number and counts how many times the unit vector  along the magnetic \nmoment  wraps the unit sphere with the form [26] \n1,,4mmQ qdxdy q mxy     \n                                                      (6) \nwhere q is the topological density. Figure s 3(c) and (d) show the topological densities corresponding to the magnet ic \nskyrmions shown in  Figs.  3(a) and (b), respectivel y. It can be seen that Q = 1 in Fig.  3(c) and Q = 1 in Fig.  3(d), i.e., Q \n= W when the spins point down in the central region and  point up in the boundary region.  \nB. Skyrmion size  and d issipative force tensor  \nThe diameter of twisted skyrmion size ( d) is usually defined as the distance from in -plane to in -plane magnetization, \ni.e., the distance between the region mz = 0, as shown in the inset of Fig.  4. The dissipative force tensor  D is used to \ndescribe the effect of the dissipative forces on the moving skyrmion [46-48]. For a single twisted skyrmion, D is given by  6 \n \n0 14 , ,0 4mmdxdyxx    \nDDDD                                                    (7) \nwher e D is the diagonal element of the dissipative tensor  and also called dissipative  parameter. The dissipative  parameter \nD is determined by t he diameter  and domain wall width of the twisted skyrmion. Therefore, both d and D are affected by  \nDMI strength as shown in Fig.  4. With  the increase in DDMI from 2.5  to 3.5 mJ/m2, d increases from 7.9 to 34.8 nm and D \nincreases from 1.0577 to 1.961, respectively, for the twisted skyrmion.  \nDynamics of twisted skyrmion driven by the STT  \nTo understand the STT-induced motion of the twisted skyrmion s, we first use the Thiele equation [41] to describe the \ndynamics  of the four kinds of skyrmions  mentioned above  by casting  the L LG Eqs. (1) and (2)  to the following \nequation [46, 47] : \ns d s d( - ) ( ) 0,v v v v      D G\n                                                             (8) \nwhere G is the gyrovector with the form G = (0 0 G) = (0 0 4Q), and vd is the drift velocity of the skyrmion. When the \nvelocity of the conduction electrons vs applied along the x direction, vd = (vx, vy) is derived from Eq.  (8) as  \n2\nxs 2 2 2\nys 2 2 2()+,()\n().QvvQ\nv Q vQ  \n  \n\n     D\nD\nD\n                                                              (9) \nIt can be seen that the direction of the skyrmion deviates from the direction of the conduction elect rons when, and \nthis phenomenon  is called the skyrmion Hall effect and can be further defined  by the skyrmion Hall angle  \n \nx\nSky y22\nxy= sign( ) arccos( ),vv\nvv \n                                                          (10) \nwhich defines the angle in the range from 180o to 180o. For the situation of STT -induced skyrmion motion, the sign of \nthe vx is always  the same with vs, i.e., the skyrmion Hall angle is in the range of (-90o, 90o), and therefore the Eq. (10) can \nbe reduced to \nSky 22()= arctan( )Q\nQ\nD\nD .  \nThe trajectories  of the f our types of skyrmion driven by the in -plane STT with  vs = 100 m/s,  = 0.4,  = 0.2 and 7 \n DDMI = 3 mJ/m2 is shown in Fig.  5. The positions of the skyrmions are obtain ed by solving  the guiding center ( Rx, Ry) \nwith the form [49, 50]  \nxy ,xqdxdy yqdxdy\nR = , R =\nqdxdy qdxdy \n \n                                                         (11) \nwhere q is the topological density. One can see that the antiskyrmion deflects to the y direction, while for Bloc h \nskyrmion, Né el skyrmion  and twisted skyrmion  deflect to the y direction, i.e., θSky of the skyrmions with Q =1 \n(antiskyrmion) and Q =1 (Bloch,  Né el and twisted skyrmion ) equal to 12.89o and 12.89o, respectively. Following we \nfocus  on the STT -induced motion of twisted skyrmion with differen t conditions, as shown in Fig.  6. Figure s 6 (a) and (b) \nshow the vx and vy as a function of vs for different  with  = 0.2 and DDMI = 3 mJ/m2, respectively.  It can be seen that vx \nand vy both increase linearly with the increase in  vs for different α, it should be also note that vy is a negative value for  < \n, a positive value  for  > , and zero for  = . Then we chose the situation of vs = 100 m/s to investigate the skyrmion \nHall angle of the twisted skyrmion as a function of vs, as shown in Fig.  6(c), the skyrmion Hall angel Sky remains almost \nunchanged with the increase in vs. Figure 6(d) shows the simulation and calculation of Sky as a function of  with  = 0.2 , \nthe skyrmion Hall angle θSky decreases from 13.7 o to 12.89 o with the  increasing from 0. 01 to 0.4. According to Eq s. \n(9) and (10), both the velocity and the skyrmion Hall angle Sky are affected by the dissipative  parameter D, and the \ndissipative parameter D is determined by the DMI strength DDMI. Therefore, it is necessary to investigate the dynamics of \nthe twisted skyrmion under different DDMI, as shown in Figs.  6 (e) and (f) wit h vs = 100 m/s,  = 0.4 and  = 0.2 . vx \nincreases at first and then decreases w ith DDMI increasing from 2.5 to 3.5 mJ/m2, while vy keeps decreasing (the a bsolute \nvalue  of vy is continuously increasing ), and both simulation and calculation results support that the corresponding \nskyrmion Hall angle Sky decreases from  11.4 o to 16.8 o ( the absolute value of Sky is proportional to the DDMI). \nWe have known that the STT -induced twisted skyrmion motion is affected by the damping in the previous paragraph . \nFollowing, we investigate the dynamics of twisted skyrmion induced by the STT under a damp ing gradient, as shown in \nFig. 7. Figure 7(a) shows t he position along the y axis of the twisted skyrmion as a functio n of distance along  the x axis 8 \n with vs = 100 m/s,  = 0.4 and DDMI = 3 mJ/m2. The damping  decreases from 0.5 to 0.25 linearly from 0 to 50 nm along \nthe x axis, as indicated by the color code.  Figure 7 (b) sho ws the skyrmion Hall angle Sky of the twisted skyrmion as a \nfunction of its position along the x axis. In the region  > , the twisted skyrmion moves along the x axis direction  from 0 \nnm and deflects in the –y direction  until moving to the x axis of  20 nm , where  =  = 0.4 ; from the  region of 20 to 50 \nnm along x axis, the twisted skyrmion begins to deflect in the + y direction because of  < . Therefore , the trajectory of \ntwisted skyrmion induced by the ST T can be controlled  under a damping gradient.  \nDynamics of twisted skyrmion driven by the SHE  \nSHE -induced motion of antiskyrmion has already been studied  in Ref. [38], which demonstrates that the \nantiskyrmion Hall angle depends  on the direction of the current strongly. In thi s section, we focus on  the SHE -induced \nmotion s of the skyrmio ns whose winding number W = 1(Bloch, Né el and twisted skyrmion ). The LLG Eqs. (1) and (3)  \ncan be cast into the following form : \nd d HM 4 ( ) 0 v v B J      D GR\n                                                         (12) \nwhere G = (0 0 4) due to Q = 1, B is linked to  the SHE , and the sign of B is determined by th e SHE angle;  R(χ) is the \nin-plane rotation matrix with the form \ncos sin()sin cosR [49, 51] . When the current  JHM injected into the \nheavy metal  along the x direction, vd = (vx, vy) is derived from Eq.  (12) as  \nx HM 22\ny HM 22cos sin,1\nsin cos.1v B J\nv B J  \n\n  \n    D\nD\nD\nD\n                                                            (13) \nThe skyrmion Hall angle Sky can be obtained by the Eq.  (10), which is in the range of  180o to 180o.  \n The Eq. (13) suggests that the direction of motion of the skyrmions depends on their helicities. Therefore, we first \ninvestigate the trajectories of skyrmions driven by the SHE with  JHM = 10 ×  1010 A/m2,  = 0.2 and DDMI = 3 mJ/m2 for \ndifferent  helicities of skyrmions, as shown in Fig.  8. These  skyrmions with different helicities are achieved by changing \nthe direction of DMI vector. The simulation results in Fig.  8(a) show that the skyrmion Hall angles Sky are 150.4o, 9 \n 165.4o, 121.4o, 75.6o, 29.6o, 14.6o, 58.6o and 104.4o for the helicities χ = 0.75, 0.5, 0.25, 0, 0.25, 0.5, 0.75 \nand , respective ly. Figure 8(b) shows the skyrmion Hall angle as a function of the helicity both supported by simulations \nand calculations . Following we take the case of χ = 5 (the twisted skyrmion  shown in Fig. 1(d) ) and investigate the \nmotion induced by the SHE , as shown in Fig.  9. Figure 9(a) shows the simulation results of vx and vy of the twisted \nskyrmion as a function of JHM with  = 0.2 and DDMI = 3 mJ/m2. It can be seen that vx and vy both increase linearly with \nthe increase in  JHM, and the corresponding skyrmion Hall angle Sky is shown in Fig.  9(b). The skyrmion Hall angle  Sky \nalmost remains at 29.6o when JHM is no more than 200 ×  1010 A/m2, while for the case JHM =500 ×  1010 A/m2, the \nskyrmion Hall angle  Sky decreases to 28.9o. This is because  the size of the twisted skyrmion, i.e., the dissipative  \nparameter D, increases slightl y with JHM increasing to 500 ×  1010 A/m2, the skyrmion Hall angle  Eq. (10) can be reduced \nto  \nSky1= arctan( ).1+D\nD\n                                                                     (14) \nFor χ = 5, which indicates that the skyrmion Hall angle  Sky decreases with the increase in D. Figure 9(c) shows the \nsimulation results of vx and vy of the twisted skyrmion as a function of  with JHM =100 ×  1010 A/m2 and DDMI = 3 mJ/m2, \nvx first increases and then decreases w ith  increasing from 0.01 to 1, while vy keeps decreasing (the a bsolute value  of vy \ndecreases at first and then increases), and therefore the corresponding  skyrmion Hall angle  Sky decreases from  44.3o to \n8.2o (the trend of Sky is consistent with  vy), which also supported by  calculation, as shown in Fig.  9(d). Figure 9(e) \nshows that vx and vy both increases with DDMI increasing  from 2.5 to 3.5 mJ/m2 when JHM = 100 ×  1010 A/m2 and  = 0.2.  \nFigure 9(f) shows that the corresponding skyrmion Hall angle Sky decreases with the increase in DDMI, which is similar \nto the res ults by calculating the Eq.  (14) with the increase in D.  \nIn contrast to  the STT -induced twisted skyrmion motion under a damping gradient , we investigate th e dynamics of \ntwisted skyrmion driven  by the S HE under a damping gradient , as shown in Fig.  10. Figure 10 (a) shows the trajectory of \nthe twisted skyrmion as a functio n of its position  along  the x axis with JHM = 100 ×  1010 A/m2 and DDMI = 3 mJ/m2. The \ndamping  increases from 0.2 to 1.2 linearly from 0 to 20 0 nm along the x axis, as indicated by the color code. Figure 10 (b) 10 \n shows the corresponding skyrmion Hall angle Sky as a function of its position along the x axis. The Eq. (14)  implies that:  \nin the region 1D > 0, the twisted skyrmion moves along the x axis direction  from 0 nm and deflects in the y direction  \nuntil moving to the x axis of  114 nm where 1D = 0; in the region 1D < 0, i.e., from 114 to 200 nm along x axis, the \ntwisted skyrmion deflects in the y direction . Therefore , the trajectory of the SHE -induced motion of twisted skyrmion \ncan also be controlled  by a damping gradient.  \nConclusions  \nIn summary,  we first introduce the magnetic structure and the corresponding DMI of the twisted skyrmion in contr ast \nto that of Bloch skyrmion , Né el skyrmion  and antiskyrmion. Furthermore, we discuss and calculate the helicity, winding \nnumber, topological number, size and dissipative force tensor  of the twisted skyrmion, which pave the way for the \nfollowing study of the dynamics  of twisted skyrmion driven by the STT and the SHE . For the STT -induced motion of \ntwisted skyrmion , it is found that the skyrmion Hall angle is determined by the topological number, the dissipative force \ntensor  and the difference  between  the Gilbert damping  and the non -adiabatic factor . For the SHE -induced motion of \ntwisted skyrmion, apart from the dissipative force tensor and the Gilbert damping , the skyrmion angle depends  on the \nhelicity significantly.  At last, we demonstrate  that the trajectories of both the STT -induced and the SHE -induced motion \nof twisted skyrmion can be controlled by a Gilbert  damping gradient . These results may present guidance  for the design \nof twisted skyrmion -based  racetrack memories . \nAcknowledgments  \nThis work is supported  by National Science Fund of China  (11574121 and 51771086 ). C. J.  acknowledges the \nfunding by the China  Scholarship Council.  \nReferences  \n[1] J.C. Slonczewski, Current -driven excitation of magnetic multilayers, J. Magn. Magn. Mater.  159, L1 (1996) . \n[2] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B  54, 9353  (1996) . \n[3] S. Zhang, Z. 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We only intercept the central region of the 2D plane with the size of 50 nm × 50 nm. The red, white \nand blue represent where the z comp onent of the magnetization is positive, zero and negative, respectively. The black arrows  denote  \nthe distribution of the in -plane magnetization. (e)–(h) are the spatial profiles of the  local magnetization  corresponding to the yellow \ndotted line which marked in the Fig.  1(a). (i)–(l) are the configuration s of bulk DMI, interfacial DMI ，anisotropic DMI and twisted \nDMI , respectively. The orange arrows denote  the directions of the DMI vector.   \n \n \n14 \n  \nFIG. 2. Schematic of a general  skyrmion in two-dimensional polar co ordinates ., ,  and  () indica te the radial coordinate, \nazimuthal angle, skyrmion helicity and the angle between the magnetic moment and the radial coordinate, respectively.  \n \n \n \n \n \n \n \n \n15 \n  \nFIG. 3. (a) display s the magnetization distribution s of twisted skyrmion , Bloch skyrmion and Né el skyrmion  with W = 1. (b)  display s \nthe magnetization distribution  of anti skyrmion  with W = 1. (c) and (d) show the distribution s of topological density  corresponding to \nthe magnetizations shown in (a) and (b) with Q = 1 and Q = 1, respectively.  \n \n \n \n \n \n \n \n \n \n16 \n  \nFIG. 4. Skyrmion  diameter  (d) and the diagonal element of the dissipative tensor  (D) as a function of DMI stre ngth. The inset is the \nspatial profile of mz across the twisted skyrmion. It should be note that the twisted skyrmion exist s stably  in region of 250 nm × 250 \nnm, the diagram only show the central part of 50 nm × 50 nm . \n \n \n \n \n \n \n \n \n \n17 \n  \nFIG. 5. The trajectories of four types of skyrmion driven by the STT. The initial  position of the skyrmions is at  the center  of the 2D \nmagnetic film, the size of 2D plane is 250 nm × 250 nm, vs = 100 m/s  in x direction ,  = 0.4,  = 0.2 and DDMI = 3 mJ/m2. The big \nyellow  solid arrow and white dotted arrow s represent the direction of conduction electrons and the trajectories of skyrmions, \nrespectively. It should be note that the four types of skyrmions are enlarged with the purpose to see their helicities clearly . The actual \nsizes of the four skyrmions are almost the same as the skyrmion at the center position.  \n \n \n \n \n \n \n \n \n \n18 \n  \nFIG. 6. The STT-induced motion of the twisted skyrm ion (0.25). (a) and (b) display  the vx and vy as a function of vs for  = 0.01, \n0.1, 0.2, 0.3 and 0.4 with  = 0.2 and DDMI = 3 mJ/m2, respectively. (c) T he skyrmion Hall an gle Sky as a function of vs corresponding \nto the situation of  = 0.4 shown in Figs.  (a) and (b). (d) The skyrmion Hall a ngle Sky as a function of α corresponding to the situation \nof vs = 100 m/s shown in Figs.  6 (a) and (b). (e) and (f) display  the skyrmon velocity and the skyrmion Hall angle as a function of DDMI \nwith vs = 100 m/s,  = 0.4 and  = 0.2 , respectively.  \n \n \n \n19 \n  \nFIG. 7. The STT -induced motion of the twisted skyrmion under a damping gradient. (a) and (b) show the transverse distance  (y axis) of \nthe skyrmion  and the corresponding skyrmion Hall angle Sky as a function of radial distance ( x axis), respectively.  The initial  position \nof the skyrmions is  defined as 0 nm both in x and y axis, vs = 100 m/s,  = 0.4  and DDMI = 3 mJ/m2. The c olor code  represents that the \ndamping  decreases from 0.5 to 0.25 linearly in the region from 0 to 50 nm along the x direction. The red dotted line represents the \nposition w here  = .  \n20 \n  \nFIG. 8. The SHE -induced motion of skyrm ions with different  (a) The trajectories of eight types of  skyrmions with χ = 0.75, 0.5, \n0.25, 0, 0.25, 0.5, 0.75 and  driven by the SHE. The initial  position of the  eight  skyrmions is in the center of the 2D magnetic \nfilm whose size is 250 nm × 250 nm,  = 0.2 and DDMI = 3 mJ/m2. The big yellow  solid arrow denote s the direction of current JHM = 10 \n×  1010 A/m2. The wh ite dotted arrow s represent the trajectories of skyrmions. It also should be note here that the eight  types of \nskyrmi ons are enlarged to see their helicities clearly . The actual sizes of the eight skyrmions are almost the same as them at the center \nposition. (b) The skyrmion Hall angle Sky as a function of  the helicitiy . The black solid squares c orrespond to the e ight types of \nskyrmion in Fig.  8(a), and the black  hollow squares are calculated by the equation.  \n \n21 \n  \nFIG. 9. The SHE -induced motion of the twisted skyrm ion (0.25). (a) and (b) display  the skyrmion velocity and skyrmion Hall \nangle Sky as a function of JHM with  = 0.2 and DDMI = 3 mJ/m2, respectively. (c) and (d) denote the skyrmion velocity and skyrmion \nHall angle Sky as a function of  with JHM = 100 ×  1010 A/m2 and DDMI = 3 mJ/m2, respectively. (e ) and (f) represent the skyrmion \nvelocity and skyrmion Hall angle Sky as a function of DDMI with JHM = 100 ×  1010 A/m2 and  = 0.2, respectively.  \n \n \n \n \n22 \n  \nFIG. 10. The SHE -induced motion of the twisted skyrmion under a damping gradient. (a) and (b) show the transverse distance  (y axis) \nof the skyrmion  and the corresponding skyrmion Hall angle Sky as a function of radial distance ( x axis), respectively.  The initial  \nposition of the skyrmions is  defined as 0 nm both in x and y axis, JHM = 100 ×  1010 A/m2 and DDMI = 3 mJ/m2. The c olor code  \nrepresents that the damping  increases from 0.2 to 1.2 linearly in the region from 0 to 200 nm along the x direction. The red dotted \nline represents the position w here 1D = 0, i.e., the skyrmion Hall angle Sky = 0o. \n"    },    {        "title": "1903.05415v2.Higher_order_linearly_implicit_full_discretization_of_the_Landau__Lifshitz__Gilbert_equation.pdf",        "content": "arXiv:1903.05415v2  [math.NA]  20 Mar 2020HIGHER-ORDER LINEARLY IMPLICIT FULL DISCRETIZATION\nOF THE LANDAU–LIFSHITZ–GILBERT EQUATION\nGEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nAbstract. For the Landau–Lifshitz–Gilbert (LLG) equation of micromagnetics\nwe study linearly implicit backward difference formula (BDF) time discre tizations\nup to order 5 combined with higher-order non-conforming finite elem ent space\ndiscretizations, which are based on the weak formulation due to Alou ges but use\napproximate tangent spaces that are defined by L2-averaged instead of nodal or-\nthogonality constraints. We prove stability and optimal-order erro r bounds in the\nsituation of a sufficiently regular solution. For the BDF methods of or ders 3 to 5,\nthis requires that the damping parameter in the LLG equations be ab ove a posi-\ntive threshold; this condition is not needed for the A-stable method s of orders 1\nand 2, for which furthermore a discrete energy inequality irrespec tive of solution\nregularity is proved.\n1.Introduction\n1.1.Scope.In this paper we study the convergence of higher-order time and s pace\ndiscretizations of the Landau–Lifshitz–Gilbert (LLG) equation, wh ich is the basic\nmodel for phenomena in micromagnetism, such as in recording media [ 26, 36].\nThe main novelty of the paper lies in the construction and analysis of w hat is\napparently the first numerical method for the LLG equation that is second-order\nconvergent in both space and time to sufficiently regular solutions an d that satisfies,\nas an important robustness property irrespective of regularity, a discrete energy\ninequality analogous to that of the continuous problem.\nWe study discretization in time by linearly implicit backward difference fo rmu-\nlae (BDF) up to order 5 and discretization in space by finite elements o f arbitrary\npolynomial degree. For the BDF methods up to order 2 we prove opt imal-order\nerror bounds in the situation of a sufficiently regular solution and a dis crete energy\ninequality irrespective of solution regularity under very weak regula rity assumptions\non the data. For the BDF methods of orders 3 to 5, we prove optima l-order error\nbounds in the situation of a sufficiently regular solution under the add itional condi-\ntionthatthedampingparameter intheLLGequationbeaboveameth od-dependent\npositive threshold. However, no discrete energy inequality irrespe ctive of solution\nregularity is obtained for the BDF methods of orders 3 to 5.\nThe discretization in space is done by a higher-order non-conformin g finite ele-\nment method based on the approach of Alouges [4, 5], which uses a p rojection to\nDate: March 23, 2020.\n2010Mathematics Subject Classification. Primary 65M12, 65M15; Secondary 65L06.\nKey words and phrases. BDF methods, non-conforming finite element method, Landau–\nLifshitz–Gilbert equation, energy technique, stability.\n12 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nan approximate tangent space to the normality constraint. Contr ary to the point-\nwise orthogonality constraints in the nodes, which define the appro ximate tangent\nspace in those papers and yield only first-order convergence also f or finite elements\nwith higher-degree polynomials, we here enforce orthogonality ave raged over the\nfinite element basis functions. With these modified approximate tang ent spaces we\nproveH1-convergence of optimal order in space and time under the assump tion of\na sufficiently regular solution.\nKey issues in the error analysis are the properties of the orthogon al projection\nonto the approximate tangent space, the higher-order consiste ncy error analysis,\nand the proof of stable error propagation, which is based on non-s tandard energy\nestimates and uses both L2and maximum norm finite element analysis.\n1.2.The Landau–Lifshitz–Gilbert equation. The standard phenomenological\nmodel for micromagnetism is provided by the Landau–Lifshitz (LL) e quation\n(1.1) ∂tm=−m×Heff−αm×(m×Heff)\nwhere the unknown magnetization field m=m(x,t) takes values on the unit\nsphereS2,α >0 is a dimensionless damping parameter, and the effective mag-\nnetic fieldHeffdepends on the unknown m. The Landau–Lifshitz equation (1.1)\ncan be equivalently written in the Landau–Lifshitz–Gilbert form\n(1.2) α∂tm+m×∂tm= (1+α2)/bracketleftbig\nHeff−/parenleftbig\nm·Heff/parenrightbig\nm/bracketrightbig\n.\nIndeed, in view of the vector identity a×(b×c) = (a·c)b−(a·b)c,fora,b,c∈R3,\nwe have−m×/parenleftbig\nm×Heff/parenrightbig\n=Heff−/parenleftbig\nm·Heff/parenrightbig\nm,and taking the vector product of\n(1.1) withmand adding αtimes (1.1) then yields (1.2).\nSincem×ais orthogonal to m,for anya∈R3,it is obvious from (1.1) that\n∂tmis orthogonal to m:m·∂tm= 0; we infer that the Euclidean norm satisfies\n|m(x,t)|= 1 for allxand for allt, provided this is satisfied for the initial data.\nThe term in square brackets on the right-hand side in (1.2) can be re written as\nP(m)Heff, where (with Ithe 3×3 unit matrix)\nP(m) =I−mmT\nis the orthogonal projection onto the tangent plane to the unit sp hereS2atm.\nIn this paper we consider the situation\n(1.3) Heff=1\n1+α2/parenleftbig\n∆m+H/parenrightbig\n,\nwhereH=H(x,t) is a given external magnetic field. The factor 1 /(1 +α2) is\nchosen for convenience of presentation, but is inessential for th e theory; it can be\nreplaced by any positive constant factor.\nWiththischoiceof Heff, we arriveattheLandau–Lifshitz–Gilbert (LLG)equation\nin the form\n(1.4) α∂tm+m×∂tm=P(m)(∆m+H).\nWeconsiderthisequationasaninitial-boundaryvalueproblemonabou ndeddomain\nΩ⊂R3and a time interval 0 /lessorequalslantt/lessorequalslant¯t, with homogeneous Neumann boundaryHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 3\nconditions and initial data m0taking values on the unit sphere, i.e., the Euclidean\nnorm|m0(x)|equals 1 for all x∈Ω.\nWe consider the following weak formulation, first proposed by Alouge s [4, 5]:\nFind the solution m:Ω×[0,¯t]→S2withm(·,0) =m0by determining, at\nm(t)∈H1(Ω)3, the time derivative ∂tm(omitting here and in the following the\nargumentt) as that function in the tangent space\nT(m) :=/braceleftbig\nϕ∈L2(Ω)3:m·ϕ= 0 a.e./bracerightbig\n=/braceleftbig\nϕ∈L2(Ω)3:P(m)ϕ=ϕ}\nthat satisfies, for all ϕ∈T(m)∩H1(Ω)3,\n(1.5) α/parenleftbig\n∂tm,ϕ/parenrightbig\n+/parenleftbig\nm×∂tm,ϕ/parenrightbig\n+/parenleftbig\n∇m,∇ϕ/parenrightbig\n=/parenleftbig\nH,ϕ/parenrightbig\n,\nwhere the brackets ( ·,·) denote the L2inner product over the domain Ω. The\nnumerical methods studied in this paper are based on this weak form ulation.\n1.3.Previous work. There is a rich literature on numerical methods for Landau–\nLifshitz(–Gilbert) equations; for the numerical literature up to 20 07 see the review\nby Cimr´ ak [17].\nAlouges & Jaisson [4, 5] propose linear finite element discretizations in space and\nlinearly implicit backward Euler in time for the LLG equation in the weak fo rmula-\ntion (1.5) and prove convergence withoutrates towards nonsmooth weak solutions,\nusing a discrete energy inequality and compactness arguments. Co nvergence of this\ntype was previously shown by Bartels & Prohl [11] for fully implicit meth ods that\nare based on a different formulation of the Landau–Lifshitz equatio n (1.1). In [6],\nconvergence without rates towards weak solutions is shown for a m ethod that is\n(formally) of “almost” order 2 in time, based on the midpoint rule, for the LLG\nequation with an effective magnetic field of a more general type than (1.3).\nIn a complementary line of research, convergence withrates has been studied\nunder sufficiently strong regularity assumptions, which can, howev er, not be guar-\nanteed over a given time interval, since solutions of the LLG equation may develop\nsingularities. A first-order error bound for a linearly implicit time discr etization\nof the Landau–Lifshitz equation (1.1) was proved by Cimr´ ak [16]. Op timal-order\nerror bounds for linearly implicit time discretizations based on the bac kward Euler\nand Crank–Nicolson methods combined with finite element full discret izations for\na different version of the Landau–Lifshitz equation (1.1) were obta ined under suf-\nficient regularity assumptions by Gao [23] and An [7], respectively. In contrast to\n[4, 5, 6, 11], these methods do not satisfy an energy inequality irres pective of the\nsolution regularity.\nNumerical discretizations for the coupled system of the LLG equat ion (1.5) with\nthe eddy current approximation of the Maxwell equations are stud ied by Feischl &\nTran [21], with first-order error bounds in space and time under suffi cient regularity\nassumptions. This also yields thefirst result offirst-order conver gence ofthemethod\nof Alouges & Jaisson [4, 5].\nThere are several methods for the LLG equations that are of for mal order 2 in\ntime (thoughonlyoforder 1 inspace), e.g., [35, 31, 19], but noneof t hemcomes with\nan error analysis. Fully implicit BDF time discretizations for LLG equatio ns have4 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nbeen used successfully in the computational physics literature [37 ], though without\ngiving any error analysis.\nTo the authors’ knowledge, the second-order linearly implicit metho d proposed\nand studied here is thus the first numerical method for the LLG (or LL) equation\nthathasrigorousapriorierrorestimatesoforder2inbothspace andtimeunderhigh\nregularity assumptions and that satisfies a discrete energy inequa lity irrespective of\nregularity.\nWe conclude this brief survey of the literature with a remark: The ex isting con-\nvergence results either give convergence of a subsequence witho ut rates to a weak\nsolution(withoutimposingstrongregularityassumptions), orthey showconvergence\nwith rates towards sufficiently regular solutions (as we do here). Bo th approaches\nyield insight into the numerical methods and have their merits, and th ey comple-\nment each other. Clearly, neither approach is fully satisfactory, b ecause convergence\nwithout rates of some subsequence is nothing to observe inactual computations, and\non the other hand high regularity is at best provable for close to con stant initial con-\nditions [22] or over short time intervals. We regard the situation as a nalogous to\nthe development of numerical methods and their analysis in other fie lds such as\nnonlinear hyperbolic conservation laws: second-order methods ar e highly popular\nin that field, even though they can only be shown to converge with ve ry low order\n(1/2 or less or only without rates) for available regularity properties; s ee, e.g., [32,\nChapter 3]. Nevertheless, second-order methods arefavoredo ver first-order methods\nin many applications, especially if they enjoy some qualitative propert ies that give\nthem robustness in non-regular situations. A similar situation occur s with the LLG\nequation, where the most important qualitative property appears to be the energy\ninequality.\n1.4.Outline. InSection2we describe thenumerical methodsstudied inthispaper .\nThey use time discretization by linearly implicit BDF methods of orders u p to 5 and\nspace discretization by finite elements of arbitrary polynomial degr ee in a numerical\nscheme that is based on the weak formulation (1.5), with an approxim ate tangent\nspace that enforces the orthogonality constraint approximately in anL2-projected\nsense.\nIn Section 3 we state our main results:\n•For the full discretization of (1.5) by linearly implicit BDF methods of or ders 1\nand 2 and finite element methods of arbitrary polynomial degree we g ive optimal-\norder error bounds in the H1norm, under very mild mesh conditions, in the case\nof sufficiently regular solutions (Theorem 3.1). For these methods w e also show a\ndiscrete energy inequality that requires only very weak regularity a ssumptions on\nthe data (Proposition 3.1). This discrete energy inequality is of the s ame type as\nthe one used in [5, 11] for proving convergence without rates to a weak solution.\n•For the linearly implicit BDF methods of orders 3 to 5 and finite element m ethods\nwith polynomial degree at least 2, we have optimal-order error boun ds in theH1\nnorm only if the damping parameter αis larger than some positive threshold, which\ndepends on the order of the BDF method (Theorem 3.2). Moreover , a stronger (but\nstill mild) CFL condition τ/lessorequalslantchis required. A discrete energy inequality underHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 5\nvery weak regularity conditions is not available for the BDF methods o f orders 3\nto 5, in contrast to the A-stable BDF methods of orders 1 and 2.\nIn Section 4 we prove a perturbation result for the continuous pro blem by energy\ntechniques, as a preparation for the proofs of our error bounds for the discretization.\nIn Section 5 we study properties of the L2-orthogonal projection onto the discrete\ntangent space, which are needed to ensure consistency of the fu ll order and stability\nof the space discretization with the higher-order discrete tangen t space.\nIn Section 6 we study consistency properties of the methods and p resent the error\nequation.\nIn Sections 7 and 8 we prove Theorems 3.1 and 3.2, respectively. The higher-\norder convergence proofs are separated into consistency (Sec tion 6) and stability\nestimates. The stability proofs use the technique of energy estima tes, in an unusual\nversion where the error equation is tested with a projection of the discrete time\nderivative of the error onto the discrete tangent space. These p roofs are different\nfor the A-stable BDF methods of orders 1 and 2 and for the BDF met hods of orders\n3 to 5. For the control of nonlinearities, the stability proofs also re quire pointwise\nerror bounds, which are obtained with the help of finite element inver se inequalities\nfrom theH1error bounds of previous time steps.\nIn Section 9 we illustrate our results by numerical experiments.\nIn an Appendix we collect basic results on energy techniques for BDF methods\nthat are needed for our stability proofs.\n2.Discretization of the LLG equation\nWe now describe the time and space discretization that is proposed a nd studied\nin this paper.\n2.1.Time discretization by linearly implicit BDF methods. We shall dis-\ncretize the LLG equation (1.5) in time by the linearly implicit k-step BDF methods,\n1/lessorequalslantk/lessorequalslant5, described by the polynomials δandγ,\nδ(ζ) =k/summationdisplay\nℓ=11\nℓ(1−ζ)ℓ=k/summationdisplay\nj=0δjζj, γ(ζ) =1\nζ/bracketleftbig\n1−(1−ζ)k/bracketrightbig\n=k−1/summationdisplay\ni=0γiζi.\nWe lettn=nτ, n= 0,...,N,be a uniform partition of the interval [0 ,¯t] with\ntime stepτ=¯t/N.For thek-step method we require kstarting values mifor\ni= 0,...,k−1. Forn/greaterorequalslantk, we determine the approximation mntom(tn) as\nfollows. We first extrapolate the known values mn−k,...,mn−1to a preliminary\nnormalized approximation /hatwidermnattn,\n(2.1) /hatwidermn:=k−1/summationdisplay\nj=0γjmn−j−1/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1/vextendsingle/vextendsingle/vextendsingle.\nTo avoid potentially undefined quantities, we define /hatwidermnto be an arbitrary fixed\nunit vector if the denominator in the above formula is zero.6 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nThe derivative approximation ˙mnand the solution approximation mnare related\nby the backward difference formula\n(2.2) ˙mn=1\nτk/summationdisplay\nj=0δjmn−j,i.e.,mn=/parenleftig\n−k/summationdisplay\nj=1δjmn−j+τ˙mn/parenrightig\n/δ0.\nWe determine mnby requiring that for all ϕ∈T(/hatwidermn)∩H1(Ω)3,\n(2.3)α/parenleftbig\n˙mn,ϕ/parenrightbig\n+/parenleftbig\n/hatwidermn×˙mn,ϕ/parenrightbig\n+/parenleftbig\n∇mn,∇ϕ/parenrightbig\n=/parenleftbig\nH(tn),ϕ/parenrightbig\n˙mn∈T(/hatwidermn),i.e.,/hatwidermn·˙mn= 0.\nHere we note that on inserting the formula in (2.2) for mnin the third term of (2.3),\nwe obtain a linear constrained elliptic equation for ˙mn∈T(/hatwidermn)∩H1(Ω)3of the\nform\nα/parenleftbig\n˙mn,ϕ/parenrightbig\n+/parenleftbig\n/hatwidermn×˙mn,ϕ/parenrightbig\n+τ\nδ0/parenleftbig\n∇˙mn,∇ϕ/parenrightbig\n=/parenleftbig\nfn,ϕ/parenrightbig\n∀ϕ∈T(/hatwidermn)∩H1(Ω)3,\nwherefnconsistsofknownterms. Thebilinearformontheleft-handsideis H1(Ω)3-\ncoercive on T(/hatwidermn)∩H1(Ω)3, and hence the above linear equation has a unique\nsolution ˙mn∈T(/hatwidermn)∩H1(Ω)3by the Lax–Milgram lemma. Once this elliptic\nequation is solved for ˙mn, we obtain the approximation mn∈H1(Ω)3tom(tn)\nfrom the second formula in (2.2).\n2.2.Full discretization by BDF and higher-order finite elements .For a\nfamilyofregularandquasi-uniformfiniteelement triangulationsof Ωwithmaximum\nmeshwidth h >0 we form the Lagrange finite element spaces Vh⊂H1(Ω) with\npiecewise polynomials of degree r/greaterorequalslant1. We denote the L2-orthogonal projections\nonto the finite element space by Πh:L2(Ω)→VhandΠh=I⊗Πh:L2(Ω)3→V3\nh.\nWith a function m∈H1(Ω)3that vanishes nowhere on Ω, we associate the discrete\ntangent space\n(2.4)Th(m) ={ϕh∈V3\nh: (m·ϕh,vh) = 0∀vh∈Vh}\n={ϕh∈V3\nh:Πh(m·ϕh) = 0}.\nThis space is different from the discrete tangent space used in [4, 5 ], where the\northogonality constraint m·ϕh= 0 is required to hold pointwise at the finite\nelement nodes. Here, the constraint is enforced weakly on the finit e element space,\nas is done in various saddle point problems for partial differential equ ations, for\nexample forthedivergence-free constraint inthe Stokes problem [14, 25]. Incontrast\nto that example, here the bilinear form associated with the linear con straint, i.e.,\nb(m;ϕh,vh) = (m·ϕh,vh), depends on the state m. This dependence substantially\naffects both the implementation and the error analysis.\nFollowing the general approach of [4, 5] with this modified discrete ta ngent space,\nwe discretize (1.5) in space by determining the time derivative ∂tmh(t)∈Th(mh(t))\nsuch that (omitting the argument t)\n(2.5)α/parenleftbig\n∂tmh,ϕh/parenrightbig\n+/parenleftbig\nmh×∂tmh,ϕh/parenrightbig\n+/parenleftbig\n∇mh,∇ϕh/parenrightbig\n=/parenleftbig\nH,ϕh/parenrightbig\n∀ϕh∈Th(mh),\nwhere the brackets ( ·,·) denote again the L2inner product over the domain Ω.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 7\nThe full discretization with the linearly implicit BDF method is then readily\nobtained from (2.3): determine ˙mn\nh∈Th(/hatwidermn\nh) such that\n(2.6)α/parenleftbig˙mn\nh,ϕh/parenrightbig\n+/parenleftbig/hatwidermn\nh×˙mn\nh,ϕh/parenrightbig\n+/parenleftbig\n∇mn\nh,∇ϕh/parenrightbig\n=/parenleftbig\nHn,ϕh/parenrightbig\n∀ϕh∈Th(/hatwidermn\nh),\nwhere/hatwidermn\nhand˙mn\nhare related to mn−j\nhforj= 0,...,kin the same way as in (2.1)\nand (2.2) above with mn−j\nhin place ofmn−j, viz.,\n(2.7) ˙mn\nh=1\nτk/summationdisplay\nj=0δjmn−j\nh,/hatwidermn\nh=k−1/summationdisplay\nj=0γjmn−j−1\nh/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle.\nTo avoid potentially undefined quantities, we define /hatwidermn\nhto be an arbitrary fixed\nunit vector if the denominator in the above formula is zero. (We will, ho wever, show\nthat this does not occur in the situation of sufficient regularity.)\nTo implement the discrete tangent space Th(/hatwidermn\nh), there are at least two options:\nusing the constraints Πh(m·ϕh) = 0 or constructing a local basis of Th(m).\n(a)Constraints : Letφifori= 1,...,N:= dimVhdenote the nodal basis of\nVhand denote the basis functions of V3\nhbyφi=ek⊗φifori= (i,k), where\nekfork= 1,2,3 are the standard unit vectors of R3. We denote by MandA\nthe usual mass and stiffness matrices, respectively, with entries mij= (φi,φj)L2(Ω)\nandaij= (∇φi,∇φj)L2(Ω)3. We further introduce the sparse skew-symmetric matrix\nSn= (sn\ni,j)∈R3N×3Nwithentries sn\ni,j= (/hatwidermn\nh×φi,φj)L2(Ω)3andthesparseconstraint\nmatrixCn= (cn\ni,j)∈R3N×Nbycn\ni,j= (/hatwidermn\nh·φi,φj)L2(Ω). Finally, we denote the\nmatrix of the unconstrained time-discrete problem as\nKn=αI⊗M+τ\nδ0I⊗A+Sn.\nLet ˙mn∈R3Ndenote the nodal vector of ˙mn\nh∈Th(/hatwidermn\nh). In this setting, (2.6) yields\na system of linear equations of saddle point type\nKn˙mn+(Cn)Tλn=fn,\nCn˙mn= 0,\nwhereλn∈RNis the unknown vector of Lagrange multipliers and fn∈R3Nis a\nknown right-hand side.\n(b)Local basis : It is possible to compute a local basis of Th(m) by solving small\nlocal problems. To see that, let ω⊂Ωdenote a collection of elements of the mesh\nand letω⊃ωdenote the same set plus the layer of elements touching ω(the patch\nofω). A sufficient (and necessary) condition for ϕh∈V3\nhwith supp(ϕh)⊆ωto\nbelong toTh(m) is\n(2.8) ( m·ϕh,ψh) = 0 for all ψh∈Vhwith supp(ψh)⊆ω.\nIf we denote by # ωthe number of generalized hat functions of Vhsupported in ω,\nthe space of functions in V3\nhwith support in ωis 3#ω-dimensional. On the other\nhand, the space of test functions in (2.8) is # ω-dimensional. We may choose ω\nsufficiently large (depending only on shape regularity) such that 3# ω >#ωand\nhence (2.8) has at least one solution which is then a local basis functio n ofTh(m).\nChoosing different ωto coverΩyields a full basis of Th(m).8 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nLet us denote the so obtained basis of Th(/hatwidermn\nh) by (ψn\nℓ), given via ψn\nℓ=/summationtext\niφibn\niℓ,\nand the sparse basis matrix by Bn= (bn\niℓ). Then, the nodal vector ˙ mn=Bnxnis\nobtained by solving the linear system\n(Bn)TKnBnxn= (Bn)Tfn.\nAn advantage of this approach is that the dimension is roughly halved compared\nto the formulation with constraints. However, the efficiency of one approach versus\nthe other depends heavily on the numerical linear algebra used. Suc h comparisons\nare outside the scope of this paper.\nRemark 2.1. The algorithm described above does not enforce the norm constra int\n|m|= 1 at the nodes. The user might add a normalization step in the definit ion\nofmnin (2.2). However, here we do not consider this normalized variant of the\nmethod, whose convergence properties are not obvious to derive .\nRemark 2.2. Differently to [4], we do not use the pointwise discrete tangent space\nTpw\nh(m) ={ϕh∈V3\nh:m·ϕ= 0 in every node }\n={ϕh∈V3\nh:Ih(m·ϕh) = 0}=IhP(m)V3\nh,\nwhereIh:C(¯Ω)→Vhdenotesfiniteelementinterpolationand Ih=I⊗Ih:C(¯Ω)3→\nV3\nh. It is already reported in [4, Section 4] that an improvement of the o rder with\nhigher-degree finite elements could not be observed in numerical ex periments when\nusing the pointwise tangent spaces in the discretization (2.5). Our a nalysis shows\na lack of consistency of optimal order in the discretization with Tpw\nh(m), which\noriginates from the fact that IhP(m) is not self-adjoint. The order reduction can,\nhowever, be cured by adding a correction term: in the nth time step, determine\n˙mn\nh∈Tpw\nh(/hatwidermn\nh) such that for all ϕh∈Tpw\nh(/hatwidermn\nh),\n(2.9)α/parenleftbig˙mn\nh,ϕh/parenrightbig\n+/parenleftbig/hatwidermn\nh×˙mn\nh,ϕh/parenrightbig\n+/parenleftbig\n∇mn\nh,∇ϕh/parenrightbig\n−/parenleftbig\n∇/hatwidermn\nh,∇(I−P(/hatwidermn\nh))ϕh/parenrightbig\n=/parenleftbig\nP(/hatwidermn\nh)H(tn),ϕh/parenrightbig\n,\nwith notation /hatwidermn\nhand˙mn\nhas in (2.7). With the techniques of the present paper, it\ncan be shown that like (2.6), also this discretization converges with o ptimal order\nin theH1norm under sufficient regularity conditions. Since this paper is alread y\nrather long, we do not include the proof of this result. In contrast to (2.6) for the\nfirst- and second-order BDF methods, the method (2.9) does not admit anh- and\nτ-independent bound of the energy that is irrespective of the smoo thness of the\nsolution.\n3.Main results\n3.1.Error bound and energy inequality for BDF of orders 1 and 2. For\nthe full discretization with first- and second-order BDF methods a nd finite elements\nof arbitrary polynomial degree r/greaterorequalslant1 we will prove the following optimal-order error\nbound in Sections 5 to 7.\nTheorem 3.1 (Error bound for orders k= 1,2).Consider the full discretization\n(2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time discretiza-\ntion fork/lessorequalslant2and finite elements of polynomial degree r/greaterorequalslant1from a family ofHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 9\nregular and quasi-uniform triangulations of Ω. Suppose that the solution mof the\nLLG equation is sufficiently regular. Then, there exist ¯τ >0and¯h >0such that\nfor numerical solutions obtained with step sizes τ/lessorequalslant¯τand meshwidths h/lessorequalslant¯h, which\nare restricted by the very mild CFL-type condition\nτk/lessorequalslant¯ch1/2\nwith a sufficiently small constant ¯c(independent of handτ), the errors are bounded\nby\n(3.1) /ba∇dblmn\nh−m(tn)/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr)fortn=nτ/lessorequalslant¯t,\nwhereCis independent of h,τandn(but depends on αand exponentially on ¯t),\nprovided that the errors of the starting values also satisfy such a bound.\nThe precise regularity requirements are as follows:\n(3.2)m∈Ck+1([0,¯t],L∞(Ω)3)∩C1([0,¯t],Wr+1,∞(Ω)3),\n∆m+H∈C([0,¯t],Wr+1,∞(Ω)3).\nRemark 3.1 (Discrepancy from normality ).Sincem(x,tn) are unit vectors, an\nimmediate consequence of the error estimate (3.1) is that\n(3.3) /ba∇dbl1−|mn\nh|/ba∇dblL2(Ω)/lessorequalslantC(τk+hr) fortn=nτ/lessorequalslant¯t,\nwith a constant Cindependent of n,τandh. The proof of Theorem 3.1 also shows\nthat the denominator in the definition of the normalized extrapolate d value/hatwidermn\nh\nsatisfies\n/vextenddouble/vextenddouble/vextenddouble1−/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble\nL∞(Ω)/lessorequalslantCh−1/2(τk+hr)/lessorequalslant1\n2fortn=nτ/lessorequalslant¯t,\nwhich in particular ensures that /hatwidermn\nhis unambiguously defined.\nTesting with ϕ=∂tm∈T(m) in (1.5), we obtain (only formally, if ∂tmis not\ninH1(Ω)3)\nα(∂tm,∂tm)+(∇m,∂t∇m) = (H,∂tm),\nwhich, by integration in time and the Cauchy–Schwarz and Young ineq ualities, im-\nplies the energy inequality\n/ba∇dbl∇m(t)/ba∇dbl2\nL2+1\n2α/integraldisplayt\n0/ba∇dbl∂tm(s)/ba∇dbl2\nL2ds/lessorequalslant/ba∇dbl∇m(0)/ba∇dbl2\nL2+1\n2α/integraldisplayt\n0/ba∇dblH(s)/ba∇dbl2\nL2ds.\nSimilarly, wetestwith ϕh=˙mn\nh∈Th(/hatwidermn\nh)in(2.6). Thenwecanprovethefollow-\ning discrete energy inequality, which holds under very weak regularit y assumptions\non the data.\nProposition 3.1 (Energy inequality for orders k= 1,2).Consider the full dis-\ncretization (2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time\ndiscretization for k/lessorequalslant2and finite elements of polynomial degree r/greaterorequalslant1. Then, the10 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nnumerical solution satisfies the following discrete energy inequality :forn/greaterorequalslantkwith\nnτ/lessorequalslant¯t,\nγ−\nk/ba∇dbl∇mn\nh/ba∇dbl2\nL2+1\n2ατn/summationdisplay\nj=k/ba∇dbl˙mj\nh/ba∇dbl2\nL2/lessorequalslantγ+\nkk−1/summationdisplay\ni=0/ba∇dbl∇mi\nh/ba∇dbl2\nL2+τ\n2αn/summationdisplay\nj=k/ba∇dblH(tj)/ba∇dbl2\nL2,\nwhereγ±\n1= 1andγ±\n2= (3±2√\n2)/4.\nThis energy inequality is an important robustness indicator of the nu merical\nmethod. In [5, 11], such energy inequalitys are used to prove conve rgence with-\nout rates (for a subsequence τn→0 andhn→0) to a weak solution of the LLG\nequation for the numerical schemes considered there (which have γ±= 1, but this\nis inessential in the proofs).\nAs the proof of Proposition 3.1 is short, we give it here.\nProof.TheproofreliesontheA-stabilityofthefirst-andsecond-orderB DFmethods\nvia Dahlquist’s G-stabilitytheoryasexpressed inLemma 10.1ofthe Ap pendix, used\nwithδ(ζ) =/summationtextk\nℓ=1(1−ζ)ℓ/ℓandµ(ζ) = 1. The positive definite symmetric matrices\nG= (gij)k\ni,j=1are known to be G= 1 fork= 1 and (see [27, p.309])\nG=1\n4/parenleftbigg\n1−2\n−2 5/parenrightbigg\nfork= 2,\nwhich has the eigenvalues γ±= (3±2√\n2)/4.\nWe test with ϕh=˙mn\nh∈Th(/hatwidermn\nh) in (2.6) and note/parenleftbig\n/hatwidermn\nh×˙mn\nh,˙mn\nh/parenrightbig\n= 0, so that\nα/ba∇dbl˙mn\nh/ba∇dbl2\nL2+(∇mn\nh,∇˙mn\nh) = (Hn,˙mn\nh).\nThe right-hand side is bounded by\n(Hn,˙mn\nh)/lessorequalslantα\n2/ba∇dbl˙mn\nh/ba∇dbl2\nL2+1\n2α/ba∇dblHn/ba∇dbl2\nL2.\nRecalling the definition of ˙mn\nh, we have by Lemma 10.1\n(∇mn\nh,∇˙mn\nh)/greaterorequalslant1\nτk/summationdisplay\ni,j=1gij(∇mn−i+1\nh,∇mn−j+1\nh)−1\nτk/summationdisplay\ni,j=1gij(∇mn−i\nh,∇mn−j\nh).\nWe fix ¯nwithk/lessorequalslant¯n/lessorequalslant¯t/τand sum from n=kto ¯nto obtain\nk/summationdisplay\ni,j=1gij(∇m¯n−i+1\nh,∇m¯n−j+1\nh)+1\n2ατ¯n/summationdisplay\nn=k/ba∇dbl˙mn\nh/ba∇dbl2\nL2\n/lessorequalslantk/summationdisplay\ni,j=1gij(∇mk−i\nh,∇mk−j\nh)+τ\n2α¯n/summationdisplay\nn=k/ba∇dblHn/ba∇dbl2\nL2.\nNoting that\nγ−/ba∇dbl∇m¯n\nh/ba∇dbl2\nL2/lessorequalslantk/summationdisplay\ni,j=1gij(∇m¯n−i+1\nh,∇m¯n−j+1\nh),HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 11\nk/summationdisplay\ni,j=1gij(∇mk−i\nh,∇mk−j\nh)/lessorequalslantγ+k−1/summationdisplay\ni=0/ba∇dbl∇mi\nh/ba∇dbl2\nL2,\nwe obtain the stated result. /square\n3.2.Error bound for BDF of orders 3to5.For the BDF methods of orders\n3 to 5 we prove the following result in Section 8. Here we require a stro nger, but\nstill moderate stepsize restriction in terms of the meshwidth. More importantly, we\nmust impose a positive lower bound on the damping parameter αof (1.1).\nTheorem 3.2 (Error bound for orders k= 3,4,5).Consider the full discretization\n(2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time discretiza-\ntion for3/lessorequalslantk/lessorequalslant5and finite elements of polynomial degree r/greaterorequalslant2from a family of\nregular and quasi-uniform triangulations of Ω. Suppose that the solution mof the\nLLG equation has the regularity (3.2), and that the damping parameter αsatisfies\n(3.4)α>α kwith\nαk= 0.0913,0.4041,4.4348,fork= 3,4,5,respectively.\nThen, for an arbitrary constant ¯C >0, there exist ¯τ >0and¯h >0such that for\nnumerical solutions obtained with step sizes τ/lessorequalslant¯τand meshwidths h/lessorequalslant¯hthat are\nrestricted by\n(3.5) τ/lessorequalslant¯Ch,\nthe errors are bounded by\n/ba∇dblmn\nh−m(tn)/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr)fortn=nτ/lessorequalslant¯t,\nwhereCis independent of h,τandn(but depends on αand exponentially on ¯C¯t),\nprovided that the errors of the starting values also satisfy such a bound.\nTheorem 3.2 limits the use of the BDF methods of orders higher than 2 (and more\nseverely for orders higher than 3) to applications with a large dampin g parameter α,\nsuch as cases described in [24, 39]. We remark, however, that in man y situations\nαis of magnitude 10−2or even smaller [10]. A very small damping parameter α\naffects not only the methods considered here. To our knowledge, t he error analysis\nof any numerical method proposed in the literature breaks down as α→0, as does\nthe energy inequality.\nIt is not surprising that a positive lower bound on αarises for the methods of\nordersk/greaterorequalslant3, since they are not A-stable and a lower bound on αis required also for\nthe simplified linear problem ( α+i)∂tu=∆u, which arises from (1.4) by freezing m\nin the termm×∂tmand diagonalizing this skew-symmetric linear operator (with\neigenvalues ±i and 0) and by omitting the projection P(m) on the right-hand side\nof (1.4).\nThe proof of Theorem 3.2 uses a variant of the Nevanlinna–Odeh mult iplier tech-\nnique [34], which is described in the Appendix for the convenience of th e reader.\nWhile for sufficiently large αwe have an optimal-order error bound in the case of\na smooth solution, there is apparently no discrete energy inequality under weak\nregularity assumptions similar to Proposition 3.1 for the BDF methods of orders 3\nto 5.12 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nAs in Remark 3.1, the error bounds also allow us to bound the discrepa ncy from\nnormality.\n4.A continuous perturbation result\nIn this section we present a perturbation result for the continuou s problem, be-\ncause we will later transfer the arguments of its proof to the discr etizations to prove\nstability and convergence of the numerical methods.\nLetm(t) be a solution of (1.4) for 0 /lessorequalslantt/lessorequalslant¯t, and letm⋆(t), also of unit length,\nsolve the same equation up to a defect d(t) for 0/lessorequalslantt/lessorequalslant¯t:\n(4.1)α∂tm⋆+m⋆×∂tm⋆=P(m⋆)(∆m⋆+H)+d\n=P(m)(∆m⋆+H)+r,\nwith\nr=−/parenleftbig\nP(m)−P(m⋆)/parenrightbig\n(∆m⋆+H)+d.\nThen,m⋆also solves the perturbed weak formulation\nα(∂tm⋆,ϕ)+(m⋆×∂tm⋆,ϕ)+(∇m⋆,∇ϕ) = (r,ϕ)∀ϕ∈T(m)∩H1(Ω)3,\nand the error e=m−m⋆satisfies the error equation\n(4.2)α(∂te,ϕ)+(e×∂tm⋆,ϕ)+(m×∂te,ϕ)+(∇e,∇ϕ) =−(r,ϕ)\n∀ϕ∈T(m)∩H1(Ω)3.\nBefore we turn to the perturbation result, we need Lipschitz-typ e bounds for the\northogonal projection P(m) =I−mmTapplied to sufficiently regular functions.\nLemma 4.1. The projection P(·)satisfies the following estimates, for functions\nm,m⋆,v:Ω→R3, wheremandm⋆take values on the unit sphere and m⋆∈\nW1,∞(Ω)3:\n/ba∇dbl(P(m)−P(m⋆))v/ba∇dblL2(Ω)3/lessorequalslant2/ba∇dblv/ba∇dblL∞(Ω)3/ba∇dblm−m⋆/ba∇dblL2(Ω)3,/vextenddouble/vextenddouble∇/parenleftbig\n(P(m)−P(m⋆))v/parenrightbig/vextenddouble/vextenddouble\nL2(Ω)3×3/lessorequalslant2/ba∇dblm⋆/ba∇dblW1,∞(Ω)3/ba∇dblv/ba∇dblW1,∞(Ω)3/ba∇dblm−m⋆/ba∇dblL2(Ω)3\n+6/ba∇dblv/ba∇dblL∞(Ω)3/ba∇dbl∇(m−m⋆)/ba∇dblL2(Ω)3×3.\nProof.Settinge=m−m⋆, we start by rewriting\n(P(m)−P(m⋆))v=−(mmT−m⋆mT\n⋆)v=−(meT+emT\n⋆)v.\nThe first inequality then follows immediately by taking the L2norm of both sides\nof the above equality, using the fact that mandm⋆are of unit length. The second\ninequality is proved similarly, using the product rule\n∂i(P(m)−P(m⋆))v=−∂i(eeT+m⋆eT+emT\n⋆)v\n=−(∂ieeT+e∂ieT+∂im⋆eT+m⋆∂ieT+∂iemT\n⋆+e∂imT\n⋆)v\n+(meT+emT\n⋆)∂iv,\ntheL∞bound of∂im⋆, and the fact that /ba∇dble/ba∇dblL∞/lessorequalslant/ba∇dblm/ba∇dblL∞+/ba∇dblm⋆/ba∇dblL∞/lessorequalslant2./square\nWe have the following perturbation result.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 13\nLemma 4.2. Letm(t)andm⋆(t)be solutions of unit length of (1.5)and(4.1),\nrespectively, and suppose that, for 0/lessorequalslantt/lessorequalslant¯t, we have\n(4.3)/ba∇dblm⋆(t)/ba∇dblW1,∞(Ω)3+/ba∇dbl∂tm⋆(t)/ba∇dblW1,∞(Ω)3/lessorequalslantR\nand/ba∇dbl∆m⋆(t)+H(t)/ba∇dblL∞(Ω)3/lessorequalslantK.\nThen, the error e(t) =m(t)−m⋆(t)satisfies, for 0/lessorequalslantt/lessorequalslant¯t,\n(4.4) /ba∇dble(t)/ba∇dbl2\nH1(Ω)3/lessorequalslantC/parenleftig\n/ba∇dble(0)/ba∇dbl2\nH1(Ω)3+/integraldisplayt\n0/ba∇dbld(s)/ba∇dbl2\nL2(Ω)3ds/parenrightig\n,\nwhere the constant Cdepends only on α,R,K, and¯t.\nProof.Let us first assume that ∂tm(t)∈H1(Ω)3for allt. Following [21], we test in\nthe error equation (4.2) with ϕ=P(m)∂te∈T(m). By the following argument,\nthis test function is then indeed in H1(Ω)3and can be viewed as a perturbation\nof∂te:\nϕ=P(m)∂te=P(m)∂tm−P(m)∂tm⋆\n=P(m)∂tm−P(m⋆)∂tm⋆−(P(m)−P(m⋆))∂tm⋆\n=∂tm−∂tm⋆−(P(m)−P(m⋆))∂tm⋆,\nand so we have\n(4.5)ϕ=P(m)∂te=∂te+qwithq=−(P(m)−P(m⋆))∂tm⋆.\nBy Lemma 4.1 and using (4.3) we have\n(4.6) /ba∇dblq/ba∇dblL2/lessorequalslant2R/ba∇dble/ba∇dblL2and/ba∇dbl∇q/ba∇dblL2/lessorequalslantCR/ba∇dble/ba∇dblH1.\nTesting the error equation (4.2) with ϕ=∂te+q, we obtain\nα(∂te,∂te+q)+(e×∂tm⋆,∂te+q)+(m×∂te,∂te+q)\n+(∇e,∇(∂te+q)) =−(r,∂te+q),\nwhere, by (4.1) and Lemma 4.1 with (4.3), ris bounded as\n(4.7)/ba∇dblr/ba∇dblL2/lessorequalslant/ba∇dbl/parenleftbig\nP(m)−P(m⋆)/parenrightbig\n(∆m⋆+H)/ba∇dblL2+/ba∇dbld/ba∇dblL2\n/lessorequalslant2K/ba∇dble/ba∇dblL2+/ba∇dbld/ba∇dblL2.\nBy collecting terms, and using the fact that ( m×∂te,∂te) vanishes, we altogether\nobtain\nα/ba∇dbl∂te/ba∇dbl2\nL2+1\n2d\ndt/ba∇dbl∇e/ba∇dbl2\nL2=−α(∂te,q)−(e×∂tm⋆,∂te+q)−(m×∂te,q)\n−(∇e,∇q)−(r,∂te+q).\nFor the right-hand side, the Cauchy–Schwarz inequality and /ba∇dblm/ba∇dblL∞= 1 yield\nα/ba∇dbl∂te/ba∇dbl2\nL2+1\n2d\ndt/ba∇dbl∇e/ba∇dbl2\nL2/lessorequalslantα/ba∇dbl∂te/ba∇dblL2/ba∇dblq/ba∇dblL2+R/ba∇dble/ba∇dblL2(/ba∇dbl∂te/ba∇dblL2+/ba∇dblq/ba∇dblL2)\n+/ba∇dbl∂te/ba∇dblL2/ba∇dblq/ba∇dblL2+/ba∇dbl∇e/ba∇dblL2/ba∇dbl∇q/ba∇dblL2+/ba∇dblr/ba∇dblL2(/ba∇dbl∂te/ba∇dblL2+/ba∇dblq/ba∇dblL2).14 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nYoung’s inequality and absorptions, together with the bounds in (4.6 ) and (4.7),\nyield\nα1\n2/ba∇dbl∂te/ba∇dbl2\nL2+1\n2d\ndt/ba∇dbl∇e/ba∇dbl2\nL2/lessorequalslantc/ba∇dble/ba∇dbl2\nH1+c/ba∇dbld/ba∇dbl2\nL2.\nHere, we note that\n1\n2d\ndt/ba∇dble/ba∇dbl2\nL2= (∂te,e)/lessorequalslant1\n2/ba∇dbl∂te/ba∇dbl2\nL2+1\n2/ba∇dble/ba∇dbl2\nL2,so that /ba∇dbl∂te/ba∇dbl2\nL2/greaterorequalslantd\ndt/ba∇dble/ba∇dbl2\nL2−/ba∇dble/ba∇dbl2\nL2.\nCombining these inequalities and integrating in time, we obtain\n/ba∇dble(t)/ba∇dbl2\nH1/lessorequalslantc/ba∇dble(0)/ba∇dbl2\nH1+c/integraldisplayt\n0/ba∇dble(s)/ba∇dbl2\nH1ds+c/integraldisplayt\n0/ba∇dbld(s)/ba∇dbl2\nL2ds.\nBy Gronwall’s inequality, we then obtain the stated error bound.\nFinally, if∂tm(t) is not inH1(Ω)3for somet, then a regularization and density\nargument, which we do not present here, yields the result, since th e error bound\ndoes not depend on the H1norm of∂tm. /square\n5.Orthogonal projection onto the discrete tangent space\nFor consistency and stability of the full discretization, we need to s tudy properties\nof theL2(Ω)-orthogonal projection onto the discrete tangent space Th(m), which\nwe denote by\nPh(m):V3\nh→Th(m).\nWe do not have an explicit expression for this projection, but the pr operties stated\nin Lemmas 5.1 to 5.3 will be used for proving consistency and stability. W e recall\nthat we consider a quasi-uniform, shape-regular family Thof triangulations with\nLagrange finite elements of polynomial degree r.\nThe first lemma states that the projection Ph(m) approximates the orthogonal\nprojection P(m) =I−mmTonto the tangent space T(m) with optimal order. It\nwill be used in the consistency error analysis of Section 6.\nLemma 5.1. Form∈Wr+1,∞(Ω)3with|m|= 1almost everywhere we have\n/ba∇dbl(Ph(m)−P(m))v/ba∇dblL2(Ω)3/lessorequalslantChr+1/ba∇dblv/ba∇dblHr+1(Ω)3,\n/ba∇dbl(Ph(m)−P(m))v/ba∇dblH1(Ω)3/lessorequalslantChr/ba∇dblv/ba∇dblHr+1(Ω)3,\nfor allv∈Hr+1(Ω)3, whereCdepends on a bound of /ba∇dblm/ba∇dblWr+1,∞(Ω)3.\nThe second lemma states that the projection Ph(m) has Lipschitz bounds of the\nsame type as those of the orthogonal projection P(m) given in Lemma 4.1. It will\nbe used in the stability analysis of Sections 7 and 8.\nLemma 5.2. Letm∈W1,∞(Ω)3and/tildewiderm∈H1(Ω)3with|m|=|/tildewiderm|= 1almost\neverywhere and /ba∇dblm/ba∇dblW1,∞/lessorequalslantR. There exist CR>0andhR>0such that for\nh/lessorequalslanthR, for allvh∈V3\nh,\n(i)/ba∇dbl(Ph(m)−Ph(/tildewiderm))vh/ba∇dblL2(Ω)3/lessorequalslantCR/ba∇dblm−/tildewiderm/ba∇dblLp(Ω)3/ba∇dblvh/ba∇dblLq(Ω)3,\nfor(p,q)∈ {(2,∞),(∞,2)}, and\n(ii)/ba∇dbl(Ph(m)−Ph(/tildewiderm))vh/ba∇dblH1(Ω)3/lessorequalslantCR/ba∇dblm−/tildewiderm/ba∇dblH1(Ω)3/ba∇dblvh/ba∇dblL∞(Ω)3HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 15\n+CR/ba∇dblm−/tildewiderm/ba∇dblL2(Ω)3/ba∇dblvh/ba∇dblW1,∞(Ω)3.\nThe next lemma shows the Ws,p-stability of the projection. It is actually used for\np= 2 in the proof of Lemmas 5.1 and 5.2 and will be used for p= 2 in Section 6\nand forp=∞in Sections 7 and 8.\nLemma 5.3. There exists a constant depending only on p∈[1,∞]and the shape\nregularity of the mesh such that for all m∈W1,∞(Ω)3with|m|= 1almost every-\nwhere,\n/ba∇dblPh(m)vh/ba∇dblWs,p(Ω)3/lessorequalslantC/ba∇dblm/ba∇dbl2\nW1,∞(Ω)3/ba∇dblvh/ba∇dblWs,p(Ω)3\nfor allvh∈V3\nhands∈ {−1,0,1}.\nThese three lemmas will be proved in the course of this section, in whic h we\nformulate also three more lemmas that are of independent interest but will not be\nused in the following sections.\nIn the following, we use the dual norms\n/ba∇dblv/ba∇dblW−1,q:= sup\nw∈W1,p(v,w)\n/ba∇dblw/ba∇dblW1,pfor 1/p+1/q= 1.\nThe space W−1,1(Ω) is not the dual space of W1,∞(Ω) but rather defined as the\nclosure ofL2(Ω) withrespect to thenorm /ba∇dbl·/ba∇dblW−1,1. Wealso recall that Πh:Ws,p(Ω)\n→Ws,p(Ω) is uniformly bounded for s∈ {0,1}andp∈[1,∞] (see, e.g., [20]\nfor proofs in a much more general setting). By duality, we also obta in uniform\nboundedness for s=−1 andp∈[1,∞]. A useful consequence is that for vh∈Vh,\n/ba∇dblvh/ba∇dblW−1,q= sup\nw∈W1,p(vh,Πhw)\n/ba∇dblw/ba∇dblW1,p\n/lessorequalslantsup\nw∈W1,p(vh,Πhw)\n/ba∇dblΠhw/ba∇dblW1,psup\nw∈W1,p/ba∇dblΠhw/ba∇dblW1,p\n/ba∇dblw/ba∇dblW1,p/lessorsimilarsup\nwh∈Vh(vh,wh)\n/ba∇dblwh/ba∇dblW1,p.\nLemma 5.4. There holds /ba∇dblv/ba∇dblWs,p(Ω)≃supw∈W−s,q(Ω)(v,w)\n/bardblw/bardblW−s,q(Ω)with1/p+1/q= 1\nforp∈[1,∞]ands∈ {−1,0,1}.\nProof.The interesting case is ( s,p) = (1,∞) since all other cases follow by duality.\nForv∈W1,∞(Ω), thereexists asequence offunctions qn∈C∞\n0(Ω)3with/ba∇dblqn/ba∇dblL1= 1\nsuch that\n/ba∇dbl∇v/ba∇dblL∞= lim\nn→∞(∇v,qn) = lim\nn→∞−(v,divqn)/lessorequalslantsup\nq∈W1,1(v,divq)\n/ba∇dblq/ba∇dblL1.\nMoreover, there holds\n/ba∇dbldivq/ba∇dblW−1,1/lessorequalslantsup\nw∈W1,∞(q,∇w)\n/ba∇dbl∇w/ba∇dblL∞/lessorequalslant/ba∇dblq/ba∇dblL1.\nCombining the last two estimates shows\n/ba∇dbl∇v/ba∇dblL∞/lessorequalslantsup\nw∈W−1,1(v,w)\n/ba∇dblw/ba∇dblW−1,1.16 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nSince\n/ba∇dblv/ba∇dblL∞= sup\nw∈L1(v,w)\n/ba∇dblw/ba∇dblL1/lessorequalslantsup\nw∈W−1,1(v,w)\n/ba∇dblw/ba∇dblW−1,1,\nwe conclude the proof. /square\nLetthediscretenormalspace Nh(m) :=V3\nh⊖Th(m)begivenasthe L2-orthogonal\ncomplement of Th(m) inV3\nh. We note that\n(5.1) Nh(m) ={Πh(mψh) :ψh∈Vh}\nby the definition of Th(m). The functions in the discrete normal space are bounded\nfrom below as follows.\nLemma 5.5. For everyR >0, there exist hR>0andc >0such that for all\nm∈W1,∞(Ω)3with|m|= 1almost everywhere and /ba∇dblm/ba∇dblW1,∞(Ω)/lessorequalslantRand for all\nh/lessorequalslanthR,\n/ba∇dblΠh(mψh)/ba∇dblWs,p(Ω)3/greaterorequalslantc/ba∇dblψh/ba∇dblWs,p(Ω)\nfor allψh∈Vhand(s,p)∈ {−1,0,1}×[1,∞].\nProof.(a) We first prove the result for s∈ {−1,0}. LetIh:C(Ω)→V3\nhdenote the\nnodal interpolation operator and define mh:=Ihm∈V3\nh.\nThere holds\n/ba∇dblΠh(mhψh)/ba∇dblLp/greaterorequalslant/ba∇dblmhψh/ba∇dblLp−/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp.\nMoreover, stability of ΠhinLp(Ω)3, for 1/lessorequalslantp/lessorequalslant∞, see [20], implies the estimate\n/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp/lessorequalslant(1+C) inf\nvh∈V3\nh/ba∇dblmhψh−vh/ba∇dblLp.\nIn turn, this implies\n/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp/lessorsimilar/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp\n=/parenleftig/summationdisplay\nT∈Th/ba∇dbl(I−Ih)(mhψh)/ba∇dblp\nLp(T)3/parenrightig1/p\n.\nFor each element, the approximation properties of Ihshow\n/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp(T)3/lessorsimilarhr+1/ba∇dbl∇r+1(mhψh)/ba∇dblLp(T)3\n/lessorequalslanthr+1/summationdisplay\ni+j=r+1/ba∇dbl∇min{i,r}mh/ba∇dblL∞(T)3/ba∇dbl∇min{j,r}ψh/ba∇dblLp(T)3.\nThus, multiple inverse estimates yield\n/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp(T)3/lessorsimilarh/ba∇dblmh/ba∇dblW1,∞/ba∇dblψh/ba∇dblLp(T)3.\nMoreover, we have\n/ba∇dblmhψh/ba∇dblLp/greaterorequalslant/ba∇dblmψh/ba∇dblLp−/ba∇dbl(m−mh)ψh/ba∇dblLp/greaterorequalslant1\n2/ba∇dblψh/ba∇dblLp\nprovided that /ba∇dblm−mh/ba∇dblL∞/lessorequalslant1\n2, which in view of\n/ba∇dblm−mh/ba∇dblL∞=/ba∇dbl(I−Ih)m/ba∇dblL∞/lessorsimilarh/ba∇dbl∇m/ba∇dblL∞\nis satisfied for h/lessorequalslanthRwith a sufficiently small hR>0 that depends only on R.\nAltogether, this shows\n/ba∇dblΠh(mhψh)/ba∇dblLp/greaterorsimilar/ba∇dblψh/ba∇dblLpHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 17\nforh/lessorequalslanthR. Similarly we estimate\n/ba∇dblΠh((m−mh)ψh)/ba∇dblLp/lessorsimilar/ba∇dblm−mh/ba∇dblL∞/ba∇dblψh/ba∇dblLp/lessorsimilarh/ba∇dbl∇m/ba∇dblL∞/ba∇dblψh/ba∇dblLp.\nAltogether, we obtain\n/ba∇dblΠh(mψh)/ba∇dblLp/greaterorsimilar/ba∇dblΠh(mhψh)/ba∇dblLp−/ba∇dblΠh((mh−m)ψh)/ba∇dblLp/greaterorsimilar/ba∇dblψh/ba∇dblLp\nforh/lessorequalslanthR. This concludes the proof for s= 0. Finally, for s=−1 we note that by\nusing the result for s= 0 and an inverse inequality,\n/ba∇dbl(I−Πh)(mψh)/ba∇dblW−1,p/lessorsimilarh/ba∇dblψh/ba∇dblLp\n/lessorsimilarh/ba∇dblΠh(mψh)/ba∇dblLp/lessorsimilar/ba∇dblΠh(mψh)/ba∇dblW−1,p.\nSince/ba∇dblmψh/ba∇dblW−1,p/greaterorsimilar/ba∇dblm/ba∇dbl−1\nW1,∞/ba∇dblψh/ba∇dblW−1,p, this concludes the proof for s∈ {−1,0}.\n(b) It remains to prove the result for s= 1. Note that the result follows from\nduality if we show\n(5.2) /ba∇dblΠh(m·wh)/ba∇dblW−1,q/greaterorsimilar/ba∇dblwh/ba∇dblW−1,q\nfor allwh∈Nh(m). To see this, note that (5.2) implies\n/ba∇dblΠh(mψh)/ba∇dblW1,p/greaterorequalslantsup\nwh∈Nh(m)(ψh,Πh(m·wh))\n/ba∇dblwh/ba∇dblW−1,q\n/greaterorsimilarsup\nwh∈Nh(m)(ψh,Πh(m·wh))\n/ba∇dblΠh(m·wh)/ba∇dblW−1,q= sup\nωh∈Vh(ψh,ωh)\n/ba∇dblωh/ba∇dblW−1,q≃ /ba∇dblψh/ba∇dblW1,p,\nwhereweusedinthesecondtolastequalitythatpart(a)for s= 0alreadyshowsthat\ndim(Nh(m)) = dim(Vh) and since (5.2) implies that the map Nh(m)→Vh,wh/ma√sto→\nΠh(m·wh) is injective, it is already bijective. It remains to prove (5.2). To tha t\nend, we first show for wh=Πh(mωh)∈Nh(m) for someωh∈Vh, using the reverse\ntriangle inequality, that\n/ba∇dblm·wh/ba∇dblW−1,q/greaterorequalslant/ba∇dblωh/ba∇dblW−1,q−/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q\n/greaterorsimilar/ba∇dblm/ba∇dbl−1\nW1,∞/ba∇dblwh/ba∇dblW−1,q−/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q.\nWithmh:=Ih(m)∈V3\nh, the last term satisfies\n/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q/lessorsimilarh/ba∇dblm/ba∇dblW1,∞/ba∇dbl(I−Πh)(mωh)/ba∇dblLq\n/lessorsimilarh/ba∇dblm/ba∇dblW1,∞(/ba∇dblm−mh/ba∇dblL∞/ba∇dblωh/ba∇dblLq+h/ba∇dblmh/ba∇dblW1,∞/ba∇dblωh/ba∇dblLq),\nwhere we used the same arguments as in the proof of part (a) to ge t the estimate\n/ba∇dbl(I−Πh)(mhωh)/ba∇dblLq/lessorsimilarh/ba∇dblmh/ba∇dblW1,∞/ba∇dblωh/ba∇dblLq. The fact /ba∇dblmh/ba∇dblW1,∞/lessorsimilar/ba∇dblm/ba∇dblW1,∞, the\napproximation property /ba∇dblm−mh/ba∇dblL∞/lessorsimilarh/ba∇dblm/ba∇dblW1,∞, and an inverse inequality con-\nclude\n(5.3) /ba∇dblm·wh/ba∇dblW−1,q/greaterorsimilar/ba∇dblwh/ba∇dblW−1,q\nwith (hidden) constants depending only on /ba∇dblm/ba∇dblW1,∞and shape regularity of the\nmesh.18 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nToprove(5.2), itremainstoboundtheleft-handsideaboveby /ba∇dblΠh(m·wh)/ba∇dblW−1,q.\nTo that end, we note\n/ba∇dbl(I−Πh)(m·wh)/ba∇dblW−1,q/lessorsimilarh/ba∇dblwh/ba∇dblLq=hsup\nv∈Lp(wh,v)\n/ba∇dblv/ba∇dblLp\n/lessorsimilarhsup\nv∈Nh(m)(wh,v)\n/ba∇dblv/ba∇dblLp=hsup\nv∈Vh(Πh(m·wh),v)\n/ba∇dblΠh(mv)/ba∇dblLp/lessorsimilarh/ba∇dblΠh(m·wh)/ba∇dblLq,\nwhere we used part (a) for s= 0 for the last inequality. An inverse inequality\nand the combination with (5.3) imply (5.2) for h >0 sufficiently small in terms of\n/ba∇dblm/ba∇dbl−1\nW1,∞. This concludes the proof. /square\nLemma 5.6. Define the matrix M∈RN×N, whereNdenotes the dimension of Vh,\nbyMij:=h−3(Πh(mφj),Πh(mφi)). Under the assumptions of Lemma 5.5, there\nexistsC >0such that for h/lessorequalslanthR,\n/ba∇dblM/ba∇dblp+/ba∇dblM−1/ba∇dblp/lessorequalslantCfor1/lessorequalslantp/lessorequalslant∞,\nwhereCdepends only on the shape regularity.\nProof.Lemma 5.5 shows for x∈RN\n(5.4) Mx·x=h−3/ba∇dblΠh(mN/summationdisplay\ni=1xiφi)/ba∇dbl2\nL2/greaterorsimilarh−3/ba∇dblN/summationdisplay\ni=1xiφi/ba∇dblL2≃ |x|2,\nwhere|·|denotes the Euclidean norm on RN. Letd(i,j) := dist(zi,zj)h−3denote\nthe metric which (approximately) measures the number of elements between the\nsupports of φiandφj, corresponding to the nodes ziandzj, and letBd(z) denote\nthe corresponding ball. In the following, we use a localization propert y of theL2-\nprojection, i.e., there exist a,b>0 such that for all ℓ∈N,\n(5.5) /ba∇dblΠh(mφi)/ba∇dblL2(Ω\\Bℓ(zi))3/lessorequalslantae−bℓ/ba∇dblmφi/ba∇dblL2.\nThe proof of this bound is essentially contained in the proof of [9, Le mma 3.1].\nSince we use the very same arguments below, we briefly recall the st rategy: First,\none observes that the mass matrix /tildewiderM∈RN×Nwith entries /tildewiderMij:=h−3(φj,φi) is\nbandedinthesense that d(i,j)/greaterorsimilar1implies /tildewiderMij= 0, anditsatisfies /tildewiderMx·x/greaterorsimilar|x|2. As\nshown below, this implies that the inverse matrix /tildewiderM−1satisfies|(/tildewiderM−1)ij|/lessorsimilare−bd(i,j)\nfor someb >0 independent of h >0. Note that each entry of the vector field\nΠh(mφi)∈V3\nhcan be represented by/summationtextN\nj=1xk,jφj,k= 1,2,3,and is computed by\nsolving/tildewiderMxk=gk∈RNwithm= (m1,m2,m3)Tandgk,j:= (mkφi,φj). Hence, the\nexponential decay of /tildewiderM−1directly implies (5.5).\nFrom the decay property (5.5), we immediately obtain\n|Mij|/lessorequalslant/tildewideae−/tildewidebd(i,j)\nfor all 1/lessorequalslanti,j/lessorequalslantNand some /tildewidea,/tildewideb>0. This already proves /ba∇dblM/ba∇dblp/lessorequalslantC. We follow\nthe arguments from [28] to show that also M−1decays exponentially. To that end,HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 19\nnote that (5.4) implies the existence of c >0 such that /ba∇dblI−cM/ba∇dbl2=:q <1 and\nhence\n(5.6) M−1=c(I−(I−cM))−1=c∞/summationdisplay\nk=0(I−cM)k.\nClearly,I−cMinherits the decay properties from Mand therefore\n|((I−cM)k+1)ij|/lessorequalslant/tildewideak+1N/summationdisplay\nr1,...,rk=1e−/tildewideb(d(i,r1)+···+d(rk,j))\n/lessorequalslant/tildewideak+1/parenleftig\nmax\ns=1,...,NN/summationdisplay\nr=1e−/tildewidebd(s,r)/2/parenrightigk\ne−/tildewidebd(i,j)/2.\nThe value of max s=1,...,N/summationtextN\nr=1e−/tildewidebd(s,r)/2depends only on the shape regularity of the\ntriangulation and on /tildewideb, but is independent of h(it just depends on the number of\nelements contained in an annulus of thickness ≈h). This implies the existence of\n/tildewidec/greaterorequalslant1 such that\n|((I−cM)k+1)ij|/lessorequalslantmin{qk+1,/tildewideck+1e−/tildewidebd(i,j)/2}.\nThus, for /tildewideck+1/lessorequalslante/tildewidebd(i,j)/4, we have |((I−cM)k+1)ij|/lessorequalslante−/tildewidebd(i,j)/4, whereas for /tildewideck+1>\ne/tildewidebd(i,j)/4, we have |((I−cM)k+1)ij|/lessorequalslantqk+1<q/tildewidebd(i,j)/(4log(/tildewidec)). Altogether, we find some\n/tildewideb>0 (we reuse the symbol), independent of hsuch that\n|((I−cM)k+1)ij|/lessorequalslantq(k+1)/2|((I−cM)k+1)ij|1/2/lessorsimilarq(k+1)/2e−/tildewidebd(i,j).\nPlugging this into (5.6), we obtain\n|(M−1)ij|/lessorsimilar∞/summationdisplay\nk=0q(k+1)/2e−/tildewidebd(i,j)/lessorsimilare−/tildewidebd(i,j).\nThis yields the stated result. /square\nWe are now in a position to prove Lemma 5.3.\nProof of Lemma 5.3. (a) We first consider the case s= 0. In view of (5.1), we write\n(I−Ph(m))vh∈Nh(m) as\n(I−Ph(m))vh=h−3/2N/summationdisplay\ni=1xiΠh(mφi)\nfor some coefficient vector x∈RNand letbi:=h−3/2(vh,mφi) fori= 1,...,N.\nThen, there holds Mx=bwith the matrix Mfrom Lemma 5.6. This lemma and\ntheLp-stability of the L2-orthogonal projection Π h[20] imply that for p∈[1,∞],\n/ba∇dbl(I−Ph(m))vh/ba∇dblLp=/ba∇dblΠhh−3/2N/summationdisplay\ni=1ximφi/ba∇dblLp/lessorsimilar/ba∇dblh−3/2N/summationdisplay\ni=1ximφi/ba∇dblLp\n/lessorsimilarh−3/2/parenleftigN/summationdisplay\ni=1h3|xi|p/parenrightig1/p\n=h3/p−3/2|x|p=h3/p−3/2|M−1b|p/lessorsimilarh3/p−3/2|b|p.20 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nWith|bi|/lessorequalslanth−3/2/ba∇dblvh/ba∇dblLp(supp(φi))3h3(1−1/p)=/ba∇dblvh/ba∇dblLp(supp(φi))3h3/2−3/p, this shows\n/ba∇dblPh(m)vh/ba∇dblLp/lessorsimilar/ba∇dblvh/ba∇dblLp.\n(b) We now turn to the cases s=±1. Define the operator\n/tildewideP⊥\nh(m)vh:=Πh(mΠh(m·vh))\nandnotethat /tildewideP⊥\nh(m)vh∈Nh(m)aswellasker /tildewideP⊥\nh(m) =Th(m)(duetoLemma5.5).\nHowever, /tildewideP⊥\nh(m) is no projection. We observe for vh=Πh(mψh)∈Nh(m) that\n/ba∇dbl(I−/tildewideP⊥\nh(m))vh/ba∇dblW−1,p=/ba∇dblΠhmψh−Πh(mΠh(m·Πh(mψh)))/ba∇dblW−1,p\n/lessorsimilar/ba∇dblm/ba∇dblW1,∞/ba∇dblψh−m·Πh(mψh)/ba∇dblW−1,p\n=/ba∇dblm/ba∇dbl2\nW1,∞/ba∇dbl(I−Πh)(mψh)/ba∇dblW−1,p\n/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞h/ba∇dblψh/ba∇dblLp.\nWith Lemma 5.5 we conclude\n/ba∇dbl(I−/tildewideP⊥\nh(m))vh/ba∇dblW−1,p/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞h/ba∇dblvh/ba∇dblLp.\nSince/tildewideP⊥\nh(m)Ph(m) = 0 by definition of Th(m), we obtain with part (a) and an\ninverse inequality that for all vh∈V3\nh,\n/ba∇dbl(I−Ph(m)−/tildewideP⊥\nh(m))vh/ba∇dblW−1,p=/ba∇dbl(I−/tildewideP⊥\nh(m))(I−Ph(m))vh/ba∇dblW−1,p\n/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞h/ba∇dbl(I−Ph(m))vh/ba∇dblLp\n/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞h/ba∇dblvh/ba∇dblLp\n/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞/ba∇dblvh/ba∇dblW−1,p.\nTheW−1,p(Ω)-stability of Πhimplies/ba∇dbl/tildewideP⊥\nh(m)vh/ba∇dblW−1,p/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞/ba∇dblvh/ba∇dblW−1,pand\nthe triangle inequality concludes the proof for s=−1. The case s= 1 follows by\nduality. /square\nProof of Lemma 5.2. (a) (s= 0) The projection vh:=Ph(m)vis given by the\nequation\n(vh,ϕh) = (v,ϕh)∀ϕh∈Th(m),\nwhich in view of the definition of Th(m) is equivalent to the solution of the saddle\npoint problem (with the Lagrange multiplier λh∈Vh)\n(vh,wh)+(m·wh,λh) = (v,wh)∀wh∈V3\nh,\n(m·vh,µh) = 0 ∀µh∈Vh.\nBy the first equation, we also obtain the identity Πh(mλh) = (I−Ph(m))vh, which\nwill be used below. Furthermore, /tildewidevh:=Ph(/tildewiderm)vis given by the same system with\n/tildewidermin place ofm, yielding a corresponding Lagrange multiplier /tildewideλh. Hence, the\ndifferenceseh:=vh−/tildewidevhandδh:=λh−/tildewideλhsatisfy\n(eh,wh)+(m·wh,δh) =−(wh,(m−/tildewiderm)/tildewideλh)∀wh∈V3\nh,\n(m·eh,µh) = −((m−/tildewiderm)·/tildewidevh,µh)∀µh∈Vh.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 21\nThe classical results on saddle-point problems (see [13, Proposition 2.1]) require two\ninf-sup conditions to be satisfied. First,\ninf\nqh∈Vhsup\nvh∈V3\nh(m·vh,qh)\n/ba∇dblvh/ba∇dblHs/ba∇dblqh/ba∇dblH−s>0\nholds uniformly in hdue to Lemma 5.5. Second,\ninf\nwh∈Th(m)sup\nvh∈Th(m)(vh,wh)\n/ba∇dblvh/ba∇dblHs/ba∇dblwh/ba∇dblH−s>0\nholdsuniformlyin hduetothestabilityestimatesfromLemma5.3(notingthat vh=\nPh(m)vhandwh=Ph(m)whforvh,wh∈Th(m)). For the above saddle-point\nproblems, these bounds for s= 0 give us an L2bound foreh=Ph(m)v−Ph(/tildewiderm)v:\nFrom [13] we obtain\n/ba∇dbl/tildewidevh/ba∇dblL2+/ba∇dbl/tildewideλh/ba∇dblL2/lessorsimilar/ba∇dblv/ba∇dblL2\nand\n/ba∇dbleh/ba∇dblL2+/ba∇dblδh/ba∇dblL2/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblL2+/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblL2.\nWith the stability from Lemma 5.3 and Lemma 5.5, we also obtain\n/ba∇dbl/tildewidevh/ba∇dblL∞+/ba∇dbl/tildewideλh/ba∇dblL∞/lessorsimilar/ba∇dblPh(/tildewiderm)v/ba∇dblL∞+/ba∇dbl(I−Ph(/tildewiderm))v/ba∇dblL∞/lessorsimilar/ba∇dblv/ba∇dblL∞.\nAltogether, this implies\n/ba∇dbleh/ba∇dblL2+/ba∇dblδh/ba∇dblL2/lessorsimilar/ba∇dblm−/tildewiderm/ba∇dblLp/ba∇dblv/ba∇dblLq\nfor (p,q)∈ {(2,∞),(∞,2)}.\n(b) (s= 1) For the H1(Ω)-estimate, we introduce the Riesz mapping Jhbetween\nVh⊂H1(Ω) and its dual Vh⊂H1(Ω)′, i.e., the isometry defined by\n(vh,Jhψh)H1=/a\\}b∇acketle{tvh,ψh/a\\}b∇acket∇i}ht ∀vh∈Vh, ψh∈Vh.\nByJh:=I⊗Jhwe denote the corresponding vector-valued mapping on V3\nh. We\nconsider the bilinear form on V3\nh×V3\nhdefined by\nah(vh,wh) =/a\\}b∇acketle{tvh,J−1\nhwh/a\\}b∇acket∇i}ht,vh,wh∈V3\nh,\nand reformulate the saddle-point problem for ( vh,λh)∈V3\nh×Vh⊂H1(Ω)3×H1(Ω)′\nas\nah(vh,wh)+/a\\}b∇acketle{tm·J−1\nhwh,λh/a\\}b∇acket∇i}ht=a(v,wh)∀wh∈V3\nh,\n/a\\}b∇acketle{tm·vh,J−1\nhµh/a\\}b∇acket∇i}ht = 0 ∀µh∈Vh.\nAs in the case s= 0 (algebraically it is the same system), we have vh=Ph(m)v\nandΠh(mλh) = (I−Ph(m))v. The system for eh=vh−/tildewidevhandδh=λh−/tildewideλh\nreads\nah(eh,wh)+/a\\}b∇acketle{tm·J−1\nhwh,δh/a\\}b∇acket∇i}ht=−/a\\}b∇acketle{t(m−/tildewiderm)·J−1\nhwh,/tildewideλh/a\\}b∇acket∇i}ht ∀wh∈V3\nh,\n/a\\}b∇acketle{tm·eh,J−1\nhµh/a\\}b∇acket∇i}ht =−/a\\}b∇acketle{t(m−/tildewiderm)·/tildewidevh,J−1\nhµh/a\\}b∇acket∇i}ht ∀µh∈Vh.\nThe above inf-sup bounds for s= 1 ands=−1 are precisely the inf-sup condi-\ntions that need to be satisfied for these generalized saddle-point p roblems (see [15,\nTheorem 2.1]), whose right-hand sides are bounded by\n|ah(v,wh)|/lessorequalslant/ba∇dblv/ba∇dblH1/ba∇dblJ−1\nhwh/ba∇dblH−1≃ /ba∇dblv/ba∇dblH1/ba∇dblwh/ba∇dblH122 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nand\n|/a\\}b∇acketle{t(m−/tildewiderm)·J−1\nhwh,/tildewideλh/a\\}b∇acket∇i}ht|/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblH1/ba∇dblwh/ba∇dblH1,\n|/a\\}b∇acketle{t(m−/tildewiderm)·/tildewidevh,J−1\nhµh/a\\}b∇acket∇i}ht|/lessorequalslant/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblH1/ba∇dblµh/ba∇dblH1.\nAs in the case s= 0, we obtain from Lemma 5.3 and Lemma 5.5 that\n/ba∇dbl/tildewidevh/ba∇dblW1,∞+/ba∇dbl/tildewideλh/ba∇dblW1,∞/lessorsimilar/ba∇dblPh(/tildewiderm)v/ba∇dblW1,∞+/ba∇dbl(I−Ph(/tildewiderm))v/ba∇dblW1,∞\n/lessorsimilar/ba∇dblv/ba∇dblW1,∞.\nHence, we obtain from [15, Theorem 2.1], for ( p,q)∈ {(2,∞),(∞,2)},\n/ba∇dbleh/ba∇dblH1/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblH1+/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblH1\n/lessorsimilar1/summationdisplay\ns′=0/parenleftig\n/ba∇dblm−/tildewiderm/ba∇dblH1/ba∇dbl/tildewideλh/ba∇dblW1−s′,q+/ba∇dblm−/tildewiderm/ba∇dblWs′,p/ba∇dbl/tildewidevh/ba∇dblW1−s′,q/parenrightig\n/lessorsimilar1/summationdisplay\ns′=0/ba∇dblm−/tildewiderm/ba∇dblWs′,p/ba∇dblv/ba∇dblW1−s′,q.\nThis implies the H1(Ω)3estimate and hence concludes the proof. /square\nProof of Lemma 5.1. SincePh(m)vis the Galerkin approximation of the saddle\npoint problem for P(m)v(as in the previous proof), the C´ ea lemma for saddle-\npoint problems (see [13, Theorem 2.1]) shows in L2\n/ba∇dbl(Ph(m)−P(m))v/ba∇dblL2\n/lessorsimilarinf\n(wh,µh)∈V3\nh×Vh/parenleftig\n/ba∇dblP(m)v−wh/ba∇dblL2+/ba∇dblm·v−µh/ba∇dblL2/parenrightig\n/lessorsimilarhr+1/ba∇dblm/ba∇dblWr+1,∞/ba∇dblv/ba∇dblHr+1\nand similarly in H1, using [15, Theorem 2.1],\n/ba∇dbl(Ph(m)−P(m))v/ba∇dblH1\n/lessorsimilarinf\n(wh,µh)∈V3\nh×Vh/parenleftig\n/ba∇dblP(m)v−wh/ba∇dblH1+/ba∇dblm·v−µh/ba∇dblH1/parenrightig\n/lessorsimilarhr/ba∇dblm/ba∇dblWr+1,∞/ba∇dblv/ba∇dblHr+1.\nThis concludes the proof. /square\n6.Consistency error and error equation\nTo study the consistency errors, we find it instructive to separat e the issues of\nconsistency for the time and space discretizations. Therefore, w e first show defect\nestimates for the semidiscretization in time, and then turn to the fu ll discretization.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 23\n6.1.Consistency error of the semi-discretization in time. The order of both\nthe fully implicit k-step BDF method, described by the coefficients δ0,...,δ kand 1,\nandtheexplicit k-step BDFmethod, thatis themethoddescribed by thecoefficients\nδ0,...,δ kandγ0,...,γ k−1,isk,i.e.,\n(6.1)k/summationdisplay\ni=0(k−i)ℓδi=ℓkℓ−1=ℓk−1/summationdisplay\ni=0(k−i−1)ℓ−1γi, ℓ= 0,1,...,k.\nWe first rewrite the linearly implicit k-step BDF method (2.3) in strong form,\n(6.2) α˙mn+/hatwidermn×˙mn=P(/hatwidermn)(∆mn+Hn),\nwith Neumann boundary conditions.\nThe consistency error dnof the linearly implicit k-step BDF method (6.2) for the\nsolutionmis the defect by which the exact solution misses satisfying (6.2), and is\ngiven by\n(6.3) dn=α˙mn\n⋆+/hatwidermn\n⋆×˙mn\n⋆−P(/hatwidermn\n⋆)(∆mn\n⋆+Hn)\nforn=k,...,N, where we use the notation mn\n⋆=m(tn) and\n(6.4)/hatwidermn\n⋆=k−1/summationdisplay\nj=0γjmn−j−1\n⋆/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\n⋆/vextendsingle/vextendsingle/vextendsingle,\n˙mn\n⋆=P(/hatwidermn\n⋆)1\nτk/summationdisplay\nj=0δjmn−j\n⋆∈T(/hatwidermn\n⋆).\nNotethat thedefinition of ˙mn\n⋆contains theprojection P(/hatwidermn\n⋆), while ˙mnwasdefined\nwithout a projection (see the first formula in (2.2)), since ˙mn=P(/hatwidermn)˙mnis\nautomatically satisfied due to the constraint in (2.3).\nThe consistency error is bounded as follows.\nLemma 6.1. If the solution of the LLG equation (1.4)has the regularity\nm∈Ck+1([0,¯t],L2(Ω)3)∩C1([0,¯t],L∞(Ω)3)and∆m+H∈C([0,¯t],L∞(Ω)3),\nthen the consistency error (6.3)is bounded by\n/ba∇dbldn/ba∇dblL2(Ω)3/lessorequalslantCτk\nforn=k,...,N.\nProof.We begin by rewriting the equation for the defect as\n(6.5)dn=α˙mn\n⋆+/hatwidermn\n⋆×˙mn\n⋆−P(mn\n⋆)(∆mn\n⋆+Hn)\n−/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn).\nIn view of (1.4), we have\nP(mn\n⋆)(∆mn\n⋆+Hn) =α∂tm(tn)+mn\n⋆×∂tm(tn),\nand can rewrite (6.5) as\ndn=α/parenleftbig\n˙mn\n⋆−∂tm(tn)/parenrightbig\n+/parenleftbig\n/hatwidermn\n⋆×˙mn\n⋆−mn\n⋆×∂tm(tn)/parenrightbig\n−/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn),24 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\ni.e.,\ndn=α/parenleftbig˙mn\n⋆−∂tm(tn)/parenrightbig\n+(/hatwidermn\n⋆−mn\n⋆)×˙mn\n⋆+mn\n⋆×/parenleftbig˙mn\n⋆−∂tm(tn)/parenrightbig\n−/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn).\nTherefore,\n(6.6)dn=α˙dn+/hatwidedn×˙mn\n⋆+mn\n⋆×˙dn−/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn),\nwith\n(6.7) ˙dn:=˙mn\n⋆−∂tm(tn),/hatwidedn:=/hatwidermn\n⋆−mn\n⋆.\nNow, in view of the first estimate in Lemma 4.1, we have\n/ba∇dbl/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn)/ba∇dblL2/lessorequalslantC/ba∇dbl/hatwidermn\n⋆−mn\n⋆/ba∇dblL2,\ni.e.,\n(6.8) /ba∇dbl/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn)/ba∇dblL2/lessorequalslantC/ba∇dbl/hatwidedn/ba∇dblL2.\nTherefore, it suffices to estimate ˙dnand/hatwidedn.\nTo estimate /hatwidedn, we shall proceed in two steps. First we shall estimate the extrap-\nolation error\n(6.9)k−1/summationdisplay\nj=0γjmn−j−1\n⋆−mn\n⋆\nand then /hatwidedn.\nBy Taylor expanding about tn−k,the leading terms of order up to k−1 cancel,\ndue to the second equality in (6.1), and we obtain\n(6.10)k−1/summationdisplay\ni=0γimn−i−1\n⋆−mn\n⋆=1\n(k−1)!/bracketleftiggk−1/summationdisplay\nj=0γj/integraldisplaytn−j−1\ntn−k(tn−j−1−s)k−1m(k)(s)ds\n−/integraldisplaytn\ntn−k(tn−s)k−1m(k)(s)ds/bracketrightigg\n,\nwithm(ℓ):=∂ℓm\n∂tℓ,whence\n(6.11)/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\ni=0γimn−i−1\n⋆−mn\n⋆/vextenddouble/vextenddouble/vextenddouble\nL2/lessorequalslantCτk.\nNow, for a normalized vector aand a non-zero vector b,we have\na−b\n|b|= (a−b)+1\n|b|(|b|−|a|)b,\nwhence/vextendsingle/vextendsinglea−b\n|b|/vextendsingle/vextendsingle/lessorequalslant2|a−b|.\nTherefore, (6.11) yields\n(6.12) /ba∇dbl/hatwidedn/ba∇dblL2/lessorequalslantCτk.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 25\nTo bound ˙dn,we use the fact that P(m(tn))∂tm(tn) =∂tm(tn)∈T(m(tn)), so\nthat we have\n˙dn=P(/hatwidermn\n⋆)1\nτk/summationdisplay\nj=0δjm(tn−j)−∂tm(tn)\n=P(/hatwidermn\n⋆)/parenleftig1\nτk/summationdisplay\nj=0δjm(tn−j)−∂tm(tn)/parenrightig\n+/parenleftbig\nP(/hatwidermn\n⋆)−P(m(tn))/parenrightbig\n∂tm(tn).\nBy Lemma 4.1 and (6.12), we have for the last term\n/ba∇dbl/parenleftbig\nP(/hatwidermn\n⋆)−P(m(tn))/parenrightbig\n∂tm(tn)/ba∇dblL2/lessorequalslantCτk.\nBy Taylor expanding the first term about tn−k,we see that, due to the order condi-\ntions of the implicit BDF method, i.e., the first equality in (6.1), the leadin g terms\nof order up to k−1 cancel, and we obtain\n(6.13)1\nτk/summationdisplay\nj=0δjm(tn−j)−∂tm(tn) =1\nk!/bracketleftigg\n1\nτk/summationdisplay\nj=0δj/integraldisplaytn−j\ntn−k(tn−j−s)km(k+1)(s)ds\n−k/integraldisplaytn\ntn−k(tn−s)k−1m(k+1)(s)ds/bracketrightigg\n,\nwhence\n(6.14) /ba∇dbl˙dn/ba∇dblL2/lessorequalslantCτk,\nprovided the solution mis sufficiently regular. Now, (6.6), (6.8), (6.14), and (6.12)\nyield\n(6.15) /ba∇dbldn/ba∇dblL2/lessorequalslantCτk.\nThis isthe desired consistency estimate, which isvalidfor BDFmethod s of arbitrary\norderk. /square\n6.2.Consistency error of the full discretization. Wedefine theRitzprojection\nRh:H1(Ω)→Vhcorresponding to the Poisson–Neumann problem via/parenleftbig\n∇Rhϕ,∇ψ/parenrightbig\n+/parenleftbig\nRhϕ,1/parenrightbig/parenleftbig\nψ,1/parenrightbig\n=/parenleftbig\n∇ϕ,∇ψ/parenrightbig\n+/parenleftbig\nϕ,1/parenrightbig/parenleftbig\nψ,1/parenrightbig\nfor allψ∈Vh, and we denote Rh=I⊗Rh:H1(Ω)3→V3\nh. We denote again\ntheL2-orthogonal projections onto the finite element space by Πh:L2(Ω)→Vh\nandΠh=I⊗Πh:L2(Ω)3→V3\nh. As in the previous section, we write Ph(m) for\ntheL2-orthogonal projection onto the discrete tangent space at m. We insert the\nfollowing quantities, which are related to the exact solution,\nmn\n⋆,h=Rhm(tn),\n/hatwidermn\n⋆,h=k−1/summationdisplay\nj=0γjmn−j−1\n⋆,h/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\n⋆,h/vextendsingle/vextendsingle/vextendsingle, (6.16)\n˙mn\n⋆,h=Ph(/hatwidermn\n⋆,h)1\nτk/summationdisplay\nj=0δjmn−j\n⋆,h∈Th(/hatwidermn\n⋆,h),26 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nintothelinearlyimplicit k-stepBDFmethod(2.6)andobtainadefect dn\nh∈Th(/hatwidermn\n⋆,h)\nfrom\n(6.17)α/parenleftbig\n˙mn\n⋆,h,ϕh/parenrightbig\n+/parenleftbig\n/hatwidermn\n⋆,h×˙mn\n⋆,h,ϕh/parenrightbig\n=−/parenleftbig\n∇mn\n⋆,h,∇ϕh/parenrightbig\n+/parenleftbig\nHn,ϕh/parenrightbig\n+/parenleftbig\ndn\nh,ϕh/parenrightbig\nfor allϕh∈Th(/hatwidermn\n⋆,h). By definition, there holds ( Rhϕ,1) = (ϕ,1) (this can be seen\nby testing with ψ= 1) and hence/parenleftbig\n∇mn\n⋆,h,∇ϕ/parenrightbig\n=/parenleftbig\n∇m(tn),∇ϕ/parenrightbig\n=−/parenleftbig\n∆m(tn),ϕ/parenrightbig\n.\nThus, we obtain the consistency error for the full discretization b y\n(6.18)dn\nh=Ph(/hatwidermn\n⋆,h)Dn\nhwithDn\nh=α˙mn\n⋆,h+/hatwidermn\n⋆,h×˙mn\n⋆,h−∆m(tn)−H(tn)\nforn=k,...,N. The consistency error is bounded as follows.\nLemma 6.2. If the solution of the LLG equation (1.4)has the regularity\nm∈Ck+1([0,¯t],L2(Ω)3)∩C1([0,¯t],Wr+1,∞(Ω)3)and\n∆m+H∈C([0,¯t],Wr+1,∞(Ω)3),\nthen the consistency error (6.18)is bounded by\n/ba∇dbldn\nh/ba∇dblL2(Ω)3/lessorequalslantC(τk+hr)\nfornwithkτ/lessorequalslantnτ/lessorequalslant¯t.\nProof.We begin by defining\nDn:=α∂tm(tn)+m(tn)×∂tm(tn)−∆m(tn)−H(tn)\nand note that P(mn\n⋆)Dn= 0. Here we denote again mn\n⋆=m(tn) and in the\nfollowing we use also the notations ˙mn\n⋆and/hatwidermn\n⋆as defined in (6.4). With this, we\nrewrite the equation for the defect as\ndn\nh=Ph(/hatwidermn\n⋆,h)Dn\nh−P(mn\n⋆)Dn\n=Ph(/hatwidermn\n⋆,h)/parenleftbig\nDn\nh−Dn/parenrightbig\n+/parenleftbig\nPh(/hatwidermn\n⋆,h)−Ph(/hatwidermn\n⋆)/parenrightbig\nDn\n+/parenleftbig\nPh(/hatwidermn\n⋆)−P(/hatwidermn\n⋆)/parenrightbig\nDn+/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\nDn\n≡I+II+III+IV.\nFor the term IVwe have by Lemma 4.1\n/ba∇dblIV/ba∇dblL2/lessorequalslant2/ba∇dbl/hatwidermn\n⋆−mn\n⋆/ba∇dblL2/ba∇dblDn/ba∇dblL∞,\nwhere the last term /hatwidermn\n⋆−mn\n⋆has been bounded in the L2norm byCτkin the proof\nof Lemma 6.1.\nThe term IIIis estimated using the first bound from Lemma 5.1, under our\nregularity assumptions, as\n/ba∇dblIII/ba∇dblL2/lessorequalslantChr.\nFor the bound on IIwe use Lemma 5.2 ( i) (withp= 2 andq=∞), to obtain\n/ba∇dblII/ba∇dblL2/lessorequalslantCR/ba∇dbl/hatwidermn\n⋆,h−/hatwidermn\n⋆/ba∇dblL2/ba∇dblDn/ba∇dblL∞,\nwhere, using (7.11), we obtain\n/ba∇dbl/hatwidermn\n⋆,h−/hatwidermn\n⋆/ba∇dblL2/lessorequalslant2/ba∇dbl/summationtextk\ni=1γi(Rh−I)mn−i\n∗/ba∇dblL2\nmin/vextendsingle/vextendsingle/summationtextk\ni=1γimn−i\n∗/vextendsingle/vextendsingle/lessorequalslantChr.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 27\nThe denominator is bounded from below by 1 −Cτk, because |mn\n∗|= 1 and\n|/summationtextk\ni=1γimn−i\n∗−mn\n∗|/lessorequalslantCτk. For the first term we have\n/ba∇dblI/ba∇dblL2/lessorequalslant/ba∇dblDn−Dn\nh/ba∇dblL2\n/lessorequalslantα/ba∇dbl∂tm(tn)−˙mn\n⋆,h/ba∇dblL2+/ba∇dblm(tn)×∂tm(tn)−/hatwidermn\n⋆,h×˙mn\n⋆,h/ba∇dblL2.\nThe terms /ba∇dbl∂tm(tn)−˙mn\n⋆/ba∇dblL2and/ba∇dblmn\n⋆×∂tm(tn)−/hatwidermn\n⋆×˙mn\n⋆/ba∇dblL2can be handled\nas in the proof of Lemma 6.1. Standard error estimates for the Ritz projectionRh\n(we do not exploit the Aubin–Nitsche duality here) imply\n/ba∇dbl(I−Rh)˙mn\n⋆/ba∇dblL2/lessorequalslantchr/ba∇dbl˙mn\n⋆/ba∇dblHr+1.\nTogether this yields, under the stated regularity assumption,\n/ba∇dblI/ba∇dblL2/lessorequalslantC(τk+hr),\nand the result follows. /square\n6.3.Error equation. We recall, from (2.6), the fully discrete problem with the\nlinearly implicit BDF method: find ˙mn\nh∈Th(/hatwidermn\nh) such that for all ϕh∈Th(/hatwidermn\nh),\n(6.19) α(˙mn\nh,ϕh)+(/hatwidermn\nh×˙mn\nh,ϕh)+(∇mn\nh,∇ϕh) = (H(tn),ϕh).\nThen, similarly as we have done in Section 4, we first rewrite (6.17): fo r all\nϕh∈Th(/hatwidermn\nh),\n(6.20) α(˙mn\n⋆,h,ϕh)+(/hatwidermn\n⋆,h×˙mn\n⋆,h,ϕh)+(∇mn\n⋆,h,∇ϕh) = (rn\nh,ϕh)\nwith\n(6.21) rn\nh=−(Ph(/hatwidermn\nh)−Ph(/hatwidermn\n⋆,h))(∆m⋆(tn)+H(tn))+dn\nh.\nThe erroren\nh=mn\nh−mn\n⋆,hsatisfies the error equation that is obtained by sub-\ntracting (6.20) from (6.19). We use the notations\n/hatwideen\nh=/hatwidermn\nh−/hatwidermn\n⋆,h, (6.22)\n˙en\nh=˙mn\nh−˙mn\n⋆,h=1\nτk/summationdisplay\nj=0δjen−j\nh+sn\nh, (6.23)\nwithsn\nh= (I−Ph(/hatwidermn\n⋆,h))1\nτk/summationdisplay\nj=0δjmn−j\n⋆,h.\nWe have the following bound for sn\nh.\nLemma 6.3. Under the regularity assumptions of Lemma 6.2, we have\n(6.24) /ba∇dblsn\nh/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr).\nProof.We use Lemmas 5.1 and 5.3, and the bounds in the proof of Lemma 6.2.\nWe start by subtracting ( I−P(/hatwidermn\n⋆,h))∂tmn\n⋆= 0, and obtain (with ∂τmn\n⋆,h:=\n1\nτ/summationtextk\nj=0δjmn−j\n⋆,h)\nsn\nh= (I−Ph(/hatwidermn\n⋆,h))∂τmn\n⋆,h−(I−P(/hatwidermn\n⋆,h))∂tmn\n⋆\n= (∂τmn\n⋆,h−∂tmn\n⋆)−/parenleftbig\nPh(/hatwidermn\n⋆,h)∂τmn\n⋆,h−P(/hatwidermn\n⋆,h)∂tmn\n⋆/parenrightbig\n.28 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nThe first term above is bounded as O(τk+hr) via the techniques of the consistency\nproofs, Lemma 6.1 and 6.2. For the second term we have\nPh(/hatwidermn\n⋆,h)∂τmn\n⋆,h−P(/hatwidermn\n⋆,h)∂tmn\n⋆\n=Ph(/hatwidermn\n⋆,h)(∂τmn\n⋆,h−∂tmn\n⋆)+/parenleftbig\nPh(/hatwidermn\n⋆,h)−P(/hatwidermn\n⋆,h)/parenrightbig\n∂tmn\n⋆,\nwhere the first term is bounded as O(τk+hr), using Lemma 5.3 and the previous\nestimate, while the second term is bounded as O(hr) by theH1estimate from\nLemma 5.1. Altogether, we obtain the stated H1bound forsn\nh. /square\nWe then have the error equation\n(6.25)α(˙en\nh,ϕh)+(/hatwideen\nh×˙mn\n⋆,h,ϕh)+(/hatwidermn\nh×˙en\nh,ϕh)+(∇en\nh,∇ϕh) =−(rn\nh,ϕh),\nfor allϕh∈Th(/hatwidermn\nh), which is to be taken together with (6.21)–(6.23).\n7.Stability of the full discretization for BDF of orders 1 and 2\nFor the A-stable BDF methods (those of orders 1 and 2) we obtain t he follow-\ning stability estimate, which is analogous to the continuous perturba tion result\nLemma 4.2.\nLemma 7.1 (Stability for orders k= 1,2).Consider the linearly implicit k-step\nBDF discretization (2.6)fork/lessorequalslant2with finite elements of polynomial degree r/greaterorequalslant1.\nLetmn\nhandmn\n⋆,h=Rhm(tn)satisfy equations (2.6)and(6.17), respectively, and\nsuppose that the exact solution m(t)is bounded by (4.3)and/ba∇dblH(t)/ba∇dblL∞/lessorequalslantMfor\n0/lessorequalslantt/lessorequalslant¯t. Then, for sufficiently small h/lessorequalslant¯handτ/lessorequalslant¯τ, the erroren\nh=mn\nh−mn\n⋆,h\nsatisfies the following bound, for kτ/lessorequalslantnτ/lessorequalslant¯t,\n(7.1)/ba∇dblen\nh/ba∇dbl2\nH1(Ω)3/lessorequalslantC/parenleftigk−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nH1(Ω)3+τn/summationdisplay\nj=k/ba∇dbldj\nh/ba∇dbl2\nL2(Ω)3+τn/summationdisplay\nj=k/ba∇dblsj\nh/ba∇dbl2\nH1(Ω)3/parenrightig\n,\nwhere the constant Cis independent of h,τandn, but depends on α,R,K,M , and¯t.\nThis estimate holds under the smallness condition that the r ight-hand side is bounded\nbyˆchwith a sufficiently small constant ˆc(note that the right-hand side is of size\nO((τk+hr)2)in the case of a sufficiently regular solution ).\nCombining Lemmas 7.1, 6.2 and 6.3 yields the proof of Theorem 3.1 : These lem-\nmas imply the estimate\n/ba∇dblen\nh/ba∇dblH1(Ω)3/lessorequalslant/tildewideC(τk+hr)\nin the case of a sufficiently regular solution. Since then /ba∇dblRhm(tn)−m(tn)/ba∇dblH1(Ω)3/lessorequalslant\nChrand because of mn\nh−m(tn) =en\nh+(Rhm(tn)−m(tn)), this implies the error\nbound (3.1).\nThe smallness condition imposed in Lemma 7.1 is satisfied under the very mild\nCFL condition, for a sufficiently small ¯ c>0 (independent of h,τandn),\nτk/lessorequalslant¯ch1/2.\nTaken together, this proves Theorem 3.1.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 29\nProof.(a)Preparations. The proof of this lemma transfers the arguments of the\nproof of Lemma 4.2 to the fully discrete situation, using energy estim ates obtained\nby testing with (essentially) the discrete time derivative of the erro r, as presented\nin the Appendix, which is based on Dahlquist’s G-stability theory.\nHowever, testing the error equation (6.25) directly with ˙en\nhis not possible, since\n˙en\nhis not in the tangent space Th(/hatwidermn\nh). Therefore, as in the proof of Lemma 4.2, we\nagain start by showing that the test function ϕh=Ph(/hatwidermn\nh)˙en\nh∈Th(/hatwidermn\nh)∩H1(Ω)3\nis a perturbation of ˙en\nhitself:\nϕh=Ph(/hatwidermn\nh)˙en\nh=Ph(/hatwidermn\nh)˙mn\nh−Ph(/hatwidermn\nh)˙mn\n⋆,h\n=Ph(/hatwidermn\nh)˙mn\nh−Ph(/hatwidermn\n⋆,h)˙mn\n⋆,h+(Ph(/hatwidermn\n⋆,h)−Ph(/hatwidermn\nh))˙mn\n⋆,h.\nHere we note that Ph(/hatwidermn\nh)˙mn\nh=˙mn\nh∈Th(/hatwidermn\nh) by construction of the method(2.6),\nandPh(/hatwidermn\n⋆,h)˙mn\n⋆,h=˙mn\n⋆,h∈Th(/hatwidermn\n⋆,h) by the definition of ˙mn\n⋆,hin (6.4). So we have\nϕh=˙mn\nh−˙mn\n⋆,h−(Ph(/hatwidermn\nh)−P(/hatwidermn\n⋆,h))˙mn\n⋆,h,\nand hence\n(7.2)ϕh=Ph(/hatwidermn\nh)˙en=˙en\nh+qn\nhwithqn\nh=−(Ph(/hatwidermn\nh)−P(/hatwidermn\n⋆,h))˙mn\n⋆,h.\nTheproofnowtransferstheproofofthecontinuousperturbat ionresultLemma4.2\nto the discrete situation with some notable differences, which are em phasized here:\n(i) Instead of using the continuous quantities it uses their spatially d iscrete coun-\nterparts, in particular the discrete projections Ph(/hatwidermn\nh) andPh(/hatwidermn\n⋆,h), defined and\nstudied in Section 5. In view of the definition (2.1) and (6.16) of /hatwidermn\nhand/hatwidermn\n⋆,h, re-\nspectively, thisrequiresthat/summationtextk−1\nj=0γjmn−j−1\nh(x)and/summationtextk−1\nj=0γjmn−j−1\n⋆,h(x)arebounded\naway from zero uniformly for all x∈Ω.\n(ii) Instead of Lemma 4.1 we use Lemma 5.2 (with /hatwidermn\nhand/hatwidermn\n⋆,hin the role of /tildewiderm\nandm, respectively) to bound the quantity qn\nh. This requires that /hatwidermn\n⋆,hand˙mn\n⋆,h\nare bounded in W1,∞independently of h.\nAd(i): In order to show that |/summationtextk−1\nj=0γjmn−j−1\nh(x)|stays close to 1 for all x∈Ω,\nwe need to establish an L∞bound for the errors en−j−1\nh=mn−j−1\nh−mn−j−1\n⋆,h.\nWe use an induction argument and assume that for some time step nu mber ¯n\nwith ¯nτ/lessorequalslant¯twe have\n(7.3) /ba∇dblen\nh/ba∇dblL∞/lessorequalslantρ,for 0/lessorequalslantn<¯n,\nwhere we choose ρsufficiently small independent of handτ. (In this proof it suffices\nto chooseρ/lessorequalslant1/(4Cγ), whereCγ=/summationtextk−1\nj=0|γj|= 2k−1.)\nNote that the smallness condition of the lemma implies that (7.3) is satis fied\nfor ¯n=k, because for the L∞errors of the starting values we have by an inverse\ninequality, for i= 0,...,k−1,\n/ba∇dblei\nh/ba∇dblL∞/lessorequalslantCh−1/2/ba∇dblei\nh/ba∇dblH1/lessorequalslantCh−1/2(ˆch)1/2=Cˆc1/2/lessorequalslantρ,\nprovided that ˆ cis sufficiently small (independent of τandh), as is assumed.\nWe will show in part (b) of the proof that with the induction hypothes is (7.3) we\nobtain also /ba∇dble¯n\nh/ba∇dblL∞/lessorequalslantρso that finally we obtain (7.3) for all¯nwith ¯nτ/lessorequalslant¯t.30 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nUsing reverse and ordinary triangle inequalities, the error bound of [12, Corol-\nlary 8.1.12] (noting that m(t)∈W2,∞(Ω) under our assumptions) and the L∞\nboundedness of ∂tm, and the bound (7.3), we estimate\n(7.4)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle−1/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle−|mn\n⋆|/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞/lessorequalslant/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γjmn−j−1\nh−mn\n⋆/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞\n/lessorequalslant/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γjen−j−1\nh/vextenddouble/vextenddouble/vextenddouble\nL∞+/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γj(Rhmn−j−1\n⋆−mn−j−1\n⋆)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞+/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γj(mn−j−1\n⋆−mn\n⋆)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞\n/lessorequalslant/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γjen−j−1\nh/vextenddouble/vextenddouble/vextenddouble\nL∞+Ch+Cτ/lessorequalslantk−1/summationdisplay\nj=0|γj| ·ρ+Ch+Cτ/lessorequalslant1\n2,\nprovided that handτare sufficiently small. The same argument also yields that/vextenddouble/vextenddouble|/summationtextk−1\nj=0γjmn−j−1\n⋆,h|−1/vextenddouble/vextenddouble\nL∞/lessorequalslant1\n2, and so we have\n(7.5)1\n2/lessorequalslant/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh(x)/vextendsingle/vextendsingle/vextendsingle/lessorequalslant3\n2and1\n2/lessorequalslant/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\n⋆,h(x)/vextendsingle/vextendsingle/vextendsingle/lessorequalslant3\n2\nfor allx∈Ω. In particular, it follows that /hatwidermn\nhand/hatwidermn\n⋆,hare unambiguously defined.\nAd(ii): The required W1,∞bound formn\n⋆,h=Rhm(tn) follows from the W1,∞-\nstability of the Ritz projection: by [12, Theorem 8.1.11] and by the as sumedW1,∞\nbound (4.3) for m(t),\n(7.6) /ba∇dblmn\n⋆,h/ba∇dblW1,∞/lessorequalslantC/ba∇dblm(tn)/ba∇dblW1,∞/lessorequalslantCR.\nThe bounds (7.5) and (7.6) for n/lessorequalslant¯nimply that also\n(7.7) /ba∇dbl/hatwidermn\n⋆,h/ba∇dblW1,∞/lessorequalslantCR\nforn/lessorequalslant¯n(with a different constant C). Using this bound in Lemma 5.3 and the\nassumedW1,∞bound (4.3) for ∂tm(t), we obtain with δ(ζ)/(1−ζ) =/summationtextk\nℓ=1(1−\nζ)ℓ−1/ℓ=:/summationtextk−1\nj=0µjζjthat\n/ba∇dbl˙mn\n⋆,h/ba∇dblW1,∞=/ba∇dblPh(/hatwidermn\n⋆,h)1\nτk/summationdisplay\nj=0δjmn−j\n⋆/ba∇dblW1,∞\n=/ba∇dblPh(/hatwidermn\n⋆,h)k−1/summationdisplay\nj=0µj1\nτ(mn−j\n⋆−mn−j−1\n⋆)/ba∇dblW1,∞\n=/ba∇dblPh(/hatwidermn\n⋆,h)k−1/summationdisplay\nj=0µj1\nτ/integraldisplaytn−j\ntn−j−1∂tm(t)dt/ba∇dblW1,∞\n/lessorequalslantCR/ba∇dblk−1/summationdisplay\nj=0µj1\nτ/integraldisplaytn−j\ntn−j−1∂tm(t)dt/ba∇dblW1,∞HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 31\n/lessorequalslantCRk−1/summationdisplay\nj=0|µj|R.\nWe can now establish a bound for qn\nhas defined in (7.2), using Lemma 5.2 together\nwith the above W1,∞bounds for /hatwidermn\n⋆,hand˙mn\n⋆,hto obtain\n(7.8) /ba∇dblqn\nh/ba∇dblL2/lessorequalslantc/ba∇dbl/hatwideen\nh/ba∇dblL2and/ba∇dbl∇qn\nh/ba∇dblL2/lessorequalslantc/ba∇dbl/hatwideen\nh/ba∇dblH1.\nWith theW1,∞bound of /hatwidermn\n⋆,hwe also obtain a bound of rn\nhdefined in (6.21). Using\nLemma 5.2 ( i) and recalling the L∞bound of∆m+Hof (4.3), we find that rn\nhis\nbounded by\n(7.9)/ba∇dblrn\nh/ba∇dblL2/lessorequalslant/ba∇dbl(Ph(/hatwidermn\nh)−Ph(/hatwidermn\n⋆,h))(∆mn\n⋆+Hn)/ba∇dblL2+/ba∇dbldn\nh/ba∇dblL2\n/lessorequalslantc/ba∇dbl/hatwideen\nh/ba∇dblL2+/ba∇dbldn\nh/ba∇dblL2.\n(b)Energy estimates. Forn/lessorequalslant¯nwith ¯nof (7.3), we test the error equation (6.25)\nwithϕh=˙en\nh+qn\nhand obtain\nα(˙en\nh,˙en\nh+qn\nh)+(/hatwideen\nh×˙mn\n⋆,h,˙en\nh+qn\nh)+(/hatwidermn\nh×˙en\nh,˙en\nh+qn\nh)\n+(∇en\nh,∇(˙en\nh+qn\nh)) =−(rn\nh,˙en\nh+qn\nh).\nBy collecting the terms, and using the fact that ( /hatwidermn\nh×˙en\nh,˙en\nh) = 0, we altogether\nobtain\nα/ba∇dbl˙en\nh/ba∇dbl2\nL2+(∇en\nh,∇˙en\nh) =−α(˙en\nh,qn\nh)−(/hatwideen\nh×˙mn\n⋆,h,˙en\nh+qn\nh)\n−(/hatwidermn\nh×˙en\nh,qn\nh)−(∇en\nh,∇qn\nh)−(rn\nh,˙en\nh+qn\nh).\nWe now estimate the term ( ∇en\nh,∇˙en\nh) on the left-hand side from below using\nDahlquist’s Lemma 10.1, so that the ensuing relation (10.2) yields\n(∇en\nh,∇˙en\nh)/greaterorequalslant1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n+(∇en\nh,∇sn\nh),\nwhereEn\nh= (en−k+1\nh,...,en\nh) and theG-weighted semi-norm is given by\n/ba∇dbl∇En\nh/ba∇dbl2\nG=k/summationdisplay\ni,j=1gij(∇en−k+i\nh,∇en−k+j\nh).\nThis semi-norm satisfies the relation\n(7.10) γ−k/summationdisplay\nj=1/ba∇dbl∇en−k+j\nh/ba∇dbl2\nL2/lessorequalslant/ba∇dbl∇En\nh/ba∇dbl2\nG/lessorequalslantγ+k/summationdisplay\nj=1/ba∇dbl∇en−k+j\nh/ba∇dbl2\nL2,\nwhereγ−andγ+are the smallest and largest eigenvalues of the positive definite\nsymmetric matrix G= (gij) from Lemma 10.1.\nThe remaining terms are estimated using the Cauchy–Schwarz inequ ality and\n/ba∇dbl/hatwidermn\nh/ba∇dblL∞= 1; we altogether obtain\nα/ba∇dbl˙en\nh/ba∇dbl2\nL2+1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n/lessorequalslantα/ba∇dbl˙en\nh/ba∇dblL2/ba∇dblqn\nh/ba∇dblL2+/ba∇dbl/hatwideen\nh/ba∇dblL2(/ba∇dbl˙en\nh/ba∇dblL2+/ba∇dblqn\nh/ba∇dblL2)\n+/ba∇dbl˙en\nh/ba∇dblL2/ba∇dblqn\nh/ba∇dblL2+/ba∇dbl∇en\nh/ba∇dblL2(/ba∇dbl∇qn/ba∇dblL2+/ba∇dbl∇sn\nh/ba∇dblL2)+/ba∇dblrn\nh/ba∇dblL2(/ba∇dbl˙en\nh/ba∇dblL2+/ba∇dblqn\nh/ba∇dblL2).32 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nWe now show an L2error bound for /hatwideen\nhin terms of (en−j−1\nh)k−1\nj=0. Using the fact that\nfora,b∈R3\\{0},\n(7.11)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea\n|a|−b\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(|b|−|a|)a+|a|(a−b)\n|a| |b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant2|a−b|\n|b|,\nand the lower bounds in (7.5) for both |/summationtextk−1\nj=0γjmn−j−1\nh|and|/summationtextk−1\nj=0γjmn−j−1\n⋆,h|, we\ncan estimate\n(7.12)/ba∇dbl/hatwideen\nh/ba∇dblL2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationtextk−1\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle/summationtextk−1\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle−/summationtextk−1\nj=0γjmn−j−1\n⋆,h/vextendsingle/vextendsingle/vextendsingle/summationtextk−1\nj=0γjmn−j−1\n⋆,h/vextendsingle/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/lessorequalslantCk−1/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nL2.\nTo show a similar bound for /ba∇dbl∇/hatwideen\nh/ba∇dblL2we need the following two observations: First,\ntheW1,∞bounds formn−j−1\n⋆,hfrom (7.6) imply W1,∞boundedness for /hatwidermn\n⋆,hby\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j/parenleftbiggb\n|b|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂jb\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleb(∂jb,b)\n|b|3/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nSecond, similarly we have/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j/parenleftbigga\n|a|−b\n|b|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja\n|a|−∂jb\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsinglea(∂ja,a)|b|3−b(∂jb,b)|a|3\n|a|3|b|3/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja\n|a|−∂jb\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+||a|3−|b|3||∂jb|\n|a|3|b|+|a(∂ja,a)−b(∂jb,b)|\n|b|3\n/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja\n|a|−∂jb\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|a−b|(|b|2+|b||a|+|a|2)|∂jb|\n|a|3|b|\n+|a|2|∂ja−∂jb|\n|b|3+|a||∂jb||a−b|\n|b|3+|a−b||∂jb|\n|b|2.\nCombining these two observations, again with mhandm⋆,hin the role of aandb,\nrespectively, and the upper and lower bounds from (7.5) altogethe r yield\n(7.13) /ba∇dbl∇/hatwideen\nh/ba∇dbl2\nL2/lessorequalslantCk−1/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nH1.\nWe estimate further using Young’s inequality and absorptions into th e term\n/ba∇dbl˙en/ba∇dbl2\nL2, together with the bounds in (7.8) and (7.9), to obtain\nα1\n2/ba∇dbl˙en\nh/ba∇dbl2\nL2+1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n/lessorequalslantck/summationdisplay\nj=0/ba∇dblen−j\nh/ba∇dbl2\nH1+c/ba∇dbldn\nh/ba∇dbl2\nL2+c/ba∇dbl∇sn\nh/ba∇dbl2\nL2.\nMultiplying both sides by τ, summing up from kton/lessorequalslant¯n, and using an absorption\nyield\nα1\n2τn/summationdisplay\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2+/ba∇dbl∇En\nh/ba∇dbl2\nG\n/lessorequalslant/ba∇dbl∇Ek−1\nh/ba∇dbl2\nG+cτn/summationdisplay\nj=k/ba∇dblej\nh/ba∇dbl2\nH1+cτn/summationdisplay\nj=k/parenleftbig\n/ba∇dbldj\nh/ba∇dbl2\nL2+/ba∇dblsj\nh/ba∇dbl2\nH1/parenrightbig\n+ck−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nL2.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 33\nWe then arrive, using (7.10), at\n(7.14)α1\n2τn/summationdisplay\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2+/ba∇dbl∇en\nh/ba∇dbl2\nL2/lessorequalslantcτn/summationdisplay\nj=k/ba∇dblej\nh/ba∇dbl2\nH1+cτn/summationdisplay\nj=k/parenleftbig\n/ba∇dbldj\nh/ba∇dbl2\nL2+/ba∇dblsj\nh/ba∇dbl2\nH1/parenrightbig\n+ck−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nL2,\nwithcdepending on α.\nSimilarly as in the time continuous case in the proof of Lemma 4.2, we con nect\n/ba∇dblen\nh/ba∇dbl2\nL2andτ/summationtextn\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2. We rewrite the identity\n1\nτk/summationdisplay\nj=0δjen−j\nh=˙en\nh−sn\nh, n/greaterorequalslantk,\nas\n1\nτn/summationdisplay\nj=kδn−jej\nh=˙ehn−sn\nh−gn\nh, n/greaterorequalslantk,\nwithδℓ= 0 forℓ>kand where\ngn\nh:=1\nτk−1/summationdisplay\ni=0δn−iei\nh\ndepends only on the starting errors and satisfies gn\nh= 0 forn/greaterorequalslant2k. With the inverse\npower series of δ(ζ),\nκ(ζ) =∞/summationdisplay\nn=0κnζn:=1\nδ(ζ),\nwe then have, for n/greaterorequalslantk,\nen\nh=τn/summationdisplay\nj=kκn−j(˙ej\nh−sj\nh−gj\nh).\nBy the zero-stability of the BDF method of order k/lessorequalslant6, the coefficients κnare\nuniformly bounded: |κn|/lessorequalslantcfor alln/greaterorequalslant0. Therefore we obtain via the Cauchy–\nSchwarz inequality\n/ba∇dblen\nh/ba∇dbl2\nL2/lessorequalslant2τ2/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nj=kκn−j(˙ehj−sj\nh)/vextenddouble/vextenddouble/vextenddouble2\nL2+2τ2/vextenddouble/vextenddouble/vextenddouble2k−1/summationdisplay\nj=kκn−jgj\nh/vextenddouble/vextenddouble/vextenddouble2\nL2\n/lessorequalslant(2nτ)τc2n/summationdisplay\nj=k/ba∇dbl˙ehj−sj\nh/ba∇dbl2\nL2+2τ2c2k2k−1/summationdisplay\nj=k/ba∇dblgj\nh/ba∇dbl2\nL2\n/lessorequalslantCτn/summationdisplay\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2+Cτn/summationdisplay\nj=k/ba∇dblsj\nh/ba∇dbl2\nL2+Ck/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nL2.34 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nInserting this bound into (7.14) then yields\nα/ba∇dblen\nh/ba∇dbl2\nL2+/ba∇dbl∇en\nh/ba∇dbl2\nL2/lessorequalslantcτn/summationdisplay\nj=k/ba∇dblej\nh/ba∇dbl2\nH1+cτn/summationdisplay\nj=k/parenleftbig\n/ba∇dbldj\nh/ba∇dbl2\nL2+/ba∇dblsj\nh/ba∇dbl2\nH1/parenrightbig\n+ck−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nL2,\nand a discrete Gronwall inequality implies the stated stability result fo rn/lessorequalslant¯n. It\nthen follows from this stability bound, the smallness condition of the le mma and the\ninverse estimate from H1toL∞that (7.3) is satisfied also for ¯ n+1. This completes\nthe induction step for (7.3) and proves the stated error bound. /square\n8.Stability of the full discretization for BDF of orders 3 to 5\nStability for full discretizations using the BDF methods of orders 3 t o 5 can be\nshown under additional conditions on the damping parameter αand the stepsize τ.\nLemma 8.1 (Stability for orders k= 3,4,5).Consider the linearly implicit k-step\nBDF discretization (2.6)for3/lessorequalslantk/lessorequalslant5with finite elements of polynomial degree\nr/greaterorequalslant2. Letmn\nhandmn\n⋆,hsatisfy(2.6)and(6.17), respectively, and suppose that the\nregularity assumptions of Lemma 7.1 hold. Furthermore, ass ume that the damping\nparameterαsatisfies\n(8.1) α>α k:=ηk\n1−ηk\nwith the multiplier ηkof Lemma 10.2, and that τandhsatisfy the mild CFL-type\ncondition, for some ¯c>0,\n(8.2) τ/lessorequalslant¯ch.\nThen, for sufficiently small h/lessorequalslant¯handτ/lessorequalslant¯τ, the erroren\nh=mn\nh−mn\n⋆,hsatisfies\nthe following bound, for kτ/lessorequalslantnτ/lessorequalslant¯t,\n(8.3)/ba∇dblen\nh/ba∇dbl2\nH1(Ω)3/lessorequalslantC/parenleftigk−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nH1(Ω)3+τn/summationdisplay\nj=k/ba∇dbldj\nh/ba∇dbl2\nL2(Ω)3+τn/summationdisplay\nj=k/ba∇dblsj\nh/ba∇dbl2\nH1(Ω)3/parenrightig\n,\nwhere the constant Cis independent of τ,handn, but depends on α,R,K,M , and\nexponentially on ¯c¯t. This estimate holds under the smallness condition that the\nright-hand side is bounded by ˆch3with a constant ˆc(note that the right-hand side is\nof sizeO((τk+hr)2)in the case of a sufficiently regular solution ).\nTogether with the defect bounds of Section 6, this stability lemma pr oves Theo-\nrem 3.2. We remark that the thresholds αk>0 defined here are the same as those\nappearing in Theorem 3.2.\nProof.The proof of this lemma combines the arguments of the proof of Lem ma 7.1\nwith a nonstandard variant of the multiplier technique of Nevanlinna a nd Odeh, as\noutlined in the Appendix. Since the size of the parameter αdetermines which BDF\nmethods satisfy the stability estimate, the dependence on αwill be carefully traced\nall along the proof.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 35\n(a)Preparations. As in the previous proof, we make again the induction hypoth-\nesis (7.3) for some ¯ nwith ¯nτ/lessorequalslant¯t, but this time with ρ=c0hfor some positive\nconstantc0:\n(8.4) /ba∇dblen\nh/ba∇dblL∞/lessorequalslantc0h, n< ¯n.\nBy an inverse inequality, this implies that /ba∇dblen\nh/ba∇dblW1,∞has anh- andτ-independent\nbound, and hence also /ba∇dblmn\nh/ba∇dblW1,∞forn<¯n. Together with (7.5), this implies\n(8.5) /ba∇dbl/hatwidermn\nh/ba∇dblW1,∞/lessorequalslantC\nand further\n(8.6) /ba∇dbl/hatwideen\nh/ba∇dblL∞/lessorequalslantCh.\nAs in the Appendix, we aim to subtract ηktimes the error equation for time\nstepn−1 from the error equation for time step n, and then to test with ϕh=\nPh(/hatwidermn\nh)˙en\nh∈Th(/hatwidermn\nh) (similarly as in the proof of Lemma 7.1). However, this is not\npossible directly due to the different test spaces at different time st eps:\nα(˙en\nh,ϕh)+(/hatwideen\nh×˙mn\n⋆,h,ϕh)\n+(/hatwidermn\nh×˙en\nh,ϕh)+(∇en\nh,∇ϕh) =−(rn\nh,ϕh),(8.7a)\nfor allϕh∈Th(/hatwidermn\nh), and\nα(˙en−1\nh,ψh)+(/hatwideen−1\nh×˙mn−1\n⋆,h,ψh)\n+(/hatwidermn−1\nh×˙en−1\nh,ψh)+(∇en−1\nh,∇ψh) =−(rn−1\nh,ψh),(8.7b)\nfor allψh∈Th(/hatwidermn−1\nh).\nAs in (7.2), we have\n(8.8)ϕh=Ph(/hatwidermn\nh)˙en\nh=˙en\nh+qn\nh,withqn\nh=−(Ph(/hatwidermn\nh)−Ph(/hatwidermn\n⋆,h))˙mn\n⋆,h,\nwhereqn\nhis bounded by (7.8).\nIn turn, the test function ψh=Ph(/hatwidermn−1\nh)˙en\nh∈Th(/hatwidermn−1\nh) is a perturbation of\nϕh=˙en\nh+qn\nh, since using (8.8) we obtain\nψh=Ph(/hatwidermn−1\nh)˙en\nh\n=Ph(/hatwidermn\nh)˙en\nh−(Ph(/hatwidermn\nh)−Ph(/hatwidermn−1\nh))˙en\nh\n=˙en\nh+qn\nh+pn\nhwithpn\nh=−(Ph(/hatwidermn\nh)−Ph(/hatwidermn−1\nh))˙en\nh.\nThe perturbation pn\nhis estimated using the second bound in Lemma 5.2 ( i) with\np=∞,q= 2, and noting (8.5). We obtain\n/ba∇dblpn\nh/ba∇dblL2/lessorequalslant/ba∇dbl(Ph(/hatwidermn\nh)−Ph(/hatwidermn−1\nh))˙en\nh/ba∇dblL2\n/lessorequalslantc/ba∇dbl˙en\nh/ba∇dblL2/ba∇dbl/hatwidermn\nh−/hatwidermn−1\nh/ba∇dblL∞\n/lessorequalslantc/ba∇dbl˙en\nh/ba∇dblL2/parenleftig\n/ba∇dbl/hatwideen\nh/ba∇dblL∞+/ba∇dbl/hatwidermn\n⋆,h−/hatwidermn−1\n⋆,h/ba∇dblL∞+/ba∇dbl/hatwideen−1\nh/ba∇dblL∞/parenrightig\n/lessorequalslantc/ba∇dbl˙en\nh/ba∇dblL2/parenleftig\n/ba∇dbl/hatwideen\nh/ba∇dblL∞+k−1/summationdisplay\nj=0|γj|/integraldisplaytn−j−1\ntn−j−2/ba∇dblRh∂tm(t)/ba∇dblL∞dt+/ba∇dbl/hatwideen−1\nh/ba∇dblL∞/parenrightig\n.36 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nWe have /ba∇dblRh∂tm(t)/ba∇dblL∞/lessorequalslantc/ba∇dbl∂tm(t)/ba∇dblW1,∞by [12, Theorem 8.1.11]. In view of (8.6)\nwe obtain, for τ/lessorequalslant¯Ch,\n(8.9) /ba∇dblpn\nh/ba∇dblL2/lessorequalslantCh/ba∇dbl˙en\nh/ba∇dblL2,\nand by an inverse estimate,\n(8.10) /ba∇dbl∇pn\nh/ba∇dblL2/lessorequalslantC/ba∇dbl˙en\nh/ba∇dblL2.\nWe also recall the bound (7.9) for /ba∇dblrn\nh/ba∇dblL2.\n(b)Energy estimates. By subtracting (8.7a) −ηk(8.7b) with the above choice of\ntest functions, we obtain\n(8.11)α(˙en\nh−ηk˙en−1\nh,˙en\nh+qn\nh)+(/hatwideen\nh×˙mn\n⋆,h−ηk/hatwideen−1\nh×˙mn−1\n⋆,h,˙en\nh+qn\nh)\n+(/hatwidermn\nh×˙en\nh−ηk/hatwidermn−1\nh×˙en−1\nh,˙en\nh+qn\nh)+(∇en\nh−ηk∇en−1\nh,∇(˙en\nh+qn\nh))\n−ηk/bracketleftbig\nα(˙en−1\nh,pn\nh)+(/hatwideen−1\nh×˙mn−1\n⋆,h,pn\nh)\n+(/hatwidermn−1\nh×˙en−1\nh,pn\nh)+(∇en−1\nh,∇pn\nh)/bracketrightbig\n=−(rn\nh−ηkrn−1\nh,˙en\nh+qn\nh)−ηk(rn−1\nh,pn\nh).\nWe estimate the terms of the error equation (8.11) separately and track carefully\nthe dependence on ηkandα.\nThe termα(˙en\nh−ηk˙en−1\nh,˙en\nh) is bounded from below, using Young’s inequality and\nabsorptions, by\nα(˙en\nh−ηk˙en−1\nh,˙en\nh)/greaterorequalslantα/parenleftbig\n1−1\n2ηk/parenrightbig\n/ba∇dbl˙en\nh/ba∇dbl2\nL2−α\n2ηk/ba∇dbl˙en−1\nh/ba∇dbl2\nL2,\nwhile the term ( ∇en\nh−ηk∇en−1\nh,∇˙en\nh) is bounded from below, via the relation (10.2)\nand (6.23), by\n(∇en\nh−ηk∇en−1\nh,∇˙en\nh)/greaterorequalslant1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n+(∇en\nh−ηk∇en−1\nh,∇sn\nh),\nwithEn\nh= (en−k+1\nh,...,en\nh), and where the G-weighted semi-norm is generated by\nthe matrix G= (gij) from Lemma 10.1 for the rational function δ(ζ)/(1−ηkζ).\nThe remaining terms outside the rectangular bracket are estimate d using the\nCauchy–Schwarz and Young inequalities (the latter often with a suffi ciently small\nbut fixedh- andτ-independent weighting factor µ >0) and/ba∇dbl/hatwidermn\nh/ba∇dblL∞= 1 and\northogonality. We obtain, with varying constants c(which depend on αand are\ninversely proportional to µ)\nα(˙en\nh−ηk˙en−1\nh,qn\nh)+(/hatwideen\nh×˙mn\n⋆,h−ηk/hatwideen−1\nh×˙mn−1\n⋆,h,˙en\nh+qn\nh)\n+(/hatwidermn\nh×˙en\nh−ηk/hatwidermn−1\nh×˙en−1\nh,˙en\nh+qn\nh)+(∇en−ηk∇en−1\nh,∇qn\nh)\n/lessorequalslant/parenleftbig\nαµ+µ+1\n2ηk/parenrightbig\n/ba∇dbl˙en\nh/ba∇dbl2\nL2+/parenleftbig\nαµηk+1\n2ηk/parenrightbig\n/ba∇dbl˙en−1\nh/ba∇dbl2\nL2\n+c/parenleftbig\n/ba∇dblqn\nh/ba∇dblL2+/ba∇dbl/hatwideen\nh/ba∇dbl2\nL2+/ba∇dbl/hatwideen−1\nh/ba∇dbl2\nL2/parenrightbig\n+1\n2/parenleftbig\n/ba∇dbl∇en\nh/ba∇dbl2\nL2+η2\nk/ba∇dbl∇en−1\nh/ba∇dbl2\nL2+/ba∇dbl∇qn\nh/ba∇dblL2/parenrightbig\n/lessorequalslant/parenleftbig\nαµ+µ+1\n2ηk/parenrightbig\n/ba∇dbl˙en\nh/ba∇dbl2\nL2+/parenleftbig\nαµηk+1\n2ηk/parenrightbig\n/ba∇dbl˙en−1\nh/ba∇dbl2\nL2+ck/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nH1,\nwhere in the last inequality we used (7.12) and (7.13) to estimate /hatwideen\nh.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 37\nThe terms inside the rectangular bracket are bounded similarly, usin g (8.9) and\n(8.10) and the condition τ/lessorequalslant¯Ch, by\nα(˙en−1\nh,pn\nh)+(/hatwideen−1\nh×˙mn−1\n⋆,h,pn\nh)+(/hatwidermn−1\nh×˙en−1\nh,pn\nh)+(∇en−1\nh,∇pn\nh)\n/lessorequalslantµ/ba∇dbl˙en\nh/ba∇dbl2\nL2+ch/ba∇dbl˙en−1\nh/ba∇dbl2\nL2+c/parenleftbig\n/ba∇dbl/hatwideen−1\nh/ba∇dbl2\nL2+/ba∇dbl∇en−1\nh/ba∇dbl2\nL2/parenrightbig\n/lessorequalslantµ/ba∇dbl˙en\nh/ba∇dbl2\nL2+ck/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nH1.\nHereµis an arbitrarily small positive constant (independent of τandh), andc\ndepends on the choice of µ.\nIn view of (7.9), the terms with the defects rn\nhare bounded by\n−(rn\nh−ηkrn−1\nh,˙en\nh+qn\nh)−ηk(rn−1\nh,pn\nh)\n/lessorequalslantµ/ba∇dbl˙en\nh/ba∇dbl2\nL2+c/parenleftbig\n/ba∇dblrn\nh/ba∇dbl2\nL2+/ba∇dblrn−1\nh/ba∇dbl2\nL2+/ba∇dblqn\nh/ba∇dbl2\nL2/parenrightbig\n/lessorequalslantµ/ba∇dbl˙en\nh/ba∇dbl2\nL2+ck/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nL2+c1/summationdisplay\nj=0/ba∇dbldn−j\nh/ba∇dbl2\nL2.\nCombination of these inequalities yields\n/parenleftig\nα(1−1\n2ηk)−1\n2ηk−µ/parenrightig\n/ba∇dbl˙en\nh/ba∇dbl2\nL2−/parenleftig\nα\n2ηk+1\n2ηk+µαηk/parenrightig\n/ba∇dbl˙en−1\nh/ba∇dbl2\nL2\n+1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n/lessorequalslantck/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nH1+c1/summationdisplay\nj=0/ba∇dbldn−j\nh/ba∇dbl2\nL2+c/ba∇dbl∇sn\nh/ba∇dbl2\nL2.\nUnder condition (8.1) we have\nω:=α(1−ηk)−ηk>0.\nMultiplying both sides by τand summing up from ktonwithn/lessorequalslant¯nyields, for\nsufficiently small µ,\n1\n2ωτn/summationdisplay\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2+/ba∇dbl∇En\nh/ba∇dbl2\nG\n/lessorequalslantcτ/ba∇dbl˙ek−1\nh/ba∇dbl2\nL2+/ba∇dbl∇Ek−1\nh/ba∇dbl2\nG+cτn−1/summationdisplay\nj=0/ba∇dblej\nh/ba∇dbl2\nH1+cτn/summationdisplay\nj=k/ba∇dbldj\nh/ba∇dbl2\nL2+cτn/summationdisplay\nj=k/ba∇dbl∇sj\nh/ba∇dbl2\nL2.\nThe proof is then completed using exactly the same arguments as in t he last part of\ntheproofofLemma7.1,byestablishinganestimatebetween /ba∇dblen\nh/ba∇dbl2\nL2andτ/summationtextn\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2\nand using a discrete Gronwall inequality, and completing the induction step for\n(8.4). /square\n9.Numerical experiments\nTo obtain significant numerical results, we prescribe the exact solu tionmon\ngiven three-dimensional domains Ω:= [0,1]×[0,1]×[0,L] withL∈ {1/100,1/4}.38 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nThe discretizations of these domains will consist of a few layers of ele ments inz-\ndirection (one layer for L= 1/100 and ten layers for L= 1/4) and a later specified\nnumber of elements in xandydirections. This mimics the common case of thin\nfilm alloys as for example in the standard problems of the Micromagnet ic Modeling\nActivityGroupatNISTCenterforTheoreticalandComputational MaterialsScience\n(ctcms.nist.gov ). Moreover, this mesh structure helps to keep the computationa l\nrequirements reasonable and allow us to compute the experiments o n a desktop\nPC. We are aware that these experiments are only of preliminary nat ure and are\njust supposed to confirm the theoretical results. A more thorou gh investigation\nof the numerical properties of the developed method is needed. Th is will require\nus to incorporate preconditioning, parallelization of the computatio ns, as well as\nlower order energy contributions in the effective field (1.3) to be able to compare to\nbenchmark results from computational physics. This, however, is beyond the scope\nof this paper, and will be the topic of a subsequent work.\nWe consider the time interval [0 ,¯t] with¯t= 0.2 and define two different exact\nsolutions. Since within our computational budget either the time disc retization\nerror or the space discretization error dominates, we construct the solutions such\nthat the first oneis harder to approximate inspace, while thesecon d oneis harder to\napproximate in time. Both solutions are constant in z-direction as is often observed\nin thin-film applications.\n9.1.Implementation. The numerical experiments were conducted using the finite\nelement package FEniCS ( www.fenicsproject.org ) on a desktop computer. As al-\nreadydiscussed inSection2.2, thereareseveral ways toimplement thetangent space\nrestriction. We decided to solve a saddle point problem (variant (a) in Section 2.2)\nfor simplicity of implementation. For preconditioning, we used the blac k-box AMG\npreconditioner that comes with FEniCS. Although this might not be th e optimal\nsolution, it keeps the number of necessary iterative solver steps w ithin reasonable\nbounds. Assuming perfect preconditioning, the cost per time-ste p is then propor-\ntional to the number of mesh-elements. We observed this behavior approximately,\nalthough further research beyond the scope of this work is requir ed to give a definite\nconclusion.\n9.2.Exact solutions. We choose the damping parameter α= 0.2 and define\ng(t) := (¯t+0.1)/(¯t+0.1−t) as well as d(x) := (x1−1/2)2+(x2−1/2)2, which is\nthe squared distance of the projection of xto [0,1]×[0,1] and the point (1 /2,1/2).\nFor some constant C= 400 (a choice made to have pronounced effects), define\n(9.1)m(x,t) :=\nCe−g(t)\n1/4−d(x)(x1−1/2)\nCe−g(t)\n1/4−d(x)(x2−1/2)/radicalig\n1−C2e−2g(t)\n1/4−d(x)d(x)\nifd(x)/lessorequalslant1\n4andm(x,t) :=\n0\n0\n1\nelse.\nIt iseasy to check that |m(x,t)|= 1for all ( x,t)∈Ω×[0,¯t]. Moreover, ∂nm(x,t) =\n0 for allx∈∂Ω. We may calculate the time derivative of min a straightforwardHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 39\nfashion, i.e., ∂tm(x,t) = 0 ford(x)>1/4 and\n∂tm(x,t) =\n−g′(t)\n1/4−d(x)Ce−g(t)\n1/4−d(x)(x1−1/2)\n−g′(t)\n1/4−d(x)Ce−g(t)\n1/4−d(x)(x2−1/2)\ng′(t)\n1/4−d(x)C2e−2g(t)\n1/4−d(x)d(x)\nm3(x,t)\nifd(x)/lessorequalslant1\n4.\nHere,m3denotes the third component of mas defined above.\nThe second exact solution is defined via\n(9.2) /tildewiderm(x,t) :=\n−(x3\n1−3x2\n1/2+1/4)sin(3πt/¯t)/radicalbig\n1−(x3\n1−3x2\n1/2+1/4)2\n−(x3\n1−3x2\n1/2+1/4)cos(3πt/¯t)\n.\nDue to the polynomial nature in the first and the third component, a nd the well-\nbehaved square-root, the space approximation error does not d ominate the time\napproximation.\n9.3.The experiments. We now may compute the corresponding forcings Hresp.\n/tildewiderHto obtain the prescribed solutions by inserting into (1.4), i.e.,\nH=α∂tm+m×∂tm−∆m.\n(Note that we may disregard the projection P(m) from (1.4) since we solve in\nthe tangent space anyway.) We compute Hnumerically by first interpolating m\nand∂tmand then computing the derivatives. This introduces an additional e rror\nwhich is not accounted for in the theoretical analysis. However, th e examples below\nconfirm the expected convergence rates and hence conclude tha t this additional\nperturbation is negligible. Figure 9.1 shows slices of the exact solution at different\ntime steps. Figure 9.2 shows the convergence with respect to the t ime step size τ,\nwhile Figure 9.3 shows convergence with respect to the spatial mesh sizeh. All the\nexperiments confirm the expected rates for smooth solutions.\nFinally, we consider an example with nonsmooth initial data and consta nt right-\nhand side. The initial data are given by\n(9.3)m0(x) :=\nx1−1/2\nx2−1/2/radicalbig\n1−d(x)\nifd(x)/lessorequalslant1\n4andm0(x) :=\n0\n0\n1\nelse.\nWith the constant forcing field H:= (0,1,1)Twe compute a numerical approxima-\ntion to the unknown exact solution. Note that we do not expect any smoothness of\nthe solution (even the initial data is not smooth). Figure 9.4 neverth eless shows a\nphysically consistent decay of the energy /ba∇dbl∇m(t)/ba∇dblL2(Ω)3over time as well as a good\nagreement between different orders of approximation. Moreover , the computed ap-\nproximation shows little deviation from unit length as would be expecte d for smooth\nsolutions.40 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nFigure 9.1. Thefirstrowshowstheexactsolution m(x,t)from(9.1)\nforx∈[0,1]×[0,1]× {0}andt∈ {0,0.05,¯t}(from left to right),\nwhereas the second row shows the exact solution /tildewiderm(x,t) from (9.2)\nforx∈[0,1]×[0,1]×{0}andt∈ {0,0.2/6,0.2/3}(from left to right).\nWhile the problems are three-dimensional, the solutions are constan t\ninz-direction and we only show one slice of the solution.PSfrag replacements\n10−810−610−410−2100\n10−310−210−1k= 1\nk= 2\nk= 3\nk= 4\ntimestep τPSfrag replacements\n10−610−510−410−310−210−1100101\n10−310−210−1k= 1\nk= 2\nk= 3\nk= 4\ntimestep τ\nFigure 9.2. The plots show the error between computed solutions\nand exact solution /tildewidermfor a given time stepsize with a spatial poly-\nnomial degree of r= 2 and a spatial mesh size 1 /40 which results\nin≈6·104degrees of freedom per time step in the left plot. In the\nright plot we use a thicker domain D= [0,1]×[0,1]×[0,1/4] with 10\nelements in z-direction. This results in ≈4·105degrees of freedom\nper timestep. We use the k-step methods of order k∈ {1,2,3,4}and\nobserve the expected rates O(τk) indicated by the dashed lines. The\ncoarse levels of the higher order methods are missing because the kth\nstep is already beyond the final time ¯t.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 41\nPSfrag replacements\nr= 1\nr= 2\nr= 3\nr= 4\nmeshsize h10−410−310−210−1100\n10−1\nFigure 9.3. The plot shows convergence in meshsize hwith respect\nto the exact solution mfrom (9.1) on the domain D= [0,1]×[0,1]×\n[0,1/100]withonelayerofelementsin z-direction. Weusedthesecond\norder BDF method with τ= 10−3and spatial polynomial degrees\nr∈ {1,2,3,4}. The mesh sizes range from 1 /2 to 1/32. We observe\nthe expected rates O(hr) indicated by the dashed lines. The finest\nmesh-size for r= 4 does reach the expected error level. This is due to\nthe fact that the time-discretization errors start to dominate in t hat\nregion.\n10.Appendix: Energy estimates for backward difference formula e\nThe stability proofs of this paper rely on energy estimates, that is, on the use\nof positive definite bilinear forms to bound the error ein terms of the defect d.\nThis is, of course, a basic technique for studying the time-continuo us problem and\nalso for backward Euler and Crank–Nicolson time discretizations (se e, e.g., Thom´ ee\n[38]), but energy estimates still appear to be not well known for bac kward difference\nformula (BDF) time discretizations of order up to 5, which are widely u sed for\nsolving stiff ordinary differential equations. To illustrate the basic me chanism, we\nhere just consider the prototypical linear parabolic evolution equa tion in its weak\nformulation, given by two positive definite symmetric bilinear forms ( ·,·) anda(·,·)\non Hilbert spaces HandVwith induced norms |·|and/ba∇dbl·/ba∇dbl, respectively, and with\nVdensely and continuously embedded in H. The problem then is to find u(t)∈V\nsuch that\n(10.1) ( ∂tu,v)+a(u,v) = (f,v)∀v∈V,\nwith initial condition u(0) =u0. Ifu⋆is a function that satisfies the equation up to\na defectd, that is,\n(∂tu⋆,v)+a(u⋆,v) = (f,v)+(d,v)∀v∈V,42 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICHPSfrag replacements\n0.15\n0.14\n0.13\n0.12\n0.11\n0.1\n0.09\n0.08\n0.07\n0.06\n0.050 0.05 0.1 0.15 0.2\ntimer=k= 1\nr=k= 2\nr=k= 3\nr=k= 4r=k= 1\nr=k= 2\nr=k= 3\nr=k= 4PSfrag replacements10−1\n10−2\n10−3\n0 0 .05 0.1 0.15 0.2\ntime\nFigure 9.4. Left plot: Decay of energies /ba∇dbl∇m(t)/ba∇dblL2(Ω)3for the ap-\nproximations to the unknown solution with m0andHgiven in (9.3)\nand one line after (9.3). We plot four approximations of the k-step\nmethod with polynomial degree rforr=k∈ {1,2,3,4}. The spa-\ntial mesh-size is 1 /40 and the size of the timesteps is 10−3(blue) and\n10−2(red). Right plot: Deviation from unit length /ba∇dbl1−|m(t)|2/ba∇dblL∞(Ω)\nplotted over time for step sizes τ= 10−2(blue),τ= 10−3(red), and\nτ= 10−4(green). The solid lines indicate k= 1, whereas the dashed\nlines indicate k= 2. The spatial mesh-size is 1 /40 withr= 1.\nthen the error e=u−u⋆satisfies, in this linear case, an equation of the same form,\n(∂te,v)+a(e,v) = (d,v)∀v∈V,\nwith initial value e0=u0−u⋆\n0. Testing with v=eyields\n1\n2d\ndt|e|2+/ba∇dble/ba∇dbl2= (d,e).\nEstimating the right-hand side by ( d,e)/lessorequalslant/ba∇dbld/ba∇dbl⋆/ba∇dble/ba∇dbl/lessorequalslant1\n2/ba∇dbld/ba∇dbl2\n⋆+1\n2/ba∇dble/ba∇dbl2, with the dual\nnorm/ba∇dbl·/ba∇dbl⋆, and integrating from time 0 to tresults in the error bound\n|e(t)|2/lessorequalslant|e(0)|2+/integraldisplayt\n0/ba∇dbld(s)/ba∇dbl2\n⋆ds.\nOn the other hand, testing with v=∂teyields\n|∂te|2+1\n2d\ndt/ba∇dble/ba∇dbl2= (d,∂te),\nwhich leads similarly to the error bound\n/ba∇dble(t)/ba∇dbl2/lessorequalslant/ba∇dble(0)/ba∇dbl2+/integraldisplayt\n0|d(s)|2ds.\nThis procedure is all-familiar, but it is not obvious how to extend it to tim e dis-\ncretizations beyond the backward Euler and Crank–Nicolson metho ds. The use of\nenergy estimates for BDF methods relies on the following remarkable results.\nLemma 10.1. (Dahlquist [18]; see also [8] and [27, Section V.6]) Letδ(ζ) =δkζk+\n···+δ0andµ(ζ) =µkζk+···+µ0be polynomials of degree at most k(and at leastHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 43\none of them of degree k)that have no common divisor. Let (·,·)be an inner product\nwith associated norm |·|.If\nReδ(ζ)\nµ(ζ)>0for|ζ|<1,\nthen there exists a positive definite symmetric matrix G= (gij)∈Rk×ksuch that\nforv0,...,v kin the real inner product space,\n/parenleftigk/summationdisplay\ni=0δivk−i,k/summationdisplay\nj=0µjvk−j/parenrightig\n/greaterorequalslantk/summationdisplay\ni,j=1gij(vi,vj)−k/summationdisplay\ni,j=1gij(vi−1,vj−1).\nIn combination with the preceding result for the multiplier µ(ζ) = 1−ηkζ,the\nfollowing property of BDF methods up to order 5 becomes important .\nLemma 10.2. (Nevanlinna & Odeh [34]) Fork/lessorequalslant5,there exists 0/lessorequalslantηk<1such\nthat forδ(ζ) =/summationtextk\nℓ=11\nℓ(1−ζ)ℓ,\nReδ(ζ)\n1−ηkζ>0for|ζ|<1.\nThe smallest possible values of ηkare\nη1=η2= 0, η3= 0.0836, η4= 0.2878, η5= 0.8160.\nPrecise expressions for the optimal multipliers for the BDF methods of orders 3,4\nand 5 are given by Akrivis & Katsoprinakis [1].\nAn immediate consequence of Lemma 10.2 and Lemma 10.1 is the relation\n(10.2)/parenleftigk/summationdisplay\ni=0δivk−i,vk−ηkvk−1/parenrightig\n/greaterorequalslantk/summationdisplay\ni,j=1gij(vi,vj)−k/summationdisplay\ni,j=1gij(vi−1,vj−1)\nwith a positive definite symmetric matrix G= (gij)∈Rk×k; it is this inequality that\nplays a crucial role in our energy estimates, and the same inequality f or the inner\nproducta(·,·).\nThe errorequationfortheBDFtimediscretization ofthelinear para bolicproblem\n(10.1) reads\n(˙en,v)+a(en,v) = (dn,v)∀v∈V,where ˙en=1\nτk/summationdisplay\nj=0δjen−j,\nwith starting errors e0,...,ek−1. When we test with v=en−ηken−1, the first term\ncan be estimated from below by (10.2), the second term is bounded f rom below by\n(1−1\n2ηk)/ba∇dblen/ba∇dbl2−1\n2ηk/ba∇dblen−1/ba∇dbl2, and the right-hand term is estimated from above by the\nCauchy-Schwarz inequality. Summing up from ktonthen yields the error bound\n(10.3) |en|2+τn/summationdisplay\nj=k/ba∇dblej/ba∇dbl2/lessorequalslantCk/parenleftigk−1/summationdisplay\ni=0/parenleftbig\n|ei|2+τ/ba∇dblei/ba∇dbl2/parenrightbig\n+τn/summationdisplay\nj=k/ba∇dbldj/ba∇dbl2\n⋆/parenrightig\n,\nwhereCkdepends only on the order kof the method. This kind of estimate for\nthe BDF error has recently been used for a variety of linear and non linear parabolic\nproblems [33, 3, 2, 30].44 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nOn the other hand, when we first subtract ηktimes the error equation for n−1\nfrom the error equation with nand then test with ˙ en, we obtain\n(˙en−ηk˙en−1,˙en)+a(en−ηken−1,˙en) = (dn−ηkdn−1,˙en).\nHere, thesecond termis boundedfrombelow by (10.2)withthe a(·,·)inner product,\nthe first term is bounded from below by (1 −1\n2ηk)|˙en|2−1\n2ηk|˙en−1|2, and the right-\nhand term is estimated from above by the Cauchy–Schwarz inequalit y. Summing\nup fromktonthen yields the error bound\n(10.4) /ba∇dblen/ba∇dbl2+τn/summationdisplay\nj=k|˙ej|2/lessorequalslantCk/parenleftigk−1/summationdisplay\ni=0/ba∇dblei/ba∇dbl2+τn/summationdisplay\nj=k|dj|2/parenrightig\n.\nIt is this type of estimate that we use in the present paper for the n onlinear problem\nconsidered here. It has previously been used in [29].\nAcknowledgment. The work of Michael Feischl, Bal´ azs Kov´ acs and Christian Lu-\nbichissupportedbyDeutscheForschungsgemeinschaft –projec t-id 258734477–SFB\n1173.\nReferences\n1. G. Akrivis and E. Katsoprinakis, Backward difference formulae :new multipliers and stability\nproperties for parabolic equations , Math. Comp. 85(2016) 2195–2216.\n2. G. Akrivis, B. Li, and C. Lubich, Combining maximal regularity and energy estimates for time\ndiscretizations of quasilinear parabolic equations , Math. Comp. 86(2017) 1527–1552.\n3. G. Akrivis and C. Lubich, Fully implicit, linearly implicit and implicit–explicit b ackward dif-\nference formulae for quasi-linear parabolic equations , Numer. Math. 131(2015) 713–735.\n4. F. Alouges, A new finite element scheme for Landau–Lifshitz equations , Disc. Cont. Dyn. Syst.\nSer. S.1(2008) 187–196.\n5. F. Alouges and P. 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Methods Eng. 15(2008) 277–309.\n18. G. Dahlquist, G-stability is equivalent to A-stability , BIT18(1978) 384–401.\n19. G. Di Fratta, C. M. Pfeiler, D. Praetorius, M. Ruggeri, and B. S tiftner,Linear second-order\nIMEX-type integrator for the (eddy current )Landau–Lifshitz–Gilbert equation , IMA J. Numer.\nAnal. (2019) doi:10.1093/imanum/drz046\n20. J. Douglas Jr., T. Dupont, and L. Wahlbin, The stability in Lqof theL2-projection into finite\nelement function spaces , Numer. Math. 23(1974/75) 193–197.\n21. M. Feischl and T. Tran, The eddy current–LLG equations :FEM–BEM coupling and a priori\nerror estimates , SIAM J. Numer. Anal. 55(2017) 1786–1819.\n22. M. Feischl and T. Tran, Existence of regular solutions of the Landau-Lifshitz-Gil bert equation\nin 3D with natural boundary conditions , SIAM J. Math. Anal. 49(2017) 4470–4490.\n23. H. Gao, Optimal error estimates of a linearized backward Euler FEM f or the Landau–Lifshitz\nequation, SIAM J. Numer. Anal. 52(2014) 2574–2593.\n24. I. 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Prohl, Computational Micromagnetism , Teubner, Stuttgart, 2001.\n37. D. Suess, V. Tsiantos, T. Schrefl, J. Fidler, W. Scholz, H. Fors ter, R. Dittrich, and J. J. Miles,\nTime resolved micromagnetics using a preconditioned time i ntegration method , J. Magn. Magn.\nMater.248(2002) 298–311.\n38. V.Thom´ ee, Galerkin Finite Element Methods for Parabolic Problems , 2nded., Springer–Verlag,\nBerlin, 2006.\n39. D. Thonig and J. Henk, Gilbert damping tensor within the breathing Fermi surface m odel:\nanisotropy and non-locality , New J. Phys. 16(2014) 013032.46 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nDepartment of Computer Science & Engineering, University o f Ioannina, 45110\nIoannina, Greece, and Institute of Applied and Computation al Mathematics, FORTH,\n70013 Heraklion, Crete, Greece\nE-mail address :akrivis@cse.uoi.gr\nInstitute for Analysis and Scientific Computing (E 101), Te chnical University\nWien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria\nE-mail address :michael.feischl @kit.edu\nE-mail address :michael.feischl @tuwien.ac.at\nMathematisches Institut, Universit ¨at T¨ubingen, Auf der Morgenstelle, D-72076\nT¨ubingen, Germany\nE-mail address :kovacs@na.uni-tuebingen.de\nMathematisches Institut, Universit ¨at T¨ubingen, Auf der Morgenstelle, D-72076\nT¨ubingen, Germany\nE-mail address :lubich@na.uni-tuebingen.de"    },    {        "title": "1903.08395v2.Nonlinear_magnetization_dynamics_driven_by_strong_terahertz_fields.pdf",        "content": "Nonlinear magnetization dynamics driven by strong terahertz \felds\nMatthias Hudl,1Massimiliano d'Aquino,2Matteo Pancaldi,1See-Hun Yang,3Mahesh G. Samant,3Stuart\nS. P. Parkin,3, 4Hermann A. D urr,5Claudio Serpico,6Matthias C. Ho\u000bmann,7and Stefano Bonetti1, 8,\u0003\n1Department of Physics, Stockholm University, 106 91 Stockholm, Sweden\n2Department of Engineering, University of Naples \\Parthenope\", 80143 Naples, Italy\n3IBM Almaden Research Center, San Jose CA 95120, USA\n4Max-Planck Institut f ur Mikrostrukturphysik, 06120 Halle, Germany\n5Department of Physics and Astronomy, Uppsala University, 751 20 Uppsala, Sweden\n6DIETI, University of Naples Federico II, 80125 Naples, Italy\n7SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA\n8Department of Molecular Sciences and Nanosystems,\nCa' Foscari University of Venice, 30172 Venezia Mestre, Italy\n(Dated: September 26, 2019)\nWe present a comprehensive experimental and numerical study of magnetization dynamics in a\nthin metallic \flm triggered by single-cycle terahertz pulses of \u001820 MV/m electric \feld amplitude\nand\u00181 ps duration. The experimental dynamics is probed using the femtosecond magneto-optical\nKerr e\u000bect, and it is reproduced numerically using macrospin simulations. The magnetization\ndynamics can be decomposed in three distinct processes: a coherent precession of the magnetiza-\ntion around the terahertz magnetic \feld, an ultrafast demagnetization that suddenly changes the\nanisotropy of the \flm, and a uniform precession around the equilibrium e\u000bective \feld that is relaxed\non the nanosecond time scale, consistent with a Gilbert damping process. Macrospin simulations\nquantitatively reproduce the observed dynamics, and allow us to predict that novel nonlinear mag-\nnetization dynamics regimes can be attained with existing table-top terahertz sources.\nSince Faraday's original experiment [1] and until two\ndecades ago, the interaction between magnetism and\nlight has been mostly considered in a unidirectional way,\nin which changes to the magnetic properties of a mate-\nrial cause a modi\fcation in some macroscopic observable\nof the electromagnetic radiation, such as polarization\nstate or intensity. However, the pioneering experiment of\nBeaurepaire et al. [2], where femtosecond optical pulses\nwere shown to quench the magnetization of a thin-\flm\nferromagnet on the sub-picoseconds time scales, demon-\nstrated that intense laser \felds can conversely be used\nto control magnetic properties, and the \feld of ultrafast\nmagnetism was born. Large research e\u000borts are nowadays\ndevoted to the attempt of achieving full and deterministic\ncontrol of magnetism using ultrafast laser pulses [3{10], a\nfundamentally di\u000ecult problem that could greatly a\u000bect\nthe speed and e\u000eciency of data storage [11].\nRecently, it has been shown that not only femtosecond\nlaser pulses, but also intense single-cycle terahertz (THz)\npulses [12] can be used to manipulate the magnetic order\nat ultrafast time scales in di\u000berent classes of materials\n[13{18]. The main peculiarity of this type of radiation,\ncompared with more conventional femtosecond infrared\npulses, is that the interaction with the spins occurs not\nonly through the overall energy deposited by the radia-\ntion in the electronic system, but also through the Zee-\nman torque caused by the magnetic \feld component of\nthe intense THz pulse. This is a more direct and e\u000ecient\nway of controlling the magnetization, and to achieve the\nfastest possible reversal [19, 20]. However, an accurate\ndescription of the magnetization dynamics triggered by\nstrong THz pulses is still missing.In this Letter, we present a combined experimental and\nnumerical study of the magnetization dynamics triggered\nby linearly polarized single-cycle THz pulses with peak\nelectric (magnetic) \felds up to 20 MV/m (67 mT). We\ninvestigate not only the fast time scales that are com-\nparable to the THz pulse duration ( \u00181 ps), but also\nthe nanosecond regime, where ferromagnetic resonance\n(FMR) oscillations are observed. Moreover, we write\nan explicit form of the Landau-Lifshitz-Gilbert (LLG)\nequation [21, 22] suitable to analyze terahertz-driven dy-\nnamics, that we use to predict yet-unexplored nonlin-\near magnetization dynamics regimes uniquely achievable\nwith this type of excitation mechanism.\nExperimental details. Room temperature experi-\nmental data is obtained from a time-resolved pump-\nprobe method utilizing the magneto-optical Kerr e\u000bect\n(MOKE), Refs. [23, 24]. A sketch of the experimental\nsetup is presented in Fig. 1 (a). Strong THz radiation is\ngenerated via optical recti\fcation of 4 mJ, 800 nm, 100\nfs pulses from a 1 kHz regenerative ampli\fer in a lithium\nniobate (LiNbO 3) crystal, utilizing the tilted-pulse-front\nmethod [25]. In contrary to the indirect (thermal) cou-\npling present in visible- and near-infrared light-matter in-\nteraction, THz radiation can directly couple to the spin\nsystem via magnetic dipole interaction (Zeeman inter-\naction) [26]. In this respect, a fundamental aspect is\nthe orientation of the THz polarization, which is con-\ntrolled using a set of two wire grid polarizers, one vari-\nably oriented at \u000645\u000eand a second one \fxed to +90\u000e\n(or -90\u000e) with respect to the original polarization direc-\ntion of ETHz. As depicted in Fig. 1 (b), the magnetic\n\feld component of the THz pulse HTHzis \fxed alongarXiv:1903.08395v2  [cond-mat.mes-hall]  25 Sep 20192\n(a)\n(e) (f)(b)\n+HTHz\n-HExtOUTWPLNO OAPM\nEM+SP P\nMPump\nProbe±45° ±90°\n0 2.5 5.0 7.5 10−100−50050100MOKE signal (a.u.)\n0 250 500 750−100−50050100\n+HExt\n-HExt\n0 2.5 5.0 7.5 10\nTime (ps)−100−50050MOKE signal (a.u.)\n0 250 500 750\nTime (ps)−100−50050\n+HTHz\n-HTHz\n(a)\n(c)\n(e)(d)\n(f)(b)\n+HTHz\n-HExtOUTWPLNO OAPM\nEM+SP P\nMPump\nProbe±45° ±90°\nFIG. 1. (Color online) (a) Schematic drawing of the exper-\nimental setup: BD - Balanced detection using two photodi-\nodes and a lock-in ampli\fer, WP - Wollaston prism, EM + S\n- Electromagnet with out-of-plane \feld and sample, OAPM\n- O\u000b-axis parabolic mirror, P - Wire grid polarizer, LNO -\nLithium niobate, and M - Mirror. (b) Sample geometry: The\nelectric \feld component of the THz pulse at the sample po-\nsition is oriented parallel to the y-axis direction, ETHzky,\nand the magnetic \feld component parallel to the x-axis direc-\ntion,HTHzkx. A static magnetic \feld HExtis applied along\nthez-axis direction. (c-f) Experimental MOKE data showing\nthe in\ruence of reversing the external magnetic \feld ( \u0006HExt,\nHTHz= const.) on the THz-induced demagnetization (c) and\non the FMR oscillations (d). The in\ruence of reversing the\nTHz magnetic \feld pulse ( \u0006HTHz,HExt.= const.) on the\nTHz-induced demagnetization and on the FMR oscillations is\nshown in (e) and (f), respectively. (The shaded green area is\na guide to the eye.)\nthex-axis direction and is therefore \ripped by 180\u000eby\nrotating the \frst polarizer. An amorphous CoFeB sample\n(Al2O3(1.8nm)/Co 40Fe40B20(5nm)/Al 2O3(10nm)/Si\nsubstrate) is placed either in the gap of a \u0006200 mT elec-\ntromagnet or on top of a 0.45 T permanent magnet. In\nboth cases, the orientation of the externally applied \feld\nHExtis along the z-axis direction, i.e. out of plane with\nrespect to the sample surface. However, a small compo-\nnent of this external bias \feld lying in the sample plane\nparallel to the y-axis direction has to be taken into ac-count, due to a systematic (but reproducible) small mis-\nalignment in positioning the sample. The THz pump\nbeam, with a spot size \u001f\u00191 mm (FWHM), and the 800\nnm probe beam, with a spot size \u001f\u0019200\u0016m (FWHM),\noverlap spatially on the sample surface in the center of\nthe electromagnet gap. Being close to normal incidence,\nthe MOKE signal is proportional to the out-of-plane com-\nponent M zof the magnetization, i.e. polar MOKE geom-\netry [27]. The probe beam re\rected from the sample sur-\nface is then analyzed using a Wollaston prism and two\nbalanced photo-diodes, following an all-optical detection\nscheme [4].\nResults and discussion. The experimental data demon-\nstrating THz-induced demagnetization and the magnetic\n\feld response of the spin dynamics is shown in Fig. 1\n(c-f) when \u00160HExt= 185 mT. For short timescales on\nthe order of the THz pump pulse, \u001c\u00181 ps, the polar\nMOKE is sensitive to the coherent response of the mag-\nnetization, i.e. its precession around the THz magnetic\n\feld, as shown in Fig. 1 (c)+(e). Within \u001c\u0018100 fs af-\nter time zero (t 0\u00185 ps) a sudden demagnetization step\nof the order of 0.1-0.2% of the total magnetization vec-\ntor is observed. The demagnetization step is followed by\na 'fast' relaxation process, \u001c\u00181 ps, and subsequently\nby a 'slow' recovery of the magnetization on a longer\ntimescale,\u001c\u0018100 ps. During and after magnetization\nrecovery, a relaxation precession (corresponding to the\nFMR) is superimposed, see Fig. 1 (d)+(f). The e\u000bect of\nreversing the externally applied magnetic \feld HExton\nthe MOKE measurements is illustrated in Fig. 1 (c)+(d).\nThis data shows that all the di\u000berent processes just iden-\nti\fed (demagnetization, coherent magnetization response\nin the range t 0<t.t0+ 2 ps, and FMR response)\nare indeed magnetic, as they all reverse their sign upon\nreversal of the sign of the bias magnetic \feld. Fig. 1\n(e)+(f) instead depict the e\u000bect of reversing the THz \feld\npolarity, while keeping the externally applied \feld con-\nstant. This data illustrate the symmetry of the magnetic\nresponse with respect to the THz magnetic \feld. The\ndemagnetization and FMR response, whose amplitude\nscales quadratically with the THz magnetic \feld HTHz,\nremain unchanged upon reversal of the terahertz mag-\nnetic \feld, while the coherent magnetization response,\nlinear in HTHz, reverses its sign.\nThe dependence of the THz-induced FMR response on\nthe magnitude of the magnetic bias \feld ( HExt) is shown\nin Fig. 2 (a). Oscillation amplitude and resonance fre-\nquency are summarized in Fig. 2 (b). The oscillation\namplitude is obtained from \ftting a damped sinusoidal\nfunction to the data shown in Fig. 2 (a), and the reso-\nnance frequency follows from subsequent Fourier trans-\nformation of that \ftting function. The relationship be-\ntween magnetic resonance \feld Hrand FMR frequency\nfFMR can be described by the phenomenological Kittel3\n0 200 400 600 800\nTime (ps)-1000100200300400500600700800MOKE signal (a.u.)2.0 2.2 2.4 2.6\nFrequency (GHz)0204060FMR amp. (a.u.)\n0102030405060\nMagnetic field (mT)02468Frequency (GHz)(a) (b)\n(c)\nFIG. 2. (Color online) Experimental data showing the mag-\nnetic \feld dependence of the FMR. (a) MOKE signal for\ndi\u000berent out-of-plane \felds \u00160HExt- 20, 50, 75, 100, 125,\n150, 175 and 185 mT in ascending order. The solid line is a\ndamped sine function \ftted to the experimental data. Both\n\ft and data curve are displaced on the y axis by a constant\no\u000bset. (b) Fourier transformation of the data shown in (a)\nvs. FMR amplitude. (c) FMR frequency as a function of the\nin-plane component of HExt. The solid line corresponds to\nKittel equation, Eq. (1).\nequation [28]\nfFMR =\r\u00160\n2\u0019p\nHr(Hr+Me\u000b) (1)\nwith gyromagnetic ratio \rand e\u000bective magnetization\nMe\u000b=MS\u0000H?\nK, where H?\nKis the perpendicular\nanisotropy \feld. From room-temperature magnetization\nmeasurements (see the Supplemental Material), a satu-\nration magnetization \u00160MS= 1.84 T and an anisotropy\n\feld\u00160H?\nK= 0.76 T are obtained. The value of H rrep-\nresents the small in-plane component of HExt(i.e. along\nthey-axis direction), which can be inferred from the mea-\nsurements in Fig. 2 (a) in order to get good agreement\nwith the experimental data, as shown in Fig. 2 (c).\nNumerical modelling. A detailed picture of the THz-\ninduced demagnetization and FMR response for an exter-\nnal magnetic \feld generated by a 0.45 T permanent mag-\nnet and a THz pulse of E THz= 18 MV/m ( \u00160HTHz\u001860\nmT) is presented in Fig. 3. Experimental data has been\nrecorded both on a short timescale after arrival of the\npump pulse showing coherent precession and demagneti-\nzation, Fig. 3 (a), and on long timescales showing mag-\nnetization recovery and FMR at a frequency of about 7.2\nGHz, Fig. 3 (b). By knowing the FMR frequency, the\nactual applied magnetic \feld can be deduced from the\ndata shown in Fig. 2 (a) and Eq. (1), as described in the\nprevious paragraph. Following this approach, we \fnd a\nvalue of 450 mT for the out-of-plane component and 56\nmT for the in-plane component of HExt, corresponding\nto the right-most point in Fig. 2 (c).\nThis complete set of good-quality data has been used\n(b)\n012345678\nTime (ps)53.853.954.054.154.254.354.4Mz/M 0(%)\n0 200 400 600 800\nTime (ps)(a) (b)FIG. 3. (Color online) Comparison of experimental data\n(open markers) and macrospin simulations (solid lines). (a)\nTHz-induced demagnetization for an external applied \feld of\n\u00160HExt\u0018450 mT and a THz pulse of E THz= 18 MV/m,\n(\u00160HTHz\u001860 mT). The blue markers/lines correspond to\n+HTHz, and the orange markers/lines to \u0000HTHz. (b) Exper-\nimental data and macrospin simulations under similar condi-\ntions as (a) showing FMR on a longer timescale.\nfor validating all the assumptions considered in our\nmacrospin simulations, whose results are shown as solid\nlines in Fig. 3. Indeed, magnetization dynamics of a uni-\nform ferromagnetic material can be modeled and under-\nstood from macrospin simulations where the description\nof the magnetic state is simpli\fed by a single-domain ap-\nproximation. The phenomenological LLG equation [22]\ncan be used to obtain a \frst approximate description of\nthe magnetization continuum precession and relaxation.\nSince the LLG equation preserves the length of the mag-\nnetization vector ( jMj= const.) [29], it does not account\nfor laser-induced demagnetization and it ignores relevant\nphysical phenomena such as scattering e\u000bects [11]. A\npossibility to account for optically- or thermally-induced\ndemagnetization is the use of the Landau-Lifshitz-Bloch\n(LLB) equation [30], which has been shown to describe\nultrafast demagnetization processes [31]. However, the\nparameters involved in the LLB equation depend on\ntemperature, and a description of their temporal evo-\nlution due to the interaction with ultrafast pulses is\nneeded. Such a description is often provided using a two-\ntemperature model [8, 31], which then has to be coupled\nto the LLB equation. However, such a two-temperature\nmodel relies on several phenomenological material pa-\nrameters, and little insight on the physics is gained by\nthis approach, in particular at longer time scales. Since\nhere we are interested in these time scales, we simplify\nthe problem by just assuming a phenomenological LLG\nequation with non-constant magnetization magnitude, a\nphysical quantity which can be readily measured.\nOur version of the LLG equation reads (see the Sup-4\nplemental Material)\ndM\ndt=\u0000\r0\u0002\nM\u0002He+\u000b\nMS(M\u0002(M\u0002He))\u0003\n\u0000c(M;He)M;\n(2)\nwith\r0=\r=(1 +\u000b2), where the gyromagnetic ratio is\nj\r=2\u0019j \u001928.025 GHz/T, and M= Mm, where M( t)\nis the magnetization amplitude and m(t) the unit vec-\ntor. The e\u000bective magnetic \feld Heincludes the exter-\nnal \feld HExt, THz \feld HTHzand demagnetization \feld\nHD. The Gilbert damping \u000b= 0:01 was independently\nmeasured with conventional FMR spectroscopy, as shown\nin the Supplemental Material. In Eq. (2), the \frst and\nsecond terms on the right-hand side describe coherent\nspin precession and the macroscopic spin relaxation, re-\nspectively, whereas the third term describes the fast de-\nmagnetization along the mdirection controlled by the\nfunctionc(M;He). A non-constant magnetization mag-\nnitude is also needed for reproducing the observed FMR\noscillation, as discussed in the Supplemental Material.\nHence, we propose a semi-empirical approach to iden-\ntifyc(M;He) and, consequently, the time evolution of\nthe magnetization vector length M( t) from experimental\ndemagnetization data. For this purpose, the change in\nM(t) as a function of time is calculated from the cumu-\nlative integral of the incident THz pulse [16]:\n\u0001M(t)/e\u0000t=\u001cRZt\n\u00001HTHz(\u0018)2d\u0018; (3)\nwhere the exponential term describes the recovery of the\nmagnetization on a timescale of \u0018100 ps for CoFeB. The\nshape of the incident THz pulse (in the time domain)\nis proportional to ETHz(t), which can be measured via\nelectro-optical sampling in GaP [32]. We notice that the\nTHz magnetic \feld HTHz(t) inside the material, to be\nused in Eq. (3), can be derived from the measured free-\nspace ETHz(t) after taking into account the whole sample\nstack, e.g. by means of the transfer matrix method [33].\nEventually, the function c(M;He) is obtained from Eq.\n(3) after re-scaling it with experimentally obtained de-\nmagnetization values (see Ref. [16] for more details).\nAfter determining c(M;He), all the terms in Eq. (2)\nare known, and the equation can be used to compute\nthe dynamics of the magnetization orientation in terms\nof the spherical angles \u0012(t); \u001e(t). Indeed, Eq. (2) is nu-\nmerically solved in spherical coordinates (M ;\u0012;\u001e) (see\nthe Supplemental Material for details) using a custom-\nmade Python solver based on the 4th-order Runge-Kutta\nmethod, which provides a simple implementation and\ngood accuracy of the numerical solution [29].\nIt is now interesting to explore the expected response of\nthe magnetization to THz \felds with strength larger than\nthe ones considered in this work, but nowadays accessi-\nble with table-top sources. For simplicity and generality,\nonly the free-space value of the THz peak electric \feld is\nreported in the below discussion. In case of THz \felds\n0 250 500 750 1000\nTime (ps)-0.4-0.200.2∆M y,z(%)\n0 5 10 15 20\nFrequency (GHz)0246Amplitude (a.u.)×10−9\nMz\nMy\n0 250 500 750 1000\nTime (ps)-40-20020∆M y,z(%)\n0 5 10 15 20\nFrequency (GHz)0246Amplitude (a.u.)×10−7\nMz\nMy(a)\n(c)(b)\n(d)FIG. 4. (Color online) Comparison of macrospin simulation\ndata for moderate (20 MV/m) and high (200 MV/m) THz-\npeak-\feld stimulus. The external magnetic \feld is set to\n\u00160HExt= 450 mT, i.e. replicating the experimental condi-\ntions from Fig. 3. Panel (a) shows the relative change of the\nMy(blue line) and M z(light blue line) magnetization com-\nponent for a E THzpeak \feld of 20 MV/m and (b) shows the\ncorresponding Fourier spectrum. (c) and (d) show the simu-\nlation data and corresponding Fourier spectrum for an E THz\npeak \feld of 200 MV/m, respectively.\nsmaller than E THz\u0018100 MV=m, the demagnetization\nfollows the square of the amplitude of the THz \feld, see\nRef. [16]. Lacking detailed experimental data, it is rea-\nsonable to assume that THz-induced demagnetization for\nhigher THz peak \felds (E THz>100 MV=m) can be de-\nscribed by the positive section of an error function, allow-\ning for a quadratic behavior for small demagnetization\nand a saturation for large demagnetization approaching\n100% [17]. From our experimental data, we derive a func-\ntional description of the demagnetization as a function of\nthe THz \feld such as Demag = f(E) = erf( A\u0001E2), with\nTHz peak \feld E and \ftting parameter A\u00196:0\u000110\u00006\nm2V\u00002. (See the Supplemental Material for further de-\ntails.)\nWith this assumption, the macrospin simulation re-\nsults for THz \felds E THz= 20 MV=m and E THz= 200\nMV=m are presented in Fig. 4 (a-b) and Fig. 4 (c-d),\nrespectively. For E THz = 200 MV =m, a clear nonlin-\near response of the magnetization to the THz \feld is\nfound, illustrated by the second harmonic oscillation in\nthe Mycomponent of the magnetization. The simulated\nTHz-induced demagnetization for E THz= 200 MV=m is\non the order of \u0001M z\u001820%. In Fig. 4 (b)+(d), the\nFourier spectrum of the FMR oscillation for the M yand\nMzcomponents of the magnetization at THz pump peak\n\felds of 20 MV/m and 200 MV/m are depicted. The\nFourier data of the 200 MV/m simulation shown in Fig. 4\n(c) clearly shows a second harmonic peak at \u001814 GHz,\npresent for M ybut not for M z. A similar behavior was5\nobserved recently by performing FMR spectroscopy of\nthin \flms irradiated with femtosecond optical pulses in-\nducing either ultrafast demagnetization [34], by exciting\nacoustic waves [35], and by two-dimensional THz mag-\nnetic resonance spectroscopy of antiferromagnets [36]. In\nour case, the high-harmonic generation process is solely\ndriven by the large amplitude of the terahertz magnetic\n\feld that is completely o\u000b-resonant with the uniform pre-\ncession mode. This would allow for exploring purely mag-\nnetic dynamics in regimes that are not accessible with\nconventional FMR spectroscopic techniques, where high-\namplitude dynamics are prevented by the occurrence of\nso-called Suhl's instabilities, i.e. non-uniform excitations\ndegenerate in energy with the uniform mode. Such non-\nresonant, high THz magnetic \felds are within the capa-\nbilities of recently developed table-top THz sources [37],\nand can also be generated in the near-\feld using meta-\nmaterial structures as described by Refs. [38{40].\nIn summary, we investigated magnetization dynam-\nics induced by moderate THz electromagnetic \felds in\namorphous CoFeB, in particular the ferromagnetic reso-\nnance response as a function of applied bias and THz\nmagnetic \felds. We demonstrate that semi-empirical\nmacrospin simulations, i.e. solving the Landau-Lifshitz-\nGilbert equation with a non-constant magnitude of the\nmagnetization vector to incorporate THz-induced de-\nmagnetization e\u000bect, are able to describe all the details\nof the experimental results to a good accuracy. Exist-\ning models of terahertz spin dynamics and spin pumping\nwould need to be extended to include the evidence pre-\nsented here [41, 42]. Starting from simulations describ-\ning experimental data for THz-induced demagnetization,\nwe extrapolate that THz \felds one order of magnitude\nlarger drive the magnetization into a nonlinear regime.\nIndeed, macrospin simulations with THz \felds on the or-\nder E THz\u0018200 MV=m (\u00160HTHz\u0018670 mT) predict a sig-\nni\fcant demagnetization of \u0001M z\u001820%, and a marked\nnonlinear behavior, apparent from second harmonic gen-\neration of the uniform precessional mode. We anticipate\nthat our results will stimulate further theoretical and ex-\nperimental investigations of nonlinear spin dynamics in\nthe ultrafast regime.\nM.H. gratefully acknowledges support from the\nSwedish Research Council grant E0635001, and the\nMarie Sk lodowska Curie Actions, Cofund, Project INCA\n600398s. The work of M.d'A. was carried out within the\nProgram for the Support of Individual Research 2017\nby University of Naples Parthenope. M.C.H. is sup-\nported by the U.S. Department of Energy, O\u000ece of Sci-\nence, O\u000ece of Basic Energy Sciences, under Award No.\n2015-SLAC-100238-Funding. M.P. and S.B. acknowledge\nsupport from the European Research Council, Starting\nGrant 715452 \\MAGNETIC-SPEED-LIMIT\".\u0003stefano.bonetti@fysik.su.se\n[1] M. Faraday, Philosophical Transactions of the Royal So-\nciety of London 136, 1 (1846).\n[2] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y.\nBigot, Phys. Rev. Lett. 76, 4250 (1996).\n[3] B. Koopmans, M. Van Kampen, J. T. 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Lett. 118, 257202 (2017)."    },    {        "title": "1904.05950v2.Measurement_of_spin_mixing_conductance_in_Ni___81__Fe___19____α__W_and_Ni___81__Fe___19____β__W_heterostrucutures_via_ferromagnetic_resonance.pdf",        "content": "Measurement of spin mixing conductance in Ni 81Fe19/\u000b-W and Ni 81Fe19/\f-W\nheterostrucutures via ferromagnetic resonance\nW. Cao,1,a)J. Liu,1A. Zangiabadi,1K. Barmak,1and W. E. Bailey1,b)\nMaterials Science and Engineering, Department of Applied Physics and\nApplied Mathematics, Columbia University, New York, New York 10027,\nUSA\nWe present measurements of interfacial Gilbert damping due to the spin pumping\ne\u000bect in Ni 81Fe19/W heterostructures. Measurements were compared for heterostruc-\ntures in which the crystallographic phase of W, either \u000b(bcc)-W or \f(A15)-W, was en-\nriched through deposition conditions and characterized using X-ray di\u000braction (XRD)\nand high-resolution cross-sectional transmission electron microscopy (HR-XTEM).\nSingle phase Ni 81Fe19/\u000b-W heterostructures could be realized, but heterostructures\nwith\f-W were realized as mixed \u000b-\fphase. The spin mixing conductances (SMC)\nfor W at interfaces with Ni 81Fe19were found to be signi\fcantly lower than those\nfor similarly heavy metals such as Pd and Pt, but comparable to those for Ta, and\nindependent of enrichment in the \fphase.\na)Electronic mail: wc2476@columbia.edu\nb)Electronic mail: web54@columbia.edu\n1arXiv:1904.05950v2  [cond-mat.mtrl-sci]  4 Sep 2019I. INTRODUCTION\nThe heavy metals Ta, W and Pt have drawn attention as charge-to-spin-current-\nconverters using spin Hall and related e\u000bects1{4. Beta phase W, \f-W, with the topologically\nclose-packed A15 structure5, possesses a \\giant\" spin Hall angle of \u0012SH\u00190.3{0.43,6. The\nspin transport properties of \f-W, such as the spin Hall angle \u0012SHand spin di\u000busion length\n\u0015SD, have been characterized by di\u000berent methods3,6{8. In these studies, the metastable\n\f-W layers were deposited directly on the substrate, were only stable for small W thickness,\nand were presumably stabilized through residual water vapor or oxygen on the substrate\nsurface; thicker W \flms typically revert to the stable (bcc) \u000bphase.\nRecently, some of us9{11have optimized a di\u000berent method to stabilize the metastable-\n\f-phase, using the introduction of N 2gas12while sputtering at low power. Relatively thick\n(over 100 nm) monophase \f-W \flms could be stabilized this way, when deposited on glass\nsubstrates. This technique has allowed deposition of majority \fphase W for 14 nm W\n\flms on CoFeB, as CoFeB/W(14 nm), and of minority \fphase for 14 nm W \flms on\nNi and Ni 81Fe19(\\Py\"), as Ni/W(14 nm) and Py/W(14 nm). In the present work, we\nhave prepared both monophase Py/ \u000b-W (here Py/\\ \u000b\"-W) and mixed phase Py/( \u000b+\f)-\nW (here Py/\\ \f\"-W) heterostructures using our optimized sputtering technique to enrich\nthe fraction of \f-W. Crystallographic phases of W were characterized by X-ray di\u000braction\n(XRD), and high-resolution cross-sectional transmission electron microscopy (HR-XTEM);\nsecondary structural information was provided by electrical resistivity measurements at room\ntemperature. We note that our measurements cannot distinguish between purely metallic,\nA15\f-W and A15 W oxide or nitride (e.g. W 3O); the identity of \f-W as a purely metallic\nphase or a compound is a longstanding controversy12,13.\nIn ferromagnet (FM)/normal metal (NM) heterostructures, pure (chargeless) spin cur-\nrents can be injected from the FM into the NM by exciting ferromagnetic resonance (FMR)\nin the FM layer, \\pumping\" out spin current14,15. If the spin current is absorbed in the\nNM layer, the in\ruence of \\spin pumping\" can be observed through the increase in the\nlinewidth of the resonance, proportional to frequency !as Gilbert damping, due to the\nloss of angular momentum from the precessing spin system14,15. The e\u000eciency of the spin\npumping e\u000bect for a given interface is characterized through the spin mixing conductance\n(SMC)g\"#\nFM=NM. The SMC is also an important parameter for the interpretation of inverse\n2spin Hall e\u000bect (ISHE) measurements4,16, in which the spin Hall angle \u0012SHis measured by\npumping chargeless spin current into the NM by FMR and measuring spin-to-charge current\nconversion through the generated charge current. Measurements of spin mixing conductance\nfor Py/\u000b-W and Py/ \f-W have not been reported previously, although some measurements\nhave been reported for W oxide8. For these measurements, the simplest way to isolate the\ncontribution of the FM/NM interface to the damping, and thus the spin pumping e\u000bect and\nspin mixing conductance g\"#\nFM=NM, is to deposit the FM on the bottom and the NM on top,\nso that comparison structures without the NM layer have nearly identical microstructure.\nThe ability to deposit enriched \f-W on Py rather than on an insulating substrate is thus im-\nportant for the measurement of spin mixing conductance of Py/ \f-W. In this manuscript, we\nreport measurements of spin mixing conductances for Py/\\ \u000b\"-W and Py/\\ \f\"-W interfaces\nusing variable-frequency, swept-\feld FMR, as in our previous work17{19.\nII. SAMPLE PREPARATION\nUltrahigh vacuum (UHV) magnetron sputtering was used to deposit substrate/Ta(5\nnm)/Cu(5 nm)/Ni 81Fe19(Py)/W/Cu(5 nm)/Ta(5 nm) heterostructures on both oxidized\nSi and glass substrates at room temperature, with base pressure better than 2 \u000210\u00008Torr.\nThe samples consist of two thickness series in \\ \u000b\"-W and \\\f\"-W for a total of four se-\nries. In the \frst thickness series, the thickness of Py ( tPy= 5 nm) was \fxed and the\nthickness of W was varied, with tW= 2, 5, 10, 30 nm, for both \\ \u000b\"-optimized and \\ \f\"-\noptimized conditions. This thickness series was used for resistivity measurements, X-ray\ndi\u000braction (XRD) ( tW= 10, 30 nm), high-resolution cross-sectional transmission electron\nmicroscope (HR-XTEM) ( tW= 30 nm) and FMR characterization. In the second thickness\nseries, the thickness of W ( tW= 10 nm) was \fxed and the thickness of Py was varied, with\ntPy= 3, 5, 10, 20 nm, also for both \\ \u000b\" and \\\f' conditions. This thickness series was used\nonly for FMR characterization. The same stacks without W layers, Py(3, 5, 10, 20 nm),\nwere deposited as reference samples for FMR measurements. One heterostructure with re-\nverse depostion order, \\ \u000b\"-W(10 nm)/Py(5 nm), was deposited in the absence of N 2gas\nand characterized by XRD and FMR; this was not possible for \\ \f\"-W because the \fphase\ncannot be stabilized on Cu underlayers10.\nThe W layers in all samples were deposited with 10 W power, nearly constant deposition\n3rate (<0:1\u0017A/s), and Ar pressure of 3 \u000210\u00003Torr. Nitrogen gas, with 1 :2\u000210\u00005Torr\npressure measured by a residual gas analyzer, was introduced to promote the growth of \f\nphase W10.\nIII. STRUCTURAL CHARACTERIZATION\nCrystalline phases of W in the Py/W heterostructures were characterized primarily by\nXRD (Section A), with supporting measurements by HR-XTEM (Section B), and \fnally with\nsome indirect evidence in room-temperature electrical resistivity measurements (Section C).\nOur basic \fndings are that \flms deposited without N 2, optimized for \\ \u000b\"-W, are nearly\nsingle-phase \u000bin Py/\\\u000b\"-W, while in the Py/\\ \f\"-W optimized heterostructures, deposited\nin the presence of N 2, the W layers are mixed \u000b+\fphase, with a roughly 50%{50% mixture\nof\u000b-W and\f-W averaged over a 10 nm \flm. The phase composition within the \frst 5 nm\nof the interface may have a slightly greater fraction of \u000b-W, but\f-W could be positively\nidenti\fed here as well.\nA. X-ray di\u000braction\nBoth symmetric ( \u0012-2\u0012) and grazing-incidence, \fxed sample angle X-ray di\u000braction (XRD)\nscans were carried out on Py(5 nm)/W(10 nm) and Py(5 nm)/W(30 nm) heterostructures\ndeposited on glass substrates. The scans are compared for \\ \u000b\"-W and \\\f\"-W depositions.\nScans were recorded using Cu K\u000bradiation and a commercial di\u000bractometer.\nThe symmetric ( \u0012-2\u0012) scans, with scattering vector perpendicular to \flm planes, are\npresented \frst. We point out some obvious features of the symmetric XRD spectra, shown\nin Figures 1a) and 1b). For the Py/\\ \u000b\"-W(30 nm) \flm in Fig. 1a), all peaks can be indexed\nto the close-packed planes, Cu(111)/Py(111) (fcc) and \u000b-W(110) (bcc). The small peak\nat 2\u0012= 36\u000ecan be indexed to the re\rection of a small amount of Cu K\fradiation from\n\u000b-W(110). Moving to the thinner \u000bphase \flm in Fig. 1b), Py/\\ \u000b\"-W(10 nm), it is still\nthe case that all re\rections can be indexed to the close-packed Cu(111)/Py(111) and \u000b-\nW(110) planes. However, there is greater structure in these re\rections, presumably due to\n\fnite-size oscillations (Laue satellites), expected to be more evident in thinner \flms. Nearly\nidentical spectra are recorded for the 10 nm \\ \u000b\"-W \flms regardless of deposition order:\n4Py(5 nm)/\\ \u000b\"-W(10 nm) and \\ \u000b\"-W(10 nm)/Py(5 nm) \flms scatter X-rays very similarly,\nas shown in Fig. 1b). We should note that Cu deposited on Ta has strong f111gtexture in\nour \flms. Py (Ni 81Fe19) deposited on Cu also has strong f111gtexture; growth of Py on Cu\nand vice-versa is found to be largely coherent within grains. Both layers are fcc with similar\nlattice parameters: aCu\u00193:61\u0017A for Cu10,20andaPy\u00193:55\u0017A for Py10,21, with a small\nmis\ft strain of \u000f=jaCu\u0000aPyj=aCu\u00192%. The XRD peaks for (111)-re\rections in bulk\nphases, broadened by \fnite-size e\u000bects ( FWHM\u00191:7\u000efor 5 nm \flms, using the Scherrer\nequation22,23), are very close to each other, at 44 :2\u000e(Py) and 43 :4\u000e(Cu) respectively, so we\nexpect (and have observed) one averaged peak for Cu and Py.\nThe nominal \\ \f\"-W \flms (red lines) clearly show the presence of the \fphase through\nthe unique\f-W(200) re\rection at 2 \u0012'36\u000e. This unique re\rection is very strong in the \\ \f\"-\nW(30 nm) heterostructure (Fig. 1a) but weaker as a proportion of the total intensity in the\nthinner \\\f\"-W(10 nm) heterostructure (Fig. 1b). In Fig. 1a), experimental \f-W(200) and\n\f-W(210) re\rections have intensities in a ratio similar to the theoretical scattering intensity\nratios for randomly-oriented \fgrains. This is not the case for the thinner \\ \f\"-W(10 nm)\nheterostructure in Fig. 1b); here the unique \f-W(200) peak is less intense than expected.\nWe interpret the relative weakness of \f(200) as the presence of a large fraction of \u000bgrains\nin the nominal Py/\\ \f\"-W(10 nm) heterostructure.\nIn order to quantify the amount of \u000b-W in the nominal \\ \f\"-W \flm, we have carried out\ngrazing incidence measurements of Py(5 nm)/W(10 nm) samples (20\u000e\u00142\u0012\u0014100\u000e) on the\nsame di\u000bractometer, as illustrated in Fig. 1c). The samples were measured at a \fxed source\nposition of 5\u000ewith 0:1\u000estep size, 0:25\u000e\fxed slit and the 15 mm beam mask. From the TEM\nmeasurements in Fig. 2b), we \fnd that the deposited \\ \u000b\"-W \flms havef110gtexture, i.e.,\nthe hexagonal arrangement (60\u000eangles) of thef011g\u000b-W re\rections away from the surface\nnormal. Thus with the grazing incidence geometry, in which the scattering vector does not\nremain perpendicular to the \flm plane, the relative intensities of the peaks will not match\ntheoretical calculations (vertical lines) based on randomly-oriented, untextured \flms. For\nexample, the \u000b-W(200) peak (blue, \u001858\u000ein 2\u0012) almost vanishes in the XRD scan here, due\nto thef011g\u000b-W texture.\nHere we focus on the \u000b-W(211) peaks near 2 \u0012= 72\u000e, observed in both Py/\\ \u000b\"-W and\nPy/\\\f\"-W samples. As shown in the Fig. 1c) inset, the \u000b-W(211) peaks (60\u000e\u00142\u0012\u001485\u000e),\nwere \ftted as the sum of Lorentzian peak and identical background, assumed quadratic in\n52\u0012, for both Py/\\ \u000b\"-W and Py/\\ \f\"-W samples. First we \ft the \u000b-W(211) peak (blue)\nin the Py/\\ \u000b\"-W sample to the summed function to determine the Lorentzian peak and\nquadratic background parameters. Next, we use this \ftted background in the \ft to the\n\u000b-W(211) peak (red) in the Py/\\ \f\"-W sample. The two \ftted \u000b-W(211) peaks are shown\nas blue (for Py/\\ \u000b\"-W) and red (for Py/\\ \f\"-W) dashed lines in the Fig. 1c) inset. The \fts\nreproduce the experimental data well in the \ftted region. The integrated \u000b-W(211) peak\n(i.e., the 2\u0012-integrated area between the measured data and the \ftted background) for the\nPy/\\\f\"-W sample has roughly half the intensity of the integrated peak for the Py/\\ \u000b\"-W\nsample. Assuming that the nominal \u000b-W is 100% \u000bphase and that the \u000bgrains in mixed\nphase \\\f\"-W have similar f110gtexture, as is supported by the HR-XTEM measurements\nin Figures 2 and 3, we conclude that the Py/\\ \f\"-W(10 nm) \flm is roughly 50% \u000b-W and\n50%\f-W.\nB. Transmission electron microscopy\nThe phases of the nominal Py(5 nm)/\\ \u000b\"-W(30 nm) and the nominal Py(5 nm)/\\ \f\"-\nW(30 nm) samples deposited on oxidized Si substrates were characterized in high-resolution\ncross-sectional imaging, selected-area di\u000braction, and focused-beam nanodi\u000braction, by\ntransmission electron microscopy (for details see the endnote1).\nFig. 2 shows a cross-sectional image and di\u000braction pattern for the nominal Py/\\ \u000b\"-\nW(30 nm) heterostructure. First, one can see from the mass contrast between W and the 3d\ntransition metal elements (Ni, Fe, Cu) that the Py/W and W/Cu interfaces are relatively\n\rat and sharp on the scale of the image resolution of \u00183 nm, presumably broadened\nby topographic variation through the thickness of the TEM foil. Second, based on (less\npronounced) di\u000braction contrast parallel to the interface, the grains appear to be columnar,\nin many cases extending through the \flm thickness, with an average (lateral) grain diameter\nof 10{20 nm. The selected-area di\u000braction (SAD) pattern can be indexed according to unique\n(111)Py//(011) \u000b-W \fber texture, as shown by the hexagonal arrangement (60\u000eangles) of\nthef011gre\rections in \u000b-W, and the arrangement of f111gre\rections in Py,\u001870:5\u000eaway\nfrom the (vertical) \fber axis. The calculated di\u000braction spots based on f111gPy//f011g\u000b-\nW \fber texture with 1-fold rotational symmetry about the \flm-normal axis are shown in\nFig. 2 b), inset; good agreement is found.\n6Cross-sectional images and di\u000braction patterns for the Py/\\ \f\"-W(30 nm) heterostructure\nare shown in Fig. 3. Here again, in Fig. 3 a), the mass contrast shows similarly well-\nde\fned interfaces, but the topographic variations have a shorter wavelength, due presumably\nto smaller, more equiaxed grains in the mixed-phase \\ \f\"-W. Circles indicate areas where\nconvergent nanobeam electron di\u000braction (CBED) patterns were taken. The di\u000braction\npatterns over these small regions can be indexed to single phases: fcc Ni 81Fe19(Py) in\ngreen, bcc\u000b-W in blue, and A15 \f-W in red.\nThe CBED patterns in Fig. 3 a) con\frm that the nominal \\ \f\"-W \flm is mixed-phase \u000b-W\nand\f-W. The critical question for distinguishing the spin mixing conductances of \u000b-W and\n\f-W in Py/W is the identity of the W phase located within the \frst several nanometers of\nthe interface with Py: the pumped spin current is ejected through the interface and absorbed\nover this region; see the x-axis of Fig. 6. We have addressed this question locally using high\nresolution imaging (see Fig. 3 b) and over a larger area using frequency analysis (see Fig. 3\nc) of the image, roughly equivalent to SAD. In Fig. 3 b), a 10 nm area (red box) shows what\nappears to be a single-crystal region with (1 \u001611)[110]Py//(011)[1 \u001611]\u000b-W//(002)[200] \f-W,\nindicating that the \fcrystals may nucleate on top of the \u000bcrystals; however, this is contrary\nto our previous observations10and not distinguishable in the image from the superposition\nof grains through the foil, with nucleation of \fat the Py/W interface. The discrete spatial\nFourier transform (FT) of this region shows that the four vertically/horizontally circled \f-\nWf002gspots are similar in intensity to the six \u000b-Wf011gspots, supporting a similar \f-W\ncontent in this region. Carrying out a spatial FT of the full selected region within 5 nm of\nthe interface (dotted box) in Fig. 3 a), we can con\frm that \f-W is indeed present adjacent\nto the interface, as indicated by the \f-Wf002gFT spots in Fig. 3 c), although these appear\nto be somewhat less intense than the \u000b-Wf011gspots.\nC. Resistivity\nFour-point probe van der Pauw resistivity measurements were performed at room tem-\nperature on the \frst thickness series of samples ( tPy= 5 nm \fxed, variable tW) deposited\non 25\u000225 mm square glass substrates, i.e., glass substrate/Ta(5 nm)/Cu(5 nm)/Py(5\nnm)/W(tW)/Cu(5 nm)/Ta(5 nm). Two point probes for current and two point probes\nfor voltage were placed at the four corners of the square coupons. For square samples, the\n7voltage-to-current ratios were converted to resistance per square using the known geomet-\nrical factor \u0019=ln 2\u00194:5324. To isolate the W resistances, we plot the thickness-dependent\nsheet conductance and \ft according to:\n1\nRtotal=Gtotal=G0+4:53\n\u001aWtW (1)\nwhereRtotal(Gtotal) is the total resistance (conductance) of the sample, \u001aWandtWare\nthe resistivity and the thickness of the W layer, and G0is the parallel conductance of other\nlayers in the stack.\nWe have veri\fed Ohmic response by \ftting the proportional dependence of voltage V\non current Iover the range 2 mA \u0014I\u001410 mA for each sample. Fig. 4 summarizes the\ntotal conductance Gtotal= 1=Rtotalas a function of W thickness tWfor all Py(5 nm)/W( tW)\nheterostructures. Solid lines represent linear \fts for the W resistivity \u001aW, assumed constant\nas a function of W thickness for \\ \u000b\"-W and \\\f\"-W samples. The extracted resistivity for\n\\\u000b\" phase W \u001a\u000bis found to be\u001835\u0016\ncm and for \\ \f\" phase W \u001a\f\u0018148\u0016\ncm. The\nresistivity for \\ \f\"-W more than four times greater than that for \\ \u000b\"-W, is due in large part\nto the much smaller grain size for \f-W and is typically observed in prior studies25. Here the\nresistivity for \\ \u000b-W\" is larger by a factor of 2{3 than \flms deposited at room temperature\nand postannealed in previous work26, also attributable to a smaller grain size in these \flms\ndeposited at ambient temperature. The resistivity measurements for these thin \flms might\nbe taken as indirect evidence for the presence of the \fphase in the nominal \\ \f\"-W layers.\nIV. FERROMAGNETIC RESONANCE MEASUREMENTS\nThe four thickness series of Py( tPy)/W(tW) \flms, for \\ \u000b\"-W and \\\f\"-W, as described\nin Section II were characterized using variable-frequency \feld-swept FMR using a coplanar\nwaveguide (CPW) with center conductor width of 300 \u0016m. The bias magnetic \feld was\napplied in the \flm plane ( pc-FMR, or parallel condition). For details, see e.g., our prior\nwork in Ref. [20].\nFig. 5 summarizes half-power FMR linewidth \u0001 H1=2as a function of frequency !=2\u0019\nfor Py(5 nm), Py(5 nm)/\\ \u000b\"-W(10 nm) and Py(5 nm)/\\ \f\"-W(10 nm) samples. The mea-\nsurements were taken at frequencies from 3 GHz to above 20 GHz. Solid lines are linear\nregression of the variable-frequency FMR linewidth \u0001 H1=2= \u0001H0+ 2\u000b!=\r , where \u0001H1=2\n8is the full-width at half-maximum, \u0001 H0is the inhomogeneous broadening, \u000bis the Gilbert\ndamping,!is the resonance frequency and \ris the gyromagnetic ratio. Good linear \fts\nwere obtained with resonance frequency !=2\u0019for experimental linewidths \u0001 H1=2(!) of all\nthe samples measured.\nFor the \frst sample thickness series Py(5 nm)/W( tW), we plot damping parameters \u000b\nextracted from the linear \fts, as a function of W thickness in Fig. 6. Standard deviation\nerrors in the \ft for \u000bare\u00182\u000210\u00004. The Gilbert damping \u000bsaturates quickly as a function\noftWfor both \\\u000b\"-W and \\\f\"-W, with almost all of the e\u000bect realized with the \frst 2 nm\nof W. Loosely speaking, this fast saturation implies a short spin di\u000busion length \u0015SD\u00142\nnm, so the identity of the W phase ( \u000bor\f) over this length scale near the interface is\nmost relevant. The averaged damping, \u000bPy=\\\u000b\"\u0000Wand\u000bPy=\\\f\"\u0000W, are shown as horizontal\ndashed lines in the \fgure. \u000bPy=\\\u000b\"\u0000Wis slightly smaller than \u000bPy=\\\f\"\u0000W, but this may be\nwithin experimental error. Due to spin pumping, the damping is enhanced with the addition\nof W layers \u0001 \u000b=\u000bPy=W\u0000\u000bPy, normalized to the Gilbert damping \u000bPyof the reference\nsample without W layers. The e\u000bective SMC g\"#\neffat the Py/W interfaces can be calculated\nfollowing:\n\u0001\u000b=\r\u0016hg\"#\neff\n(4\u0019MS)tPy(2)\nwhere\ris the gyromagnetic ratio, \u0016 his the reduced Planck constant, and 4 \u0019MS\u001910\nkG is the saturation inductance of Py. In this series of samples, the e\u000bective SMC at the\nPy/\\\u000b\"-W interface g\"#\nPy=\\\u000b\"\u0000W\u00197:2\u00060:3 nm\u00002and the e\u000bective SMC at the Py/\\ \f\"-\nW interface g\"#\nPy=\\\f\"\u0000W\u00197:4\u00060:2 nm\u00002. These values are signi\fcantly lower than those\nreported in Ref.8for CoFeB/W (20{30 nm\u00002), as measured by spin-torque FMR.\nFor the second sample thickness series Py( tPy)/W(10 nm), we plot the extracted Gilbert\ndamping\u000band damping enhancement \u0001 \u000b=\u000bPy=W\u0000\u000bPyas a function of Py thickness\nin Fig. 7. The enhanced damping is normalized to the Gilbert damping \u000bPyof reference\nsamples with the same Py thickness tPy. The result is in good agreement with the inverse\nthickness dependence of contributed damping predicted from Equation 2. The experimental\ndata is \ftted with Equation 2 to extract the e\u000bective SMC. In this series of samples, the\ne\u000bective SMC at the Py/\\ \u000b\"-W interface g\"#\nPy=\\\u000b\"\u0000W\u00196:7\u00060:1 nm\u00002and the e\u000bective SMC\nat the Py/\\ \f\"-W interface g\"#\nPy=\\\f\"\u0000W\u00197:4\u00060:3 nm\u00002.\nPrevious studies on W have shown that the formation of \u000b-W is preferred, for thicker\n9W layers (e.g. 10 nm)3,26. We also prepared the sample \\ \u000b\"-W(10 nm)/Py(5 nm) with\nreverse deposition order, with the same seed and cap layers, on an oxidized Si substrate.\nHere the top surface of the 10 nm thick \u000b-W layer is pure \u000bphase, as shown by XRD in Fig.\n1 a). We performed the same FMR measurement on the reverse order sample; its Gilbert\ndamping enhancement \u0001 \u000bis plotted as the green dot in Fig. 7. This point almost overlaps\nwith the measurement for the normal order sample Py(5 nm)/\\ \u000b\"-W(10 nm), indirectly\nsupporting the conclusion that the phase of the Py/\\ \u000b\"-W interface is similar to the phase\nof the \\\u000b\"-W/Py interface, i.e., almost 100% \u000bphase W. Note that it was not possible to\ndeposit a reverse-order \fphase sample because no \fphase W could be stabilized on Cu\nusing our technique10.\nThe FMR measurements of spin mixing conductance g\"#for Py/\\\u000b\"-W and Py/\\ \f\"-W\nare new in this study. We \fnd that the value is similar to that measured for Ta27(g\"#\u0018\n10 nm\u00002) regardless of the enriched phase. First-principles-based calculations including\nrelativistic e\u000bects28forg\"#at Py/NM interfaces have shown that Ta, next to W in the\nperiodic table, is a good spin sink due to its large spin-orbit coupling (SOC), but has a\nrelatively small g\"#\u00188{9 nm\u00002. The e\u000ecient absorption of spin current can be connected\nwith a large SOC from the large atomic number, and the low SMC can be connected to\nrelatively poor band matching across the Py/W interface, compared with that for Py/Cu\nor Py/Pt28. The conclusion for Ta is consistent with our experimental results for the Py/W\nsystem, i.e., the rapid saturation of Gilbert damping within the \frst 2 nm of W, indicating\nW is also a good spin sink, with a similarly low g\"#\u00187 nm\u00002.\nV. DISCUSSION\nWe have found very little di\u000berence between the spin scattering properties (spin mixing\nconductance and spin di\u000busion length) of \u000b-W and mixed phase ( \u000b+\f)-W. The simplest\ninterpretation is that both spin mixing conductances and spin di\u000busion lengths are nearly\nequal for the two phases. However, despite our development of an optimized technique9{11\nto stabilize the \fphase, our control over the amounts of deposited \u000band\fphases is less\nthan complete, particularly near the Py/W interface.\nThe \\\u000b\"-structure we deposited, Py/ \u000b-W, is nearly\u0018100%\u000bphase. We observed no\nstrong\f-W peaks in the XRD scans, and neither crystalline structure nor di\u000braction patterns\n10for the\fphase in HR-XTEM characterization. According to our previous work10,26,29,\nwe know that ionically and covalently bonded substrates/underlayers are favorable for the\nformation of some \f-W, whereas metallic underlayers promote \u000b, so on Py even at a thickness\nof 2 nm, the nominally \u000b-W \flm is fully \u000bif deposited in the absence of nitrogen.\nIn the thinnest \\ \f\"-structure which we can characterize by XRD, Py/\\ \f\"-W(10 nm),\nwe identify a roughly 50%-50% mixture of \u000band\fphases. If this balance persists at the\ninterface as well, the SMC cannot di\u000ber by more than 10-20% for the two phases. While\nthe measurement of the 5 nm region near the interface seems to show somewhat less than\n50%\fphase, there is still a substantial population of \f-W in this region, and it would seem\nthat a strong di\u000berence in SMC for \u000b-W and\f-W should be resolvable if present. Given\nthat the measured values are very similar, we conclude that the \u000band\fphases do not di\u000ber\nstrongly in this spin transport study.\nOne might ask why the spin mixing conductance, in contrast to the spin Hall angle3, does\nnot di\u000ber much for the two phases of W. The spin mixing conductance (SMC) g\"#\nFM=NMis a\nproperty of the FM/NM interface, rather than a bulk property of the NM layer. The SMC\nmay be approximated (in a single-band, free-electron model) as g\"#\u0019\u0014k2\nFA=4\u00192, wherekF\nis the Fermi wave number for the NM, \u0014represents the number of scattering channels in\nunits of one channel per interface atom, and A is the total surface area of the interface30.\nDespite the possibility that bulk \f-W has a stronger e\u000bective spin-orbit coupling and spin\nHall e\u000bect due to its A15 structure, \f-W could have similar numbers of conducting channels\nper atom at the FM/NM interface as \u000b-W, which could lead to the similar values of SMC\nmeasured here.\nAnother possibility is that the spin di\u000busion length \u0015SDmay vary along the W layer\nthickness, due to nonuniformly distributed \u000b-W and\f-W phases in \\ \f\"-W samples. If this\nis true, \ftting a single spin di\u000busion length for spin pumping into very thin W layers will\nbe problematic31. However, because we have observed a very rapid saturation of Gilbert\ndamping over the \frst 2 nm of W for both \\ \u000b\"-W (almost pure \u000bphase) and \\ \f\"-W (mixed\nphase) in Fig. 6, we can only assign an upper bound for \u0015SD, similarly short in the two\nphases.\n11VI. CONCLUSIONS\nIn summary, we report measurements of spin mixing conductances of Py/W \flms with\ncontrolled amounts of \u000band\fphase W, measured by Gilbert damping through ferromag-\nnetic resonance (FMR). We \fnd no strong di\u000berences in the spin mixing conductances of\nPy/\u000b-W and Py/ \f-W, measured as g\"#= 6.7{7.4 nm\u00002, although control of the \fphase is\nseen to be more di\u000ecult near the interface with Py. Our experimental results also indicate\nthat W, no matter of which phase, is a good spin sink, but with relatively small spin mixing\nconductance in Ni 81Fe19(Py)/W, similar to Ta in Py/Ta.\nVII. ACKNOWLEDGEMENTS\nThe authors thank Daniel Paley of Columbia Nano Initiative for the grazing incidence\nXRD scans and Kadir Sentosun of Columbia University for the satellite peak calculations.\nThis work is supported by the US NSF-DMR-1411160.\nNOTES\n1Focused ion-beam (FIB) and FEI Helios NanoLab 660 were used to prepare foils for TEM studies. To\nprotect the heterostructures against the ion-beam damage during sample preparation, amorphous Platinum\n(1.5\u0016mthick) was sputtered on the surface of the wafers by electron and ion beam, respectively. TEM and\nhigh-resolution cross-sectional TEM (HR-XTEM) analyses were performed by image Cs-corrected FEI Titan\nThemis 200 at an accelerating voltage of 200 kV. Nano-beam electron di\u000braction pattern (DP) technique and\nFourier transform (FT) analysis of the HRTEM have been utilized to identify the nature of each phase at\nthe scale of 1{2 nm wide. The nano-beam DPs were obtained by FEI Talos TEM operating at 200 kV. 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Toney, Journal of Vacuum Science & Technology A:\nVacuum, Surfaces, and Films 29, 051512 (2011).\n27S. Mizukami, Y. Ando, and T. Miyazaki, Journal of Magnetism and Magnetic Materials\n226-230 , 1640 (2001), proceedings of the International Conference on Magnetism (ICM\n2000).\n28Y. Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov, and P. J. Kelly, Phys. Rev. Lett.\n113, 207202 (2014).\n29D. Choi, X. Liu, P. K. Schelling, K. R. Co\u000bey, and K. Barmak, Journal of Applied Physics\n115, 104308 (2014).\n30Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).\n31E. Montoya, P. Omelchenko, C. Coutts, N. R. Lee-Hone, R. H ubner, D. Broun, B. Heinrich,\nand E. Girt, Phys. Rev. B 94, 054416 (2016).\n14FIGURES\n15FIG. 1. X-ray di\u000braction (XRD) measurements for Py(5 nm)/\\ \u000b\"-W(tW) (blue) and Py(5\nnm)/\\\f\"-W(tW) (red) deposited on glass substrates. (a) tW= 30 nm; (b) tW= 10 nm. Solid\nvertical lines show the calculated re\rections and intensities for \u000b-W and\f-W peaks. (c) Grazing-\nincidence XRD measurements for Py(5 nm)/\\ \u000b\"-W(10 nm) and Py(5 nm)/\\ \f\"-W(10 nm) samples.\nThe inset shows the \u000b(211) re\rections observed in both samples. The blue and the red dashed\nlines refer to the \fts for Py/\\ \u000b\"-W and Py/\\ \f\"-W, respectively. The black dashed line refers to\nthe identical quadratic background.\n16FIG. 2. (a) High-resolution cross-sectional transmission electron microscopy (HR-XTEM) image\nof SiO 2/Ta(5 nm)/Cu(5 nm)/Py(5 nm)/\\ \u000b\"-W(30 nm)/Cu(5 nm)/Ta(5 nm) heterostructure. The\n\u000b-W grains are columnar with lateral radius of 10{20 nm, with larger grain size in the growth\ndirection. (b) Selected-area di\u000braction (SAD) pattern of the heterostructure, showing the preferred\ntexture of\u000b-W grains on Py layer, f111gPy//f011g\u000b-W (see calculated pattern in the inset). No\nsign of\f-W was detected in this heterostructure.\n17FIG. 3. (a) HR-XTEM image of SiO 2/Ta(5 nm)/Cu(5 nm)/Py(5 nm)/\\ \f\"-W(30 nm)/Cu(5\nnm)/Ta(5 nm), showing mixed-phase \u000b-W and\f-W. Convergent nanobeam electron di\u000braction\n(CBED) patterns, bottom, reveal the co-existence of separated \u000b-W,\f-W, and fcc Py. (b) Close-\nup of one region near the Py/W interface in (a), with discrete spatial Fourier Transform (FT). The\nFT is consistent with a single-crystal pattern of (1 \u001611)[110]Py//(011)[1 \u001611]\u000b-W//(002)[200] \f-W, as\nshown in the calculated pattern (bottom right). (c) FT of interface region (dotted box), showing\nco-existence of \u000b-W and\f-W in the \frst 5 nm W adjacent to the Py/W interface.\n18FIG. 4. The total conductance Gtotal= 1=Rtotalas a function of W thickness. Blue dots refer\nto Py(5 nm)/\\ \u000b\"-W(tW) samples and red dots refer to Py(5 nm)/\\ \f\"-W(tW) samples. The solid\nlines are linear \fts.\n19FIG. 5. Half-power FMR linewidth \u0001 H1=2spectra of reference sample Py(5 nm) (black), Py(5\nnm)/\\\u000b\"-W(10 nm) (blue) and Py(5 nm)/\\ \f\"-W(10 nm) (red) samples. The solid lines are linear\n\fts.\n20FIG. 6. Gilbert damping \u000bof the reference sample Py(5 nm) (black), Py(5 nm)/\\ \u000b\"-W(tW) (blue)\nand Py(5 nm)/\\ \f\"-W(tW) (red) samples. The blue and red dash lines refer to averaged enhanced\ndamping for Py(5 nm)/\\ \u000b\"-W(tW) and Py(5 nm)/\\ \f\"-W(tW), respectively.\n21FIG. 7. Damping enhancement \u0001 \u000b=\u000bPy=W\u0000\u000bPyof Py(tPy)/\\\u000b\"-W(10 nm) (blue), Py( tPy)/\\\f\"-\nW(10 nm) (red) and \\ \u000b\"-W(10 nm)/Py(5 nm) (green) samples, normalized to the Gilbert damping\nof reference samples \u000bPywith the same Py thickness. Solid lines refer to \ftting with Equation 2.\nInset: Gilbert damping \u000bof the reference sample Py( tPy) (black), Py( tPy)/\\\u000b\"-W(10 nm) (blue),\nPy(tPy)/\\\f\"-W(10 nm) (red) and \\ \u000b\"-W(10 nm)/Py(5 nm) (green) samples.\n22"    },    {        "title": "1904.10197v2.Ultrafast_depinning_of_domain_wall_in_notched_antiferromagnetic_nanostructures.pdf",        "content": " \n \nUltrafast  depinning of domain wall  in notched antiferromagnet ic nanostructures  \n \nZ. Y . Chen1, M. H. Qin1,*, and J. –M. Liu2 \n1Institute for Advanced Materials, South China Academy of Advanced Optoelectronics and \nGuangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, \nSouth China Normal University, Guangzhou 510006, China  \n2Laboratory of Solid State Microstructures and Innovative Center for Advanced \nMicrostructu res, Nanjing University, Nanjing 210093, China  \n \n[Abstract]  The pinning/ depinning  of antiferromagnetic (AFM) domain  wall is certainly the \ncore issue of AFM spintronics. I n this work, we study theoretically the Néel-type domain wall \npinning and depinning at a notch  in an antiferromagnetic  (AFM)  nano -ribbon . The depinning \nfield depending on the notch  dimension and intrinsic physical parameters are  deduced  and \nalso numerically  calculated . Contrary to conventional conception , it is revealed that the \ndepinning field is remarkably dependent of the damping constant  and the time -dependent \noscillation  of the domain wall position in the weakly damping regime benefits to the wall \ndepinning, resulting in a gradual  increase of the depinning field up to a s aturation value with \nincreas ing damping constant . A one-dimensional model  accounting of the internal dynamics \nof domain wall  is used to explain perfectly the simulated  results. It is demonstrated that the \ndepinning mechanism of an AFM domain wall differ s from ferromagnetic domain wall  by \nexhibiting a depinning speed t ypically three orders of magnitude faster than the latter , \nsuggesting  the ultrafast dynamics of an AFM system.  \n \nKeywords: antiferromagnetic dynamics, domain wall, lattice defect, pinning effect  \n \n  \n                                                             \nEmail: qinmh@scnu.edu.cn  Antiferromagnetic (AFM) materials are promising for next generation of spintronic  \ndevices and attract substantial a ttention especially because they have  strong  anti-interference \ncapability and promised ultrafast magnetic dynamics.1-8 As a frontier and highly concerned \nissue for advanced spintronics, the domain wall (DW) dynamics of antiferromagnets is under \nextensive investigat ion. Specifically , several  stimuli have been  propose d to drive the domain \nwall motion , including the Néel spin-orbit torques,9-10 spin waves,11-12 temperature \ngradients13-15 and so on.16-18 These works provide useful information fo r future AFM storage \ndevice design.  \nNevertheless, most of these works discuss models on perfect samples and the wall \npinning caused by disorder and local defects  is neglected . As a matter of fact, the wall pinning \nmay play an important role in magnetic dynamics. On one hand, for a realistic spintronic \ndevice where inhomogeneity and lattice defects are inevitable, the wall dynami cs could be \nsignificantly affected  and the wall pinning/depinning becomes the limited step for device \noperation . For example, it was reported that electrical current induced switching of AFM \ndomains in CuMnAs  occurs only in localized regions, strongly suggesting the important role \nof wall pinning.19 Given these reasons, a clarification of the underlying mechanisms for wall \npinning/depinning becomes essential . On the other hand, artificial lattice defects such as \nnotches with proper shape  could be used in discretizing domain wall position and enhancing \nits stability against thermal fluctuations and stray fields in potential race -track memory and \nlogic devices.20-24 Therefore, the dynamics of AFM domain wall pinning/depinning  appears to \nbe one of the core issues for application potential s and basic research  of AFM spintronics .  \nFortunately, the domain wall pinning in ferromagnetic systems  have been extensively \ninvestigated, and the accumulated experience can be partially transferred to the study of AFM \ndomain dynamics.25-32 For a ferromagnetic domain wall, the depinning field can be \nanalytically  obtained by minimizing the total energy , demonstrating the critical role of notch \ngeometry in pinning the wall .26 More interestingly, the dependence of dep inning field on the \nGilbert damping for a ferromagnetic system has been revealed in microm agnetic simulations , \nand the damping constant , if small, can reduce the depinning field, contrary to the general \nexpectation that they should be independent of each other.27 This phenomenon not only \nreveals the complexity of domain wall pinning, but more importantly provides a method of domain wall manipulation.  However, as far as we know, few work on the pinning /depinning  \nof an AFM domain wall has been available, while this issue is certainly more important than \nand distinctly different from the case of ferromagnetic wall . \nIn proceeding, we may discuss the domain wall pinning/depinning for an AFM \nnanostructure with a notch, wi thout losing the generality , while  the calculation methods and \nmain conclusions apply to antiferromagnets with other lattice defects. For simplicity \nconsideration, such a notch has a rectangular  section, as shown in Fig. 1(a). We can derive the \ndepinning  field hdep as a function of the notch size  and uniaxial anisotr opy in a simplified \nframework and the theory agrees well with  numerical simulations  in large damping systems . \nMoreover, it will be shown that the depinning field gradually increases to a saturation value  \nwith increasing damping constant,  and this prediction allows one to modulate the damping \nconstant through elaborately material design , so that the domain wall depinning can be i n turn \neffectively control led. In order to understand the unde rlying physics better, we perform the \nanaly tical calculation based on the one-dimensional model  which reveals the important role of  \nthe internal domain wall dynamics . Our work also proposes a depinning mechanism for an \nAFM wall different from ferromagnetic wall. This new mechanism allows the depinning \nspeed to be typically three  order s of magnitude faster than that for a ferromagnetic wall \ndepinning . \nWe start from the domain wall pinning at a rectangular notch for an AFM nano ribbon . \nThis nanoribbon is geometrically defined by length l along the z-axis, width w, and thickness \ntl, as shown in Fig. 1. We discuss the scenario of current induced Néel spin -orbit torques (or \nstaggered effective field) , as demonstrated in CuMnAs and Mn 2Au for driving the domain \nwall motion , i.e. the wall is typically of the N éel type .6,8-9 For this scenario , the model \nHamiltonian is given by33-34 \n22 0\n02 2 2z\nzzA K AH L n hn          m n n m n\n,        (1) \nwhere A0 = 4JS2/a is the homogeneous exchange constant  with AFM coupling  J > 0, spin \nlength S and lattice constant a, m is the total magnetization m = (m1 + m2)/2S with m1 and m2 \nthe AFM sublattice magnetizations , A = 2aJS2 is the inhomogeneous exchange constant,  n is \nthe staggered magnetization n = (m1  m2)/2S, L0 = 2JS2 is the parity -breaking  parameter , Kz = 2K0S2/a is the anisotropy constant along  the z-axis in the continuum model  with anisotropy \nconstant K0 in the discrete model , γ is the gyromagnetic ratio,   = S/a is the density of the \nstaggered spin angular momentum per unit cell,  h is the staggered  effective field  and nz is the \nz component of  n. Here , the notch has its width  d and depth wN, as depicted  in Fig. 1 (a). \nNoting that  m is just a slave variable of n,33 and we eliminate m by m = L0n/A0 and \nobtain  \n*\n2\n22z\nzzK AH n hn       nn\n,            (2) \nwhere  A* = A  L2 \n0/A0 is the effective exchange constant.  As shown in the Supplementary \nMaterial s for the detailed derivation , the depinning field  hdep, based on this  Hamiltonian \nmodel, can be solved strictly  after a similar derivation26  \n02/\n2 / 1S\ndep\nNKhww\n,               (3) \nwhere S is the saturation moment. It’s noted that for an ultra-thin nano ribbon , the depinning \nfield is independent of  thickness.  As clearly indicated i n Eq.  (3), hdep depends on several \nparameters including the anisotropy constant K0 and the w/wN ratio. Thus, the devices with \nvarious depin ning field s could be designed through modulating ratio w/wN and/or choosing \nappropriate materials.  \nIn order to check the validity of Eq. (3), we also perform  the numerical simulation s based \non the atomistic Landau -Lifshitz -Gilbert  (LLG ) equation ,14 \n21i\ni i i it\n    SS H S H\n,          (4) \nwhere Si is the normalized atomic spin  at site i,  is the damping constant , Hi = μ-1 \nS∂H/∂Si is \nthe effective field.  Without loss of generality , l = 120 a, tl = a, w = 8a, K0 = 0.02 J, d = 4a, wN = \n2a and  = 0.02 are selected , as shown in Fig. 1 . \nFig. 1 presents  the spin structures of the nano ribbon  for various h. Here, the N éel-type \nAFM domain wall  is clearly pinned at the notch at h = 0 and the spin configuration  is \nsymmetric around the notch  due to the absence of chirality, as shown in Fig. 1(a). The spins \non the wall mid -plane are aligned in parallel to the x-axis and perpendicular to those spins inside the AFM domains aside .  \nWhen a small h is applied  along the z-axis, the wall slightly s hifts toward the right side , \nas seen from the delicate change of the spin configuration . With increasing h, those spins on \nthe left side of the notch mid -plane tend to rotate towards the negative z-axis while those on \nthe right side of the notch mid -plane tend to rotate towards the x-axis, as shown in Fig. 1(b) \nand 1(c) , a consequence of the wall depinning from the notch . The wall depinning becomes \nclear in Fig. 1(c) where the wall mid -plane deviates clearly from the notch mid -plane. The \nspin configuration after the full wall depinning from the notch is shown in Fig. 1(d).  \nSubsequently, we investigate the dependence s of hdep on the notch geometry and several \nphysical parameters including the anisotropy and damping constants. The calculated curves \n(analytical) from Eq. (3) plus the simulated results (numerical) based on the LLG dynamics, \nEq. (4), for differe nt valu es of notch depth wN, nanoribbon thickness w, anisotropy constant \nK0, and damping constant ( ) are plo tted in Fig. 2(a) ~ (d) respectively. Several features \ndeserve highlighting here. First, the model calculat ed curves  and numerical ly simulat ed data \non dependences hdep(wN), hdep(w), and hdep(K0) respectively show qualitatively similar \ntendencies, suggesting that Eq. (3) can describe roughly these dependences  although \nquantitative difference between the model and simulation appears for each dependence. \nSecond, qualitative difference between the model and simulation appears for function hdep(), \nas shown in Fig. 2(d). While the model suggests independence of hdep on damping constant , \nthe numerical simulation reveals that hdep is remarkably dependent of  in the small  regime. \nhdep shows a gradual growth with  until the large  regime where hdep becomes saturated, i.e. \nindependent of  in the large  regime. T he difference between Eq. (3) and simulat ed results \nfor hdep() is understandable since the LLG damping is a time -dependent effect. It is noted \nthat the internal dynamics of domain wall is completely neglected  in deriv ing Eq. (3), while \nthis dynamics becomes particularly remarkable in the small  regime where the \ntime-dependent spin oscillation can be significant  due to the weak damping . Therefore, the \nmodel prediction Eq. (3) becomes invalid and the underlying physics should be rec onsidered.  \nIn order to uncover the intriguing physics, we need to track the domain wall evolution. In \nproceeding, we first define the position of a domain wall. Similar to the well -studied \nskyrmions, the position of a domain wall  is estimated by  q(t)35 \n1\n1z\nzz n dxdz\nq\nn dxdz\n\n\n,               (5) \nwhere q is the coordinate of the wall mid -plane. Given this definition, one starts with the \none-dimensional model with inclusion of the internal dynamics of domain wall motion .28-29 \nThe Hamiltonian density for this model reads33 \n 2 22 0\n102 2 2z\nD z z z zA K AH L n hn V z          m n m n\n,      (6) \nwhere the pinning effect from the notch is described by  potential energy V(z).  \nSubsequently, we study  the Lagrangian density  L = K – H1D with K =  m∙(ṅ × n) is the \nkinetic energy term introduced by the Berry phase, and ṅ represents the derivative with \nrespect to time .33,36 -37 Then , we eliminate m with m = ( ṅ  n  L0∂zn)/A0,33 and obtain  \n 2*\n2 22\n02 2 2z\nz z zK AL n hn V zA       nn\n,         (7) \nIt is noted that the Rayleigh function density R =  ṅ2/2 is introduced into the \nLagrangian formalism  in order to describe  the dissipative dynamics .36-37 Following the earlier \nwork , we assume a robust domain wall structure which can be  described by n = [sech(( z  \nq)/)cosΦ, sech(( z  q)/)sinΦ, tanh(( z  q)/)],36 where the azimuthal angle Φ of the wall is  \nintroduced as the collective coordinates.  After substituting the domain wall  ansatz  and \napplying the Euler -Lagrange equation, we obtain the equation of motion for variables q and \nΦ, \n2\n00dq q hA dq     \n,             (8) \nand \n2\n00A \n,               (9) \nIt is noted that  the first term in Eq. (8) describes the wall inertia and other  terms represent \nthe forces exerted respectively by the damping  , pinning potential ε(q), and current -induced \neffective magnetic field h. By substituting the initial condition Φ(0) = d Φ/dt|t = 0 = 0 into Eq.  \n(9), one obtains Φ(t) = 0 , consistent with the fact that  an AFM domain wall is confined in the \neasy plane due to the antiparallel arrangement of neighboring spins.   For simplicity, we assume a parabolic potential23,29 \n\n2\n2/2\n/2N N\nNN NqL Kqq\nKL qL   \n,             (10) \nwhere KN is the elastic constant and LN is the radius  of the potential well. After substitutions \nand necessary simplification,  the equation of motion for q is updated to  \n20NN q Gq q h   \n,              (11) \nwhere  G = A0/, hN = γA0h/, and N = (A0KN/2)1/2 is the natural angular frequency of the \nfree harmonic oscillator. Here, we can see the existence of domain wall oscillation if damping \nconstant  is small. This oscillation is the major reason for the invalid prediction of the \ndepinning field by Eq. (3).  \nNoting that Eq. (11) describes the damping oscillation of a domain  wall, one has the \nsolution for  < c = 22aN/JA0 representing the under -damped oscillation :  \n 2\n12cos sin /Gt\np p N N q t e C t C t h     \n,         (12) \nwher e p = (2 \nN  G2/4)1/2 is the oscillating angular frequency of the wall, and C1, C2 are \nintegral constants depending on the initial condition.  \nFor better illustration, the simulated  q(t) curves  based on the LLG equation  at various \ndamping constant  are plotted in Fig. 3(a), benefiting to discussion . For  > 0.005,  one \nobserves the domain wall oscillat ion around the equilibrium position with an attenuat ing \namplitude.  Moreover,  the oscillation amplitude is enhanced with the decreasing . Finally , for \n < 0.005,  when the maximum displacement  of the wall oscillation, defined as |Δq|max = |q(t) - \nq(0)| max, exceeds the height of the  pinning  potential,29 the wall would successfully d epin from \nthe notch  and propagates freely along the nano ribbon . \nAs demonstrated  in Eq. (12), the displacement of the wall oscillation consists of  the \noscillatory  part (AS) and stationary part (qeq),29 and its maximum value is  approximately  given \nby \n 21 arctan / / 2 2 2\n12 max+/p G C C\nS eq N N q A q e C C h    \n,       (13) \nwhere p ≈ N is obtained for  < c. In this case,  since  |Δq|max decreases exponentially with \n, larger external field is required to generate  the wall displacement  for the wall depinning . \nAs |Δq|max > LN, the wall eventually depins  from the notch .  Noting  that the pinning potential parameters including  KN and LN are unknown , we need a \nreasonable estimation of them b y fitting the simulat ed results based on  Eq. (13). As shown in \nFig. 3(b)  where the simulated furthest position  of the domain wall, qmax, as a function of , is \nplotted. The excellent fitting of the simulated data by E q. (13) on the other hand  further \nconfirm s the validity of our theory.  \nSince the oscillating amplitude s C1 and C2 are proportional to  external or current induced \nfield h, one can introduce the field-independent parameters  c1 = C2/C1, c2 = (C2 \n1+C2 \n2)1/2/h for \nbrevity. Subsequently,  the depinning field under the condition |Δq|max = LN is obtained:  \n1 arctan /\n2 /pN\ndep Gc\nNLh\ne c K\n\n,             (14) \nSimilar fitting approach can be  used t o estimate  LN. As shown in Fig. 2(d) , the simulated \nresults coincide very well with Eq. (14) with one adjustable variable LN, demonstrating  the \nimportant role of the domain wall oscillation in the domain wall depinning. Such an \noscillation  behavior is one character of the internal dynamics  for a AFM nanoribbon with a \nnotch.  \nFinally, we would like to address the significance of the present results. It is known that  \nthe performance of domain wall based race-track memory not only depends on the wall \nmotion velocity, but also relies on the wall depinning  time. It is clearly shown here that an \nAFM domain wall depinning is distinctly different from that of a ferromagnetic domain wall. \nFor a ferromagnetic nanoribb on, the wall oscillation  is related to the wall internal angle  which  \nis mainly  determined  by the internal fields including  magnetocrystalline anisotropy an d \nDzyaloshinskii -Moriya  (DM) exchange .27 Generally, the depinning time is inversely \nproportional to the magnitude of internal fields  and has a typical value  of ~ 1.0 ns .22-24,27 \nHowever,  for an AFM system , the wall o scillation stem s from the second -order derivative of \nDW position q with respect to time rather than the azimuthal angle of the DW , as clearly \nillustrated in Eq. (8). Since the derivative originates from the strong AFM exchange \ninteraction between two sublattices  which is about three orders larger than the anisotropy  and \nDM exchange , one is sure that the depinning time  for such an AFM domain wall should be \nthree orders of magnitude shorter than a ferromagnetic one. It implies a surprisingly short \ndepinning time of ~ 0.001  ns for CuMnAs with  the N éel temperature  TN ≈ 480 K, a ≈ 3.8 Å and μS ≈ 3.6 μB,38 where μB is the Bohr magneton . While it is well believed that the AFM \ndomain switching is faster than ferromagnetic domain switching, the present work presents a \nquantitative estimation of the domain wall depinning time, direct evidence with this \nwell-believed but not yet  well-evidenced claim.  \nIn conclusion , we study theoretically the domain wall pinning and depinning at  a notch  in \nan AFM  nano -ribbon . The depinning field depending on the notch dimension and intrinsic \nphysical parameters are  derived theoretically  and also simulated  based on the LLG equation . \nContrary to the conventional conception , the remarkable dependence of the depinning field  on \nthe damping constant  is revealed , which  attributes to  the time -dependent oscillation  of the \nDW position in the small  damping region . A one-dimensional model  considering the internal \ndynamics of DW is investigated theoretically to explain perfectly the simulations . More \nimportantly,  our work also demonstrate s the different  depinning mechanism of an AFM DW \nfrom FM DW which may result  in a depinning speed t ypically three orders faster than the \nlatter , demonstrating  again  the ultrafast dynamics of an AFM system.  \n \nAcknowledgment  \nWe sincerely  appreciate  the insightful discussions with Zhengren Yan , Yilin Zhang  and \nHuaiyang Yuan. The work is supported by the National Key Projects for Basic Research of \nChina (Grant No. 2015CB921202), and the Natural Science Foundation of China (No. \n11204091), and the Science and Technology Planning Project of Guangdong Province (Grant \nNo. 2015B09092700 6), and the Natural Science Foundation of Guangdong Province (Grant \nNo. 2016A030308019).  References:  \n \n1. O. Gomonay, V . Baltz, A. Brataas , and Y . Tserkovnyak , Nat. Phys. 14, 213 (2018) . \n2. A. V . Kimel, B. A. Ivanov, R. V . Pisarev, P. A. Usachev, A. Kirilyuk , and Th. Rasing, Nat. \nPhys. 5, 727  (2009) . \n3. N. T. Kü hn, D . Schick, N . Pontius, C . Trabant, R . Mitzne r, K. Holldack, H . Zabel, A . \nFö hlisch,  and C . S. Langeheine , Phys. Rev. Lett.  119, 197202 (2017) . \n4. M. J. Grzybowski  et al., Phys. Rev. Lett.  118, 057701 (2017) . \n5. I. Fina, X. Mart í, D. Yi, J. Liu, J. H. Chu, C. R. Serrao, S. Suresha, A. B. Shick, J. \nŽelezn ý, T. Jungwirth, J. Fontcuberta, and R. Ramesh, Nat . 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Parkin, Science 315, \n1553 (2007) . \n25. M. Hayashi, L . Thomas, C . Rettner, R . Moriya, X . Jiang, and S . S. P. Parkin, Phys. Rev. \nLett. 97, 207205 (2006) . \n26. H. Y . Yuan and X. R. Wang, Phys. Rev.  B 89, 054423 (2014) . \n27. S. Moretti, M . V oto, and E . Martinez, Phys. Rev.  B 96, 054433 (2017) . \n28. E. Martinez, L. Lopez -Diaz, O. Alejos, L. Torres, and C. Tristan , Phys. Rev. Lett.  98, \n267202 (2007) . \n29. E. Martinez, L. Lopez -Diaz, O. Alejos, and L. Torres,  Phys. Rev. B 77, 144417 (2008) . \n30. D. Ravelosona, D. Lacour, J. A. Katine, B. D. Terris, and C. Chappert , Phys. Rev. Lett.  95, \n117203 (2005) . \n31. D. Bedau, M. Klä ui, M. T. Hua, S. Krzyk, U. Rüdiger, G. Faini , and L. Vila , Phys. Rev. \nLett. 101, 256602 (2008) . \n32. H. Y . Yuan and X. R. Wang, Phys. Rev.  B 92, 054419 (2015) . \n33. E. G. Tveten, T. Muller, J. Linder, and A. Brataas, Phys. Rev. B 93, 104408 (2016) . \n34. K. M. Pan, L. D. Xing,  H. Y . Yuan , and W. W.  Wang, Phys . Rev. B 97, 184418  (2018).  \n35. J. J. Liang, J. H. Yu, J. Chen, M. H. Qin, M. Zeng, X. B. Lu, X. S. Gao, and J. –M. Liu , \nNew J. Phys. 20 , 053037  (2018) . \n36. K.-J. Kim, S . K. Kim et al., Nat. Mater. 16, 1187 (2017) . \n37. S. K. Kim, Y . Tserkovnyak, and O . Tchernyshyov , Phys. Rev. B 90, 104406  (2014).  \n38. P. Wadley et al., Sci. Rep. 5, 17079 (2015).  \n \n  FIGURE CAPTIONS  \n \nFig.1 . (color online)  Equilibrium s pin structures  around the notch in the AFM nano ribbon  \nwith lattice sizes  l × w × tl under  (a) h = 0, (b) h = 0.002 J/μS, (c) h = 0.004 J/μS, and (d) h = \n0.004 58J/μS. The color represents the magnitude  of the z component of  the staggered \nmagnetization  nz, and the position of the DW center is depicted by the black dashed lines.   \n \nFig.2. (color online)  Numerical (empty circles) and analytical ( blue solid line) calculated  \ndepinning field as a function of (a)  the depth of the notch wN, (b) the width of the nano ribbon  \nw, (c) the anisotropy constant K0, and (d) the damping constant . The red solid  line in (d) is  \nthe fitting results based on  Eq. (14). \n \nFig.3. (color online)  (a) The DW position as a function of time  for various damping constants  \nunder h = 0.0039 J/μS. (b) Numerical (empty circles) and analytical (solid line) calculated  \nmaximum displacement of the DW as a function of  under h = 0.0039 J/μS. \n   \n \nFig.1. (color online) Equilibrium s pin structures around the notch in the AFM nano ribbon  \nwith lattice sizes  l × w × tl under  (a) h = 0, (b) h = 0.002 J/μS, (c) h = 0.004 J/μS, and (d) h = \n0.00458 J/μS. The color represents the magnitude of the z component of  the staggered \nmagnetization nz, and the position of the DW center is depicted by the black dashed lines.  \n  \n \nFig.2. (color online) Numerical (empty circles) and analytical (blue solid line) calculated  \ndepinning field as a function of (a) the depth of the notch wN, (b) the width of the nano ribbon  \nw, (c) the anisotropy constant K0, and (d) the damping constant . The red solid line in (d) is \nthe fitting results based on  Eq. (1 4). \n  \n \nFig.3. (color online) (a) The DW position as a function of time  for various damping constants \nunder h = 0.0039 J/μS. (b) Numerical (empty circles) and analytical (solid line) calculated  \nmaximum displacement of the DW as a function of  under h = 0.0039 J/μS. \n \n \nSupplementary m aterial  for  \n“Depinning of domain walls in notched antiferromagnet ic nano structures ” \n \nZ. Y. Chen1, M. H. Qin1,*, and J. –M. Liu2 \n1Institute for Advanced Materials, South China Academy of Advanced Optoelectronics and \nGuangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, \nSouth China Normal University, Guangzhou 510006, China  \n2Laboratory of Solid State Microstructures and Innovative Center for Adva nced \nMicrostructures, Nanjing University, Nanjing 210093, China  \n \nA. Derivation  of the  depinning field  \nThe model Hamiltonian density reads  \n22 0\n02 2 2z\nzzA K AH L n hn        m n n m n\n.                              (1) \nAfter eliminating m with m = L0n / A0, we obtain  \n*\n2\n22z\nzzK AH n hn       nn\n.                                           (2)  \nIn the following, we use the same method  with Ref. 24  to derive the depinning field for AFM \nDWs.  At low temperatures,  we introduce the Lagrange multiplier ξ to take into accou nt the \nconstraint condition n·n = 1, and then construct a new function  \n*\n12otAF dV f       n n n n\n,                                    (3)  \nwhere fot is the sum of the anisotropy and Zeeman energy.  Using the variational method, we \nobtain  \n2220ot\nex i i\nifl n nn    \n,                                                (4) \nwhere lex = (aA* / J)1/2 is the exchange length in AFM systems , ni is the xi component of n (xi \n                                                 \n*qinmh@scnu.edu.cn  = x，y，z). To eliminate ξ, we take the product of Eq. 4 and sum over i and obtain  \n220i ot\nex i\njjnflnxx   \n.                                                  (5) \nTransform ing Eq. 5 with the identity  \n2 2 1\n2j j jggg g gx x x          \n,                                      (6) \nand we obtain  \n221\n2i\nex i i ot ex i\njjnl n n f l nxx           \n.                                   (7) \nTo eliminate the space -dependent variables, we take the summation over the whole regions of \nthe sample Ω, \n2 2 21\n2ii\nex i i ot ex i ex i\nj j jnndV l n n f dVl n d l nx x x                    S\n,          (8) \nwhere ∂Ω is the boundary of Ω. Considering  the boundary condition ni = 0, we have  \n2211022ex i i ot ex i i otGH EF CD ABdV l n n f dxdy l n n fz                   \n.          (9) \nSubstitut ing the configuration of the system into Eq. 9 and we obtain  \n2122l ex i i ot lCD EFt dy l n n f ht w\n   \n.                                   (10) \nThen the magnitude of the current -induced effective field is given by  \n 21\n2\n2/l ex i i anCD EF\nl N N zEF zCDt dy l n n f\nht w w w n n  \n,                                       (11) \nwhere < nzEF>, <nzCD> are the average z components of n on surfaces EF and CD, respectively.   \nThe depinning field represents the minimum field to move a DW , and in other  words , the \nmaximum field that Eq. 11 has a stationary solution . Thus, critical condition is the key to \nderiv ing the depinning field.  Similar to  the earlier work , we consider the critical  condition \n<nzEF> = 0, < nzCD> = 1  in our derivation, whose validity is confi rmed in Fig. 1(c) in the \nmanuscript.  After substitutions and simplifications , we obtain the depinning field of AFM DWs  \n2/\n2 / 1zS\ndep\nNdhww\n.                                                     (12)  \n \n \n "    },    {        "title": "1905.10804v2.Influence_of_field_like_torque_in_synchronization_of_spin_torque_oscillators.pdf",        "content": "arXiv:1905.10804v2  [nlin.AO]  22 Jun 2020IEEE TRANSACTIONS ON MAGNETICS JOURNALS 1\nInfluence of field-like torque in synchronization of spin torque\noscillators\nR. Arun2, R. Gopal1V . K. Chandrasekar1, and M. Lakshmanan2\n1Centre for Nonlinear Science and Engineering, School of Ele ctrical and Electronics Engineering, SASTRA Deemed Univer sity,\nThanjavur - 613 401, India\n2Department of Nonlinear Dynamics, School of Physics, Bhara thidasan University, Tiruchirapalli - 620 024, India\nThe magnetization dynamics of two parallelly coupled spin t orque oscillators, destabilization of steady states and re moval of\nmultistability, are investigated by taking into account th e influence of field-like torque. It is shown that the existenc e of such torque\ncan cancel the effect of damping and can, therefore, cause th e oscillators to exhibit synchronized oscillations in resp onse to direct\ncurrent. Further, our results show that the presence of field -like torque enhances the power and Q-factor of the synchron ized\noscillations. The validity of the above results is confirmed by numerical and analytical studies based on the stochastic Landau-\nLifshitz-Gilbert-Slonczewski equation.\nIndex Terms —nonlinear dynamics,spintronics,synchronization\nI. I NTRODUCTION\nSynchronization phenomenon in spin torque oscillators\n(STOs) has been the subject of active research in recent year s\ndue to its potential applications to generate microwave pow er\nin nanoscale devices [1]–[5]. A number of significant effort s\nhave been made to study magnetization dynamics and synchro-\nnization of STOs driven by spin polarized current [6], injec tion\nlocking [7], external ac excitation [8], [9], spin waves [10 ],\nmagnetic fields [11]–[13], electrical couplings [14], [15] and\nthrough self-emitted microwave currents [16]. The synchro -\nnization of STOs greatly enhances the output microwave\npower when compared with the low output power of an\nindividual STO. Also it is more desirable for an enhancement\nof efficiency, quality factor and oscillation frequency of t he\npractical STO devices such as high density microwave sig-\nnal processors and chip-to-chip communication system [14] ,\n[17]–[20]. Moreover, synchronization of STOs has also been\nidentified in new applications such as wireless communicati on,\nbrain-inspired computing and microwave assisted magnetic\nreading [21]–[25]. In particular, it has been observed that\nan STO with the configuration of perpendicularly magnetized\nfree layer and in-plane magnetized pinned layer is suitable\nfor high emission of power, narrow line width and wide\nfrequency tunability [20], [26], [27]. The oscillation pro perties\nof this STO have also been investigated both experimentally\nand theoretically in Refs. [27], [28]. Further, the existen ce\nand stability of the synchronized state and the conditions t o\nsynchronize the individual precessions have also been stud ied\nin an array of Nserially connected identical STOs coupled\nthrough current has been demonstrated in Ref. [29]. Recentl y,\nthe mutual synchronization between two parallelly connect ed\nSTOs, coupled by current, has also been identified [30].\nIn this connection, some of the important issues in under-\nstanding the nonlinear dynamics of the system of coupled\nSTOs are the formation of steady states, multistability and\nthe decrease of frequency with respect to current. The oc-currence of steady states and multistable states prevents t he\nsystem to exhibit stable synchronized oscillations for all initial\nconditions. Removing these steady states and multistabili ty be-\nhaviour and making the system to exhibit stable synchronize d\noscillations for all initial conditions are important task s and\nhave not yet been fully clarified as far our understanding goe s.\nAlso, a decrease in the frequency of synchronized oscillati ons\nwhile increasing the current limits the enhancement of fre-\nquency beyond some specific value which is also a problem\nto be overcome with minimal efforts.\nIn this paper, we study the existence of steady states and\nmultistable states in the absence of field-like torque, thei r\nremoval and the mutual synchronization of the macrospin\ndynamics of a system of two parallelly coupled STOs in the\npresence of field-like torque [31]–[39]. By solving the asso ci-\nated stochastic Landau-Lifshitz-Gilbert-Slonczewski(s LLGS)\nequation with the configuration of perpendicularly magneti zed\nfree layer and in-plane magnetized pinned layer(as introdu ced\nin Sec.II), the analytical formula for the frequency of syn-\nchronized oscillations is derived in Sec.III. The existenc e of\nsteady states and multistable states is confirmed and the imp act\nof field-like torque on the STOs for various strengths of cou-\npling is observed. In the absence of field-like torque the two\nSTOs show the existence of steady states and synchronized\noscillations. The presence of field-like torque removes the\nsteady states and makes the system to oscillate with in-phas e\nsynchronization(Sec. III and Appendix). Further, the freq uency\nof synchronized oscillations is also enhanced in the presen ce\nof field-like torque. The onset of steady states in the absenc e of\nsuch a torque and the onset of stable synchronized oscillati ons\ndue to it are also analytically verified.\nII. M ODEL DESCRIPTION OF TWO PARALLELLY COUPLED\nSTO S\nWe consider a system that consists of two parallelly coupled\nspin torque oscillators. The schematic diagram of the syste mIEEE TRANSACTIONS ON MAGNETICS JOURNALS 2\nFig. 1. The schematic view of two parallely coupled spin torq ue oscillators.\nthat consists of two parallely coupled spin torque oscillat ors is\nshown in Fig.1. Each oscillator consists of a perpendicular ly\nmagnetized free layer, where the direction of magnetizatio n\nis allowed to change and an in-plane magnetized pinned\nlayer where the direction of magnetization is fixed along the\npositive x-direction. Both free and pinned layers are separ ated\nby a nonmagnetic conducting layer. The two free layers are\nlabeled as j= 1,2and the material parameters of the two\noscillators are kept identical for simplicity. The unit vec tor\nalong the direction of free layer’s magnetization is given b y\nmj= (mjx,mjy,mjz). The z axis is kept perpendicular to the\nplane of the free layer and ex,eyandezare unit vectors along\npositive x,y and z directions respectively. The unit vector along\nthe direction of magnetization of the pinned layers is given by\nP(=ex). The magnetization of the free layers (j= 1,2)is\ngoverned by the following sLLGS equation,\ndmj\ndt=−γmj×Heff,j+αmj×dmj\ndt\n+γHSjmj×(mj×P)+γβHSjmj×P, j= 1,2.(1)\nHereHeff,j is the effective field, given by Heff,j=\n[Ha+ (Hk−4πMs)mjz]ez+Hth,j, which includes ex-\nternally applied field Ha, crystalline anisotropy field Hk,\nshape anisotropy field (or demagnetizing field) 4πMsand the\nthermal noise given by [40]–[42]\nHth,j=√\nDGj, D=2αkBT\nγMsµ0V△t(2)\nIn the above, Gjis the Gaussian random number generator\nvector of the jthoscillator with components (Gjx,Gjy,Gjz),\nwhich satisfies the statistical properties < Gjm(t)>= 0 and\n< Gjm(t)Gjn(t′)>=δmnδ(t−t′)for allm,n=x,y,z .\nMsis the saturation magnetization, γis the gyromagnetic\nratio,αis the Gilbert damping parameter, βis the strength\nof the field-like torque, kBis the Boltzmann constant, Tis\nthe temperature, Vis the volume of the free layer, △tis the\nstep size of the time scale used in the simulation, µ0is the\nmagnetic permeability at free space and HSjis the strength\nof the spin-transfer torque, given by\nHSj=¯hηIj\n2eMsV(1+λmj.P). (3)In Eq.(3) ¯h=h/2π(h- Planck’s constant), eis the electron\ncharge,ηandλare dimensionless parameters which determine\nthe magnitude and the angular dependence of the spin transfe r\ntorque respectively. Ijis the total current passing through the\nfree layer which is given by [30]\nIj=I0+Icoupling\nj=I0+I0χ[mjx(t)−mj′x(t)],(4)\nwherej,j′= 1,2, j/negationslash=j′andIcoupling\nj is the current\ninjected from the free layer j′toj. In Eq.(4) I0is the current\nflowing through the free layer when there is no coupling\nbetween the oscillators. The second term in Eq.(4) describe s\nthe current flowing through the connection between the two\nSTOs and χis the coupling strength which characterizes the\nenergy loss in the connector. The oscillating electric curr ent\ngenerated by the STO is proportional to [2Vi/(RP+RAP)][1+\n△R(mj.P)/(RP+RAP)]as pointed out in [30], which\nimplies that the electric current generated by the oscillat or\ndepends upon the component of the free layer’s magnetizatio n\nalong the pinned layer’s magnetization direction. Here Viis\nthe external voltage, RPandRAP=RP+△Rare the\nresistances of the STO when the magnetization of the free\nlayer is parallel and antiparallel to the magnetization of t he\npinned layer, respectively.\nIII. E FFECT OF FIELD -LIKE TORQUE\nA. Destabilization of steady state due to aribitrary initia l\nconditions (covering both the hemispheres of magnetizatio n)\nby field-like torque\nTo understand the dynamics of the magnetization of the free\nlayer, Eq.(1) is numerically solved by Runge-Kutta 4th orde r\nstep-halving method for the material parameters [27], [28] ,\n[30]Ms= 1448.3emu/c.c., Hk= 18.6kOe,η= 0.54,λ\n=η2,γ= 17.64 Mrad/(Oe s), α= 0.005, µ0= 1 and V=\nπ×60×60×2nm3. Throughout our study HaandTare\nfixed as 2.0 kOe and 300 K respectively.\nTo study the dependance of the nature of the evolution of\nm1andm2on the initial conditions on the sphere formed by\nthe unit vector maround the origin, we have plotted the time\nevolution of m1x,m2xandm1zandm2zin Figs.2(a,c,e) and\n(b,d,f) respectively for I0= 5.0 mA and χ= 0.6. Figs.2(a)\nand (b) confirm the oscillations of m1andm2around the\npositive z-direction in the absence of field-like torque. Th e\ninitial conditions of the two STNOs have been chosen from\nthe northern hemisphere ( 0.99< m1z,m2z<1.00). The\nrandom fluctuations in Fig.2(b) is due to the thermal noise.\nNext, when the initial conditions of the two STNOs are taken\nfrom the two different hemispheres( 0.99< m1z<1.00,\n−0.99> m2z>−1.00), the system shows steady state\nmotion which we can observe from Figs.2(c) and (d) in the\nabsence of field-like torque. This is due to the fact that when\nthe two magnetization vectors evolve in the two different\nhemispheres the term I0χ[mjx(t)−mj′x(t)](see Eq.(4))\ncan become negative and consequently the current passing\nthrough the oscillators gets reduced. On the other hand,\nwhen field-like torque is additionally present(as shown in\nFigs.2(e) and (f) with χ=0.6), even with initial conditions taken\nfrom two different hemispheres, both the oscillators exhib itIEEE TRANSACTIONS ON MAGNETICS JOURNALS 3\nFig. 2. (Color online) Time evolution of m1x,m2x(a) andm1z,m2z(b) for the initial conditions from same hemispheres( 0.99< m1z,m2z<1.00).\nTime evolution of m1x,m2xandm1z,m2zwhenβ= 0(c,d) and β= 0.6(e,f) for the initial conditions from different hemisphere s(0.99< m1z<1.00,\n−0.99> m2z>−1.00). HereI0= 5.0 mA, T= 300 K and χ= 0.6. The inset figures in (a) and (e) show the synchronizatio n ofm1x(black solid line)\nandm2x(red solid circle). Similarly, the inset in (b) and (f) show t he synchronization of m1z(black solid line) and m2z(red solid line).\nsynchronized oscillations. The synchronization between t he\ntwo oscillators is shown in the insets of Figs.2(e) and (f).\nIn addition to the above, the LLGS equation with random\ntorque is solved for 200 trials in order to average the dynami cs.\nFor this purpose, we have also plotted the averaged values of\nmagnetization components < m1x>,< m 2x>,< m 1z>\nand< m2z>in Figs.3. Figs.3(a) and (b) show the averaged\ndynamics of the xandzcomponents of the magnetizations\nin the absence of field-like torque for the initial condition s\nfrom the same hemisphere. Due to the randomness of the\nphase, the average value of mxbecomes close to zero and\nthis clearly shows the significance of LLGS equation with\nthermal noise at finite temperature. The averaged dynamics\ncorresponding to steady state motion of the two oscillators\nfor the inital conditions from different hemispheres have b een\nplotted in Figs.3(c) and (d) in the absence of field-like torq ue.\nFurther, Figs.3(e) and (f) show the average dynamics of thesynchronized oscillations between the two oscillators due to\nthe presence of field-like torque corresponding to the initi al\nconditions similar to Figs.3(c) and (d). Thus when the initi al\nconditions are taken from different hemispheres, Figs.3(c ) and\n(d) imply that synchronized oscillations are not possible a nd\nonly steady states can exist in the absence of field-like torq ue,\nwhile Figs.3(e) and (f) confirm that synchronized oscillati ons\nindeed can be induced by the presence of field-like torque.\nB. Probability of synchronizations and steady state for ar-\nbitrary initial conditions\nThe dynamics of the coupled spin torque oscillators is more\ncomplicated than that of a single oscillator. In Appendix we\nshow that the dynamics of the two oscillators can be altered\nwhen there is a lack of simultaneity between the currents\npassing through the individual oscillators and the externa lIEEE TRANSACTIONS ON MAGNETICS JOURNALS 4\nFig. 3. (Color online) Averaged time evolution of m1x,m2x(a) andm1z,m2z(b) from 200 distinct initial conditions from same hemisphe res(0.99<\nm1z,m2z<1.00). Average time evolution of m1x,m2xandm1z,m2zwhenβ= 0(c,d) and β= 0.6(e,f) from 200 distinct initial conditions taken from\ndifferent hemispheres( 0.99< m1z<1.00,−0.99> m2z>−1.00). HereI0= 5.0 mA, T= 300 K and χ= 0.6.\nmagnetic field when they are switched off at different times\nwith even nanosecond differences.\nFrom the above studies we understand that there is a\ndefinite probability for the oscillators to reach steady sta tes\nin different hemispheres, and therefore it is essential to v erify\ntheir existence and the possibility of their removal by suit able\nmeans. Here by probability we mean only the possibility of\ninitial conditions reaching a particular final state(synch ronized\nstate/steady state) and we do not associate this with the\nprobability concept related to the randomness of the therma l\nfield. Hence, Eq.(1) is numerically solved for 100 numbers\nof randomly chosen initial conditions, chosen from both the\nhemispheres, and the corresponding probability to reach st eady\nstate (SS) and synchronized oscillation(SYN) state are com -\nputed. The values of SS and SYN are plotted against current in\nFigs. 4(a) and 4(b) for β= 0andβ= 0.61respectively, when\nχ= 0.5. Fig.4(a) shows that in the absence of field-like torque,\nthere is a nonzero probability of existence for both the stea dystate and synchronized oscillations beyond a critical curr ent\nstrength, whereas in the presence of positive field-like tor que\nthe system exhibits synchronized oscillations only, as sho wn in\nFig.4(b). Also we wish to point out here that by multistabili ty\nwe imply in this paper the possibility of the coexistence of\nsteady states and synchronized oscillatory states for arbi trary\nglobal initial conditions. In order to understand the impac t of\nfield-like torque, in Figs.4(c) and 4(d), we have depicted th e\nbifurcation diagrams of the system corresponding to Eq.(1)\nin the absence and presence of field-like torque respectivel y.\nIn the absence of field-like torque ( β= 0) the system shows\n(Fig.4(c)) multistability when the current I0exeeds the critical\ncurrentIc\n0. In the multistable region both the steady state and\nsynchronized oscillatory state are stable. Now by introduc ing\nthe field-like torque, we have plotted the bifurcation diagr am\nas a function of I0in Fig.4(d) for β= 0.61. It shows that the\nfield-like torque facilitates the emergence of stable synch ro-\nnized oscillatory state by destabilizing the steady-state throughIEEE TRANSACTIONS ON MAGNETICS JOURNALS 5\n 0 0.5 1\n02468(a)Probability\nI0(mA)SYN\nSS\n 0 0.5 1\n02468(b)Probability\nI0(mA)SYN\nSS\n-1-0.5 0 0.5 1\n 0 1 2 3 4 5 6 7 8χ = 0.5(c)\nHBm1x , m2x \nI0 (mA)-0.8-0.4 0 0.4 0.8\n 0 1 2 3 4 5 6 7 8χ = 0.5 (d)\nHBm1x , m2x \nI0 (mA)\nFig. 4. (Color online) Probabilities of synchronized oscil lations (PSOs) and\nsteady state (PSS) in the (a) absence ( β= 0) and (b) presence ( β= 0.61) of\nfield-like torque. The bifurcation diagrams of the system sp ecified by Eq.(1)\nare plotted in (c) the absence ( β= 0) and (d) presence ( β= 0.61) of field-\nlike torque. The red line( m1x) and black open circle( m2x) represent the\nmaxima(m1x,m2x>0) and minima( m1x,m2x<0) of the stable synchro-\nnized oscillatory state and the blue line( m1x) and the magenta square( m2x)\nindicate the stable steady state. ‘HB’ represents the Hopf b ifurcation point.\nThe other parameters are χ= 0.5 and T= 300 K.\nHopf bifurcation. By increasing the strength of the current , the\nexistence of the monostable synchronized oscillatory stat e can\nbe seen in Fig.4(d) for I0> Ic\n0.\nC. Removal of steady state by field-like torque\nTo analyze the impact of field-like torque on coupling\nstrength, we plot the SYN and SS for 100 randomly chosen\ninitial conditions for I0= 8mA. Figure 5(a) shows that in\nthe absence of field-like torque the probability fo SYN(SS)\nreduces(increases) from 1(0) when the coupling strength is\nincreased. This evidences that the system does not exhibit\nsynchronized oscillations for all initial conditions beyo nd\nsome critical value of coupling strength in the absence of fie ld-\nlike torque. From Fig.5(b) it is observed that the oscillato rs do\nnot get synchronized for all initial conditions in the absen ce\nof field-like torque. However, beyond certain critical valu e\nof positive field-like torque both the oscillators oscillat e syn-\nchronously for all initial conditions, which is confirmed fr om\nFig.5(b) where the SYN reaches 1 when the strength of field-\nlike torque is increased beyond the critical value ( βc= 0.33).\nWe have also depicted the bifurcation diagram with respect\ntoβforχ= 0.6 and I0= 8 mA in Fig.5(c). It is evident\nfrom the figure that the field-like torque term destabilizes t he\nsteady state and leads to only the synchronized oscillatory\nstate when β > β c. The magnetization trajectories of the\nsystem underlying Eq.(1) corresponding to β= 0 and 0.34\nare plotted as Figs.5(d) and 5(e) respectively. These figure s\nconfirm the existence of a stable steady state and the out-\nof-plane synchronized oscillatory state in the absence and\npresence of field-like torque respectively.\nFig. 5. (a) Probabilities of synchronized oscillations(re d) and steady state\n(black) against coupling strength in the absence of field-li ke torque at I0\n= 8 mA. (b) Probabilities of synchronized oscillations(red ) and steady state\n(black) against field-like torque at χ= 0.6,I0= 8 mA and T= 300 K.\nThe vertical lines correspond to the critical values χc= 0.29 and βc= 0.33\nobtained from Eq.(8) and Eq.(9) respectively. (c) The bifur cation diagram of\nthe system corresponding to Eq.(1) with χ= 0.6 and I0= 8 mA. The red\nline(m1x) and black open circle( m2x) represent the maxima and minima\nof the stable synchronized oscillatory state and the blue li ne(m1x) and the\nmagenta square( m2x) indicate the stable steady state. The magnetization\ntrajectories of the two oscillators are shown for (d) β= 0 and (e) β= 0.34.\nD. Steady states and critical values of and βandχfor\nsynchronized oscillations\nThe Eq.(1) can be transformed into spherical\npolar coordinates using the transformations mj=\n(sinθjcosφj,sinθjsinφj,cosθj)as follows:\n(1+α2)dθj\ndt=\n−2παFsinθj+√\nDGjx(αcosφjcosθj−sinφj)\n+√\nDGjy(αsinφjcosθj+cosφj)\n−γHSj[(α−β)sinφj+(1+αβ)cosθjcosφj], (5)\n(1+α2)sinθjdφj\ndt=\n2πFsinθj−√\nDGjx(αsinφj+cosθjcosφj)\n+√\nDGjy(αcosφj−cosθjsinφj)\n+γHSj[(1+αβ)sinφj−(α−β)cosθjcosφj], (6)\nwhereF= (γ/2π)[Ha+√\nDGz+(Hk−4πMs)cosθj].\nThe steady state solution of the system (1) is found around\nφ∗\n1=φ∗\n2≈3π/2, and\nθ∗\n1≈sin−1/parenleftbiggHS0\nHa+P/parenrightbigg\n, θ∗\n2≈π−sin−1/parenleftbiggHS0\nHa−P/parenrightbigg\n,\nwhereHS0= ¯hηI0/2eMsVandP=Hk−4πMs. Here, the\nthermal noise is not included for simplicity. From the linea r\nstability analysis, in the absence of field-like torque the s teady\nstate is found to be stable when [45]\n2/summationdisplay\ni=1/parenleftbigg∂fi\n∂θi+∂gi\n∂φi/parenrightbigg\nθ∗\n1,θ∗\n2,φ∗\n1,φ∗\n2<0. (7)\nHere,fiandgiare derived from Eqs.(5) and (6) as ˙θi=\nfi(θ1,θ2,φ1,φ2),˙φi=gi(θ1,θ2,φ1,φ2), i= 1,2.From the\ncondition (7), the critical value of coupling strength χcaboveIEEE TRANSACTIONS ON MAGNETICS JOURNALS 6\nwhich the system exhibits stable steady state solution in th e\nabsence of field-like torque ( β= 0), is derived as\nχc=λ+α\n2HS0[2PU−2Haτ−−Pτ+], (8)\nwhereτ±= (/radicalbig\n1−T+±/radicalbig\n1−T−),T±=H2\nS0/(Ha±P)2\nandU= (T++T−−1).\nHowever, in the presence of field-like torque, the critical\nvalue of βcabove which the steady state loses the stability,\nso that the synchronized state is the only stable state, can b e\nfound to be\nβc=αP[τ+−2U]−2HS0(λ−χ)+2Haατ−\n2HS0α(λ−χ)+Haτ−+Pτ+.(9)\nThe values of χcandβcmatch well with the numerical\nvalues, as confirmed by the vertical lines in Figs. 5(a,b).\nE. Stability of synchronized oscillations in the presence o f\nfield-like torque\nIn the absence of field-like torque and thermal noise the\nstability of the synchronized oscillations has already bee n\nconfirmed by Taniguchi et al [30]. However, here(Eq.(11)),\nin the presence of positive field-like torque and thermal noi se\nthe stability of the synchronized oscillations is confirmed by\nperturbing φ1asφ1=φ2+δφafter synchronization is reached,\nand the time evolution of δφis analysed over nperiods of\noscillations. By substituting φ1= 2πft+δφ(t), φ2= 2πft\nandθ1=θ2=θin Eq.(6) and after averaging over n\noscillations we can obtain [30]\n1\nnT/integraldisplaynT\n0dδφ\ndtdt≈ −χγHS0(1+αβ)\n(1+α2)nT/integraldisplaynT\n0δφ. (10)\nThe solution of Eq.(10) is given by\nδφ(t)≈δφ(0)exp/parenleftbigg\n−χγHS0(1+αβ)nT\n(1+α2)/parenrightbigg\n, (11)\nindicating the small deviation (δφ)betweenφ1andφ2expo-\nnentially decreases to zero as the number of oscillations( n)\nincreases. This implies that the presence of field-like torq ue\nand thermal noise do not affect the stability of the synchro-\nnized oscillatory state of the two parallelly coupled spin t orque\noscillators as long as (1 +αβ)>0. Further, from Eq.(11)\none may also note that when n→ ∞ the phase difference\nbetween oscillations of the two oscillators approaches zer o\ncorresponding to in-phase synchronized oscillations. Thi s has\nalso been verified numerically by using the algorithm given i n\nRef. [43].\nF . Frequency, power and Q-factor of synchronized oscilla-\ntions\nThe in-phase synchronization and its stability between the\ntwo oscillators in the presence of field-like torque have bee n\nconfirmed in Figs.2(c), 3(c) & 4(b) and Eq.(11) respectively .\nIn the synchronized state, the values of θ1andθ2are the\nsame and can be approximated to a constant value [30], [44]\nsince the amplitude of the oscillations of m1zandm2zare\nsmall as shown in Figs.2(d) and 3(d). Also, φ1= 2πft andφ1−φ2= 2nπ,n=0,±1,±2. . . . Here, fis the frequency of\nthe synchronized oscillations derived from Eq.(6) as\nf(θ) =/parenleftbigg1\n1+α2/parenrightbigg\n/bracketleftBigg\nF+(β−α)γ¯hηI0cosθ\n4πeMsVλsin2θ/parenleftBigg\n1−1/radicalbig\n1−λ2sin2θ/parenrightBigg/bracketrightBigg\n.(12)\nThe frequency and power spectral density(PSD) of the syn-\nchronized oscillations against current in the absence ( β= 0)\nand presence ( β= 0.61andβ=−0.61) of field-like torque\nhave been plotted in Figs. 6(a) and 6(b) respectively for χ=\n0.5. The solid line in Fig.6(a) corresponds to numerically c om-\nputed frequency and the open circles correspond to analytic ally\ncomputed frequency from Eq.(12). From Fig.6(a) it is observ ed\nthat the frequency of the synchronized oscillations is enha nced\nby positive field-like torque and decreased by negative field -\nlike torque. Fig. 6(a) shows that the frequency obtained fro m\nthe analytical expression(open circles) and numerical com pu-\ntation(solid lines) matches well and this evidently sugges ts the\nvalidity of the analytical results. The small deviation app earing\nin the frequency for positive field-like torque at about 3 mA\nis due to the drop in the mean value of θjaround 3 mA.\n 4 5 6 7\n2 4 6 7.5(a)\nβ = 0.61\nβ = 0\nβ = −0.61f (GHz)\nI0 (mA)04812\n 6  6.2  6.4  6.6  6.8β = 0β = -0.61\nβ = 0.61(b)PSD (arb. unit)\nf (GHz)\nFig. 6. (Color online) (a) The frequency of synchronized osc illations in the\nabsence and presence of field-like torque when χ= 0.5 and T= 300 K.\nThe solid line and open circle correspond to the frequency co mputed by\nnumerical and analytical (Eq.(12)) calculations, respect ively. (b) The power\nspectral density of the oscillations in the absence ( β= 0 ) and presence\n(β=0.61 and -0.61) of field-like torque when I0= 2.0 mA, T= 300 K and\nχ= 0.5.\nIn order to elucidate the experimental consequences of\nenhancement of the frequency and power of synchronized\noscillations due to field-like torque, we have plotted the\nspectral power in the frequency domain in Fig.6(b) for β= 0,\nβ= 0.61andβ=−0.61whenI0= 2.0 mA, χ= 0.5 and\nT= 300 K. It is evident from Fig.6(b) that the frequency\nof the synchronized oscillations is enhanced by the positiv e\nfield-like torque. Also, the power and Q-factor are enhanced\nby negative field-like torque. For instance the frequency is\nincreased by 0.241 GHz when βis increased from 0 to 0.61.\nThe power is enhanced by more than 2.5 times when βis\nnegatively increased from 0 to -0.61 along with the incremen t\nof Q-factor from 447.51( β= 0) to 672.61 ( β= -0.61). On the\nother hand the power is decreased by increasing the value of\nβfrom 0 to 0.61 with a slight decrement in Q-factor from\n447.51(β= 0) to 411.33 ( β= 0.61). Thus, the negative field-\nlike torque enhances the power with large increment in Q-\nfactor and positive field-like torque increases the frequen cy\nwith slight decrement in Q-factor.IEEE TRANSACTIONS ON MAGNETICS JOURNALS 7\nFig. 7. (Color online) Time evolution of m1x,m2x(a & c) and m1z,m2z(b & d) for β=0 (a & b) and β=0.2 (c & d) when the currents passing through\nthe first, second oscillators and the applied field are instan taneously switched off at 500 ns and switched on at 1500 ns. He reI0= 2.0 mA, T= 300 K and\nχ= 0.5. The inset figures show the synchronization of m1x(black solid circle) and m2x(red solid line).\nIV. C ONCLUSION\nIn conclusion, the existence of steady state and its removal\nin the system of two parallelly coupled spin torque oscillat ors\nby field-like torque has been investigated theoretically, w ith\na physical configuration of perpendicularly magnetized fre e-\nlayer and in-plane magnetized pinned layer. The numerical\nsimulation of the LLGS equation has revealed that the exis-\ntence of field-like torque can cancel out the damping effect\nand thus can induce synchronized oscillations with respect to\napplied current. One can also note that the existence of stea dy\nstate behavior in coupled STOs can be efficiently removed\nby introducing the field-like torque. The frequency of the\nsynchronized oscillations gets enhanced by positive field- like\ntorque. Also, the power and Q-factor are enhanced by the\nnegative-field like torque.\nV. A PPENDIX\nA. Destabilization of steady staets due to small time delays\nin switching of current and field\nIn this appendix we wish to point out even when the initial\nconditions are chosen in the same hemisphere, multistable\nstates can arise due to nanoscale level time delays in switch ing\noff the current and field. Investigations on pulse fields by\nKikuchiet al. [46] and Flovik et al. [47] suggest that the out-\nof-plane magnetic field can be produced on magnetic layers fo rthe duration of nano and picco second by nonsized coil using\ncurrent or by laser pulses through inverse Faraday effect. A s\nan example, in this Appendix, we consider a situation where\ninitially the currents to the first and second oscillators ar e\nswitched on at τon\nI1,1andτon\nI2,1, respectively, and the field at\nτon\nHa,1. After the oscillators attain synchronized oscillations, the\ncurrents and field are switched off at τoff\nI1,1,τoff\nI2,1andτoff\nHa,1.\nAfter some time they are again switched on at τon\nI1,2,τon\nI2,2and\nτon\nHa,2, respectively. Figs.7 & 8 show the time evolution of mx\nandmzcomponents of the two oscillators in the presence\nof thermal noise field for the initial conditions chosen for\n0.99< m1z,m2z<1.00, whenI0= 2.0 mA, χ= 0.5. To\nconfirm the synchronized oscillations, the m1xandm2xare\nplotted as inset figures for small time window. Figs.7(a,b) &\n(c,d) have been plotted for β=0 andβ=0.2, respectively, when\nτon\nI1,1=τon\nI2,1=τon\nHa,1= 0s,τoff\nI1,1=τoff\nI2,1=τoff\nHa,1= 500ns\nandτon\nI1,2=τon\nI2,2=τon\nHa,2= 1500ns. Figs.7(a) & (c) show\nthat irrespective of whether the field-like torque is presen t\nor not, both the oscillators reach steady state after 500 ns\nand regain synchronized oscillations after 1500 ns. The fina l\nsynchronized oscillations are similar as in Ref. [30] for β= 0.\nTo show the impact of field-like torque on retrieving the\nmagnetizations from steady states to synchronized oscilla tions\nFigs.8 are plotted for τon\nI1,1=τon\nI2,1=τon\nHa,1= 0s,\nτoff\nI1,1= 504ns, τoff\nI2,1= 500ns, τoff\nHa,1= 496nsandIEEE TRANSACTIONS ON MAGNETICS JOURNALS 8\nFig. 8. (Color online) Time evolution of m1x,m2x(a & c) and m1z,m2z(b & d) for β=0 (a & b) and β=0.2 (c & d) when the currents passing through\nthe first oscillator, second oscillator and applied field are cut off at 504 ns, 500 ns and 496 ns respectively and switched o n simultaneously at 1500 ns. Here\nI0= 2.0 mA, T= 300 K and χ= 0.5. The inset figures show the synchronization of m1x(black solid circle) and m2x(red solid line).\nτon\nI1,2=τon\nI2,2=τon\nHa,2= 1500 ns. It is observed that in\nthe absence of field-like torque, the oscillations of the two\noscillators damp out after 500 ns to the steady states at\ndifferent hemispheres, formed by the unit vector maround the\norigin, and continue in the same steady states even after the\ncurrents and field are applied at 1500 ns as shown in Figs.8(a)\nand (b). For the present case, m1zandm2zreach steady states\nat north and south poles respectively. Occasionally, the th ermal\nnoise leads both the oscillators to steady states at norther n\nhemisphere and they exhibit synchronized oscillations aft er the\ncurrents and field are switched on at 1500 ns. This is shown in\nFig.9, where we can observe that in the presence of thermal\nnoisem2zreturns to north pole after 500 ns and oscillates\nafter 1500 ns. In the absence of thermal noise m2zmoves\nto the steady state at south pole after 500 ns and continues\nthere even after the currents and field are switched on at 1500\nns. On the other hand in the presence of positive field-like\ntorque, both the oscillators always attain the steady state at the\nnorthern hemisphere after the currents and field are switche d\noff around 500 ns and reach synchronized oscillations after\nthe currents and field are switched on at 1500 ns as shown in\nFigs.8(c) and (d). From Figs.7(a) & 8(a) it is understood tha t\nthe lack of simultaneity in switching off the currents and fie ld\ntransforms the system from getting synchronized oscillato ry\nstate to steady state. In realistic applications the couple d\noscillators may be switched off and on many times. Every time\nthe system is switched off, the currents passing through theindividual oscillators might be cut off at slightly differn t times\nwith at least few nanosecond differences between them due to\nvarious disturbances or at the same time. In these situation s,\nthe system of coupled oscillators may exhibit synchronized\noscillations or steady state motion as shown in Figs.7(a) &\n8(a) respectively. However, the presence of field-like torq ue\ndestabilizes the steady state at the southern hemisphere an d\nmakes the magnetization vectors of the two oscillators to\nstay in the northern hemisphere and exhibit synchronized\noscillations after the currents and field are switched on as\nconfirmed in Figs.8(c) & (d). It has also been verified that\nthe positive field-like torque destabilizes the steady stat e and\nmakes synchronized oscillations even when τon\nI1,1,τon\nI2,1,τon\nHa,1\nandτon\nI1,2,τon\nI2,2,τon\nHa,2differ by nanoseconds. From Figs.8(b) &\n(d) it is also verified that the system reaches steady state wh en\nthe magnetization vectors evolve in opposite hemispheres a nd\nthat the thermal noise has no impact on it.\nTo prove the strong destabilization of steady states by\nfield-like torque the average values of the zcomponents of\nmagnetizations are plotted in Figs.10(a) and (b) from 100 tr ials\nin the absence and presence of field-like torque respectivel y.\nFrom Fig.10(a) it can be understood that when the field\nand currents are switched off at 500 ns with nanoscale time\ndifference between them, some of the magnetizations of the\nfirst and second oscillators are settled near north pole of th e\nsphere and the remaining magnetizations of the two oscillat ors\nsettle near south pole. When the currents and field are switch edIEEE TRANSACTIONS ON MAGNETICS JOURNALS 9\nFig. 9. (Color online) Time evolution of m2zwith(blue) and without(black)\nthermal noise in the absence of field-like torque when I0= 2.0 mA, T= 300\nK andχ= 0.5.\nFig. 10. (Color online) Averaged time evolution of m1z,m2zwhenβ= 0\n(a) andβ= 0.2(b) from 100 trials for the same initial conditon taken for\nthe Figs.8(b) and (d) when the currents passing through the fi rst oscillator,\nsecond oscillator and applied field are cut off at 504 ns, 500 n s and 496 ns\nrespectively and switched on simultaneously at 1500 ns.. He reI0= 2.0 mA,\nT= 300 K and χ= 0.5.\non simultaneously at time 1500 ns, the values of < m1z>\nand< m2z>slightly increase from their corresponding\nvalues between 500 ns and 1500 ns. This is due to the fact\nthat the thermal fluctuations occasionally drive both of the\nmagnetizations into the northern hemisphere and make them\nto oscillate synchronously after 1500 ns (Refer Fig.9). Hen ce,\nfew of the cases from out of the 100 trials make synchronized\noscillations after 1500 ns which tend to increase the values of\n< m1z>and< m2z>after 1500 ns. On the other hand when\nthe field-like torque is present the magnetizations of the tw o\noscillators are driven into the northern hemisphere and exh ibit\nsynchronized oscillations for all the 100 trials as shown in\nFig.10(b). Also, we checked that a negative field-like torqu e\ndoes not produce synchronized oscillations when the curren ts\nand field are switched on again after switching off at differe nt\ntimes.\nWe also wish to point out that two things can happen in\nthe absence of field-like torque as seen from Figs.7 and 8.\nFirst, due to the lack of simultaneity in switching off/on th e\ncurrents passing through the individual oscillators and fie ld the\nmagnetizations of the two oscillators are driven into the st eady\nstates near the poles at opposite hemispheres(occasionall y the\nmagnetizations are kept in the northern hemisphere due to\nthermal flucutation) formed by mand continue there even\nafter the currents and field are switched on again. Second, if\nthe magnetizations are settled in the steady states at diffe rent\nhemispheres, the synchronized oscillations are not possib le by\napplying field Haand currents I1andI2. When the field-\nlike torque is additionally present, the magnetizations ar e keptin the northern hemispheres only and avoid steady states at\nopposite hemisperes. Also, even if the magnetizations are\nin steady states at opposite hemispheres the synchronized\noscillations can be induced by the field-like torque.\nACKNOWLEDGEMENTS\nThe work of V .K.C. forms part of a research project\nsponsored by CSIR Project No. 03/1444/18/EMR II. M.L.\nwishes to thank the Department of Science and Technology\nfor the award of a SERB Distinguished Fellowship under\nGrant No.SB/DF/04/2017 in which R. Arun is supported by\na Research Associateship.\nREFERENCES\n[1] J. Grollier, V . Cros, and A. Fert, Phys. Rev. B 73, 060409(R) (2006).\n[2] S. Urazhdin, P. Tabor, V . Tiberkevich and A. Slavin, Phys . Rev. Lett.\n105, 104101 (2010).\n[3] F.B. Mancoff, N. D. Rizzo, B. N. Engel and S. Tehrani, Natu re437,\n393 (2005).\n[4] S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russ ek, and J.\nA. 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Onbaşlı1,2,* \n \n1 Graduate School of Materials Science and Engineering, Koç University , Sarıyer, 34450 Istanbul, \nTurkey . \n2 Department of Electrical and Electronics Engineering, Koç University, Sarıyer, 34450 Istanbul, \nTurkey.  \n* Corresponding Author: monbasli@ku.edu.tr  \n \nAbstract:  \nPerpendicular magnetic anisotropy (PMA) is a necessary condition for many spintronic \napplications like spin -orbit  torques  switching , logic and memory devices. An important class of \nmagnetic insulators with low Gilbert damping at room temperature are iron garnets, which only \nhave a few PMA types such as terbium  and samarium iron garnet. More and stable PMA garnet \noptions are necessary for researchers to be able to investigate new spintronic phenomena. In this \nstudy, we predict 20 new substrate/magnetic iron garnet film pairs with stable PMA at room \ntemperature. The effective anisotropy energies of 10 different garnet films that are lattice -matched \nto 5 different commercially available garnet substrates have been calculated using shape, \nmagnetoelastic and magnetocrystalline anisotropy  terms . Strain type, tensile o r compressive \ndepending on substrate choice, as well as the sign and the magnitude of the magnetostriction \nconstants of garnets determine if a garnet film may possess PMA. We show the conditions in which \nSamarium, Gadolinium, Terbium, Holmium, Dysprosium a nd Thulium garnets may possess PMA \non the investigated garnet substrate types . Guidelines for obtaining garnet films with low damping \nare presented. New PMA garnet film s with tunable saturation moment and field may improve spin -\norbit torque memory and compensated magnonic thin film devices.  \n \n \n \n \n 2 \n Introduction  \nWith the development of sputtering and pulsed laser deposition of high -quality iron garnet thin \nfilms with ultralow Gilbert damping, researchers have been able t o investigate a wide variety of \nmagnetization switching and spin wave phenomena1-3. The key enabler in many of these studies \nhas been Yttrium iron garnet (Y 3Fe5O12, YIG)4 which has a very low Gilbert damping allowing \nspin waves to propagate over multiple millimeters across chip. YIG thin films are useful for spin \nwave device applications, but since their easy axes lie along film plane, their utility cannot b e \nextended to different mechanisms such as spin -orbit torques, Rashba -Edelstein effect, logic \ndevices, forward volume magnetostatic spin waves1. At the same time, to have reliable and fast \nresponse using low current densities as in spin -orbit torque switching, magnetization orientation \nneeds to be perpendicular to the surface plane5. The possibility of having Dzyaloshinskii –Moriya \ninteraction (DMI) in TmIG/GGG may enable stabilizing skyrmions and help drive skyrmion \nmotion with pure spin currents6. \n \nThere is a number of studies on tuning anisotropy or obtaining perpendicular magnetic anisotropy \nin insulator thin films7-12. Among the materials studied, insulating magnetic garnet s whose \nmagnetic pr operties can be tuned have been a matter of interest over the past decades13-15 due to \ntheir low damping and high magnetooptical Faraday rotation. In order to obtain perpendicular \nmagnetic anisotropy in magnetic garnets, one needs to engineer the anisotropy terms that give rise \nto out -of-plane easy axis. Angular dependence of total m agnetization energy density is called \nmagnetic anisotropy energy and consists of contributions from shape anisotropy, strain -induced \n(magnetoelastic)  and magnetocrystalline anisotropy. A magnetic material preferentially relaxes its \nmagnetization vector tow ards its easy axis, which is the least energy axis , when there is no external \nfield bias . Such energy minimization process drives magnetic switching rates  as well as the stability \nof total magnetization v ector . Controlling magnetic anisotropy in thin film garnets not only offers \nresearchers different testbeds for experimenting new PMA -based switching phenomena, but also \nallows  the investigation of anisotropy -driven ultrafast dynamic magnetic response in thin film s and \nnanostructures.  \nThe most extensively studied garnet thin film is Yttrium Iron Garnet ( YIG). YIG films display in -\nplane easy axis because of their large shape anisotropy  and negligible magnetocrystalline \nanisotropy3. Although PMA  of ultrathin epitaxial YIG films has been reported16,17, the tolerance 3 \n for fabrication condition variations for PMA YIG is very limited and strain effects were found to \nchange magnetocrystalline anisotropy in YIG. Strain -controlled anisotr opy has been observed in \npolycrystalline ultrathin YIG films17,18. In case of YIG thin film grown on Gadolinium Gallium \nGarnet (GGG), only partial anisotropy control has been possible through significant change in \noxygen stoichiometry19, which increases damping. Since the fabrication of high -quality and highly \nPMA YIG films  is not easy for practical thicknesses on gadolinium gallium garnet substrates \n(GGG), researchers have explored tuning magnetic anisotropy by substituting Yttrium sites with \nother rare earth elements20,21. New garnet thin films that can exhibit PMA with different \ncoercivities, saturation fields, compensation points and tunable Gilbert damping values must be \ndeveloped  in order to evaluate the effect of these p arameters on optimized spintronic insulator \ndevices . \nSince the dominant anisotropy energy term is s hape anisotropy  in thin film YIG , some studies focus \non reducing the shape anisotropy contribution by micro and nanopatterning22-24. Continuous YIG \nfilms were etched  to form rectangular nanostrips with nanometer -scale thickness es, as \nschematic ally shown on Fig. 1(a) . Thus, least magnetic saturation field is needed along the longest \ndimension of YIG nanostrips . By growing ultrathin YIG, magnetoelastic strain contributions lead \nto a negative anisotropy field and thus PMA in YIG film s24. As the length -to-thickness ratio \ndecreases , the effect of shape anisotropy is reduced and in-plane easy axis is converted to PMA17.  \nWhile reducing the effect of shape anisotropy is necessary, one also needs to use m agnetoelastic \nanisotropy  contribut ion to reorient magnetic easy axis towards out of film plane , as schematically \nshown on Fig. 1 (b). Strain-induced perpendicular magnetic anisotropy in rare earth (RE) iron \ngarnets, especially in YIG, has been demonstrated to overcome shape anisotropy16,17,25,26. If \nmagnetoelastic anisotropy term induced by crystal lattice mismatch is large r than shape anisotropy \nand has opposite sign, then magnetoelastic anisotropy overcome s shape . Thus, the easy axis of the \nfilm becomes perpendicular to the film plane and the hyster esis loop becomes square -shaped with \nlow saturation field8. One can also achieve PMA in other RE magnetic iron garnets due to their \nlattice parameter  mismatch with their substrates . PMA has previously been achieved using \nSubstituted Gadolinium Gallium Garnet (SGGG) as substrate and a Samarium Gallium Garnet \n(SmGG) ultrathin film as buffer layer under (and on) YIG16. In case of thicker YIG films (40nm), \nthe magnetic easy axis becomes in -plane again. An important case shown by Kubota et.al19 \nindicates that increasing in-plane strain (ε ||) or anisotropy field (H a) helps achieve perpendicular 4 \n magnetic anisotropy. In ref.8,19, they reported that if magnetostriction coefficient (λ 111) is negative \nand large eno ugh to overcome shape anisotropy, and tensile strain is introduced to the thin film \nsample (ε ||>0), the easy axis becomes perpendicular to the sample plane  as in the case of Thul ium \niron garnet (Tm 3Fe5O12, TmIG ). \nA different form of magnetoelastic anisotropy effect can be induced by using porosity in garnet \nthin films. Mesoporous  Holmium Iron Garnet (Ho 3Fe5O12, HoIG) thin film on Si (001)27 exhibit s \nPMA due to reduced shape anisotropy, increased magnetostrictive and growth -induced anisotropy \neffects. Such combined effects lead to PMA in HoIG. In this porous thin film, the PMA was found \nto be independent of the substrate used, because the mechanical s tress does not result from a lattice \nor thermal expansion mismatch between the substrate and the film. Instead, the pore -solid \narchitecture itself imposes an intrinsic strain on the solution processed garnet film.  This example \nindicates that the film struc ture can be engineered in addition to the substrate choices in order to \novercome shape anisotropy in thin film iron garnets.  \nAnother key method for controlling anisotropy is strain doping through substitutional elements and \nusing their growth -induced aniso tropy effects , as schematically shown on Fig. 1(c) . Bi-doped \nyttrium iron garnet (Bi:YIG and Bi:GdIG) has been reported to possess perpendicular magnetic \nanisotropy due to the chemical composition change as the result of increased annealing \ntemperature21,28. Another reason for PMA in these thin films is strain from GGG substrate29. \nDoping of oxides by Helium implantation was shown to reversibly and locally tune magnetic \nanisotropy30. For TbxY3-xFe5O12 (x=2.5, 2.0, 1.0, 0.37 ) samples  grown by spontaneous nucleation \ntechnique31, magnetic easy axis was found to change from [111] to [100] direction as Tb \nconcentration was decreased. The first -order anisotropy constant K 1 undergoes a change of sign \nnear 190K . Another temperature -dependent lattice distortion effect that caused anisotropy chang e \nwas also reported for YIG films32. These results indicate  that temperature also plays an important \nrole in both magnetic compensation, lattice distortion and change in anisotropy.   \nIn this study, we systematically calculate the anisotropy  energ ies of  10 different types of lattice -\nmatched iron garnet compounds epitaxially -grown as thin films (X 3Fe5O12, X = Y, Tm, Dy, Ho, \nEr, Yb, Tb, Gd, Sm, Eu) on commercial ly available (111) -oriented garnet substrates (Gd 3Ga5O12-\nGGG , Y 3Al5O12-YAG , Gd 3Sc2Ga3O12 -SGGG, Tb 3Ga5O12 –TGG, Nd 3Ga5O12 -NGG).  Out of the \n50 different film/substrate pairs, we found that 20 cases are candidates for room temperature PMA. \nOut of these 20 cases, 7 film/substrate pairs were experimentally tested and shown to exhibit 5 \n characteristics originating from PMA. The remaining 13 pairs, to the best of our knowledge, have \nnot been tested for PMA experimentally.  We indicate through systematic anisotropy calculations \nthat large strain -induced magnetic anisotropy terms may overcome shape when the films are highly \nstrained . We use only the room temperature values of λ 11133 and only report predictions for room \ntemperature (300K) . Throughout the rest of this study, the films are labelled as XIG (X = Y, Tm, \nDy, Ho , Er, Yb, Tb, Gd, Sm, Eu), i.e. TbIG (Terbium iron garnet) or SmIG (Samarium iron garnet) \netc. to distinguish them based on the rare earth element.  Our model could accurately predict the \nmagnetic easy axis in almost all experimentally tested garnet film/substrate cases provided that the \nactual film properties are entered in the model and that the experimental film properties satisfy \ncubic lattice mat ching condition to the substrate.  6 \n  \nFigure 1.  Methods to achieve perpendicular magnetic anisotropy in iron garnet thin films. \n(a) Micro/nano -patterning reduces shape anisotropy and magnetoelastic anisotropies overcome \nshape.  (b) Large strain -induced anisotropy must over come  shape anisotropy to yield out -of-plane \neasy axis.  (c) Substitutional doping in garnets overcome shape anisotropy by enhancing \nmagnetocrystalline, growth -induced or magnetoelastic terms.   \n \nAnisotropy energy den sity calculation s \nTotal anisotropy energy density contains three main contributions; according the Equation 1, shape \nanisotropy ( Kshape ), first order cubic magnetocrystalline anisotropy ( K1), and strain -induced  \n7 \n (magnetoelastic)  anisotropy ( Kindu) parameters determine the total effective anisotropy energy \ndensity16. \nKeff=Kindu +Kshape +K1   (1) \nIn case of garnet film magnetized along [111] direction  (i.e. on a 111 substrate) , the magneto -elastic \nanisotropy energy density, resulting from magneto -elastic coupling is calculated by Equation 2: \nKindu =−3\n2λ111σ||      (2) \nwhere λ111 is magnetostriction constant along [111] direction  and it is usually  negative  at room \ntemperature34. In Eqn. 2,  σ|| is the in -plane  stress induced in the material  as a result of lattice \nmismatch between film and the substrate , and the in -plane stress is calculated from Equation 335: \nσ||=Y\n1−νε||      (3) \nwhere Y is elastic mo dulus, and ν is P oisson’s ratio36. \nFor calculation of in -plane strain, lattice parameter values obtaine d from the XRD characterization \nof the thin films are used . Equation 4  shows the strain relation as the lattice constant difference \nbetween the bulk form of the film and that of the substrate divided by the lattice constant  for the \nbulk form of the film16. \nε||=afilm −abulk\nafilm       (4) \nAssuming the lattice parameter of the thin film matches with that of the substrate, the lattice \nconstant of substrate can be used as the lattice constant of thin film for  calculation of strain in \nEquation 537: \nε||=asub−afilm\nafilm       (5) \nThe lattice constants used for the films and substrates examined for this study are presented on \nTable 1.  \nShape anisotropy energy density depends  on the geometry  and the intrinsic saturation magnetic \nmoment of the iron garnet material. Shape anisotropy has a demagnetizing effect on the total 8 \n anisotropy energy density. These significant anisotropy effects can be observed in magnetic \nhysteresis loop s and FMR measurements24.  \nThe most common anisotropies in magnetic materials are shape anisotropy and magneto -crystalline \nanisotropy38,39. Considering that the film is continuous, the shape anisotropy is calculated as16 \nKshape =2πMs2      (6) \nBy obtaining the values of M s for rare earth iron garnets as a function of temperature40,41, the value \nfor shape anisotropy energy density have been calculated using Equation 6.   \nIntrinsic magnetic anisotropy42, so called magnetocrystalline anisotropy, has the weakest \ncontribution to anisotropy energy densit y compared  to shape, and strain -induced \nanisotropies9,11,16,19. The values for the first order magnetocrystalline anisotropy is calculated and \nreported previously for rare earth iron garnets at different temperatures43. A key consideration in \nmagnetic thin films is saturation field. In anisotropic magnetic thin films, the anisotropy fiel ds have \nalso been calculated using equation 7 as a measure of how much field the films need for magnetic \nsaturation along the easy axis:  \nHA=2Keff\nMs       (7) \nTable 1. List of magnetic iron garnet thin films and garnet substrates available off -the-shelf used \nfor this study. The fourth column shows the lattice constants used for calculating the magnetoelastic \nanisotropy values of epitaxial garnets on the given substrates.  \nGarnet \nmaterial  Chemical \nformula  Purpose  Bulk \nlattice \nconstant  \n(Å) \nGGG  Gd3Ga5O12 Substrate  12.383  \nYAG  Y3Al5O12 Substrate  12.005  \nSGGG  Gd3Sc2Ga3O12 Substrate  12.480  \nTGG  Tb3Ga5O12 Substrate  12.355  \nNGG  Nd3Ga5O12 Substrate  12.520  \nYIG Y3Fe5O12 Film  12.376  \nTmIG  Tm 3Fe5O12 Film  12.324  \nDyIG  Dy3Fe5O12 Film  12.440  \nHoIG  Ho3Fe5O12 Film  12.400  \nErIG  Er3Fe5O12 Film  12.350  \nYbIG  Yb3Fe5O12 Film  12.300  9 \n TbIG  Tb3Fe5O12 Film  12.460  \nGdIG  Gd3Fe5O12 Film  12.480  \nSmIG  Sm 3Fe5O12 Film  12.530  \nEuIG  Eu3Fe5O12 Film  12.500  \n \nResults and Discussion  \nTable s 1 and 2 list in detail the parameters used and the calculated anisotropy energy density terms \nfor magnetic rare earth iron garnets  at 300K . These tables show only the cases predicted to be PMA  \nout of a total of 50 film/substrate pairs investigated . The extended version of Tables  1 and 2 for all \ncalculated anisotropy energy density terms for all combinations of the 50 film/subst rate pairs are \nprovided in the s upplementary tables. The tabulated values for saturation magnetization40,41 and \nlattice parameters44 have been  used for the calculations . In this study , we assumed the value of \nYoung’s modulus and Poisson ratio as 2.00×1012 dyne ·cm-2 and 0.29 for all garnet types , \nrespectively, based on ref.36. We also assume that the saturation magnetization, used for calculation \nof shape anisotropy, does not change with the film thickness. The saturation magnetization (Ms) \nvalues and  shape anisotropy  for iron garnet  films  are presented in the third  and fourth  columns , \nrespectively. The stress values for  fully lattice -matched film s σ calculated using equation 3 and \nmagnetostriction constants  of the films , λ111, are presented  on column s 6 and 7. Magnetoelastic \nanisotropy K indu, magnetocrystalline  anisotropy energy density K 1, and the total magnetic \nanisotropy energy density K eff are calculated  and listed on columns 8 , 9 and 10, respectively . H A \non column 11  is the anisotropy field ( the fields required to saturate the film s). \n \nTable 2 . Anisotropy energy density parameters calculation results. Rare earth iron garnets on GGG \n(as=12.3 83Å), YAG ( Y3Al5O12, as=12.005Å) , SGGG (a s=12.48Å) and TGG (Tb 3Ga5O12, \nas=12.355Å), and NGG ( Nd3Ga5O12, as=12.509Å)  substrates, with K eff < 0, are presented.   \nFilm  Substr ate Ms \n(emu·cm-\n3) Kshape \n(erg·cm-\n3) (× 103) ε \n(× \n10-3) σ \n(dyn·cm-\n2) \n(×1010) λ111 \n(×10-\n6) Kindu \n(erg·cm-\n3) (×104) K1 \n(300K) \n(erg·cm-\n3) \n(× 103) Keff \n(erg·cm-\n3) \n(× 103) HA \n(Oe) \n(× \n103) \nDyIG  GGG  31.85  6.37 -4.58 -1.29 -5.9 -11.4 -5.00 -113 -7.09 \nHoIG  GGG  55.73  19.5 -1.37 -0.386  -4 -2.3 -5.00 -8.66 -0.311  \nGdIG  GGG  7.962  0.398  -7.77 -2.19 -3.1 -10.2 -4.10 -106 -26.5 \nSmIG  GGG  140 123 -11.7 -3.30 -8.6 -42.6 -17.4 -321 -4.58 \nYIG YAG  141.7  126 -30.0 -8.44 -2.4 -30.4 -6.10 -184 -2.60 10 \n TmIG  YAG  110.9  77.2 -25.9 -7.29 -5.2 -56.9 -5.80 -497 -8.97 \nDyIG  YAG  31.85  6.37 -35.0 -9.85 -5.9 -87.2 -5.00 -870 -54.7 \nHoIG  YAG  55.73  19.5 -31.9 -8.97 -4 -53.8 -5.00 -524 -18.8 \nErIG  YAG  79.62  39.8 -27.9 -7.87 -4.9 -57.8 -6.00 -545 -13.7 \nYbIG  YAG  127.3  102 -24.0 -6.76 -4.5 -45.6 -6.10 -360 -5.66 \nGdIG  YAG  7.962  0.398  -38.1 -10.7 -3.1 -49.9 -4.10 -502 -126 \nSmIG  YAG  140 123 -41.9 -11.8 -8.6 -152.3  -17.4 -1420  -20.2 \nTbIG  SGGG  15.92  1.59 1.61 0.452  12 -8.14 -8.20 -88.0 -11.1 \nGdIG  SGGG  7.962  0.398  0.00 0.00 -3.1 0.00 -4.10 -3.70 -0.930  \nSmIG  SGGG  140 123 -3.99 -1.12 -8.6 -14.5 -17.4 -39.3 -0.562  \nDyIG  TGG  31.85  6.37 -6.83 -1.92 -5.9 -17.0 -5.00 -169 -10.6 \nHoIG  TGG  55.73  19.5 -3.63 -1.02 -4 -6.13 -5.00 -46.8 -1.68 \nGdIG  TGG  7.962  0.398  -10.0 -2.82 -3.1 -13.1 -4.10 -135 -33.9 \nSmIG  TGG  140 123 -14.0 -3.93 -8.6 -50.8 -17.4 -402 -5.74 \nTbIG  NGG  15.92  1.59 3.93 1.11 12 -19.9 -8.20 -206 -25.9 \n \nIn this study, we take the same sign convention as in ref. 16 and the film s exhibit PMA when Keff \n< 0. So for obtaining PMA, negative and large values for anisotropy energy density are desired. As \nall the garnets (except TbIG ) possess negative magnetostriction constant s at room temperature, the \nsign of the strain  (tensile or compressive)  determines whether the induced anisotropy is negative \nor positive . In the literature16,20,45,46 however , we observe that PMA was defined for either positive \nor ne gative effective anisotropy energy density (K eff). This inconsistency may cause confusion \namong researchers . Thermodynamically, a higher energy means an unstable state with respect to \nlower energy cases. Easy axis, by definition, is the axis along which the magnetic material can be \nsaturated with lowest external field or lowest total energy. A magnetic material would thus \nspontaneously minimize its energy and reorient it s magnetic moment along the easy axis. As a \nresult, we use here Keff < 0 for out-of-plane easy axis . Due to the thermodynamic arguments \nmentioned above, we suggest researchers to use K eff < 0 definition for PMA.  \n \nEffect of Substrate on Anisotropy Energy Density  \nChanging the substrate alters the strain in the film, which  also changes strain -induced anisotropy \nin the film. Figure 2 show s the calculated anisotropy energy density of rare earth iron garnet  thin \nfilms  grown on five commercially available differ ent substrates : Gadolinium Gallium Garnet \n(Gd3Ga5O12, GGG), Yttrium Aluminum Garnet ( Y3Al5O12, YAG), Substituted Gadolinium 11 \n Gallium Garnet ( Gd3Sc2Ga3O12, SGGG), Terbium Gallium Garnet (Tb 3Ga5O12, TGG ), and \nNeodymium Gallium Garnet (Nd 3Ga5O12, NGG) . As shown on Fig. 2(a), when gr own on GGG \nsubstrate ; Dysprosium Iron Garnet (DyIG), Holmium Iron Garnet (HoIG), Gadolinium Iron Garnet \n(GdIG), and Samarium Iron Garnet (SmIG) possess compressive strain  (afilm>asubstrate ). Considering \nthe large negative magnetostriction constant (λ 111) for each case, the strain -induced anisotropy \nenergy densit ies are estimated to cause negative total effective anisotropy energy density . As a \nresult, DyIG, HoIG, GdIG, and SmIG on GGG are predicted to be PMA cases.  \nBased o n the shape, magnetoelastic and magnetocrystalline anisotropy terms (room temperature \nK1), Thulium iron garnet (TmIG) on GGG  (111)  is estimated to be in -plane easy axis although \nunambiguous experimental evidence indicates that TmIG grows with PMA on GGG (111)  [1,19] . \nThe fact that only considering shape, magnetocrystalline and magnetoelastic anisotropy terms does \nnot verify this experimental result suggests that the PMA in TmIG/GGG (111) may originate from \na different anisotropy term such as surface anisotropy , growth -induced  or stoichiometry -driven \nanisotropy. Since the films used in the experiments are less than 10 nm or 5 -8 unit cells thick, \nsurface effects may become more significant and may require density functional theory -based \npredictions to account for surface anisotropy effects .  12 \n  \nFigure 2.  Calculated effective  anisotropy values for each rare earth iron garnet thin film when they \nare epitaxially grown on ( a) GGG, ( b) YAG, ( c) SGGG, ( d) TGG, (e) NGG  substrates . Note that \nthe scales of the axes are different in each part.  \n13 \n Yttrium Aluminum Garnet (YAG) is a substrate with smaller lattice parameter than all the rare \nearth iron garnet films  considered . With a substrate lattice parameter of as=12.005Å, YAG causes \nsignificant and varying amounts of strain on  YIG (af=12.376Å) , TmIG (af=12.324Å) , DyIG \n(af=12.440Å) , HoIG (af=12.400Å) , ErIG  (af=12.350Å) , YbIG (af=12.300Å) , TbIG (af=12.460Å) , \nGdIG (af=12.480Å) , SmIG (af=12.530Å)  and EuIG (a f=12.500Å) . Strain from YAG substrate \nyields negative strain -induced anisotropy energy density for these films . The strain -induced \nanisotropy term overcomes the shape anisotropy in these garnets when they are grown on YAG. \nConsequently , effective anisotropy energy densit ies become negative and these garnet  films are \nestimated to  possess perpendicular magnetic anisotropy . In the e xceptional case s of Terbium Iron \nGarnet (TbIG)  and Europium Iron Garnet (EuIG) , compressive strain is not enough to induce \nnegative strain anisotropy  because  the magnetostriction coefficient s of TbIG and EuIG are positive. \nSo the strain -induced anisotropy term s are also positive for both TbIG (Kindu(TbIG)  = 1.85×106 \nerg·cm-3) and EuIG (K indu(EuIG)  = 3.01×105 erg·cm-3) and do not yield PMA.  \nOther potential PMA garnets as a film on SGGG substrate are GdIG , TbIG , and SmIG. TbIG and \nGdIG cases are particularly interesting as growth conditions of these materials can be further \noptimized to achieve room temperature compensation and zero saturation magnetization. This \nproperty enables PMA garnet -based room temperature terahertz magnonics. The lattice parameters \nof GdIG (af=12.480Å) and SGGG (as=12.48Å) match exactly, so the in -plane strain value is zero \nand the effect of strain -induced anisotropy is eliminated completely. Consequently, due to small \nvalue for saturation magnetizat ion of GdIG, shape anisotropy (3.98 ×102 erg·cm-3) cannot compete \nwith magnetocrystalline anisotropy ( -4.1×103 erg·cm-3). In other words, in this case, the influence \nof magnetocrystalline anisotropy is not negligible compared to the other anisotropy terms.  \nConsequently , the anisotropy energy density is negative for GdIG when grown on SGGG due to \nthe influence of magnetocrystalline anisotropy energy density.  \nOne other candidate for a PMA rare earth iron garnet on SGGG substrate is SmIG. Since the film \nlattice parameter is greater than that of the substrate, compressive strain ( -3.99 ×10-3) is induced in \nthe film such that the resulting anisotropy energy density possesses a negative value of an order of \nmagnitude ( -1.45×105 erg·cm-3) comparable to the sh ape anisotropy energy density (1.23×105 \nerg·cm-3). With its relatively large magnetocrystalline anisotropy energy density ( -1.74×104 \nerg·cm-3), SmIG has a perpendicular magnetic anisotropy due to negative value for effective 14 \n anisotropy energy density (-4.80×105 erg·cm-3). TbIG film on SGGG substrate is a PMA candidate \nwith positive strain  and this film was also recently experimentally demonstrated to have PMA47.  \nSince TbIG’s lattice constant is smaller than that of the substrate, the film becomes subject to tensile \nstrain ( +1.61  ×10-3). Since TbIG also has a positive λ 111 (in contrast to that of SmIG), the film’s \nmagnetoelastic anisotropy term becomes large  and negative  and overcomes the shape anisotropy. \nIn case of TbIG, magnetocrystalline anisotropy alone overcomes shape and renders the film PMA \non SGGG. With the additio nal magnetoelastic anisotropy contribution ( -8.14×104 erg·cm-3), \nsignificant stability of PMA can be achieved.  \nTerbium Gallium  Garnet (TGG) is a substrate with lattice parameter (as=12.355Å ) such that it can \ninduce tensile strain on TmIG, ErIG and YbIG  and it induces compressive strain on the rest of the \nrare earth iron garnets (YIG, DyIG, HoIG, TbIG, GdIG, SmIG, EuIG) . In none of the tensile -\nstrained cases, PMA can be achieved since the sign of the magnetoelastic anisotropy is positive \nand has the same  sign as the shape anisotropy.  Among the compressively  strained cases, YIG, TbIG \nand EuIG are found to have weak magnetoelastic anisotropy terms which cannot overcome shape. \nAs a result, YIG, TbIG and EuIG on TGG substrate are expected to have in -plane easy axis. DyIG, \nHoIG, GdIG and SmIG films on TGG achieve large and negative effective  total anisotropy energy \ndensities  due to their negative λ 111 values . In addition, since the materials have compressive strain, \nthe signs cancel and lead to large magnetoelastic anisotropy energy terms that can overcome shape \nin these materials. So these cases are similar to the conditions explained for GGG substrate,  on \nwhich  only DyIG, HoIG, GdIG, and SmIG films with compressive strain can gain a large negative \nstrain -induced anisotropy energy density which can overcome shape a nisotropy . \nNeodymium gallium garnet  (NGG)  is a substrate 45 used for growing garnet thin films by pulsed \nlaser deposition method. NGG has large lattice constant compared with the rest of the bulk rare \nearth garnets and yield compressive strain in all rare earth garnets investigated except for Samarium \niron garnet (SmIG). For all cases other than SmIG, the sign of the magnetoelastic a nisotropy term \nis determined by the respective λ 111 for each rare earth iron garnet. YIG, TmIG, DyIG, HoIG, ErIG, \nYbIG cases have positive magnetoelastic anisotropy terms, which  lead to easy axes along the ir film \nplane s. For SmIG and EuIG on NGG, magnetoel astic strain and anisotropy terms are not large \nenough to overcome large shape anistropy. For GdIG, the weak tensile strain on NGG substrates \nactually causes in -plane easy axes as magnetoelastic strain offsets the negative magnetocrystalline 15 \n anisotropy. The only rare earth iron garnet that can achieve PMA on NGG is Terbium Iron Garnet \n(TbIG) due to its large negative strain -induced anisotropy energy density ( -2.44×105 erg·cm-3). Its \nlarge and negative magnetoelastic anisotropy can offset shape (1.59×103 erg·cm-3) and first order \nmagnetocrystalline anisotropy term ( -8.20×103 erg·cm-3), leading to a large negative effective \nmagnetic anisotropy energy density ( -2.51×105 erg·cm-3). Consequently, we predict that  growing \nTbIG on NGG substrate may yield PMA . \n \nFigure 3 show s the substrates on which one may expect PMA rare earth iron garnet films (or \nnegative Keff) due to strain only. Figures  3(a)-(d) compare the calculated effective energy densities \nas a function of strain type and sign for YIG, TmIG, YbIG, TbIG. For comparing the calculation \nresults presented here with the experimentally reported values for the anisotropy energy density of \nTmIG, we added the K eff directly from the experimental data in20 to Fig. 3(b). As shown on Fig. \n3(b), the experimental TmIG thin film shows positive K eff as the result of tensile in -plane strain \nand large negative magnetostriction constant.  \nFigure 4(a) -(f) shows the calculated effective energy densities as a function of strain type and sign \nfor GdIG, SmIG, EuIG, HoIG, DyIG, ErIG, respectively. K eff may get a positive or a negative value \nin both compressive and tensile strain cases due to vary ing signs of λ 111 constants of rare earth iron \ngarnets. In almost all cases that yield PMA on the given substrates, PMA iron garnets form under \ncompressive lattice strain. The only exception s in which tensile strain can yield PMA in garnet thin \nfilms is Tb IG on SGGG  and TbIG on NGG . In both of those cases, a small tensile strain enhances \nPMA but the magnetocrystalline anisotropy could already overcome shape and yield PMA without \nlattice strain. Therefore, experimental studies should target compressive latti ce strain.  \n 16 \n  \nFigure 3.  Effect of substrate strain on the effective anisotropy energy densities of ( a) YIG, ( b) \nTmIG, the data inserted on the graph, with red square symbol , GGG (Exp.) , is the experimental \nvalue of effective anisotropy energy of TmIG  on GGG based on ref.20 (c) YbIG, ( d) TbIG. Note \nthat the axes scales are different in each part.  \n17 \n  \nFigure 4. (a) GdIG, ( b) SmIG, ( c) EuIG, ( d) HoIG, ( e) DyIG, ( f) ErIG films on GGG, YAG, \nSGGG, TGG, and NGG substrates. Note that the axes scales are different in each part.  \n \n18 \n Based on Fig. 2 -4, the calculations  in this paper  numerically match with the reported values in the \nexperimental demonstrations in literature  both in sign and order of magnitude . However, since \nthere are inconsistent sign conventions for predictin g the magnetic anisotropy state  of the iron \ngarnet samples in the literature so far, some of the previous studies draw dif ferent conclusions on \nthe anisotropy despite the similar K eff. \nIn case of TmIG, as shown in Fig. 3(b), our model p redicts that there is a tensile strain -induced \nanisotropy resulting from the difference in film and substrate lattice parameters  and the film’ s \nnegative magnetostriction constant. The experimental results of magnetic anisotropy in \nTmIG/GGG8,20,48 are consistent with our model predictions . Previous studies identify PMA , if the \nfilm Keff is positive. A shortcoming of this approach is that such a definition would also identify \nYIG/GGG as PMA although its in-plane easy axis behavior has been experimentally de monstrated \nin numerous studies3,32,49. Keff < 0 for PMA definition  would be thermodynamically more \nappropriate and would also accurately explain almost  all cases including YIG/GGG.  Further \nexplanation about TmIG exceptional case is included in the Supplementary Information.   \nSensitivity of Anisotropy to Variations in Saturation Magnetic Moment and Film Relaxation  \nThe films with predicted effective anisotropy energy and field  may come out  differently when \nfabricated due to unintentional variability  in fabrication process conditions , film stoichiometry \n(rare earth ion to iron ratio and oxygen deficiency) as well as process -induced non -equilibrium \nphases in the garnet films.  These changes may partially or completely relax the films or increase \nstrain further due to secondary crystallin e phases. Practical minor changes in strain may \ndramatically alter both the sign and the magnitude of magnetoelastic ani sotropy energy and may \ncause a film predicted as PMA to come out with in -plane easy axis.  On the other hand, o ff-\nstoichiometry may cause reduction in saturation magnetic moment.  Reduction in saturation \nmagnetic moment decreases shape anisotropy term  quadratically (Kshape = 2πM s2), which implies \nthat a 1 0% reduction in Ms leads to a 19% decrease in K shape and anisotropy field may increase  (HA \n= 2K eff/Ms). Therefore, sample fabrication issues and the consequent changes in anisotropy terms \nmay weaken or completely eliminate the PMA of a film/substrate pair  and alter anisotropy field . \nWhile these effects may arise unintentionally, one can also use these effects deliberately for \nengineering garnet films for devices.  Therefore , the sensitivity of anisotropy properties of garnet 19 \n thin films such as anisotropy field and effective anisotropy energy density need s to be evaluated \nwith respect to changes in film strain and saturation magnetic moment.  \nFigure 5 shows the sensitivity of  the effective magnetic anisotropy energy density to deviation of \nboth strain and saturation magnetization, M s for five PMA film/substrate  combinations:  (a) \nHoIG/GGG, (b) YIG/YAG, (c) SmIG/ SGGG, (d) HoIG/TGG , and (e) SmIG/NGG ). The negative \nsign of effective magnetic anisotropy energy indicates PMA . The change of anisotropy energy from \nnegative to positive indicates a transition from PMA to in -plane easy axis . In these plots, calculated \nanisotropy energies are presented for saturation moments and strains scanned from 60% to 140% \nof tabulated bulk garnet Ms and of the strain s of fully lattice -matched films on the substrate s. The \ncolor scale indicates the anisotropy energy density in erg·cm-3. Although magnetocrystalline \nanisotropy energies are negative for all of the thin film rare earth garnets considered here, these \nterms are negligible with respect to shape anisotropy (K 1(300 K) ~ -5% of K shape). Therefore, \nmagnetoelastic anisotropy term must be large enough to overcome shape anisotropy. A derivation \nof anisotropy energy  as a function of Ms and strain in equations ( 8)-(10) shows that the negative \nλ111 values and negative strain states (compressive strain) for garnet films in Fig. 5(a)-(e) (HoIG, \nYIG, SmIG) enable these films to have PMA . To retain PMA state; λ 111 must be negative and large  \nassuming elastic moduli and the Poisson’s ratio are constant . The necessary condition for \nmaintai ning PMA is shown in equation 11 . \nKeff=Kindu +Kshape +K1    (8) \nKeff=−3\n2λ111Y\n1−vε||+2πMs2+K1   (9) \nKeff=3\n2λ111Y\n1−v|ε|||+2πMs2+K1   (10) \nKeff<0    if   |3\n2λ111Y\n1−v|ε|||+K1|>2πMs2  (11) \nRelaxing each fully strained and lattice -matched thin film towards unstrained state (ε → 0  or \nmoving from left  to right on each plot in Fig. 5  causes  the magnetoelastic anisotropy energy term \nto decrease in magnitude and gradually vanish . The total anisotropy energy decreases  in intensity \nfor decreasing strain and constant M s. When M s increases, shape anisotropy term also increases \nand overcomes magnetoelastic anisotropy term. As a result, higher M s for relaxed films (i.e. \nrelatively thick and iron -rich garnets) may lose PMA. Therefore, one needs to optimize the film 20 \n stoichiometry and deposition process conditions, especially growth temp erature, oxygen partial \npressure  and film thickness, to ensure that the films are strained and stoichiometric. Since strains \nare less than 1% in Fig. 5(a), 5(c)-(e), these samples are predicted to be exp erimentally more \nreproducible. For YIG on YAG, as shown in Fig. 5 (b), the strains are around 3%, which may be \nchallenging to reproduce.  The cases presented in Fig. 5 (a)-(e) are the only cases among 50 \nfilm/substrate pairs where reasonable changes in M s and strain may lead to complete loss of PMA. \nThe rest of the cases have not been found as sensitive to strain and M s variability and those \npredicted to be PMA are estimated to have stable anisotropy.  Effective a nisotropy energy plots \nsimilar to Fig. 5(a) -(e) are presented in the supplementary figures for all 50 film/substrate pairs.   \nWhile PMA is a useful metric for garnet films, the effective anisotropy energy of the films should \nalso not be too high ( < a few 105 erg·cm-3) otherwise the saturation fields for these films would \nreach or exceed 0.5 Tesla (5000 Oe). Supplementary figures present the calculated  effective \nanisotropy energy and anisotropy field values for all 50 film/substrate pairs for changing strain and \nMs values. These figures indicate that one can span anisotropy fields of about 300  Oersteds up to \n12.6 Tesla in PMA garnets. For practical integrated magnonic  devices, the effective anisotropy \nenergy should be large enough to have robust PMA although it shoul d not be too high such that \neffective anisotropy fields (i.e. saturation fields) would still be small and feasible.  Engineered strain \nand M s through controlled oxygen stoichiometry may help keep anisotropy field low while \nretaining PMA.  In addition, according to the recently published paper  on magnetic anisotropy of \nHoIG50, the lattice matching in case of the thick samples becomes  challenging to sustain, and the \nstrain relaxes inside the film. Thus, the decrease in the anisotropy field is one consequences of the \nlower strain state, which is an advantage for magnonics or spin -orbit torque devices. Below a \ncritical thickness, HoIG gr own on GGG has PMA. However, as the film reaches this critical \nthickness, the 40% or more strain relaxation is expected and the easy axis becomes in -plane. So \nthinner films are preferred to be grown in integrated device applications.  21 \n  \nFigure 5. Effect of partial film relaxation and saturation magnetic moment variability on the \neffective anisotropy  energy density of the films . Variation of effective magnetic anisotropy energy \ndensities for (a) HoIG on GGG, ( b) YIG on YAG, ( c) SmIG on SGGG, ( d) HoIG on TGG and ( e) \nSmIG on NGG are presented when strain relaxation and magnetic saturation moments change \nindependently. Film strain may vary from a completely lattice -matched state to the substrate to a \nrelaxed state or a highly strained state due to microparticle nucleation . Strain variability alter s \nmagnetoelastic  anisotropy and cause a PMA film become in -plane easy axis.  On the other hand, \n22 \n magnetic saturation moments may deviate from the tabulated values because of process -induced \noff-stoich iometry in the films (i.e. rare earth ion to iron ratio or iron deficiency or excess , oxygen \ndeficiency) . Relaxing the films reduces the magnetoelastic anisotropy term and diminishes PMA. \nIncreasing M s strengthens shape anisotropy and eliminates PMA for lo w enough strains for all five \ncases presented.  \nMinimizing Gilbert damping coefficient in garnet thin films is also an important goal for spintronic \ndevice applications. First principles predictions of physical origins of Gilbert damping  51 indicate \nthat magnetic materials with lower M s tend to have lower damping. Based on this prediction, DyIG, \nHoIG and GdIG films are predicted to have lower Gilbert damping with respect to the others. Since \nthe compen sation temperatures of these films could be engineered near room temperature, one may \noptimize their damping for wide bandwidths all the way up to terahertz (THz)52 spin waves  or \nmagnons . The first principles predictions  also indicate that higher magnetic susceptibility (χm) in \nthe films helps reduce damping (i.e. lower saturation field). Therefore, the PMA garnet films with \nlower anisotropy fields are estimated to have lower Gilbert dampi ng parameters with respect to \nPMA garnets with higher anisotropy fields . \n \nConclusion  \nShape, magnetoelastic and magnetocrystalline m agnetic anisotropy energy terms have been \ncalculated for ten different garnet thin films  epitaxially grown on five different garnet substrates. \nNegative K eff (effective magnetic anisotropy energy) corresponds to perpendicular magnetic \nanisotropy  in the convention used here .  By choosing a substrate with a lattice parameter smaller \nthan that of the film, one can  induce compressive strain in the films  to the extent that one can \nalways overcome shape anisotropy  and achieve PMA  for large and negative λ 111. Among the PMA \nfilms predicted, SmIG possesses  a high anisotropy energy density and this film is estimated to be \na robus t PMA when grown on all five different substrates.  \nIn order to obtain PMA, magnetoelastic anisotropy term must be large enough to overcome shape \nanisotropy. Magnetoelastic anisotropy overcomes shape anisotropy when the strain type \n(compressive or tensile) and magnetoelastic anisotropy constants λ 111 of the garnet film  have the \ncorrect signs (not necessarily opposite or same) and the magnetoelastic anisotropy term has a \nmagnitude larger than shape anisotropy. Both compressive and tensile -strained films can , in \nprinciple, become  PMA as long as shape anisotropy can be overcome with large magnetoelastic 23 \n strain effects.  Here, in almost all cases that yield PMA on the given substrates, PMA iron garnets \nform under compressive lattice strain, except TbIG on SGGG and  TbIG on NGG . These two cases \nhave tensile strain and relatively large magnetocrystalline anisotropy, which could already \novercome shape anisotropy without strain. Experiments are therefore suggested to target mainly \ncompressive lattice strain.  \n20 different garnet film/substrate pairs have been predicted to exhibit PMA and their properties are \nlisted on Table 2 . For 7 of these 20 potential PMA cases, we could find unambiguous experimental \ndemonstration of PMA. Among the  20 PMA cases, HoIG/GGG, YIG/Y AG, SmIG/SGGG, \nHoIG/ TGG and SmIG/ NGG  cases have been found to be sensitive to fabrication process or \nstoiochiometry -induced variations in M s and strain.  In order to control  effective anisotropy in rare \nearth iron garnets (RIGs), shape anisotropy could be tuned  by doping garnet film s with Ce 53, Tb \n31 and Bi 54 or by micro/nano -patterning . Saturation magnetization could also be increased \nsignificantly by doping, which results in increasing the shape anisotropy in the ma gnetic thin films.  \nAmong the cases predicted to possess PMA , anisotropy fields ranging from 310 Oe (0.31 T) to \n12.6 T have b een calculated . Such a wide anisotropy field range could be spanned and engineered \nthrough strain state, stoichiometry as we ll as substrate choice. For integrated magnonic devices and \ncircuits, garnets with low M s and lower anisotropy field s (HA < 0.5 T)  would require less energy \nfor switching and would be more appropriate due to their lower estimated Gilbert damping.  \nMethods  \nCalculation of anisotropy energy density. We used  Keff = K indu + K shape + K 1 equation to calculate \nthe total anisotropy energy density for each thin film rare earth iron garnet/substrate pair. Each \nanisotropy term consist of the following parameters:  Keff=−3\n2λ111Y\n1−vε||+2πMs2+K1. The \nenergy density is calculated based on the parameters reported in previous  references16,34,36,40,41,43. \nFirst order magnetocrystalline anisotropy, K 1, is an intrinsic, temperature -dependent constant \nreported for each REIG material. Young’s modulus  (Y), poison ratio  (ν) and magnetostriction \nconstant (λ111) parameters evolving in the magnetoelastic anisotropy energy density term  (first \nterm)  are considered to be constant according to the values previous ly reported . For shape \nanisotropy energy calculations (second term), bulk s aturation magnetization (Ms) for each film was \nused. Since each film may exhibit variability in M s with respect to bulk, the model presented here \nyields the most accurate predictions when the actual film Ms, λ 111, Y, ν and K 1, and in -plane strain 24 \n are enter ed for each term. 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Optical Materials \nExpress  7, 2248 -2259 (2017).  \n Acknowledgments  \nM.C.O. acknowledges BAGEP 2017 Award and TUBITAK Grant No. 117F416.  \nCompeting interests  \nThere is no financial and non -financial competing interest among the authors.  \nAuthor contributions  \nM.C.O. designed the study. S.M.Z. performed the calculations and evaluated and analyzed the \nresults with M.C.O. Both authors discussed the results and wrote the manuscript together.  \n \n "    },    {        "title": "1905.13262v1.Sub_nanosecond_switching_in_a_cryogenic_spin_torque_spin_valve_memory_element_with_a_dilute_permalloy_free_layer.pdf",        "content": "Sub-nanosecond switching in a cryogenic spin-torque spin-valve memory element with\na dilute permalloy free layer\nL. Rehm,1,\u0003V. Sluka,1,yG. E. Rowlands,2M.-H. Nguyen,2T. A. Ohki,2and A. D. Kent1,z\n1Center for Quantum Phenomena, Department of Physics,\nNew York University, New York, NY 10003, USA\n2Raytheon BBN Technologies, Cambridge, MA 02138, USA\n(Dated: July 13, 2021)\nWe present a study of the pulsed current switching characteristics of spin-valve nanopillars with\nin-plane magnetized dilute permalloy and undiluted permalloy free layers in the ballistic regime at\nlow temperature. The dilute permalloy free layer device switches much faster: the characteristic\nswitching time for a permalloy free (Ni 0.83Fe0.17) layer device is 1.18 ns, while that for a dilute\npermalloy ([Ni 0.83Fe0.17]0.6Cu0.4) free layer device is 0.475 ns. A ballistic macrospin model can\ncapture the data trends with a reduced spin-torque asymmetry parameter, reduced spin polarization\nand increased Gilbert damping for the dilute permalloy free layer relative to the permalloy devices.\nOur study demonstrates that reducing the magnetization of the free layer increases the switching\nspeed while greatly reducing the switching energy and shows a promising route toward even lower\npower magnetic memory devices compatible with superconducting electronics.\nThere is a growing interest in spin-transfer devices that\nwork in a cryogenic environment, such as for use in super-\nconducting logic and circuits [1]. While past low temper-\nature memory e\u000borts combined, for example, Josephson\nand complementary metal-oxide semiconductor devices\nin hybrid circuits or explored circuits that stored mag-\nnetic \rux quanta in superconducting loops [2, 3], these\napproaches did not simultaneously o\u000ber high speed, low\npower, and scalability. Spin-transfer torque (STT) driven\nmagnetic memory elements are known to be non-volatile,\nfast, and energy e\u000ecient [4, 5], but so far, they are almost\nexclusively being developed and tested for commercial\napplications [6], which require operation at and above\nroom temperature. Cryogenic operation with supercon-\nducting circuits change device and material requirements.\nFor example, the magnetic anisotropy energy barrier that\nstabilizes the magnetic states and permits long-term data\nretention can be greatly reduced. Large magnetoresis-\ntance also may not be essential given the sensitivity of\nsuperconducting circuits and the reduced thermal noise\nat low temperature. This makes it promising to study\nall metallic spin-valve structures, both due to their low\nimpedance and potential for fast switching [7, 8].\nSpin-transfer induced magnetization switching is fun-\ndamentally based on the transfer of angular momentum\nbetween itinerant electrons and background magnetiza-\ntion. Switching thus requires that the number of elec-\ntrons that \row through a circuit to be of order of the\nnumber of elemental magnetic moments (or spins) in the\nfree layer [9]. This requirement sets the order of magni-\ntude of the product of the current and the switching time\n(which is proportional to the total number of charges\ntransmitted) in what is known as the ballistic limit, the\n\u0003laura.rehm@nyu.edu\nyvolker.sluka@julumni.fz-juelich.de\nzandy.kent@nyu.edushort-pulse-time limit (typically pulse durations less than\nseveral nanoseconds) in which thermal energy has a min-\nimal e\u000bect on the switching dynamics [10]. Reducing\nthe magnetization density is thus expected to reduce the\nswitching current. It is also expected to increase the\nswitching speed and thus reduce the switching energy,\nwhich is a product of the power supplied and the time\nthe device is energized.\nIn this article we test this hypothesis by compar-\ning the switching characteristics of spin-valve nanopil-\nlars with in-plane magnetized dilute permalloy and undi-\nluted permalloy free layers, but otherwise the same lay-\ners, nanopillar shape and size. In both cases the layer\nstacks are deposited on a Niobium (Nb) bottom electrode\nto show that integration with superconducting materials\nis practical. We characterize the pulsed current switch-\ning thresholds in the ballistic regime for both composition\nfree layers and \fnd a signi\fcant decrease in the character-\nistic time scale from 1.18 ns for permalloy to 0.475 ns for\nthe dilute permalloy free layer. A macrospin model was\nused to \ft the switching time data with a reduced spin-\ntorque asymmetry parameter, reduced spin polarization\nand increased Gilbert damping for the dilute permalloy\nfree layer.\nWe investigated two sets of spin-valve nanopillar de-\nvices. One with an undiluted permalloy (Ni 0.83Fe0.17,\ndenoted as Py) free layer and another with a diluted\npermalloy ([Ni 0.83Fe0.17]0.6Cu0.4, denoted as PyCu) free\nlayer. The layer stacks consist of a Nb(50)/Al(8)\nbottom electrode layer, a CoFe(3) reference layer\n(RL) which is part of a synthetic antiferromagnet\n(SAF) CoFe(3)/Ru(0.8)/CoFe(3), and a 3 nm thick\nPy or PyCu free layer (FL): Nb(50)/Al(8)/IrMn\n(10)/SAF/Co(0.2)/Cu(3.5)/Co(0.2)/FL, as shown in\nFig. 1. The numbers in brackets are the layer thicknesses\nin nm. The Nb bottom electrode enables the integration\nwith superconducting circuitry, while the Al interlayer is\ncrucial for the properties of the magnetic stack: it wets\nthe surface of the Nb layer and creates a smoother sur-arXiv:1905.13262v1  [physics.app-ph]  30 May 20192\nLoops\n1\nDCBias TeePulse\nPyorPyCu\nSAF\nAl\nNb\nFIG. 1. Schematic of a spin-valve nanopillar device with an\nundiluted permalloy (Py) or diluted permalloy (PyCu) free\nlayer. The write pulses Iware applied through the capacitive\nport of a bias tee while the inductive port is used to read out\nthe state of the device.\nface, reducing the e\u000bect of N\u0013 eel \\orange peel\" coupling\nbetween layers [11] and e\u000bects of roughness on the mag-\nnetic switching characteristics.\nFollowing the deposition, the wafers were annealed at\n230\u000eC and 1 T to set the magnetization orientation of\nthe SAF. The annealed wafers were pattered into ellip-\ntically shaped nanopillars of various sizes using e-beam\nlithography and ion-milling. Here, we present results on\ndevices with a 50 nm \u0002110 nm cross-section. The de-\nvices are characterized by measuring their \feld and cur-\nrent pulse resistance hysteresis loops at 3.2 K. The state\nof the device is recorded using a lock-in technique. Small\nAC currents of 20 and 40 \u0016A are applied for the PyCu\nand Py free layer device, respectively. Figures 2a) and\n2c) show the minor loops of the Py and PyCu free layer\ndevice, respectively. The Py sample exhibits a resistance\nchange between the antiparallel (AP) and parallel (P)\nmagnetic con\fguration of 190 m\n, while the PyCu free\nlayer device exhibits a \u0001 Rof around 120 m\n. Both de-\nvices show a well-centered hysteresis with a small o\u000bset\n\feld of 6 mT. Both samples also show a bistable region\naround zero applied current and current-induced switch-\ning with 10 ns duration current pulse with pulse ampli-\ntudes of 403 \u0016A (for AP!P switching) and -523 \u0016A (for\nP!AP switching) of the PyCu free layer (Fig. 2b)) and\n480\u0016A (AP!P) and -868 \u0016A (P!AP ) for the Py free\nlayer sample (Fig. 2d)). A di\u000berence in the P !AP and\nAP!P switching current magnitude is often observed in\nspin-valves and associated with spin-torque asymmetries,\nas discussed in Refs. [12{14].\nIn order to explore high speed spin-torque switching,\nshort current pulses with durations of less than 5 ns were\nused. Pulses are applied using a pulse generator (Pi-\ncosecond Pulse Labs 10,070A) as well as an arbitrary\nwaveform generator (AWG, Keysight M8190A). The \frst\ngenerator provides the short pulses to explore the ballis-\ntic regime, while the second generator is used to apply\nlonger (20 ns) duration pulses to reset the magnetization\ndirection of the free layer. To increase the pulse ampli-\ntude resolution (below the 1 dB resolution of the pulse\nLoops_bigger\n0\nc)a)\nd)b)Py Py\nPyCu PyCuFIG. 2. Field- and current-induced magnetization switching\nof Py and PyCu free layer device at 3.2 K. Panels a) and c)\nshow \feld-induced switching of devices with Py and PyCu free\nlayer, respectively. The hysteresis loop shown in panel c) does\nnot fully close due to drift in the measurement setup. The\nexternal \feld is applied along the easy axis of the elliptically\nshaped nanopillar. Panels b) and d) display current-induced\nswitching for 10 ns long pulses of the same set of devices. No\ndata was taken along the dashed lines.\ngenerator's internal step attenuator) a voltage controlled\nvariable attenuator (RFMD RFSA2113SB) is employed.\nThe state of the device is determined by applying a small\nAC current and using a lock-in ampli\fer to determine the\ndevice resistance. The lock-in ampli\fer is operated at a 4\nkHz baseband. We use a bias-tee (Picosecond Pulse Labs\n5575A) to combine low-frequency measurement and high-\nfrequency switching pulses (see Fig. 1). Two 0 dB attenu-\nators at the 4 K and 50 K stage are utilized to thermalize\nthe center conductor of a ground signal ground (GSG)\nprobe. A small external \feld (6 mT) applied along the\nlong axis of the ellipse is used to conduct these pulse\nstudies at the midpoint of the free layer hysteresis loop.\nThe measurement procedure thus consists of applying\ntwo square pulses (reset IRSTand write Iwpulses) with\nopposite pulse amplitudes and reads after each pulse. We\nstart by applying a reset pulse to bring the device to a\nknown state, either P or AP. We then veri\fed the desired\nstate by measuring the resistance of the device. The sub-\nsequent write pulse is applied by the pulse generator and\nthe end state is again determined by measuring the de-\nvice resistance. The whole procedure is repeated about\n64 times for each write pulse amplitude and duration to\ndetermine the switching probability. We vary amplitude\nand duration of the write pulse to create the phase dia-\ngrams shown in Fig. 3. All the pulse measurements were\nperformed at 3.2 K.\nFigure 3 shows the switching phase diagrams for3\nExtented Fit_small\n2\n0.0 0.2 0.4 0.6 0.8 1.0ProbabilityAmplitude (mA) Amplitude (mA)\na)Py: AP→P1.2\n1.0\n0.8\n0.6\n0.4\nPy: P→AP\nb)1.6\n1.4\n1.2\n1.0\n0.8\nc)PyCu : AP→P0.9\n0.8\n0.7\n0.6\n0.40.5\nPyCu : P→AP\nd)0.9\n0.8\n0.7\n0.6\n0.40.5\nPulse Duration ( ns)1 2 3 4 5\nPulse Duration ( ns)1 2 3 4 5\nFIG. 3. Nanosecond pulsed current switching results at 3.2 K.\nSwitching phase diagrams of a device with Py free layer, a)\nAP!P and b) P!AP, and a PyCu free layer, c) AP !P and\nd) P!AP. The color in the plot represents the switching prob-\nability, where red corresponds to 0% and black is 100%. The\nblue points represent the 50% switching probability and the\nsolid cyan line shows the \ft to the macrospin model described\nin the main text.\nAP!P (left panels) and P !AP transitions (right panels)\nfor Py (Figs. 3(a) and (b)) and PyCu free layer device\n(Fig. 3(c) and (d)). The results from these samples di\u000ber\nsigni\fcantly. For longer pulse durations, \u00185 ns, switching\nof the PyCu free layer device occurs for lower pulse am-\nplitudes, especially for the P !AP transition (Figs. 3(d)).\nThe PyCu free layer device also switches with high prob-\nability for shorter duration pulses than the device with\nthe Py free layer, as seen by form of the switching bound-\naries (blue points in Fig. 3) for pulse durations less than\n1 ns. For the P!AP direction comparatively longer pulse\ndurations are required for switching, as discussed further\nbelow.\nIn order to understand the data trends in Fig. 3 we\nconsider a macrospin model, a simple model that pro-\nvides analytic expressions for the switching times in the\nballistic limit and how they vary with material and de-\nvice parameters [9, 15]. Since the devices are metallic\nspin-valves (in contrast to magnetic tunnel junctions),\nthe spin-transfer torque angular dependence is expected\nto be asymmetrical, to be di\u000berent for angular devia-\ntions from the P and AP states, and characterized by a\nparameter \u0003 [12], with a ratio of threshold currents given\nbyIcP!AP/IcAP!P= \u00032. Incorporating this asym-\nmetry into a model for switching of biaxial anisotropy\nmacrospins, and following the approach of Ref. [15], we\nderive an approximate formula relating the switching\nspeed 1/\u001c(\u001cbeing the switching time) to the overdrive\ncurrentI\u0000Ic. Due to the spin-torque asymmetry, P !AP\nand AP!P switching di\u000ber. While the relation for theformer case remains the same as in Ref. [15],\n\u001c\u00001=\r~P\n4e\u00160MsV1\nln\u0010\n\u0019\n2\u00120\u0011\u0000\nI\u0000IP!AP\nc\u0001\n; (1)\nfor the other switching direction we have\n\u0000\n\u00032\u001c\u0001\u00001=\r~P\n4e\u00160MsV1\nln\u0010\n\u0019\n2\u00120\u0011\u0000\nI\u0000IAP!P\nc\u0001\n;(2)\nwhere all currents are taken as positive. In these expres-\nsionsPis the spin polarization of the current, Msthe free\nlayer saturation magnetization, Vthe free layer volume,\n\rthe gyromagnetic ratio, \u00160the vacuum permeability,\n~the reduced Planck's constant and ethe magnitude of\nthe electron charge (i.e., e >0).\u00120is the initial angu-\nlar deviation of the free layer's magnetization from the\neasy axis, the deviation the moment the current pulse is\napplied, discussed further below.\nThe threshold currents for switching are (c.f. [15]):\nIP!AP\nc =4e\n~P\u00160MsV\u000b(Hk+Ms=2) (3)\nIAP!P\nc =4e\n~P\u00160MsV\u000b(Hk+Ms=2)=\u00032; (4)\nwhere\u000bis the damping and Hkis the easy axis\nanisotropy \feld. Important for our analysis, Eqs. 1 and 2\nare each are of the form\nI\u0000Ic=Ic\u001c0\n\u001c; (5)\nwhere\u001c0= ln(\u0019=(2\u00120))=(\r\u000b(Hk+Ms=2)) is independent\nof the switching direction. We therefore \ft the experi-\nmental data in Fig. 3 with Eq. 5 under the constraint that\n\u001c0is the same for both P !AP and AP!P switching di-\nrections. The \fts are displayed as cyan lines in Fig. 3\nand the corresponding \ft parameters are listed in Table\n1.\nFrom this analysis we draw the following conclusions.\nFirst, taking the ratios of \ft parameters IP!AP\nc to\nIAP!P\nc we \fnd that the spin-transfer torque asymme-\ntry is signi\fcantly reduced by diluting the free layer with\nCu: for the Py case \u0003 = 1.44, while in the PyCu free\nlayer device \u0003 = 1.16. Next, we consider the e\u000bect of the\ndilution on the P!AP switching currents and determine\nwhat this implies for the device's material parameters.\nTo this end we note that the uniaxial in-plane anisotropy\nin our samples can be assumed to be entirely due to the\ndevice shape, which is designed to be the same for each\ndevice (up to fabrication-induced sample to sample vari-\nations, of course). Comparing the P !AP switching cur-\nrents between the two devices, we obtain the relation\nIP!AP;u\nc\nIP!AP;d\nc=\u001f2Pd\u000bu\nPu\u000bd; (6)\nwhere\u001fdenotes the ratio of the saturation magnetiza-\ntionsMu\ns=Md\ns. The labels uanddstand for undiluted4\nTABLE I. Fit parameters from pulsed switching measurements in the ballistic regime and corresponding spin-torque asymmetry\nparameter \u0003 of PyCu and PyCu free layer devices. Saturation magnetization Msfor Py and PyCu layers at 3.2 K was determined\nby VSM measurements.\nSample Ic(\u0016A) \u0003 \u001c0(ns) \u00160Ms,3.2 K (mT)\nAP!P P!AP\nPyCu FL 395 \u00062 532\u00062 1.16 0.475 \u00060.007 240\nPy FL 432 \u00062 902\u00063 1.44 1.18 \u00060.01 860\nand diluted, respectively. Vibrating sample magnetome-\ntry (VSM) measurements give \u001f= 3:6 (see Table 1) and\nthus from Eq. 6 we \fndPu\u000bd\nPd\u000bu= 7:6.\nFurther analysis gives estimates of the ratio of the spin\npolarizations in the di\u000berent free layer devices and also\nan estimate of the ratio of the damping parameters. This\ncan be achieved by observing that\nIP!AP;u\nc\nIP!AP;d\nc\u001cu\n0\n\u001cd\n0\u0019Pdln\u0010\n\u0019\n2\u0012u\n0\u0011\nPuln\u0010\n\u0019\n2\u0012d\n0\u0011\u001f; (7)\nwhere the\u0019, refers to the assumption that the gyro-\nmagnetic ratios do not vary between the devices. The\nleft-hand side of Eq. 7 is obtained from the \fts to the\nexperimental data and is approximately equal to 4.21.\nThe two initial angles \u0012u(d)\n0are expectation values that\ndepend on the device shape, the respective saturation\nmagnetizations, and the temperature. With the satura-\ntion magnetizations in Table 1, the right-hand side of\nEq. 7 can be used to estimate the ratio of the spin po-\nlarizations Pd=Pu. To make this estimate, we assume a\nBoltzmann distribution of the initial magnetization state\nof the free layer to obtain\nh\u0012u(d)\n0i\u0019vuut\u0019DkT\n2\u00160\u0010\nMu(d)\ns\u00112\nV; (8)\nwherekis the Boltzmann constant and D=Mu(d)\ns\nHu(d)\nk\u001919.6\nonly depends on the device shape and is assumed to be\nsu\u000eciently similar for the two samples. The same applies\nto the device volume V. Inserting Eq. 8 into Eq. 7, we ob-\ntainPd=Pu= 0:85 forT= 3:2 K. The value depends only\nweakly on the assumed temperature, ranging from 0.85\nat 3.2 K to about 0.82 at 10 K. As a consistency check, we\ncalculateh\u0012u(d)\n0i= 0.015 (0.054) which are small enough\nfor Eq. 8 to be a good approximation. The above range of\nvalues is consistent with the reduced magnetoresistance\nobserved in the dilute free layer devices (c.f. Fig. 2).\nFinally, we can revisit Eq. 6 to estimate\u000bd\n\u000bu\u00196:5, in-\ndicating about a six-fold increase of the damping due to\nthe dilution. This is a large increase, but not entirely\nunexpected. Mathias et al. [16] found a factor of three\nincrease in the damping with 40% Cu dilution of Py at\nroom temperature. Also, Rantschler et al. [17] found\nthat the damping of Py at room temperature increasesby 0.2\u000210-3per atomic percent of Cu. The analysis and\nparticularly the very large apparent increase in damping\nmay also be associated with the lower magnetization and\nexchange sti\u000bness in the PyCu opening other dissipation\nchannels in spin-torque switching, such as the excitation\nof spin-waves, or the formation micromagnetic structure\nin the switching process.\nIn summary, we have studied nanosecond switching\nphase diagrams for spin-valve nanopillars with in-plane\nmagnetized PyCu and Py free layers at low temperature.\nThe PyCu free layer sample exhibits reduced switching\ncurrents for the parallel to antiparallel con\fguration and\nsigni\fcant speed-up of the characteristic switching time\ncompared to the Py free layer device. This results in\ngreatly reduced switching energies ( E=RI2\u001c) for the\nPyCu free layer device. While the switching energy for\nthe antiparallel to parallel con\fguration is reduced from\n53 to 18 fJ, the switching energy for the opposite switch-\ning direction shows over a seven-fold decrease for the di-\nluted sample from 230 fJ to 32 fJ.\nThe clear reduction in the energy consumption of the\nPyCu free layer device as well as its speed-up in the\nswitching characteristics makes it especially interesting\nas a low-energy data storage solution for superconduct-\ning computing. Further, our modeling suggest a means\nto further signi\fcant reductions in the switching energy\nand increases in device performance metrics. Foremost,\nlarger reductions in switching energy require low mag-\nnetization density materials with larger spin polarization\nand lower damping (for example, Heusler alloys [18{20]),\nwhich would have the added bene\ft of increasing the de-\nvice magnetoresistance while reducing the switching cur-\nrent.\nACKNOWLEDGMENTS\nWe thank Jamileh Beik Mohammadi for comments on\nthe manuscript. We thank Canon ANELVA for providing\nthe layer stacks and Spin Memory for patterning the de-\nvices. The research is based on work supported by the Of-\n\fce of the Director of National Intelligence (ODNI), Intel-\nligence Advanced Research Projects Activity (IARPA),\nvia contract W911NF-14-C0089. The views and conclu-\nsions contained herein are those of the authors and should\nnot be interpreted as necessarily representing the o\u000ecial\npolicies or endorsements, either expressed or implied, of5\nthe ODNI, IARPA, or the U.S. Government. The U.S.\nGovernment is authorized to reproduce and distribute\nreprints for Governmental purposes notwithstanding any\ncopyright annotation thereon. This document does notcontain technology or technical data controlled under ei-\nther the U.S. International Tra\u000ec in Arms Regulations\nor the U.S. Export Administration Regulations.\n[1] S. Holmes, A. L. Ripple, and M. A. Manheimer, IEEE\nTransactions on Applied Superconductivity 23, 1701610\n(2013).\n[2] T. Van Duzer, L. Zheng, S. R. Whiteley, H. Kim, J. Kim,\nX. Meng, and T. Ortlepp, IEEE Transactions on Applied\nSuperconductivity 23, 1700504 (2013).\n[3] S. Nagasawa, Y. Hashimoto, H. Numata, and S. Tahara,\nIEEE Transactions on Applied Superconductivity 5, 2447\n(1995).\n[4] J. C. Slonczewski, Journal of Magnetism and Magnetic\nMaterials 159, L1 (1996).\n[5] L. Berger, Physical Review B 54, 9353 (1996).\n[6] A. D. Kent and D. C. Worledge, Nature Nanotechnology\n10, 187 (2015).\n[7] D. Bedau, H. Liu, J.-J. Bouzaglou, A. D. 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Aeschli-\nmann, M. M. Murnane, and H. C. Kapteyn, Proceedings\nof the National Academy of Sciences 109, 4792 (2012).\n[17] J. O. Rantschler, R. D. McMichael, A. Castillo, A. J.\nShapiro, W. F. Egelho\u000b Jr, B. B. Maranville, D. Pu-\nlugurtha, A. P. Chen, and L. M. Connors, Journal of\nApplied Physics 101, 033911 (2007).\n[18] T. Kubota, S. Tsunegi, M. Oogane, S. Mizukami,\nT. Miyazaki, H. Naganuma, and Y. Ando, Applied\nPhysics Letters 94, 122504 (2009).\n[19] M. J. Carey, S. Maat, S. Chandrashekariaih, J. A. Ka-\ntine, W. Chen, B. York, and J. R. Childress, Journal of\nApplied Physics 109, 093912 (2011).\n[20] S. Andrieu, A. Neggache, T. Hauet, T. Devolder, A. Hal-\nlal, M. Chshiev, A. M. Bataille, P. Le F\u0012 evre, and\nF. Bertran, Phys. Rev. B 93, 094417 (2016)."    },    {        "title": "1906.01042v1.Magnon_phonon_interactions_in_magnetic_insulators.pdf",        "content": "Magnon-phonon interactions in magnetic insulators\nSimon Streib,1Nicolas Vidal-Silva,2, 3, 4Ka Shen,5and Gerrit E. W. Bauer1, 5, 6\n1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Departamento de Física, Universidad de Santiago de Chile, Avda. Ecuador 3493, Santiago, Chile\n3Center for the Development of Nanoscience and Nanotechnology (CEDENNA), 917-0124 Santiago, Chile\n4Departamento de Física, Facultad de Ciencias Físicas y Matemáticas,\nUniversidad de Chile, Casilla 487-3, Santiago, Chile\n5Department of Physics, Beijing Normal University, Beijing 100875, China\n6Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\n(Dated: June 4, 2019)\nWe address the theory of magnon-phonon interactions and compute the corresponding quasi-\nparticle and transport lifetimes in magnetic insulators with focus on yttrium iron garnet at inter-\nmediate temperatures from anisotropy- and exchange-mediated magnon-phonon interactions, the\nlatter being derived from the volume dependence of the Curie temperature. We find in general weak\neffects of phonon scattering on magnon transport and the Gilbert damping of the macrospin Kittel\nmode. The magnon transport lifetime differs from the quasi-particle lifetime at shorter wavelengths.\nI. INTRODUCTION\nMagnons are the elementary excitations of magnetic\norder, i.e. the quanta of spin waves. They are bosonic\nandcarryspinangularmomentum. Ofparticularinterest\nare the magnon transport properties in yttrium iron gar-\nnet (YIG) due to its very low damping ( \u000b<10\u00004), which\nmakes it one of the best materials to study spin-wave or\nspin caloritronic phenomena [1–6]. For instance, the spin\nSeebeck effect (SSE) in YIG has been intensely studied\nin the past decade [7–13]. Here, a temperature gradi-\nent in the magnetic insulator injects a spin current into\nattached Pt contacts that is converted into a transverse\nvoltage by the inverse spin Hall effect. Most theories ex-\nplain the effect by thermally induced magnons and their\ntransport to and through the interface to Pt [7, 14–19].\nHowever, phonons also play an important role in the SSE\nthrough their interactions with magnons [20–22].\nMagnetoelastic effects in magnetic insulators were ad-\ndressed first by Abrahams and Kittel [23–25], and by\nKaganov and Tsukernik [26]. In the long-wavelength\nregime, the strain-induced magnetic anisotropy is the\nmost important contribution to the magnetoelastic en-\nergy, whereas for shorter wavelengths, the contribution\nfrom the strain-dependence of the exchange interaction\nbecomes significant [27–29]. Rückriegel et al.[28] com-\nputed very small magnon decay rates in thin YIG films\ndue to magnon-phonon interactions with quasi-particle\nlifetimes\u001cqp?480 ns;even at room temperature. How-\never, these authors do not consider the exchange interac-\ntion and the difference between quasi-particle and trans-\nport lifetimes.\nRecently, it has been suggested that magnon spin\ntransport in YIG at room temperature is driven by\nthe magnon chemical potential [3, 30]. Cornelissen et\nal. [3] assume that at room temperature magnon-\nphonon scattering of short-wavelength thermal magnons\nis dominated by the exchange interaction with a scat-\ntering time of \u001cqp\u00181 ps, which is much faster than\nthe anisotropy-mediated magnon-phonon coupling con-sidered in Ref. [28] and efficiently thermalizes magnons\nand phonons to equal temperatures without magnon de-\ncay. Recently, the exchange-mediated magnon-phonon\ninteraction [31] has been taken into account in a Boltz-\nmann approach to the SSE, but this work underestimates\nthe coupling strength by an order of magnitude, as we\nwill argue below.\nIn this paper we present an analytical and numeri-\ncal study of magnon-phonon interactions in bulk ferro-\nmagnetic insulators, where we take both the anisotropy-\nand the exchange-mediated magnon-phonon interactions\ninto account. By using diagrammatic perturbation the-\nory to calculate the magnon self-energy, we arrive at a\nwave-vector dependent expression of the magnon scat-\ntering rate, which is the inverse of the magnon quasi-\nparticle lifetime \u001cqp. The magnetic Grüneisen parameter\n\u0000m=@lnTC=@lnV[32, 33], where TCis the Curie tem-\nperature and Vthe volume of the magnet, gives direct\naccess to the exchange-mediated magnon-phonon inter-\naction parameter. We predict an enhancement in the\nphonon scattering of the Kittel mode at the touching\npoints of the two-magnon energy (of the Kittel mode and\na finite momentum magnon) and the longitudinal and\ntransverse phonon dispersions, for YIG at around 1:3 T\nand4:6 T. We also emphasize the difference in magnon\nlifetimesthatbroadenlightandneutronscatteringexper-\niments, and the transport lifetimes that govern magnon\nheat and spin transport.\nThe paper is organized as follows: in Sec. II we briefly\nreview the theory of acoustic magnons and phonons in\nferro-/ferrimagnets, particularly in YIG. In Sec. III we\nderive the exchange- and anisotropy-mediated magnon-\nphonon interactions for a cubic Heisenberg ferromagnet\nwith nearest neighbor exchange interactions in the long-\nwavelength limit. In Sec. IV we derive the magnon decay\nrate from the imaginary part of the magnon self-energy\nin a diagrammatic approach and in Sec. V we explain\nthe differences between the magnon quasi-particle and\ntransport lifetimes. Our numerical results for YIG are\ndiscussed in Sec. VI. Finally in Sec. VII we summarizearXiv:1906.01042v1  [cond-mat.str-el]  3 Jun 20192\nand discuss the main results of the present work. The va-\nlidity of our long-wavelength approximation is analyzed\nin Appendix A and in Appendix B we explain why sec-\nond order magnetoelastic couplings may be disregarded.\nIn Appendix C we briefly discuss the numerical methods\nused to evaluate the k-space integrals.\nII. MAGNONS AND PHONONS IN\nFERROMAGNETIC INSULATORS\nWithout loss of generality, we focus our treatment on\nyttrium iron garnet (YIG). The magnon band structure\nof YIG has been determined by inelastic neutron scatter-\ning [34–36] and by ab initio calculation of the exchange\nconstants [37]. The complete magnon spectral function\nhasbeencomputedforalltemperaturesbyatomisticspin\nsimulations [38], taking all magnon-magnon interactions\ninto account, but not the magnon-phonon scattering.\nThe pure phonon dispersion is known as well [29, 39]. In\nthe following, we consider the interactions of the acoustic\nmagnons from the lowest magnon band with transverse\nand longitudinal acoustic phonons, which allows a semi-\nanalytic treatment but limits the validity of our results\nto temperatures below 100 K. Since the low-temperature\nvalues of the magnetoelastic constants, sound velocities,\nand magnetic Grüneisen parameter are not available for\nYIG, we use throughout the material parameters under\nambient conditions.\nA. Magnons\nSpinsinteractwitheachotherviadipolarandexchange\ninteractions. We disregard the former since at the energy\nscaleEdip\u00190:02 meV [28] it is only relevant for long-\nwavelength magnons with wave vectors k.6\u0002107m\u00001\nand energies Ek=kB.0:2 K, which are negligible for\nthe thermal magnon transport in the temperature regime\nwe are interested in. The lowest magnon band can then\nbe described by a simple Heisenberg model on a course-\ngrained simple cubic ferromagnet with exchange interac-\ntionJ\nHm=\u0000J\n2X\nhi6=jiSi\u0001Sj\u0000X\nig\u0016BBSz\ni;(2.1)\nwhere the sum is over all nearest neighbors and ~Siis the\nspin operator at lattice site Ri. The lattice constant of\nthe cubic lattice or YIG is a= 12:376\u0017Aand the effective\nspin per unit cell ~S=~Msa3=(g\u0016B)\u001914:2~at room\ntemperature [28] ( S\u001920forT.50 K[40]), where the\ng-factorg\u00192,\u0016Bis the Bohr magneton and Msthe sat-\nuration magnetization. The parameter Jis an adjustable\nparameter that can be fitted to experiments or computed\nfrom first principles. Bis an effective magnetic field that\norients the ground-state magnetization vector to the z\naxis and includes the (for YIG small) magnetocrystallineanisotropyfield. The 1=Sexpansionofthespinoperators\nin terms of Holstein-Primakoff bosons reads [41],\nS+\ni=Sx+iSy\u0019p\n2S[bi+O(1=S)];(2.2)\nS\u0000\ni=Sx\u0000iSy\u0019p\n2Sh\nby\ni+O(1=S)i\n;(2.3)\nSz\ni=S\u0000by\nibi; (2.4)\nwhereby\niandbiare the magnon creation and annihilation\noperators with boson commutation ruleh\nbi;by\nji\n=\u000ei;j.\nThen\nHm!X\nkEkby\nkbk; (2.5)\nwhere the magnon operators by\nkandbkare defined by\nbi=1p\nNX\nkeik\u0001Ribk; (2.6)\nby\ni=1p\nNX\nke\u0000ik\u0001Riby\nk; (2.7)\nandNthe number of unit cells. The dispersion relation\nEk=g\u0016BB+ 4SJX\n\u000b=x;y;zsin2(k\u000ba=2) (2.8)\nbecomes quadratic in the long-wavelength limit ka\u001c1:\nEk=g\u0016BB+Eexk2a2; (2.9)\nwhereEex=SJ. WithEex=kB\u000240 K = 3:45 meV\nthe latter is a good approximation up to k0= 1=a\u0019\n8\u0002108m\u00001[34]. The effective exchange coupling is\nthenJ\u00190:24 meV. The lowest magnon band does not\ndepend significantly on temperature [38], which implies\nthatEex=SJdoes not depend strongly on temper-\nature. The temperature dependence of the saturation\nmagnetization and effective spin Sshould therefore not\naffect the low-energy exchange magnons significantly. By\nusing Eq. (2.9) in the following, our theory is valid for\nk.k0(see Fig. 1) or temperatures T.100 K. In this\nregime the cut-off of an ultraviolet divergence does not\naffect results significantly (see Appendix A). We disre-\ngard magnetostatic interactions that affect the magnon\nspectrum only for very small wave vectors since at low\ntemperatures the phonon scattering is not significant.\nB. Phonons\nWe expand the displacement Xiof the position riof\nunit cellifrom the equilibrium position Ri\nXi=ri\u0000Ri; (2.10)\ninto the phonon eigenmodes Xq\u0015,\nX\u000b\ni=1p\nNX\nq;\u0015e\u000b\nq\u0015Xq\u0015eiq\u0001Ri;(2.11)3\nwhere\u000b2 fx;y;zgandqa wave vector. We define\npolarizations \u00152f1;2;3gfor the elastic continuum [42]\neq1= (cos\u0012qcos\u001eq;cos\u0012qsin\u001eq;\u0000sin\u0012q);(2.12)\neq2=i(\u0000sin\u001eq;cos\u001eq;0); (2.13)\neq3=i(sin\u0012qcos\u001eq;sin\u0012qsin\u001eq;cos\u0012q);(2.14)\nwhere the angles \u0012qand\u001eqare the spherical coordinates\nof\nq=q(sin\u0012qcos\u001eq;sin\u0012qsin\u001eq;cos\u0012q);(2.15)\nwhich is valid for YIG up to 3 THz(12 meV) [29, 39].\nThe phonon Hamiltonian then reads\nHp=X\nq\u0015\u0014P\u0000q\u0015Pq\u0015\n2m+m\n2~2\"2\nq\u0015X\u0000q\u0015Xq\u0015\u0015\n;\n=X\nq\u0015\"q\u0015\u0012\nay\nq\u0015aq\u0015+1\n2\u0013\n; (2.16)\nwherethecanonicalmomenta Pq\u0015obeythecommutation\nrelations [Xq\u0015;Pq0\u00150] =i~\u000eq;\u0000q0\u000e\u0015\u00150and the mass of the\nYIG unit cell m=\u001aa3= 9:8\u000210\u000024kg[27]. The phonon\ndispersions for YIG then read\n\"q\u0015=~c\u0015jqj; (2.17)\nwherec1;2=ct= 3843 m=sis the transverse sound ve-\nlocity andc3=cl= 7209 m=sthe longitudinal velocity\nat room temperature [27]. In terms of phonon creation\nand annihilation operators\nXq\u0015=aq\u0015+ay\n\u0000q\u0015p\n2m\"q\u0015=~2; P q\u0015=1\nirm\"q\u0015\n2\u0010\naq\u0015\u0000ay\n\u0000q\u0015\u0011\n;\n(2.18)\nandh\naq\u0015;ay\nq0\u00150i\n=\u000eq;q0\u000e\u0015;\u00150.\nIn Fig. 1 we plot the longitudinal and transverse\nphonon and the acoustic magnon dispersion relations for\nYIG at zero magnetic field. The magnon-phonon inter-\naction leads to an avoided level crossing at points where\nmagnon and phonon dispersion cross, as discussed in\nRefs. [27] and [28].\nIII. MAGNON-PHONON INTERACTIONS\nWe derive in this section the magnon-phonon interac-\ntions due to the anisotropy and exchange interactions for\na cubic lattice ferromagnet.\nA. Phenomenological magnon-phonon interaction\nIn the long-wavelength/continuum limit ( k.k0) the\nmagnetoelastic energy to lowest order in the deviations\nof magnetization and lattice from equilibrium reads [23–\n26, 28]\n0.0 0.5 1.0 1.5\nk[109m−1]024681012Ek[meV]magnon model\nparabolic approximation\nlongitudinal acoustic phonon\ntransverse acoustic phononFigure 1. Dispersion relations of the acoustic phonons and\nmagnons in YIG at zero magnetic field.\nEme=n\nM2sZ\nd3rX\n\u000b\f[B\u000b\fM\u000b(r)M\f(r)\n+B0\n\u000b\f@M(r)\n@r\u000b\u0001@M(r)\n@r\f\u0015\nX\u000b\f(r);(3.1)\nwheren= 1=a3. The strain tensor X\u000b\fis defined in\nterms of the lattice displacements X\u000b,\nX\u000b\f(r) =1\n2\u0014@X\u000b(r)\n@r\f+@X\f(r)\n@r\u000b\u0015\n;(3.2)\nwith, for a cubic lattice [28],\nB\u000b\f=\u000e\u000b\fBk+ (1\u0000\u000e\u000b\f)B?; (3.3)\nB0\n\u000b\f=\u000e\u000b\fB0\nk+ (1\u0000\u000e\u000b\f)B0\n?: (3.4)\nB\u000b\fis caused by magnetic anisotropies and B0\n\u000b\fby the\nexchange interaction under lattice deformations. For\nYIG at room temperature [27, 33]\nBk=kB\u000247:8 K = 4:12 meV;(3.5)\nB?=kB\u000295:6 K = 8:24 meV;(3.6)\nB0\nk=a2=kB\u00022727 K = 235 meV ;(3.7)\nB0\n?=a2\u00190: (3.8)\nWe discuss the values for B0\nkandB0\n?in Sec. IIIC.\nB. Anisotropy-mediated magnon-phonon\ninteraction\nThe magnetoelastic anisotropy (3.1) is described by\nthe Hamiltonian [28],4\nHan\nmp=X\nq\u0015\u0002\n\u0000q\u0015b\u0000qXq\u0015+ \u0000\u0003\n\u0000q\u0015by\nqXq\u0015\u0003\n+1p\nNX\nq;k;k0\u000ek\u0000k0\u0000q;0X\n\u0015\u0000an\nkk0;\u0015by\nkbk0Xq\u0015\n+1p\nNX\nq;k;k0\u000ek+k0+q;0X\n\u0015\u0000bb\nkk0;\u0015bkbk0Xq\u0015\n+1p\nNX\nq;k;k0\u000ek+k0\u0000q;0X\n\u0015\u0000\u0016b\u0016b\nkk0;\u0015by\nkby\nk0Xq\u0015;(3.9)\nwith interaction vertices\n\u0000q\u0015=B?p\n2Sh\niqzex\nq\u0015+qzey\nq\u0015\n+ (iqx+qy)ez\nq\u0015\u0003\n; (3.10)\n\u0000an\nkk0;\u0015=Uk\u0000k0;\u0015; (3.11)\n\u0000bb\nkk0;\u0015=V\u0000k\u0000k0;\u0015; (3.12)\n\u0000\u0016b\u0016b\nkk0;\u0015=V\u0003\n\u0000k\u0000k0;\u0015; (3.13)\nand\nUq;\u0015=iBk\nSh\nqxex\nq\u0015+qyey\nq\u0015\u00002qzez\nq\u0015i\n;(3.14)\nVq;\u0015=iBk\nSh\nqxex\nq\u0015\u0000qyey\nq\u0015i\n+B?\nSh\nqyex\nq\u0015+qxey\nq\u0015i\n: (3.15)\nThe one magnon-two phonon process is of the same\norder in the total number of magnons and phonons as\nthe two magnon-one phonon processes, but its effect on\nmagnon transport is small, as shown in Appendix B.\nC. Exchange-mediated magnon-phonon interaction\nThe exchange-mediated magnon-phonon interaction is\nobtained under the assumption that the exchange inter-\nactionJijbetween two neighboring spins at lattice sites\nriandrjdepends only on their distance, which leads to\nthe expansion to leading order in the small parameter\n(jri\u0000rjj\u0000a)\nJij=J(jri\u0000rjj)\u0019J+J0\u0001(jri\u0000rjj\u0000a);(3.16)\nwhereais the equilibrium distance and J0=@J=@a.\nWith ri=Ri+XRi;the Heisenberg Hamiltonian (2.1)\nis modulated by\nHex\nmp=\u0000J0X\niX\n\u000b=x;y;z\u0000\nX\u000b\nRi+ae\u000b\u0000X\u000b\nRi\u0001\nSRi\u0001SRi+ae\u000b;\n(3.17)where e\u000bis a unit vectors in the \u000bdirection. Expanding\nthe displacements in terms of the phonon and magnon\nmodes\nHex\nmp=1p\nNX\nq;k;k0\u000ek\u0000k0\u0000q;0X\n\u0015\u0000ex\nkk0;\u0015by\nkbk0Xq\u0015;(3.18)\nwith interaction\n\u0000ex\nkk0;\u0015= 8iJ0SX\n\u000be\u000b\nk\u0000k0;\u0015sin\u0012k\u000ba\n2\u0013\nsin\u0012k0\n\u000ba\n2\u0013\n\u0002sin\u0012(k\u000b\u0000k0\n\u000b)a\n2\u0013\n\u0019iJ0a3SX\n\u000be\u000b\nk\u0000k0;\u0015k\u000bk\u000b0(k\u000b\u0000k0\n\u000b);(3.19)\nwhere the last line is the long-wavelength expansion. The\nmagnon-phonon interaction\n\u0000\u0016bb\nk;k0;\u0015= \u0000ex\nk;k0;\u0015+ \u0000an\nk;k0;\u0015 (3.20)\nconserves the magnon number, while (3.12) and (3.13) do\nnot. Phonon numbers are not conserved in either case.\nThe value of J0for YIG is determined by the magnetic\nGrüneisen parameter [32, 33]\n\u0000m=@lnTC\n@lnV=@lnJ\n@lnV=J0a\n3J;(3.21)\nwhereV=Na3is the volume of the magnet. The only\nassumption here is that the Curie temperature TCscales\nlinearly with the exchange constant J[43]. \u0000mhas been\nmeasured for YIG via the compressibility to be \u0000m=\n\u00003:26[32], and via thermal expansion, \u0000m=\u00003:13[33],\nso we set \u0000m=\u00003:2. For other materials the magnetic\nGrüneisen parameter is also of the order of unity and in\nmany cases \u0000m\u0019\u000010=3[32, 33, 44]. A recent ab initio\nstudy of YIG finds \u0000m=\u00003:1[45].\nComparing the continuum limit of Eq. (3.17) with the\nclassical magnetoelastic energy (3.1)\nB0\nk= 3\u0000mJS2a2=2; (3.22)\nwhereforYIG B0\nk=a2\u0019235 meV . Wedisregard B0\n?since\nit vanishes for nearest neighbor interactions by cubic lat-\ntice symmetry.\nThe coupling strength of the exchange-mediated\nmagnon-phonon interaction can be estimated from the\nexchange energy SJ0a\u0019Eex=SJ[31, 46] following\nAkhiezer et al.[47, 48]. Our estimate of SJ0a= 3\u0000mSJ\nis larger by 3\u0000m, i.e. one order of magnitude. Since the\nscattering rate is proportional to the square of the in-\nteraction strength, our estimate of the scattering rate is\na factor 100larger than previous ones. The assumption\nJ0a\u0019Jis too small to be consistent with the experi-\nmental Grüneisen constant [32, 33]. Ref. [3] educatedly\nguessedJ0a\u0019100J;which we now judge to be too large.5\nFigure2. Feynmandiagramsofinteractionsbetweenmagnons\n(solid lines) and phonons (dashed lines). The arrows indicate\nthe energy-momentum flow. (a) magnon-phonon interconver-\nsion, (b) magnon number-conserving magnon-phonon inter-\naction, (c) and (d) magnon number non-conserving magnon-\nphonon interactions.\nD. Interaction vertices\nThe magnon-phonon interactions in the Hamiltonian\n(3.9) are shown in Fig. 2 as Feynman diagrams. Fig. 2(a)\nillustrates magnon and phonon interconversion, which\nis responsible for the magnon-phonon hybridization and\nlevel splitting at the crossing of magnon and phonon dis-\npersions [27, 28]. The divergence of this diagram at the\nmagnon-phonon crossing points is avoided by either di-\nrect diagonalization of the magnon-phonon Hamiltonian\n[42] or by cutting-off the divergence by a lifetime param-\neter [31]. This process still generates enhanced magnontransport that is observable as magnon polaron anoma-\nlies in the spin Seebeck effect [22] or spin-wave excitation\nthresholds [49, 50], but these are strongly localized in\nphase space and disregarded in the following, where we\nfocus on the magnon scattering rates to leading order in\n1=Sof the scattering processes in Fig. 2(b)-(d).\nIV. MAGNON SCATTERING RATE\nHere we derive the magnon reciprocal quasi-particle\nlifetime\u001c\u00001\nqp=\ras the imaginary part of the wave vector\ndependent self-energy, caused by acoustic phonon scat-\ntering [28],\n\r(k) =\u00002\n~Im\u0006(k;Ek=~+i0+):(4.1)\nThis quantity is in principle observable by inelastic neu-\ntron scattering. The total decay rate\n\r=\rc+\rnc+\rother(4.2)\nis the sum of the magnon number conserving decay rate\n\rcand the magnon number non-conserving decay rate\n\rnc, which are related to the magnon-phonon scattering\ntime\u001cmpand the magnon-phonon dissipation time \u001cmr\nby\n\u001cmp=1\n\rc; \u001cmr=1\n\rnc: (4.3)\n\rotheris caused by magnon-magnon and magnon disorder\nscattering, thereby beyond the scope of this work.\nThe self-energy to leading order in the 1=Sexpansion\nis of second order in the magnon-phonon interaction [28],\n\u00062(k;i!) =1\nNX\nk0\u0015~2\f\f\f\u0000\u0016bb\nk;k0;\u0015\f\f\f2\n2m\"k\u0000k0;\u0015\u0014nB(\"k\u0000k0;\u0015)\u0000nB(Ek0)\ni~!+\"k\u0000k0;\u0015\u0000Ek0+1 +nB(\"k\u0000k0;\u0015) +nB(Ek0)\ni~!\u0000\"k\u0000k0;\u0015\u0000Ek0\u0015\n\u00001\nNX\nk0\u0015~2\f\f\f\u0000bb\nk;k0;\u0015\f\f\f2\n2m\"k\u0000k0;\u0015\u00141 +nB(\"k+k0;\u0015) +nB(Ek0)\ni~!+\"k+k0;\u0015+Ek0+nB(\"k+k0;\u0015)\u0000nB(Ek0)\ni~!\u0000\"k+k0;\u0015+Ek0\u0015\n; (4.4)\nwhere the magnon number conserving magnon-phonon\nscattering vertex \u0000\u0016bb\nk;k0;\u0015= \u0000ex\nk;k0;\u0015+ \u0000an\nk;k0;\u0015and the\nPlanck (Bose) distribution function nB(\") = (e\f\"\u00001)\u00001\nwith inverse temperature \f= 1=(kBT). The Feynman\ndiagrams representing the magnon number conserving\nand non-conserving contributions to the self-energy areshown in Fig. 3.\nWe write the decay rate in terms of four contributions\n\r(k) =\rc\nout(k) +\rnc\nout(k)\u0000\rc\nin(k)\u0000\rnc\nin(k);(4.5)\nwhereoutandindenote the out-scattering and in-\nscattering parts. The contributions to the decay rate\nread [28]6\nk\nqk-q\nk kk\nqq-k\nk k(a) (b)\nFigure 3. Feynman diagrams representing the self-energy\nEq. (4.4) due to (a) magnon number-conserving magnon-\nphonon interactions and (b) magnon number non-conserving\nmagnon-phonon interactions.\n\rc\nout(k) =\u0019~\nmNX\nq;\u0015\f\f\f\u0000\u0016bb\nk;k\u0000q;\u0015\f\f\f2\n\"q\u0015[(1 +nB(Ek\u0000q))nB(\"q\u0015)\u000e(Ek\u0000Ek\u0000q+\"q\u0015)\n+ (1 +nB(Ek\u0000q))(1 +nB(\"q\u0015))\u000e(Ek\u0000Ek\u0000q\u0000\"q\u0015)]; (4.6)\n\rc\nin(k) =\u0019~\nmNX\nq;\u0015\f\f\f\u0000\u0016bb\nk;k\u0000q;\u0015\f\f\f2\n\"q\u0015[nB(Ek\u0000q)(1 +nB(\"q\u0015))\u000e(Ek\u0000Ek\u0000q+\"q\u0015)\n+nB(Ek\u0000q)nB(\"q\u0015)\u000e(Ek\u0000Ek\u0000q\u0000\"q\u0015)]; (4.7)\n\rnc\nout(k) =\u0019~\nmNX\nq;\u0015\f\f\f\u0000bb\nk;q\u0000k;\u0015\f\f\f2\n\"q\u0015[nB(Eq\u0000k)(1 +nB(\"q\u0015))\u000e(Ek+Eq\u0000k\u0000\"q\u0015)]; (4.8)\n\rnc\nin(k) =\u0019~\nmNX\nq;\u0015\f\f\f\u0000bb\nk;q\u0000k;\u0015\f\f\f2\n\"q\u0015[(1 +nB(Eq\u0000k))nB(\"q\u0015)\u000e(Ek+Eq\u0000k\u0000\"q\u0015)]; (4.9)\nwhere the sum is over all momenta qin the Brillouin\nzone. Here the magnon/phonon annihilation rate is pro-\nportional to the Boson number nB, while the creation\nratescaleswith 1+nB. Forexample,intheout-scattering\nrate\rc\nout(k)theincomingmagnonwithmomentum kgets\nscattered into the state k\u0000qand a phonon is either ab-\nsorbedwithprobability \u0018nBoremittedwithprobability\n\u0018(1 +nB). The out- and in-scattering rates are related\nby the detailed balance\n\rc\nin(k)=\rc\nout(k) =\rnc\nin(k)=\rnc\nout(k) =e\u0000\fEk:(4.10)\nFor high temperatures kBT\u001dEk, we may expand the\nBose functions nB(Ek)\u0018kBT=E kand we find \rin\u0018\n\rout\u0018T2and\r=\rout\u0000\rin\u0018T. For low temperatures\nkBT\u001cEk, the out-scattering rate \rout!const:and\nthe in-scattering rate \rin\u0018e\u0000\fEk!0. The scattering\nprocesses (c) and (d) in Fig. 2 conserve energy and linear\nmomentum, but not angular momentum. A loss of an-gular momentum after integration over all wave vectors\ncorresponds to a mechanical torque on the total lattice\nthat contributes to the Einstein-de Haas effect [51].\nV. MAGNON TRANSPORT LIFETIME\nInthissectionwecomparethetransportlifetime \u001ctand\nthe magnon quasi-particle lifetime \u001cqpthat can be very\ndifferent [52–54], but, to the best of our knowledge, has\nnot yet been addressed for magnons. The magnon decay\nrate is proportional to the imaginary part of self energy,\nas shown in Eq. (4.1). On the other hand, the transport\nis governed by transport lifetime \u001ctin the Boltzmann\nequation that agrees with \u001cqponly in the relaxation time\napproximation. The stationary Boltzmann equation for7\nthe magnon distribution can be written as [3, 42]\n@fk(r)\n@r\u0001@Ek\n@(~k)= \u0000in[f]\u0000\u0000out[f];(5.1)\nwherefk(r)is the magnon distribution function. The in\nandoutcontributions to the collision integral are related\nto the previously defined in- and out-scattering rates by\n\u0000in[f] = (1 +fk)\rin[f]; (5.2)\n\u0000out[f] =fk\rout[f]; (5.3)\nwhere the equilibrium magnon distribution nB(Ek)is re-\nplaced by the non-equilibrium distribution function fk.\nThe factor (1 +fk)corresponds to the creation of a\nmagnon with momentum kin the in-scattering process\nand the factor fkto the annihilation in the out-scattering\nprocess. The phonons are assumed to remain at thermal\nequilibrium, so we disregard the phonon drift contribu-\ntion that is expected in the presence of a phononic heat\ncurrent.\nMagnon transport is governed by three linear response\nfunctions, i.e. spin and heat conductivity and spin See-\nbeck coefficient [42]. These can be obtained from the ex-\npansion of the distribution function in terms of temper-\nature and chemical potential gradients and correspond\nto two-particle Green functions with vertex corrections,\nthat reflect the non-equilibrium in-scattering processes,\ncaptured by a transport lifetime \u001ctthat can be different\nfrom the quasi-particle (dephasing) lifetime \u001cqpdefined\nby the self-energy. We define the transport life time of\na magnon with momentum kin terms of the collision\nintegral\n\u0000out[f]\u0000\u0000in[f] =1\n\u001ck;t[f](fk(r)\u0000f0;k);(5.4)\nwithf0;k=nB(Ek)and we assume a thermalized quasi-\nequilibrium distribution function\nfk(r) =nB\u0012Ek\u0000\u0016(r)\nkBT(r)\u0013\n; (5.5)\nwhere\u0016is the magnon chemical potential. We linearize\nthe function fkin terms of small deviations \u000efkfrom\nequilibrium f0;k,\n\u000efk=fk\u0000f0;k: (5.6)\nleading to [3]\n\u000efk=\u001ck;t[f]@f0;k\n@Ek@Ek\n@(~k)\u0001\u0012\nr\u0016+Ek\u0000\u0016\nTrT\u0013\n;(5.7)\nwhere the gradients of chemical potential r\u0016and tem-\nperature rTdrive the magnon current. In the relax-\nation time approximation we disregard the dependence\nof\u001ck;t[f]on\u000efand recover the quasi-particle lifetime\n\u001ck;t!\u001ck;qp.Tofirstorderinthephononoperatorsandsecondorder\nin the magnon operators the collision integral for magnon\nnumber non-conserving processes,\n\u0000nc\nout[f]\u0000\u0000nc\nin[f]\n=\u0019~\nmNX\nq\u0015j\u0000bb\nk;q\u0000k;\u0015j2\n\"q\u0015\u000e(Ek+Eq\u0000k\u0000\"q\u0015)\n\u0002[(1 +nq\u0015)fkfq\u0000k\u0000nq\u0015(1 +fq\u0000k)(1 +fk)];\n(5.8)\nwhere the interaction vertex \u0000bb\nk;k0;\u0015is given by Eq. (3.12)\nandnq\u0015=nB(\"q\u0015). By using the expansion (5.6) in the\ncollision integral that vanishes at equilibrium,\n\u0000out[f0]\u0000\u0000in[f0] = 0; (5.9)\nwe arrive at\n1\n\u001cnc\nk;t=\u0019~\nmNX\nq\u0015j\u0000bb\nk;q\u0000k;\u0015j2\n\"q\u0015\u000e(Ek+Eq\u0000k\u0000\"q\u0015)\n\u0002\u0014\nnB(Ek\u0000q)\u0000nq\u0015+\u000efq\u0000k\n\u000efk(nB(Ek)\u0000nq\u0015)\u0015\n:\n(5.10)\nFor the magnon number conserving process the deriva-\ntion is similar and we find\n1\n\u001cc\nk;t=\u0019~\nmNX\nq\u0015j\u0000\u0016bb\nk;k\u0000q;\u0015j2\n\"q\u0015\"\n\u000e(Ek\u0000Ek\u0000q+\"q\u0015)\n\u0002\u0012\nnq\u0015\u0000nB(Ek\u0000q)\u0000\u000efk\u0000q\n\u000efk(nB(Ek) +nq\u0015+ 1)\u0013\n+\u000e(Ek\u0000Ek\u0000q\u0000\"q\u0015)\n\u0002\u0012\n1 +nB(Ek\u0000q) +nq\u0015+\u000efk\u0000q\n\u000efk(nB(Ek)\u0000nq\u0015)\u0013#\n;\n(5.11)\nwith interaction vertex \u0000\u0016bb\nk;k0;\u0015given by Eq. (3.20). Due\nto the\u000efk\u0000q=\u000efkterm this is an integral equation. It\ncan be solved iteratively to generate a geometric series\nreferred to as vertex correction in diagrammatic theo-\nries. By simply disregarding the in-scattering with terms\n\u000efk\u0000q=\u000efkthetransportlifetimereducestothethequasi-\nparticle lifetime of the self-energy. We leave the general\nsolution of this integral equation for future work, but\nargue in Sec. VID that the vertex corrections are not\nimportant in our regime of interest.\nVI. NUMERICAL RESULTS\nA. Magnon decay rate\nIn the following we present and analyze our results\nfor the magnon decay rates in YIG. We first consider8\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nk[109m−1]01020304050γc(k) [106s−1]\n(100)\n(001)\n(110)\n(111)\n(011)0.0 0.5 1.0−0.10.00.10.2\nFigure 4. Magnon decay rate in YIG due to magnon-phonon\ninteractions for magnons propagating along various directions\natT= 50 KandB= 0. We denote the propagation direction\nby(lmn), i.e.lex+mey+nez. The inset shows the relative\ndeviation\u000e\rc=\rcfrom the (100) direction.\n0.00 0.01 0.02 0.03 0.04 0.05\nkx[109m−1]0.000.050.100.150.200.250.300.350.40γc(k) [103s−1]γc, total\nγc, anisotropy\nγc, exchange\nFigure 5. Comparison of the contributions from exchange-\nmediated and anisotropy-mediated magnon-phonon interac-\ntions to the magnon number conserving scattering rate \rcat\nT= 50 KandB= 0.\nthe case of vanishing effective magnetic field ( B= 0)\nand discuss the magnetic field dependence in Sec. VIC.\nSince our model is only valid in the long-wavelength ( k<\n8×108m\u00001) and low-temperature ( T.100 K) regime,\nwe focus first on T= 50 Kand discuss the temperature\ndependence in Sec. VIB.\nInFig.4weshowthemagnonnumberconservingdecay\nrate\rc(k), which is on the displayed scale dominated by\nthe exchange-mediated magnon-phonon interaction and\nis isotropic for long-wavelength magnons.\nIn Fig. 5 we compare the contribution from the\nexchange-mediated magnon-phonon interaction ( \rc\u0018\nk4) and from the anisotropy-mediated magnon-phonon\ninteraction ( \rc\u0018k2). We observe a cross-over at\nk\u00194\u0002107m\u00001: for much smaller wave numbers, the\nexchange contribution can be disregarded and for larger\nwave numbers the exchange contribution becomes domi-\nnant.\nThe magnon number non-conserving decay rate \rncin\nFig. 6 is much smaller than the magnon-conserving one.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nk[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](100)\n(001)\n(110)\n(111)\n(011)Figure 6. Magnon decay rate in YIG due to magnon num-\nber non-conserving magnon-phonon interactions for magnons\npropagating along various directions at T= 50 KandB= 0.\nThis is consistent with the low magnetization damping\nof YIG, i.e. the magnetization is long-lived. We observe\ndivergent peaks at the crossing points (shown in Fig. 1)\nwith the exception of the (001) direction. These diver-\ngences occur when magnons and phonons are degenerate\natk= 0:48\u0002109m\u00001(1:2 meV) andk= 0:9\u0002109m\u00001\n(4:3 meV), respectively, at which the Boltzmann formal-\nism does not hold; a treatment in the magnon-polaron\nbasis [42] or a broadening parameter [31] would get rid\nof the singular behavior. The divergences are also sup-\npressed by arbitrarily small effective magnetic fields (see\nSec. VIC). There are no peaks along the (001) direc-\ntion because in the (001) direction the vertex function\nVq;\u0015(see Eq. (3.15)) vanishes for q= (0;0;kz). For\nk >~cl=(D(p\n8\u00002)) = 1:085\u0002109m\u00001the decay rate\n\rncvanishes because the decay process does not conserve\nenergy (\u000e(Ek+Eq\u0000k\u0000\"q\u0015) = 0).\nB. Temperature dependence\nAbove we focused on T= 50 Kand explained that we\nexpect a linear temperature dependence of the magnon\ndecay rates at high, but not low temperatures. Fig. 7\nshowsourresultsforthetemperaturedependenceat kx=\n108m\u00001. Deviations from the linear dependence at low\ntemperatures occurs when quantum effects set in, i.e. the\nRayleigh-Jeans distribution does not hold anymore,\n1\ne\"=(kBT)\u000016\u0019kBT\n\": (6.1)\nC. Magnetic field dependence\nThe numerical results presented above are for a mono-\ndomain magnet in the limit of small applied magnetic\nfields. A finite magnetic field Balong the magnetization\ndirectioninducesanenergygap g\u0016BBinthemagnondis-\npersion, which shifts the positions of the magnon-phonon9\n0 2 4 6 8 10\nT[K]0.00.10.20.30.40.50.60.7γ[103s−1]γc\nγnc\nFigure7. Temperaturedependenceofthemagnondecayrates\n\rncand\rcatB= 0,kx= 108m\u00001andky=kz= 0, i.e.\nalong (100).\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nkx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1]B= 0 T\nB= 0.1 T\nB= 0.5 T\nB= 1 T\nB= 2 T\nFigure 8. Magnetic field dependence of the magnon number\nnon-conserving magnon decay rate in YIG at T= 50 Kwith\nmagnon momentum along (100).\ncrossingpointsto longerwavelengths. Themagneticfield\nsuppresses the (unphysical) sharp peaks at the crossing\npoints (see Fig. 8) that are caused by the divergence of\nthePlanckdistributionfunctionforavanishingspinwave\ngap.\nIn the magnon number conserving magnon-phonon\ninteractions, the magnetic field dependence cancels in\nthe delta function and enters only in the Bose func-\ntion vianB(magnetic freeze-out). Fig. 9 shows that\nthe magnetic field mainly affects magnons with energies\n.2g\u0016BB= 0:23(B=T) meV.\nAs shown in Fig. 10 the magnon decay by phonons\ndoes not vanish for the k= 0Kittel mode, but only\nin the presence of a spin wave gap E0=g\u0016BB. Both\nmagnon conserving and non-conserving scattering pro-\ncesses contribute. The divergent peaks at B\u00191:3 Tand\nB\u00194:6 Tin\rncare caused by energy and momentum\nconservation in the two-magnon-one-phonon scattering\nprocess,\n\u000e(Ek=0+Eq\u0000\"q\u0015) =\u000e(2g\u0016BB+Eexq2a2\u0000~c\u0015q);(6.2)\nwhen the gradient of the argument of the delta function\n0.0 0.1 0.2 0.3 0.4 0.5\nkx[109m−1]0246810δγc/γcB= 1 T\nB= 10 TFigure 9. Relative deviation \u000e\rc=\rcfrom theB= 0result of\nthe magnon number conserving magnon decay rate in YIG at\nT= 50 Kwith magnon momentum along (100).\nvanishes,\nrq(Ek=0+Eq\u0000\"q\u0015) = 0; (6.3)\ni.e., the two-magnon energy Ek=0+Eqtouches either the\ntransverse or longitudinal phonon dispersion \"q\u0015. The\ntotal energy of the two magnons is equivalent to the en-\nergy of a single magnon with momentum qbut in a field\n2B, resulting in the divergence at fields that are half of\nthose for the magnon-polaron observed in the spin See-\nbeck effect [31, 42]. The two-magnon touching condition\ncan be satisfied in all directions of the phonon momen-\ntumq, which therefore contributes to the magnon decay\nrate when integrating over the phonon momentum q. For\nk6= 0this two-magnon touching condition can only be\nfulfilled for phonons along a particular direction and the\ndivergence is suppressed.\nThe magnon decay rate is related to the Gilbert damp-\ning\u000bkas~\rk= 2\u000bkEk[55]. We find that phonons\ncontribute only weakly to the Gilbert damping, \u000bnc\n0=\n~\rnc\n0=(2E0)\u001810\u00008atT= 50 K, which is much smaller\nthan the total Gilbert damping \u000b\u001810\u00005in YIG, but\nthe peaks at 1:3 Tand4:6 Tmight be observable. The\nphonon contribution to the Gilbert damping scales lin-\nearly with temperature, so is twice as large at 100 K. At\nlow temperatures ( T.100 K) Gilbert damping in YIG\nhas been found to be caused by two-level systems [56]\nand impurity scattering [40], while for higher tempera-\ntures magnon-phonon [57] and magnon-magnon scatter-\ning involving optical magnons [34] have been proposed to\nexplain the observed damping. Enhanced damping as a\nfunction of magnetic field at higher temperatures might\nreveal other van Hove singularities in the joint magnon-\nphonon density of states.\nD. Magnon transport lifetime\nWe do not attempt a full solution of the integral equa-\ntions (5.10) and (5.11) for the transport lifetime. How-\never, we can still estimate its effect by the observation10\n0 1 2 3 4 5 6\nMagnetic field [T]01020304050γ[103s−1]γc(k= 0)\nγnc(k= 0)\nFigure 10. Magnetic field dependence of the magnon decay\nrates in YIG at k= 0andT= 50 K.\nthat the ansatz \u001c\u00001\nk;t\u0018kncan be an approximate solu-\ntion of the Boltzmann equation with in-scattering.\nOur results for the magnon number conserving interac-\ntion are shown in Fig. 11 (for rT= 0and finite r\u0016jjex),\nwhere\rt=\u001c\u00001\nt. We consider the cases n= 0;2;4,\nwheren= 0or\u001ck;t= const:would be the solution for\na short-range scattering potential. For very long wave-\nlengths (k.4\u0002107m\u00001) the inverse quasi-particle life-\ntime\u001c\u00001\nk;qp\u0018k2and for shorter wavelengths \u001c\u00001\nk;qp\u0018k4.\nn= 2is a self-consistent solution only for very small\nk.4\u0002107m\u00001, while\u001c\u00001\nk;qp\u0018k4is a good ansatz up\ntok.0:3\u0002109m\u00001. We see that the transport life-\ntime approximately equals the quasi-particle lifetime in\nthe regime of the validity of the n= 4power law.\nFor the magnon number non-conserving processes in\nFig. 12 the quasi-particle lifetime behaves as \u001c\u00001\nk;qp\u0018k2.\nThe ansatz n= 2turns out to be self-consistent and we\nsee deviations of the transport lifetime from the quasi-\nparticlelifetimefor k&5\u0002107m\u00001. Theplotonlyshows\nour results for k<1\u0002108m\u00001because our assumption\nof an isotropic lifetime is not valid for higher momenta\nin this case.\nWe conclude that for YIG in the long-wavelength\nregime the magnon transport lifetime (due to magnon-\nphonon interactions) should be approximately the same\nas the quasi-particle lifetime, but deviations at shorter\nwavelengths require more attention.\nVII. SUMMARY AND CONCLUSION\nWe calculated the decay rate of magnons in YIG\ninduced by magnon-phonon interactions in the long-\nwavelength regime ( k.1\u0002109m\u00001). Our model\ntakes only the acoustic magnon and phonon branches\ninto account and is therefore valid at low to intermedi-\nate temperatures ( T.100 K). The exchange-mediated\nmagnon-phonon interaction has been recently identified\nas a crucial contribution to the overall magnon-phonon\ninteraction in YIG at high temperatures [3, 29, 45]. We\nemphasize that its coupling strength can be derived from\n0.0 0.1 0.2 0.3 0.4 0.5\nkx[109m−1]0.00.51.01.52.0γc\nt(k) [106s−1]quasi-particle\n1/τ∼k0\n1/τ∼k2\n1/τ∼k4Figure 11. Inverse of the magnon transport lifetime in YIG\n(with magnon momentum along (100)) due to magnon num-\nber conserving magnon-phonon interactions at T= 50 Kand\nB= 0for magnons along the (100) direction.\n0.00 0.02 0.04 0.06 0.08 0.10\nkx[109m−1]0.00.20.40.60.81.01.21.41.6γnc\nt(k) [103s−1]quasi-particle\n1/τ∼k0\n1/τ∼k2\n1/τ∼k4\nFigure 12. Inverse of the magnon transport lifetime in YIG\n(with magnon momentum along (100)) due to magnon num-\nber non-conserving interactions at T= 50 KandB= 0.\nexperimental values of the magnetic Grüneisen parame-\nter\u0000m=@lnTC=@lnV[32, 33]. In previous works this\ninteraction has been either disregarded [28], underesti-\nmated [29, 46], or overestimated [3].\nIn the ultra-long-wavelength regime the wave vector\ndependent magnon decay rate \r(k)is determined by the\nanisotropy-mediated magnon-phonon interaction with\n\r(k)\u0018k2, while for shorter wavelengths k&4\u0002107m\u00001\nthe exchange-mediated magnon-phonon interaction be-\ncomes dominant, which scales as \r(k)\u0018k4. The magnon\nnumber non-conserving processes are caused by spin-\norbit interaction, i.e., the anisotropy-mediated magnon-\nphonon interaction, and are correspondingly weak.\nIn a finite magnetic field the average phonon scatter-\ningcontribution, fromthemechanismunderstudy, tothe\nGilbert damping of the k= 0macrospin Kittel mode is\nabout three orders of magnitude smaller than the best\nvalues for the Gilbert damping \u000b\u001810\u00005. However, we\npredict peaks at 1:3 Tand4:6 T, that may be experi-\nmentally observable in high-quality samples.\nThe magnon transport lifetime, which is given by the\nbalance between in- and out-scattering in the Boltz-11\nmann equation, is in the long-wavelength regime approx-\nimately the same as the quasi-particle lifetime. However,\nthe magnon quasi-particle and transport lifetime differ\nmore significantly at shorter wavelengths. A theory for\nmagnon transport at room temperature should therefore\ninclude the “vertex corrections”.\nA full theory of magnon transport at high temperature\nrequires a method that takes the full dispersion relations\nof acoustic and optical phonons and magnons into ac-\ncount. This would also require a full microscopic descrip-\ntion of the magnon-phonon interaction, since the magne-\ntoelastic energy used here only holds in the continuum\nlimit.\nACKNOWLEDGMENTS\nN. V-S thanks F. Mendez for useful discussions. This\nwork is part of the research program of the Stichting voor\nFundamenteel Onderzoek der Materie (FOM), which is\nfinancially supported by the Nederlandse Organisatie\nvoor Wetenschappelijk Onderzoek (NWO) as well as a\nGrant-in-Aid for Scientific Research on Innovative Area,\n”Nano Spin Conversion Science” (Grant No. 26103006),\nCONICYT-PCHA/Doctorado Nacional/2014-21140141,\nFondecyt Postdoctorado No. 3190264, and Fundamen-\ntal Research Funds for the Central Universities.\nAppendix A: Long-wavelength approximation\nThe theory is designed for magnons with momen-\ntumk < 0:8\u0002109m\u00001and phonons with momen-\ntumq < 2:5\u0002109m\u00001(corresponding to phonon en-\nergies/frequencies \u001412 meV/3 THz), but relies on high-\nmomentum cut-off parameters kcbecause of the assump-\ntion of quadratic/linear dispersion of magnon/phonons.\nWe see in Fig. 13 that the scattering rates only weakly\ndepend onkc.\nThe dependence of the scattering rate on the phonon\nmomentum cut-off qcis shown in Fig. 14. qc= 3:15\u0002\n109m\u00001corresponds to an integration over the whole\nBrillouin zone, approximated by a sphere. From these\nconsiderations we estimate that the long-wavelength ap-\nproximation is reliable for k.8\u0002108m\u00001. Opti-\ncal phonons (magnons) that are thermally excited for\nT?100 K (300 K) are not considered here.\nAppendix B: Second order magnetoelastic coupling\nThe magnetoelastic energy is usually expanded only to\nfirst order in the displacement fields. Second order terms\ncan become important e.g. when the first order terms\nvanish. Thisisthecaseforone-magnontwo-phononscat-tering processes. The first order term\nX\nq\u0015\u0002\n\u0000q\u0015b\u0000qXq\u0015+ \u0000\u0003\n\u0000q\u0015by\nqXq\u0015\u0003\n(B1)\nonlycontributeswhenphononandmagnonmomentaand\nenergies cross, giving rise to magnon polaron modes [42].\nIn other areas of reciprocal space the second order term\nshould therefore be considered. Eastman [58, 59] derived\nthe second-order magnetoelastic energy and determined\nthecorrespondingcouplingconstantsforYIG.Inmomen-\ntum space, the relevant contribution to the Hamiltonian\nis of the form\nH2p1m=1p\nNX\nk;q1;\u00151;q2;\u00152\u0000\n\u000eq1+q2+k;0\u0000b\nq1\u00151;q2\u00152Xq1\u00151Xq2\u00152bk\n+\u000eq1+q2\u0000k;0\u0000\u0016b\nq1\u00151;q2\u00152Xq1\u00151Xq2\u00152by\nk\u0011\n;(B2)\nwhere the interaction vertices are symmetrized,\n\u0000b\nq1\u00151;q2\u00152=1\n2\u0010\n~\u0000b\nq1\u00151;q2\u00152+~\u0000b\nq2\u00152;q1\u00151\u0011\n;(B3)\nand obey\n\u0000b\nq1\u00151;q2\u00152=\u0010\n\u0000\u0016b\n\u0000q1\u00151;\u0000q2\u00152\u0011\u0003\n: (B4)\nThe non-symmetrized vertex function is\n~\u0000b\nq1\u00151;q2\u00152=1\na2p\n2S[B144(iI1\u0000I1;x$y)\n+B155(iI2\u0000I2;x$y)\n+B456(iI3\u0000I3;x$y)]; (B5)\nwith\nI1=a2ex\nq1\u00151qx\n1h\ney\nq2\u00152qz\n2+ez\nq2\u00152qy\n2i\n;(B6)\nI2=a2h\ney\nq1\u00151qy\n1+ez\nq1\u00151qz\n1i\n\u0002h\ney\nq2\u00152qz\n2+ez\nq2\u00152qy\n2i\n; (B7)\nI3=a2\u0002\nex\nq1\u00151qz\n1+ez\nq1\u00151qx\n1\u0003\n\u0002h\nex\nq2\u00152qy\n2+ey\nq2\u00152qx\n2i\n; (B8)\nandx$ydenotes an exchange of xandy. The relevant\ncoupling constants in YIG are [58, 59]\nB144=\u00006\u000648 meV; (B9)\nB155=\u000044\u00066 meV; (B10)\nB456=\u000032\u00068 meV: (B11)\nThe magnon self-energy (see Fig. 15) reads\n\u00062p1m(k;i!) =\u00002\nNX\nq1;\u00151;q2;\u001521\n\fX\n\n\u000eq1+q2+k;0\n\u0002\f\f\u0000b\nq1\u00151;q2\u00152\f\f2F\u00151(q1;\n)F\u00152(q2;\u0000\n\u0000!):\n(B12)12\n0.0 0.2 0.4 0.6 0.8 1.0\nkx[109m−1]0102030405060γc(k) [106s−1](a)\nkc= 3.15×109m−1\nkc= 0.8×109m−1\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nkx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](b)\nkc= 3.15×109m−1\nkc= 0.8×109m−1\nFigure 13. Dependence the magnon decay rate along (100) on the high magnon momentum cut-off kcfor the (a) magnon\nnumber conserving ( \rc) and (b) non-conserving ( \rnc) contributions at T= 50 KandB= 0.\n0.0 0.2 0.4 0.6 0.8 1.0\nkx[109m−1]0102030405060γc(k) [106s−1](a)\nqc= 3.15×109m−1\nqc= 2.5×109m−1\nqc= 2×109m−1\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nkx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](b)\nqc= 3.15×109m−1\nqc= 2×109m−1\nFigure 14. Dependence the magnon decay rate along (100) on the high phonon momentum cut-off qcfor the (a) magnon number\nconserving ( \rc) and (b) non-conserving ( \rnc) contributions at T= 50 KandB= 0.\nwith phonon propagator\nF\u0015(q;\n) =~2\nm1\n~2\n2+\"2\nq\u0015: (B13)\nand leads to a magnon decay rate\n\rnc\n2p(k) =\u00002\n~Im\u0006 2p1m(k;i!!Ek=~+i0+)\n=\u0019~3\nm2NX\nq1;\u00151;q2;\u00152\u000eq1+q2+k;01\n\"1\"2\f\f\u0000b\nq1\u00151;q2\u00152\f\f2\n\u0002f2\u000e(Ek+\"1\u0000\"2) [n1\u0000n2]\n+\u000e(Ek\u0000\"1\u0000\"2) [1 +n1+n2]g; (B14)\nwhere\nn1=nB(\"q1\u00151); n2=nB(\"q2\u00152); (B15)\n\"1=\"q1\u00151; \"2=\"q2\u00152: (B16)\nThe first term in curly brackets on the right-hand-side\nof Eq. (B14) describes annihilation and creation of a\nphonon as a sum of out-scattering minus in-scattering\ncontributions,\nn1(1 +n2)\u0000(1 +n1)n2=n1\u0000n2;(B17)while the second term can be understood in terms of\nout-scattering by the creation of two phonons and the\nin-scattering by annihilation of two phonons,\n(1 +n1)(1 +n2)\u0000n1n2= 1 +n1+n2:(B18)\nFor this one-magnon-two-phonon process the quasi-\nparticle and the transport lifetimes are the same,\n\u001ct=\u001cqp; (B19)\nsince this process involves only a single magnon that is\neither annihilated or created. The collision integral is\nthen independent of the magnon distribution of other\nmagnons and the transport lifetime reduces to the quasi-\nparticle lifetime.\nThe two-phonon contribution to the magnon scatter-\ning rate in YIG at T= 50 Kand along (100) direction\nas shown in Fig. 16 is more than two orders of magni-\ntude smaller than that from one-phonon processes and\ntherefore disregarded in the main text. The numerical\nresults depend strongly on the phonon momentum cutoff\nqc, even in the long-wavelength regime, which implies\nthat the magnons in this process dominantly interact13\nk k\nq'q\nFigure 15. Feynman diagram representing the self-energy\nEq. (B12) due to one-magnon-two-phonon processes.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nkx[109m−1]0.00.10.20.30.40.5γnc\n2p(k) [103s−1]qc= 3.15×109m−1\nqc= 2.5×109m−1\nqc= 2×109m−1\nFigure 16. Two-phonon contribution to the magnon number\nnon-conserving magnon scattering rate with magnon momen-\ntum along (100) for different values of the phonon momentum\ncutoffqcatT= 50 KandB= 0.\nwith short-wavelength, thermally excited phonons. In-\ndeed, the second order magnetoelastic interaction (B5) is\nquadratic in the phonon momenta, which favors scatter-\ningwithshort-wavelengthphonons. Ourlong-wavelength\napproximation therefore becomes questionable and the\nresults may be not accurate at T= 50 K, but this should\nnot change the main conclusion that we can disregard\nthese diagrams.\nOur finding that the two-phonon contributions are\nso small can be understood in terms of the dimension-\nful prefactors of the decay rates (Eqs. (4.8-4.9) and\n(B14)): The one-phonon decay rate is proportional to\n~=(ma2)\u00197\u0002106s\u00001, while the two-phonon decay\nrate is proportional to ~3=(m2a4\")\u001933 s\u00001, where\n\"\u00191 meVis a typical phonon energy. The coupling con-\nstants for the magnon number non-conserving processes\nareBk;?\u00185 meVwhile the strongest two phonon cou-\npling which enhances the two-phonon process by about\na factor 100, but does not nearly compensate the pref-\nactor. The two phonon process is therefore three orders\nof magnitudes smaller than the contribution of the one\nphonon process. The physical reason appears to be the\nlarge mass density of YIG, i.e. the heavy yttrium atoms.\nAppendix C: Numerical integration\nThe magnon decay rate is given be the weighted den-\nsity of statesI=Z\nBZd3qf(q)\u000e(\"(q)); (C1)\nthat contain the Dirac delta function \u000e(\")that can be\neliminated to yield\nI=X\nqiZ\nAid2qf(q)\njr\"(q)j; (C2)\nwhere the qiare the zeros of \"(q)andAithe surfaces\ninside the Brillouin zone with \"(q) =\"(qi). The calcu-\nlation these integrals is a standard numerical problem in\ncondensed matter physics.\nFor aspherical Brillouin zone of radius qcand spherical\ncoordinates (q;\u0012;\u001e ),\nI=Z\u0019\n0d\u0012Z2\u0019\n0d\u001eZqc\n0dqq2sin(\u0012)f(q;\u0012;\u001e )\u000e(\"(q;\u0012;\u001e )):\n(C3)\nWhen\"(qi;\u0012;\u001e) = 0\n\u000e(\"(q;\u0012;\u001e )) =X\nqi(\u0012;\u001e)\u000e(q\u0000qi(\u0012;\u001e))\nj\"0(qi(\u0012;\u001e);\u0012;\u001e)j;(C4)\nwhere\"0=@\"=@qand\nI=Z\u0019\n0d\u0012Z2\u0019\n0d\u001eX\nqi(\u0012;\u001e)<qcq2\ni(\u0012;\u001e) sin(\u0012)\n\u0002f(qi(\u0012;\u001e);\u0012;\u001e)\nj\"0(qi(\u0012;\u001e);\u0012;\u001e)j;(C5)\nwhich is particularly useful when the zeros of \"(q;\u0012;\u001e )\ncan be calculated analytically for linear and quadratic\ndispersion relations.\nWe can also evaluate the integral Ifully numerically\nby broadening the delta function [60] e.g. replacing it by\na Gaussian [60],\n\u000e(\")!1p\u0019\u001bexp\u0012\n\u0000\"2\n\u001b2\u0013\n; (C6)\nwhere\u001bis the broadening parameter. 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Peremohy, Kyiv, 03056, Ukraine\n(Dated: 5 May 2020)\nWe propose and analyze theoretically a class of energy-efficient magneto-elastic devices for analogue signal processing.\nThe signals are carried by transverse acoustic waves while the bias magnetic field controls their scattering from a\nmagneto-elastic slab. By tuning the bias field, one can alter the resonant frequency at which the propagating acoustic\nwaves hybridize with the magnetic modes, and thereby control transmission and reflection coefficients of the acoustic\nwaves. The scattering coefficients exhibit Breit-Wigner/Fano resonant behaviour akin to inelastic scattering in atomic\nand nuclear physics. Employing oblique incidence geometry, one can effectively enhance the strength of magneto-\nelastic coupling, and thus countermand the magnetic losses due to the Gilbert damping. We apply our theory to discuss\npotential benefits and issues in realistic systems and suggest routes to enhance performance of the proposed devices.\nOptical and, more generally, wave-based computing\nparadigms gain momentum on a promise to replace and com-\nplement the traditional semiconductor-based technology.1The\nenergy savings inherent to non-volatile memory devices has\nspurred the rapid growth of research in magnonics,2,3in which\nspin waves4are exploited as a signal or data carrier. Yet, the\nprogress is hampered by the magnetic loss (damping).5,6In-\ndeed, the propagation distance of spin waves is rather short in\nferromagnetic metals while low-damping magnetic insulators\nare more difficult to structure into nanoscale devices. In con-\ntrast, the propagation distance of acoustic waves is typically\nmuch longer than that of spin waves at the same frequencies.7\nHence, their use as the signal or data carrier could reduce the\npropagation loss to a tolerable level. Notably, one could con-\ntrol the acoustic waves using a magnetic field by coupling\nthem to spin waves within magnetostrictive materials.8–10To\nminimize the magnetic loss, the size of such magneto-acoustic\nfunctional elements should be kept minimal. This implies\ncoupling propagating acoustic waves to confined spin wave\nmodes of finite-sized magnetic elements. As we show below\nthis design idea opens a route towards hybrid devices combin-\ning functional benefits of magnonics2,3with the energy effi-\nciency of phononics.7,11,12\nThe phenomena resulting from interaction between coher-\nent spin and acoustic waves have already been addressed\nin the research literature: the spin wave excitation of prop-\nagating acoustic waves7,13–15and vice versa,8,16–18acous-\ntic parametric pumping of spin waves,19–21magnon-phonon\ncoupling in cavities22–24and mode locking,25magnonic-\nphononic crystals,26,27Bragg scattering of spin waves from a\nsurface acoustic wave induced grating,28–30topological prop-\nerties of magneto-elastic excitations,15,31acoustically driven\nspin pumping and spin Seebeck effect,32,33and optical excita-\ntion and detection of magneto-acoustic waves.34–40However,\nstudies of the interaction between propagating acoustic waves\nand spin wave modes of finite-sized magnetic elements, which\nare the most promising for applications, have been relatively\nscarce to date.10,34,36,39\nHere, we explore theoretically the class of magneto-\na)Electronic mail: V .V .Kruglyak@exeter.ac.uk\nθHB\nM\nI\nTω\nRω\nM NM NM\nFIG. 1: The prototypical magneto-elastic resonator is a thin\nmagnetic slab (M) of width d, biased by an external field HB,\nand embedded into a non-magnetic (NM) matrix. The\nacoustic wave with amplitude Iincident at angle qinduces\nprecession of the magnetisation vector Mvia the\nmagneto-elastic coupling. As a result, the wave is partly\ntransmitted and reflected, with respective amplitudes Tw\nandRw.\nacoustic devices in which the signal is carried by acoustic\nwaves while the magnetic field controls its propagation via\nthe magnetoelastic interaction in thin isolated magnetic in-\nclusions as shown in Fig. 1. By changing the applied mag-\nnetic field, one can alter the frequency at which the incident\nacoustic waves hybridize with the magnetic modes of the in-\nclusions. Thereby, one can control the acoustic waves by the\nresonant behaviour of Breit-Wigner and Fano resonances in\nthe magnetic inclusion.41We find that the strength of the res-\nonances is suppressed by the ubiquitous magnetic damping\nin realistic materials, but this can be mitigated by employing\noblique incidence geometry. To compare magneto-acoustic\nmaterials for such devices, we introduce a figure of merit. The\nmagneto-elastic Fano resonance is identified as most promis-\ning in terms of frequency and field tuneability. To enhance res-\nonant behaviour, we explore the oblique incidence as a means\nby which to enhance the figure of merit.\nWe consider the simplest geometry in which magneto-arXiv:1906.07297v2  [physics.app-ph]  4 May 2020Controlling acoustic waves using magneto-elastic Fano resonances 2\nelastic coupling can affect sound propagation. A ferromag-\nnetic slab (\"magnetic inclusion\") of thickness d, of the or-\nder of 10 nm, is embedded within a non-magnetic medium\n(Fig.1). The slab is infinite in the Y\u0000Zplane, has satu-\nration magnetization Ms, and is biased by the applied field\nHB=HBˆz. Due to the magneto-elastic coupling, this equilib-\nrium configuration is perturbed by shear stresses in the xz- and\nyzplanes associated with the incident acoustic wave.\nTo derive the equations of motion, we represent the mag-\nnetic energy density Fof the magnetic material as a sum of the\nmagneto-elastic FMEand purely magnetic FMcontributions.42\nTaking into account the Zeeman and demagnetizing energies,\nwe write FM=\u0000m0HBM+m0\n2(NxM2\nx+NyM2\ny), where Nx(y)\nare the demagnetising coefficients, Nx+Ny=1,Mis the\nmagnetization and m0is the magnetic permeability. In a crys-\ntal of cubic symmetry, the magnetoelastic contribution takes\nthe form43\nFME=B\nM2så\ni6=jMiMjui j+B0\nM2så\niM2\niuii;i;j=x;y;z;(1)\nwhere B0and Bare the linear isotropic and anisotropic\nmagneto-elastic coupling constants, respectively.44The strain\ntensor is ujk=1\n2(¶jUk+¶kUj), where Ujare the displace-\nment vector components. To maximize the effect of the cou-\npling B, we consider a transverse acoustic plane wave incident\non the slab from the left and polarized along the bias field, so\nthatUx=Uy=0,Uz=U(x;y;t). The non-vanishing com-\nponents of the strain tensor are uxz=1\n2¶xUanduyz=1\n2¶yU,\nandFMEis linear in both MandU:\nFME=B\nMs(Mxuxz+Myuyz): (2)\nThe magnetization dynamics in the slab is due to the effec-\ntive magnetic field, m0Heff=\u0000dF=dM. We define mas\nthe small perturbation of the magnetic order, i.e. jmj\u001cMs.\nLinearizing the Landau-Lifshitz-Gilbert equation,4we write\n\u0000¶mx\n¶t=gm0(HB+NyMs)my+gB¶U\n¶y+a¶my\n¶t;(3)\n¶my\n¶t=gm0(HB+NxMs)mx+gB¶U\n¶x+a¶mx\n¶t;(4)\nwhere gis the gyromagnetic ratio and ais the Gilbert\ndamping constant. To describe the acoustic wave, we in-\nclude the magneto-elastic contribution to the stress, s(ME)\njk=\ndFME=dujk, into the momentum balance equation:\nr¶2U\n¶t2=¶\n¶x\u0012\nC¶U\n¶x+B\nMsmx\u0013\n+¶\n¶y\u0012\nC¶U\n¶y+B\nMsmy\u0013\n;\n(5)\nwhere C=c44is the shear modulus and ris the mass density.\nThe non-magnetic medium is described by Eq.(5) with B=0.\nSince the values of C,B, and Nx;yare constant within each\nindividual material, we shall seek solutions of the equations in\nthe form of plane waves U;mx(y)µexp[i(kw;xx+kw;yy\u0000wt)].\nFrom herein, we consider all variables in the Fourier domain.For the magnetization precession in the magnetic layer driven\nby the acoustic wave, we thus obtain\nmx=gB(wkw;y+iewykw;x)\nw2\u0000ewxewyU; (6)\nmy=igB(ewxkw;y+iwkw;x)\nw2\u0000ewxewyU; (7)\nwhere we have denoted wx(y)=gm0(HB+Nx(y)Ms)and\newx(y)=wx(y)\u0000iwa. The complex-valued wave number kw;x\nis given by the dispersion relation\nk2\nw;x=r\nCw2\u0000\nw2\u0000ewxewy\u0001\n\u0000k2\nw;y\u0010\nw2\u0000ewxewy+gB2\nMsCewx\u0011\nh\nw2\u0000ewxewy+gB2\nMsCewyi ;\n(8)\nwhere kw;yis equal to that of the incident wave, and the\nbranch with Im kw;x>0 describes a forward wave decaying\ninto the slab. Eq. (8) describes the hybridization between\nacoustic waves and magnetic precession at frequencies close\nto ferromagnetic resonance (FMR) at frequency wFMR, with\nlinewidth GFMR. The frequency at which the precession am-\nplitudes (Eqs. (6) and (7)) diverge is given by the condition\n(wFMR+iGFMR=2)2=ewxewy. In the limit of small a, this\nyields wFMR=wxwyandGFMR=a(wx+wy). Away from\nthe resonance, Eq. (8) gives the linear dispersion of acous-\ntic waves. In the non-magnetic medium ( B=0), one finds\nk2\n0=w2r0=C0. Here and below, the subscript ’0’ is used to\nmark quantities pertaining to the non-magnetic matrix.\nTo calculate the reflection and transmission coefficients, Rw\nandTw, for a magnetic inclusion, we introduce the mechanical\nimpedance as Z=isxz=wUw. Solution of the wave matching\nproblem can then be expressed via the ratio of load ( ZME) and\nsource ( Z0) impedances. For impedances in the forward (F)\nand backward (B) directions in the magnetic slab, we find\nZ(F=B)\nw;ME=Ckw;x\nw0\n@1+gB2\nCM sewy\u0007iwkw;y\nkw;x\nw2\u0000ewxewy1\nA: (9)\nHere, the ‘-’ and ‘+’ signs correspond to (F) and (B), re-\nspectively. For the non-magnetic material, Eq. (9) recov-\ners the usual acoustic impedance45Z0=cosqpr0C0. Due\nto magnon-phonon hybridization, Re Z(F=B)\nw;MEdiverges at wFMR\nand vanishes at a nearby frequency wME. For a=0, the latter\nis given by\nwME=s\nwxwy\u0000gB2\nMsCwy: (10)\nReflection Rwand transmission Twcoefficients are then\nfound via the well-known relations45as\nRw=(ehw+1)(1\u0000hw)sin(kw;xd)\n(ehwhw+1)sin(kw;xd)+i(hw+ehw)cos(kw;xd);\n(11)\nTw=i(hw+ehw)\n(ehwhw+1)sin(kw;xd)+i(hw+ehw)cos(kw;xd);\n(12)Controlling acoustic waves using magneto-elastic Fano resonances 3\nwhere dis the thickness of the magnetic inclusion, hw=\nZ(F)\nME=Z0andehw=Z(B)\nME=Z0.46In close proximity to the res-\nonance, the impedances changes rapidly. Expanding Eq. (11)\nnearwMEin the limit kwd\u001c1, we obtain\nRw=iGR=2\n(w\u0000wME)+iGR=2eif+R0; (13)\nf=\u00002 arctan\u0014C\nC0rwx\nwytanq\u0015\n;\nwhere R0represents a smooth non-resonant contribution due\nto elastic mismatch at the interfaces, while frepresents a res-\nonant phase, which is non-zero for finite qand approaches p\nrapidly. In a system with no magnetic damping, the hybridiza-\ntion yields a resonance of finite linewidth GR,\nGR=gB2\n2MsC2cosqp\nr0C0\u0012\nwycos2q+C2\nC2\n0wxsin2q\u0013\nd:\n(14)\nThe origin of this linewidth can be explained as follows. Due\nto the magneto-elastic coupling incident propagating acoustic\nmodes can be converted into localised magnon modes. These\nmodes in turn either decay due to the Gilbert damping or are\nre-emitted as phonons. The rates of these transitions are pro-\nportional to GFMR andGR, respectively, and the total decay\nrate is G=GR+GFMR. This is similar to resonant scattering\nin quantum theory47, such that GRandGFMRare analogous to\nthe the elastic (Ge)and inelastic (Gi)linewidths respectively.\nWhen a=0,GFMRvanishes, and G=GR.\nAcoustic waves in the geometry of Fig. 1 can be scattered\nvia several channels. E.g. in a non-magnetic system ( B=0),\nelastic mismatch can yield Fabry-Pérot resonance due to the\nquarter wavelength matching of dand the acoustic wave-\nlength. However, this occurs at very high frequencies, which\nwe do not consider here. To understand the resonant magneto-\nelastic response, it is instructive to consider first the case of\nnormal incidence ( q=0), when the demagnetising energy\ntakes a simplified form due to the lack of immediate inter-\nfaces to form surface poles in ythe direction, so that Nx=1\nandNy=0. Including magneto-elastic coupling ( B6=0), we\nplot the frequency dependence of RwandTwusing Eq. (11)\nand (12) in Fig.2. To gain a quantitative insight, we analysed\na magnetic inclusion made of cobalt ( r=8900kgm\u00003,B=\n10MPa, C=80GPa, g=176GHzT\u00001,M=1MAm\u00001), em-\nbedded into a non-magnetic matrix ( r0=3192kgm\u00003;C0=\n298GPa). To highlight the resonant behaviour, we first sup-\npress ato 10\u00004. The reflection coefficient exhibits an asym-\nmetric non-monotonic dependence, shown as a black curve in\nFig.2(a), characteristic of Fano resonance.27,41This line shape\ncan be attributed to coupling between the discrete FMR mode\nof the magnetic inclusion and the continuum of propagating\nacoustic modes in the surrounding non-magnetic material.41\nIf the two materials had matching elastic properties, Rwwould\nexhibit a symmetric Breit-Wigner lineshape.47The transmis-\nsion shown in Fig.2(b) exhibits an approximately symmetric\ndip near the resonance.48The absorbancejAwj2=1\u0000jRwj2\u0000\njTwj2, shown in Fig.2(c) exhibits a symmetric peak, since theacoustic waves are damped in our model only due to the cou-\npling with spin waves.\nTo consider how the magneto-elastic resonance is affected\nby the damping, we also plot the response for aof 10\u00003and\n10\u00002, red and blue curves in Fig.2, respectively. An increase\nofafrom 10\u00004to 10\u00003significantly suppresses and broad-\nens the resonant peak. For a more common, realistic value of\n10\u00002the resonance is quenched entirely. A stronger magne-\ntoelastic coupling (i.e. high values of B) could, in principle,\ncountermand this suppression. This, however, is also likely\nto enhance the phonon contribution to the magnetic damping,\nleading to a correlation between Bandaobserved in realistic\nmagnetic materials.49\nTo characterise the strength of the Fano resonance, we note\nthat the fate of the magnon excited by the incident acoustic\nwave is decided by the relation between the emission rate GR,\nsee Eq. (14), and absorption rate GFMR. Hence, we introduce\nthe respective figure of merit as ¡=GR=GFMR. This quan-\ntity depends upon the material parameters, device geometry,\nand bias field. As seen from the first terms on the l.h.s. of\nEqs. (6) and (7), the relation between the dynamic magnetisa-\ntion components mx;yare determined by the quantities wxand\nwy. Equating these terms, one finds mxµmyp\nwy=wx, i.e. the\nprecession of mis highly elliptical,50due to the demagnetis-\ning field along x. This negatively affects the phonon-magnon\ncoupling for normal incidence ( ky=0): the acoustic wave\ncouples only to mx, as given by the second term in Eqs. (6)\nand (7). One way to mitigate this is to increase HB, mov-\ning the ratio wy=wxcloser to 1 and thus improving the figure\nof merit. To compare different magneto-elastic materials, the\ndependence on the layer thickness dand elastic properties of\nthe non-magnetic matrix (i.e. r0andC0) can be eliminated by\ncalculating a ratio of the figures of merit for the compared ma-\nterials. The comparison can be performed either at the same\nvalue of the bias field, or at the same operating frequency. The\nlatter situation is more appropriate for a device application,\nbut to avoid unphysical parameters, we present our results for\nthe same m0HB. An example of such comparisons for yttrium\niron garnet (YIG), cobalt (Co) and permalloy (Py) is offered\nin Table I.\nAnother way to improve ¡is to employ the oblique inci-\ndence ( q6=0), in which the acoustic mode is also coupled to\nthe magnetisation component my. The latter is not suppressed\nby the demagnetisation effects if Ny\u001c1. The resulting en-\nhancement in ¡is reflected in the full equation by the inclu-\nsion of wxandwyfromGR,\n¡=GR\nGFMR=gdB2\n2p\nr0C0\u0010\nHBcos2q+C2\nC2\n0Mssin2q\u0011\naC2M2scosq;(15)\nwhere wx\u001dwyandHB\u001cMsis assumed. For small q, the\napproximation Nx'1 and Ny'0 still holds. As a result, non-\nzeroqincreases peak reflectivity, as seen in Fig.3. The evolu-\ntion of the curves in Fig.3 with qis explained by the variation\nof the phase fof the resonant scattering relative to that of the\nnon-resonant contribution R0. The latter changes its sign at in-\ncidence angle of about 30\u000e, which yields a nearly symmetric\ncurve (blue), and an inverted Fano resonance at larger anglesControlling acoustic waves using magneto-elastic Fano resonances 4\n7.10 7.12 7.14 7.16 7.180.000.050.100.150.200.25|R(f)|(a)Damping, α:\n10−4\n10−3\n10−2\n7.10 7.12 7.14 7.16 7.18\nFrequency, f (GHz)0.750.800.850.900.951.00|T(f)|(b)\n7.10 7.12 7.14 7.16 7.180.00.10.20.3|A(f)|2(c)\nFIG. 2: The frequency dependence of the absolute values of (a) reflection and (b) transmission coefficients and (c) absorbance\nis shown for a 20nm thick magnetic inclusion. The vertical dashed and solid black lines represent the ferromagnetic resonance\nfrequency wFMRand magneto-elastic resonance frequency wMErespectively. The non-magnetic and magnetic materials are\nassumed to be silicon nitride and cobalt, respectively, with parameters given in the text. The bias field is m0HB=50mT, which\nleads to fME\u00197:138 GHz.\n(green). Although larger incidence angles may be hard to im-\nplement in a practical device, the resonant scattering is still\nenhanced at smaller angles.\nAbove, we have focused on the simplest geometry that ad-\nmits full analytic treatment. To implement our idea exper-\nimentally, particular care should be taken about the acous-\ntic waves polarization and propagation direction relative to\nthe direction of the magnetization. Indeed, our choice max-\nimises magnetoelastic response. If however, the polariza-\ntion is orthogonal to the bias field HB, i.e. Uz=0, the cou-\npling would be second-order in magnetization components\nmx;y, and would not contribute to the linearized LLG equation.\nFurthermore, we have neglected the exchange and magneto-\ndipolar fields that could arise due to the non-uniformity of the\nmagnetization. To assess the accuracy of this approximation,\nwe note that the length scale of this non-uniformity is set by\nthe acoustic wavelength l, of about 420nm for our parame-\nters rather than by the magnetic slab thickness d. The asso-\n7.00 7.05 7.10 7.15 7.20 7.25\nFrequency, f (GHz)0.010.020.030.040.050.06|R(f)|0◦\n15◦\n30◦45◦\nFIG. 3: Peak R(f)is enhanced and slightly shifted to the left\nin the oblique incidence geometry ( q>0\u000e). Coloured curves\nrepresent specific incidence angles sweeping from 0\u000eto 45\u000e.\nModerate Gilbert damping of a=10\u00003is assumed. The\ndashed vertical line corresponds to the magnetoelastic\nresonance frequency.\n0 10 20 30 40\nAngle, θ (deg.)0.000.020.040.06Figure of Merit, Υ\n0.050.100.150.200.25\nΓR(10−1)/FMR(ns)−1ΓFMR\nΓRΥFIG. 4: Figure of merit ¡and radiative linewidth GRare both\nenhanced in the oblique incidence geometry ( q>0\u000e).\nFerromagnetic linewidth GFMRremains unchanged. Co is\nassumed with a=10\u00003:\nciated exchange field is m0Ms(klex)2'9mT. The k-dependent\ncontributions to the magneto-dipole field vanish at normal in-\nTABLE I: Comparison of the figure of merit ¡for different\nmaterials, assuming d=20nm, m0HB=50mT and\nC0=298GPa.\nParameters YIG Co Py\n¡(q=0\u000e) 4:3x10\u000021:7x10\u000032:7x10\u00004\nGR(ns\u00001) 1 :9x10\u000047:5x10\u000032:0x10\u00004\nGFMR (ns\u00001) 4 :4x10\u000034.3 0.74\n¡(q=30\u000e) 4:1x10\u000022:5x10\u000032:8x10\u00004\nGR(ns\u00001) 1 :8x10\u000041:1x10\u000022:1x10\u00004\nGFMR (ns\u00001) 4 :4x10\u000034.3 0.74\nfME=wME=2p(GHz) 2.97 7.14 6.26\nB(MJm\u00003) 0.55 10 -0.9\nC(GPa) 74 80 50\nr(kgm\u00003) 5170 8900 8720\na 0:9x10\u000041:8x10\u000024:0x10\u00003\nMs(kAm\u00001) 140 1000 760Controlling acoustic waves using magneto-elastic Fano resonances 5\ncidence but may become significant at oblique incidence, giv-\ningm0Mskyd'98mT at q=15\u000e. In principle, these could\nincrease the resonant frequency of the slab by a few GHz but\nwould complicate the theory significantly. 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Smith1, Claudia Mewes2,3, Sujan Budhathoki2,3, Adam J. Hauser2,3, \nMin Gao4, Jie-Fang Li4, Dwight D. Viehland4, Zijian Jiang1, Jean J. Heremans1, Prasanna  V. \nBalachandran5,6, Tim Mewes2,3, Satoru Emori1* \n1 Department of Physics, Virginia Tech , VA 24061, U.S.A  \n2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA  \n 3 Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa, \nAL 35487, U .S.A. \n4  Department  of Material Science and Engineering, Virginia Tech ,  \n Blacksburg, VA 24061, U.S.A . \n5  Department of Material Science and Engineering, University of Virginia,  \n Charlottesville, VA 22904, U.S.A . \n6  Department of Mechanical and Aerospace Engineering , Univer sity of Virginia,  \n Charlottesville, VA 22904, U.S.A.  \n*email: semori@vt.edu  \n \nConfirming  the or igin of Gilbert damping by experiment has remained  a challenge  for \nmany decades , even for simple  ferromagnetic metals . In this Letter, we experimentally \nidentify  Gilbert damping that increases with decreasing electronic scattering  in epitaxial \nthin films of pure Fe . This observation of  conductivity -like damping, which cannot be  \naccounted for by  classical eddy current loss , is in excellent quantitative agreement with \ntheoretical  predictions of Gilbert damping due to intraband scatte ring. Our results  resolve 2 \n the longstanding question about a fundamental damping mechanism  and offer  hints for \nengineering low -loss magnetic metals for cryogenic spintronic s and quantum devices.    \n \nDamping determines how fast the magnetization relaxes towards the effective magnetic \nfield and plays a  central  role in many aspects  of magnetization dynamics  [1,2] . The magnitude of \nviscous Gilbert damping governs the  threshold current for spin -torque magnetic switching  and \nauto-oscillations  [3,4] , mobility of magnetic domain walls  [5,6] , and  decay leng ths of diffusive \nspin waves and superfluid -like spin current s [7,8] . To enable spintronic technologies with low \npower dissipation , there is currently  much interest in  minimizing Gilbert damping in thin films of \nmagnetic m aterials  [9–13], especially ferromagnetic metals  [14–23] that are compatible with \nconventional device fabrication schemes . Despite the fundamental and technological importance \nof Gilbert damping, its physical mechanisms  in various magnetic materials have yet to be \nconfirmed by experiment .   \nGilbert damping is generally attributed to spin-orbit coupling that ultimately dissipates  \nthe energy of the magnetic system  to the lattice  [1,2] . Kambersky’s torque correlation model  [24] \nqualitatively captures  the temperature dependence of damping in some experiments  [25–28] by \npartitioning Gilbert damping into two mechanisms due to spin -orbit coupling, namely interband \nand intraband scattering mechanisms, each with a distinct dependence  on the elect ronic \nmomentum scattering tim e e. For the interband scattering mechanism  where magnetization \ndynamics can excite electron -hole pairs across dif ferent bands, the resulting Gilbert damping is \n“resistivity -like” as its magnitude scales with e-1, i.e., increased  electronic scattering results in \nhigher damping  [29,30] . By contrast, the intraband scattering mechanism  is typically understood \nthrough the breathing Fermi surface mode l [31], where electron -hole pairs are excited in the 3 \n same band , yielding  “conductivity -like” Gilbert damping  that scales with  e, i.e., reduced \nelectronic scattering  results in  higher damping.  \nConductivity -like Gilbert damping was reported  experimentally more than  40 years ago \nin bulk crystals of pure Ni and Co at low temperatures , but surprisingly not in pure Fe [25]. The  \napparent absence  of co nductivity -like damping in Fe has been  at odds with many theoretical \npredictions that intraband scattering should dominate at low temperatures  [32–38], although \nsome theoretical studies have suggested that intraband scattering may be  absent alt ogether in \npure metals  [39,40] . To date, no experimental work has conclusively addressed  the role of \nintraband scattering in  pure Fe1. There thus remains  a significant gap in the fundamental \nunderstanding of damping in one of the simplest ferromagnetic metals.  Intrinsic conductivity -\nlike Gilbert damping in Fe is also technologically relevant,  since  minimizing damping in \nferromagnetic metals at low temperatures is crucial for cryogenic superconducting spintronic \nmemories  [41,42]  and quantum information  transduction schemes  [43,44] .  \nIn this Letter, we experimentally demonstrate the presence of conductivity -like Gilbert \ndamping due to intr aband scattering  in epitaxial thin films of body -centered -cubic (BCC) Fe. By \ncombining broadband ferromagnetic resonance (FMR) measurements with characterization of \nstructural and transport properties of these model -system thin films, we show that conductivity -\nlike Gilbert damping dominates at lo w temperatures in epitaxial Fe . These experimental  results \n                                                           \n1 Ref.  [36] includes experimental data that suggest the presence of conductivity -like Gilbert damping in an ultrathin \nFe film, although no detailed information is given about the sample and t he experimental results deviate \nconsiderably from the calculations. An earlier study by Rudd et al. also suggests an increase in Gilbert damping with \ndecreasing temperature  [27], but quantification of the Gilbert damping parameter in this experiment is difficult.  \n 4 \n agree remarkably well with the magnitude of Gilbert damping derived from first -principles \ncalculations  [32,33,36] , thereby  providing evidence for intraband scatterin g as a key mechanism \nfor Gilbert damping in pure BCC Fe. Our experiment  thus resolves the longstanding  question  \nregarding the origin of damping in the prototypical ferromagnetic metal . Our results also confirm  \nthat – somewhat counterintuitively – disorder can partially suppress intrinsic damping at low \ntemperatures in ferromagnetic metals, such that optimally disordered films  may be well suited \nfor cryogenic spintronic and quantum applications  [41–44].   \nEpitaxial BCC Fe thin films were sputter deposited on (001) -oriented MgAl 2O4 (MAO) \nand MgO single crystal substrates. The choices of substrates were inspired by the recent \nexperiment by Lee et al. [20], where epitaxial growth is enabled with t he [100] axis of a B CC \nFe-rich alloy oriented 45o with respect to  the [100] axis of MAO or MgO. MAO with a lattice \nparameter of a MAO /(2√2) = 0.2858  nm exhibits a lattice mismatch of less than 0.4% with Fe (a Fe \n≈ 0.287 nm) , whereas the lattice mismatch between MgO ( aMgO/√2 = 0.2978  nm) and Fe is of the \norder  4%. Here , we focus on 25 -nm-thick Fe films that were grown  simultaneously on MAO and \nMgO by confocal DC magnetron sputtering  [45].  In the Supplemental Material  [45], we report \non additional films depos ited by off -axis magnetron sputtering.  \nWe verified the crystalline quality of the epitaxial Fe films by X -ray diffraction, as s hown \nin Fig. 1( a-c). Only (00X )-type peaks of the substrate and film are found in each 2θ-ω scan, \nconsistent with the single -phase  epitaxial growth of the Fe films. The 2θ-ω scans reveal a larger \namplitude of film peak for MAO/Fe, suggesting higher crystalline quality  than that of MgO/Fe. \nPronounced Laue oscillations, indicative of atomically smooth film interfaces, are o bserved \naround  the film peak of MAO/Fe, whereas they are absent for MgO/Fe. The high crystalline \nquality of MAO/Fe is also evidenced  by its narrow film -peak rocking curve with a FWHM of 5 \n only 0.02o, comparable to the rocking curve F WHM of the substrate2. By contrast, the film -peak \nrocking curve of MgO/Fe has a FWHM of  1o, which indicates substantial mosaic  spread in the \nfilm due to the large  lattice mismatch with the MgO substrate.  \nResults of 2θ -ω scans for different film thicknesses  [45] suggest  that the 25 -nm-thick Fe \nfilm may be  coherently strained to the MAO substrate , consistent with  the smooth interfaces and \nminimal mosaic spread of MAO/Fe . By contrast, i t is likely that 25 -nm-thick Fe on MgO is \nrelaxed to accommodate the large film-substrate lattice mismatch. Static magnetometry provides \nfurther evidence that Fe is strained  on MAO a nd relaxed  on MgO  [45]. Since  strained MAO/Fe \nand relaxed MgO/Fe exhibit  distinct  crystalline quality,  as evidenced by an approximately 50 \ntimes narrower rocking FWHM for  MAO/Fe , we have two model systems that enable \nexperimental investigation of the impact of structural disorder on Gilbert damping.   \nThe residual electrical resistivity also reflects  the structural quality  of metal s. As shown \nin Fig. 1(d ), the residual resistivity is 20 % lower for MAO/Fe compared to MgO/Fe, which \ncorroborates the lower defect density in MAO/Fe.  The resistivity increases  by nearly an order of \nmagnitude with increasing  temperature, reaching  1.1×10-7  m for both samples  at room \ntemperature , consistent with behavior expected for pure metal  thin film s.  \nWe now examine how the difference in crystalline quality correlates with  magnetic \ndamping in MAO/Fe and MgO /Fe. Broadband FMR measurements were performed at room \ntemperature up to 65 GHz with a custom spectrometer that empl oys a coplanar waveguide \n(center conductor width  0.4 mm ) and an electromagnet (maximum field < 2 T) . For each \nmeasurement at a fixed excitat ion frequency, an external bias magnetic field was swept parallel \nto the film plane along the [110] axis of Fe , unless otherwise noted. I n the Supplemental \n                                                           \n2 The angular resolu tion of the diffractometer is 0.0068o.  6 \n Material  [45], we show similar results with the  field applied along the [110] and [100] axes of \nFe; Gilbert damping is essentially isotropic within the film plane for our epitaxial Fe films , in \ncontrast to a recent report of anisotropic damping in ultrathin epitaxial Fe  [22]. \nFigure 2  shows that the peak -to-peak FMR linewidth Hpp scales linearly with frequency \nf, enabling a precise  determination of  the measured  Gilbert damping parameter  𝛼𝑚𝑒𝑎𝑠  from the \nstandard equation,  \n𝜇0∆𝐻𝑝𝑝=𝜇0∆𝐻0+2\n√3𝛼𝑚𝑒𝑎𝑠\n𝛾′𝑓,    (1) \nwhere  Hpp,0 is the zero -frequency linewidth and 𝛾′=𝛾/2𝜋≈29.5 GHz/T is the reduced \ngyromagnetic ratio . Despite the difference  in crystalline  quality , we find essentially the same \nmeasured  Gilbert damp ing parameter of  𝛼𝑚𝑒𝑎𝑠  ≈ 2.3×10-3 for MAO/Fe and MgO/Fe. We note \nthat t his value of 𝛼𝑚𝑒𝑎𝑠  is comparable to the lowest damping parameters reported for epitaxial Fe \nat room temperature  [15–17]. Our results indicate  that Gilbert  damping at room temperature is \ninsensitive to the strain state or structural disorder  in epitaxial Fe.3  \n The measured damping parameter 𝛼𝑚𝑒𝑎𝑠  from  in-plane FMR can generally include a \ncontribution from non-Gilbert  relaxation , namely two -magnon scattering  driven by defects  [46–\n49]. However,  two-magnon scattering is suppressed when the film is magnetized out-of-\nplane  [19,48] . To isolate any two -magnon scattering contribution to d amping, we performed out-\nof-plane  FMR measurements under  a sufficiently large magnetic field (>4 T) for complete \nsaturation of  the Fe film, using a custom W-band shorted waveguide  combined with a \n                                                           \n3 However, the crystallographic texture of Fe has significant impact on damping; for example, non -epitaxial Fe films \ndeposited directly on amorphous SiO 2 substrates exhibit an order of magnitude wider linewidths, due to much more \npronounced non -Gilbert damping (e.g., two -magnon scattering), compared to (001) -oriented epitaxial Fe films.  \n 7 \n superconducting magnet. As shown in Fig. 2, the out -of-plane and in -plane FMR data yield  the \nsame slope  and hence 𝛼𝑚𝑒𝑎𝑠  (Eq. 1)  to within <  8%. This finding indicates  that two -magnon \nscattering is negligible and that frequency -dependent magnetic relaxation is dominated by \nGilbert damping in epitaxial Fe examined here.  \nThe insensitivity of Gilbert damping to disorder found in Fig. 2 can be explained by the \ndominance of the interband (resistivity -like) mechanis m at room temperature, with phonon \nscattering dominating over defect scattering. Indeed, since MAO/Fe and MgO/Fe have the same \nroom -temperature resistivity  (Fig. 1(d )), any contributions to Gilbert damping from  electronic \nscattering should be identical for  both samples at room temperature. Moreover, according to our \ndensity functional theory calculations  [45], the density of states of BCC Fe at the Fermi energy, \nD(EF), does not depend significantly on  the strain state of the crystal. Therefore, i n light of the \nrecent reports  that Gilbert damping is proportional to  D(EF) [18,50,51] , the different strain states \nof MAO/Fe and MgO/Fe are not expe cted to cause a significant  difference in Gilbert damping.   \n However, since MAO/Fe and MgO/Fe exhibit distinct resistivities (electronic scattering \ntimes e) at low temperatures, one might expect  to observe distinct temperature dependence in \nGilbert damping for these two samples. To this end, we performed variable -temperature FMR \nmeasurements using  a coplanar -waveguide -based spectrometer (maximum frequency 40 GHz, \nfield < 2 T)  equipped with  a clos ed-cycle cryostat4. Figure 3(a,b) shows that meas is enhanced \nfor both samples at lower temperatures. Notably, this damping enhancement with decreasing \ntemperature is significantl y greater for MAO/Fe . Thus, at low temperatures, we find a \n                                                           \n4 The W -band spectrometer for out -of-plane FMR (Fig. 2) could not be  cooled below room temperature  due to its \nlarge thermal mass , limiting us to in -plane FMR measurements at low temperatures.  8 \n conductivity -like damping increase that is evidently more pronounced in epitaxial Fe with less \nstructural disorder.  \nWhile this increased damping at low temperatures is reminiscent of intrinsic Gilbert \ndamping from intraband scattering  [31–38], we first consider other possible contributions. One \npossibility is two -magnon scattering  [46–49], which we have ruled out at room temperature (Fig. \n2) but could  be present in our low -temperature in-plane FMR measurements . From Fig. 3(a,b), \nthe zero -frequency linewidth   H0  (Eq. 1 ) – typically attributed to magnetic inhomogeneity – is \nshown to increase  along with meas at low temperatures  [45], which might  point to  the emergence \nof two -magnon scattering  [48,49] . However, our mean -field model calculations (see \nSupplemental Material  [45]) shows that H0  correlates with  meas due to  interactions among \ndifferent regions  of the inhomogeneous film  [52]. The increase of H0 at low temperatures is \ntherefore readily accounted for  by increased  Gilbert damping , rather than two -magnon scattering .  \nWe are also not aware of  any mechanism that enhance s two-magnon scattering with \ndecreasing temperature, particularly given that  the saturation magnetization (i.e., dipolar \ninteractions) is constant across  the measured temperature range  [45]. Moreover, the isotropic in -\nplane damping found in our study is inconsistent with typically anisotropic two-magnon \nscattering tied to the crystal symmetry of  epitaxial films  [46,47] , and the film thickness in our \nstudy (e.g., 25 nm) rules out t wo-magnon scattering of interfacial origin  [49]. As such, we \nconclude  that two -magnon scattering does not play any essential  role in our experimental \nobservations.  \n Another  possible contribution  is dissipation due to classical eddy current s, which \nincrease s proportionally with the increasing conductivity  𝜎 at lower temperatures . We estimate \nthe eddy current contribution to the measured Gilbert damping with [15,53]   9 \n 𝛼𝑒𝑑𝑑𝑦 =𝜎\n12𝛾𝜇02𝑀𝑠𝑡𝐹2,    (2) \nwhere  𝜇0𝑀𝑠≈2.0 T is the saturation magnetization  and tF is the film thickness . We find that  \neddy curr ent damping accounts  for only ≈20% (≈ 30%) of  the total measured  damping of  \nMAO/Fe (MgO/Fe)  even at the lowest measured temperature  (Fig. 3(c)) . Furthermore, a s shown \nin the Supplemental Material  [45], thinner MAO/Fe film s, e.g., tF = 11 nm , with negligible eddy \nstill exhibit a significant increase in damping with decreasing temperature. Our results thus \nindicate a substantial contribution to conductivity -like Gilbert damping that is not accounted for \nby classica l eddy current damping.  \n For further discussion , we subtract the eddy -current damping from the measured damping \nto denote the Gilbert damping parameter attributed to intrinsic spin-orbit coupling as \n𝛼𝑠𝑜= 𝛼𝑚𝑒𝑎𝑠 − 𝛼𝑒𝑑𝑑𝑦. To correlate electronic transport and magnetic damping across the entire \nmeas ured temperature range, we perform a phenomenological fit of the temperature dependence \nof Gilbert damping with  [26]  \n𝛼𝑠𝑜=𝑐𝜎(𝑇)\n𝜎(300  𝐾)+𝑑𝜌(𝑇)\n𝜌(300  𝐾),   (3) \nwhere  the conductivity -like (intraband) and resistivity -like (interband) terms are scaled by \nadjustable parameters c and d, respectively. As shown in Fig.  4(a),(b), t his simple \nphenomenological model  using the experimental transport results (Fig. 1(d))  agrees remarkably \nwell with the temperature dependence of Gilbert damping for both MAO/Fe and MgO/Fe.  \nOur fi nding s that Gilbert damping can  be phenomenologically partit ioned  into two \ndistinct contributions (Eq. 3 ) are in line with  Kambersky’s  torque correlation model . We \ncompare our experimental resul ts to first-principles calculations by Gilmore et al. [32,33]  that \nrelate  electronic momentum scattering rate e-1 and Gilbert damping through Kambersky’s torque \ncorrelation model. We use the  experimentally measured resistivity ρ (Fig. 1(d)) to convert the 10 \n temperature to e-1 by assuming the constant conversion factor ρ e = 1.30×10-21  m s [33]. To \naccount for the difference in electronic scattering time for the minority spin  and majority spin \n, we take the calculated curve from Gilmore et al. with / = 4 [33], which is close to the \nratio of D(EF) of the spin-split bands  for BCC Fe , e.g., derived from our density functional \ntheory calculations  [45]. For explicit comparison with Refs.  [32,33] , the Gilbert damping \nparameter in Fig. 4(c) is converted to the magnetic relaxation rate 𝜆= 𝛾𝛼𝑠𝑜𝜇0𝑀𝑠. The \ncalculated prediction is in excellent  quantitative agreement with our experimental results for both \nstrained MAO/Fe and relaxed MgO/Fe  (Fig. 4(c)) , providing additional experimental evidence \nthat intraband scattering predominately contribute s to Gilber t damping at low temperatures.   \n We also compare our experimental results to a more recent first -principles calculation \nstudy by Mankovsky et al., which  utilizes the linear response formalism  [36]. This approach  \ndoes not  rely on a phenomenological electronic scattering rate and instead  allows for explicitly \nincorporating thermal  effects  and structural disorder . Figure 4(d ) shows the calculated \ntemperature dependence of the Gilbert damping parameter for BCC Fe with a small density of \ndefects, i.e., 0.1% vacancies , adapted from Ref.  [36]. We again find good quantitative agreement \nbetween the ca lculations and our experimental results for MAO/Fe. On the other hand, the \nGilbert damping parameter s at low temperatures  for relaxed MgO/Fe are significantly below the \ncalculated values . This  is consistent with  the reduction  of intraband scattering  due to enhanced \nelectronic scattering (enhanced e-1) from defects in relaxed MgO/Fe .  \n Indeed, significant defect -mediated electronic scattering may explain  the absence of \nconductivity -like Gilbert damping for crystalline Fe in prior  experiments. For example, Ref.  [25] \nreports an upper limit of  only a two -fold increase of the estimated Gilbert damping parameter \nfrom T = 300 K to 4 K . This relatively small damping enhancement is similar to that for MgO/Fe 11 \n in our study  (Fig. 4(b)) , suggesting that intraband scattering may have been suppressed in Fe in \nRef. [25] due to a similar degree of structural disorder  to MgO/Fe. We therefore conclude that \nconductivity -like Gilbert damping from  intraband scattering is highly sensitive to disorder in \nferromagnetic metals.  \nMore generally , the presence of defects  in all real metals – evidenced by finite residual \nresistivity  – ensures that the Gilbert damping parameter is finite even in the zero -temperature \nlimit . This circumvents the theoretical deficiency of Kambersky’s  torque correlation model \nwhere Gilbert damping would diverge in a perfectly clean ferromagnetic metal at T  0 [39,40] . \nWe also remark that a fully quantum mechanical many -body theory of magnetization dynami cs \nyields finite Gilbert damping even in the clean, T = 0 limit  [54].  \n In summary, we have demonstrated the dominance of conductiv ity-like Gilbert damping \ndue to intraband scattering at low temperatures in high-quality epitaxial Fe . Our experimental \nresults  also validate  the longstanding theoretical prediction of  intraband scattering as an essential  \nmechanism for Gilbert damping in pure ferromagnetic metals  [32–38], thereby advancing the \nfundamental understanding of magnetic relaxation in real materials . Moreover, we have \nconfirmed that, at low temperatures, a ma gnetic metal with imperfect crystallinity can exhibit \nlower Gilbert damping (sp in decoherence) than its cleaner  counterpart. This somewhat \ncounterintuitive finding suggests that  magnetic thin film s with optimal structural or chemical \ndisorder may be useful for cryogenic spintronic memories  [41,42]  and spin-wave -driven  \nquan tum information systems  [43,44] .   \n \n \n 12 \n Acknowledgements  \nThis research was funded in part by 4 -VA, a collaborative partnership for advancing the  \nCommonwealth of Vir ginia , as well as by the ICTAS Junior Faculty Award . A. Sapkota and C. \nMewes would like to acknowledge support by NSF-CAREER  Award No. 1452670 , and A. \nSrivastava would like to acknowledge support by NASA Award No. CAN80NSSC18M0023 . \nWe thank M. D. Stiles , B. K. Nikolic , and F. Mahfouzi  for helpful discussions on theoretical \nmodels for computing Gilbert damping , as well as R. D. McMichael for his input on the mean -\nfield modeling of interactions in inhomogeneous ferromagnetic films.    \n \n \n1.  B. 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B \n96, 214421 (2017).  \n \n  19 \n  \nFigure 1. (a,b) 2θ -ω X-ray diffraction scans of MAO/Fe and MgO/Fe (a) over a wide angle range \nand (b) near the BCC Fe (002) film peak. (c) Rocking curve scans about the film peak. (d) \nTemperature dependence of resistivity plotted on a log -log scale.  \n \n \nFigure 2. Frequency dependence of FMR linewidth Hpp for MAO/Fe and MgO/Fe at room \ntemperature. Linewidths measured under in -plane field are shown as open symbols, whereas \nthose measured under out -of-plane (OP) field are shown as filled symbols .  \n62 64 66 68log(intensity) [a.u.]\n2q [deg.]\n30 40 50 60 70log(intensity) [a.u.]\n2q [deg.]\n-1.0 -0.5 0.0 0.5 1.0intensity [a.u.]\nw002 [deg.]\n10 10010-810-7r [ m]\nT [K] MgO/Fe\n MAO/Fe\nMAO (004)MgO (002)\nFe (002)MAO/Fe\nMgO /Fe\n( 4)(a) (b) (c) (d)MAO/Fe\nMgO /Fe\n(a)\n0 20 40 60 80 100 120024681012 MAO/Fe (OP)\nMgO/Fe\nMAO/Fem0Hpp [mT]\nf [GHz]20 \n  \nFigure 3. (a,b) Frequency dependence of FMR linewidth for MA O/Fe and MgO/Fe at (a) T = 100 \nK and (b) T = 10 K. (c) Temperature dependence of measured  Gilbert damping parameter meas \nand estimated eddy -current damping parameter eddy. \n \n \n0 50 100 150 200 250 3000246810\n meas MAO/Fe  \n eddy estimate\n meas MgO/Fe\n eddy estimatemeas, eddy [10-3]\nT [K]\n0 10 20 30 40051015\nMAO/Fe\nMgO/Fem0Hpp (mT)\nf (GHz)T = 100 K\n0 10 20 30 40051015m0Hpp (mT)\nf (GHz)T = 10 K(c)(a) (b)21 \n   \nFigure 4. (a,b) Temperature dependence of the spin-orbit -induced Gilbert damping parameter \nso, fit phenomenologically with the experimentally measured resistivity for (a) MAO/Fe and (b) \nMgO/Fe. The dashed and dotted curves indicate the conductivity -like and resistivity -like \ncontributions, respectively; the solid curve represents the fit curve for the total spin -orbit -induced \nGilbert damping parameter. (c,d) Comparison of our experimental results with calculated Gilbert \ndamping parameters by (c) Gilmore et al. [32,33]  and (d) Mankovsky et al. [36]. \n \n \n0 100 200 30002468\nr-liker-likeso [10-3]\nT [K]s-likeMAO/Fe\n0 100 200 30002468\nMgO/Fe\nr-likes-likeso [10-3]\nT [K](a) (b)\n0 100 200 30002468\n MAO/Fe\n MgO/Fe\n calculated [Mankovsky]so [10-3]\nT [K]\n0 50 1000123\nr-like MAO/Fe\n MgO/Fe\n calculated [Gilmore]l [109 s-1]\ne-1 [1012 s-1]s-like0.0 0.5 1.0\n02468\nso [10-3]r [10-7 m]\n(c)\n(d)"    },    {        "title": "1907.01045v1.Magnon_decay_theory_of_Gilbert_damping_in_metallic_antiferromagnets.pdf",        "content": "Magnon decay theory of Gilbert damping in metallic antiferromagnets\nHaakon T. Simensen, Akashdeep Kamra, Roberto E. Troncoso, and Arne Brataas\nCenter for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: July 3, 2019)\nGilbert damping is a key property governing magnetization dynamics in ordered magnets. We present a\ntheoretical study of intrinsic Gilbert damping induced by magnon decay in antiferromagnetic metals through\ns-dexchange interaction. Our theory delineates the qualitative features of damping in metallic antiferromagnets\nowing to their bipartite nature, in addition to providing analytic expressions for the damping parameters. Magnon-\ninduced intraband electron scattering is found to predominantly cause magnetization damping, whereas the Néel\nfield is found to be damped via disorder. Depending on the conduction electron band structure, we predict that\nmagnon-induced interband electron scattering around band crossings may be exploited to engineer a strong Néel\nfield damping.\nIntroduction.— The dynamical properties of a harmonic\nmode are captured by its frequency and lifetime [ 1,2]. While\nthe eigenfrequency is typically determined by the linearized\nequations of motion, or equivalently by a non-interacting de-\nscription of the corresponding quantum excitation, the lifetime\nembodies rich physics stemming from its interaction with one\nor more dissipative baths [ 1,3]. Dissipation plays a central\nrole in the system response time. In the context of magnetic\nsystems employed as memories, the switching times decrease\nwith increasing damping thereby requiring a stronger dissi-\npation for fast operation [ 4–6]. The dissipative properties of\nthe system also result in rich phenomena such as quantum\nphase transitions [ 7–10]. Furthermore, the formation of hybrid\nexcitations, such as magnon-polarons [ 11–18] and magnon-\npolaritons [ 19–24], requires the dissipation to be weak with\nrespect to the coupling strengths between the two participating\nexcitations [ 25]. Therefore, in several physical phenomena that\nhave emerged into focus in the recent years [ 12,16,26–30],\ndamping not only determines the system response but also the\nvery nature of the eigenmodes themselves. Understanding,\nexploiting and controlling the damping in magnets is thus a\nfoundational pillar of the field.\nThe success of Landau-Lifshitz-Gilbert (LLG) phenomenol-\nogy [ 31,32] in describing ferromagnetic dynamics has inspired\nvigorous e \u000borts towards obtaining the Gilbert damping param-\neter using a wide range of microscopic theories. The quantum\nparticles corresponding to magnetization dynamics - magnons\n- provide one such avenue for microscopic theories and form\nthe central theme in the field of magnonics [ 33,34]. While\na vast amount of fruitful research has provided a good under-\nstanding of ferromagnets (FMs) [ 35–54], analogous studies on\nantiferromagnets (AFMs) are relatively scarce and have just\nstarted appearing [ 55,56] due to the recently invigorated field\nof antiferromagnetic spintronics [ 57–62]. Among the ongoing\ndiscoveries of niches borne by AFMs, from electrically and\nrapidly switchable memories [ 63], topological spintronics [ 60],\nlong range magnonic transport [ 64] to quantum fluctuations\n[65], an unexpected surprise has been encountered in the first\nprinciples evaluation of damping in metallic AFMs. Liu and\ncoworkers [ 56] and another more recent first-principles study\n[66] both found the magnetization dissipation parameter to bemuch larger than the corresponding Néel damping constant,\nin stark contrast with previous assumptions, exhibiting richer\nfeatures than in FMs. An understanding of this qualitative\ndi\u000berence as well as the general AFM dissipation is crucial\nfor the rapidly growing applications and fundamental novel\nphenomena based on AFMs.\nHere, we accomplish an intuitive and general understanding\nof the Gilbert damping in metallic AFMs based on the magnon\npicture of AFM dynamics. Employing the s-d, two-sublattice\nmodel for a metallic AFM, in which the dandselectrons\nconstitute the magnetic and conduction subsystems, we derive\nanalytic expressions for the Gilbert damping parameters as\na function of the conduction electron density of states at the\nFermi energy and s-dexchange strength. The presence of spin-\ndegenerate conduction bands in AFMs is found to be the key\nin their qualitatively di \u000berent damping properties as compared\nto FMs. This allows for absorption of AFM magnons via s-\ndexchange-mediated intraband conduction electron spin-flip\nprocesses leading to strong damping of the magnetization as\ncompared to the Néel field [ 67]. We also show that interband\nspin-flip processes, which are forbidden in our simple AFM\nmodel but possible in AFMs with band crossings in the conduc-\ntion electron dispersion, result in a strong Néel field damping.\nThus, the general qualitative features of damping in metallic\nAFMs demonstrated herein allow us to understand the Gilbert\ndamping given the conduction electron band structure. These\ninsights provide guidance for engineering AFMs with desired\ndamping properties, which depend on the exact role of the\nAFM in a device.\nModel.— We consider two-sublattice metallic AFMs within\nthes-dmodel [ 35,36,44]. The delectrons localized at lat-\ntices sites constitute the magnetic subsystem responsible for\nantiferromagnetism, while the itinerant selectrons form the\nconduction subsystem that accounts for the metallic traits. The\ntwo subsystems interact via s-dexchange [Eq. (3)]. For ease of\ndepiction and enabling an understanding of qualitative trends,\nwe here consider a one-dimensional AFM (Fig. 1). The re-\nsults within this simple model are generalized to AFMs with\nany dimensionality in a straightforward manner. Furthermore,\nwe primarily focus on the uniform magnetization dynamics\nmodes.arXiv:1907.01045v1  [cond-mat.mes-hall]  1 Jul 20192\nFIG. 1: Schematic depiction of our model for a metallic AFM.\nThe red and blue arrows represent the localized delectrons\nwith spin up and down, respectively, thereby constituting the\nNéel ordered magnetic subsystem. The green cloud illustrates\nthe delocalized, itinerant selectrons that forms the conduction\nsubsystem.\nAt each lattice site i, there is a localized delectron with spin\nSi. The ensuing magnetic subsystem is antiferromagnetically\nordered (Fig. 1), and the quantized excitations are magnons\n[68,69]. Disregarding applied fields for simplicity and as-\nsuming an easy-axis anisotropy along the z-axis, the magnetic\nHamiltonian, Hm=˜JP\nhi;jiSi\u0001Sj\u0000KP\ni(Sz\ni)2, wherehi;ji\ndenotes summation over nearest neighbor lattice sites, is quan-\ntized and mapped to the sublattice-magnon basis [69]\nHm=X\nqh\nAq\u0010\nay\nqaq+by\nqbq\u0011\n+By\nqay\nqby\nq+Bqaqbqi\n; (1)\nwhere we substitute ~=1,Aq=(2˜J+2K)SandBq=\n˜JS e\u0000iq\u0001aP\nh\u000eieiq\u0001\u000e, where S=jSij,ais the displacement\nbetween the two atoms in the basis, and h\u000eidenotes sum-\nming over nearest neighbor displacement vectors. aqandbq\nare bosonic annihilation operators for plane wave magnons\non the A and B sublattices, respectively. We diagonalize\nthe Hamiltonian [Eq. 1] through a Bogoliubov transforma-\ntion [ 69] toHm=P\nq!q\u0010\n\u000by\nq\u000bq+\fy\nq\fq\u0011\n;with eigenenergies\n!q=q\nA2q\u0000jBqj2. In the absence of an applied field, the\nmagnon modes are degenerate.\nTheselectron conduction subsystem is described by a tight-\nbinding Hamiltonian that includes the “static” contribution\nfrom the s-dexchange interaction [Eq. (3)] discussed below:\nHe=\u0000tX\nhi;jiX\n\u001bcy\ni\u001bcj\u001b\u0000JX\ni(\u00001)i\u0010\ncy\ni\"ci\"\u0000cy\ni#ci#\u0011\n:(2)\nHere ci\u001bis the annihilation operator for an selectron at site\niwith spin\u001b.t(>0)is the hopping parameter, and J(>0)\naccounts for s-dexchange interaction [Eq. (3)]. The (\u00001)i\nfactor in the exchange term reflects the two-sublattice nature of\nthe AFM. The conduction subsystem unit cell consists of two\nbasis atoms, similar to the magnetic subsystem. As a result,\nthere are four distinct electron bands: two due to there being\ntwo basis atoms per unit cell, and twice this due to the two\npossible spin polarizations per electron. Disregarding applied\nfields, these bands constitute two spin-degenerate bands. We\nlabel these bands 1 and 2, where the latter is higher in energy.The itinerant electron Hamiltonian [Eq. (2)] is diagonalized\ninto an eigenbasis (c1k\u001b;c2k\u001b)with eigenenergies \u000f1k=\u0000\u000fk\nand\u000f2k= +\u000fk, where\u000fk=p\nJ2S2+t2j\rkj2, where\rk=P\nh\u000eie\u0000ik\u0001\u000e. The itinerant electron dispersion is depicted in Fig.\n2.\nThe magnetic and conduction subsystems interact through\ns-dexchange interaction, parametrized by J:\nHI=\u0000JX\niSi\u0001si; (3)\nwhere si=P\n\u001b\u001b0cy\ni\u001b\u001b\u001b\u001b0ci\u001b0is the spin of the itinerant elec-\ntrons at site i, where \u001bis the vector of Pauli matrices. The term\nwhich is zeroth order in the magnon operators, and thus ac-\ncounts for the static magnetic texture, is already included in He\n[Eq. (2)]. To first order in magnon operators, the interaction\nHamiltonian can be compactly written as\nHe\u0000m=X\n\u0015\u001aX\nkk0qcy\n\u0015k\"c\u001ak0#\u0010\nWA;\u0015\u001a\nkk0qay\n\u0000q+WB;\u0015\u001a\nkk0qbq\u0011\n+h.c.;(4)\nwhere\u0015and\u001aare summed over the electron band indices. As\ndetailed in the Supplemental material, WA;\u0015\u001a\nkk0qandWB;\u0015\u001a\nkk0q, both\nlinear in J, are coe \u000ecients determining the amplitudes for\nscattering between the itinerant electrons and the aqandbq\nmagnons, respectively. Specifically, when considering plane\nwave states, WA=B;\u0015\u001a\nkk0qbecomes a delta function, thereby enforc-\ning the conservation of crystal momentum in a translationally\ninvariant lattice. Inclusion of disorder or other many-body\ne\u000bects results in deviation of the eigenstates from ideal plane\nwaves causing a wave vector spread around its mean value [ 2].\nThe delta function, associated with an exact crystal momentum\nconservation, is thus transformed to a peaked function with\nfinite width (\u0001k). The\u0015\u001acombinations 11and22describe\nintraband electron scattering, while 12and21describe in-\nterband scattering. Intraband scattering is illustrated in Fig.\n2. Interband scattering is prohibited within our model due to\nenergy conservation, since the uniform q=0magnon energy\nis much smaller than the band gap.\nThe scattering described by He\u0000m[Eq. (4)] transfers spin\nangular momentum between the magnetic and conduction sub-\nsystems. The itinerant electrons are assumed to maintain a\nthermal distribution thereby acting as a perfect spin sink. This\nis consistent with a strong conduction electron spin relaxation\nobserved in metallic AFMs [ 70,71]. As a result, the magnetic\nsubsystem spin is e \u000bectively damped through the s-dexchange\ninteraction.\nGilbert damping.— In the Landau-Lifshitz-Gilbert (LLG)\nphenomenology for two-sublattice AFMs, dissipation is ac-\ncounted via a 2\u00022 Gilbert damping matrix [ 72]. Our goal here\nis to determine the elements of this matrix in terms of the\nparameters and physical observables within our microscopic\nmodel. To this end, we evaluate the spin current “pumped”\nby the magnetic subsystem into the sconduction electrons,\nwhich dissipate it immediately within our model. The angu-\nlar momentum thus lost by the magnetic subsystem appears\nas Gilbert damping in its dynamical equations [ 72,73]. The3\n-\n/2 -\n /4 0\n /2\n /4\ne\ne\nkF,1a/epsilon1=µ1\nm\ne\ne\nm\n/epsilon1=µ2\nkF,2a\nFIG. 2: The selectron dispersion in metallic AFM model,\nwhere the red and blue dispersions depict electron bands 1 and\n2, respectively. Illustrations of intraband electron-magnon\nscattering at two di \u000berent Fermi levels, \u00161and\u00162, are added.\nThe depicted momentum transfer is exaggerated for clarity.\nsecond essential ingredient in identifying the Gilbert damping\nmatrix from our microscopic theory is the idea of coherent\nstates [ 74,75]. The classical LLG description of the magne-\ntization is necessarily equivalent to our quantum formalism,\nwhen the magnetic eigenmode is in a coherent state [ 74–76].\nDriving the magnetization dynamics via a microwave field,\nsuch as in the case of ferromagnetic resonance experiments,\nachieves such a coherent magnetization dynamics [73, 77].\nThe spin current pumped by a two-sublattice magnetic sys-\ntem into an electronic bath may be expressed as [78]\nIz=Gmm(m\u0002˙m)z+Gnn(n\u0002˙n)z\n+Gmn\u0002(m\u0002˙n)z+(n\u0002˙m)z\u0003;(5)\nwhere mand nare the magnetization and Néel field nor-\nmalized by the sublattice magnetization, respectively. Here,\nGi j=\u000bi j\u0002(M=j\rj), where\u000bi jare the Gilbert damping co-\ne\u000ecients,\ris the gyromagnetic ratio of the delectrons\nandMis the sublattice magnetization. Considering the uni-\nform magnetization mode, Izis the spin current operator\nIz=i[He\u0000m;Sz][79], where Sz=P\niSz\ni. We get\nIz=iX\n\u0015\u001aX\nkk0qcy\n\u0015k\"c\u001ak0#\u0010\nWA;\u0015\u001a\nkk0qay\n\u0000q+WB;\u0015\u001a\nkk0qbq\u0011\n\u0000h.c.:(6)\nThe expectation value of this operator assuming the uniform\nmagnetization mode to be in a coherent state corresponds to\nthe spin pumping current [Eq. (5)].\nIn order to evaluate the spin pumping current from Eq. (6),\nwe follow the method employed to calculate interfacial spin\npumping current into normal metals in Refs. [ 73,77,78], and\nthe procedure is described in detail therein. Briefly, this method\nentails assuming the magnetic and conduction subsystems to\nbe independent and in equilibrium at t=\u00001, when the mu-\ntual interaction [Eq. (4)] is turned on. The subsequent timeevolution of the coupled system allows evaluating its physical\nobservables in steady state. The resulting coherent spin-current\ncorresponds to the classical spin current Izthat can be related\nto the motion of the magnetization and the Néel field [Eq. (5)].\nAs a last step, we identify expressions for (m\u0002˙m)z,(m\u0002˙n)z\nand(n\u0002˙n)zin terms of coherent magnon states, which enables\nus to identify the Gilbert damping coe \u000ecients\u000bmm,\u000bnnand\n\u000bmn.\nResults.— Relegating the detailed evaluation to Supplemen-\ntal Material, we now present the analytic expression obtained\nfor the various coe \u000ecients [Eq. (5)]. A key assumption that\nallows these simple expressions is that the electronic density of\nstates in the conduction subsystem does not vary significantly\nover the magnon energy scale. Furthermore, we account for a\nweak disorder phenomenologically via a finite scattering length\nlassociated with the conduction electrons. This results in an\ne\u000bective broadening of the electron wavevectors determined by\nthe inverse electron scattering length, (\u0001k)=2\u0019=l. As a result,\nthe crystal momentum conservation in the system is enforced\nonly within the wavevector broadening. By weak disorder we\nmean that the electron scattering length is much larger than\nthe lattice parameter a. Ifkandk0are the wave vectors of the\nincoming and outgoing electrons, respectively, we then have\n(k\u0000k0)a=(\u0001k)a\u001c1. This justifies an expansion in the wave\nvector broadening (\u0001k)a. The Gilbert damping coe \u000ecients\nstemming from intraband electron scattering are found to be\n\u000bmm=\u000b0(\u0018J)\u0000\u000b0(\u0018J)\n40BBBBBBBB@1+\u00182\nJ\u0010\n\u00182\nJ+8\u00004 cos2(kFa)\u0011\n\u0010\n\u00182\nJ+4 cos2(kFa)\u001121CCCCCCCCA[(\u0001k)a]2;\n\u000bnn=\u000b0(\u0018J)\n40BBBBBB@1+sin2(kFa)\ncos2(kFa)\u00182\nJ\u0010\n\u00182\nJ+4 cos2(kFa)\u00111CCCCCCA[(\u0001k)a]2:\n(7)\nwhere\u0018J=JS=t,kFis the Fermi momentum and ais the lattice\nparameter, and where\n\u000b0(\u0018J)=\u0019v2J2\n8g2(\u0016)j˜Vj24 cos2(kFa)\n\u00182\nJ+4 cos2(kFa): (8)\nHere, vis the unit cell volume, g(\u000f)is the density of states\nper unit volume, \u0016is the Fermi level, and !0is the energy of\ntheq=0magnon mode. ˜Vis a dimensionless and generally\ncomplex function introduced to account for the momentum\nbroadening dependency of the scattering amplitudes. It satisfies\n˜V(0)=1and0\u0014j˜V(\u0001k)j\u00141within our model. These analytic\nexpressions for the Gilbert damping parameters constitute one\nof the main results of this letter.\nDiscussion.– We straightaway note that \u000bnn=\u000bmm\u0018\n[(\u0001k)a]2\u001c1.\u000bnnis strictly dependent upon (\u0001k)a, and is non-\nzero only if there is disorder and a finite electron momentum\nbroadening. \u000bmmis large even when considering a perfectly\nordered crystal. This latter result is in good accordance with\nrecent first-principles calculations in metallic AFMs [ 56,66].\nWe moreover observe that both \u000bmmand\u000bnnare quadratic\ninJandg(\u0016). This result is shared by Gilbert damping ow-\ning to spin-pumping in insulating ferrimagnet |normal metal4\ne\n e\nm\nkFa/epsilon1=µ\nFIG. 3: A schematic depiction of magnon-induced interband\nscattering in a band crossing at the Fermi level.\n(NM) and AFM |NM bilayers with interfacial exchange cou-\npling [ 78]. Metallic AFMs bear a close resemblance to these\nbilayer structures. There are however two main di \u000berences:\nThes-dexchange coupling exists in the bulk of metallic AFMs,\nwhereas it is localized at the interface in the bilayer structures.\nAdditionally, the itinerant electron wave functions are qual-\nitatively di \u000berent in metallic AFMs and NMs, owing to the\nmagnetic unit cell of the AFM. Indeed, these di \u000berences turn\nout to leave prominent signatures in the Gilbert damping in\nmetallic AFMs.\nThe uniform mode magnon energy is much smaller than the\nelectron band gap within our simple model. Interband scat-\ntering is thus prohibited by energy conservation. However,\nin real AFM metals, the electron band structure is more com-\nplex. There may for instance exist band crossings [ 80–82].\nIn such materials, magnon-induced interband electron scatter-\ning should also contribute to Gilbert damping, depending on\nthe position of the Fermi surface. Motivated by this, we now\nconsider Gilbert damping stemming from interband scattering,\nwhile disregarding the energy conservation for the moment,\nlabeling the coe \u000ecients\u000bI\nmmand\u000bI\nnn. We then find the same\nexpressions as in Eq. (7) with the roles of \u000bI\nmm;nninterchanged\nwith respect to \u000bmm;nn. This implies that \u000bI\nnnis large and inde-\npendent of electron momentum broadening, whereas \u000bI\nmmis\nproportional to the electron momentum broadening squared.\nAlthough arriving at this result required disregarding the en-\nergy conservation constraint, the qualitative e \u000bect in itself is\nnot an artifact of this assumption. In materials with a band\ncrossing, as depicted in Fig. 3, \u000bI\nnn=\u000bI\nmm> \u000b nn=\u000bmmis a gen-\neral result. This generic principle derived within our simple\nmodel provides valuable guidance for designing materials with\nan engineered Gilbert damping matrix.\nWe now provide a rough intuitive picture for the damping\ndependencies obtained above followed by a more mathemati-\ncal discussion. Consider a conventional di \u000braction experiment\nwhere an incident probing wave is able to resolve the two\nslits only when the wavelength is comparable to the physical\nseparation between the two slits. In the case at hand, the wave-\nfunctions of electrons and magnon participating in a scatteringprocess combine in a way that the wavenumber by which the\nconservation of crystal momentum is violated becomes the\nprobing wavenumber within a di \u000braction picture. Therefore,\nthe processes conserving crystal momentum have vanishing\nprobing wavenumber and are not able to resolve the opposite\nspins localized at adjacent lattice sites. Therefore, only the aver-\nage magnetization is damped leaving the Néel field una \u000bected.\nWith disorder, the probing wavenumber becomes non-zero and\nthus also couples to the Néel field. The interband scattering,\non the other hand, is reminiscent of Umklapp scattering in a\nsingle-sublattice model and the probing wavenumber matches\nwith the inverse lattice spacing. Therefore, the coupling with\nthe Néel field is strong.\nThe Gilbert damping in metallic AFMs here considered is\ncaused by spin pumping from the magnetic subsystem into\nthesband. This spin pumping induces electron transitions\nbetween spin\"/#states among the selectrons. The Gilbert\ndamping coe \u000ecients depend thus on transition amplitudes pro-\nportional to products of itinerant electron wave functions such\nas y\n\u0015k\"(x) \u001ak0#(x). The damping e \u000bect on sublattice A depends\non this transition amplitude evaluated on the A sublattice, and\nequivalently for the B sublattice. Assuming without loss of gen-\nerality that site i=0belongs to sublattice A, we find in the one-\ndimensional model that the damping on sublattice A is a func-\ntion ofP\njcos2\u0010\u0019xj\n2a\u0011\n y\n\u0015k\"(xj) \u001ak0#(xj), whereas the damping\non sublattice B is a function ofP\njsin2\u0010\u0019xj\n2a\u0011\n y\n\u0015k\"(xj) \u001ak0#(xj).\nEquivalently, by straightforwardly using that m=(mA+mB)=2\nandn=(mA\u0000mB)=2, this analysis predicts that \u000bmmis a\nfunction ofP\nj y\n\u0015k\"(xj) \u001ak0#(x), whereas\u000bnnis a function of\nP\njcos\u0010\u0019xj\na\u0011\n y\n\u0015k\"(xj) \u001ak0#(x). Assuming plane wave solutions\nof the electron wave functions, and if we consider intraband\nscattering only, we more concretely find that \u000bmmis a function\nof(1\u0000i(\u0001k)a), where iis the imaginary unit, whereas \u000bnnis a\nfunction of ( \u0001k)a. This coincides well with Eq. (7).\nAbove, we presented a discussion of interband scattering in\nthe minimal model where the band gap artificially was set to\nzero. In this limit, the upper electron band is a continuation\nof the lower band with a \u0006\u0019=amomentum shift. We may then\nwrite 2k\u001b= 1;k+\u0019=a;\u001b. Under the assumption of a disappear-\ning band gap, momentum-conserving interband scattering at\nmomentum kis therefore equivalent to intraband scattering be-\ntween kandk\u0006\u0019=a. This is the exact phase shift which results\nin a small\u000bmmand a large \u000bnnconsistent with the discussion\nabove. In real metallic AFMs with complex band structures,\nthe exact wave function relations unveiled above do not apply.\nHowever, interband transition amplitudes will undoubtedly\ncarry a position dependent phase. This position dependence\nresults in a dephasing of transition amplitudes at neighboring\nlattice sites, which gives rise to a non-negligible \u000bnn. The pre-\ncise damping coe \u000ecients in real metallic AFMs depend on the\ndetailed electron wave functions. We may however generally\nconclude that \u000bI\nnn=\u000bI\nmm>\u000b nn=\u000bmm.\nConclusion.— We have provided a microscopic derivation\nof Gilbert damping resulting from magnon decay through s-d\nexchange interaction in metallic antiferromagnets. Analytic5\nexpressions for Gilbert damping coe \u000ecients resulting from in-\ntraband electron scattering are presented, while Gilbert damp-\ning resulting from interband electron scattering is discussed on\na conceptual level. We find that intraband electron scattering\ngives rise to a large magnetization damping and a negligible\nNéel field damping. The intraband Néel field damping is pro-\nportional to the inverse electron scattering length squared, and\ndisappears exactly if there is no crystal disorder. 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Landau Institute for Theoretical Physics RAS, 119334 M oscow, Russia\n4ITMO University, Saint Petersburg 197101, Russia\nSpin-transfer torques (STT), Gilbert damping (GD), and effe ctive spin renormalization (ESR) are\ninvestigated microscopically in a 2D Rashba ferromagnet wi th spin-independent Gaussian white-\nnoise disorder. Rashba spin-orbit coupling-induced aniso tropy of these phenomena is thoroughly\nanalysed. For the case of two partly filled spin subbands, a re markable relation between the\nanisotropic STT, GD, and ESR is established. In the absence o f magnetic field and other torques\non magnetization, this relation corresponds to a current-i nduced motion of a magnetic texture with\nthe classical drift velocity of conduction electrons. Fina lly, we compute spin susceptibility of the\nsystem and generalize the notion of spin-polarized current .\nPossibility to efficiently manipulate magnetic order by\nmeansofelectriccurrenthasgainedalotofattentionover\nthe past decades1,2. Potential applications include race\ntrackmemory3,4, spin torquemagnetization switching5,6,\nskyrmion-based technology7,8, and other promising con-\ncepts. Spintronic logic and memory devices based on\ncurrent-driven magnetization dynamics are believed to\nachieve high speed, low volatility, outstanding durabil-\nity, and low material costs with promises to outperform\ncharge-trapping solid-state memory devices9.\nIn the light of recent detection of fast domain wall\n(DW) motion in magnetic films10,11and predictions of\nevenhigherDWvelocitiesinantiferromagnets12, current-\ninduced dynamics of domain walls, skyrmions, and other\nmagnetic textures remain an important research subject\nin the field of spintronics. Such dynamics is mainly de-\ntermined by the interplay of the two phenomena: Gilbert\ndamping (GD) and spin torques13–16.\nIn the absence of spin-orbit coupling (SOC), spin\ntorques emerge only in the systems with nonuniform\nmagnetization profiles and are most often referred to as\nspin-transfer torques (STT). At the same time, the clas-\nsification of spin torques usually gets more complicated\nif coupling between spin and orbital degrees of freedom\nbecomes pronounced. Moreover, the debate on the mi-\ncroscopic origin of spin torques in the latter case remains\nongoing17,18. Below, we regard STT, in the continuum\nlimit, as a contribution to the total torque on magnetiza-\ntion that is linearwith respect to both the electricfield E\nand the first spatial derivatives of the unit vector of mag-\nnetization direction n. We note that, in the absence of\nSOC, physics of STT is well understood15,16.\nIn a similar fashion, Gilbert damping may be gener-\nally associated with the terms of the Landau-Lifshitz-\nGilbert (LLG) equation that are odd under time reversal\nand linear with respect to the time derivative of n. In\nthe most simplistic approach, GD is modeled by a sin-\ngle phenomenological term αn×∂tnthat corresponds to\n“isotropic” damping.\nHowever, it has been known for quite a while that GD\nmay exhibit anisotropic behaviour19–27. Or, to be more\nprecise, that the scalar damping constant α, in general,should be replaced by a damping matrix with the com-\nponents depending on the orientation of n. These two\nmanifestations of anisotropy may be referred to as rota-\ntional and orientational anisotropy, respectively22. Ex-\nperimental observation of the orientational anisotropy\nof Gilbert damping has been reported very recently in\na metal ferromagnet (FM)/semiconductor interface of\nFe/GaAs(001)28and in epitaxial CoFe films29. The au-\nthors of Ref. [28] argued that the measured anisotropy\nrooted in the interplay of interfacial Rashba and Dressel-\nhaus spin-orbit interaction.\nGiven the equal importance of GD and STT in the\ncontext of current-induced magnetization dynamics and\nthe significant progressmade in the understanding of the\nanisotropic nature of Gilbert damping, we find it surpris-\ning that the anisotropyof spin-transfer torques has so far\nonly been addressed phenomenologically24,30.\nIn the present paper, we consider a 2D Rashba FM\nwithspin-independent electronscattering. Amicroscopic\nanalysis, performed for an arbitrarymagnetization direc-\ntion, allows us to quantify the rotational as well as the\norientationalanisotropyofboth STT and GD induced by\nRashba SOC. Our results indicate that, for a Rashba FM\nsystem, spin-transfer torques TSTTand Gilbert damp-\ningTGDentering the LLG equation\n∂tn=γn×Heff+TSTT+TGD+... (1)\nnaturally acquire the following forms:\nTSTT=ξ0∂vn−ξ/bardbl[n×∂vn/bardbl]−ξ⊥[n×∂vn⊥],(2a)\nTGD=ξ0∂tn−ξ/bardbl[n×∂tn/bardbl]−ξ⊥[n×∂tn⊥],(2b)\nwhereξi=ξi(n), the operator ∂v= (vd·∇) is expressed\nvia the classical electron drift velocity vd=eE/planckover2pi1τ/m,\nandn/bardbl/⊥stands for the in-plane/perpendicular-to-the-\nplane component of the vector field n:\nn=n/bardbl+n⊥,n⊥=eznz=ezcosθ.(3)\nFor convenience, we have included the term ξ0∂tninto\nthe definition of TGD. This term, being even under time\nreversal, leads to a renormalization of spin in the LLG2\nequation16and does not contribute to damping. In what\nfollows, we refer to such renormalization as effective spin\nrenormalization (ESR).\nThe rotational and orientational anisotropy arising in\nEqs. (2) appear to be a natural consequence of the fact\nthat the Rashba spin-orbit interaction singles out the di-\nrection perpendicular to the electron 2D plane. The ori-\nentationalanisotropyofthedimensionlessfunctions ξi(n)\nis determined by all space symmetries of the system and,\nfor a general Rashba FM, may turn out to be rather\ncomplex. However, for the particular interface model of\ntheC∞vsymmetry class, which we consider below, one\nsimply finds ξi=ξi(n2\nz).\nBefore we proceed, let us describe at least two impor-\ntant outcomes of Eqs. (2). First, according to the usual\nconvention, STT consist of two contributions: the adi-\nabatic torque ∝(js·∇)nand the nonadiabatic torque\n∝n×(js·∇)n, wherejsdenotes a spin-polarized cur-\nrent. For vanishing SOC, the adiabatictorque has aclear\nphysical meaning. As far as spins of conduction elec-\ntrons adiabatically follow local magnetization direction,\nthe corresponding change of their angular momentum is\ntransferred to the magnetic texture. Since ↑and↓spins\npoint in the opposite directions along n, the transfer rate\nis proportional to ( js·∇)n, where js=j↑−j↓. In\nthe presence of SOC, however, conduction spins are no\nlonger aligned with the direction of nand, thus, the en-\ntire concept of spin-polarized current becomes somewhat\nvague. For the particular Rashba model, our results re-\nveal an important relation between the adiabatic torque\nand ESR, providing steps toward better understanding\nof the former for systems with SOC.\nAnother remarkable property of Eqs. (2) is a simple\nand exact relation between the nonadiabatic torque and\nGD, which has an important implication for current-\ninduced motion of magnetic textures (e.g., domain walls\nor skyrmions). Indeed, by transforming Eq. (1) into the\nmoving reference frame31r′=r−vdt, one immedi-\nately observes that both components of the nonadiabatic\ntorqueareexactlycancelledbythe correspondingGilbert\ndamping terms. Therefore, if the effect of other driving\ntorques on the motion of a magnetic texture is negligible,\nthen its terminal velocity, in the moving reference frame,\nshall vanish for mediate currents32,33(in the absence of\nmagnetic field). This implies that, in the laboratory ref-\nerence frame, the texture moves with the universal elec-\ntron drift velocity vd. Certainly, in the presence of, e.g.,\nspin-orbit torques, which can assist motion of domain\nwalls and skyrmions10,34, the resulting dynamics might\ndiffer. In any case, the analysisofsuch dynamics can still\nbe performed in the moving reference frame, where the\neffect of the nonadiabatic spin-transfer torque is conve-\nniently absent.\nHaving outlined our main results, we skip further dis-\ncussion until Sec VII. The rest of the paper is organized\nas follows. In Sec. I we introduce the model and use\nan expansion in spatial gradients to reduce the analysis\nto a study of a homogeneous system. Self-energy andKubo formulas are addressed in Sec. II. A general re-\nlation between STT, GD, and ESR (in the considered\nmodel) is obtained in Sec. III, while in Sec. IV we estab-\nlish the exact vector structures of these quantities. Some\nanalytical insight into our general results is provided in\nSec. V and Sec. VI. An extensive Discussion of Sec VII\nis followed by Conclusions (and seven Appendices).\nI. MODEL\nA. Generalized torque in s-dmodel\nIn whatfollows, weadoptthe ideologyofthe s-dmodel\nby performing a decomposition of a FM into a system of\nlocalized spins Siand a system of noninteracting con-\nduction electrons. Despite being rather simplistic, this\napproach has proven to describe very well the key prop-\nerties of current-induced magnetization dynamics in fer-\nromagnetic systems35–38.\nIf the value of |Si|=Scan be assumed sufficiently\nlarge, then it is natural to treat the localized spins clas-\nsically by means of the unit vector n(ri) =Si/S, which\npoints in the opposite to local magnetization direction.\nIn this case, the s-d-like local exchange interaction be-\ntween the localized spins and conduction electrons is\ngiven, in the continuum limit, by\nHsd=JsdSn(r,t)·σ, (4)\nwithJsdquantifying the strength of the exchange and\nPauli matrices σrepresenting the spins of conduction\nelectrons.\nIt is known16that interaction of the form of Eq. (4),\nleads to the following LLG equation for the dynamics of\nthe vector n:\n∂tn=γn×Heff+JsdA\n/planckover2pi1[s(r,t)×n(r,t)],(5)\nwhereγisthebaregyromagneticratio, Heffdescribesthe\neffective magnetic field, Adenotes the areaof the magnet\nunit cell, and s(r,t) stands for the nonequilibrium spin\ndensity of conduction electrons39. The second term on\nthe right hand side of Eq. (5) represents the generalized\ntorque on magnetization\nT=JsdA\n/planckover2pi1[s(r,t)×n(r,t)]. (6)\nAssuming slow dynamics of n(r,t) on the scale of elec-\ntron scattering time and smoothness of magnetization\nprofile on the scale of electron mean free path, one may\nexpand the generalized torque in time and space gradi-\nents ofn. In this paper, we consider twoparticularterms\nof such expansion,\nT=TSTT+TGD+..., (7)\nignoring all other contributions (such as, e.g., spin-orbit\ntorques). In Eq. (7) and below, we identify spin-transfer3\ntorquesTSTTas a double response of Tto the electric\nfieldEandtothespatialgradientsof n, whiletheGilbert\ndamping vector TGD(which also includes the ESR term)\nis defined as a response to the time derivative of n,\nTSTT\nα=/summationdisplay\nβγδTSTT\nαβγδEβ∇γnδ, (8a)\nTGD\nα=/summationdisplay\nδTGD\nαδ∂tnδ. (8b)\nMicroscopic analysis of the tensors TSTTandTGDis the\nmain subject of the present work.\nB. Single particle problem\nAccording to Eqs. (8), the vectors TSTTandTGDrep-\nresent linear response to the time derivative of magne-\ntization direction and to the time derivative of vector\npotential, respectively. Hence, computation of both vec-\ntors can be performed with the help of Kubo formulas\nthat make use of Green’s functions of the correspond-\ning time-independent problem. We choose the latter to\noriginate in the 2D Rashba model40with the effective\ns-d-type term of Eq. (4),\nH=p2/2m+αR[p×σ]z+JsdSn(r)·σ,(9)\nwhereαRcharacterizes the strength of Rashba coupling\nandmis the effective electron mass.\nThe Hamiltonian of Eq. (9) should be supplemented\nwith a momentum relaxation mechanism since both STT\nandGDtensors,similarlytotheconductivitytensor,con-\ntain essentially dissipative components. We assume that\nmomentum relaxation in the system is provided by scat-\ntering on a spin-independent Gaussian white-noise dis-\norder potential Vdis(r). Thus, the full Hamiltonian of a\nsingle conduction electron reads\nHdis=H+Vdis(r), (10)\nwhere the disorder potential is characterized by the zero\naverage∝an}b∇acketle{tVdis(r)∝an}b∇acket∇i}ht= 0 and the pair correlator\n∝an}b∇acketle{tVdis(r)Vdis(r′)∝an}b∇acket∇i}ht= (/planckover2pi12/mτ)δ(r−r′).(11)\nThe angular brackets in Eq. (11) stand for the averaging\nover the disorder realizations, τis the mean scattering\ntime measured in the inverse energy units.\nOne can readily observe from Eq. (6) that the general-\nized torque Tcan be understood as a spatial density of a\ndisorder-averagedmeanvalueoftheoperator( JsdA//planckover2pi1)ˆT,\nwhere we refer to\nˆT=σ×n(r), (12)\nas the dimensionless torque operator.C. Expansion in spatial gradients\nComputation of STT involves the expansion of the\nHamiltonian Hof Eq. (9) and the corresponding Green’s\nfunction\nGR,A= (ε−H±i0)−1(13)\nin the first spatial gradients of nup to the linear terms.\nWe obtain the latter utilizing the Taylor expansion\nn(r) =n(r∗)+/summationdisplay\nγ(r−r∗)γ∇γn(r∗),(14)\nat some particular point r∗.\nWith the help of Eq. (14), Hcan be, then, approxi-\nmated as\nH=H+JsdS/summationdisplay\nγ(r−r∗)γ∇γn(r∗)·σ,(15)\nwhere the Hamiltonian\nH=p2/2m+αR[p×σ]z+JsdSn(r∗)·σ(16)\ndescribes the homogeneouselectronic system with a fixed\ndirection of magnetization set by n(r∗).\nSimilarly, we approximate the Green’s function GR,A,\nemploying the Dyson series\nGR,A(r,r′) =GR,A(r−r′)+JsdS/integraldisplay\nd2r′′GR,A(r−r′′)\n×/bracketleftig/summationdisplay\nγ(r′′−r∗)γ∇γn(r∗)·σ/bracketrightig\nGR,A(r′′−r′) (17)\nand the Green’s function\nGR,A= (ε−H±i0)−1(18)\nthat corresponds to the homogeneous system. Note that,\nin Eq. (17), we kept only the terms that are linear in the\ngradients of n, as prescribed.\nD. Spectrum of the homogeneous system\nThe spectrum of Hincorporates two spectral branches\nε±(p) =p2/2m±/radicalig\n∆2\nsd+(αRp)2−2ςαR∆sdpsinθsinϕ,\n(19)\nwhere the angle θstands for the polar angle of nwith\nrespecttothe zaxis[seealsoEq.(3)], while ϕisthe angle\nbetween the momentum pand the in-plane component\nof the vector n:ϕ=φp−φn. We have also introduced\nthe notations\n∆sd=|Jsd|S, ς = signJsd, (20)\nwhere ∆ sdhas a meaning of half of the exchange\ninteraction-induced splitting (in the absence of SOC).4\nFIG. 1. Guide for an eye: spectrum of the homogeneous sys-\ntem of conduction electrons with a fixed direction of magne-\ntization. Note that the actual spectrum is not isotropic, an d\nthe two subbands may even touch each other. We restrict the\nanalysis to the case of ε >∆sd. For the latter, both subbands\nare always partly filled.\nIf the chemical potential εexceeds the value of ∆ sd,\nboth subbands are always partly filled41. Below, we fo-\ncus solely on the latter case, which is schematically illus-\ntrated in Fig. 1. Note that the spectrum is not isotropic.\nMoreover, for finite values of sin θ, separation of the\ntwo subbands diminishes and they may even touch each\nother.\nIn what follows, we also find it convenient to intro-\nduce the energy scale ∆ so=|αR|√\n2mε, which is equal\nto half of the spin-orbit coupling-induced splitting of the\nbranches (for vanishing ∆ sd).\nE. Roots of dispersion relation\nNow let us analyze the roots of the dispersion of\nEq. (19). Using, for example, Ref. [42], one can show\nthat, under the assumption ε >∆sd, the quartic func-\ntion (ε+(p)−ε)(ε−(p)−ε) of the absolute value of mo-\nmentum palways has four real roots: two positive and\ntwonegative. The former twodefine the angle-dependent\nFermi momenta p±corresponding to ε±branches. The\nfour roots are distinct in all cases, except one. Namely,\nwhenn⊥= 0 (i.e., when sin θ= 1) and ∆ so= ∆sd, the\nsubbands touch each other. We will not consider this\nparticular case.\nUsing the notation p±,negfor the negative roots, we\nhave\np−> p+>0> p+,neg> p−,neg, (21)\nwhere\np∓=1\n2/parenleftbigg√\n2u±/radicalig\n−2u−2q−r/radicalbig\n2/u/parenrightbigg\n,(22a)\np±,neg=1\n2/parenleftbigg\n−√\n2u±/radicalig\n−2u−2q+r/radicalbig\n2/u/parenrightbigg\n,(22b)\nu >0 is the largest root of the resolvent cubic\nu3+qu2−(s−q2/4)u−r2/8, (23)while the parameters q,s, andrare given by\nq=−4m(ε+mα2\nR), s= (2m)2(ε2−∆2\nsd),(24a)\nr= 8m2αRς∆sdsinθsinϕ. (24b)\nIt is straightforward to see, from Eqs. (24), that the\ndependence on the momentum angle enters Eq. (23) only\nvia the parameter r2. As a result, the quantity umay\nonly depend on sin2ϕand other parameters of the model\nthat areϕindependent. This will play an important role\nbelow.\nForαR= 0 (vanishing SOC), ∆ sd= 0 (nonmagnetic\nlimit), or n=n⊥(perpendicular-to-the-plane magneti-\nzation) situation with the rootsbecomes less complex. In\nthese cases, ( ε+(p)−ε)(ε−(p)−ε) is biquadratic (with\nrespect to p) andp±=−p±,neg, as one can also see di-\nrectly from Eqs. (22). Furthermore, the Fermi momenta\np±, then, are angle independent, while their values yield\nthe relations\np2\n±= 2m[ε∓∆sd],forαR= 0,(25a)\np2\n±= 2m/bracketleftbig\nε+mα2\nR∓λ(0)/bracketrightbig\n,for ∆sd= 0,(25b)\np2\n±= 2m/bracketleftbig\nε+mα2\nR∓λ(∆sd)/bracketrightbig\n,forn=n⊥,(25c)\nwhereλ(Υ) =/radicalbig\nΥ2+2εmα2\nR+m2α4\nR.\nII. DISORDER AVERAGING\nHaving analysed the spectrum of the “clean” homoge-\nneous system, we can proceed with the inclusion of the\ndisorder. In what follows, we assume ε0τ≫1, where\nε0is the difference between the Fermi energy εand the\nclosest band edge. We start with a calculation of the\nself-energy in the first Born approximation.\nA. Self-energy\nAccording to Eq. (11), the self-energy is defined as\nΣR,A(r) = (/planckover2pi12/mτ)GR,A(r,r), (26)\nwith the Green’s function GR,Aof Eq. (13). It should be\nexplicitly pronounced that ΣR,A(r) may have a spatial\ndependenceoriginatinginthespatialdependenceof n(r).\nHowever, as we are about to see, the first spatial gradi-\nents of magnetization do not affect the self-energy in the\nmodel under consideration.\nDisregarding the “real” part of the self-energy that\nshould be included in the renormalized value of the\nchemical potential, we focus only on the calculation of\nImΣ(r) =−i[ΣR(r)−ΣA(r)]/2. By substituting the\nexpansion of Eq. (17) into Eq. (26), switching to mo-\nmentum representation, and symmetrizing the result we\nobtain\nImΣ(r) = Σ(0)+/summationdisplay\nγδ/braceleftig\n(r−r∗)γΣ(1)\nδ+Σ(2)\nγδ/bracerightig\n∇γnδ(r∗),\n(27)5\nwith\nΣ(0)=1\n2imτ/integraldisplayd2p\n(2π)2/parenleftbig\nGR−GA/parenrightbig\n,(28a)\nΣ(1)\nδ=ς∆sd\n2imτ/integraldisplayd2p\n(2π)2/parenleftig\nGRσδGR−GAσδGA/parenrightig\n,(28b)\nΣ(2)\nγδ=ς∆sd/planckover2pi1\n4mτ/integraldisplayd2p\n(2π)2/parenleftig\nGRσδGRvγGR−\nGRvγGRσδGR+h.c./parenrightig\n,(28c)\nwhere “h.c.” denotes Hermitian conjugate, GR,Ais the\nGreen’s function of Eq. (18) in momentum representa-\ntion,\nGR,A=ε−p2/2m+αR[p×σ]z+ς∆sdn(r∗)·σ\n(ε−ε+(p)±i0)(ε−ε−(p)±i0),(29)\nandv=∂H/∂pis the velocity operator. In Eqs. (28),\nΣ(0)defines the scattering time (for uniform magneti-\nzation), Σ(1)corresponds to the renormalization of the\ngradient term on the right hand side of Eq. (15), while\nΣ(2)determinesthe possible dependence ofthe scattering\ntime on the first spatial gradients of magnetization.\nTo proceed, we take advantage of the additional sym-\nmetrization of the integrands with respect to the trans-\nformation43ϕ→π−ϕand observe that, in the first\nBorn approximation, integration over the absolute value\nof momentum, in Eqs. (28), is reduced to a calculation\nof residues at p=p±. Using Eqs. (22), we, then, get\nΣ(0)=−1\n2τ/integraldisplay2π\n0dϕ\n2π/bracketleftbig\n1+rW1+rW2n(r∗)·σ\n+W3n/bardbl(r∗)·σsinϕ/bracketrightbig\n,(30)\nwhereWi=Wi/parenleftbig\nr2,u(r2)/parenrightbig\nare some functions of the pa-\nrameterr2andϕ-independent parameters of the model.\nSincer∝sinϕand, obviously, all integrals of the form/integraltext2π\n0W(sin2ϕ)sinϕdϕvanish for arbitrary function W,\nwe obtain a particularly simple result for the constant\npart of the self-energy,\nΣ(0)=−1/2τ. (31)\nSimilar, but more lengthy, analysis shows that each\ncomponent of Σ(1)and Σ(2)is equal to zero. Therefore,\nthereexistsnorenormalizationofthegradienttermofthe\nHamiltonian Has well as no scattering time dependence\non the first magnetization gradients. The self-energy, in\nthe first Born approximation, is found as\nΣR,A(r) =∓i/2τ. (32)\nB. Kubo formula for STT\nAs was outlined in Sec. IB, the generalized torque\nT(r0) of Eq. (6), at a certain position r0in space, is de-\nfined as a disorder-averaged mean value of the operatorFIG.2. Diagrammatic representationoftheSTTtensor TSTT\nαβγδ\nof Eq. (34). Solid lines correspond to the disorder-average d\nGreen’s functions gR,A. Vertex corrections (impurity ladders)\nare represented by green fillings.\n(JsdA//planckover2pi1)δr0ˆT, whereδr0=δ(r−r0). At zero tempera-\nture, thelinearresponse44ofTα(r0)tothezerofrequency\nelectric field Eis given by the standard Kubo expression\ne/planckover2pi1\n2πJsdA\n/planckover2pi1/angbracketleftig\nTr/bracketleftig\nGAδr0ˆTαGRv/bracketrightig\nE/angbracketrightig\n,(33)\nwherev=∂H/∂pis the velocity operator, Tr stands for\nthe operator trace, and angular brackets represent the\ndisorder averaging.\nFrom Eq. (33), we can further deduce the Kubo for-\nmula for spin-transfer torques. In order to do that, we\nsubstitutetheexpansionofEq.(17)intoEq.(33)andcol-\nlect all terms proportional to ∇γnδ(r∗). Then we switch\ntomomentum representationandperformspatialaverag-\ning of torque on the scale of transport mean free path in\nthe vicinity of r=r0. In the noncrossingapproximation,\nthis leads to the general formula for the STT tensor,\nTSTT\nαβγδ=e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2\n×itr/bracketleftig\ngAσδgAvγgAˆTvc\nαgRvvc\nβ−h.c./bracketrightig\n,(34)\nwhere the superscript “vc” marks the vertices corrected\nwith the impurity ladders, the notation tr refers to the\nmatrix trace, and\ngR,A=∝an}b∇acketle{tGR,A∝an}b∇acket∇i}ht= (ε−H±i/2τ)−1(35)\nis the disorder-averaged Green’s function of the homoge-\nneous system. In Eq. (35), we have used the result for\nthe self-energy obtained in Sec. IIA.\nThe expression of Eq. (34) is represented diagrammat-\nically in Fig. 2. We note that similar diagrams have been\nused in Ref. [45] to compute STT in a 3D FM, in the\nabsence of SOC, and in Ref. [46] to study STT for the\nmodel of massive Dirac fermions.\nC. Kubo formula for GD and ESR\nSimilarly, from the zero frequency linear response44of\nTα(r0) to the time derivative of n,\nJsdS/planckover2pi1\n2πJsdA\n/planckover2pi1/angbracketleftig\nTr/bracketleftig\nGAδr0ˆTαGRσ/bracketrightig\n∂tn/angbracketrightig\n,(36)6\nonemayderivethe formulaforthe GDtensorofEq.(8b),\nTGD\nαδ=∆2\nsdA\n2π/planckover2pi12S/integraldisplayd2p\n(2π)2tr/bracketleftig\ngAˆTvc\nαgRσδ/bracketrightig\n,(37)\nwhere, according to the definition of TGD, spatial depen-\ndence of nis completely disregarded.\nNote that n,∇γnδ, and∂tnin Eqs. (8), (34), and (37)\nare all taken at r=r0. From now on, we consistently\nomit the argument of all these functions.\nD. Relation between TGDand vertex corrections\nto the torque operator ˆT\nVertex corrected torque operator that enters both\nEqs. (34) and (37) can be expressed with the help of\nvertex corrected Pauli matrices. One can infer the latter\nfrom the “matrix of one dressing” M, whose elements\nMij=1\n2mτ/integraldisplayd2p\n(2π)2tr/bracketleftig\ngAσigRσj/bracketrightig\n(38)\narethe coordinates(in the basis {σx,σy,σz}) ofthe oper-\natorσidressed with a single impurity line. We note that,\nin the model considered, vertex corrected Pauli matrices\nσvc\niappear to have zero trace if ε >∆sd. This is a direct\nconsequence of the fact that the self-energy in Eq. (32) is\nscalar. Hence, {σx,σy,σz}is, indeed, a proper basis for\nthe operators σvc\ni.\nMatrixrepresentationofthe operator ˆT=σ×n, with\nrespect to this basis, is defined as\nˆTi=/summationdisplay\njUijσj, U=\n0nz−ny\n−nz0nx\nny−nx0\n.(39)\nSince, obviously,\nˆTvc\ni=/summationdisplay\njUijσvc\nj, (40)\nwe can see, from Eq. (38), that the geometric series\nT=U(M+M2+···) =UM(I−M)−1,(41)\nprovides the matrix representation of vertex corrections\nto the torque operator. Moreover, from Eq. (37), it is\nevident that the GD tensor is, in fact, determined by the\nsame matrix T,\nTGD\nαδ=∆2\nsdAmτ\nπ/planckover2pi12STαδ. (42)\nE. Crossing diagrams\nIt has been demonstrated recently that the diagrams\nwith two crossing impurity lines may contribute to such\nquantitiesasthe anomalousHall effect47–49, the spinHalleffect50, and the Kerr effect51in the same leading or-\nder with respect to the small parameter ( ε0τ)−1, as the\nconventionalnoncrossingapproximationdoes. Scattering\nmechanisms associated with these diagrams, in general,\nshould affect spin torques and damping as well.\nIn the presentstudy we, however,completely disregard\nthe crossing diagrams, as being significantly more diffi-\ncult to calculate. At the same time, preliminary anal-\nysis shows that the related additional contributions to\nSTT, GD, and ESR are parametrically different from the\npresent resultsand that, for ε≫∆sd, they are negligible.\nIII. RELATION BETWEEN STT, GD, AND ESR\nA. Symmetrization of STT diagrams\nCalculation of spin-transfer torques can be performed\nwith the help of Eq. (34) directly. Such brute-force cal-\nculation has been originally performed by us. We have,\nhowever, subsequently found a shortcut that makes it\npossible not only to obtain the same results in a much\nmore concise manner but also to establish a general re-\nlation between TSTTandTGDtensors. This alternative\napproach takes a reformulation of the result of Eq. (34)\nin a more symmetric form.\nWe apply the identity gAvγgA=∂gA/∂pγin Eq. (34)\nand perform integration by parts. Then, we take a half-\nsum of the result obtained and the original expression of\nEq. (34). This leads to the formula\nTSTT\nαβγδ=δTSTT\nαβγδ+e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2\n×i\n2tr/bracketleftbigg\n−gAσδgAˆTvc\nαgR∂vvc\nβ\n∂pγ−h.c./bracketrightbigg\n,(43)\nwhere the first term on the right-hand side\nδTSTT\nαβγδ=e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2\ni\n2tr/bracketleftig\ngAσδgAvγgAˆTvc\nαgRvvc\nβ−gAvγgAσδgAˆTvc\nαgRvvc\nβ\n−gAσδgAˆTvc\nαgRvγgRvvc\nβ−h.c./bracketrightig\n.(44)\nis illustrated schematically in Fig. 3 by a group of en-\ncircled diagrams. The remaining two diagrams in Fig. 3\ncorrespond to the second term on the right-hand side of\nEq.(43). We will seebelowthat, in fact, the entiretensor\nδTSTTdoes vanish.\nB. Relation between TSTTand vertex corrections\nto the torque operator ˆT\nAs was argued in Ref. 52 on the basis of perturbative\nexpansions, the velocity operator v=p/m−αR[ez×σ],7\nFIG. 3. Another diagrammatic representation of the STT tens orTSTT\nαβγδ, given by Eq. (43). Six diagrams encircled by the\ndashed line define the δTSTT\nαβγδtensor of Eq. (44) that vanishes for any direction of nprovided ε >∆sd. Solid lines correspond\nto the disorder-averaged Green’s functions gR,A. Vertex corrections (impurity ladders) are represented by green fillings.\ncorrectedbyanimpurityladder,hasaparticularlysimple\nform in the present model,\nvvc=p/m. (45)\nA formal proof of this statement that does not refer to\nany perturbative expansion is presented in Appendix A.\nInterestingly, Eq. (45) also allows to make a spin-orbit\ntorque (SOT) calculation extremely concise. We provide\na brief discussion of this matter in the same Appendix A.\nIt is important that the momentum operator p, as well\nasvvc, commutes with the Green’s function gR,A. In Ap-\npendix B, we demonstrate that this is sufficient for the\nentire tensor δTSTTto vanish. As a result, TSTTis de-\ntermined by the second term on the right hand side of\nEq. (43) alone. Computation of the this term is facili-\ntated by the relation\n∂vvc\nβ/∂pγ=δβγ/m, (46)\nwhereδq1q2is Kronecker delta. With the help of the\nabove, the STT tensor of Eq. (43) readily simplifies to\nTSTT\nαβγδ=δβγe∆2\nsdA\n2π/planckover2pi1Sm/integraldisplayd2p\n(2π)2\n×i\n2tr/bracketleftig\n−gAσδgAˆTvc\nαgR−h.c./bracketrightig\n,(47)\nsince, as we have mentioned, δTSTT= 0.\nEmploying the Hilbert’s identity for the Green’s func-\ntions of Eq. (35),\ngA−gR=gR(i/τ)gA, (48)\nwe can further reduce44Eq. (47) to the formula\nTSTT\nαβγδ=δβγe∆2\nsdAτ\n2π/planckover2pi1Sm/integraldisplayd2p\n(2π)2tr/bracketleftig\ngAˆTvc\nαgRσδ/bracketrightig\n,(49)which resembles very closely the formula of Eq. (37) for\nthe GD tensor. The result of Eq. (49) can also be ex-\npressed in terms of the matrix Tas\nTSTT\nαβγδ=δβγe∆2\nsdAτ2\nπ/planckover2pi1STαδ, (50)\nwhere we have again used the argumentationof Sec. IID.\nC. Relation between TSTTandTGD\nIt can now be seen that both TSTTandTGDvectors\nturn out to be fully defined by the matrix of vertex cor-\nrectionsTto the torque operator. Moreover, comparison\nof Eq. (42) and Eq. (50) reveals a remarkable direct con-\nnection between the STT and GD tensors,\nTSTT\nαβγδ=δβγe/planckover2pi1τ\nmTGD\nαδ, (51)\nwhich is one of the central results of the paper.\nAccordingtothe definitionsofEqs.(8), the established\nrelation between the two tensors indicates that all quan-\ntities of interest (STT, GD, and ESR) may be related to\nthe action of a single linear operator Ξ,\nTSTT= Ξ[∂vn],TGD= Ξ[∂tn],(52)\non one of the vectors, ∂vnor∂tn. We remind here the\nshort-handed notations for the directional spatial deriva-\ntive53∂v= (vd·∇) and for the classical drift velocity of\nconduction electrons vd=eE/planckover2pi1τ/m.\nThe matrix of the operator Ξ coincides with the ma-\ntrixTGD, being also proportional to the matrix T[see\nEqs. (8b), and (42)]. In the next section we obtain the\ngeneral form of the latter and then use it to derive the\nexact vector forms of TSTTandTGD.8\nIV. VECTOR FORMS\nA. Matrix gauge transformation\nIn order to establish the structure of the operator Ξ,\nit should be first noted that the constraint n2≡1 is\nresponsibleforanessentialfreedominthedefinition of T.\nFor an arbitrary operator of differentiation ∂, we have\n1\n2∂n2=/summationdisplay\nδnδ∂nδ= 0. (53)\nTherefore, the left hand sides of\nTSTT\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδTαδ∂vnδ,(54a)\nTGD\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδTαδ∂tnδ,(54b)\nremain invariant under the addition of the matrix row\nR= (nx,ny,nz), with an arbitrary coefficient, to any of\ntherowsofthematrix T. Inotherwords,thetransforma-\ntionT → T Xdoes not change TSTTandTGD, provided\nTX=T+XR, (55)\nwith any matrix column X= (X1,X2,X3)T.\nB. Vector structure of TSTTandTGD\nThe matrix Tis defined in Eq. (41) with the help of\nthe matrix M. The latter is determined by the disorder-\naveraged Green’s function which, in momentum repre-\nsentation, takes the form\ngR,A=ε±i/2τ−p2/2m+αR[p×σ]z+ς∆sdn·σ\n(ε−ε+(p)±i/2τ)(ε−ε−(p)±i/2τ).\n(56)\nUsing Eq. (56), one can prove that M, in general, is\nexpressed as a linear combination of six matrices,\nI, P, U, U2, P UP, P U2P, (57)\nwhereUis introducedin Eq.(39) and P= diag(1 ,1,0)is\na diagonal matrix. In Appendix C, we demonstrate how\nthe components of this decomposition can be calculated\nforn∝ne}ationslash=n⊥.\nThen, in Appendix D, we show that any power of M\nretains the same structure. It immediately follows that\nthe matrix T=U(M+M2+···) can be represented as\nT=c1U+c2UP+c3U2+c4U3+c5UP UP+c6UP U2P,\n(58)\nwhereciare some dimensionless scalar functions.\nThe representation of Eq. (58) can be substantially\nsimplified with the use of the matrix gauge transforma-\ntion described in the previous section. Namely, by takingadvantage of the directly verifiable relations\nU2=RTR−I, U3=−U, (59a)\nUP UP= (I−P)RTR−n2\nzI, (59b)\nUP U2P=UPRTR−UP+n2\nzU(I−P) (59c)\nwe find that the choice of the gauge\n/tildewideX=−[c3I+c5(I−P)+c6UP]RT,(60)\nfor the transformation T → T /tildewideX≡/tildewideT, leads to\n/tildewideT=t0I+t/bardblUP+t⊥U(I−P),(61)\nor, more explicitly, to\n/tildewideT=\nt0nzt/bardbl−nyt⊥\n−nzt/bardblt0nxt⊥\nnyt/bardbl−nxt/bardblt0\n, (62)\nwhere the quantities tiare related to the matrix Tby\nmeans of the relations\nt0=−c3−c5n2\nz, (63a)\nt/bardbl=c1+c2−(c4+c6), (63b)\nt⊥=c1−c4+c6n2\nz. (63c)\nReplacing Twith/tildewideTin Eqs. (54),\nTSTT\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδ/tildewideTαδ∂vnδ,(64a)\nTGD\nα=∆2\nsdAmτ\nπ/planckover2pi12S/summationdisplay\nδ/tildewideTαδ∂tnδ,(64b)\nwe observe that the operator Ξ in Eq. (52) is represented\nby three dimensionless quantities ξ0,ξ/bardbl,ξ⊥, such that\nξi=∆2\nsdAmτ\nπ/planckover2pi12Sti, (65)\nwhile the vector structure of TSTTandTGDis, indeed,\nprovided by the formulas\nTSTT=ξ0∂vn−ξ/bardbl[n×∂vn/bardbl]−ξ⊥[n×∂vn⊥],\nTGD=ξ0∂tn−ξ/bardbl[n×∂tn/bardbl]−ξ⊥[n×∂tn⊥],\nannouncedin the introductorypart. With someremarks,\nthey remain valid for n=n⊥as well. We consider this\nspecific case separately, in Sec. VB.\nIn the next section, we derive closed-form results for\nξ0,ξ/bardbl, andξ⊥, in two particular regimes. Afterwards, we\nfind asymptotic expansions of these functions in either\nsmallαRor in small ∆ sd. All the obtained results are\ncollected in Table I and represented in Fig. 4 alongside\nwith the corresponding numerical curves.9\nV. CLOSED-FORMS\nThe analysis of TSTTandTGDtensors, as has been\npointed out, reduces to integration in Eq. (38) and sub-\nsequent matrix arithmetics. Unfortunately, for arbitrary\ndirection of magnetization, the results cannot be ex-\npressed in terms of elementary functions. For example,\nforn⊥= 0, Eq. (38) already involves elliptic integrals.\nThe complexity is caused, primarily, by the angle de-\npendence of the dispersion relation roots p±,p±,negof\nEqs. (22). Additional complications arise due to the fact\nthat all four roots are distinct.\nOn the other hand, if the parameter rdefined in\nEq. (24b) vanishes, then the angle dependence of p±,\np±,negis absent and, furthermore, p±=−p±,neg(see\nalso Sec. IE). In this case, angle integration in Eq. (38)\nis trivial, while integration over the absolute value pof\nmomentum can be replaced with an integration over p2.\nFor such integrals, we can extend the integration contour\nto−∞and close it through the upper half-plane. Then\nthe value of the integral is given by a sum of residues at\nthep2\n±poles of Eqs. (25) that acquire finite imaginary\nparts due to a ε→ε+i/2τshift.\nHence, computation of the matrix Mis straightfor-\nward when αR= 0, ∆ sd= 0, orn=n⊥. In this section,\nwe calculate ξ0,ξ/bardbl, andξ⊥, for the first and third cases.\nIn the next section, we use the first two cases as reference\npoints for perturbative analysis of these functions.\nA. Vanishing spin-orbit coupling\nWewillstudythecaseof αR= 0first. Inthe absenceof\nSOC, conservation of spin brings a technical difficulty to\nthe calculation of T. Namely, at zero frequency and zero\nmomentum, the matrix of disorder-averaged advanced-\nretarded spin-spin correlators M(I− M)−1that enters\nEq. (41) cannot be finite. Indeed, using the formulas of\nAppendix C with αR= 0, one finds\nM=I−2ςτ∆sd\n1+(2τ∆sd)2U(I−2ςτ∆sdU),(66)\nso thatI− Mis proportional to U. But det U= 0\nand, therefore, M(I− M)−1=∞. Physically, this di-\nvergence is caused by the absence of linear response of\nelectron spins polarized along nto time-dependent ho-\nmogeneous perturbations of Jsd(cf. Sec. 8.3 in Ref. 54).\nNevertheless, even in the limit of zero momentum and\nzero frequency, STT, GD, and ESR remain finite, since\nthe series\nT=UM+UM2+UM3+... (67)\nactually converges.\nThe sum in Eq. (67) is most easily calculated in the\ndiagonal representation of U,\nU=VUdiagV†, U diag= diag(i,−i,0),(68)which is defined by the unitary matrix\nV=\niny−nxnz√\n2(n2x+n2y)−iny+nxnz√\n2(n2x+n2y)nx\n−inx+nynz√\n2(n2x+n2y)inx−nynz√\n2(n2x+n2y)ny√\nn2x+n2y√\n2√\nn2x+n2y√\n2nz\n.(69)\nIntroducing MU=V†MVand making use of the rela-\ntionUdiag=UdiagP, to take care of the potential diver-\ngence, we can rewrite Eq. (67) as\nT=VUdiag(PMU+PM2\nU+PM3\nU+...)V†,(70)\nwhere, according to Eqs. (66) and (68),\nPMk\nU= diag/parenleftig\n[1+2iςτ∆sd]−k,[1−2iςτ∆sd]−k,0/parenrightig\n.\n(71)\nSummation in Eq. (70) is trivially performed, leading to\nT=−ς\n2τ∆sdVU2\ndiagV†=−ς\n2τ∆sdU2=\nς\n2τ∆sd/parenleftbig\nI−RTR/parenrightbig\n=/tildewideT −/tildewideXR,(72)\nwhere/tildewideT= (ς/2τ∆sd)Irepresents the gauge of Eq. (61)\nand we have used the first identity of Eq. (59a).\nThe above result clearly corresponds to t0=ς/2τ∆sd\nandt/bardbl=t⊥= 0, or\nξ0=ς∆sdAm\n2π/planckover2pi12S, ξ /bardbl=ξ⊥= 0. (73)\nHence, Gilbert damping and the nonadiabatic spin-\ntransfertorqueareboth absentwhen αR= 0, asit should\nbe in the model with no SOC, spin-dependent disorder,\nor other sources of spin relaxation.\nThe parameter ξ0defines the effective spin renormal-\nization(duetoconductionelectrons)intheLLGequation\nas16ξ0=−δSeff/S. In fact, for αR= 0, the effective spin\nrenormalization coincides with actual spin renormaliza-\ntion. Indeed, without SOC, all electrons are polarized\nalong±n, and, for the calculation of the total electron\nspin in a unit cell,\nδS=δS↑−δS↓=ς\n2(N+−N−) =\nςA\n8π2/planckover2pi12\n/integraldisplay\nε+(p)≤εpdpdφ p−/integraldisplay\nε−(p)≤εpdpdφ p\n,(74)\none may use ε±(p)≤ε⇔p2≤2m(ε∓∆sd) to obtain\nδS=−ς∆sdAm\n2π/planckover2pi12. (75)\nThus,δS=−ξ0S=δSeffin this case.\nIn Appendix E, we compute spin susceptibility of the\nsystem for αR∝ne}ationslash= 0 and demonstrate that the spin renor-\nmalization does not depend on the SOC strength. At the10\nsametime, the effectivespinrenormalizationdoes. More-\nover, the identity δSeff=δSis, in fact, a very specific\ncase. It holds either for vanishing spin-orbit interaction,\noratsomeparticularvalueof∆ so≈∆sd, asonecanlearn\nfrom Table I and Fig. 4 (we recall that ∆ so=|αR|√\n2mε\ncharacterizes the SOC-induced splitting of the spectral\nbranches).\nB. Perpendicular-to-the-plane magnetization\nNow we turn to the n=n⊥regime. The formulas of\nAppendix C arenot applicable in this case. Nevertheless,\none can perform the integration in Eq. (38) directly, uti-\nlizing the expression for the Green’s function of Eq. (56)\nwith sinθ= 0 (and n=ezcosθ). It follows that\nM=/bracketleftbig\n1+4τ2(∆2\nsd+∆2\nso)/bracketrightbig−1/parenleftig/bracketleftbig\n1+2(τ∆so)2/bracketrightbig\nP\n+/bracketleftbig\n1+4(τ∆sd)2/bracketrightbig\n(I−P)−2ςτ∆sdP UP/parenrightig\n(76)\nand, after some arithmetic,\nT=ς\n2τ∆sd/bracketleftigg\n1−/parenleftbig\nτ∆2\nso/parenrightbig2\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\nP\n+1\n2/bracketleftigg\n∆2\nso/bracketleftbig\n1+2τ2(2∆2\nsd+∆2\nso)/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\nP UP.(77)\nSubstitution of this result into Eqs. (54) shows that, in\nthis case, both TSTTandTGDare represented as linear\ncombinations of two vector forms: ∂n/bardblandn⊥×∂n/bardbl.\nSincen=n⊥and, thus, ∂n⊥= 0, the coefficients in\nfront of these forms should be recognized as t0andt/bardbl,\nrespectively. With the help of Eq. (75), we, therefore,\nfind\nξ0=−δS\nS/bracketleftigg\n1−/parenleftbig\nτ∆2\nso/parenrightbig2\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\n,(78a)\nξ/bardbl=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n∆2\nso/bracketleftbig\n1+2τ2(2∆2\nsd+∆2\nso)/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\n.(78b)\nFor a fixed n=n⊥, however, one cannot directly de-\nfineξ⊥. Indeed, the latter function, in this case, is a\nprefactor in front of the vanishing vector form n×∂n⊥\nand, in principle, can be even taken arbitrary. The only\nway to assign a clear meaning to ξ⊥, here, is to consider\nitsasymptoticbehaviouratsmallvaluesofsin θ. Namely,\none should expand the integrands in Eq. (38) up to sin2θ\nand, after the integration, compute the coefficients of the\ndecomposition of Eq. (58) with the same accuracy. Ap-\nplication of a sin θ→0 limit in Eq. (63c), afterwards,\nwill lead to\nξ⊥=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n1\n2∆2\nso/bracketleftbig\n1+(2τ∆sd)2/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2/bracketrightigg\n.(79)One may use Eqs. (78b) and (79) to evaluate the\nstrengthoftherotationalanisotropyofGDandthenona-\ndiabatic STT, given n≈n⊥. We see, for example, that,\nfor small sin θ, the ratio\nξ/bardbl/ξ⊥= 2+∆2\nso\n∆2\nsd+(1/2τ)2+O(sin2θ),(80)\nexceeds 2, making the rotational anisotropy considerable\neven if SOC is weak. At the same time, for strong spin-\norbit coupling, ξ/bardblcan potentially be orders of magnitude\nlarger than ξ⊥(see also Fig. 4).\nFor the perpendicular-to-the-plane magnetization, GD\nwas analyzed previously in Ref. [55] under an additional\nassumption of large chemical potential. Our result for\nthe Gilbert damping coefficient ξ/bardbl, given by Eq. (78b),\ncoincides with the expression on the right hand side of\nEq. (25) of Ref. [55], to an overall factor that we were\nunable to identify (most likely, it is equal to 4). The\nτ→ ∞limit of the same expression was derived recently\nin Ref. [56] (with another overall factor). This paper also\nmentions the role of the diagonal terms of the GD tensor\non ESR.\nA separate study of the nonadiabatic STT (also lim-\nited to the n=n⊥case) was reported in Ref. [57]. As\nwe have shown above, this torque should be fully de-\ntermined by the very same function ξ/bardblas is GD. The\nauthors, however, ignored vertex corrections, and, as it\nseems, overlooked this fact. In any case, their results\ndiffer from those of Eq. (78b).\nVI. ASYMPTOTIC EXPANSIONS\nWe proceed with a calculation of the ξiexpansions\nin either small αRor small ∆ sd. To perform such cal-\nculation, one should expand the integrands in Eq. (38)\nor, alternatively, in Eqs. (C2), with respect to the corre-\nspondingvariable. Thentheresultcanbeintegratedover\nthe poles, provided by Eqs. (25a) and (25b), respectively\n(whereεshould be replaced with ε+i/2τ).\nA. Weak spin-orbit coupling\nKeeping the notation of Sec. VA for the matrices M\nandTin the absence of SOC, below we use the symbols\nδMandδTtorepresenttherespectivecontributionspro-\nvided by finite αR.\nSinceδM ∝ne}ationslash= 0, the result of matrix inversion in\nT+δT=U(M+δM)(I−M−δM)−1(81)\nis finite, making the analysis straightforward yet rather\ncumbersome. Retaining only proportionalto α2\nRterms in\nδM(see Appendix F for explicit formulas), we obtain\nδT=δc2P+δc3U+δc4U2+..., (82)11\nξ0/(−δS\nS) orδSeff/δS ξ/bardbl/(|δS\nS|τ∆sd) ξ⊥/(|δS\nS|τ∆sd)\nαR= 0 1 0 0\nO(∆2\nso)1+2(τ∆so)2\n1+(2τ∆sd)21−n2\nz\n1+n2z(∆so/∆sd)2\n1+(2τ∆sd)2/bracketleftbigg\n(2τ∆sd)2+2\n1+n2z/bracketrightbigg(∆so/∆sd)2\n1+(2τ∆sd)21+(2nzτ∆sd)2\n1+n2z\n∆sd→0/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftbigg\n4n2\nz+1+n2\nz\n2(τ∆so)2/bracketrightbigg\n2+1\n(τ∆so)21\n2(τ∆so)2\nn=n⊥1−(τ∆2\nso)2\n∆2\nsd+τ2(2∆2\nsd+∆2so)2∆2\nso/bracketleftbig\n1+2τ2(2∆2\nsd+∆2\nso)/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)21\n2∆2\nso/bracketleftbig\n1+(2τ∆sd)2/bracketrightbig\n∆2\nsd+τ2(2∆2\nsd+∆2so)2\nTABLE I. Closed-form results and asymptotic expansions for the dimensionless functions ξ0,ξ/bardbl, andξ⊥that define anisotropic\nspin-transfer torques, Gilbert damping, and effective spin renormalization. The results are expressed in terms of the e nergy\nscales ∆ sd=|Jsd|Sand ∆ so=|αR|√\n2εmthat describe, respectively, the exchange and spin-orbit- induced splitting. The second\nrow shows the expansion up to the second order in ∆ so. The third row provides the leading order terms of the expans ion with\nrespect to small ∆ sd. Spin renormalization is defined in Eq. (75) by δS=−JsdSAm/2π/planckover2pi12.\nwhere dots represent terms that do not contribute to the\nδ/tildewideTgauge in the α2\nRorder and\nδc2=∆2\nso\n2∆2\nsd1\n1+n2z, (83a)\nδc3=−τ∆2\nso\nς∆sd/bracketleftig\n1+(2τ∆sd)2/bracketrightig1−n2\nz\n1+n2z,(83b)\nδc4=−∆2\nso\n2∆2\nsd1+(2nzτ∆sd)2\n1+(2τ∆sd)21\n1+n2z.(83c)\nThen, utilizingEqs.(63)with cireplacedby δci,wearrive\nat the second-orderexpansionsin small SOC strength for\nthe functions ξi. Those are collected in the second row\nof Table I.\nWe may again use the obtained results to quantify the\nrotational anisotropy of GD and the nonadiabatic STT\nby computing the ratio\nξ/bardbl/ξ⊥= 2+1−n2\nz\nn2z+1/(2τ∆sd)2+O(∆2\nso).(84)\nFor weak spin-orbit coupling, the rotational anisotropy\nis minimal when magnetization is perpendicular to the\nplane and increases for the magnetization approaching\nthe in-plane direction.\nWe also note that the asymptotic expansions up to the\norderα2\nRallowustoestimatetheorientationalanisotropy\nofξi. Employing the notation ξi=ξi(n2\nz), we find\nξ0(0)−ξ0(1) =2(τ∆so)2\n1+(2τ∆sd)2, (85a)\nξ/bardbl(0)−ξ/bardbl(1) =1\n1+(2τ∆sd)2∆2\nso\n∆2\nsd, (85b)\nξ⊥(0)−ξ⊥(1) =1−(2τ∆sd)2\n1+(2τ∆sd)2∆2\nso\n2∆2\nsd,(85c)\nfor weak SOC. Clearly, ξ0andξ/bardblare both maximal for\nn⊥= 0. On the other hand, the expression on the righthand side of Eq. (85c) can change sign, depending on the\nvalue of τ∆sd. Therefore, the orientational anisotopy of\nξ⊥in a “clean” system ( τ∆sd≫1) differs from that in a\n“dirty” one (Fig. 4 corresponds to the case of a “clean”\nsystem).\nInterestingly, at αR= 0 the matrix function δTturns\nout to be discontinuous. Namely, its elements have fi-\nnite limits for αR→0. This discontinuity has, however,\nno physical consequences, since the matrix δTitself is\nnot gauge invariant. In the δ/tildewideTgauge, the discontinuity\nis removed and, thus, it does not affect the physically\nrelevant quantities ξ0,ξ/bardbl, andξ⊥. This property demon-\nstrates the importance of full analysis of all components\nof the STT and GD tensors.\nB. Weak exchange interaction\nUp to the linear order in ∆ sd, we have\nM=I+2(τ∆so)2P\n1+4(τ∆so)2−2ςτ∆sdU+4(τ∆so)2P UP\n[1+4(τ∆so)2]2.\n(86)\nThis corresponds to the following coefficients of the de-\ncomposition of Eq. (58),\nc1=1\n4(τ∆so)2, c2= 1+1\n4(τ∆so)2, (87a)\nc3=−ςτ∆sd\n4(τ∆so)4, c4= 0, (87b)\nc5=−ςτ∆sd/bracketleftbig\n1+8(τ∆so)2/bracketrightbig\n4(τ∆so)4, c6= 0.(87c)\nSubstituting the latter expressions into Eqs. (63), one\nobtains the leading-order contributions to ξiin the limit\nof small ∆ sd. The respective results are presented in the\nthird row of Table I. Using them, we can find yet another\nexpression for the ratio\nξ/bardbl/ξ⊥= 2+(2τ∆so)2+O(∆2\nsd).(88)12\nFIG. 4. Dimensionless functions ξ0,ξ/bardbl, andξ⊥that define anisotropic spin-transfer torques, Gilbert dam ping, and effective spin\nrenormalization as functions of the spin-orbit coupling st rengthαRfor four different polar angles of magnetization ( nz= cosθ).\nThe notations coincide with those of Table I. We use the dimen sionless combinations ετ= 50,τ∆sd= 10. Since for θ= 0 it is\nimpossible to compute ξ⊥numerically, only analytical result is shown. The O/parenleftbig\n1/∆4\nso/parenrightbig\nexpansion is addressed in Appendix G.\nRemarkably, the rotational anisotropy of GD and the\nnonadiabatic STT, ξ/bardbl/ξ⊥= 2, persists to both limits\n∆sd≪∆so≪1/τand ∆ so≪∆sd≪1/τ,(89)\nin which the Fermi surfaces defined in Eq. (19) are not\nonly essentially isotropic but, at the same time, do get\nstrongly broadened by the disorder (the broadening 1 /τexceeds the splitting of the subbands).\nIt is also interesting to mention that, for small values\nof ∆sd, the nonadiabatic spin-transfer torque dominates\nover the adiabatic one: ξ/bardbl,⊥/ξ0∝1/∆sd. This agrees\nwiththeintuitivelogicthat, foraweakexchangebetween\nconduction and localized spins, the former would rather\nnot adiabatically follow the direction of the latter.13\nVII. DISCUSSION\nA. Role of vertex corrections\nWe would like to begin this final section by stressing\nthat it is the accurate consideration of vertex corrections\nthat is responsible for the established vector structures\nof anisotropic STT, GD, and ESR, as well as for the\nrelation between them. Practically none of this would be\nseen from an uncontrolled analysis that ignores vertex\ncorrections.\nFor example, if one does not apply the disorder dress-\ning to the current vertex v, the relation of Eq. (50) will\nno longer be valid. Instead, the STT tensor, in this case,\nwill contain 18 additional nonzero components of differ-\nent symmetries, which one might by mistake interpret as\nphysical torques.\nB. Renormalization of spin\nIn Sec. VA, we have demonstrated that, in the limit\nof vanishing SOC, the ESR factor δSeff=−ξ0Sdoes\ncoincide with the actual total electron spin in a unit cell\nδS=−JsdSAm/2π/planckover2pi12. On the other hand, this equality\nbreaksdown forfinite αR, and the ratio δSeff/δSstarts to\ndepend on all of the parameters of the system, including\nscattering time (see Table I and Fig. 4).\nForlargevaluesofspin-orbit-inducedsplitting∆ so, the\nquantity ξ0(which determines ESR) understandably de-\ncays due to the effective randomization of the electron\nspin direction induced by SOC. What is, however, rather\ninteresting, is that, for relatively small values of αR, the\nESRfactor δSeffexceedsδS,reachingthemaximumvalue\nat ∆so≈∆sd. We do not have an intuitive explanation\nfor such behaviour.\nC. LLG equation\nIt is instructive to compare the microscopic LLG\nEq.(1)toitsconventionalphenomenologicalcounterpart.\nIn the absence of spin-orbit, thermal, and other torques\nthat we do not consider in this study, the latter equation\nreads\n∂tn=γn×Heff+(js·∇)n\n−α[n×∂tn]−β[n×(js·∇)n],(90)\nwhere the vector quantity jsis interpreted as the phe-\nnomenological spin-polarized current, while the param-\netersαandβdefine Gilbert damping and the nonadi-\nabatic spin-transfer torque, respectively. The latter is\nalsocommonlyreferredtoasthe β-torque. Theadiabatic\nspin-transfertorqueisrepresentedbytheterm( js·∇)n,\nwhileHeffstands for effective field contributions.\nFirst, taking into account Eqs. (2), we can rewrite the\nmicroscopic LLG Eq. (1) in a form which is similar tothat of Eq. (90),\n∂tn= ¯γn×Heff+(js·∇)n\n−α/bardbl[n×∂tn/bardbl]−β/bardbl[n×(js·∇)n/bardbl]\n−α⊥[n×∂tn⊥]−β⊥[n×(js·∇)n⊥],(91)\nwhere\njs=vdξ0\n1−ξ0=−vdδSeff\nS+δSeff,(92a)\nα/bardbl,⊥=ξ/bardbl,⊥\n1−ξ0, β/bardbl,⊥=ξ/bardbl,⊥\nξ0,¯γ=γ\n1−ξ0(92b)\nand each of the quantities js,α/bardbl,⊥,β/bardbl,⊥, ¯γdepend on\nthe orientation of the vector n. For the particular 2D\nRashba FM model system considered in this paper,\njs=js(n2\nz), α /bardbl,⊥=α/bardbl,⊥(n2\nz),(93a)\nβ/bardbl,⊥=β/bardbl,⊥(n2\nz),¯γ= ¯γ(n2\nz). (93b)\nWe see that the microscopic LLG Eq. (91) is essentially\nanisotropic, in contrast with the phenomenological LLG\nEq. (90). Namely, the coefficients αandβgot split into\ntwo components each. Moreover, the new coefficients\nα/bardbl,⊥andβ/bardbl,⊥as well as the other parametersof the LLG\nequation became dependent on the direction of magneti-\nzation. We note that the splitting of the GD coefficient α\nhas been reported, for a Rashba FM, in Ref. [58].\nNext, let us comment on the microscopic definiton of\nthe spin-polarized current formulated in Eq. (92a). Nor-\nmally, ifspins ofconductionelectrons(travellingwith the\ncharacteristic velocity v) adiabatically follow the direc-\ntion ofn, one assumes js=−vδS/(S+δS), where δS\nis a contribution from conduction electrons to the total\nspin of the system. In this case, Eq. (90) can be simply\nviewedasamanifestationofthetotalangularmomentum\nconservation (for n×Heff= 0),\n(S+δS)∂tn+δS(v·∇)n= 0. (94)\nwhere−δS(v·∇)nis the rate of angular momentum\ntransfer from conduction to total spin.\nThe definition of the vector quantity js, given by\nEq. (92a), provides a perfect generalization of the above\nlogic for a system with finite Rashba SOC. Indeed, con-\nduction spins no longer follow the direction of n(due to,\ne.g., nonzero damping). Nevertheless, −δSeff(vd·∇)n\nstill has a meaning of the rate of “angular momentum\ntransfer” from the effective conduction spin δSeffto the\ntotalS+δSeff. Importantly, it was a fully controllable\naccurate microscopic treatment of the problem that led\nus to Eq. (92a). (We identified the drift velocity vd\nas a “proportionality coefficient” between the STT and\nGD tensors and observedthat the adiabatic spin-transfer\ntorque and ESR are described by the same quantity ξ0.)\nFinally, for the sake of historical integrity, let us also\nmention that the equalities α/bardbl=β/bardblandα⊥=β⊥, in\nthis system, are equivalent59to the relation\nδSeff=−S/2, (95)\nwhich appears to be rather unphysical.14\nD. Material derivative and moving reference frame\nIn the presence of the anisotropic STT and GD of\nEqs. (2), it is natural to analyse the microscopic LLG\nEq. (1) in such a frame, where the effect of the nonadia-\nbaticspin-transfertorqueisabsent. Namely, inthe frame\nthat moves with the classical drift velocity of conduction\nelectrons vd. One may use a nice analogy to continuum\nmechanics as an illustration of this fact.\nIndeed, despite the essentially anisotropic character of\nbothTSTTandTGD, their sum is conveniently expressed\nin the LLG Eq.(1) viathe operatorofmaterialderivative\nDt=∂t+(vd·∇) as\n(1−ξ0)Dtn=γn×Heff+(vd·∇)n−ξ/bardbl/bracketleftbig\nn×Dtn/bardbl/bracketrightbig\n−ξ⊥[n×Dtn⊥]+...,(96)\nwhere we have moved the term ξ0Dtnto the left hand\nside and added ( vd·∇)nto both sides. By considering\nconduction electrons as a “fluid” flowing with the drift\nvelocity vd, one may interpret the material derivatives\nof Eq. (96) as the change rates of components of nthat\nare associated with the electronic “fluid parcels”. Thus,\nin the moving (“flowing”) frame, r′=r−vdt, the ma-\nterial derivatives Dtare automatically replaced31by the\nordinary time derivatives ∂t.\nIn other words, in the movingreferenceframe, Eq. (96)\ntakes the form of the LLG equation\n(1−ξ0)∂tn=γn×Heff+(vd·∇)n−ξ/bardbl/bracketleftbig\nn×∂tn/bardbl/bracketrightbig\n−ξ⊥[n×∂tn⊥]+...(97)\nthat comprises the analogue of the adiabatic torque\n(vd·∇)n, two components of damping, and (repre-\nsentedherebydots)allotherpossibletorques. As longas\nthe latterareabsent, the dynamicsofamagnetictexture,\ngoverned by such equation (under mediate currents and\nin the absence of magnetic field), is likely to be a motion\nwith zero terminal velocity (as it is32,33, in the isotropic\ncase, for domain walls). For a general situation, current-\ninduced magnetic dynamics can differ significantly. Nev-\nertheless, it should still be more convenient to perform\nthe analysis once the effect of the nonadiabatic STT has\nbeen accounted for by switching to the “flowing” frame.\nInterestingly, any “propagating” texture of the form\nn(r,t)=ζ(r−vdt)=ζr(t)nullifiesthesum TSTT+TGD.\nHence, for such textures, the LLG Eq. (1) reads\ndζr/dt=γζr×Heff+..., (98)\nwherercan be regardedas aparameter. Ifone takesinto\naccountonlyspin-transfertorquesandfieldlike spin-orbit\ntorque, solutions of this equation will have an oscillatory\ncharacter. Note that Eq. (98) is different from the LLG\nequation\n0 =γζr×Heff (99)\nthat describes the uniform motion of the ground state in\nthe presence of the Galilean invariance [the case α=β\nin Eq. (90)]13,15,16,60.E. Response to electric current\nSo far, we have computed spin-transfer torques as a\nlinear response of the system to the external electric field\nE. In experiment, however, it is not the electric field\nbut rather the electric current jwhich is externally ap-\nplied. To relate spin torques to the latter, one should\ncompute the conductivity tensor ˆ σand, afterwards, use\nthe identity\nE= ˆσ−1j (100)\nto replace Ewithj. Importantly, the conductivity ten-\nsor has to be computed up to the linear order in first\nmagnetization gradients ∇αnβ.\nF. Relation to Edelstein effect\nIt is worth noting that some of our results can be inde-\npendently benchmarked. As it was suggested in Ref. 36,\nthereexistsaconnectionbetweensomeparticularpairsof\nquantities in the model of Eq. (9), as, e.g., between the\nDzyaloshinskii-Moriya interaction strength and the ex-\nchange stiffness, or between spin-orbit torques and spin-\ntransfer torques. The latter relation is relevant to our\nstudy.\nA general interpretation of the approach described in\nRef. 36 would be the following. Suppose there exists\na quantity F(αR) which, for the model with αR= 0,\ndepends on the gradients of n, such that\nF(0) =F(∇xn,∇yn). (101)\nThen, up to the linear order with respect to αR, one\nwould obtain61\nF(αR) =F(0)+αR/bracketleftbigg∂\n∂αRF(/tildewide∇xn,/tildewide∇yn)/bracketrightbigg\nαR=0,(102)\nwhere\n/tildewide∇in=∇in+2mαR\n/planckover2pi1[n×[ez×ei]].(103)\nLet us now choose three functions Fi(αR) to be the\ncomponents of the vector TSTT. Using the expression\nfor the quantity ξ0in the limit αR= 0 (see Table I), we\ncan write\nTSTT=eA\n2π/planckover2pi1Jsdτ(E·∇)n. (104)\nFrom Eq. (102) we, then, find another contribution to\nthe generalized torque in the ∝αRorder\nTSOT=2mαR\n/planckover2pi1eA\n2π/planckover2pi1Jsdτ[n×[ez×E]],(105)\nwhichis preciselythe expressionfor the Edelsteineffect62\ninaformofafieldliketorqueonmagnetization. Inasimi-\nlarway,vanishingofthefunctions ξ/bardblandξ⊥whenαR= 015\ncan be translated into the absence52of the antidamping\nSOT in the model of Eq. (9).\nTheresultofEq.(105)coincideswith thedirectderiva-\ntion of SOT, for the model of Eq. (9), that has been re-\nported previously52. A more compact and accurate form\nof this derivation is also presented in Appendix A. Such\nindependent consistency check adds to the credibility of\nour results.\nCONCLUSIONS\nWe have presented a thorough microscopic analysis of\nSTT, GD, and ESR, for the particular 2D FM system\nwith Rashba spin-orbit coupling and spin-independent\nGaussian white-noise disorder. Assuming arbitrary di-\nrection of magnetization, we have established the ex-\nact relation between these effects. We have intro-\nduced the notion of the matrix gauge transformation\nfor magnetization-dependent phenomena and used it to\nexpress spin-transfer torques, Gilbert damping, and ef-\nfective spin renormalization in terms of meaningful vec-\ntor forms. The latter allowed us to quantify the SOC-\ninduced anisotropy of the former. We have analysed,\nboth analytically and numerically, three dimensionless\nfunctions that fully define anisotropic STT, GD, and\nESR. We have also generalized the concept of spin-\npolarized current, computed spin susceptibility of the\nsystem, and obtained a number of other results.\nIt would be an interesting challenge to observe the\nanisotropy of STT experimentally. It might be possi-\nble to do this by measuring current-induced corrections\nto the magnon spectrum asymmetry that is normally as-\nsociated with the Dzyaloshinskii-Moriya interaction. We\nalso believe that, to some extent, the anisotropy of STT\nand GD might explain the differences in dynamics of do-\nmainwalls(andskyrmions)with differentcharacteristics.\nACKNOWLEDGMENTS\nWe would like to thank Jairo Sinova for pointing out a\nnumber of flaws in the original version of the manuscript.\nWe are also grateful to Artem Abanov, Arne Brataas,\nSergey Brener, Ivan Dmitriev, Rembert Duine, Olena\nGomonay, Andrew Kent, Alessandro Principi, Alireza\nQaiumzadeh, and Yaroslav Tserkovnyak for helpful dis-\ncussions. This research was supported by the JTC-\nFLAGERAProjectGRANSPORTandbytheDutch Sci-\nence Foundation NWO/FOM 13PR3118. M.T. acknowl-\nedges the support from the Russian Science Foundation\nunder Project 17-12-01359.Appendix A: Vertex corrections to velocity\noperator; spin-orbit torque\nIn order to compute vertex corrections to the velocity\noperator v=p/m−αR[ez×σ], we first apply a single\nimpurity line to the scalar part of the latter,\n(p/m)1×dr=1\nmτ/integraldisplayd2p\n(2π)2gR(p/m)gA.(A1)\nDuetothefactthatthemomentumoperator pcommutes\nwith the Green’s functions gR,A, the above relation can\nbe equivalently written as\n(p/m)1×dr=i\nm/integraldisplayd2p\n(2π)2(p/m)/parenleftbig\ngR−gA/parenrightbig\n,(A2)\nwhere we have used the Hilbert’s identity of Eq. (48).\nThe subsequent analysis follows the route of Sec. IIA.\nIntegration over the absolute value of momentum in\nEq. (A2) is performed by computing residues at p=p±.\nSymmetrization of the obtained result, with respect to\nthe transformation43ϕ→π−ϕ, leads to\n(p/m)1×dr=/integraldisplay2π\n0dϕ\n2π/parenleftig\nαR(1+rW4)[ez×σ]\n+(αR+rW5)/braceleftbig\nn/bardbl[σ×n]z−/parenleftbig\nn/bardbl·σ/parenrightbig\n[ez×n]/bracerightbig\ncos2ϕ\n+(W6+W7n·σ)[ez×n]sinϕ/parenrightig\n,(A3)\nwhereWi=Wi/parenleftbig\nr2,u(r2)/parenrightbig\nare some functions of the pa-\nrameterr2andϕ-independent parameters of the model.\nAgain, all terms that contain Wivanish identically after\nintegration over the angle and we conclude that\n(p/m)1×dr=αR[ez×σ]. (A4)\nNext, we observe that the corrected by an impurity\nladder velocity operator vvccan be recast in the form\nvvc=/braceleftbig\np/m−αR[ez×σ]/bracerightbigvc=\np/m+/braceleftbig\n(p/m)1×dr−αR[ez×σ]/bracerightbigvc.(A5)\nAccording to Eq. (A4), expression inside the brackets on\nthe second line vanishes, leading us to the desired result,\nvvc=p/m, (A6)\nwhich coincides with Eq. (45) of the main text. Note\nthat, since the momentum operator commutes with the\nGreen’s functions, Eq. (A6) determines both advanced-\nretardedand retarded-advancedvertexcorrectionsto the\nvelocity operator.\nOne immediate consequence of Eqs. (A4) and (A6) is\na trivial form of spin-orbit torque in the considered inter-\nface Rashba model. Indeed, it was conjectured in Ref. 52\nthat the antidamping SOT, in this model, is identically16\nabsent, while the field-like SOT is entirely isotropic. To\nprove the conjecture, we use the Kubo formula for SOT\nTSOT=eJsdA\n2π/planckover2pi12/integraldisplayd2p\n(2π)2tr/braceleftig\nˆTgR(vvc·E)gA/bracerightig\n.(A7)\nSubstituting vvc=p/mand using Eq. (A1), we immedi-\nately find\nTSOT=eJsdAmτ\n2π/planckover2pi12tr/braceleftig\nˆT/parenleftig\n(p/m)1×dr·E/parenrightig/bracerightig\n,(A8)\nFinally, with the help of Eqs. (12) and (A4), we obtain\nthe expression for spin-orbit torque,\nTSOT=eJsdAmτα R\n2π/planckover2pi12tr/braceleftbig\n[σ×n]([ez×σ]·E)/bracerightbig\n=\neJsdAmτα R\nπ/planckover2pi12[n×[ez×E]],(A9)\nwhich coincides with that of Eq. (105), as expected.\nAppendix B: Vanishing of δTSTT\nWe will now prove that the absence of the spin compo-\nnent inthe vertexcorrectedvelocityoperator vvcnullifies\nthe contribution δTSTTto the STT tensor of Eq. (44).\nUsing cyclic permutations under the matrix trace and\nthe fact that vvc=p/mcommutes with any function of\nmomentum, one can rewrite Eq. (44) as\nδTSTT\nαβγδ=−e∆2\nsdA\n2π/planckover2pi1S/integraldisplayd2p\n(2π)2pβτ\n2mtr[Λ1+Λ2] (B1)\nwith\nΛ1=/parenleftig\nvγgAˆTvc\nαgRσδ−σδgAˆTvc\nαgRvγ/parenrightiggRgA\niτ,(B2a)\nΛ2=/parenleftig\nσδgAvγgAˆTvc\nα−vγgAσδgAˆTvc\nα/parenrightiggRgA\niτ\n−/parenleftig\nˆTvc\nαgRvγgRσδ−ˆTvc\nαgRσδgRvγ/parenrightiggRgA\niτ.(B2b)\nIn Eq. (B2a), we employ the Hilbert’s indentity of\nEq. (48) to replace the factor gRgA/iτwithgR−gAand\nagain use cyclic permutations to obtain\nΛ1=ˆTvc\nαgRσδgRvγgA−ˆTvc\nαgRvγgRσδgA\n−ˆTvc\nαgRσδgAvγgA+ˆTvc\nαgRvγgAσδgA.(B3)\nA similar procedure is performed to simplify the expres-\nsion for Λ 2. We note, however, that terms with only re-\ntarded or only advanced Green’s functions, in Eq. (B2b),\nshould be disregarded44. Hence, gRgA/iτis replaced\nwithgRin the first line of Eq. (B2b) and with −gAin\nthe second line. After moving the torque operator to the\nfirst place in each term,\nΛ2=ˆTvc\nαgRσδgAvγgA−ˆTvc\nαgRvγgAσδgA\n+ˆTvc\nαgRvγgRσδgA−ˆTvc\nαgRσδgRvγgA,(B4)\nwe conclude that Λ 1+Λ2= 0 and, therefore, δTSTT= 0\nas well.Appendix C: Structure of M\nUsing Green’s function of Eq. (56) we compute the\nmatrix trace in Eq. (38) and further symmetrize the in-\ntegrandswith respect to the transformation43ϕ→π−ϕ.\nThis results in the decomposition\nM=γ1I+γ2P+γ3U+γ4U2+γ5P UP+γ6P U2P(C1)\nwhere the coefficients are given in the integral form,\nγ1= 2/bracketleftig/parenleftig\n∆2\nsd+|ε+i/2τ|2/parenrightig\nI0−2(ε+δso)I1+I2/bracketrightig\n,\n(C2a)\nγ2=−4/bracketleftigg\n2δson2\nz\n1−n2zI1+/parenleftbig\n1+n2\nz/parenrightbig\nς∆sd/radicalbig\n1−n2zJ1−1+n2\nz\n1−n2zJ2/bracketrightigg\n,\n(C2b)\nγ3=−2\nτ/bracketleftigg\nς∆sdI0−1/radicalbig\n1−n2zJ1/bracketrightigg\n, (C2c)\nγ4= 4ς∆sd/bracketleftigg\nς∆sdI0−1/radicalbig\n1−n2zJ1/bracketrightigg\n,(C2d)\nγ5=−2\nτ/radicalbig\n1−n2zJ1, (C2e)\nγ6=−4/bracketleftigg\n2δso\n1−n2zI1+ς∆sd/radicalbig\n1−n2zJ1−2\n1−n2zJ2/bracketrightigg\n,(C2f)\nwithδso=mα2\nRand\nIk=/integraldisplayd2p\n(2π)2(2mτ)−1/parenleftbig\np2/2m/parenrightbigk\n|ε−ε+(p)+i/2τ|2|ε−ε−(p)+i/2τ|2,\n(C3a)\nJk=/integraldisplayd2p\n(2π)2(2mτ)−1(αRpsinϕ)k\n|ε−ε+(p)+i/2τ|2|ε−ε−(p)+i/2τ|2.\n(C3b)\nSomeofEqs.(C2)formallybecomeinvalidwhen n=n⊥.\nHowever,structureof MandTintherespectivecasewas\nanalysed directly in Sec. VB.\nAppendix D: Structure of Mk\nWe have already demonstrated that\nM ∈spanL,L={I, P, U, U2, P UP, P U2P},(D1)\nLet us now prove that any natural power of Mbelongs\nto the same linear span,\nMk∈spanL,∀k∈N. (D2)\nTheoperationofmatrixproduct, byitself, isnotclosed\non spanL. Moreover, 14 of 36 elements of L×Ldo not17\nbelong to span L. On the other hand, a combination of\ntwo such elements (matrices P UandUP),\nP U+UP={P,U}=U+P UP, (D3)\nobviously does. Similarly, the remaining 12 “unsuit-\nable” elements of L × Ldo form 6 pairs, such that\nthe corresponding anticommutators (namely, {P,U2},\n{P UP,U},{P U2P,U},{P UP,U2},{P U2P,U2}, and\n{P UP,P U2P}) belong to span L.\nIn general, the following statement holds: operation of\nmatrix anticommutation sends elements of L × Lto a\nlinear span of L,\n{,}:L×L → spanL. (D4)\nTaking into account the fact that anticommutator is a\nbilinear map, we deduce from Eq. (D4):\n{,}: spanL×spanL →spanL.(D5)\nFinally, since for arbitrary kwe have\nMk=1\n2{M,Mk−1}, (D6)\nthe desired result, Mk∈spanL, is proven by induction.\nAppendix E: Spin susceptibility in the presence of\nSOC\nIn this Appendix, the total spin δSof conduction elec-\ntronsin a unit cell ofthe area Ais computed fora general\ncase ofαR∝ne}ationslash= 0. We use the following standard definition:\nδS=A\n2πi/integraldisplay\ndǫf(ǫ)/integraldisplayd2p\n(2π/planckover2pi1)2tr/bracketleftigσ\n2/parenleftbig\nGA−GR/parenrightbig/bracketrightig\n,(E1)\nwherefstands for the Fermi-Dirac distribution,\nf(ǫ) = (1+exp[( ǫ−ε)/T])−1, (E2)\nandGA,Rrefers to the momentum-dependent Green’s\nfunction of Eq. (29). We will first consider the in-plane\ncomponent of δS.\nMatrix trace calculation followed by an integration\noverǫ, in Eq. (E1), gives\nδSx=A/integraldisplayd2p\n(2π/planckover2pi1)2ς∆sdnx−αRpy\nε+(p)−ε−(p)(f+−f−),(E3a)\nδSy=A/integraldisplayd2p\n(2π/planckover2pi1)2ς∆sdny+αRpx\nε+(p)−ε−(p)(f+−f−),(E3b)\nwheref±=f(ε±(p)). It is convenient to introduce the\nquantity δS+=δSx+iδSy. For the latter, we find\nδS+=A\n4αR/integraldisplayd2p\n(2π/planckover2pi1)2(f+−f−)\n×/parenleftbigg\ni∂\n∂px−∂\n∂py/parenrightbigg\n[ε+(p)−ε−(p)],(E4)where we took advantage of the fact that the fractions\nin Eqs. (E3) can be expressed as the derivatives with\nrespect to the components of momentum. In the zero-\ntemperaturelimit, onecanuseGreen’stheoremtoreduce\nthe double integrals in Eq. (E4) to the integrals over the\nclosed curves C±={p|ε±(p) =ε},\nδS+=δS+\n++δS−\n+, (E5a)\nδS±\n+=±A\n4αR/integraldisplay\nC±dpx+idpy\n(2π/planckover2pi1)2[ε+(p)−ε−(p)].(E5b)\nNext, we follow the approach used by K.-W. Kim et al.\nin Ref. 41. Using the variable w=px+ipyand the\nrelationε±(p) =p2/2m±[ε+(p)−ε−(p)]/2, we find\nδS±\n+=A\n16π2/planckover2pi12αR/integraldisplay\nC±dw/parenleftbigg\n2ε−w∗w\nm/parenrightbigg\n,(E6)\nwherew∗w=p2andC±={w|ε±(w,w∗) =ε}are now\nregarded as contours in the complex w-plane. Since the\ncontours are closed, Eq. (E6) is further simplifed to\nδS±\n+=−A\n16π2/planckover2pi12mαR/integraldisplay\nC±dww∗w. (E7)\nIn order to perform integration in Eq. (E7), we solve\nthe equation ε±(w,w∗) =εforw∗and express the result\nas a function of w∈C±,\nw∗=2m\nw2/parenleftig\nw/bracketleftbig\nε+mα2\nR/bracketrightbig\n−imαRς∆sdn+±√\nR/parenrightig\n,(E8)\nwheren+=nx+inyandRis a cubic function of w.\nDifferent signs in front of the square root in Eq. (E8)\ncorrespond to two different functions w∗=w∗\n±(w) of\nw∈C±, respectively. We do not specify which sign\ncorrespondsto which function. Such ambiguity, however,\ndoes not affect the final result for δS+. Indeed, it can\nbe proven41that all three zeroes of Rare of the form\nwk=irkn+with real rk. Then, from the general relation\n[ε−ε+(w,w∗)][ε−ε−(w,w∗)] =−R\n+/parenleftbiggw∗w\n2m−/bracketleftbig\nε+mα2\nR/bracketrightbig\n+imαRς∆sdn+\nw/parenrightbigg2\n,(E9)\nwe learn that\n[ε−ε+(wk,w∗\nk)][ε−ε−(wk,w∗\nk)]≥0 (E10)\nand, thus, ε−(wk,w∗\nk)< ε⇒ε+(wk,w∗\nk)≤ε. Hence,\nall the singularities of w∗\n−that lie inside the contour C−\nare, in fact, located inside or, at most, on the contour\nC+(note that C+is inside C−). Disregarding the case63\nwk∈C±and using Cauchy integral theorem, we can\nshrink64C−in Eq. (E7) to obtain\nδS+=−A\n16π2/planckover2pi12mαR/integraldisplay\nC+dw/parenleftbig\nw∗\n++w∗\n−/parenrightbig\nw,(E11)18\nso that the terms ±√\nR, in Eq. (E8), do not contribute\ntoδS+. The only remaining singularity of the integrand\nis located at the origin and, by the residue theorem,\nδS+=−ς∆sdAm\n2π/planckover2pi12n+orδS/bardbl=−ς∆sdAm\n2π/planckover2pi12n/bardbl,(E12)\nwhich completes the computation of the in-plane compo-\nnent ofδS.\nIn order to calculate δSz, it is useful to introduce the\n“magnetization”vector M=ς∆sdn. Intermsof M,one\ncan straightforwardlyestablish the “thermodynamic” re-\nlationδSi=∂Ω/∂Mi, where Ω has a meaning of the\nelectronic grand potential in a unit cell,\nΩ =−TA\n2πi/integraldisplay\ndǫg(ǫ)/integraldisplayd2p\n(2π/planckover2pi1)2tr/bracketleftbig\nGA−GR/bracketrightbig\n,(E13a)\ng(ǫ) = log(1+exp[( ε−ǫ)/T]). (E13b)\nWe further note that, according to Eq. (E12), δSxand\nδSydo not depend on Mz. Therefore, equating the sec-\nond derivatives, we find\n∂δSz\n∂Mα=∂2Ω\n∂Mα∂Mz=∂δSα\n∂Mz= 0,(E14)\nwhereα=x,y. As a result, δSzdoes not depend on Mx\nandMyand, thus, can be computed for Mx=My= 0\n(or, equivalently, for nx=ny= 0).\nFrom Eq. (E1) we obtain\nδSz=A/integraldisplayd2p\n(2π/planckover2pi1)2ς∆sdnz\nε+(p)−ε−(p)(f+−f−),(E15)which, for nx=ny= 0, can be integrated over the mo-\nmentum angle with the result\nδSz=Aς∆sdnz\n4π/planckover2pi12∞/integraldisplay\n0pdpf+−f−/radicalig\n∆2\nsd+(αRp)2.(E16)\nAtzerotemperature, theintegrationdomaininEq.(E16)\nis reduced to a finite interval p+< p < p −, wherep±are\ngiven by Eq. (25c). After some algebraic practice, we\nfinally arrive at\nδSz=Aς∆sdnz\n4π/planckover2pi12α2\nR/radicalig\n∆2\nsd+(αRp)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglep−\np+=−ς∆sdAm\n2π/planckover2pi12nz.\n(E17)\nCombining the results of Eqs. (E12) and (E17) into a\nsingle vector form\nδS=−ς∆sdAm\n2π/planckover2pi12n, (E18)\nweseethat, onaverage,evenforfinite valuesofspin-orbit\ncoupling strength αR, spins of conduction electrons, in\ntheequilibrium, arealignedwiththe localmagnetization.\nMoreover, the spin susceptibility tensor is fully isotropic\nand is expressed by a single scalar parameter\nδS=−|δS|=−ς∆sdAm\n2π/planckover2pi12, (E19)\nwhich coincides with that given by Eq. (75) of the main\ntext.\nAppendix F: Expansion of Mup toα2\nR\nExpansion of Eqs. (C2) up to α2\nR= ∆2\nso/2εmprovides us with the coefficients\nδγ1=−/bracketleftbigg2τ∆so\n1+(2τ∆sd)2/bracketrightbigg2/bracketleftbig\n1+(2nzτ∆sd)2/bracketrightbig\n, δγ 2= 2/bracketleftbiggτ∆so\n1+(2τ∆sd)2/bracketrightbigg2/bracketleftbig\n1−(1+2n2\nz)(2τ∆sd)2/bracketrightbig\n,(F1a)\nδγ3=/bracketleftbigg4τ∆so\n1+(2τ∆sd)2/bracketrightbigg21+(2nzτ∆sd)2\n1+(2τ∆sd)2ςτ∆sd, δγ 4=−2/bracketleftbigg4τ2∆so∆sd\n1+(2τ∆sd)2/bracketrightbigg21+(2nzτ∆sd)2\n1+(2τ∆sd)2, (F1b)\nδγ5=−2/bracketleftbigg2τ∆so\n1+(2τ∆sd)2/bracketrightbigg2\nςτ∆sd, δγ 6=−/bracketleftbigg4τ2∆so∆sd\n1+(2τ∆sd)2/bracketrightbigg2\n(F1c)\nof the decomposition that we refer to in Sec. VIA: δM=δγ1I+δγ2P+δγ3U+δγ4U2+δγ5P UP+δγ6P U2P.\nAppendix G: O(1/∆4\nso)expansion of ξi(limit of strong SOC)\nThe quantities ξiare shown in the plots of Fig. 4 as functions of the spin-orbit coupling strength αR(while keeping\nbothmandεconstant). Therefore, the right “tails” of the curves can be pro perly fit using the asymptotic expansion\nwith respect to the parameter 1 /∆so. Such expansion can be obtained indirectly, from the expansion in sm all ∆sd.\nBelow, for consistency with the results of Sec. VIB, we list all the co ntributions to ξithat do not exceed the fourth19\norder in 1 /∆so,\nξ0=−δS\nS/bracketleftigg/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftigg\n4n2\nz+1+n2\nz\n2(τ∆so)2/bracketrightigg\n+6/parenleftbigg∆sd\n∆so/parenrightbigg4/bracketleftbig\n1−3n2\nz/bracketrightbig\nn2\nz/bracketrightigg\n, (G1a)\nξ/bardbl=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n2+1\n(τ∆so)2−/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftigg\n4n2\nz−1−7n2\nz\n(τ∆so)2/bracketrightigg\n−4/parenleftbigg∆sd\n∆so/parenrightbigg4/bracketleftbig\n1−3n2\nz/bracketrightbig\nn2\nz/bracketrightigg\n, (G1b)\nξ⊥=/vextendsingle/vextendsingle/vextendsingleδS\nS/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg\n1\n2(τ∆so)2+/parenleftbigg∆sd\n∆so/parenrightbigg2/bracketleftigg\n2n2\nz+1−5n2\nz\n2(τ∆so)2/bracketrightigg\n+2/parenleftbigg∆sd\n∆so/parenrightbigg4/bracketleftbig\n1−5n2\nz/bracketrightbig\nn2\nz/bracketrightigg\n. 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Lett. 95, 107204 (2005).\n61According to the definition of Eq. (9), the spin-orbit cou-\npling term has an opposite sign as compared to that used\nin Ref. 36.\n62V. M. Edelstein, Solid State Commun. 73, 233 (1990).\n63The conditions wk∈C±can only be fulfilled for some\nparticular values of ε. SinceδSis a continuous function\nofε, one may just ignore such values.\n64See Ref. 41 for important details on branch cuts."    },    {        "title": "1907.02734v1.Theory_for_shift_current_of_bosons__Photogalvanic_spin_current_in_ferrimagnetic_and_antiferromagnetic_insulators.pdf",        "content": "Theory for shift current of bosons: Photogalvanic spin current\nin ferrimagnetic and antiferromagnetic insulators\nHiroaki Ishizuka1and Masahiro Sato2\n1Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo, 113-8656, JAPAN\n2Department of Physics, Ibaraki University, Mito, Ibaraki, 310-8512, JAPAN\n(Dated: July 8, 2019)\nWe theoretically study the optical generation of dc spin current (i.e., a spin-current solar cell) in\nordered antiferromagnetic and ferrimagnetic insulators, motivated by a recent study on the laser-\ndriven spinon spin current in noncentrosymmetric quantum spin chains [H. Ishizuka and M. Sato,\nPhys. Rev. Lett. 122, 197702 (2019)]. Using a non-linear response theory for magnons, we analyze\nthe dc spin current generated by a linearly-polarized electromagnetic wave (typically, terahertz or\ngigahertz waves). Considering noncentrosymmetric two-sublattice magnets as an example, we \fnd\na \fnite dc spin current conductivity at T= 0, where no thermally-excited magnons exist; this is in\ncontrast to the case of the spinon spin current, in which the optical transition of the Fermi degenerate\nspinons plays an essential role. We \fnd that the dc spin-current conductivity is insensitive to the\nGilbert damping, i.e., it may be viewed as a shift current carried by bosonic particles (magnons).\nOur estimate shows that an electric-\feld intensity of E\u0018104\u0000106V/cm is su\u000ecient for an\nobservable spin current. Our theory indicates that the linearly-polarized electromagnetic wave\ngenerally produces a dc spin current in noncentrosymmetric magnetic insulators.\nI. INTRODUCTION\nMaterials subject to an intense incident light shows\nrich behaviors which are studied in the context of non-\nlinear response and non-equilibrium phenomena. An ex-\nample of such is electric shift current in noncentrosym-\nmetric semiconductors and ferroelectrics [1{7], where a\nnon-trivial shift of electron position during its optical\ntransition produces a macroscopic electric current. Re-\ncent studies revealed that the shift current exhibits strik-\ningly di\u000berent behaviors from the ordinary photocurrent;\nthe shift current shows unique light-position dependence\nwhen it is excited locally [8{10], and propagates faster\nthan the Fermi velocity of electrons [10{12]. On the other\nhand, in correlated materials, lower-energy excitations of-\nten emerge due to the interaction e\u000bect; a typical exam-\nple is magnetic excitations in Mott insulators. The op-\ntical transition of these emergent particles may produce\nnon-trivial phenomena, especially, transport phenomena,\nrelated to the nonlinear response of the emergent excita-\ntions.\nSeveral recent studies in opto-spintronics and magneto-\noptics [13{15] implies that the intensity and coherence of\ncurrently-available electromagnetic waves are su\u000ecient\nfor the control of magnetic excitations or magnetism.\nTypical results are the following: Magnetization switch-\ning by a circularly-polarized laser in ferrimagnets [16{19],\nlaser-driven demagnetization [20{22], the spin pumping\nby gigahertz (GHz) or terahertz (THz) waves [23, 24],\nfocused-laser driven magnon propagation [25, 26], intense\nTHz-laser driven magnetic resonance [27, 28], spin con-\ntrol by THz-laser driven electron transitions [29], dichro-\nisms driven by THz vortex beams [30], angular mo-\nmentum transfer between photons and magnons in cav-\nities [31{35], a ultrafast detection of spin Seebeck ef-\nfect [36], a phonon-mediated spin dynamics with THzlaser [37], etc. Moreover, recent theoretical works have\nproposed several ways of optical control of magnetism:\nTHz-wave driven inverse Faraday e\u000bect [38, 39], Floquet\nengineering of magnetic states such as chirality ordered\nstates [40, 41] and a spin liquid state [42], generation\nof magnetic defects with laser-driven heat [43, 44], ap-\nplications of topological light waves to magnetism [44{\n47], control of exchange couplings in Mott insulators\nwith high- [48] and low-frequency [49] waves, optical\ncontrol of spin chirality in multiferroic materials [50],\nrecti\fcation of dc spin currents in magnetic insulators\nwith electromagnetic waves [51{53]. These studies are\npartly supported by recent developments in THz laser\nscience [54, 55] which realized high-intensity light beams\nwith the photon energy comparable to those of magnetic\nexcitations. Despite these developments, the optical con-\ntrol of the current carried by magnetic excitations is lim-\nited to some theoretical proposals.\nAmong the proposals, a recent theory proposes a mech-\nanism for producing a dc spin current in quantum spin\nchains without the angular-momentum transfer [52]; it\nis distinct from the known mechanisms in which the an-\ngular momentum of photons are transferred to the mag-\nnet [23, 24, 51, 53, 56]. The mechanism in Ref. 52 is anal-\nogous to that of the shift-current photovoltaic e\u000bect [2].\nThe close relation between two phenomena are clear from\nthe Jordan-Wigner fermion representation of spin chain;\nthe ground state of the spin chain is a band insulator of\nJordan-Wigner fermions, and the photovoltaic response\nis related to the optical transition of the fermions by the\nlinearly-polarized THz light. However, the relation of\nthis mechanism to the fermion excitations casts doubt\non the generality because the low-energy excitations of\nthe ordered magnets are usually magnons, i.e., bosonic\nexcitations.\nIn this work, we theoretically show that a dc spin cur-\nrent similar to that of the spin chain [52] also appearsarXiv:1907.02734v1  [cond-mat.mes-hall]  5 Jul 20192\nin ordered antiferromagnetic (AFM) and ferrimagnetic\n(FRM) insulators by applying a linearly-polarized elec-\ntromagnetic wave. The symmetry argument in Sec. III\nshows that the creation of dc spin current with linearly-\npolarized waves is possible only if both site- and bond-\ncenter inversion symmetries are broken. AFM and FRM\ninsulators violate the bond-center inversion symmetry\nand thereby they naturally satisfy half of the required\nsymmetry condition. The staggered moment is an ad-\nvantage of considering AFM/FRM insulators for gen-\nerating a dc spin current. As an example, we con-\nsider two-sublattice models with N\u0013 eel type ground state.\nBosonic particles describe the low-energy excitations of\nthese models, i.e., magnons; the ground state is the zero-\nmagnon state. This ground state is very di\u000berent from\nthat of noncentrosymmetric S= 1=2 spin chains [52]\nwhich are described by a Fermi degenerated state of\nspinons. Despite the di\u000berence, our calculation using\na nonlinear response theory \fnds a \fnite photovoltaic\nspin current similar to that of the spinons. We dis-\ncuss that it is related to the zero-point \ructuation of\nthe quantum magnets. Our theory also indicates that\nthe magnon spin current is shift-current like, i.e., it is\ninsensitive to the magnon lifetime as in the spinon case.\nThis mechanism allows generation of spin current using\na linearly-polarized electromagnetic wave and ordinary\nAFM or FRM insulators.\nThe remaining part of the paper is organized as fol-\nlows. In Sec. II, we introduce the nonlinear-response\ntheory for two-species magnons, which we will use in the\nfollowing sections. The main results of this paper are in\nSecs. III and IV. Section III focuses on the photo-induced\nspin current in AFM and FRM insulators with a strong\none dimensionality, while we study the three-dimensional\n(3D) magnets in Sec. IV. E\u000bective experimental setups\nand signatures for investigating the proposed mechanism\nare discussed in Sec. V. Section VI is devoted to the sum-\nmary and discussions.II. NONLINEAR RESPONSE THEORY\nWe calculate the nonlinear response coe\u000ecients for\nthe photo-induced spin current by extending the linear-\nresponse theory to the quadratic order in the perturba-\ntion. A similar method for fermions is used to calculate\nthe photovoltaic current in semiconductors [57, 58] and\nthe spin current of spinons [52]. The derivation of the\nformula is summarized in Appendix A. We here summa-\nrize the outline of the derivation. We also discuss the\nphysical implications.\nWe consider a two-sublattice AFM/FRM insulator\nwith two species of magnons. The e\u000bective Hamiltonian\nfor the magnons is\nH=X\nk\"\u000b(k)\u000by\nk\u000bk+\"\f(k)\fk\fy\nk; (1)\nwhere\u000bk(\u000by\nk) and\fk(\fy\nk) are the boson annihila-\ntors (creators) for the magnons with the momentum\nk= (kx;ky;kz) and\"a(k) (a=\u000b;\f) is the energy\nof the magnons in the a=\u000b;\f branch with momen-\ntumk. We here consider a general perturbation (spin-\nelectromagnetic-wave coupling)\nH0=\u0000X\n\u0016;kZd!\n2\u0019h\u0016\n!ei!t y\nk\u0012\n(B\u0016\nk)\u000b\u000b(B\u0016\nk)\u000b\f\n(B\u0016\nk)\f\u000b(B\u0016\nk)\f\f\u0013\n k\n+ h.c.; (2)\nand spin-current operator\nJ=X\nk y\nk\u0012\n(Jk)\u000b\u000b(Jk)\u000b\f\n(Jk)\f\u000b(Jk)\f\f\u0013\n k: (3)\nHere,!is the frequency of ac light, h\u0016\n!is the spin-\nlight coupling constant for the \u0016direction, and  k=\n(\u000bk;\fy\n\u0000k)T.\nThe nonlinear conductivity is de\fned by\nhJi(\n) =X\n\u0016;\u0017Z\nd!\u001b\u0016\u0017(\n;!;\n\u0000!)h\u0016\n!h\u0017\n\n\u0000!;(4)\nwherehJi(\n)\u0011R\ndthJi(t)e\u0000i\ntis the Fourier transform\nof the expectation value of the spin current hJi(t). For\nthe two-sublattice model, the formula for nonlinear spin\ncurrent conductivity reads\n\u001b\u0016\u0017(\n;!;\n\u0000!) =1\n2\u0019X\nk;ai=\u000b;\fsgn(a3)(~\u001ak;a1sgn(a2)\u0000sgn(a1)~\u001ak;a2)(B\u0016\nk)a1a2\n!\u0000~\"a2(k) + ~\"a1(k)\u0000i=(2\u001ck)\n\u0002\u0014(B\u0017\nk)a2a3(Jk)a3a1\n\n + ~\"a1(k)\u0000~\"a3(k)\u0000i=(2\u001ck)\u0000(Jk)a2a3(B\u0017\nk)a3a1\n\n + ~\"a3(k)\u0000~\"a2(k)\u0000i=(2\u001ck)\u0015\n; (5)\nwhere\n~\"a(k) =sgn(a)\"a(k); (6)\nsgn(a) =\u001a\n1 (a=\u000b)\n\u00001 (a=\f); (7)\n~\u001ak;a=\u001ah\u000by\nk\u000bki0(a=\u000b)\nh\f\u0000k\fy\n\u0000ki0(a=\f): (8)The relaxation time of magnons, \u001ck, was introduced in3\n(a) (b)\nω ω\n001234\n0\nE\nπ −π −π π(c) (d)\nk kαk, βkαk+ β −k\nαkαk+ β −k\nβk\nω\nFIG. 1. (Color online) Schematic pictures of the noncen-\ntrosymmetric magnets. A quasi-one-dimensional magnet\nconsisting of weakly-coupled spin chains (a) and a three-\ndimensional magnet with two-sublattice order (b). Each sub-\nlattice (blue and orange) has a di\u000berent environment, e.g., dif-\nferentgfactors, uniaxial anisotropy, etc., and with the bond\ndimerization (shown by the thick bond). The two-sublattice\norder and bond dimerization respectively breaks the inversion\nsymmetry on the bond center and sites. Magnetic excitation\nand nonlinear spin current conductivity of the spin chain. The\nmagnon band dispersions of the model in Eq. (11a) for (c)\nh+= 0 and (d) h+= 1=100. Parameters h\u0006are de\fned in\nEq. (17). When h+= 0, two magnon dispersions are degen-\nerate. The THz light produces two magnons, one on each\nbranch as schematically shown in panel (d).\nEq. (5), andh\u0001\u0001\u0001i 0is the expectation value of \u0001\u0001\u0001in the\nequilibrium state of the Hamiltonian in Eq. (1). The\nconductivity for dc spin current corresponds to the \n = 0\ncase,\u001b\u0016\u0017(0;!;\u0000!). In the rest of this work, we focus\non the case B\u0016\nk=B\u0017\nk=Bkbecause we are interested\nin the response to a linearly polarized light. Hence, we\nabbreviate the subscripts in the nonlinear conductivity,\n\u001b\u0016\u0017(0;!;\u0000!) =\u001b(0;!;\u0000!).\nWe note that the conductivity in Eq. (5) remains non-\nzero atT= 0. The substitutions of ~ \u001ak;\u000b= 0 and ~\u001ak;\f= 1\nin Eq. (5) reduce the formula to\n\u001b(0;!;\u0000!) =X\nk\n\u00001\n\u0019\u0014(1 +i2\u001ck!)j(Bk)\f\u000bj2((Ak)\u000b\u000b+ (Ak)\f\f)\n(!\u0000i=2\u001ck)2\u0000(\"\u000b(k) +\"\f(k))2\u0015\n+1\n2\u0019(Bk)\f\u000b(Ak)\u000b\f((Bk)\f\f+ (Bk)\u000b\u000b)\n(!\u0000i=2\u001ck\u0000\"\u000b(k)\u0000\"\f(k))(\"\u000b(k) +\"\f(k) +i=2\u001ck)\n+1\n2\u0019(Bk)\u000b\f(Ak)\f\u000b((Bk)\f\f+ (Bk)\u000b\u000b)\n(!\u0000i=2\u001ck+\"\u000b(k) +\"\f(k))(\"\u000b(k) +\"\f(k)\u0000i=2\u001ck):\n(9)Because of ~ \u001ak;\f= 1, the terms involving the o\u000b-diagonal\ncomponent of Bkremains at T= 0. In other words, the\ntwo-magnon creation/annihilation process plays a crucial\nrole as shown in Fig. 1(d). We focus on the T= 0 case in\nthe rest of this paper as this process is dominant in the\nlow temperature limit.\nFrom a di\u000berent viewpoint, Eq. (9) implies the zero-\npoint \ructuation plays a key role in the photovoltaic\nresponse of magnons. In our formalism, the zero-point\n\ructuation is manifested in the Bogoliubov transforma-\ntion of Holstein-Primakov bosons. This transformation\ncreates\fk\fy\nkand\u000by\nk\fy\n\u0000kterms which contribute to the\nphotovoltaic response in the ground state. ~ \u001ak;\f= 1 is\nanother consequence of the Bogoliubov transformation.\nThe importance of the zero-point \ructuation resembles\nthe spinon spin current [52], in which the Fermi degen-\neracy of spinons represents the quantum \ructuation of\nspins. A crucial di\u000berence in the current case is the\nabsence of Fermi degeneracy. However, in the case of\nthe AFMs/FRMs, the condensate of Holstein-Primakov\nbosons plays a similar role to the Fermi degeneracy. The\npair-creation process represented by \u000by\nk\fy\nkgenerates pho-\ntovoltaic response of the magnons which is manifested\nin the denominator of Eq. (9); the sum of eigenener-\ngies,\"\u000b(k) +\"\f(k), represents creation/annihilation of a\nmagnon pair. These features implies that the zero-point\n\ructuation is necessary for the shift current response at\nT= 0.\nThe \frst term in Eq. (9) vanishes when the ground\nstate has a certain symmetry. For example, collinear\nmagnetic orders with the moments parallel to Szaxis\nare often symmetric with respect to G=TMs\nx, which is\nthe product of time-reversal operation ( T) and the mir-\nror operation for the spin degrees of freedom about xaxis\n(Ms\nx). In this case, the real part of \u001b(0;!;\u0000!) reads\nRe [\u001b(0;!;\u0000!)] =\n\u00001\n\u0019X\nkRe\u001a(Bk)\f\u000b(Ak)\u000b\f((Bk)\f\f+ (Bk)\u000b\u000b)\n!2\u0000(\"\u000b(k) +\"\f(k) +i=2\u001ck)2\u001b\n:\n(10)\nThe conductivities for the models considered in the fol-\nlowing sections are calculated using this formula.\nIII. SPATIALLY-ANISOTROPIC MAGNET\nIn this section, we apply the above formula to a spin\nchain with AFM or FRM order, which corresponds to\na quasi-one-dimensional (quasi-1D) magnetic compound\nwith a negligible inter-chain interaction. The spins are\ncoupled to the electromagnetic wave through the Zee-\nman coupling. To make the problem theoretically well-\nde\fned, we consider a model which conserves the spin\nangular momentum Sz; the model has an easy axis and\nthe applied ac magnetic \feld is parallel to the ordered\nmoments. The conservation of Szallows us to unam-\nbiguously de\fne the spin current operator from the con-4\ntinuity equation. This setup is in contrast to those of\nusual magnetic resonances and spin pumping [23, 24], in\nwhich the ac \feld is perpendicular to the magnetic mo-\nment. We use the standard spin-wave approximation to\ndescribe magnetic excitations (magnons).\nA. Model\nWe consider an ordered noncentrosymmetric spin chain\nwith a two-sublattice unit cell [Fig. 1(a)], whose Hamil-\ntonian is given by\nHtot=H0+H(!)\nZ; (11a)\nH0\u0011X\nry;rzH1D(ry;rz); (11b)\nH1D(ry;rz)\u0011X\nrxJ(1 +\u000e)SA(r)\u0001SB(r)\n+J(1\u0000\u000e)SA(r+ ^x)\u0001SB(r)\n\u0000(D+Ds) [Sz\nA(r)]2\u0000(D\u0000Ds) [Sz\nB(r)]2\n\u0000h[gASz\nA(r) +gBSz\nB(r)]; (11c)\nH(!)\nZ=\u0000(h!ei!t+ h.c.)X\nrgASz\nA(r) +gBSz\nB(r);\n(11d)\nwhereH1Dis the spin-chain Hamiltonian with the stag-\ngered nearest-neighbor exchange interaction (i.e., dimer-\nization) along the xdirection,H0is the bundle of all\nthe chains, and H(!)\nZis the Zeeman coupling between\nthe spins and the external electromagnetic wave. Here,\nSa(r)\u0011(Sx\na(r);Sy\na(r);Sz\na(r)) (a=A;B) is the spin- Sa\noperator on the asublattice of the unit cell at position\nr= (rx;ry;rz). Symbols ^ x, ^y, and ^zstand for the unit\nvectors along the x,y, andzdirections, respectively. The\nparameters in the Hamiltonian H1Dare as follows: J >0\nis the antiferromagnetic exchange interaction along the\nspin-chain ( x) direction, \u000eis the dimerization, D > 0\n(Ds) is the uniform easy-axis (staggered) anisotropy, gA\n(gB) is thegfactor for the spins on A(B) sublattice,\nandhis the external static magnetic \feld along the Sz\naxis. In the spin-light coupling H(!)\nZ,jh!jand arg(h!)\nare respectively the magnitude and the phase of the ac\nmagnetic \feld of the linearly-polarized electromagnetic\nwave. We assume jDsj<D, andj\u000ej<1.\nWhenSA6=SB, the ground state of the model in\nEq. (11c) is a FRM-ordered state with magnetization\njSA\u0000SBjper a unit cell [59, 60]. The ground state is a\nN\u0013 eel ordered when SA=SB. The classical ground state\nofH0has a collinear order with spins pointing along the\nSzaxis because of the easy axis anisotropy D[Fig. 1(a)].\nThe anisotropy also produces the spin gap in the excita-\ntion spectrum [Fig. 2(a) and 1(b)]. We discuss the e\u000bect\nof the gap and its relation to the frequency dependence\nof the nonlinear spin conductivity in the next section.\nHere we de\fne the spin current for Sz. Since\nthe model H0conserves the zcomponent of to-tal spin angular momentum, the spin current for\nSzcan be de\fned from the continuity equations\n@tSz\nA=Jz\nx(rx\u00001;B;rx;A)\u0000Jz\nz(rx;A;rx;B) and\n@tSz\nB=Jz\nx(rx;A;rx;B)\u0000Jz\nz(rx;B;rx+ 1;A), in which\nJ\u000b\n\f(r;a;r0;b) is the local spin- S\u000bcurrent operator be-\ntween two neighboring sites ( r;a) and (r0;b) and it \rows\nalong the\fdirection. The above continuity equation is\nobtained from Heisenberg equation of motion for local\nspins. With these procedures, we \fnd the uniform cur-\nrent operator for H1Dreads\nJz\nx=J\n2NX\nr(1 +\u000e)fSx\nB(r)Sy\nA(r)\u0000Sy\nB(r)Sx\nA(r)g\n+ (1\u0000\u000e)fSx\nA(r+ ^x)Sy\nB(r)\u0000Sy\nA(r+ ^x)Sx\nB(r)g;\n(12)\nwhereNis the total number of unit cells.\nB. Linear spin-wave approximation\nHereafter, we assume that in the ground state of H1D,\nthe spins on the Asublattice points up while those on\nBsublattice are down [see Fig. 1 (a)]. The low-energy\nexcitations of H0is calculated by linear spin-wave ap-\nproximation. Using the Holstein-Primakov bosons, the\nspin operators are given by\nSz\nA=SA\u0000^nA(r); (13a)\nS+\nA(r) =p\n2SA\u0012\n1\u0000^nA(r)\n2SA\u00131\n2\na(r); (13b)\nS\u0000\nA=p\n2SAay(r)\u0012\n1\u0000^nA(r)\n2SA\u00131\n2\n; (13c)\nfor theAsublattice and\nSz\nB=^nB(r)\u0000SB; (14a)\nS+\nB(r) =p\n2SBby(r)\u0012\n1\u0000^nB(r)\n2SB\u00131\n2\n; (14b)\nS\u0000\nB=p\n2SB\u0012\n1\u0000^nB(r)\n2SB\u00131\n2\nb(r); (14c)\nfor theBsublattice. Up to the linear order in SAand\nSB,H0reads\nH0\u0018X\nk\u0012ak\nby\n\u0000k\u0013y\u0012\nh0\nk+hz\nkhx\nk\u0000ihy\nk\nhx\nk+ ihy\nkh0\nk\u0000hz\nk\u0013\u0012ak\nby\n\u0000k\u0013\n+ const. (15)\nwhere the wave number along the chain ( x) direction\nis simply represented by k,ak\u0011(1=p\nN)P\nra(r)eik\u0001r,\nbk\u0011(1=p\nN)P\nrb(r)eik\u0001(r+^x=2)are the Fourier trans-\nformation of Holstein-Primakov bosons. The matrix ele-\nments of the magnon Hamiltonian (15) are calculated as5\nh0\nk=h++J(SA+SB); (16a)\nhx\nk=2Jp\nSASBcos(k=2); (16b)\nhy\nk=\u00002J\u000ep\nSASBsin(k=2); (16c)\nhz\nk=h\u0000\u0000J(SA\u0000SB); (16d)\nwhere\nh+=D(SA+SB\u00001) +Ds(SA\u0000SB) +h\n2(gA\u0000gB);\n(17a)\nh\u0000=D(SA\u0000SB) +Ds(SA+SB\u00001) +h\n2(gA+gB):\n(17b)\nThe quadratic Hamiltonian (15) is diagonalized by the\nBogoliubov transformation:\nak= cosh \u0002 k\u000bk+ sinh \u0002 k\fy\n\u0000k; (18)\nby\n\u0000k= sinh \u0002 kei\bk\u000bk+ cosh \u0002 kei\bk\fy\n\u0000k; (19)\nwhere\u000bk(\u000by\nk) and\fk(\fy\nk) are bosonic annihilation (cre-\nation) operators. By choosing\nei\bk=hx\nk+ ihy\nkp\n(hx\nk)2+ (hy\nk)2(20a)\nand\ncosh(2\u0002 k) =h0\nkp\n(h0\nk)2\u0000(hx\nk)2\u0000(hy\nk)2; (20b)\nsinh(2\u0002 k) =\u0000p\n(hx\nk)2+ (hy\nk)2\np\n(h0\nk)2\u0000(hx\nk)2\u0000(hy\nk)2; (20c)the Hamiltonian becomes\nH0=X\nk\"\u000b(k)\u000by\nk\u000bk+\"\f(k)\fy\n\u0000k\f\u0000k; (21)\nwhere\n\"\u000b(k) =hz\nk+q\n(h0\nk)2\u0000(hx\nk)2\u0000(hy\nk)2; (22a)\n\"\f(k) =\u0000hz\nk+q\n(h0\nk)2\u0000(hx\nk)2\u0000(hy\nk)2: (22b)\nHere, we ignored the constant term in H0. We note that\nthe dispersions \"\u000b;\f(k) and the phases (\u0002 k;\bk) are all\nindependent of kyandkzbecause we now consider the\n1D modelH0. Using the same transformation, we \fnd\nH(!)\nZ=hX\nk(gAcosh2\u0002k\u0000gBsinh2\u0002k)\u000by\nk\u000bk\n+ (gAsinh2\u0002k\u0000gBcosh2\u0002k)\f\u0000k\fy\n\u0000k\n+gA\u0000gB\n2sinh(2\u0002 k)(\u000by\nk\fy\n\u0000k+\f\u0000k\u000bk)\n+gB+hN(gBSB\u0000gASA): (23)\nand\nJz\nx=Jp\nSASBX\nksinh(2\u0002 k)\u0012\nsink\n2cos \b k+\u000ecosk\n2sin \b k\u0013\u0010\n\u000by\nk\u000bk+\f\u0000k\fy\n\u0000k\u0011\n+\u0014\u001a\ncosh(2\u0002 k)\u0012\ncos \b ksink\n2\u0000\u000esin \b kcosk\n2\u0013\n+ i\u0012\nsin \b ksink\n2+\u000ecos \b kcosk\n2\u0013\u001b\n\u000by\nk\fy\n\u0000k+ h.c.\u0015\n:(24)\nC. Spin current conductivity\nCombining the magnon representation of ( \u000bk;\fk) with the formula (10), we compute the nonlinear dc spin-current\nconductivity for the model Htotunder the application of THz laser. We \frst study the nonlinear conductivity in the\nclean limit with in\fnite relaxation time \u001ck!1 . The analytic solution for the conductivity Re [ \u001b(0;!;\u0000!)] obtained\nfrom Eq. (10) reads\nRe [\u001b(0;!;\u0000!)] =(gA\u0000gB)2\u000e(h++J(SA+SB))\u0000\n!2\u00004(h++J(SA+SB))2\u00002J2SASB(1 +\u000e2)\u0001\n8\u0019(1\u0000\u000e2)!2p\n4J4S2\nAS2\nB(1\u0000\u000e2)2\u0000f(!=4)2+ 2J2SASB(1 +\u000e2)\u0000(h++J(SASB))2g2;(25)\nwhen!2[!c1;!c2] and zero otherwise. Here,\n!c1\u0011\"\u000b(0) +\"\f(0)\n=2p\n(h++J(SA+SB))2\u00004J2SASB; (26)corresponds to the energy for the band bottom of the6\n(a)\nσ(0;ω,−ω) (a.u.) \nω/J(b)\n-0.15-0.10-0.050.000.05\n10-410-310-210-1100101\n 0  1  2  3  4σ(ω)\nα=5x10-3\nα=1x10-2\nα=1x10-1\n0.01 0.10 1\n0.0010.0100.1001J┴=0\nJ┴=1/4\nJ┴=1δω|σ(0;ω,−ω)| (a.u.)\nFIG. 2. (Color online) Frequency dependence of the non-\nlinear spin current conductivity \u001b(0;!;\u0000!). (a) Analytic re-\nsult for the small Gilbert damping limit \u000b!0 and (b) numer-\nical results for a \fnite \u000b. The inset in (a) is the \u000e!\u0011!\u0000!c1\nfor di\u000berent J?. The calculations are done using a chain with\nN= 2048\u000032768 unit cells. All results are for J= 1,\u000e= 1=4,\nSA=SB= 1,gA= 1,gB= 1=2,h\u0000= 0, andh+= 1=100\nunless noted explicitly.\npair excitation and\n!c2\u0011\"\u000b(\u0019) +\"\f(\u0019)\n=2p\n(h++J(SA+SB))2\u00004\u000e2J2SASB;(27)\nis that for the top of the pair excitation [See Fig. 1(c)\nand (d)]. The frequency dependence of the conductivity\nis shown in Fig. 2(a).\nEquation (25) is an odd function of \u000e. This re\rects\nthe fact that the inversion-symmetry breaking is neces-\nsary for the spin current. H0has two inversion centers\nwhen\u000e= 0,Ds= 0,gA=gB, andSA=SB: one at\nthe center of the bond and the another on the site. The\ninversion center on the site is broken by the dimerization\n\u000e. To see the dependence of \u001b(0;!;\u0000!) on the model pa-\nrameters, we explicitly write down the nonlinear conduc-\ntivity as a function of the parameters, i.e., \u001b(0;!;\u0000!) =\n\u001b(!;\u000e;gA\u0000gB;Ds;m), wherem=hSz\nr2Ai\u0000hSz\nr2Biis\nthe order parameter of the AFM or FRM insulators. A\nsymmetry argument on the transport coe\u000ecient \fnds\n\u001b(!;\u000e;gA\u0000gB;Ds;m) =\u0000\u001b(!;\u0000\u000e;gA\u0000gB;Ds;m) for\nthe site-center inversion operation. This result is identi-\ncal to the spinon case in Ref. [52].\nOn the other hand, the magnetic order changes the\nparameter dependence of \u001b(0;!;\u0000!), which is related\nto the bond-center inversion operation. The inversionoperation about the center of the bonds is broken by\nthe N\u0013 eel ordering, Ds6= 0 orgA6=gB. Therefore, the\nsymmetry operation indicates \u001b(!;\u000e;gA\u0000gB;Ds;m) =\n\u0000\u001b(!;\u000e;\u0000gA+gB;\u0000Ds;\u0000m). In addition, the trans-\nlation operation about half a unit cell switches Aand\nBsublattices and m!\u0000m;\u001b(!;\u000e;gA\u0000gB;Ds;m) =\n\u0000\u001b(!;\u000e;gA\u0000gB;Ds;m). Hence, the conductivity in the\nordered phase is an even function of gA\u0000gBandDs. This\nis a di\u000berent behavior from the spinon case, in which the\nconductivity is an odd function of the staggered magnetic\n\feld (corresponds to gA\u0000gBin our case).\nThe conductivity diverges when !approaches !c1. The\nasymptotic form reads\nRe [\u001b(0;!;\u0000!)]\u0019\n\u0000(gA\u0000gB)2J2\u000eSASB(h++J(SA+SB))\n8\u0019Jf(h++J(SA+SB))2\u00004J2SASBg5\n4\n\u00021p\n(1\u0000\u000e2)SASB\u000e!; (28)\nwhere\u000e!\u0011!\u0000!c1. A similar feature is also found in\nthe spinon case, in which the divergence is related to the\nsingularity of the density of states [52]. On the other\nhand, the asymptotic form around !=!c2reads\nRe [\u001b(0;!;\u0000!)]\u0019\n(gA\u0000gB)2J2\u000eSASB(h++J(SA+SB))\n8\u0019Jf(h++J(SA+SB))2\u00004J2\u000e2SASBg5\n4\n\u00021p\n(1\u0000\u000e2)SASBj\u000e!j: (29)\nThe sign of the conductivity is the opposite of that in the\nlower frequency regime. This is in contrast to the spinon\ncase [52], in which the sign of the nonlinear conductivity\nremains the same for all frequencies !2[!c1;!c2].\nD. Relaxation-time dependence\nWe next study the damping (relaxation time) depen-\ndence of the spin current. In the study of photovoltaic\ne\u000bect, the relaxation-time dependence re\rects the micro-\nscopic mechanism behind the photovoltaic e\u000bect [2]; it\nis called shift current when \u001b(0;!;\u0000!)/\u001c0while is\nan injection current when \u001b(0;!;\u0000!)/\u001c. In bosonic\nsystems, a slight di\u000berence appears in the momentum\ndependence of the single-particle relaxation time [61]; it\nis inversely proportional to the momentum for the Gold-\nstone modes. Therefore, we assume the momentum de-\npendence of damping term as \u001ck= 1=(\u000b0\"\f(k)) so that\nthe momentum dependence is consistent with the \feld\ntheoretic requirement ( \u000b0is the damping factor). Phys-\nically, the assumed form of \u001ckcorresponds to the phe-\nnomenological Gilbert damping.\nWe substitute \u001ck= 1=(\u000b0\"\f(k)) in Eq. (10) in or-\nder to estimate the relaxation-time dependence of the7\nkzkxEk\nFIG. 3. (Color online) Schematic \fgure of the magnon disper-\nsion for the 3D model in Eq. (30). We set ky= 0. The blue\nand orange planes are the dispersions of two magnon branches\nand the green transparent plane is that of two-magnon exci-\ntation. The plot is for J= 1,J?= 1,\u000e= 1=4,SA= 1,\nSB= 1,h+= 1=100, andh\u0000= 1=10.\nspin-current conductivity. Figure 2(b) shows the \u000b0de-\npendence of \u001b(0;!;\u0000!). Our numerical result shows\n\u001b(0;!;\u0000!) is insensitive to the damping. A slight dif-\nference, however, appears in the high-frequency region,\nwhere the smearing due to the damping is more distinct\nthan that in the low-frequency region. This behavior is\nrelated to the momentum dependence of \u001ck, which is in-\nversely proportional to the energy of the magnon. The\ninsensitivity is a signature of a shift-current type photo-\ninduced current [2]; this is a similar feature to the spinon\ncase [52].\nIV. THREE-DIMENSIONAL MAGNETIC\nINSULATORS\nIn this section, we consider a three-dimensional (3D)\nmagnet which consists of coupled spin chains H1Dwith a\nnon-negligible inter-chain interaction [See Fig. 1(b)]. We\nparticularly focus on the limit in which !is close to the\nband gap of two-magnon excitations. The procedure of\nthe calculation is the same as the 1D case in the previous\nsection. The static part of the Hamiltonian reads\nH(3D)\n0\u0011X\nrJ(1 +\u000e)SA(r)\u0001SB(r)\n+J(1\u0000\u000e)SA(r+ ^x)\u0001SB(r)\n\u0000(D+Ds) [Sz\nA(r)]2\u0000(D\u0000Ds) [Sz\nB(r)]2\n\u0000J?[SA(r)\u0001SA(r+ ^y) +SA(r)\u0001SA(r+ ^z)\n+SB(r)\u0001SB(r+ ^y) +SB(r)\u0001SB(r+ ^z)]\n\u0000h[gASz\nA(r) +gBSz\nB(r)]: (30)\nThe spin chains is parallel to the xdirection, while the y\nandzdirections are perpendicular to the chains. The fer-\nromagnetic coupling J?>0 denotes the strength of theinter-chain exchange interaction. We study this model\nwithin the linear spin-wave approximation using Holsten-\nPrimakov transformation in Sec. III B. Focusing on the\nlower edge of the magnon dispersion, we \frst expand\nthe matrix elements ha\nkof the magnon Hamiltonian [See\nEq. (16)] up to second order in k:\nh0\nk'h++J(SA+SB) +J?(SA+SB)\n2(k2\ny+k2\nz);\n(31a)\nhx\nk'Jp\nSASB(2\u00001\n4k2\nx); (31b)\nhy\nk'\u0000Jp\nSASB\u000ekx; (31c)\nhz\nk'h\u0000+J(SA\u0000SB) +J?(SA\u0000SB)\n2(k2\ny+k2\nz):\n(31d)\nWe note that the magnon dispersions depend on both\nintra- and inter-chain wave numbers di\u000berently from the\n1D case. The dispersion around \u0000 point k=0is shown in\nFig. 3. Using the momentum gradient of the low-energy\nHamiltonian with h0;x;y;z\nk, we can de\fne the spin current\noperator; this approximation is essentially equivalent to\nexpanding the lattice spin-current operator in Eq. (24)\nup to the linear order in k:\nJz\nz=Jp\nSASBX\nksinh(2\u0002 k)\u0012kx\n2cos \b k+\u000esin \b k\u0013\n\u0002\u0010\n\u000by\nk\u000bk+\f\u0000k\fy\n\u0000k\u0011\n+\u001a\ncosh(2\u0002 k)\u0012\ncos \b kkx\n2\u0000\u000esin \b k\u0013\n+i\u0012kx\n2sin \b k+\u000ecos \b k\u0013\u001b\n\u000by\nk\fy\n\u0000k+ h.c.:\n(32)\nThese equations corresponds to the k\u0001pexpansion of the\nlattice model. Therefore, it should be a good approxima-\ntion for the lattice model when !is close to the gap for\ntwo-magnon excitations.\nThe spin current conductivity is calculated using the\nformula of Eq. (10). A calculation similar to the 1D\nmodel considered in Sec. III gives\nRe [\u001b(0;!;\u0000!)] =\n\u0000J2\u000eSASB(gA\u0000gB)2\n(4\u0019)22J?!2(SA+SB)\u0000\n8kx\u0000k3\nx\u0001\nkx=KX;(33)\nwhere\nKX=\"\n8(1\u0000\u000e2)\n\u00004s\n(h++J(SA+SB))2\u0000(!=2)2\nJ2SASB+\u000e2(\u000e2\u00002)3\n51\n2\n:\n(34)8\nWhen!is close to the lower edge, i.e.,\n!\u0018!c1\u00112p\n(h++J(SA+SB))2\u00004J2SASB;(35)\nKXbecomes\nKX\u0019s\n2p\n(h++J(SA+SB))2\u00004J2SASB\u000e!\n(1\u0000\u000e2)J2SASB;(36)\nwhere\u000e!=!\u0000!c1. Therefore, the asymptotic form of\nRe [\u001bABB(0;!;\u0000!)] is\nRe [\u001b(0;!;\u0000!)]\u0019\u0000(gA\u0000gB)2\u000epSASB\n16\u00192p\n1\u0000\u000e2(SA+SB)\n\u0002Jp\n\u000e!\nJ?f(h++J(SA+SB))2\u00004J2SASBg3\n4:(37)\nUnlike the 1D case, in which the conductivity diverges\nat the band edge !c1, the 3D result in Eq. (37) decreases\nproportionally top\n\u000e!when approaching !c1. The result\nis plotted in the inset of Fig. 2(a) with the results for\nthe 1D limit. This di\u000berence is a consequence of the\ndi\u000berence in the density of states: it diverges in the 1D\nmodel while it is proportional top\n\u000e!in the present 3D\ncase.\nThe approximation we used in this section is accurate\nwhen!is close to the magnon gap at the \u0000 point in the\nBrillouin zone. In our model, the band bottom for the\ntwo-magnon excitations are at \u0000 point, and the band-\nwidth of two-magnon excitation along the xandydirec-\ntions are in the order of J?and that for zdirection is in\nthe order of J. Therefore, our approximation is accurate\nwhen\u000e!\u001cJ;J?. This condition is manifested in J?\nin the denominator of Eq. (37), which implies the diver-\ngence of Re [ \u001b(0;!;\u0000!)] atJ?!0. WhenJ?is very\nsmall, we expect Re [ \u001b(0;!;\u0000!)] to behave like that of\nthe 1D case. On the other hand, Re [ \u001b(0;!;\u0000!)] looks\nlike Eq. (37) when J?is su\u000eciently large, e.g., when\nJ?\u0018J. Therefore, the 1d result and the result in this\nsection corresponds to the two limits of the 3D magnet.\nV. EXPERIMENTAL OBSERVATION\nIn this section, we discuss experimental methods for\ndetecting signatures of a directional spin current in our\nmechanism.\nA. Setup\nWe here discuss experimental setups for the observa-\ntion of the spin current generated by linearly-polarized\nlight. The mechanism studied here produces a directional\n\row of the spin current, which is a distinct feature from\nthe spin pumping [23, 24]. Therefore, the observation of\nthe directional \row should provide an evidence for our\n(a) (b)\n(c)\n(d)\nFIG. 4. (Color online) Schematic \fgure of the experimental\nsetups for measuring photo-induced spin current: All-optical\nsetup [(a) and (b)] and two-terminal setup (c). (a) The all-\noptical setup irradiates the isolated magnet using THz light.\nThe optically-induced spin current accumulates the angular\nmomentum at the end of the magnet which is depicted by the\nclouds; it produces the asymmetric distribution of the angular\nmomentum in the magnet. (b) A similar observation by at-\ntaching a thin layer of a soft ferromagnet at the two ends. The\nphotovoltaic spin current is injected to or absorbed from the\nsoft ferromagnets. (c) The two-terminal setup observes the\ndirectional \row of spin current using the inverse spin Hall ef-\nfect. The optically-induced spin current \rows along a certain\ndirection of the system. Therefore, inverse spin Hall voltage\nof the two leads has the same sign. These setups are di\u000berent\nfrom that of spin pumping of panel (d), in which a trans-\nverse AC \feld is applied to the magnet and the spin current\nis di\u000busively expanded.\nmechanism. We discuss two di\u000berent mechanisms: First\none is an all-optical setup using Kerr rotation or Fara-\nday e\u000bect, and the second is a two-terminal setup using\ninverse spin-Hall e\u000bect.\nObservation of the spatial distribution of angular mo-\nmentum in the open circuit setup provides a direct\nevidence for the optically-generated spin current [See\nFig. 4(a)]. In an isolated magnet, the spin current pro-\nduced by a THz light \rows along a direction de\fned by\nthe magnetic order and the crystal symmetry. There-\nfore, if the system becomes close enough to a laser-driven\nnon-equilibrium steady state, the angular momentum ac-\ncumulates at the two ends in an open circuit setup in\nFig. 4(a); positive angular momentum on one end and\nnegative on the other end. The angular momentum dis-\ntribution is anti-symmetric along the direction of the spin\ncurrent. This distribution is strikingly di\u000berent from the\nspin pumping case in which the distribution is symmetric\nand its di\u000berence from the equilibrium state is larger at\nthe focal area of the laser than at the ends.\nAn all-optical setup using Kerr rotation or Faraday\ne\u000bect would be a useful setup for the observation of such a9\nspatial distribution. Measurement of magnetic moments\nand its spatial distribution using the optical probe is a\ncommonly used technique for observing the spin current.\nFor instance, this method is used to observe the spin\nHall e\u000bect [62]. Similarly, observing the magnetization\nof soft magnet layers attached to the two ends is another\npossible setup for the experiment [Fig. 4(b)].\nThe observation of spin current in a two-terminal setup\nin Fig. 4(c) also enables us to see the directional \row of\nspin current and to distinguish it from the spin pumping\ne\u000bect. This setup consists of a noncentrosymmetric mag-\nnetic insulator which is sandwiched between two metallic\nleads; the two leads detect spin current via inverse spin\nHall e\u000bect [63{65]. In the photovoltaic mechanism, the\nspin current in the two leads \rows toward the same di-\nrection. Therefore, the inverse spin Hall voltage of the\ntwo leads has the same sign. In contrast, in the spin\npumping, the spin current di\u000busively \rows outward from\nthe magnet; the inverse spin Hall voltage is positive on\none side and negative on the other. Therefore, the rela-\ntive sign of the inverse spin Hall voltage of the two leads\ncan make a distinction between the spin pump and our\nmechanism.\nFinally, we shortly comments on heating e\u000bect of ap-\nplied electromagnetic waves. When we try to detect the\nphotovoltaic spin current with the above setups, spin\npumping might also occur due to the heating e\u000bect of the\napplied laser. For such a case, extracting the asymmet-\nric part of the angular-momentum distribution or inverse\nspin Hall voltage is important to detect an evidence for\nour mechanism.\nB. Required intensity of AC \feld\nWe next estimate the required ac electromagnetic \feld\nfor generating an observable spin current. We here as-\nsume a spin current of Js= 10\u000016J/cm2is observable.\nThis estimate is based on a Boltzmann theory calcula-\ntion for spin Seebeck e\u000bect in a ferromagnet [52, 69].\nThe details of the estimate is brie\ry explained in Ap-\npendix B. We use the following parameters as a typical\nvalue for 1D insulating magnets: J= 100kBJ,\u000e= 0:1,\nSA=SB= 1,gA\u0000gB= 0:1\u0016BJ/T,h+= 10kBJ, and a\nthe light with a frequency which is ~\u000e!= 6\u0019~\u00021011Hz\nabove the band gap. Here, ~is the Planck constant.\nWith these parameters, the conductivity for the 1D\nAFM/FRM chain is Re [ \u001b(0;!;\u0000!)]\u001810\u000014J/(cm2T2).\nTherefore, the required magnitude of oscillating magnetic\n\feld to produce a spin current of Js= 10\u000016J/cm2is\nB\u0018q\nJs\njRe[\u001b(0;!;\u0000!)]j\u00180:1 T. This corresponds to the\nelectric \feld E=cB\u0018104\u0000105V/cm under the as-\nsumption of c= 108m/s which is a typical value of\nspeed of light in insulators. Similar estimate for the\n3D magnet with J= 100kBJ,J?= 10kBJ,\u000e= 0:1,\nSA=SB= 1,gA\u0000gB= 0:1\u0016BJ/T,h+= 10kBJ,\nand!= 2\u0019\u00021012Hz gives Re [ \u001b(0;!;\u0000!)]\u001810\u000011J/(cm2T2) andE=cB\u0018105\u0000106V/cm. Our estimate\npredicts that the photovoltaic spin current is experimen-\ntally observable by using a moderate-intensity THz light.\nC. Candidate material\nWe believe the photovoltaic spin current should be seen\ngenerically in noncentrosymmetric magnets. In a recent\nwork [52], the authors \fnd three kinds of spin-light cou-\nplings induce the spin current in a spin chain, and this\nwork presents photovoltaic spin current in ordered mag-\nnets. These results imply the generation of photovoltaic\nspin current is a universal phenomenon in noncentrosym-\nmetric magnetic insulators. One such material is ferri-\nmagnetic diamond chains [66{68]. These materials often\nhave a distortion associated with trimerization, which\nbreaks the inversion symmetry [66]. Also, a large den-\nsity of states for the magnon excitations is expected in\nthis material because it is a quasi-1D magnet. Thus the\nferrimagnetic phase of the diamond chain is a promising\ncandidate for studying the spin current.\nVI. SUMMARY AND DISCUSSION\nTo summarize, we studied the spin current genera-\ntion through the shift current mechanism in ferrimag-\nnetic/antiferromagnetic insulators. Our theory uses a\nnonlinear response theory, which is a natural generaliza-\ntion of the linear response theory. Based on this method,\nwe \fnd that the illumination of a linearly-polarized\nlight produces the magnon current in noncentrosymmet-\nric magnets with antiferromagnetic/ferrimagnetic order.\nThe photovoltaic spin current appears even at the zero\ntemperature where no magnon excitation exists in the\nequilibrium; the current is related to the two-magnon\nexcitation process and not to the optical transition of\nexisting (thermally-excited) magnons. We stress that\nthe photo-induced spin current in our mechanism is car-\nried by electrically-neutral particles. The relaxation-time\ndependence of the spin current indicates that our pho-\ntovoltaic e\u000bect is a \\shift current\", i.e., the nonlinear\nconductivity is insensitive to the damping. Our theory\nclearly shows that the shift current mechanism, which is\nwell known in electron (fermion) systems, is also relevant\nto systems with bosonic excitations whose the ground\nstate is the vacuum of bosons (zero boson state).\nOur result implies the zero-point quantum \ructua-\ntion is a key for the shift-current type photocurrent. In\nthe spinon spin current [52], the optical transition of a\nfermionic excitation plays a crucial role for the photocur-\nrent. In contrast to these cases, the ground state of the\nordered magnets is the zero-magnon state. Therefore,\nno optical transition of the existing magnons. Despite\nthe crucial di\u000berence, we \fnd a \fnite photovoltaic spin\ncurrent at the zero temperature. The magnon photocur-\nrent we found is ascribed to the optical transition of the10\n\\condensed\" Holstein-Primakov bosons. In the antiferro-\nmagnets/ferrimagnets, the ground state is a condensate\nof Holstein-Primakov bosons, which is technically rep-\nresented by the Bogoliubov transformation. The optical\ntransition of the condensed Holstein-Primakov bosons al-\nlows generation of the shift-current type photocurrent\neven at the zero temperature. On the other hand, we\n\fnd that the nonlinear conductivity is zero at T= 0 for\nthe ferromagnetic version of the model considered here.\nFrom this viewpoint, the two-magnon creation is similar\nto the particle-hole pair creation in semiconductors; the\noptical transition of fermions from the valence band to\nthe conduction band is equivalent to the pair creation.\nAs the condensation of the Holstein-Primakov bosons is\na manifestation of zero-point \ructuation, the zero-point\n\ructuation is the essence for the shift-current type pho-\ntovoltaic e\u000bects in the magnetic insulators.\nOur results implies that the dc spin current generation\nusing linearly polarized light is generally possible in the\nmagnets without inversion symmetry.\nAppendix A: Derivation of Kraut-von Baltz formula\nfor Bosons\nHere, we shortly explain the derivation of the nonlin-\near conductivity in two-band boson systems. We used\nthe formula in Eq. (A10) for the analytic calculations\nand Eq. (A5) for numerical results with a \fnite Gilbert\ndamping.\nWe calculate the nonlinear response coe\u000ecients us-\ning a formalism similar to the linear response theory.\nWe assume a system with a time-dependent perturba-\ntionH0=\u0000P\n\u0016^B\u0016F\u0016(t), where ^B\u0016is an operator and\nF\u0016(t) is a time-dependent \feld; the Hamiltonian reads\nH=H0+H0. The expectation value of an observable\n^Areadsh^Ai(t) = Trh\n^\u001a(t)^Ai\n=Z;where\u001a(t) is the density\nmatrix at time tandZ\u0011Tr\u001a(t). By expanding \u001a(t) up\nto the second order in F\u0016(t), the Fourier transform of\nhAi(t),hAi(\n), reads\nhAi(\n) =X\n\u0016;\u0017Z\nd!\u001b\u0016\u0017(\n;!;\n\u0000!)F\u0016(!)F\u0017(\n\u0000!);\n(A1)\nwith the nonlinear conductivity\n\u001b\u0016\u0017(\n;!;\n\u0000!) =1\n2\u0019X\nn;m;l(\u001an\u0000\u001am)(B\u0016)nm\n!\u0000Em+En\u0000i=(2\u001cmn)\n\u0002\u0014(B\u0017)mlAln\n\n +En\u0000El\u0000i=(2\u001cmn)\u0000Aml(B\u0017)ln\n\n +El\u0000Em\u0000i=(2\u001cmn)\u0015\n:\n(A2)\nHere,Enis the eigenenergy of the many-body eigenstaten,\u001cmnis the relaxation time, and Onm(O=A;B\u0016;B\u0017)\nis the matrix element of ^Oin the eigenstate basis of H0.\nWe here consider a periodic free-boson system in which\nall matrices A,B\u0016, andB\u0017have the following form:\n^O=X\nk\u0000\n\u000by\nk\f\u0000k\u0001\nOk\u0012\u000bk\n\fy\n\u0000k\u0013\n; (A3)\n=X\nk\u0000\n\u000by\nk\f\u0000k\u0001\u0012\n(Ok)\u000b\u000b(Ok)\u000b\f\n(Ok)\f\u000b(Ok)\f\f\u0013\u0012\u000bk\n\fy\n\u0000k\u0013\n;\n(A4)\nwhere\u000bk(\u000by\nk) and\fk(\fy\nk) are the annihilation (creation)\noperators of the boson eigenstates with momentum k,\nandOk=Ak;B\u0016\nk;B\u0017\nk. The theory for spinwave exci-\ntations of many antiferromagnetic models with a N\u0013 eel-\ntype order reduces to the above form by using Holstein-\nPrimakov and Bogoliubov transformations.\nFor the two-band system, we can express Eq. (A2) us-\ning single-particle eigenstates. We note that A,B\u0016, and\nB\u0017for the two-band system above do not conserve the\nparticle number. However, all operators are quadratic in\nthe annihilation/creation operators and consists of only\nfor terms: \u000by\nk\u000bk,\f\u0000k\fy\n\u0000k,\f\u0000k\u000bk, and\u000by\nk\fy\n\u0000k. There-\nfore, only few terms out of the possible Wick decomposi-\ntion remain nonzero, similar to that of the systems with\nconserved particle number. Using these features, we \fnd\n\u001b(\n;!;\n\u0000!) =\n1\n2\u0019X\nk;ai=\u000b;\fsgn(a3)(~\u001ak;a1sgn(a2)\u0000sgn(a1)~\u001ak;a2)(B\u0016\nk)a1a2\n!\u0000~\"a2(k) + ~\"a1(k)\u0000i=(2\u001ck)\n\u0002\u0014(B\u0017\nk)a2a3(Ak)a3a1\n\n + ~\"a1(k)\u0000~\"a3(k)\u0000i=(2\u001ck)\n\u0000(Ak)a2a3(B\u0017\nk)a3a1\n\n + ~\"a3(k)\u0000~\"a2(k)\u0000i=(2\u001ck)\u0015\n:(A5)\nHere,\nsgn(a) =\u001a\n1 (a=\u000b)\n\u00001 (a=\f); (A6)\n~\"a(k) =sgn(a)\"a(k); (A7)\n~\u001ak;a=\u001ah\u000by\nk\u000bki0(a=\u000b)\nh\f\u0000k\fy\n\u0000ki0(a=\f); (A8)\nand we assumed the relaxation time only depends on k.\nIt is worth noting that the conductivity remains \fnite at\nT= 0 despite there are no excitations. Technically, this\nis a consequence of ~ \u001ak;\f, which is 1 at T= 0. Physically,\nthis is because the pair creation/annihilation processes\ncontribute to the spin current even at T= 0.\nWe here focus on the T= 0 limit. In this limit, ~ \u001ak;\u000b=\n0 and ~\u001ak;\f= 1. Using these results, we obtain11\n\u001b(0;!;\u0000!) =\u00001\n\u0019X\nk;ai=\u000b;\f\u0014(1 +i2\u001c!)jB\f\u000bj2(A\u000b\u000b+A\f\f)\n(!\u0000i=2\u001ck)2\u0000(\"\u000b(k) +\"\f(k))2\u0015\n+1\n2\u0019X\nk;ai=\u000b;\f\u001a(Bk)\f\u000b(Ak)\u000b\f((Bk)\f\f+ (Bk)\u000b\u000b)\n(!\u0000i=2\u001ck\u0000\"\u000b(k)\u0000\"\f(k))(\"\u000b(k) +\"\f(k) +i=2\u001ck)\n+(Bk)\u000b\f(Ak)\f\u000b((Bk)\f\f+ (Bk)\u000b\u000b)\n(!\u0000i=2\u001ck+\"\u000b(k) +\"\f(k))(\"\u000b(k) +\"\f(k)\u0000i=2\u001ck)\u001b\n: (A9)\nAs we discussed in the main text, certain symmetries\nrestricts the \frst term to be zero; this is the case for the\nmodels we consider in the main text. Assuming the \frst\nterm vanishes, we \fnd\nRe [\u001b(0;!;\u0000!)] =\n\u00001\n\u0019Re\u001a(Bk)\f\u000b(Ak)\u000b\f((Bk)\f\f+ (Bk)\u000b\u000b)\n!2\u0000(\"\u000b(k) +\"\f(k) +i=2\u001c)2\u001b\n:\n(A10)\nWe used this formula for the calculation of nonlinear con-\nductivity in the main text.\nAppendix B: Boltzmann theory for spin Seebeck\ne\u000bect\nThe magnitude of spin current Js= 10\u000016J/cm2is\nthe estimate for the spinon spin current produced by the\nspin Seebeck e\u000bect in a recent experiment [69]. We here\nsummarize the method and result discussed in a supple-\nmental material of a recent work [52].\nThe spin current is estimated from the Seebeck e\u000bect\nof magnons whose dispersion is given by\n\"(k) =JSk2+ 2DS+h: (B1)\nnear the \u0000 point of k=0. This magnon dispersion corre-\nsponds to that of a ferromagnetic heisenberg model with\nexchange interaction J, uniaxial anisotropy D, and the\nmagnetic \feld hparallel to the anisotropy. The current\nis calculated using the semiclassical Boltzmann theory,\nin which the current reads\nJs(r) =~Zdk\n(2\u0019)3vzfk(r): (B2)\nHere,fk(r) is the density of magnons with momentum\nkat position randvz\u0011@kz\"(k) is the group velocity of\nmagnons.fk(r) is calculated from the Boltzmann equa-\ntion with temperature gradient\nvk\u0001rrfk(r) =\u0000fk(r)\u0000f(0)\nk(r)\n\u001ck; (B3)wheref(0)\nk(r) is the density at the equilibrium. Here, the\nrelaxation-time approximation is used to simplify the cal-\nculation of collision integral on the right hand side. The\nspin current induced by the spin Seebeck e\u000bect is esti-\nmated by substituting the solution of fk(r) in Eq. (B3)\ninto the current formula in Eq. (B2)\nIn the Boltzmann theory, the spin current by the spin\nSeebeck e\u000bect reads\nJs(r)\u00183(6\u00192)2\n3J2\nHS2\n2\u000bkBaT(r)\u0001T\nT(r)F\u0012JHSa2\u00032\n2kBT(r);2DS+h\n2kBT(r)\u0013\n;\n(B4)\nwhere \u0003 = (6 \u00192)1=3=ais the cuto\u000b for magnon dispersion\nand\nF(a;b) =Z1\n0x4csch2(ax2+b)dx: (B5)\nUsing a set of typical parameters S= 1,JH= 100kBJ,\nD= 0 J,h=\u0016BJ,a= 4\u000210\u000010m,\u000b= 10\u00002,T= 100\nK, \u0001T= 3\u0002104K/m, we \fnd Js\u001810\u000012J/cm2for the\nferromagnet. We assume this value as the typical spin\ncurrent density in the insulating ferromagnets.\nA recent experiment on quasi-one-dimensional mag-\nnets observed a spin current which is 10\u00004of what is\ntypically observed in a ferromagnetic phase [69]. There-\nfore, we assume Js\u001810\u000016J/cm2as the experimental\nresolution for the spin current.\nACKNOWLEDGMENTS\nWe thank Ryosuke Matsunaga and Youtarou Taka-\nhashi for fruitful discussions. 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Saitoh, Nature\nPhys. 13, 30 (2017)."    },    {        "title": "1907.04540v1.Temperature_dependence_of_magnetic_resonance_in_ferrimagnetic_GdFeCo_alloys.pdf",        "content": "1 \n Temperature dependence of magnetic resonance in ferrimagnetic \nGdFeCo alloys  \nTakaya  Okuno1, Se Kwon Kim2,3†, Takahiro  Moriyama1, Duck-Ho Kim1, Hayato  \nMizuno1,4, Tetsuya  Ikebuchi1, Yuushou  Hirata1, Hiroki  Yoshikawa5, Arata Tsukamoto5, \nKab-Jin Kim6, Yoichi  Shiota1, Kyung -Jin Lee7,8, Teruo Ono1,9† \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 611 -0011, Japan  \n2Department of Physics and Astronomy, University of California  Los Angeles, California \n90095, USA  \n3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, \nUSA \n4Institute for Solid State Physics, University of Tokyo, Kashiwa  277-8581 , Japan  \n5College of Science and Technology, Nihon University, Funabashi, Chiba 274 -8501, Japan  \n6Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea  \n7Department of Materials Science & Engineering, Korea University, Seoul 02841, \nRepublic of Korea  \n8KU-KIST Graduate School of Converging Science and Technology, Korea University, \nSeoul 02841, Republic of Korea  \n9Center for Spintronics Research Network (CSRN), Graduate School of Engineering \nScience, Osaka University, Osaka 560 -8531, Japan  \n†E-mail: kimsek@missouri.edu , ono@scl.kyoto -u.ac.jp  \n \n  2 \n We provide a macroscopic theory and experimental results for magnetic resonances of \nantiferromagnetically -coupled ferrimagnets. Our theory, which interpolates the dynami cs of \nantiferromag nets and ferromagnets smoothly, can describe ferrimagnetic resonance s across \nthe angular momentum compensation point . We also present experimental results for spin-\ntorque induced ferrimag netic resonance at several temperatures. The spectral analysis based \non our theory reveals that the Gilbert damping parameter, which has been considered to be \nstrongly temperature dependent, is insensitive to temperature. We env ision that our work \nwill facilitate further investigation of ferrimagnetic dynamics by providing a theoretical \nframework  suitable for a broad range of temperatures .  \n \n  3 \n Antiferromagnets have been gaining much attention in spintronics because of their \npotential utility for high -speed ultra -dense spintronic devices.1-4) Due to the antiparallel \nalignment of adjacent spins , their dynamics  is different  from that of ferromagnets.5) One \nemerging material platform for studying antiferromagnetic dynamics is \nantiferromagnetically -coupled ferrimagnets ,6-11) for which we can use conventional \ntechniques for ferromagnets owing to small but finite magnetizations . Indeed, r ecent \nexperiments in such ferrimagnets have found that both field -driven and current -driven \ndomain -wall dynamics are fastest at the angular momentum compensation point 𝑇A where \nthe magnetic dynamics are antiferromagnetic .12-15) However, the magnetic resonance \nphenomenon of ferrimagnets (FiMR) has not been fully clarified so far because of \ninsufficient experimental investigations. In the literature, Stanciu et al . have studied the \nlaser-induced precession and its decay to equilibrium in ferrimagnets and concluded that the \neffective Gilbert damping parameter 𝛼 , which governs the dissipation rate of angular \nmomentum,  is strongly temperature dependent and increases signif icantly  at 𝑇A.6) However, \nsome recent studies have provided a new perspective on 𝛼  of ferrimagnets by full y \nconsidering the antiferromagnetic dynamics in ferrimagnets . Kamra et al. have theoretically \nrevealed that the temperature dependence of FiMR occurs because of the temperature \ndependence of magnetic dynamics , not because of temperature dependence of  𝛼.16) Kim et \nal. have also reported the temperature -insensitive 𝛼 of ferrimagnets  through the  DW motion \nexperiment s.17) In this paper, we provide an additional evidence of temperature -insensitive \n𝛼 of ferrimagnets by performing the FiMR experiment analyzed by our macroscopic FiMR \ntheory.  \nFirst, we derive the equations for FiMR in a ferrimagnet consisting of two \nantiferromagnetically -coupled sublattices. Throughout the manuscript, we will focus on the \nregime where the ferrimagnet is away from the magnetization compensation temperature 𝑇M  \nso that th e magnetization is finite and well defined. Our experiments are also performed well \nwithin the considered regime  as detailed below.  To this end, we  expand the L andau -Lifshitz -\nGilbert -like (LLG -like) equation  for ferrimagnet films12,18-20) at the uniform ground state \nalong the positive in-plane z directio n to linear order in the small  fluctuations  |𝑛𝑥|,|𝑛𝑦|≪\n1, where the unit vector 𝒏 represents the N éel order parameter . The resultant equations are \ngiven by  \n𝑠net𝑛̇𝑥−𝛼𝑠total𝑛̇𝑦−𝜌𝑛̈𝑦=𝑀𝐻ext𝑛𝑦,     (1) 4 \n  \n𝑠net𝑛̇𝑦+𝛼𝑠total 𝑛̇𝑥+𝜌𝑛̈𝑥=−𝑀(𝐻ext+𝐻ani)𝑛𝑥,     (2) \n \nwhere 𝑠net=𝑠1−𝑠2 is the net spin density  of two sublattices  𝑠1>0 and 𝑠2>0, 𝛼 is the \nGilbert damping parameter , 𝑠total =𝑠1+𝑠2 is the sum of the magnitudes of the two spin \ndensities , 𝜌>0 is the  moment of inertia  for the dynamics  (which is inversely proportional \nto the microscopic exchange field between the two sublattices  and describes the \nantiferromagnetic dynamics of the magnet ),2) 𝐻ext is the external  field along the z direction,  \n𝐻ani  is the effective  anisotropy field along the x direction perpendicular to the film \n(including the effect of the demagnetizing field),  and 𝑀 is the magnetization . Here, we are \nneglecting the terms that are quadratic or higher order in 𝐻ext and the time derivative of the \norder parameter. The damping term is added by considering the Rayleigh dissipation \nfunction  𝑅=𝛼𝑠total ∫𝑑𝑉 𝒏̇2/2, which is the half of the ener gy dissipation rate through the \nmagnetic dynamics.20) Note that the Rayleigh function is defined in terms of 𝑠total, not in \nterms of 𝑠net, so that it is well defined even in the vicinity of 𝑇A where 𝑠net vanishes.17)  \nTo the zeroth -order in the damping paramete r 𝛼 , the resonance  frequencies for \nmonochromatic solutions to the above equations are given by  \n𝑓±2\n=𝑠net2+𝜌𝑀(2𝐻ext+𝐻ani)±√𝑠net4+2𝜌𝑀(𝑠net)2(2𝐻ext+𝐻ani)+𝜌2𝑀2𝐻ani2\n8𝜋2𝜌2,     (3) \n \nwhere 𝑓+ and 𝑓− are the frequencies for  higher and lower resonance  frequencies for the given \nfield. Far away from 𝑇A, where the  net spin density |𝑠net| is suf ficiently large, Eq. (1) and \nthe corresponding dynamics are dominated by the first -order time derivative term and thus \nwe can neglect the second -order term by setting 𝜌=0 . In that ferromagnetic limit, the \nexpression for the lower frequency is reduced to that for the ferromagnet resonance \nfrequency :21) \n         𝑓FiM= 𝑀\n2𝜋|𝑠net|√𝐻ext(𝐻ext+𝐻ani).     (4) \nNote that 𝑀/|𝑠net|  is the effective gyromagnetic ratio  𝛾eff  of the ferrimagnets. As the \ntemperature approaches 𝑇A , the net spin density | 𝑠net|  decreases and thus  the resonance \nfrequency  is expected to increase. However, this formula cannot be used in the vicinity of 5 \n 𝑇A, where 𝑠net vanishes and thus the second -order term cannot be neglected. Exactly at 𝑇A, \nthe net spin density vanishes 𝑠net=0, which reduces the obtained resonance  frequencies \n[Eq. (3)] to  \n   𝑓+= 1\n2𝜋√𝑀(𝐻ext+𝐻ani)\n𝜌,    𝑓−= 1\n2𝜋√𝑀𝐻ext\n𝜌.  (5) \nInclusion of the second -order time derivative term ∝𝜌 in the LLG -like equations [Eq. ( 1) \nand Eq. ( 2)] is necessary to obtain finite resonance  frequencies  at 𝑇A; otherwise, the LLG -\nlike equations lack in the reactive dynamic term ∝𝑠net at 𝑇A and becom e unable to describe \nthe ferrimagnetic dynamics properly therein.   \n Since our experimental results, which are presented below, are performed away from \n𝑇A, let us derive the resonance linewidth for ferrimagnets in the ferromagnetic regime. When \nwe include the Gilbert damping term, the resultant linewidth of  ferrimagnets ∆𝐻  (half -\nwidth -half-maximum) is given by  \n∆𝐻≈2𝜋𝛼\n𝛾eff 𝑠total\n|𝑠net| 𝑓FiM.     (6) \n \nTherefore 𝛼 in ferrimagnet is given by  \n𝛼FiM≈(𝛾eff\n2𝜋)|𝑠net|\n𝑠total (∆𝐻\n𝑓FiM).     (7) \nNote that b oth 𝑠total  and 𝑠net  appear in the lin ewidth expression because 1) the energy \ndissipation rate is proportional to 𝑠total since two lattices contribute additively and 2) the \nresonance frequency is inversely  proportional to 𝑠net. On the other hand, in conventional \nexpressions for ferromagnetic  resonance, the two spin -density parameters are assumed to be \nidentical , 𝑠total =𝑠net, and the corresponding expre ssion  𝛼FM≈(𝛾eff\n2𝜋)(∆𝐻\n𝑓FiM) was used to \nanalyze the magnetic resonance of ferrimagnets  in the previous reports.6,7) Below, these two \nexpressions for the Gilbert damping parameters, 𝛼FiM and 𝛼FM, will be compared based on  \nour experimental results.  \nWe experimentally  investigate d the FiMR in the GdFeCo compounds  by using  the \nhomodyne technique22-24) as shown in Fig 1 . For this study, we used a 5-nm SiN/10 -nm \nGd25.0Fe65.6Co9.4/5-nm Pt/100 -nm SiN /Si substrate  film. The film was patterned into a 10 -\nµm-wide  and 10 -µm-long strip pattern structure using optical lithography and Ar ion milling. \nA coplanar waveguide made of 100 -nm Au /5-nm Ti  were de posited at the ends of the strip.  6 \n The measurements were performed by sweeping an external magnetic field  𝐻ext at a fixed \nrf current  𝐼rf  (frequency 𝑓=4−18  GHz). 𝐻ext  was applied  in-plane  45°  away from the \nlong axis of the strip.  \nFigure 2a shows the FiMR  spectra at several temperatures 𝑇 between  220 K and  295 \nK. Although a single peak was clearly observed at 295 K, a second peak was also observed \nat 𝐻ext≈50 mT when 𝑇 is lower than 240 K. Note that the spontaneous magnetization lies \nin the sample plane at 𝑇=295  K while  it becomes perpendicular to the plane when 𝑇≤\n240  K . Thus, the two resonan ce peaks when 𝑇≤240  K  originat e from the magnetic \nresonance of perpendicular  (𝐻ext≈50 mT ) and in -plane (higher field) magnetization s, \nrespectively . Here we focus on the resonan ce peak originating from in -plane magnetization, \nso we cut off the low -field regime to exclude the resonance peak  from perpendicular \nmagnetization  and fit those spectra in Fig. 2a by the combination of symmetric and anti -\nsymmetric Lorentzian  function s, from which the resonan ce parameters are obtained .22,23)  \nFigure s 2b and 2c  show the resonance frequency  𝑓res as a function of  the resonance  \nfield 𝐻res and the spectral linewidth  ∆𝐻 (half -width -half-maximum) as a function of 𝑓res, \nrespectively . Firstly, we analyze th ese data using the conventional expressions of \nferromagnetic resonance,21,25) \n𝑓res=𝑔eff𝜇B\nℎ√𝐻res(𝐻res+𝐻ani),     (8) \n \n∆𝐻=𝛼FM\n(𝑔eff𝜇Bℎ⁄)𝑓res+∆𝐻0.     (9) \n \nHere, 𝑔eff  is the  effective  Landé  g-factor, 𝜇B  is the Bohr magneton,  ℎ  is the Planck ’s \nconst ant, 𝐻ani is the effe ctive  anisotropy field including  the demagnetization field, 𝛼FM is \nthe effective Gilbert damping parameter  defined as in Ref. 6, and ∆𝐻0  is a frequency -\nindependent linewidth known as the inhomogeneous broadening, which originates from \nmagnetic non -uniformity.25) Equation (8) can be matched with Eq. (4) once we identify \n𝑔eff𝜇B/ℏ as the effective gyrom agnetic ratio  𝑀/|𝑠net| (ℏ=ℎ2𝜋⁄ is the reduced Planck ’s \nconstant ) and 𝐻res  as 𝐻ext . The 𝐻res  vs 𝑓res  shown in Fig. 2b are well fitted b y Eq. (8), \nindicated by the solid lines, and  𝑔eff and 𝐻ani are obtained as the fitting parameters. Figures \n3a and 3b show 𝑔eff and 𝐻ani as a function of 𝑇, respectively.  It is found that 𝑔eff remarkably \nincreases as 𝑇 decreases. Since the 𝑇A of the device is estimated  to be 160 K (see below the 7 \n estimation method) , the result shows that 𝑔eff increases as 𝑇 approaches 𝑇A. Note that the \ndrastic decrease in 𝐻ani with decreasing 𝑇 (Fig. 3b) is attributed to the change in magnetic \nanisotropy from in -plane (295 K) to perpendicular (220 K) direction as mentioned above. \nThe 𝑓res vs ∆𝐻 show n in Fig. 2c are well fitted by Eq. (9), indicated by the solid lines, and  \n𝛼FM and ∆𝐻0 are obtained as the fitting parameters. Figures 3c and 3d show 𝛼FM and ∆𝐻0 \nas a function of 𝑇, respectively.  It is found that 𝛼FM increases significantly as 𝑇 decreases, \ni.e. as 𝑇 approaches 𝑇A. The 𝑇 dependences of 𝑔eff and 𝛼FM are in good agreement with the \nprevious paper s.6,7,2 7) According to the previous paper s,6,7) the 𝑇  dependences of 𝑔eff  and \n𝛼FM are understood in terms of that of the net angular momentum 𝑠net; both 𝑔eff𝜇Bℏ⁄=\n𝑀net 𝑠net⁄  and 𝛼FM [from Eq. (9)] are ill -defined  at 𝑇A where 𝑠net vanishes , which makes \nthe theory based on ferromagnets invalid therein . However, as shown in the discussion of \nour theory  for FiMR , by defining the Gilbert damping parameter in the Rayleigh dissipation \nfunction 𝑅=𝛼𝑠total ∫𝑑𝑉 𝒏̇2/2 in such a way that the damping parameter is always well -\ndefined, the resonance frequency  and the li newidth of FiMR can be described properly \nacross 𝑇A . In order to test whether our theory can explain the experimental results, we \nanalyze those data in Figs. 2b and 2c based on our theory.  \n As mentioned in the theory part, the ferrimagnetic resonance frequency  in the \nferromagnetic limit is reduced to the conventional ferromagnetic case, while the spectral \nlinewidth is modified by including the additional term 𝑠net 𝑠total⁄  . Therefore, the Gilb ert \ndamping parameter [Eq. (7)] in our theory [Eq. (1) and Eq. (2)] for the dynamics of \nferrimagnets can be obtained by the following expression:  \n𝛼FiM=𝛼FM|𝑠net\n𝑠total|.     (10) \nTo obtain 𝛼FiM from Eq. (7) and Fig. 2c, 𝑠net 𝑠total⁄  needs to be acquired.  Although the net \nspin density 𝑠net is easy to obtain from the effective gyromagnetic ratio, the total spin density \n𝑠total  is not straightforward to obtain.  To solve this problem , we perform the following \nanalysis . The effective net gyromagnetic ratio satisfies the following equation;6,7) \n𝑔eff𝜇B\nℏ=𝑀net\n𝑠net=𝑀FeCo −𝑀Gd\n𝑀FeCo\n(𝑔FeCo 𝜇Bℏ⁄)−𝑀Gd\n(𝑔Gd𝜇Bℏ⁄).     (11) \n \nHere, 𝑀FeCo (𝑀Gd) is the magnetizations of transition metal (rare -earth metal), and 𝑔FeCo \n(𝑔Gd) is the Landé  g-factor  of transition metal (rare -earth metal) sublattice. 𝑔eff is shown in 8 \n Fig. 3a, 𝑔FeCo  and 𝑔Gd  are obtained from literature  (𝑔FeCo ~2.2  and 𝑔Gd~2.0 ).28-30) Two \nquantities can be measured directly : 𝑀net is independently  measured  by SQUID  as shown \nin Fig. 4a  and 𝑠net  can be obtained from the effective gyromagnetic ratio  when the \nferrimagnet is well within the ferromagnetic regime.  With the measured values of 𝑀net  = \n𝑀FeCo −𝑀Gd and 𝑠net= 𝑀FeCo\n(𝑔FeCo 𝜇Bℏ⁄)−𝑀Gd\n(𝑔Gd𝜇Bℏ⁄), we can obtain the magnetizations of two \nsublattices,  𝑀FeCo  and 𝑀Gd , and also the spin densities of two sublattices,  𝑠FeCo  and 𝑠Gd .  \nFrom these results, we can obtain the total spin density 𝑠total =𝑠FeCo +𝑠Gd. Figures 4a and \n4b show 𝑀FeCo and 𝑀Gd, and 𝑠net=𝑠FeCo −𝑠Gd and 𝑠total =𝑠FeCo +𝑠Gd as a function of \n𝑇, respectively . Note that the 𝑇M (110 K ) determined by SQUID ( Fig. 4a ) and the 𝑇A (160 \nK) roughly estimated from the 𝑇  dependence of  𝑠net  (Fig. 4b) are clearly different,12,31) \nwhich supports the validity of this analysis. Finally, by substituting 𝑠net and 𝑠total into Eq. \n(10), the damping parameter  𝛼FiM is obtained as shown in Fig. 4c. It can be clearly seen  that \n𝛼FiM(≈0.01) is insensitive to 𝑇, in sharp contrast to 𝛼FM which significantly increases as 𝑇 \napproaches 𝑇A. Note that Eq. (10) is valid only in the ferromagnetic limit and, therefore,  it \nis necessary to confirm that the measured temperature range ( 220 – 295 K) is in the deep \nferromagne tic regime.  This would be guaranteed by the facts that 1) Fig. 4b shows \n𝑇A~160  K , which is  far below the lowest 𝑇  in our measurement s (220 K ), and 2) the \nresonance frequency  at 𝑇A is expected to  be similar to or larger than about 50  GHz  under \n300 mT ,6) which is much larger than the experimentally obtained resonance frequency  at 220 \nK (12 GHz under 300 mT).  \nThe observation that 𝛼FiM is insensitive to 𝑇 indicates that the 𝑇 dependence of the \nspectral linewidth in FiMR is attributed to the 𝑇  dependence of the net spin density 𝑠net \ninstead of that of the effective Gilbert damping parameter. This conclusion is consistent with \nsome recent papers,16,17) but it is in sharp contrast to the interpretation of the previous \nreport s,6,7) where the 𝑇 dependence of the spectral linewidth in FiMR was attributed to the \nchange of the effective Gilbert damping parameter. Our results provide an additional and \nclear evidence that properly defined Gilbert damping parameter 𝛼FiM  of ferrimagnets  is \ninsensitive to temperature, supporting the validity of these papers .16,17) Here, we would like \nto mention that, e ven though Fig. 4c is an evidence for the temperature -insensitive 𝛼FiM of \nferrimagnets, it lacks the information of 𝛼FiM in the vicinity of 𝑇A. However, obtaining  𝛼FiM \nin the vicinity of 𝑇A based on FiMR  experiment s is challenging  because 1) experimental \nobservation of ferrima gnetic resonance at 𝑇A which was measured to be larger than 5 0 GHz 9 \n for certa in ferrimagnets6) is expected to be difficult with homodyne detection technique (40 \nGHz at maximum in our measurement system) and 2) obtaining the necessary parameter \n𝑠total  is difficult because the net spin density 𝑠net  cannot be obtained from the effective \ngyromagnetic ratio in the vicinity of 𝑇A. Therefore, we believe that Fig. 4c serves as a good \nexperimental evidence to conclude that 𝛼FiM of ferrimagnets is insensitive to temperature.  \nIn conclusion, we have provided the macroscopic theoretical description of \nferrimagnetic resonance and experimental results that support it. Our theory shows that the \nresonance frequency  and the spectral linewidth of ferrimagnetic resonance can be described  \nwell across the angular momentum compensation point , by adding the antiferromagnetic -\nlike inertial term to the equations of motion and by defining the Gilbert damping parameter \nproperly through the Rayleigh dissi pation function. Moreover, we performed the spin -torque \ninduced ferrimagnetic resonances at various temperatures and successfully observed that the \nresonance frequency  and the linewidth depend on temperature. By analyzing the spectrum \nbased on our theory, we found that the Gilbert damping parameter in ferrimagnets is \ninsensitive to temperature, which has been considered to be strongly temperature -dependent. \nOur work introduces a new framework for studying ferrimagnetic resonance that allows us \nto interpret the ferrimagnetic dynamics for a wide range of temperatures .  \n \nAcknowledgments  \nThis work was supported by the JSPS KAKENHI (Grants  No. 15H05702,  No. 17H04924 , \nNo. 17H05181 , No. 26103002, and No. 26103004),  Collaborative Research Program of the \nInstitute for  Chemical Research, Kyoto University, and R & D project  for ICT Key \nTechnology of MEXT from the Japan Society  for the Promotion of Science (JSPS). This \nwork was  partly supported by The Cooperative Research Project  Program of the Researc h \nInstitute of Electrical  Communication, Tohoku University. S. K. K. w as supported by the \nstartup fund at the University of Missouri . D. H. K. was supported  as an Overseas \nResearcher under the Postdoctoral Fellowship  of JSPS (Grant No. P16314). K. J. L. w as \nsupported by the National Research Foundation of Korea  (2017R1A2B2006119) . K. J. K.  \nwas supported by  the KAIST -funded Global Singularity Research Program for 2019 .  10 \n  \nReferences  \n1) T. Jungwirth , X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol.  11, 231 \n(2016).  \n2) V . Baltz, A. Manchon , M. Tsoi , T. Moriyama , T. Ono , and Y . Tserkovnyak , Rev. Mod. \nPhys.  90, 015005 (2018).  \n3) J. Železn ý, P. Wadley, K. Olejník, A. Hoffmann, and H. Ohno , Nat. Phys.  14, 220 \n(2018).  \n4) M. B.  Jungfleisch, W. Zhang,  and A.  Hoffmann, Phys. Lett. A  382, 865 (2018).  \n5) O. Gomonay, V . Baltz, A. Brataas,  and Y . Tserkovnyak, Nat. Phys . 14, 213 (2018).  \n6) C. D.  Stanciu, A. V . Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and Th. \nRasing , Phys. Rev. B  73, 220402(R) (2006).  \n7) M. Binder, A. Weber, O. Mosendz, G. 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B  97, \n220403(R) (2018).  \n \n  12 \n  \nFigure Captions  \nFig. 1.  The schematic illustration of the device and the measurement  setup . The direction \nof the external magnetic field  𝐻ext and the AC current  𝐼rf are indicated. 𝐻ext was applied  \nin-plane  45° away from the long axis of the strip . \n \nFig. 2.  (a) The ferrimagnetic resonance spectra as a function of the external magnetic field \n𝐻ext at several temperatures from 220 -295 K. The emerging peak at 𝐻ext≈50 mT below \n240 K is attributed to the out -of-plane resonan ce peak and are neglected in this study. (b) \nThe resonance frequency  𝑓res as a function of the resonance  magnetic field  𝐻res. The solid \nlines are the fitting results by Eq. (8). (c) The spectral linewidth ∆𝐻 as a function of 𝑓res. \nThe solid lines are the fitting results by Eq. (9).  \n \nFig. 3. Resonance parameters as a function of temperature extracted by the fitting in Fig. \n(2). (a) The effective Landé  g-factor  𝑔eff. (b) The effective  anisotropy field 𝐻ani. (c) The \neffective Gilbert damping parameter 𝛼FM. (d) The f requency -independent linewidth ∆𝐻0. \n \nFig. 4. (a) The net magnetization 𝑀net and the magnetizations  of two sublattices  𝑀FeCo and \n𝑀Gd as function s of temperature . (b) The net spin density  𝑠net, the spin densities of two \nsublattices 𝑠FeCo and 𝑠Gd, and the sum of the magnitudes of the two spin densit ies 𝑠total as \nfunction s of temperature . (c) The effective Gilbert damping parameter 𝛼FM and the \nproperly defined Gilbert damping parameter of ferrimagnets  𝛼FiM as function s of \ntemperature . \n \n \n \n  13 \n  \nFig. 1 . \n  \n14 \n Fig. 2.  \n  \n0 5 10 15 200204060 295 K\n 260 K\n 240 K\n 220 KDH [mT]\nfres [GHz]数式 y = a + b*x\n重み 機械的\n残差平方和 6.89093\nピ ア ソ ン の r 0.99995\n--\n補正Ｒ 二乗 0.99989\n値 標準誤差\nΔH切片 38.04028 0.57739\n傾き 18.31533 0.07106\n数式 y = a + b*x\n重み 機械的\n残差平方和 3.53715\nピアソンの r 0.99996\n--\n補正Ｒ 二乗 0.9999\n値 標準誤差\nΔH切片 47.96301 0.69263\n傾き 21.8284 0.08274\n数式 y = a + b*x\n重み 機械的\n残差平方和 2.37794\nピ ア ソ ン の r 0.99992\n--\n補正Ｒ 二乗 0.9998\n値 標準誤差\nΔH切片 53.44711 1.23634\n傾き 24.77597 0.13079\n0 100 200 300 400 50005101520\n 295 K\n 260 K\n 240 K\n 220 Kfres [GHz]\nHres [mT]\n050100150200\n f = 4 GHz\n 6 GHz\n 10 GHz\n 14 GHz\n 18 GHzT = 295 K\n050100150200\nT = 280 K\n050100150200\nT = 260 K\n050100150200\nT = 240 K\n050100150200\nT = 230 K\n0 100 200 300 400 500 600 700 800050100150200V [mV]\nHext [mT]T = 220 K\n(a) \n(c) \n(b) 15 \n Fig. 3.  \n \n  \n220 240 260 280 3004681012DH0 [mT]\nT [K]\n220 240 260 280 300-120-80-400Hani [mT]\nT [K]\n220 240 260 280 3000.050.100.150.20\naFM\nT [K]\n220 240 260 280 3002.02.53.03.54.04.5geff\nT [K]\n(a) \n(c) \n(b) \n(d) 16 \n Fig. 4.  \n \n \n \n100 150 200 250 30001234Spin density [ ´10-6 J s/m3]\nT [K] snet\n sFeCo\n sGd\n stotal\nTA~160K\n0 50 100 150 200 250 300-101234Magnetization [ ´105 A/m]\nT [K] Mnet (by SQUID)\n MFeCo\n MGd\nTM » 110 K\n150 200 250 3000.000.050.100.150.20\na\nT [K] aFM\n aFiM\nTA~160K\n(a) \n(c) \n(b) "    },    {        "title": "1907.04577v2.The_superior_role_of_the_Gilbert_damping_on_the_signal_to_noise_ratio_in_heat_assisted_magnetic_recording.pdf",        "content": "The superior role of the Gilbert damping on the signal-to-noise ratio in\nheat-assisted magnetic recording\nO. Muthsam,1,a)F. Slanovc,1C. Vogler,1and D. Suess1\nUniversity of Vienna, Physics of Functional Materials, Boltzmanngasse 5, 1090 Vienna,\nAustria\n(Dated: 25 September 2019)\nIn magnetic recording the signal-to-noise ratio (SNR) is a good indicator for the quality of\nwritten bits. However, a priori it is not clear which parameters have the strongest in\ruence\non the SNR. In this work, we investigate the role of the Gilbert damping on the SNR. Grains\nconsisting of FePt like hard magnetic material with two di\u000berent grain sizes d= 5 nm and\nd= 7 nm are considered and simulations of heat-assisted magnetic recording (HAMR) are\nperformed with the atomistic simulation program VAMPIRE. The simulations display that\nthe SNR saturates for damping constants larger or equal than 0.1. Additionally, we can\nshow that the Gilbert damping together with the bit length have a major e\u000bect on the SNR\nwhereas other write head and material parameters only have a minor relevance on the SNR.\nI. INTRODUCTION\nThe next generation recording technology to increase\nthe areal storage density of hard drives beyond 1.5 Tb/in2\nis heat-assisted magnetic recording (HAMR)1{6. Higher\nareal storage densities (ADs) require smaller recording\ngrains. These grains need to have high anisotropy to be\nthermally stable. HAMR uses a heat pulse to locally\nenhance the temperature of the high anisotropy record-\ning medium beyond the Curie temperature. Due to the\nheating, the coercivity of the grain drops and it can be\nwritten with the available head \felds. After the grain\nis written, the medium is cooled and the information is\nsafely stored. A good indicator for the quality of the\nwritten bits is the so-called signal-to-noise ratio (SNR)\nwhich gives the power of the signal over the power of the\nnoise7. To achieve high areal storage densities, record-\ning materials that show good magnetic properties even\nat small grain sizes and thus yield high SNR values are\nneeded. However, a priori it is not clear which parame-\nters have the strongest in\ruence on the SNR.\nIn this work, we investigate the e\u000bect of a varying damp-\ning constant on the SNR. HAMR simulations with the\natomistic simulation program VAMPIRE8are performed\nfor cylindrical recording grains with two di\u000berent diame-\ntersd= 5 nm and d= 7 nm and a height h= 8 nm. The\nmaterial parameters of FePt like hard magnetic record-\ning media according to the Advanced Storage Technol-\nogy Consortium (ASTC)9are used. Damping constants\nbetween\u000b= 0:01 and\u000b= 0:5 are considered. Addition-\nally, we present an equation to include the in\ruence of\nthe bit length to the SNR. With this we can explain a\nSNR decrease of about 8.25 dB for 5- nm grains, which\nresults when changing the material and writing parame-\nters in the HAMR simulations from those used in former\nsimulations10{12to those according to the Advanced Stor-\nage Technology Consortium9, with the damping constant\nand the bit length only.\nThe structure of this paper is as follows: In Section II,\nthe HAMR model is introduced and it is explained how\na)Electronic mail: olivia.muthsam@univie.ac.atthe SNR is determined. In Section III, the results are\npresented and in Section IV they are discussed.\nII. HAMR MODEL\nCylindrical recording grains with height h= 8 nm and\ndiametersd= 5 nm and d= 7 nm are considered. One\ngrain can be interpreted as one grain of a state-of-the-\nart granular recording medium. A simple cubic crystal\nstructure is used. The exchange interaction Jijand the\ne\u000bective lattice parameter aare adjusted so that the sim-\nulations lead to the experimentally obtained saturation\nmagnetization and Curie temperature13,14. In the simu-\nlations, only nearest neighbor exchange interactions be-\ntween the atoms are included. A continuous laser pulse\nwith Gaussian shape and the full width at half maximum\n(FWHM) of 60 nm is assumed in the simulations. The\ntemperature pro\fle of the heat pulse is given by\nT(x;y;t ) = (Twrite\u0000Tmin)e\u0000x2+y2\n2\u001b2+Tmin (1)\n=Tpeak(y)\u0001e\u0000x2\n2\u001b2+Tmin (2)\nwith\n\u001b=FWHMp\n8 ln(2)(3)\nand\nTpeak(y) = (Twrite\u0000Tmin)e\u0000y2\n2\u001b2: (4)\nv= 15 m/s is the speed of the write head. xandylabel\nthe down-track and the o\u000b-track position of the grain,\nrespectively. In our simulations both the down-track po-\nsitionxand the o\u000b-track position yare variable. The\nambient and thus minimum temperature of all simula-\ntions isTmin= 300 K. The applied \feld is modeled as\na trapezoidal \feld with a \feld duration of 0.57 ns and a\n\feld rise and decay time of 0.1 ns, resulting in a bit length\nof 10.2 nm. The \feld strength is assumed to be +0 :8 T\nand\u00000:8 T inz-direction. Initially, the magnetization of\neach grain points in + z-direction. The trapezoidal \feldarXiv:1907.04577v2  [physics.app-ph]  24 Sep 20192\ntries to switch the magnetization of the grain from + z-\ndirection to\u0000z-direction. At the end of every simulation,\nit is evaluated if the bit has switched or not.\nThe material and write head parameters according to the\nAdvanced Storage Technology Consortium9are shown\nTable I.\nA. Determination of SNR\nTo calculate the signal-to-noise ratio, the read-back\nsignal of a written bit pattern has to be determined. To\nwrite the bit pattern and get the read-back signal from it,\nthe following procedure is used. First, a switching prob-\nability phase diagram is needed for the writing process of\nthe bit pattern. Since it is very time consuming to com-\npute a switching probability phase diagram with atom-\nistic or micromagnetic simulations, an analytical model\ndeveloped by Slanovc et al15is used in this work. The\nmodel uses eight input parameters (the maximum switch-\ning probability Pmax, the down-track jitter \u001bdown;the o\u000b-\ntrack jitter \u001bo\u000b;the transition curvature c, the bit length\nb, the half maximum temperature F50, the position p2of\nthe phase diagram in Tpeakdirection and the position p3\nof the phase diagram in down-track direction) to deter-\nmine a switching probability phase diagram. Slanovc et\nalshowed that the maximum switching probability Pmax\nand the down-track jitter \u001bdown are the input parameters\nwith the strongest in\ruence on the SNR. Note, that the\nbit lengthbalso has a strong in\ruence on the SNR. In\nthe further course of this work, an equation to include\nthe bit length to the SNR calculations is shown. Thus,\nthe bit length can be assumed constant during the SNR\ndetermination. The transition curvature cdid not show\nstrong in\ruence on the SNR for the used reader model\nand the o\u000b-track jitter \u001bo\u000bis neglectable since the reader\nwidth is with 30 :13 nm smaller than the track width with\n44:34 nm and thus does not sense the o\u000b-track jitter. p2\nandp3only shift the bit pattern and can thus be \fxed\nfor comparability. For this reason, it is reasonable to \fx\nthe input parameters, except for the maximum switch-\ning probability Pmaxand the down-track jitter \u001bdown.\nThe \fxed input parameters are determined by a least\nsquare \ft from a switching probability phase diagram\ncomputed with a coarse-grained Landau-Lifshitz-Bloch\n(LLB) model16for pure hard magnetic grains with mate-\nrial parameters given in Table I. The \ftting parameters\nare summarized in Table II for grain diameters d= 5 nm\nandd= 7 nm.\nFurther, it is necessary to compute the down-track jitter\n\u001bdown and the maximal switching probability Pmaxfor\nthe considered set of material and write head parameters,\nsee Table I. In the simulations, the switching probability\nof a recording grain at various down-track positions xat\na peak temperature Tpeak=Tc+ 60 K is calculated with\nthe atomistic simulations program VAMPIRE8, yielding\na down-track probability function P(x). To get the down-\ntrack jitter and the maximum switching probability, the\nswitching probability curve is \ftted with a Gaussian cu-mulative distribution function\n\b\u0016;\u001b2=1\n2(1 + erf(x\u0000\u0016p\n2\u001b2))\u0001Pmax (5)\nwith\nerf(x) =2p\u0019Zx\n0e\u0000\u001c2d\u001c; (6)\nwhere the mean value \u0016, the standard deviation \u001band\nthe mean maximum switching probability Pmax2[0;1]\nare the \ftting parameters. The standard deviation \u001b,\nwhich determines the steepness of the transition function,\nis a measure for the transition jitter and thus for the\nachievable maximum areal grain density of a recording\nmedium. The \ftting parameter Pmaxis a measure for\nthe average switching probability at the bit center. Note,\nthat the calculated jitter values \u001bdown only consider the\ndown-track contribution of the write jitter. The so-called\na\u0000parameter is given by\na=q\n\u001b2\ndown+\u001b2g (7)\nwhere\u001bgis a grain-size-dependent jitter contribution17.\nThe write jitter can then be calculated by\n\u001bwrite\u0019ar\nS\nW(8)\nwhereWis the reader width and S=D+Bis the grain\nsize, i.e. the sum of the grain diameter Dand the non-\nmagnetic boundary B15,18.\nFor each\u001bdown andPmaxcombination a switching prob-\nability phase diagram is computed with the analytical\nmodel. With the resulting phase diagram, the writing\nprocess of a certain bit pattern is simulated on granu-\nlar recording medium15. Here, the switching probabil-\nity of the grain is set according to its position in the\nphase diagram. The writing process is repeated for 50\ndi\u000berent randomly initialized granular media. Finally,\nthe read-back signal is determined with a reader model\nwhere the reader width is 30.13 nm and the reader res-\nolution in down-track direction is 13 :26 nm. The SNR\ncan then be computed from the read-back signal with\nthe help of a SNR calculator provided by SEAGATE19.\nThe resulting SNR value is given in dB (SNR dB). In the\nfollowing, the SNR dBis simply called SNR unless it is\nexplicitly noted di\u000berent.\nIII. RESULTS\nA. SNR Dependency on Damping\nFirst, the in\ruence of the damping constant on the\nSNR is investigated in more detail. The damping con-\nstant is varied from \u000b= 0:01 to\u000b= 0:5 for two di\u000berent\ngrain sizesd= 5 nm and d= 7 nm. All other parameters\nare taken from Table I. The bit length in the simula-\ntions is 10:2 nm and the track width is 44 :34 nm. The\ndown-track jitter curves are computed at Tpeak= 760 K\nand \ftted with eq. (5). In Figure 1, the SNR over the3\nCurie temp.\nTC[K]Damping\u000bUniaxial anisotropy\nku[J/link]Jij[J/link] \u0016s[\u0016B]v[m/s]\feld duration\n(fd) [ns]FWHM [nm]\n693.5 0.02 9:124\u000210\u0000236:72\u000210\u0000211.6 15 0.57 60\nTABLE I. Material and write head parameters of a FePt like hard magnetic granular recording medium accoring to the Advanced\nStorage Technology Consortium.\ngrain size\n5 nm 7 nm\n\u001bo\u000b[K] 22.5 14.4\nPmax 0.995 0.997\nF50[K] 602 628\nb[nm] 10.2 10.2\nc[10\u00004nm/K2]3.88 4.89\np2[K] 839 830\np3[nm] 27.5 25.8\nTABLE II. Reference parameters that are evaluated via least\nsquare \ft of the simulated phase diagrams for grain sizes 5 nm\nand 7 nm. Details of the parameters can be found in15.\ndamping constants for both grain sizes is visible. Note\nthat the SNR is proportional to the number of grains,\nmeaning that the number of grains per bit has to be kept\nconstant to determine a nearly constant SNR20. How-\never, since the dimensions of the granular media used for\nthe writing and reading process are \fxed, less grains form\none bit ford= 7 nm. Thus, the SNR values for the larger\ngrain size are smaller than for the small grains.\nThe results show that changing the damping constant\nfrom\u000b= 0:01 to\u000b= 0:02 already increases the SNR by\n3.66 dB for 5 nm-grains. For d= 7 nm, the SNR gain is\n1.65 dB. For 5 nm-grains, damping constants \u000b\u00150:1 lead\nto the best results with a total improvement of 6 dB com-\npared to\u000b= 0:01. Surprisingly, enhancing the damping\nconstant beyond 0 :1 does not show any further improve-\nment, the SNR saturates. This behavior is the same for\nthe 7 nm-grains. However, the total betterment of the\nSNR is only 2.24 dB for the larger grains. The SNR sat-\nuration results from the fact that Pmax= 1 for\u000b\u00150:1.\nSimultaneously, the down-track jitter \u001bdown varies only\nmarginally for \u000b\u00150:1 (see Table III) such that it does\nnot alter the SNR. The correlation between the SNR and\nthe maximum switching probability Pmaxis shown in Fig-\nure 2. It shows that the \ftted SNR curve reproduces the\ndata very well.\nBy further studying the switching dynamics of a 5 nm-\ngrain, one can show that the assumed pulse duration of\nthe heat pulse and the applied \feld strength are crucial\nfor the saturation of the SNR. In Figure 3, it is displayed\nhow the duration of the heat pulse in\ruences the maxi-\nmum switching probability and with it the SNR. During\nthe duration of the heat pulse the \feld is considered to\nconstantly point in \u0000z\u0000direction. The results demon-\nstrate that Pmaxdoes not saturate for small pulse du-\nrations. If longer pulse durations \u00150:5 ns are assumed,\naPmaxsaturation can be seen. A similar e\u000bect can be\nseen for a change of the \feld strength when the pulse\nduration is assumed to be 0 :5 ns (see Figure 4). For a\nsmall head \feld with a strength of 0 :5 T,Pmaxshows no\nsaturation whereas it does for larger head \felds. From\nFIG. 1. Resulting SNR for various damping constants \u000bfor\ngrains with two di\u000berent diameters d= 5 nm and d= 7 nm.\nFIG. 2. SNR and Pmaxdepending on the damping constant\n\u000bfor grain size with a diameter of 5 nm.\nthe simulations with varying duration of the heat pulse\nand \feld strength, it can also be seen that the SNR can\nbe improved for smaller damping constants if the dura-\ntion of the heat pulse is increased due to a smaller head\nvelocity or the \feld strength are enhanced.\nB. SNR Dependency on bit length\nThe in\ruence of the bit length on the SNR was al-\nready studied by Slanovc et al15. In this work, the fol-4\n5 nm 7 nm\n\u000b\u001bdown[nm]Pmax\u001bdown[nm]Pmax\n0.01 2.0 0.917 1.13 0.955\n0.02 0.9475 0.974 0.83 0.99\n0.05 0.7 0.989 0.549 1.0\n0.1 0.688 1.0 0.442 1.0\n0.3 0.495 1.0 0.48 1.0\n0.5 0.64 1.0 0.636 1.0\nTABLE III. Resulting down-track jitter parameters and mean maximum switching probability values for pure hard magnetic\nmaterial with di\u000berent damping constants \u000b.\nFIG. 3. Maximum switching probability Pmaxover damping\n\u000bfor di\u000berent pulse lengths of the heat pulse. A \feld strength\nof\u00000:8 T for grains with diameter d= 5 nm is assumed.\nlowing calculation is important. For the SNR calcula-\ntions a bit length b1= 10:2 nm is assumed since this is\nthe bit length resulting from the ASTC parameters. The\ntrack width in the simulations is again 44 :34 nm. How-\never, the bit length can change due to a variation of the\nwrite head parameters (\feld duration and head velocity).\nTherefore, the bit length for the former parameters10{12\nis 22 nm. To write a bit pattern with larger bit lengths\n(b >12 nm) the simulations of new granular media are\nrequired. This is computationally very expensive. Thus,\na di\u000berent approach is needed to qualitatively investi-\ngate the in\ruence of the bit length. For the SNR with\nSNR dB= 10 log10(SNR), there holds18\nSNR/\u0012b\na\u00132\u0012T50\nb\u0013\u0012W\nS\u0013\n(9)\nwith the bit length band the read-back pulse width T50\nwhich is proportional to the reader resolution in down-\ntrack direction. The ratio T50=bis called user bit density\nand is usually kept constant18. Further, the reader width\nWand the grain size Sare constant. Since the aim is to\nqualitatively describe the SNR for a bit length b2from\nSNR calculations with a bit length b1;the a-parameter a\nis also assumed to be constant. The SNR dBfor a di\u000berent\nbit lengthb2can then be calculated by\nFIG. 4. Maximum switching probability Pmaxover damping\n\u000bfor di\u000berent \feld strengths. The durations of the heat pulse\nof 0:5 ns for grains with diameter d= 5 nm is assumed.\nSNR dB(b2)\u0000SNR dB(b1)\n= 10 log10(SNR(b2))\u000010 log10(SNR(b1))\n= 10 log10(b2\n2)\u000010 log10(b2\n1) = 20 log10(b2\nb1) (10)\nsince all other parameters are the same for both bit\nlengths. Thus, one can compute the SNR dBvalue for\na varied bit length b2via the SNR dBof the bit length b1\nby\nSNR dB(b2) = SNR dB(b1) + 20 log10(b2\nb1): (11)\nThe curve achieved by eq. (11) with b1= 10:2 nm agrees\nqualitatively very well with the SNR(bit length) data\nfrom Slanovc et al15. It is thus reasonable to use this\nequation to include the bit length to the SNR.\nC. Combination of damping and bit length5\nCurie temp.\nTC[K]Damping\u000bUniaxial anisotropy\nku[J/link]Jij[J/link] \u0016s[\u0016B]v[m/s]\feld duration\n(fd) [ns]FWHM [nm]\n536.6 0.1 9:12\u000210\u0000235:17\u000210\u0000211.7 20 1.0 20\nTABLE IV. Material and write head parameters of a FePt like hard magnetic granular recording medium that were used in\nformer works10{12.\nParameter set diameter [nm] Tpeak[K]bit length [nm] Pmax\u001bdown [nm] SNR [dB]\nASTC 5 760 10.2 0.974 0.95 17.51\nParameters of former\nworks10{12 5 600 22 0.984 0.384 25.76\nASTC 7 760 10.2 0.99 0.83 15.35\nParameters of former\nworks10{12 7 600 22 1.0 0.44 22.75\nTABLE V. Resulting Pmax; \u001bdown and SNR values for the simulations with ASTC parameters and those used in former\nsimulations.\nThe simulations with write head and material parameters\naccording to the ASTC are compared to simulations with\nparameters used in former works10{12. Main di\u000berences\nto the currently used parameters are the bit length, the\ndamping constant, the height of the grain, the exchange\ninteraction, the atomistic spin moment, the full width at\nhalf maximum, the head velocity and the \feld duration.\nThese former parameters are summarized in Table IV.\nComparing the SNR values of both parameter sets shows\nthat ford= 5 nm the SNR is about 8.25 dB larger for the\nformer used parameters than for the ASTC parameters\nand ford= 7 nm it is\u00187:4 dB larger. The question is\nif the damping and bit length variation can fully explain\nthis deviation.\nIncreasing the damping constant from \u000b= 0:02 to\u000b=\n0:1, yields about +2 :25 dB ford= 5 nm and +0 :72 dB for\nd= 7 nm. Additionally, with the calculations from Sec-\ntion III B, one can show that by changing the bit length\nfromb1= 10:2 nm tob2= 22 nm gives\nSNR dB(b2) = SNR dB(b1) + 6:85 dB: (12)\nCombined, this shows that the di\u000berence in the SNR\ncan be attributed entirely to the damping and the bit\nlength enhancement. Moreover, simulations where the\nother material and write head parameters are changed\none by one con\frm this \fndings. The other write head\nand material parameters that are changed in the simu-\nlations have only minor relevance on the SNR compared\nto the damping constant and the bit length.\nIV. CONCLUSION\nTo conclude, we investigated how the damping con-\nstant a\u000bects the SNR. The damping constant was varied\nbetween\u000b= 0:01 and\u000b= 0:5 for two di\u000berent grain sizes\nd= 5 nm and d= 7 nm and the SNR was determined.\nIn practice, the damping constant of FePt might be in-\ncreased by enhancing the Pt concentration21,22. Another\noption would be to use a high/low Tcbilayer structure23\nand increase the damping of the soft magnetic layer by\ndoping with transition metals24{28. An interesting \fnd-\ning of the study is the enormous SNR improvement of6 dB that can be achieved for 5 nm-grains when enhanc-\ning the damping constant from \u000b= 0:01 to\u000b= 0:1\nand beyond. It is reasonable that the SNR improves\nwith larger damping. This results from the oscillatory\nbehavior of the magnetization for small damping dur-\ning switching. In fact, smaller damping facilitates the\n\frst switching but with larger damping it is more likely\nthat the grain will switch stably during the cooling of the\nthermal pulse29. This leads to a smaller switching time\ndistribution for larger damping constants and in the fur-\nther course to higher SNR values. However, an increase\nof the duration of the heat pulse due to a smaller head\nvelocity or an increase of the \feld strength can improve\nthe SNR even for smaller damping constant.\nFurthermore, the results display a SNR saturation for\ndamping constants \u000b\u00150:1. This SNR saturation can be\nexplained with the saturation of the maximum switching\nprobability and the only marginal change of the down-\ntrack jitter for \u000b\u00150:1. Indeed, one can check that for\nshorter pulse widths and smaller \feld strength, the be-\nhavior is di\u000berent and the SNR does not saturate. In\nthis case, the SNR rises for increasing damping constants.\nSummarizing, the SNR saturation for a varying damping\nconstant depends strongly on the used \feld strength and\nthe duration of the heat pulse.\nThe qualitative behavior for 7 nm-grains is the same. In-\nterestingly, the SNR change for a varying damping con-\nstant is not as signi\fcant as for grains with d= 5 nm.\nThis results from the higher maximum switching proba-\nbility and the smaller down-track jitter \u001bdown for 7 nm-\ngrains even for small damping constants. This is as\nexpected since larger grain sizes lead to an elevated\nmaximum switching probability11and smaller transition\njitter7compared to smaller grain sizes. This limits the\npossible increase of the recording performance in terms\nofPmaxand\u001bdown and thus the possible SNR gain. Ad-\nditionally, the SNR saturation value is smaller for 7 nm-\ngrains since one bit consists of fewer grains.\nThe overall goal was to explain the decrease of the SNR\nby about 8:25 dB and 7 :4 dB ford= 5 nm and d= 7 nm,\nrespectively, when changing from recording parameters\nused in former simulations10{12to the new ASTC pa-\nrameter. Indeed, together with the bit length variation,\nthe SNR variation could be fully attributed to the damp-6\ning enhancement. The other changed parameters like the\natomistic spin moment, the system height, the exchange\ninteraction and the full width at half maximum have only\na minor relevance compared to the in\ruence of the damp-\ning\u000band the bit length.\nIn fact, the variation of the bit length gave the largest\nSNR change. However, since an increase of the bit length\nis not realistic in recording devices, the variation of the\nmaterial parameters, especially the increase of the damp-\ning constant, is a more promising way to improve the\nSNR.\nV. ACKNOWLEDGEMENTS\nThe authors would like to thank the Vienna Sci-\nence and Technology Fund (WWTF) under grant No.\nMA14-044, the Advanced Storage Technology Consor-\ntium (ASTC), and the Austrian Science Fund (FWF)\nunder grant No. I2214-N20 for \fnancial support. The\ncomputational results presented have been achieved us-\ning the Vienna Scienti\fc Cluster (VSC).\n1L. Burns Jr Leslie and others. Magnetic recording system . Google\nPatents, December 1959.\n2G. W. Lewicki and others. Thermomagnetic recording and\nmagneto-optic playback system . Google Patents, December 1971.\n3Mark H. Kryder, Edward C. Gage, Terry W. McDaniel,\nWilliam A. Challener, Robert E. Rottmayer, Ganping Ju, Yiao-\nTee Hsia, and M. Fatih Erden. Heat assisted magnetic recording.\nProceedings of the IEEE , 96(11):1810{1835, 2008.\n4Robert E. Rottmayer, Sharat Batra, Dorothea Buechel,\nWilliam A. Challener, Julius Hohlfeld, Yukiko Kubota, Lei Li,\nBin Lu, Christophe Mihalcea, Keith Mount\feld, and others.\nHeat-assisted magnetic recording. IEEE Transactions on Mag-\nnetics , 42(10):2417{2421, 2006.\n5Hiroshi Kobayashi, Motoharu Tanaka, Hajime Machida, Takashi\nYano, and Uee Myong Hwang. Thermomagnetic recording .\nGoogle Patents, August 1984.\n6C. Mee and G. Fan. 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Journal of Applied Physics ,\n117(16):163913, 2015.\n12O. Muthsam, C. Vogler, and D. Suess. Noise reduction in heat-\nassisted magnetic recording of bit-patterned media by optimizing\na high/low Tc bilayer structure. Journal of Applied Physics ,\n122(21):213903, 2017.\n13Oleg N Mryasov, Ulrich Nowak, K Yu Guslienko, and Roy W\nChantrell. Temperature-dependent magnetic properties of fept:\nE\u000bective spin hamiltonian model. EPL (Europhysics Letters) ,\n69(5):805, 2005.\n14O Hovorka, S Devos, Q Coopman, WJ Fan, CJ Aas, RFL Evans,\nXi Chen, G Ju, and RW Chantrell. The curie temperature dis-\ntribution of fept granular magnetic recording media. Applied\nPhysics Letters , 101(5):052406, 2012.15Florian Slanovc, Christoph Vogler, Olivia Muthsam, and Dieter\nSuess. Systematic parameterization of heat-assisted magnetic\nrecording switching probabilities and the consequences for the\nresulting snr. arXiv preprint arXiv:1907.03884 , 2019.\n16Christoph Vogler, Claas Abert, Florian Bruckner, and Dieter\nSuess. Landau-Lifshitz-Bloch equation for exchange-coupled\ngrains. Physical Review B , 90(21):214431, December 2014.\n17Xiaobin Wang, Bogdan Valcu, and Nan-Hsiung Yeh. Transi-\ntion width limit in magnetic recording. Applied Physics Letters ,\n94(20):202508, 2009.\n18Gaspare Varvaro and Francesca Casoli. Ultra-High-Density Mag-\nnetic Recording: Storage Materials and Media Designs . CRC\nPress, March 2016.\n19S. Hernndez, P. Lu, S. Granz, P. Krivosik, P. Huang, W. Eppler,\nT. Rausch, and E. Gage. Using Ensemble Waveform Analysis to\nCompare Heat Assisted Magnetic Recording Characteristics of\nModeled and Measured Signals. IEEE Transactions on Magnet-\nics, 53(2):1{6, February 2017.\n20Roger Wood. The feasibility of magnetic recording at 1 terabit\nper square inch. IEEE Transactions on magnetics , 36(1):36{42,\n2000.\n21Satoshi Iihama, Shigemi Mizukami, Nobuhito Inami, Takashi Hi-\nratsuka, Gukcheon Kim, Hiroshi Naganuma, Mikihiko Oogane,\nTerunobu Miyazaki, and Yasuo Ando. Observation of Preces-\nsional Magnetization Dynamics in L10-FePt Thin Films with Dif-\nferent L10 Order Parameter Values. Japanese Journal of Applied\nPhysics , 52(7R):073002, June 2013.\n22Ji-Wan Kim, Hyon-Seok Song, Jae-Woo Jeong, Kyeong-Dong\nLee, Jeong-Woo Sohn, Toshiyuki Shima, and Sung-Chul Shin.\nUltrafast magnetization relaxation of L10-ordered Fe50pt50 al-\nloy thin \flm. Applied Physics Letters , 98(9):092509, February\n2011.\n23D. Suess and T. Schre\r. Breaking the thermally induced write er-\nror in heat assisted recording by using low and high Tc materials.\nApplied Physics Letters , 102(16):162405, April 2013.\n24W. Zhang, S. Jiang, P. K. J. Wong, L. Sun, Y. K. Wang, K. Wang,\nM. P. de Jong, W. G. van der Wiel, G. van der Laan, and Y. Zhai.\nEngineering Gilbert damping by dilute Gd doping in soft mag-\nnetic Fe thin \flms. Journal of Applied Physics , 115(17):17A308,\nMay 2014.\n25S. Ingvarsson, Gang Xiao, S. S. P. Parkin, and R. H. Koch. Tun-\nable magnetization damping in transition metal ternary alloys.\nApplied Physics Letters , 85(21):4995{4997, November 2004.\n26J. Fassbender, J. von Borany, A. Mcklich, K. Potzger, W. Mller,\nJ. McCord, L. Schultz, and R. Mattheis. Structural and magnetic\nmodi\fcations of Cr-implanted Permalloy. Physical Review B ,\n73(18), May 2006.\n27W. Bailey, P. Kabos, F. Manco\u000b, and S. Russek. Control of mag-\nnetization dynamics in Ni/sub 81/Fe/sub 19/ thin \flms through\nthe use of rare-earth dopants. IEEE Transactions on Magnetics ,\n37(4):1749{1754, July 2001.\n28J. O. Rantschler, R. D. McMichael, A. Castillo, A. J. Shapiro,\nW. F. Egelho\u000b, B. B. Maranville, D. Pulugurtha, A. P. Chen,\nand L. M. Connors. E\u000bect of 3d, 4d, and 5d transition metal\ndoping on damping in permalloy thin \flms. Journal of Applied\nPhysics , 101(3):033911, February 2007.\n29Roger W Wood, Jim Miles, and Terry Olson. Recording tech-\nnologies for terabit per square inch systems. IEEE Transactions\non Magnetics , 38(4):1711{1718, 2002.\n30R. F. L. Evans, Roy W. Chantrell, Ulrich Nowak, Andreas Ly-\nberatos, and H.-J. Richter. Thermally induced error: Den-\nsity limit for magnetic data storage. Applied Physics Letters ,\n100(10):102402, 2012."    },    {        "title": "1907.05027v1.Improving_the_Signal_to_noise_Ratio_for_Heat_Assisted_Magnetic_Recording_by_Optimizing_a_High_Low_Tc_bilayer_structure.pdf",        "content": "Improving the Signal-to-noise Ratio for Heat-Assisted Magnetic Recording by\nOptimizing a High/Low Tc bilayer structure\nO. Muthsam,1,a)F. Slanovc,1C. Vogler,1and D. Suess1\nUniversity of Vienna, Physics of Functional Materials, Boltzmanngasse 5, 1090 Vienna,\nAustria\n(Dated: 12 July 2019)\nWe optimize the recording medium for heat-assisted magnetic recording by using a high/low\nTcbilayer structure to reduce AC and DC noise. Compared to a former work, small Gilbert\ndamping\u000b= 0:02 is considered for the FePt like hard magnetic material. Atomistic simu-\nlations are performed for a cylindrical recording grain with diameter d= 5 nm and height\nh= 8 nm. Di\u000berent soft magnetic material compositions are tested and the amount of hard\nand soft magnetic material is optimized. The results show that for a soft magnetic material\nwith\u000bSM= 0:1 andJij;SM= 7:72\u000210\u000021J/link a composition with 50% hard and 50% soft\nmagnetic material leads to the best results. Additionally, we analyse how much the areal\ndensity can be improved by using the optimized bilayer structure compared to the pure hard\nmagnetic recording material. It turns out that the optimized bilayer design allows an areal\ndensity that is 1 Tb/in2higher than that of the pure hard magnetic material while obtaining\nthe same SNR.\nI. INTRODUCTION\nHeat-assisted magnetic recording (HAMR) [1{7] is a\npromising recording technology to further increase the\nareal storage densities (ADs) of hard disk drives. Con-\nventional state-of-the-art recording technologies are not\nable to overcome the so-called recording trilemma [8]:\nHigher ADs require smaller grains. These grains need\nto have high uniaxial anisotropy to be thermally sta-\nble. However today's write heads are not able to pro-\nduce \felds that are strong enough to switch these high\nanisotropy grains. In the HAMR process a heat pulse\nis included in the recording process to locally heat the\nrecording medium. This leads to a drop of the coercivity,\nmaking the high anisotropy recording medium writeable.\nThe medium is then quickly cooled and the information\nreliably stored.\nTo reach high linear densities it is necessary to reduce\nAC and DC noise in recording media [9]. AC noise de-\ntermines the distance between neighboring bits in bit-\npatterned [10{12] media or the transition between grains\nin granular media. DC noise restricts the maximum\nswitching probability of grains away from the transition.\nIt has been shown, that pure hard magnetic grains do\nnot switch reliably [13] if bit-patterned media are con-\nsidered whereas non-optimized exchange coupled bilayer\nstructures [14{19] of hard and soft magnetic material ex-\nperience high AC noise [20]. A work to reduce noise\nin recording media by optimizing a high/low Tcbilayer\nstructure (see Ref. [21]) showed that an optimial bilayer\nstructure consists of 80% hard magnetic and 20% soft\nmagnetic material. However, in the former work the\nGilbert damping was assumed to be \u000bHM= 0:1 which\nis hard to achieve in a FePt like hard magnetic material\nin reality. In realistic hard magnetic recording materi-\nals, the damping constant is \u000b= 0:02, according to the\nAdvanced Storage Technology Consortium (ASTC) [22].\na)Electronic mail: olivia.muthsam@univie.ac.atSince it has been shown that the damping constant has\na strong in\ruence on the maximum switching probabil-\nity and the down-track jitter, we follow the optimization\napproach and optimize a bilayer structure for the ASTC\nparameters. After the optimization, we study how the\noptimized material di\u000bers from that with \u000bHM= 0:1.\nAdditionally, we investigate how much the areal storage\ndensity (AD) can be improved when using the optimized\nrecording material instead of the pure hard magnetic one.\nThis is done with the help of the signal-to-noise ratio\n(SNR), which gives the power of the signal over the power\nof the noise and is a good indicator for the quality of writ-\nten bits.\nThe structure of this work is as follows: In Section II,\nthe HAMR model and the material parameters are pre-\nsented. In Section III, the results are shown and they are\ndiscussed in Section IV.\nII. HAMR MODEL\nThe optimization simulations are performed with the\natomistic simulation program VAMPIRE [23] which\nsolves the stochastic Landau-Lifshitz-Gilbert (LLG)\nequation. In the simulations, a cylindrical recording\ngrain with a diameter d= 5 nm and a height h= 8 nm\nis used. It can be considered as one recording bit in\nbit-patterned media. A simple cubic crystal structure is\nused and only nearest neighbor interactions are consid-\nered. The e\u000bective lattice parameter aand the exchange\ninteraxtion Jijare adjusted in order to lead to the exper-\nimentally obtained saturation magnetization and Curie\ntemperature. [24; 25]. The write head is assumed to\nmove with a velocity of v= 15 m/s. A continuous laser\npulse is assumed with the Gaussian temperature pro\fle\nT(x;y;t ) = (Twrite\u0000Tmin)e\u0000x2+y2\n2\u001b2+Tmin (1)\n=Tpeak(y)\u0001e\u0000x2\n2\u001b2+Tmin (2)arXiv:1907.05027v1  [physics.app-ph]  11 Jul 20192\nwith\n\u001b=FWHMp\n8 ln(2): (3)\nThe full width at half maximum (FWHM) is assumed\nto be 60 nm. Both, the down-track position xand\nthe o\u000b-track position yare variable in the simulations.\nThe initial and \fnal temperature is Tmin= 300 K. The\napplied \feld is modeled as a trapezoidal \feld with\na write \feld duration of 0.57 ns and a \feld rise and\ndecay time of 0.1 ns. The \feld is applied at an angle of\n22 deg with respect to the normal. The \feld strength is\nassumed to be +0.8 T and -0.8 T in z-direction. Initially,\nthe magnetization of each grain points in + z-direction.\nThe trapezoidal \feld tries to switch the magnetization\nof the grain from + z-direction to\u0000z-direction. At the\nend of every simulation, it is evaluated if the bit has\nswitched or not.\nA. Material parameters\nThe material parameters for the hard magnetic\nmaterial can be seen in Table I. For the soft magnetic\nmaterial, the atomistic spin moment is assumed to be\n\u0016s= 1:6\u0016Bwhich corresponds to a saturation polariza-\ntionJs= 1:35 T. The uniaxial anisotropy constant ku;SM\nin the soft magnetic layer is initially set to 0 but later\nvaried. The Gilbert damping \u000bSMand the exchange\ninteraction Jij;SM within the soft magnetic material are\nvaried. Experimentally, it is possible to increase the\ndamping constant by doping the soft magnetic material\nwith transition metals like Gd or Os [26{30]. Thus, also\nenhanced damping constants \u000bSMlarger than 0 :02 are\nconsidered in the simulations.\nIII. RESULTS\nA. Hard magentic grain\nFirst, a switching probability phase diagram for the\npure hard magnetic material is computed where the\nswitching probability is depending on the down-track po-\nsitionxand the o\u000b-track position y. With eq. (2) each\no\u000b-track position ycan be transformed into an unique\npeak temperature Tpeak, if the write temperature Twriteis\n\fxed, and vice versa. Thus, the switching probability in\nFigure 1 is shown as a function of the down-track position\nxand the peak temperature Tpeakthat corresponds to y.\nThe resolution of the phase diagram in down-track direc-\ntion is \u0001x= 1:5 nm and that in temperature direction\nis \u0001Tpeak= 25 K. In each phase point, 128 trajectories\nare simulated with a simulation length of 1 :5 ns. Thus,\nthe phase diagram contains more than 30.000 switching\ntrajectories. From the phase diagram it can be seen that\nFIG. 1. Switching probability phase diagram of a pure FePt\nlike hard magnetic grain. The contour lines indicate the\ntransition between areas with switching probability less than\n1% (red) and areas with switching probability higher than\n99.2% (blue). The dashed lines mark the switching probabil-\nity curves of Figure 2.\nthe pure hard magnetic grain shows only two small ar-\neas with switching probability larger than 99 :2%. This\nthreshold is used, since 128 simulations per phase point\nare performed and a switching probability of 100% corre-\nsponds to a number of successfully switched trajectories\nlarger than 1\u00001=128 = 0:992.\nTo determine the down-track jitter \u001b, a down-track\nswitching probability curve P(x) for\u000020 nm\u0014x\u00146 nm\nat a \fxed temperature Tpeak= 760 K is determined for\npure hard magnetic material (see Figure 2). The switch-\ning probability curve is \ftted with a Gaussian cumulative\nfunction\n\b\u0016;\u001b2=1\n2(1 + erf(x\u0000\u0016p\n2\u001b2))\u0001P (4)\nwith\nerf(x) =2p\u0019Zx\n0e\u0000\u001c2d\u001c; (5)\nwhere the standard deviation \u001b, the mean value \u0016and\nthe mean maximum switching probability P2[0;1] are\nthe \ftting parameters. The standard deviation \u001bdeter-\nmines the steepness of the transition function and is a\nmeasure for the transition jitter. In the further course\nit will be called \u001bdown:The \ftting parameter Pis a\nmeasure for the average switching probability for su\u000e-\nciently high temperatures. The resulting \ftting parame-\nters of the hard magnetic material can be seen in Table V.\nNote, that the calculated jitter values only consider the\ndown-track contribution of the write jitter. The so-called\na\u0000parameter is given by3\nCurie temp. TC[K] Damping\u000bUniaxial anisotropy. ku\n[J/link]Jij[J/link] \u0016s[\u0016B]\n693.5 0.02 9:124\u000210\u0000236:72\u000210\u0000211.6\nTABLE I. Material parameters of a FePt like hard magnetic granular recording medium.\n−15 −10 −5 000.20.40.60.81\ndown-track x[nm]switching probability\nHM\nFIG. 2. Down-track switching probability curve P(x) at a\npeak temperature Tpeak = 760 K for a pure hard magnetic\ngrain.\na=q\n\u001b2\ndown+\u001b2g (6)\nwhere\u001bgis a grain-size-dependent jitter contribution\n[31]. The write jitter can then be calculated by\n\u001bwrite\u0019ar\nS\nW(7)\nwhereWis the reader width and S=D+Bis the\ngrain diameter, i.e. the sum of the particle size Dand\nthe nonmagnetic boundary B[32; 33].\nB. Media Optimization\nTo \fnd the best soft magnetic material composition,\ndown-track switching probability curves P(x) similar to\nFigure 2 are computed for 50/50 bilayer structures with\ndi\u000berent damping constants \u000bSMand di\u000berent exchange\ninteractions Jij;SM. The range in which the parameters\nare varied can be seen in Table II. Note, that P(x) is\ncomputed at di\u000berent peak temperatures for the di\u000berent\nexchange interactions, since there holds\nJij=3kBTC\n\u000fz; (8)\nwherekBis the Boltzmann constant, z is the number\nof nearest neighbors and \u000fis a correction factor from the\nmean-\feld expression which is approximately 0.86 [23].\nThe temperature at which P(x) is calculated is chosen\nto beTC+ 60 K. The down-track switching probability\nFIG. 3. Down-track jitter \u001bdown as a function of the damp-\ning constant and the exchange interaction. The contour line\nindicates the transition between areas with down-track jitter\nlarger than 0.5 nm (light red, blue) and areas with down-track\njitter smaller than 0.5 nm (dark red).\ncurves are then \ftted with eq. (4). The down-track jitter\nparameters as a function of the damping constant and\nthe exchange interaction can be see in Figure 3. The\nmaximum switching probability is 1 for \u000b\u00150:1.\nFrom the simulations it can be seen that a Gilbert\ndamping\u000bSM= 0:1 together with Jij;SM= 7:72\u0002\n10\u000021J/link leads to the best results with the smallest\ndown-track jitter \u001bdown = 0:41 nm and a switching proa-\nbilityP= 1.\nThe last soft magnetic parameter that is varied, is the\nuniaxial anisotropy ku;SM. It is known that the small-\nest coercive \feld in an exchange spring medium can be\nachieved if KSM= 1=5KHM[34; 35]. Here\nKi=natku;i\na3i2fSM;HMg (9)\nare the macroscopic anisotropy constants in J/m3\nwith the unit cell size a= 0:24 nm and the number of\natomsnatper unit cell. ku;SMis varied between 0 and\n1=2ku;HM= 4:562\u000210\u000023J/link. The damping constant4\nParameter min. value max.value\n\u000bSM 0.02 0.5\nJij;SM[J/link] 5:72\u000210\u0000219:72\u000210\u000021\nku;SM[J/link] 0 1=2ku;HM= 4:562\u000210\u000023\nTABLE II. Range in which the di\u000berent soft magnetic material parameters are varied.\nku;SM\u000210\u000023[J/link]\u001bdown[nm]P\n0 0.41 1.0\n0:562 0.919 1.0\n1:8428 [= 1=5ku;HM] 1.04 1.0\n3:124 0.898 1.0\n4:562 [= 1=2ku;HM] 1.01 1.0\nTABLE III. Resulting down-track jitter parameters and mean maximum switching probability values for soft magnetic materials\nwith di\u000berent uniaxial anisotropy constants ku;SM.\nis\u000bSM= 0:1. The resulting \ftting parameters are sum-\nmarized in Table III. It can be seen that the switching\nprobability is one for all varied ku;SM. However, the\ndown-track jitter increases for higher ku;SM. Since for\nku;SM= 0 J/link the jitter is the smallest, this value is\nchosen for the optimal material composition.\nIn conclusion, the material parameters of the optimized\nsoft magnetic material composition can be seen in Ta-\nble IV.\nNext, simulations for di\u000berent ratios of hard and soft\nmagnetic material are performed. Down-track switching\nprobability curves P(x) are computed for di\u000berent ratios\natTpeak= 780 K and the down-track jitter and the mean\nmaximum switching probability are determined. The re-\nsults are listed in Table V.\nIt can be seen that a structure with 50% hard magnetic\nand 50% soft magnetic materials leads to the smallest\njitter and the highest switching probability. This result\ndi\u000bers from the optimized material composition in Ref.\n[21], where the optimal composition consists of 80% hard\nmagnetic and 20% soft magnetic materials. In Figure 4,\na switching probability phase diagram of the optimized\nbilayer structure with 50% hard and 50% soft magnetic\nmaterial can be seen.\nIt is visible that the switching probability of the\nstructure is larger than 99 :2% for a bigger area of down-\ntrack positions and peak temperatures. This shows the\nreduction of DC noise in the optimized structure.\nC. Areal Density\nTo analyse the possible increase of areal density by us-\ning the optimized bilayer structure instead of the pure\nhard magnetic recording medium, the signal-to-noise ra-\ntio is calculated. With the help of an analytical model of\na phase diagram developed by Slanovc et al [33] it is pos-\nsible to calculate a switching probability phase diagram\nfrom eight input parameters. The input parameters are\nPmax,\u001bdown;the o\u000b-track jitter \u001bo\u000b;the transition cur-\nvature, the bit length, the half maximum temperature\nand the position of the phase diagram in Tpeak direc-\ntion and the position of the phase diagram in down-track\ndirection. The \u001bdown andPmaxvalues are those result-\nFIG. 4. Switching probability phase diagram of recording\ngrain consisting of a composition of 50% hard magnetic ma-\nterial and 50% soft magnetic material with ku;SM= 0 J/link\nandJij;SM= 7:72\u000210\u000021J/link. The contour lines indicate\nthe transition between areas with switching probability less\nthan 1% (red) and areas with switching probability higher\nthan 99.2% (blue).\ning from the simulations for pure hard magnetic material\nand the optimized bilayer structure. All other model in-\nput parameters are obtained by a least square \ft from\na switching probability phase diagram computed with\na coarse-grained LLB model [36]. The phase diagram\nis mapped onto a granular recording medium where the\nswitching probability of the grain corresponds to its po-\nsition. The writing process is repeated for 50 di\u000berent\nrandomly initialized granular media. The SNR is then\ncomputed from the read-back process with the help of a\nSNR calculator provided by SEAGATE [37].\nThe SNR is analysed for areal densities of 2 to 5 Tb/in2.\nFor the bitsize ( bs) at a certain areal density, there are\ndi\u000berent track width and bit length combinations ( t;b)5\nDamping\u000bSMUniaxial anisotropy. ku\n[J/link]Jij[J/link] \u0016s[\u0016B]\n0.1 0 7:72\u000210\u0000211.6\nTABLE IV. Resulting material parameters for the optimal soft magnetic material composition.\nHM/SM\u001bdown[nm]P\nHM 0.974 0.95\n90/10 1.06 0.969\n80/20 0.813 0.998\n70/30 0.6 0.988\n60/40 0.8 0.999\n50/50 0.41 1.0\nTABLE V. Resulting down-track jitter parameters and mean maximum switching probability values for hard magnetic material\nand three di\u000berent hard/soft bilayer structures with di\u000berent damping constants in the soft magnetic material.\nthat yield\nbs=t\u0001b: (10)\nTo compute the SNR for a certain ( t;b) combination,\nthe reader was scaled in both the down-track and the o\u000b-\ntrack direction according to the bit length and the track\nwidth, respectively. The reader resolution Rin down-\ntrack direction is scaled by\nR=R0\u0001b\nb0(11)\nwherebis the bit length, R0= 13:26 nm is the initial\nreader resolution and b0= 10:2 nm denotes the mean ini-\ntial bit length according to ASTC. In o\u000b-track direction,\nthe reader width is scaled to the respective track width\nt. The initial track width is 44 :34 nm. In Figure 5(a) and\n(b) the SNR is shown as a function of the bit length and\nthe track width for pure hard magnetic material and the\noptimized bilayer structure, respectively. Additionally,\nthe phase plots include the SNR curves for ( t;b) combina-\ntions that yield areal densities from 2 to 5 Tb/in2. From\nthe phase diagram it is visible that higher SNR values can\nbe achieved for the optimized structures than for the pure\nhard magnetic material in the same bit length \u0000track\nwidth range. For example, the SNR for an areal density\nof 2 Tb/in2for the bilayer structure is larger than 15 dB\nwhereas it is between 10 dB and 15 dB for pure hard mag-\nnetic material. For each AD there is a ( t;b) combination\nfor which the SNR is maximal and which is marked by\na dot in the phase plot. In Figure 6 the maximum SNR\nover the areal density is displayed for both structures.\nThe results show that the SNR that can be achieved with\nthe optimized structure is around 2 db higher than that\nof the hard magnetic material, if the same areal density\nis assumed. To get the same SNR, the optimized design\nallows for an areal density that is 1 Tb/in2higher than\nfor the hard magnetic one. Summarizing, the bit length\n\u0000track width combinations at which the maximum SNR\nis achieved are given in Table VI.\nFIG. 5. Signal-to-noise ratio (in dB) as a function of the bit\nlength and the track width for (a) pure hard magnetic ma-\nterial and (b) the optimized hard/soft bilayer structure. The\nred lines indicate the bit length \u0000track width combinations\nthat yield 2, 3, 4 and 5 Tb/in2areal density. The dots indicate\nthe combination at which the SNR is maximal.\nIV. CONCLUSION\nTo conclude, we optimized a recording medium with\nhigh/lowTCgrains for heat-assisted magnetic record-\ning with a low Gilbert damping in the hard magnetic\npart\u000bHM= 0:02. The simulations for a cylindrical\nrecording grain with d= 5 nm and h= 8 nm were6\nAD [Tb/in2]Max. SNR [dB] (HM) x[nm] (HM) y[nm] (HM) Max. SNR [dB] (HM/SM) x[nm] (HM/SM) y[nm] (HM/SM)\n2 13.85 10.0 32.26 16.08 8.06 40.02\n3 11.07 6.23 34.52 13.37 5.37 37.53\n4 9.46 5.0 32.26 11.55 5.0 32.26\n5 7.16 4.3 30.01 9.16 4.69 27.51\nTABLE VI. Resulting bit length xand track width ycombinations for the maximum SNR at di\u000berent areal densities (AD) for\npure hard magnetic material (HM) and the optimized bilayer structure (HM/SM).\nFIG. 6. Maximum SNR for di\u000berent areal densities for pure\nhard magnetic material and the optimized bilayer structure.\nperformed with the atomistic simulation program VAM-\nPIRE. The damping constant of the soft magnetic mate-\nrial was assumed to be enhanced by doping the soft mag-\nnetic material with transition metals. The simulations\nshowed that larger damping constants lead to smaller jit-\nter and higher switching probabilities. A damping con-\nstant\u000bSM= 0:1, in combination with an exchange in-\nteractionJij;SM= 7:72\u000210\u000021J/link and an uniaxial\nanisotropy constant ku;SM= 0 J/link, led to the best re-\nsults in terms of small down-track jitter and high switch-\ning probability in a wide range of down-track and o\u000b-\ntrack positions. Interestingly, the soft magnetic com-\nposition is almost the same as for the structure with\n\u000bHM= 0:1 obtained in a previous work [21].\nIn further simulations the amount of hard and soft\nmagnetic material was varied. Surprisingly, the results\nshowed that a higher amount of soft magnetic material\nleads to smaller down-track jitter. This is not as expected\nsince for\u000bHM= 0:1 an increase of the soft magnetic ma-\nterial led to larger AC noise [21]. However, it can be\neasily explained why a higher amount of soft magnetic\nmaterial leads to better jitter results. Studying the in-\n\ruence of the damping constant on the down-track jitter\nshows that an increase of the damping constant from 0 :02\nto 0:1 reduces the down-track jitter by almost 30%. Ad-\nditionally,the maximum switching probability increases\nto 1. Since it can be seen that higher damping leads to\nsmaller jitter and higher maximum switching probability,\nit is reasonable that a higher amount of soft magnetic\nmaterial with \u000bSM= 0:1 leads to a better recording per-formance. In the former work the improved performance\ndue to higher damping was not an issue since the damp-\ning constant was 0.1 in both layers. This explains the\ndi\u000berent ratios of hard and soft magnetic material.\nFurthermore, we analyzed the increase of the areal den-\nsity can be improved if the optimized bilayer structure\nis used instead of pure hard magnetic recording mate-\nrial. This was done by analyzing the signal-to-noise ra-\ntio (SNR). The results showed that the areal density of\nthe optimized bilayer structure could be increased by\n1 Tb/in2to achieve the same SNR as for the pure hard\nmagnetic structure. In other words, that means that at\na certain areal density, the SNR was increased by 2 dB\nby using the optimized structure. Concluding, the opti-\nmized bilayer structure is a promising design to increase\nthe areal storage density by just modifying the recording\nmaterial.\nV. ACKNOWLEDGEMENTS\nThe authors would like to thank the Vienna Sci-\nence and Technology Fund (WWTF) under grant No.\nMA14-044, the Advanced Storage Technology Consor-\ntium (ASTC), and the Austrian Science Fund (FWF)\nunder grant No. I2214-N20 for \fnancial support. The\ncomputational results presented have been achieved us-\ning the Vienna Scienti\fc Cluster (VSC).\n1Hiroshi Kobayashi, Motoharu Tanaka, Hajime Machida, Takashi\nYano, and Uee Myong Hwang. Thermomagnetic recording .\nGoogle Patents, August 1984.\n2C. Mee and G. Fan. 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Noise reduction in heat-\nassisted magnetic recording of bit-patterned media by optimizing\na high/low Tc bilayer structure. Journal of Applied Physics ,\n122(21):213903, 2017.\n22ASTC jIDEMA. http://idema.org/?cat=10 .\n23Richard FL Evans, Weijia J. Fan, Phanwadee Chureemart,\nThomas A. Ostler, Matthew OA Ellis, and Roy W. Chantrell.\nAtomistic spin model simulations of magnetic nanomaterials.\nJournal of Physics: Condensed Matter , 26(10):103202, 2014.\n24Oleg N Mryasov, Ulrich Nowak, K Yu Guslienko, and Roy W\nChantrell. Temperature-dependent magnetic properties of fept:\nE\u000bective spin hamiltonian model. EPL (Europhysics Letters) ,\n69(5):805, 2005.\n25O Hovorka, S Devos, Q Coopman, WJ Fan, CJ Aas, RFL Evans,\nXi Chen, G Ju, and RW Chantrell. The curie temperature dis-\ntribution of fept granular magnetic recording media. Applied\nPhysics Letters , 101(5):052406, 2012.\n26W. Zhang, S. Jiang, P. K. J. Wong, L. Sun, Y. K. Wang, K. Wang,\nM. P. de Jong, W. 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E\u000bect of 3d, 4d, and 5d transition metal\ndoping on damping in permalloy thin \flms. Journal of Applied\nPhysics , 101(3):033911, February 2007.\n31Xiaobin Wang, Bogdan Valcu, and Nan-Hsiung Yeh. Transi-\ntion width limit in magnetic recording. Applied Physics Letters ,\n94(20):202508, 2009.\n32Gaspare Varvaro and Francesca Casoli. Ultra-High-Density Mag-\nnetic Recording: Storage Materials and Media Designs . CRC\nPress, March 2016.\n33Florian Slanovc, Christoph Vogler, Olivia Muthsam, and Dieter\nSuess. Systematic parameterization of heat-assisted magnetic\nrecording switching probabilities and the consequences for the\nresulting snr. arXiv preprint arXiv:1907.03884 , 2019.\n34F. B. Hagedorn. Analysis of ExchangeCoupled Magnetic Thin\nFilms. Journal of Applied Physics , 41(6):2491{2502, May 1970.\n35D. Suess. 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Rademacher‡\nDecember 12, 2019\nIn case of a spin-polarized current, the magnetization dynamics in n anowires\nare governed by the classical Landau-Lifschitz equation with Gilbertdamp-\ning term, augmented by a typically non-variational Slonczewski term. Tak-\ning axial symmetry into account, we study the existence of domain w all\ntype coherent structure solutions, with focus on one space dimen sion and\nspin-polarization, but our results also apply to vanishing spin-torqu e term.\nUsing methods from bifurcation theory for arbitrary constant ap plied fields,\nwe prove the existence of domain walls with non-trivial azimuthal pro file,\nreferred to as inhomogeneous . We present an apparently new type of do-\nmain wall, referred to as non-flat, whose approach of the axial magnetiza-\ntion has a certain oscillatory character. Additionally, we present th e leading\norder mechanism for the parameter selection of flatandnon-flat inhomoge-\nneous domain walls for an applied field below a threshold, which depends on\nanisotropy, damping, and spin-transfer. Moreover, numerical c ontinuation\nresults of all these domain wall solutions are presented.\n1 Introduction\nMagnetic domain walls (DWs) are of great interest both from a theor etical perspective\nand for applications, especially in the context of innovative magnetic storages [1]. Re-\ncent developments in controlled movement of DWs via spin-polarized c urrent pulses in\nnanomagnetic structures, in particular in nanowires, are thought to lead to a new class\nof potential non-volatile storage memories, e.g. racetrack memor y [1, 2, 3, 4]. These\ndevices make use of the fact that spin-transfer driven effects ca n change the dynamics\nin sufficiently small ferromagnetic structures (e.g. nanowires), wh ere regions of uniform\n∗Universit¨ at Bremen, lars.siemer@uni-bremen.de ; Corresponding author\n†Universit¨ at Hamburg, Lobachevsky State University of Nizhny No vgorod\n‡Universit¨ at Bremen\n1magnetization, separated by DWs, can appear [5, 6, 7]. This motivat es further studies of\nthe existence of magnetic domains and their interaction with spin-po larized currents as\na building block for the theory in this context. In this paper we take a mathematical per-\nspective and, in a model for nanomagnetic wires, rigorously study t he existence of DWs.\nThis led us to discover an apparently new kind of DWs with a certain inho mogeneous\nand oscillatory structure as explained in more detail below.\nThe description of magnetization dynamics in nanomagnetic structu res, governed by the\nLandau-Lifschitz-Gilbert (LLG) equation, is based on works by Berger and Slonczewski\nassuming a spin-polarized current [8, 9]. In the presence of a const ant applied field and\na spin-polarized current, the dynamics driven by the joint action of magnetic field and\nspin torque can be studied by adding a spin-transfer term in the dire ction of the current\n(current-perpendicular-to-plane (CPP) configuration). In cas e of a spatially uniform\nmagnetization, the resulting Landau-Lifschitz-Gilbert-Slonczewski (LLGS) equation for\nunit vector fields ( m1,m2,m3) =m=m(x,t)∈S2(cf. Figure 1) reads\n∂tm−αm×∂tm=−m×heff+m×(m×J). (LLGS)\nwith effective field heff, Gilbert damping factor α>0, and the last term is the so-called\npolarized spin transfer pseudotorque.\nNote that the above equation reduces to the LLG equation for J≡0, see§2 for more\ndetails.\nIn this paper we consider the axially symmetric case and set\nheff:=∂2\nxm+h−µm3e3,J:=β\n1+ccpm3e3, (1)\nwhereh=he3with a uniform and time-independent field strength h∈R, and m 3=\n/an}bracketle{tm,e3/an}bracketri}ht,e3∈S2. This effective field heffalso includes the diffusive exchange term ∂2\nxm,\nthe uniaxial anisotropy and demagnetization field. The specific here with parameter\nµ∈Rderives from a first order approximation in the thin film/wire limit for a u niformly\nmagnetized body [6, 10]. In the axially symmetric structure, β≥0 andccp∈(−1,1)\ndescribe the strength of the spin-transfer and the ratio of the p olarization [7, 11]. The\nspin-transfer torque term may provide energy to the system und er certain conditions\nand counterbalance dissipation associated to the Gilbert damping te rm, which gives rise\nto coherent non-variational dynamics, see e.g. [12].\nNotably, for β= 0 one obtains the LLG-equation that does not account for spin tr ansfer\neffects. Moreover, as shown in [12], this also holds up to parameter change in case\nccp= 0. Hence, solutions to the LLGS equation for β= 0 orccp= 0 are also solutions\nto the LLG equation, so that all the analytical as well as numerical r esults forccp= 0\nin this paper directly transfer to the LLG equation.\nA key ingredient for the separation of uniformly magnetized states in space are interfaces\nbetweentwomagneticdomains. Themostcoherentformofsuchint erfacesintheuniaxial\nsetting are relative equilibria with respect to translation and rotatio n symmetry of the\nform\nm(ξ,t) =m0(ξ)eiϕ(ξ,t),whereξ=x−standϕ(ξ,t):=φ(ξ)+Ωt,\n2(a)\n (b)-5 5-101\nxm1,m2\n(c)\nFigure 1: Homogeneous DW profile ( q≡0) withα= 0.5,β= 0.1,µ=−1,h= 50,ccp= 0.\n(a) (m2,m3)-profile. (b) Projection onto S2. (c) Zoom-in on m1(blue solid) and m2\n(red dashed).\nwithspeedsandfrequency Ω. Here the complex exponential acts on m0∈S2by\nrotation about the e3-axis, i.e., the azimuth, and in spherical coordinates we can choose\nm0(ξ) = (sin(θ(ξ)),0,cos(θ(ξ))) with altitude angle θ.\nWe refer to such solutions with m0(ξ)→ ±e3asξ→ ±∞orξ→ ∓∞asdomain walls .\nA first classification of DWs is based on the local wavenumber q:=φ′, which determines\nφuniquely due to the axial rotation symmetry and satisfies\nq(ξ) =/an}bracketle{t(m′\n1,m′\n2),(−m2,m1)/an}bracketri}ht\n1−m2\n3(ξ). (2)\nDefinition 1. We call a DW with constant φhomogeneous (hom) , i.e.,q≡0, and\ninhomogeneous otherwise.\nInhomogeneous DWs have a spatially inhomogeneous varying azimuth al angle, compare\nFigures 1 and 2.\nIn the case of uniaxial symmetry and the LLG case β= 0, an explicit family of homo-\ngeneous DWs was discovered in [13] for applied fields with arbitrary st rength and time\ndependence, cf. Figure 1. Furthermore, for constant applied fie lds and in case of ccp/ne}ationslash= 0\nit was shown in [12] that DWs cannot be homogeneous, and the existe nce of inhomoge-\nneous DWs was proven, whose spatial profile slowly converges to ±e3and where |s| ≫1.\nThis latter type of DWs is ‘weakly localized’ and has large ‘width’ in the se nse that the\ninverse slope of m3atm3(0) = 0 tends to infinity as |s| → ∞.\nAn apparently thus far unrecognized distinction of DWs is based on t he convergence\nbehavior of qasξ→ ±∞.\nDefinition 2. We call a DW flatif|q(ξ)|has a limit on R∪ {∞}as|ξ| → ∞and\nnon-flatotherwise.\nNote that homogeneous DWs are flat ones by definition (recall φ′=q). Moreover, for\nall DWsm0(ξ) converges to e3or−e3as|ξ| → ∞.\n3small applied field\n(a)large applied field\n(b)\n-5 5-101\nξm1\n(c)-5 5-101\nξm1,m2\n(d)\nFigure 2: Shown are profiles of inhomogeneous DWs m(ξ) computed by parameter continua-\ntion, cf.§4, inccptoccp= 0.5 with fixed α= 0.5,β= 0.1,µ=−1. (a,c) codim2case\nh= 0.5,s= 0.112027,Ω = 0.447173, (b,d) codim 0 case h= 50,s= 19.92,Ω = 40.4.\n(c) magnification of the m1-component; note the change of frequency for small vs.\nlargeξ. (d) Magnification of m1(blue solid) as well as m2(red dashed) component.\nThe main result of this paper is an essentially complete understanding of the existence\nand the type of DWs near the aforementioned explicit solution family f or a nanowire\ngeometry, i.e., µ <0. This includes the LLG case β·ccp= 0, but our focus is on the\nspintronic case β·ccp/ne}ationslash= 0 for which these results pertain 0 <|ccp| ≪1 and any value of\nthe (constant) applied field h.\nThe different types of DWs occur in parameter regimes close to ccp= 0 in the (spatial)\nODE which results from the coherent structure ansatz. Since the parameters αandµ\nare material-dependent we take the applied field strength has the primary parameter.\nIn brief, organized by stability properties of the steady states ±e3in the ODE, this leads\nto the following cases and existence results for localized DWs in nanow ires (µ<0):\n•‘codim-2’ (h∗<h<h∗) : existence of flat inhomogeneous DWs with s,Ω selected\nby the other parameters,\n•‘center’ (h=h∗orh=h∗) : existence of flat and non-flat inhomogeneous DWs,\n•‘codim-0’ (h<h∗orh>h∗) : existence of flat inhomogeneous DWs,\nwhereh∗:=β/α+2µ\nα2(1+α2) as well as h∗:=β/α−2µ\nα2(1+α2). Note that h∗<h∗always\n(recallα>0 andµ<0). Due to symmetry reasons, we mainly discuss the existence of\n4Figure 3: Stability diagram of homogeneous states ±e3inhandccpforα= 1,β= 0.5, and\nµ=−1. State + e3unstable to the left and stable to the right of Γ+,−e3stable\nto the left and unstable to the right of Γ−. Homogeneous DWs (hom) exist only on\ntheh-axis, i.e., ccp≡0. See text for further explanations. Note that also negativ e\napplied fields are shown.\nright-moving DWs close to the explicit solution family and thus focus on an applied field\nβ/α≤h(cf.§3). The main results can be directly transferred to the case of left -moving\nDWs (h≤β/α). Notably, the codim-0 case occurs for ‘large’ magnetic field habove a\nmaterial dependent threshold. In the center and codim-2 cases t here is a selection of s\nand Ω by the existence problem.\nThebasicrelationbetween thePDEandtheODEstabilitypropertiesw .r.t.handccpare\nillustrated in Figure 3 for α= 1,β= 0.5,µ=−1 fixed. Due to the fact that sand Ω are\nODE parameters only, the diagram illustrates a slice in the four dimens ional parameter\nspace with axes h,ccp,s, and Ω. Note that homogeneous DWs (hom) can occur only\non the line ccp≡0 (see [12, Theorem 5] for details). The stability regions are defined\nas follows. monostable−(blue): + e3unstable and −e3stable,bistable(shaded blue):\nboth +e3and−e3stable,monostable+(red): +e3stable and −e3unstable, unstable\n(shaded red): both + e3and−e3unstable. For a more detailed stability discussion, see\nRemark 5. Note that the transition from bistable to monostable in th e PDE does not\ncoincide with the transition, of the homogeneous family, from codim- 2 to codim-0 in\nthe ODE. In contrast, the analogous transitions occur simultaneo usly for example in the\nwell-known Allen-Cahn orNagumo equation.\nIn Figure 4 below, we present numerical evidence that inhomogeneo us DWs are indeed\nalso dynamically selected states, especially for large applied fields, als o in the LLG case\n(β,ccp= 0).\nThe understanding of DW selection by stability properties generally d epends on the exis-\ntence problem discussed in this paper, which is therefore a prerequ isite for the dynamical\nselection problem, cf. Remark 5.\n5(a)\n (b)\n(c)\nFigure 4: Direct simulation of full PDE (LLGS) for α= 0.5,β= 0.1,µ=−1,h= 50, and\nccp= 0 with dynamical selection of an inhomogeneous DW. Initial condition near\nhomogeneous DW (9) in codim-0 regime ( h∗= 10.2,s0= 19.92, and Ω 0= 40.04).\n(a) Profile at t= 20 projected onto the sphere. (b) Speed and frequency of DW\nover time with asymptotic (selected) values s= 12.5 and Ω = 78 .28. (c) Space-time\nplots of DW components (without co-moving frame), range as i n black box in (b).\nFinal profile is heteroclinic connection in (7), cf. Proposi tion 2.\nTo our knowledge, existence results of DWs for ccp/ne}ationslash= 0 are new. In more detail, the\nexistence of localised inhomogeneous, i.e. flat as well as non-flat, DW s forccp/ne}ationslash= 0 and\nespecially for ccp= 0 are new results. Indeed, the existence proof of non-flat DWs is\nthe most technical result and entails an existence proof of hetero clinic orbits in an ODE\nbetween an equilibrium and a periodic orbit. These solutions indicate th e presence of\nDWs in other regimes of spin driven phenomena and may be of interest for spin-torque\ntransfer MRAM (Magnetoresistive random-access memory) syst ems [14].\nThis paper is organized as follows. In §2, the LLGS equation and coherent structures as\nwell as first properties are discussed. Section 3 more precisely intr oduces homogeneous\nand inhomogeneous as well as flat and non-flat DWs and it also includes the main\nresults of this paper (Theorem 1, 2, and 3). The technical proofs of Theorem 2 as\nwell as Theorem 3 are deferred to Appendix 6.1 and 6.2. Section 4 pre sents results of\n6numerical continuation in parameter ccpfor the three regimes of the applied field (codim-\n2, center, and codim-0), where the center case is studied in more d etail. We conclude\nwith discussion and outlook in §5.\nAcknowledgements\nL.S. and J.R. acknowledge support by the Deutsche Forschungsge meinschaft (DFG, Ger-\nmanResearchFoundation)-Projektnummer 281474342/GRK222 4/1. J.R.alsoacknowl-\nedges support by DFG grant Ra 2788/1-1. I.O. acknowledges fund ing of a previous\nposition by Uni Bremen, where most of this paper was written, as we ll as support by\nthe recent Russian Scientific Foundation grant 19-11-00280.\n2 Model equations and coherent structure form\nThe classical model for magnetization dynamics was proposed by La ndau and Lifschitz\nbased on gyromagnetic precession, and later modified by Gilbert [15, 16]. See [17] for\nan overview. The Landau-Lifschitz-Gilbert equation for unit vector fields m(x,t)∈S2\nin one space dimension x∈Rand in terms of normalized time in dimensionless form is\n∂tm−αm×∂tm=−m×heff. (LLG)\nHerem=M/MSrepresents thenormalizedmagnetization, heff=Heff/MStheeffective\nfield, i.e.thenegativevariationalderivativeofthetotalmagneticf reeenergywithrespect\ntom, both normalized by the spontaneous magnetization MS. For gyromagnetic ratio\nγand saturation magnetization MSthe time is measured in units of ( γMS)−1, and it\nis assumed that the temperature of the magnetic body is constant and below the Curie\ntemperature [5]. Finally, Gilbertdampingα>0 turnsmtowardsheffand both vectors\nare parallel in the static solution.\nIn modern spin-tronic applications, e.g. Spin-Transfer Torque Mag netoresistive Random\nAccess Memories (MRAM), the spin of electrons is flipped using a spin- polarized current.\nTo take these effects into account, the LLG equation is supplement ed by an additional\nspin transfer torque term. Using a semiclassical approach, Sloncz ewski derived an ex-\ntended effective field\nHeff=heff−m×J,\nwhereJ=J(m) depends on the magnetization and the second term is usually called\nSlonczewski term [9]. In contrast to the LLG equation, which can be written as the\ngradient of free ferromagnetic energy, this generalized form is no longer variational and\nthe energy is no longer a Lyapunov functional.\nAs to the specific form of Heff, including a leading order form of exchange interaction,\nuniaxial crystal anisotropy in direction e3, andZeemanas well as stray-field interactions\nwith an external magnetic field, see e.g. [6], gives the well known form (1).\nIn this paper we consider a constant applied magnetic field h∈Ralonge3and uniaxial\nanisotropy with parameter µ∈R, for which the anisotropy energy density is rotationally\n7symmetric w.r.t. e3. According to the energetically preferred direction in the uniaxial\ncase, minima of the anisotropy energy density correspond to easydirections, whereas\nsaddles or maxima correspond to medium-hard orharddirections, respectively. There-\nfore, one refers to µ <0 aseasy-axis anisotropy and µ >0 aseasy-plane , both with\nregard to e3.\nAs mentioned before, the LLG equation with its variational structu re appears as a\nspecial case of (LLGS) for β= 0 orccp= 0. While our main focus is the non-variational\nspintronic case β·ccp/ne}ationslash= 0, all results contain the case β·ccp= 0 and thus carry over\nto (LLG).\nIt iswell-known that(LLGS) admitsanequivalent formasanexplicit ev olution equation\nof quasilinear parabolic type in the form, see e.g. [12],\n∂tm=∂x(A(m)∂xm)+B(m,∂xm).\nAs a starting point, we briefly note the existence of spatially homoge neous equilibrium\nsolutions of (LLGS) for which m(x,t) is constant in xandt.\nRemark 1. The only (spatially)homogeneous equilibria of (LLGS)forβ >0are the\nconstant up- and down magnetization states ±e3. Indeed, setting ∂tm=∂2\nxm= 0in\n(LLGS), forβ/ne}ationslash= 0the last equation implies that m1=m2= 0and thus the only solutions\nm∗\n±∈S2arem∗\n±= (0,0,±1)T.\nRemark 2. In caseβ= 0as well as |h/µ|<1there exist a family of additional\nhomogeneous solutions of (LLGS) given by m∗= (m1,m2,h/µ)T,withm2\n1+m2\n2=\n1−(h/µ)2. Note that similar cases occur for symmetry axis being e1ande2, respectively\n(cf. Brown’s equations ).\n2.1 Coherent structure ODE\nDuetotherotationsymmetryaroundthe e3-axisof (LLGS), itisnaturaltousespherical\ncoordinates\nm=\ncos(ϕ)sin(θ)\nsin(ϕ)sin(θ)\ncos(θ)\n,\nwhereϕ=ϕ(x,t) andθ=θ(x,t). This changes (LLGS) to\n/parenleftbigg\nα−1\n1α/parenrightbigg/parenleftbigg\n∂tϕsin(θ)\n−∂tθ/parenrightbigg\n=/parenleftbigg\n2∂xϕ∂xθcos(θ)\n−∂2\nxθ/parenrightbigg\n+sin(θ)/parenleftbigg\n∂2\nxϕ+β/(1+ccpcos(θ))\n(∂xϕ)2cos(θ)+h−µcos(θ)/parenrightbigg (3)\nNote that the rotation symmetry has turned into the shift symmet ry in the azimutal\nangleϕ, as (3) depends on derivatives of ϕonly.\n8Recall that DW solutions spatially connect the up and down magnetiza tion states ±e3\nin a coherent way as relative equilibria with respect to the translation symmetry in x\nandφ, which yields the ansatz\nξ:=x−st, θ=θ(ξ), ϕ=φ(ξ)+Ωt. (4)\nSuch solutions are generalized travelling waves that move with const ant speeds∈R\nin space and rotate pointwise with a constant frequency Ω ∈Raround the e3-axis;\nsolutions with Ω = 0 are classical travelling waves.\nAs in [12], applying ansatz (4) to (3) leads to the so-called coherent structure ODE\n/parenleftbiggα−1\n1α/parenrightbigg/parenleftbigg(Ω−sφ′)sin(θ)\nsθ′/parenrightbigg\n=/parenleftbigg2φ′θ′cos(θ)\n−θ′′/parenrightbigg\n+sin(θ)/parenleftbiggφ′′+β/(1+ccpcos(θ))\n(φ′)2cos(θ)+h−µcos(θ)/parenrightbigg\n,(5)\nwhere′=d/dξ. This system of two second-order ODEs does not depend on φand thus\nreduces to three dynamical variables ( θ,ψ=θ′,q=φ′). Following standard terminology\nfor coherent structures, we refer to qas thelocal wavenumber .\nWriting (5) as a first-order three-dimensional system gives\nθ′=ψ\nψ′= sin(θ)[h−Ω+sq+(q2−µ)cos(θ)]−αsψ\nq′=αΩ−β/(1+ccpcos(θ))−αsq−s+2qcos(θ)\nsin(θ)ψ, (6)\nand DWs in the original PDE are in 1-to-1-correspondence with the O DE solutions\nconnecting θ= 0 andθ=π.\n2.1.1 Blow-up charts and asymptotic states\nAs in [12], the singularities at zeros of sin( θ) in (6) can be removed by the singular\ncoordinate change ψ:=psin(θ), which is a blow-up transformation mapping the poles\nof the sphere ±e3to circles thus creating a cylinder. The resulting desingularized system\nreads\nθ′= sin(θ)p\np′=h−Ω−αsp+sq−(p2−q2+µ)cos(θ)\nq′=αΩ−β/(1+ccpcos(θ))−sp−αsq−2pqcos(θ).(7)\nThe coherent structure system (6) is equivalent to the desingular ized system (7) for\nθ/ne}ationslash=nπ,n∈Zand therefore also for maway from ±e3. Furthermore, the planar blow-\nup chartsθ= 0 andθ=πare invariant sets of (7), which are mapped to the single\npointse3and−e3, respectively by the blow-down transformation. System (7) has a\nspecial structure (cf. Figure 6) that will be relevant for the subs equent DW analysis. In\nthe remainder of this section we analyze this in some detail.\n9Lemma 1. Consider the equations for pandqin(7)for an artificially fixed value of θ.\nIn terms of z:=p+iqthis subsystem can be written as the complex (scalar) ODE\nz′=Az2+Bz+C, (8)\nwhereA:=−cos(θ),B:=−(α+i)s,andCθ:=h−Ω+Aµ+i/parenleftig\nαΩ−β\n1−Accp/parenrightig\n.\nForA/ne}ationslash= 0the solution with z0=z(ξ0)away from the equilibria zθ\n+=−B\n2A+ iγθ\n2Aand\nzθ\n−=−B\n2A−iγθ\n2A, withγθ=γ(θ):=√\n4ACθ−B2, reads\nz(ξ) =γθ\n2Atan/parenleftbiggγθ\n2ξ+δ0/parenrightbigg\n−B\n2A, (9)\nwhere\nδ0= arctan/parenleftbigg2Az0+B\nγθ/parenrightbigg\n−γθ\n2ξ0.\nForA= 0, the solution away from the equilibrium zπ/2=−Cπ/2/Bis given by\nz(ξ) =/parenleftbigg\nz0+Cπ/2\nB/parenrightbigg\neB(ξ−ξ0)−Cπ/2\nB.\nClearly, the solution of (8) relates only to those solutions of (7) for whichθis constant,\ni.e.,θ= 0,π. Although we are mostly interested in the dynamics on the blow-up ch arts,\nwe consider θas a parameter in order to demonstrate the special behaviour of ( 7) forθ\nartificially fixed. Notably, the equilibria zθ\n±of (8) forθ/ne}ationslash= 0,πare not equilibria in the\nfull dynamics, due to the fact that (7) is only invariant for θat the blow-up charts.\nProof.We readily verify the claimed form of the ODE and directly check the cla imed\nsolutions. /squaresolid\nRemark 3. Lemma 1 states in particular that the desingularized ODE sys tem(7)can\nbe solved explicitly on the invariant blow-up charts, where θ= 0,πand thusA=−1,1,\nrespectively. System (7)possesses two real equilibria on each blow-up chart, Z0\n±:=\n(0,p0\n±,q0\n±)TandZπ\n±:= (π,pπ\n±,qπ\n±)T. Herepθ\nσ:= Re(zθ\nσ),qθ\nσ:= Im(zθ\nσ)forθ= 0,π,\nσ=±and\nz0\n+:= 1/2(B−iγ0), z0\n−:= 1/2(B+iγ0)\nand analogously\nzπ\n+:= 1/2(−B+iγπ), zπ\n−:= 1/2(−B−iγπ),\nwhere we set\nγ0:=γ/vextendsingle/vextendsingle\nA=−1=√\n−4C0−B2andγπ:=γ/vextendsingle/vextendsingle\nA=1=√\n4Cπ−B2\nwithC0:=C/vextendsingle/vextendsingle\nA=−1andCπ:=C/vextendsingle/vextendsingle\nA=1.\nDue to the analytic solution (9), we obtain the following more detailed r esult in case\nθ/ne}ationslash=π/2 (cf. Figure 6).\n10Lemma 2. For each given 0≤θ≤πwithθ/ne}ationslash=π/2as a parameter, the fibers of (7)with\nconstantθconsists entirely of heteroclinic orbits between zθ\n−andzθ\n+in caseIm(γθ)/ne}ationslash= 0,\norγθ/ne}ationslash= 0, except for the equilibrium states. In case Im(γθ) = 0andRe(γθ)/ne}ationslash= 0, the\nfiber at fixed θis filled with periodic orbits away from the invariant line {q=s\n2A}, for\nwhich the period of solutions close to it tends to infinity.\nProof.Forθfixed in (8), consider the case Re( γθ) = 0 and also Im( γθ)/ne}ationslash= 0 forA/ne}ationslash= 0\nwhich leads to\nz(ξ) = iIm(γθ)\n2A·tan/parenleftbigg\ni/parenleftigIm(γθ)\n2ξ+Im(δ0)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:ˇξ/parenrightig\n+Re(δ0)/parenrightbigg\n−B\n2A\n=Im(γθ)\n2A·isin(2Re(δ0))−sinh/parenleftbig\n2ˇξ/parenrightbig\ncos(2Re(δ0))+cosh/parenleftbig\n2ˇξ/parenrightbig−B\n2A.\nFor Re(γθ)/ne}ationslash= 0 as well as Im( γθ)/ne}ationslash= 0, we obtain\nz(ξ) =γθ\n2Atan/parenleftbiggRe(γθ)\n2ξ+Re(δ0)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:˜ξ+iIm(γθ)\n2ξ+iIm(δ0)/parenrightbigg\n−B\n2A\n=γθ\n2A·sin(2˜ξ)+isinh/parenleftig\n2/parenleftig\nIm(γθ)\nRe(γθ)˜ξ−Im(γθ)\nRe(γθ)Re(δ0)+Im(δ0)/parenrightig/parenrightig\ncos(2˜ξ)+cosh/parenleftig\n2/parenleftig\nIm(γθ)\nRe(γθ)˜ξ−Im(γθ)\nRe(γθ)Re(δ0)+Im(δ0)/parenrightig/parenrightig−B\n2A,\nThe asymptotic states are\nIm/parenleftbig\nγθ/parenrightbig\n>0 : lim\nξ→−∞z(ξ) =−iγθ\n2A−B\n2A,lim\nξ→+∞z(ξ) = iγθ\n2A−B\n2A,\nas well as\nIm/parenleftbig\nγθ/parenrightbig\n<0 : lim\nξ→−∞z(ξ) = iγθ\n2A−B\n2A,lim\nξ→+∞z(ξ) =−iγθ\n2A−B\n2A,\nwhich simplify in case Re( γθ) = 0 to\nIm/parenleftbig\nγθ/parenrightbig\n>0 : lim\nξ→−∞z(ξ) =Im(γθ)−B\n2A,lim\nξ→+∞z(ξ) =−Im(γθ)+B\n2A,\nas well as\nIm/parenleftbig\nγθ/parenrightbig\n<0 : lim\nξ→−∞z(ξ) =−Im(γθ)+B\n2A,lim\nξ→+∞z(ξ) =Im(γθ)−B\n2A.\nNote that the asymptotic states coincide if γθ= 0.\n11The last case to consider is Re/parenleftbig\nγθ/parenrightbig\n/ne}ationslash= 0 and Im/parenleftbig\nγθ/parenrightbig\n= 0, where the solutions are\nz(ξ) =Re/parenleftbig\nγθ/parenrightbig\n2A·sin(2ˆξ)+isinh(2Im( δ0))\ncos(2ˆξ)+cosh(2Im( δ0))−B\n2A,\nwithˆξ:=Re(γθ)\n2ξ+Re(δ0) and which leads to periodic solutions of (8) iff\nIm(δ0)/ne}ationslash= 0⇔2AIm(z0)+Im(B)/ne}ationslash= 0⇔q0/ne}ationslash=s\n2A,\nwherez0=p0+iq0and recall that B=−(α+i)s. /squaresolid\nBased on Lemma 2, explicitly onthe blow-up chart θ= 0 the heteroclinic orbits are from\nz0\n−toz0\n+in case Im( −4C0−B2)>0, or Im( −4C0−B2) = 0 and Re( −4C0−B2)≤0,\nand fromz0\n+toz0\n−if Im(−4C0−B2)<0 . Forθ=π, if Im(4Cπ−B2)>0, or\nIm(4Cπ−B2) = 0 and Re(4 Cπ−B2)≤0 they are connections from zπ\n−tozπ\n+, and if\nIm(4Cπ−B2)<0 fromzπ\n+tozπ\n−.\nForA/ne}ationslash= 0, the case s= 0 is a special situation, which will be also discussed in the\ncontext of DWs in §3 later. It turns out that on the blow-up charts θ= 0 (orθ=π),\nthe solution with appropriate initial conditions has a limit as |ξ| → ∞if and only if\nIm(√\n−C0)/ne}ationslash= 0 (Im(√\nCπ)/ne}ationslash= 0). In terms of the parameters in (7) and with\nβ−:=β\n1−ccpandβ+:=β\n1+ccp,\nthis leads to the conditions for θ= 0 given by:\nΩ/ne}ationslash=β+\nαor Ω =β+\nαand Ω≤h−µ, (10)\nand forθ=πgiven by:\nΩ/ne}ationslash=β−\nαor Ω =β−\nαand Ω≥h+µ, (11)\nIn caseccp= 0, i.e. for the LLG equation, the conditions in (10) and (11) reduce to\nΩ/ne}ationslash=β\nαor Ω =β\nαand 2µ≤β\nα,\nwhere the latter inequality always holds in case of a nanowire geometr y (µ<0). Hence\nstanding domain walls in nanowires in case ccp= 0 can only connect equilibria, if they\nexist.\nLemma 2 also states that the equilibria on the blow-up charts θ∈ {0,π}are surrounded\nby periodic orbits in case Im( γ0) = 0 and Re( γ0)/ne}ationslash= 0 (Im(γπ) = 0 and Re( γπ)/ne}ationslash= 0).\nIn fact, system (7) is Hamiltonian (up to rescaling) on the blow-up ch arts for certain\nfrequencies Ω, as follows\n122p\n-11q\n(a)-5 5ξ\n-1\n2\np \u0000q\n(b)\nFigure 5: (a) Phase plane streamplot with Mathematica of (14) around the equilibrium zπ\n−,\ni.e., (7) at θ=π, forα= 0.5,β= 0.1,µ=−1,h= 10.2,s= 4,Ω = 8.2 andccp= 0,\nwhich leads to/parenleftbig\npπ\n−,qπ\n−/parenrightbigT= (1,0)T. The red solid line marks the trajectory with\ninitial condition ( p0,q0) = (7/4,0) (cf. plot of solutions in b). (b) Plot of the profile\nfor the solution highlighted in (a), where p(solid blue line) and q(dashed red line)\nare given by (9) for the parameter set as in (a).\nProposition 1. The dynamics of (7)on the invariant blow-up chart θ= 0in case\nΩ =β+\nα−s2\n2possesses the invariant line {q=−s\n2}and, after time-rescaling, for q/ne}ationslash=−s\n2\nthe Hamiltonian\nH0(p,q) =−p2+q2+αsp+sq−h+β+/α+µ\nq+s\n2\nalong solutions of (8). Analogously on the chart θ=π, in case\nΩ =β−\nα+s2\n2(12)\npossesses the invariant line {q=s\n2}and forq/ne}ationslash=s/2the Hamiltonian\nHπ(p,q) =p2+q2−αsp−sq+h−β−/α+µ\nq−s\n2. (13)\nMoreover, each half plane {θ= 0,q≤ −s\n2},{θ= 0,q≥ −s\n2}/parenleftbig\n{θ=π,q≤s\n2}and\n{θ=π,q≥s\n2}/parenrightbig\nis filled with periodic orbits encircling the equilibria at z0\n±/parenleftbig\nzπ\n±/parenrightbig\nif addi-\ntionallyΩ>h−µ+s2\n4(α2−1)/parenleftig\nΩ<h+µ+s2\n4(1+α2)/parenrightig\n.\nProof.For the sake of clarity, we will only present the computation for the blow-up chart\nθ=π; the computation on θ= 0 can be done in the same manner. With respect to the\nparameters of equation (7), the condition Im(4 C−B2) = 0 is equivalent to Ω =β−\nα+s2\n2\nand in this case the system (7) on {θ=π}is given by\np′=p2−q2−αsp+sq+h+µ−β−\nα−s2\n2\nq′= 2pq−sp−αsq+αs2\n2(14)\n13We readily compute that for solutions of this\ndHπ\ndξ=∂Hπ\n∂pp′+∂Hπ\n∂qq′=q′·p′−p′·q′\n(q−s\n2)2= 0,\nwhich shows the canonical Hamiltonian structure of (14) up to time r escaling. If addi-\ntionally Ω< h+µ+s2\n4(1+α2), it follows from Lemma 2 that each half plane is filled\nwith periodic orbits. /squaresolid\nProposition 1 concerns the special case that ccp∈(−1,1) andβ,Ω are such that (12)\nholds, whichishenceforthreferredtoasthe center-case . Inparticular, eachorbitexcept\nthe lineq≡s/2 on the blow-up chart θ=πcan by identified via the quantity (13),\nand each equilibrium zπ\n±has a neighborhood filled with periodic orbits if additionally\nh >β−\nα−µ+s2\n4(1−α2) (cf. Figure 5). Note the relation between the conditions (10)\nand (11) and the conditions in Proposition 1 in case s= 0.\nBased on Lemma 2, we also state the following uniqueness result.\nProposition 2. ForΩ<β+\nα−s2\n2/bracketleftig\nΩ>β+\nα−s2\n2/bracketrightig\n, orΩ =β+\nα−s2\n2andΩ≤h−µ+\ns2\n4(α2−1)there is a unique orbit with (θ,p,q)T(ξ)withθ(ξ)→0asξ→ −∞, and it\nholds that (p+iq)(ξ)→z0\n−/bracketleftbig\n(p+iq)(ξ)→z0\n+/bracketrightbig\nasξ→ −∞.\nProof.The conditions on Ω are equivalent to those in Lemma 2. If the statem ent were\nfalse, it nevertheless follows from Lemma 2 that ( p+iq)(ξ)→z0\n−asξ→ −∞. However,\ntransverse to the blow-up chart, the equilibrium state Z0\n−is stable for increasing ξand\nthus repelling for decreasing ξ. This contradicts the requirement θ(ξ)→0 asξ→ −∞.\nTogether with the fact that Z0\n−has a one-dimensional unstable manifold uniqueness\nfollows. Analogously in case Ω >β+\nα−s2\n2. /squaresolid\nDomain walls are heteroclinic orbits between the blow-up charts and d ecisive for their\nbifurcation structure are the dimensions (and directions) of un/s table manifolds of the\nequilibria on these charts. Hence, we next discuss the equilibria Z0\n±andZπ\n±and their\nstability.\nTransverse to the blow-up charts in θ-direction we readily compute the linearization\n∂θ(sin(θ)p) = cos(θ)p, i.e., the transverse eigenvalue is −Aθ·Re(zθ\n±) atθ= 0 andπ,\nrespectively. The eigenvalues within the blow-up charts are determ ined by±iγ. With\nσ=±, respectively, the eigenvalues for Z0\nσare\nν0\n1,σ=−σiγ0, ν0\n2,σ=ν0\n1,σ, ν0\n3,σ= Re(z0\nσ) (15)\nand forZπ\nσ\nνπ\n1,σ=σiγπ, νπ\n2,σ=νπ\n1,σ, νπ\n3,σ=−Re(zπ\nσ). (16)\nTherefore, the signs of the real parts within each blow-up chart a re opposite at Zπ\n+\ncompared to Zπ\n−and determined by the sign of Re/parenleftbig\nν0,π\n1,+/parenrightbig\n. Hence, within the blow-up\ncharts each equilibrium is either two-dimensionally stable, unstable or a linearly neutral\ncenter point.\n142p\n-11q\n(a)2p\n-11q\n(b)2p\n-11q\n(c)\n2p\n-11q\n(d)2p\n-11q\n(e)2p\n-11q\n(f)\nFigure 6: Phase plane streamplots (with Mathematica ) in blow-up charts near the equilib-\nriumzπ\n−= (1,0) forα= 0.5,β= 0.1,µ=−1,ccp= 0, i.e., the second and third\nequation of (7). (a-c) θ= 0 and (d-f) θ=π. (a,d) codim-2 regime, (b,e) center\ncase, where Ω = β/α+s2/2 holds on the chart θ=π, and (c,f) codim-0 regime.\nThe remaining parameters and equilibria in (a,d): h= 0.5,s0= 0.12, Ω0= 0.44,\nandz0\n+=−1.06−0.12i,zπ\n+=−0.94+0.12i. In (b,e): h= 10.2,s0= 4, Ω 0= 8.2,\nandz0\n+=−3−4i,zπ\n+= 1 + 4i. In (c,f): h= 50.0,s0= 19.92, Ω0= 40.04, and\nz0\n+=−10.96−19.92i,zπ\n+= 8.96+19.92i.\nFor completeness, we next notethat the equilibria onbothblow-up c harts can beneutral\ncenters simultaneously (cf. Figure 3). However, this requires a ne gative spin polarization\nand a small Gilbert damping factor, and is not further studied in this p aper.\nRemark 4. The equilibria of both blow-up charts are centers simultane ously, if and only\nifIm(±γ0,π) = 0andγ0,π/ne}ationslash= 0(compare Lemma 2). For example if α= 0.5,β= 0.1,µ=\n−1,ccp=−0.99,h= 10\ns2=3960\n199,Ω =β/α\n1−ccp+s2\n2=2000\n199,\nwe obtain\nγ0= 3.33551, γπ= 3.27469.\n3 Domain Walls\nAll domain walls between ±e3that we are aware of are of coherent structure type,\nand thus in one-to-one correspondence to heteroclinic connectio ns between the blow-up\n15charts{θ= 0}and{θ=π}in(7). Typically we expect these to beheteroclinics between\nequilibria within the charts, but this is not necessary. Based on the p revious analysis,\nthere are three options for heteroclinics between the charts: po int-to-point, point-to-\ncycle, and cycle-to-cycle. We study the first two in this section, fo r which Proposition 2\nimplies uniqueness oftheDW(uptotranslations/rotations) foragiv enset ofparameters.\nThe third case can occur at most in a relatively small set of paramete rs (see Remark 4).\nIts analysis is beyond the scope of this paper.\nNote that in case of an existing connection between an equilibrium and a periodic orbit\n(see Proposition 1), the domain wall is automatically an inhomogeneou s non-flat one.\nMoreover, via the singular coordinate change any such heteroclinic solution is hetero-\nclinic between θ= 0,πin (6) and through the spherical coordinates it is a heteroclinic\nconnection between ±e3in the sphere, possibly with unbounded ϕ.\n3.1 Homogeneous Domain Walls\nIt is known from [13] in case β= 0 and from [12] in case ccp= 0 (and arbitrary β)\nthat (7) admits for µ<0 a family of explicit homogeneous DWs m0given by\n\nθ0\np0\nq0\n=\n2arctan/parenleftbig\neσ√−µξ/parenrightbig\nσ√−µ\n0\n (17)\nand parameterized by Ω =h+αβ\n1+α2,s2=−(β−αh)2\nµ(1+α2)2>0, andσ= 1 for positive speed s\nandσ=−1 for negative s; the family extends to s= 0 in the limit h→β\nαwith scaling\nof the frequency by Ω =β\nα+√−µ\nαs. Fors= 0 (standing) fronts with both orientations\nexist simultaneously ccp= 0 and are given by\n\nθ0\np0\nq0\n=\n2arctan/parenleftbig\ne±√−µξ/parenrightbig\n±√−µ\n0\n.\nHence, the branches of left and right moving walls as parametrized b yseach have\ntermination point at s= 0 (cf. Figure 13).\nThe family of explicit DWs (17) have domain wall width√−µ, a profile independent\nof the applied field hand propagate along a nanowire ( µ <0) with velocity swhile\nprecessing with azimuthal velocity Ω. Since these are unique up to sp atial reflection\nsymmetry, the direction of motion is related to the spatial direction of connecting ±e3\nthroughσ,\nθ(−∞) = 0θ(+∞) =π⇔s>0 (wall moves to the right)\nθ(−∞) =π θ(+∞) = 0⇔s<0 (wall moves to the left) .(18)\nTo simplify some notation we will focus on the case of right-moving walls including\nstanding walls ( s≥0) and thus make the standing assumptions that h≥β/αas well as\n16µ <0. We therefore have a 1-to-1 relation of parameters ( α,β,h,µ) and right-moving\nDWs from\nm(ξ,t) =m0(ξ,t;α,β,h,µ)\nwith speed and frequency given by\ns0=s0(α,β,h,µ) :=αh−β√−µ(1+α2),Ω0= Ω0(α,β,h,µ) :=h+αβ\n1+α2(19)\nwhere the subindex 0 emphasizes that ccp= 0. Sinces0is surjective on R≥0any velocity\ncan be realised. Spatial reflection covers the case h≤β/α.\nBased on Lemma 1 as well as Remark 3 for ccp= 0 and (homogeneous) speed and\nfrequency (19), one readily finds the asymptotic states of (7) giv en by\nE0:=Z0\n−/vextendsingle/vextendsingle\n(s0,Ω0)=/parenleftbig\n0,√−µ,0/parenrightbigTandEπ:=Zπ\n−/vextendsingle/vextendsingle\n(s0,Ω0)=/parenleftbig\nπ,√−µ,0/parenrightbigT,\nwith (spatial) eigenvalues (15), (16) given by\nν0\nk,−:=−αs0−2√−µ−(−1)kis0, ν0\n3,−=√−µ,\nνπ\nk,−:=−αs0+2√−µ−(−1)kis0, νπ\n3,−=−√−µ,(20)\nwherek= 1,2. Note that the above equilibria cannot be centers simultaneously ( recall\nµ <0), hence a cycle-to-cycle connection can not exist close to it (see Remark 4 for\ndetails). For this reason, we focus on point-to-point as well as poin t-to-cycle connections.\n3.2 Inhomogeneous Domain Walls\nHomogeneous DWs exist only in case ccp= 0 [12, Theorem 5], are explicitly given\nby (17) and completely characterized by (19). By [12, Theorem 6], f ast inhomogeneous\nDW solutions with |s| ≫1 exist for any ccp∈(−1,1), but in contrast to (17), the\ngradient of these profiles is of order 1 /|s|and thus have a large ‘width’. The natural\nquestion arises what happens for any sin caseccp/ne}ationslash= 0.\nThis section contains the main results of this paper: the existence, parameter selection\nand structure of inhomogeneous DW solutions in case of small |ccp|for any value of the\napplies field h, and thus any speed s. This will be achieved by perturbing away from the\nexplicit solution m0given by (17), where the bifurcation structure is largely determine d\nby comparing the dimensions of the un/stable eigenspaces at the as ymptotic equilibrium\nstates, which are determined by (20).\nLetW0\ns/uandWπ\ns/udenote the stable and unstable manifolds associated to E0and\nEπ, respectively, and w0\ns/uas well aswπ\ns/ube the dimension of these manifolds so that\nw0\ns+w0\nu=wπ\ns+wπ\nu= 3. Notably w0\ns= 2 andw0\nu= 1 for all values of the parameters,\nandwπ\nsis either 1 or 3. Recall the standing assumption s0≥0.\nIfwπ\ns= 1, the heteroclinic connection of E0andEπgenerically has codimension-2 ,\nwhile forwπ\ns= 3 it has codimension-0 , and we refer to the transition point between\n17these cases, following the discussion in §2.1.1, as the center case . From (20) we have\nwπ\ns= 1⇔0≤s0<2√−µ\nα,wπ\ns= 3⇔s0>2√−µ\nαand the center case at s0=2√−µ\nα.\nHence, within the family of homogeneous DWs given by (17) and satisf ying (19), the\ndifferent bifurcation cases have speed and frequency relations\ncodim-0:s0>2√−µ\nαand Ω 0>β\nα−2µ\nα2,\ncenter:s0=2√−µ\nαand Ω 0=β\nα−2µ\nα2,\ncodim-2: 0 ≤s0<2√−µ\nαandβ\nα≤Ω0<β\nα−2µ\nα2.\nUsing (19) these can be written in terms of the parameters of (LLG S), which gives the\ncharacterization mentioned in the introduction §1.\nRemark 5. The case distinction is also related to the spectral stabili ty of the asymptotic\nstatesm=±e3in the dynamics of the full PDE (LLGS)which is beyond the scope\nof this paper, but see Figure 3 for an illustration. In short, it follows from, e.g., [12,\nLemma 1] that e3isL2-stable forh>β/α, while−e3isL2-stable forh<β/α −µand\nunstable for h>β/α −µ. Based on this, the stability curves in Figure 3 are defined as\nfollows\nΓ+:=β/α\nh−µ−1,Γ−:= 1−β/α\nh+µ,\nwhich intersect at\nh=β\n2α+/radicalbigg\nβ2\n4α2+µ2.\nSince the destabilisation of −e3ifβ >0corresponds to a Hopf-instability of the (purely\nessential)spectrum, it is effectively invisible in the coherent struct ure ODE, which detects\nchanges in the linearization at zero temporal eigenvalue on ly. Visible from the PDE\nstability viewpoint is a transition of absolute spectrum th rough the origin in the complex\nplane of temporal eigenmodes, cf. [18]. Now in the center cas e, the state −e3is already\nL2-unstable since α>0as well asµ<0andh>β/α implies\nh=h∗=β\nα−2µ\nα2(1+α2)>β\nα−µ\nand therefore that Γ−never intersects the line ccp≡0ath=h∗.\nMoreover, it was shown in [19] that the family of explicit hom ogeneous DWs (9)is\n(linearly) stable for sufficiently small applied fields, actually for h <−µ/2, in case\nβ= 0, hence in the bi-stable case where ±e3areL2-stable. As mentioned before, β= 0\nis equivalent to ccp= 0in the LLGS equation with an additional shift in handβ, which\nleads to the (LLG)case. We expect these DWs are also stable for small perturbat ions in\nccp, due to the properties of the operator established in [19], b ut further analysis also on\nthe transition from convective/transient to absolute inst ability will be done elsewhere.\n18With these preparations, we next state the mainresults, which con cern existence of DWs\nin the three regimes.\nTheorem 1. For any parameter set (α0,β0,h0,µ0)in the codim-0 case, i.e., µ0<0and\nh0> β0/α0−2µ0−2µ0/α2\n0, the following holds. The explicit homogeneous DWs m0\nin(9)lies in a smooth family mccpof DWs parameterized by (ccp,α,β,h,µ,s, Ω)near\n(0,α0,β0,h0,µ0,s0,Ω0)withs0,Ω0from(19)evaluated at (α0,β0,h0,µ0). Moreover,\nin caseccp= 0and(s,Ω)/ne}ationslash= (s0,Ω0)evaluated at (α,β,h,µ), orccp/ne}ationslash= 0, these are\ninhomogeneous flat DWs.\nProof.As mentioned, in the codim-0 case we have wπ\ns= 3 and for all parameters w0\nu=\n1. Due to the existence of the heteroclinic orbit (17), this means W0\nuintersectsWπ\ns\ntransversely and non-trivially for ccp= 0 in a unique trajectory. Therefore, this DW\nperturbs to a locally unique family by the implicit function theorem for p erturbations\nof the parameters in (7).\nForccp/ne}ationslash= 0 sufficiently small these are inhomogeneous DWs since the derivativ e of the\nthird equation, the q-equation, in (7) with respect to ccpis nonzero in this case; hence\nalready the equilibrium states move into the inhomogeneous regime.\nForccp= 0 but (s,Ω)/ne}ationslash= (s0,Ω0) at (α,β,h,µ), it follows from [12, Theorem 5] that\nthese DWs cannot be homogeneous. /squaresolid\nNext we consider the center case, where h=h∗=β/α−2µ−2µ/α2. We start with\na result that follows from the same approach used in the codim-2 cas e and give refined\nresults below.\nCorollary 1. The statement of Theorem 1 also holds for a parameter set in th e center\ncase if the perturbed parameters (ccp,α,β,h,µ,s, Ω)satisfyΩ> s2/2 +β−/α. IfΩ =\ns2/2 +β−/αandΩ< h+µ+s2\n4(1 +α2)the same holds except the DW is possibly\nnon-flat.\nProof.It follows from Proposition 1 and the discussion before that Ω >s2/2+β−/αfor\nthe parameter perturbation implies that the eigenvalues of the per turbed equilibrium\nZπ\n−≈Eπsatisfy Re(νπ\nk,−)<0,k= 1,2. Hence, the stable manifold at the target equi-\nlibrium is two-dimensional and lies in a smooth family with the center-sta ble manifold\nat the transition point. Then the proof is the same as in the codim-0 c ase. If the pertur-\nbation has Ω = s2/2+β−/αthen we consider as target manifold the three-dimensional\nstable manifold of a neighborhood of Zπ\n−within the blow-up chart θ= 0. This neigh-\nborhood consists of periodic orbits by Proposition 1 if Ω < h+µ+s2\n4(1 +α2). By\ndimensionality the intersection with the unstable manifold of Z0\n−persists and yields a\nheteroclinic orbit from the perturbed equilibrium at θ= 0 to the blow-up chart at θ=π.\nPerturbing ccpaway from zero moves the left-asymptotic state into the inhomoge neous\nregime and thus generates an inhomogeneous DWs. Note from Prop osition 1 that the\nright-asymptotic state is either an equilibrium with q/ne}ationslash= 0 or a periodic orbit along which\nq= 0 happens at most at two points. /squaresolid\n19Next we present a refined result in which we show that typical pertu rbations indeed give\nnon-flat DWs, i.e., heteroclinic connections with right-asymptotic st ate being a periodic\norbit. The existence of flat DWs for ccp/ne}ationslash= 0 is severely constrained, but not ruled out\nby this result. Our numerical results, such as those presented in §4, always lead to a\nselected solution with a periodic asymptotic state.\nIn addition, attempts to perform numerical continuation (see §4) of flat DWs to ccp/ne}ationslash= 0\nfailed. Here we added the constraint /tildewideH= 0 and allowed adjustment the parameters h\nands, but the continuation process did not converge, which confirms nu merically the\ngeneric selection of a periodic orbit.\nAs mentioned before, the right asymptotic state is e3in either case in the PDE coordi-\nnates; the difference between flat and non-flat lies in the finer deta ils of how the profile\napproaches e3in term ofpand alsoq, which relates to mthrough (2).\nTheorem 2. Consider the smooth family of DWs from Corollary 1 with ccp= 0for\nparameters satisfying (12)with fixedα >0,β≥0, andµ <0. Then there is a\nneighborhood (ccp,s,h)of(0,s0,h∗)suchthat the followingholds. FlatDWs occurat most\non a surface in the (ccp,s,h)-parameter space and, for β/ne}ationslash= 0, satisfy|h−h0|2+|s−s0|2=\nO(|ccp|3), more precisely (31)holds, where h0=h∗ands0= 2√−µ/α. Otherwise DWs\nare non-flat, in particular all DWs not equal to m0forccp= 0orβ= 0are non-flat.\nDue to its more technical nature, the proof of this theorem is defe rred to Appendix 6.1.\nIt remains to consider the codim-2 case.\nTheorem 3. For any parameter set (α0,β0,h0,µ0)in the codim-2 case, i.e., µ0<0and\nβ0/α0≤h0< β0/α0−2µ0−2µ0/α2\n0, the following holds. The explicit homogeneous\nDWsm0in(9)lies in a smooth family of DWs parameterized by (ccp,α,β,h,µ )near\n(0,α0,β0,h0,µ0). Here the values of (s,Ω)are functions of the parameters (ccp,α,β,h,µ )\nand lie in a neighbourhood of (s0,Ω0)from(19). This family is locally unique near m0\nand forccp/ne}ationslash= 0consists of inhomogeneous flat DWs.\nThe proof of Theorem 3 is presented in Appendix 6.2 and is based on th e Melnikov\nmethod for perturbing from m0. As the unperturbed heteroclinic orbit has codimension\ntwo, thebifurcationisstudied inathree-parametricfamilywithpert urbationparameters\nη:= (ccp,s,Ω)T, which yields a two-component splitting function M(η) that measures\nthe mutual displacement of the manifolds W0\nuandWπ\ns.\nDue to the fact that Re( νπ\nj,−)<0 also forβ= 0 in the codim-0 regime and the case\nβ= 0 is included in Theorem 2 as well as Theorem 3, we immediately get the f ollowing\nresult.\nCorollary 2. Inhomogeneous flat DWs also exist in the LLG equation (β= 0), which\ncan be flat or non-flat, respectively.\nTheorem3 completes theexistence study ofDWs. Therefore, for any valueoftheapplied\nfieldhthere exists a heteroclinic connection between the blow-up charts withccp/ne}ationslash= 0\nandq/ne}ationslash≡0, thus an inhomogeneous (typically flat) DW. Recall that we have fo cused on\n20right moving DWs, but all results are also valid for left moving walls due t o symmetry.\nTherefore inhomogeneous DWs exist with ccp/ne}ationslash= 0 for any value of the applied field\nh∈R.\n4 Numerical Results\nNumerical continuation for ordinary differential equations is an est ablished tool for bi-\nfurcation analysis in dynamical system. In this section we present c ontinuation results\nto illustrate the analytical results discussed in §3. In particular, we will focus on con-\ntinuation in the parameter ccpin the range of ( −0.5,0.5) as this perturbs away from the\nknown family m0from (17) (cf. Figure 1b) with speed and frequency determined by (19)\nfor a given applied field. Note that we also focus only on right-moving f ronts in this\nsection for reasons of clarity. All results were produced by contin uation in AUTO-07P\nand graphics were created with Mathematica as well as MATLAB .\nHeteroclinic orbits were detected as solutions to the boundary valu e problem given by\nthe desingularized system (7) plus a phase condition andboundary c onditions at ξ=−L\nandξ=Ltaken from the analytic equilibrium states in pandqon the blow-up charts\n(Remark 3). In the codim-2 case, the four required conditions are thep,qvalues at the\ncharts. In the center case, the three required conditions are: ( 1,2) the two p,qvalues\nat the left chart and (3) the energy difference determined by the f unction (13). In the\ncodim-0 case, the two required conditions are the pvalues at both charts. Moreover, we\nfoundL= 50 was sufficiently large.\nIn order to relate to (LLGS), we plot most of the profiles after blow ing down to the\nsphere rather than using the ODE phase space.\n(a)-5 5-0.04-0.02\nξq\n- 5 \u0001-101\nξ\nm1\n(b)\nFigure 7: DWs obtained from continuation of m0in system (7) in the codim-2 regime h= 0.5\n(h∗= 10.2) with initial speed and frequency s0= 0.12 as well as Ω 0= 0.44, and\n(ccp,s,Ω) = (−0.5,0.11221,0.44077). (a) Projection onto the sphere. (b) Zoom-in\nof corresponding q-profile (red) and m1component (blue).\nFollowing the standing assumption on positive speeds and using ccpas well ashas the\nmain parameters, we keep the other parameters fixed with values\nα= 0.5,β= 0.1,µ=−1.\n21(a)\n (b)\nFigure 8: DWs obtained from continuation of m0in system (7) projected onto the sphere in\nthe codim-2 regime h= 10.1 (h∗= 10.2) with initial speed and frequency s0=\n3.96 and Ω 0= 8.12. (a) ( ccp,s,Ω) = (−0.5,3.99541,8.05973). (b) ( ccp,s,Ω) =\n(0.5,4.08089,8.22402).\nThe value of the applied field for the center case, given the fixed par ameters, is h∗= 10.2\n(cf.§3.2), which leads to s0= 4.0 as well as Ω 0= 8.2 (cf. (19)).\n4.1 Codim-2 case\nThe lower boundary for values of the applied field hlies in the codim-2 regime and\nis given by h=β/α= 0.2. As a first numerical example we consider the slightly\nlarger value h= 0.5. The results upon continuation in the negative as well as positive\ndirection of ccpare presented in Figures 2a, 2c, and 7. The inhomogeneous nature of\nthese solutions ( ccp/ne}ationslash= 0) is reflected in the significantly varying azimuthal angles, also\nvisible in the oscillatory nature of the m1component in Figures 2c as well as 7b.\nThe linear part of the splitting function (33) (see Theorem 3), which predicts the direc-\ntion of parameter variation for the existence of inhomogeneous DW s (ccp/ne}ationslash= 0) to leading\norder, reads in this example\nM(ccp,s,Ω) =/parenleftbigg\n−0.00147567 −0.499245 0.245945\n−0.000577908 −0.245945 −0.499245/parenrightbigg\n·\nccp\ns\nΩ\n,\nsothatM= (0,0)Tfor(s,Ω) = (−0.00283744 ·ccp,0.000240252 ·ccp). Fortheparameter\nvalues in Figure 7 and 2a we obtain, respectively,\nM(−0.5,−0.007788,0.000771) = (0 .00481558,0.00181945)T,\nM(0.5,−0.007973,0.007173) = (0 .00648248,−0.00133122)T.\nNote that here the splitting of the (1-dimensional) unstable manifold of the left equilib-\nrium and the (1-dimensional) stable manifold of the right equilibrium diffe r, i.e., are in\nopposite directions (signs) in frequency and speed for variations in ccp.\n22In addition note the decrease in frequency in the m1component, and thus also in the\nm2as a result of the increase of the qcomponent towards zero, cf. Figure 7b. Here, the\nazimuthal angle decreases since φ=/integraltext\nqandq <0.\nAs a further example in the codim-2 regime, we consider h= 10.1<10.2 =h∗near the\nupper boundary of the codim-2 regime in terms of the applied field h. The results of the\ncontinuation in ccpare presented in Figure 8. The linear approximation of the splitting\nin this case is given by M(−0.5,0.03541,−0.06027) = ( −0.000175537,−0.00104378)Tas\nwellasM(0.5,0.12089,0.10402) = ( −0.00149519,0.0014975)Tin(a)and(b),respectively.\nNote that the direction of splitting of the two components in this cas e is also dependent\non the polarity sign, as in the previous example. In both cases ( h= 0.5 andh= 10.1),\nthecontinuationresultslookbasicallythesameinthecodim-2regime, wherethesolution\nis, roughly speaking, constantly spiraling down from the north to th e south pole.\n4.2 Center case\n(a)-20 200.0450.055\nξq\n(b)\nFigure 9: DWs obtained from continuation of m0in system (7) in the center regime h=h∗=\n10.2 withs=s0= 4,Ω = Ω 0= 8.2, andccp= 0.5. (a) Projection onto the sphere.\n(b) Profile of corresponding q-component.\nWe perform computations in the center case with applied field h=h∗= 10.2 and fixed\nfrequency Ω = β−/α+s2/2 (see Proposition 1 and its discussion details). Theorem 2\nshows that the right asymptotic state is generically a periodic orbit a nd more precisely\nthat in case ccp= 0, no constellation of handsexists, both not equal to zero, for which\nthe right asymptotic state is the (shifted) equilibrium. The results o f continuation in ccp\nprojected on the sphere look quite the same, which is why only the re sult forccp= 0.5\nis presented in Figure 9. The fact that the right asymptotic state is a periodic orbit on\nthe blow-up chart θ=πis reflected by the nearly constant oscillations in the q-profile\nforξclose to the right boundary (cf. Figure 9b).\nThat the right state is not the equilibrium in the blow-up chart is furth er corroborated\nby computing the difference in energy /tildewideHbetween this equilibrium state (see Remark 3)\nand the approximate right asymptotic state obtained from continu ation. The analytic\nprediction of this difference up to second order is given by (31), whic h reads, for the\n23-0.5 0 0.5\nccp-2-1010-6\n(a)9.2 10.2 11.2\nh-1010-4\n(b)3.5 4 4.5\ns-2010-3\n(c)\nFigure 10: Continuation of m0insystem(7)inthecenter casewithappliedfield h=h∗= 10.2\nand fixed frequency Ω = β−/α+s2/2; heres0= 4. Shown is the energy difference\n(solid blue line) between the equilibrium and asymptotic st ate from continuation\non right boundary against the continuation parameter ccpin (a),hin (b), and sin\n(c). The red dashed curve in (b) and (c) is the quadratic appro ximation (21).\nchosen parameters,\n−0.006612+0.00673s−0.00183s2−0.00134h−0.000086h2+0.00077hs.(21)\nAs this analytic prediction is independent of ccpthe dependence of /tildewideHonccp≈0 is of\ncubic or higher power. Indeed, the results plotted in Figure 10a sug gest an at least\nquartic dependence since a maximum lies at ccp= 0. The asymmetric nature of the\ngraph suggests that odd powers appear in the expansion beyond o ur analysis, but also\nnote the order of 10−6in/tildewideH. In addition to the dependence on ccp, continuations for\nccp= 0 of/tildewideHinhwith fixeds=s0= 4 and in swith fixedh=h∗= 10.2 are plotted in\nFigure 10b and 10c, respectively. Here we also plot the quadratic pr ediction (21).\n4.3 Codim-0 case\nNext, we consider an applied field h= 10.3 in the codim-0 regime, just above the applied\nfield value for the center case h∗= 10.2. The results of continuation in ccpare plotted\non the sphere in Figure 11. The azimuthal profile in φand hence in qare non-trivial as\npredicted for inhomogeneous DWs.\nIn the ODE, qpossesses an oscillating profile and has a monotonically decreasing am -\nplitude in both cases. This is a consequence of the proximity to the ce nter case and the\nconvergence to equilibria (see §3.2 for details). Recall that the speed and frequency are\nnot selected by the existence problem during continuation in ccp, but are taken as the\nfixed parameters ( s0,Ω0) defined in (19).\nThe final example is for a relatively large applied field h= 50 in the codim-0 regime,\nfar away from the center case, and the results of continuation in ccpprojected on the\nsphere are presented in Figure 2b as well as Figure 12a. Moreover, the corresponding\nm1andm2profiles for ccp= 0.5 are presented in Figure 2d, and for ccp=−0.5, in\nFigure 12b. As in the previous example, the inhomogeneous nature is visible in the\nnon-trivial azimuthal profile.\nIn summary, switching on the parameter ccpleads to a variety of inhomogeneous flat as\nwell as non-flat DW solutions, but also in case ccp= 0 there exist inhomogeneous DWs\n24(a)\n (b)\nFigure 11: DWs obtained from continuation of m0in system (7) projected onto the sphere in\nthe codim-0 regime h= 10.3 (h∗= 10.2) and initial speed and frequency s0= 4.04\nand Ω0= 8.28. (a)ccp=−0.5. (b)ccp= 0.5.\n(a)-5 5-101\nξm1,m2\n(b)\nFigure 12: DWs obtained from continuation of m0in system (7) in the codim-0 regime h= 50\n(h∗= 10.2) withinitial speedandfrequency s0= 19.92,Ω0= 40.04, andccp=−0.5.\n(a) Projection onto the sphere. (b) Zoom-in of the correspon dingm1(solid blue)\nandm2(dashed red) component.\n(cf. Figure 4) which are much more complex than the homogeneous o ne given by (9) (cf.\nFigure 1b).\nFinally, recall from §3.1 that in the explicit family (9), the right moving DWs terminate\nats= 0. The question arises what happens for ccp/ne}ationslash= 0 along the parameter s. To\nstudy this, we performed a continuation in the parameter sfor different values of ccp.\nFor decreasing swe found that numerical continuation failed at some s>0 forccp/ne}ationslash= 0\n(cf. Figure 13). The details of this apparent existence boundary a re beyond the scope of\nthis paper. Note that the special role of sis reflected in the splitting function (34). In\ncases0=s= 0, the first column of (32) is zero (see (34)) and thus the parame terization\nofccpcan not be written as a function in s.\nIn detail, we continued the analytic solution (9) in ccpaway from zero for different initial\nvalues ofβ/α < h < h∗. This led to inhomogeneous ( ccp/ne}ationslash= 0) DWs, which we in turn\ncontinued in the parameter stowards zero for different fixed values of ccpuntil the con-\ntinuation process fails to converge. Based on this, there is numeric al evidence that DWs\nwith opposite speed sign (counter-propagating fronts) can only e xist simultaneously for\n25s= 0 (standing fronts). We took a polynomial fit on these points as an approximation of\nthe existence boundary (cf. blue curve in Figure 13a). The continu ation process towards\nthe boundary is indicated by red arrows in Figure 13a for positive ccp. Additionally, the\ncorresponding results in the parameter space Ω and ccpis presented in 13b.\n0 0.10.20.40.6\nsc\nc\u0002\n(a)0.2 0.50.20.40.6\nΩc\n\u0003\u0004\n(b)\nFigure 13: Continuation results in sand Ω for fixed ccpin the codim-2 parameter regime.\nFurther parameters are α= 0.5,β= 0.1,µ=−1, andhfree. Blue solid line\nrepresents interpolated termination boundaryfor sin (a) and Ω in (b), respectively.\nRed arrows indicate continuation approach towards boundar y.\nAs a last point, we briefly describe the numerical method for time-int egration near DWs,\nincluding freezing of speed and frequency (cf. Figure 4). All calcula tions were done\nwith the (free) software package pde2path which is based on a Finite Element Method\n(FEM), cf. [20] and the references therein. Time-integration in pde2path with the so-\ncalled ‘freezing’ method is discussed in [21]. In addition to the phase co ndition for the\nspeedweaddedaphaseconditionfortherotationandtime-integra tedvia asemi-implicit\nEuler scheme.\n5 Discussion and Outlook\nWe have presented results pertaining the existence of different ty pes of domain walls for\nthe LLG as well as LLGS equation. Our main focus has been on a nonze ro polarisation\nparameterccp/ne}ationslash= 0 for any value of the applied field, including the high-field case, and\nthus for any domain wall speed. These results extend what is known in particular for\ninhomogeneously structured DWs, and we have discovered an appa rently new type of\nDWs with certain oscillatory tails, referred to as non-flat here.\nIn detail, we have provided a classification of DWs based on co-dimens ion properties\nin a reduced (spatial) coherent structure ODE, which relates to st ability and selection\nproperties that we review next. First, we have proven the existen ce of inhomogeneous\nflat DWs in case ccp= 0 as well as ccp/ne}ationslash= 0 for an applied field above a certain threshold,\nwhichismainlymaterialdepending. Toourknowledge, theonlypreviou sexistence result\nforccp/ne}ationslash= 0 with ’large’ applied fields concerns less relevant non-localized DWs [ 12]. Here\nthe existence problem does not select speed and frequency.\nSecond, we have discussed the so-called center case, which is char acterized by non-\nhyperbolic equilibria in the underlying coherent structure ODE. In th is case, we have\n26shown the existence of inhomogeneous DWs including the leading orde r selection mech-\nanism. These solutions are non-flat in case ccp= 0 and generically also non-flat for ccp\naway from zero, which was substantiated by numerical results. Th e fundamental obser-\nvation has been the existence of a Hamiltonian function in a certain pa rameter regime\nin the corresponding coherent structure ODE.\nThird, we have proven the existence of inhomogeneous DWs in the so -called codim-2\nregime, which is a range of values for the applied field in which the speed sis between\nzero and the center case speed. In this regime, each solution in cas eccp/ne}ationslash= 0 is uniquely\ndetermined by its speed as well as frequency. Here we have also pre sented the leading\norder selection function in the coherent structure ODE variables pandq, which depends\non the speed s, the frequency Ω, and is independent of ccpfor standing fronts.\nWe believe that these results are not only interesting and relevant f rom a theoretical and\nmathematical viewpoint, but also from an application viewpoint. They could help to\nbetter understand the interfaces between different magnetic do mains in nanostructures,\ne.g. in the development of racetrack memories, which are a promising prospective high\ndensity storage unit that utilize a series of DWs by shifting at high spe ed along magnetic\nnanowires through nanosecond current pulses.\nIn order to illustrate and corroborate these theoretical results , we have presented numer-\nical computations for a variety of values for the applied field in §4. On the one hand,\nthe examples in essence show that large applied fields lead to more com plex profiles of\nthe DWs in case ccp/ne}ationslash= 0. On the other hand, while in the center case the DWs projected\non the sphere appear similar to those for small applied fields, these s olutions approach\nthe poles in a qualitatively different ‘non-flat’ manner – as predicted b y our analysis.\nMoreover, we compared the numerical and analytical results of th e selection mechanism\nin the center case, showing that the analytical leading order appro ximation predicts the\neffect of small perturbations in the parameters. Notably, for app lied fields above a cer-\ntain threshold, where the existence analysis does not provide a sele ction of speed and\nfrequency, numerically the DWs selected in the PDE dynamics are in th e center case,\nboth forccp= 0 as well as ccp/ne}ationslash= 0. Hence, it might be possible to detect these solutions\nin a high-field regime in real materials.\nOne question concerning existence beyond our analysis is whether in homogeneous (flat\nor non-flat) solutions exist for any value of ccp∈(−1,1), and whether this class could\nbe utilized in applications.\nA natural step towards the understanding of domain wall motion in n anowires beyond\nthe question of existence concerns the dynamic stability. For inhom ogeneous solutions\nthere appears to be no rigorous result in this direction. In particula r for larger applied\nfields, stability results would be an essential step towards underst anding the selection\nmechanismofsolutionsintermsofspeedandfrequency; ourfirstn umericalinvestigations\nshow that solutions inthe center parameter regime areselected, i.e ., inhomogeneous non-\nflat DWs.\nMoreover, preliminary analytic results, for ccp= 0 as well as ccp/ne}ationslash= 0, show that selection\nmechanism is mainly determined by the value of the applied field, where in the bi-stable\n27case (±e3linearly stable) homogeneous DWs are selected, and in the mono-sta ble case\ninhomogeneous non-flatDWsareselected, which will bestudied indet ail inanupcoming\nwork.\nReferences\n[1] Stuart SP Parkin, Masamitsu Hayashi, and Luc Thomas. Magnetic domain-wall\nracetrack memory. Science, 320(5873):190–194, 2008.\n[2] William Reohr, Heinz Honigschmid, Raphael Robertazzi, Dietmar Gog l, Frank Pe-\nsavento, Stefan Lammers, Kelvin Lewis, Christian Arndt, Yu Lu, Ha ns Viehmann,\net al. Memories of tomorrow. IEEE circuits and devices magazine , 18(5):17–27,\n2002.\n[3] CJ Lin, SH Kang, YJ Wang, K Lee, X Zhu, WC Chen, X Li, WN Hsu, YC Ka o,\nMT Liu, et al. 45nm low power cmos logic compatible embedded stt mram u tilizing\na reverse-connection 1t/1mtj cell. In Electron Devices Meeting (IEDM), 2009 IEEE\nInternational , pages 1–4. IEEE, 2009.\n[4] Moinuddin K Qureshi, Vijayalakshmi Srinivasan, and Jude A Rivers. Scalable high\nperformance main memory system using phase-change memory tec hnology. ACM\nSIGARCH Computer Architecture News , 37(3):24–33, 2009.\n[5] IsaakDMayergoyz, GiorgioBertotti, andClaudioSerpico. Nonlinear magnetization\ndynamics in nanosystems . Elsevier, 2009.\n[6] Alex Hubert and Rudolf Sch¨ afer. Magnetic domains: the analysis of magnetic\nmicrostructures . Springer Science & Business Media, 2008.\n[7] Giorgio Bertotti. Spin-transfer-driven magnetization dynamics .Magnetic Nanos-\ntructures in Modern Technology , pages 37–60, 2008.\n[8] L Berger. Emission of spin waves by a magnetic multilayer traverse d by a current.\nPhysical Review B , 54(13):9353, 1996.\n[9] John C Slonczewski. Current-driven excitation of magnetic multila yers.Journal of\nMagnetism and Magnetic Materials , 159(1-2):L1–L7, 1996.\n[10] JA Osborn. Demagnetizing factors of the general ellipsoid. Physical review , 67(11-\n12):351, 1945.\n[11] JC Slonczewski. Currents and torques in metallic magnetic multilay ers.Journal of\nMagnetism and Magnetic Materials , 247(3):324–338, 2002.\n[12] Christof Melcher and Jens DM Rademacher. Pattern formation in axially sym-\nmetric landau–lifshitz–gilbert–slonczewski equations. Journal of Nonlinear Science ,\n27(5):1551–1587, 2017.\n28[13] Arseni Goussev, JM Robbins, and Valeriy Slastikov. Domain-wall motion in fer-\nromagnetic nanowires driven by arbitrary time-dependent fields: A n exact result.\nPhysical review letters , 104(14):147202, 2010.\n[14] Johan ˚Akerman. Toward a universal memory. Science, 308(5721):508–510, 2005.\n[15] LD Landau and EM Lifshitz. On the theory of the dispersion of ma gnetic perme-\nability in ferromagnetic bodies, phy. z. sowjetunion 8: 153. Reproduced in Collected\nPapers of LD Landau , pages 101–114, 1935.\n[16] Thomas L Gilbert. A phenomenological theory of damping in ferro magnetic mate-\nrials.IEEE Transactions on Magnetics , 40(6):3443–3449, 2004.\n[17] M Lakshmanan. The fascinating world of the landau–lifshitz–gilbe rt equation: an\noverview. Philosophical Transactions of the Royal Society of London A : Mathemat-\nical, Physical and Engineering Sciences , 369(1939):1280–1300, 2011.\n[18] Bj¨ orn Sandstede and Arnd Scheel. Absolute and convective in stabilities of waves on\nunbounded and large bounded domains. Physica D: Nonlinear Phenomena , 145(3-\n4):233–277, 2000.\n[19] Yan Gou, Arseni Goussev, JM Robbins, and Valeriy Slastikov. St ability of precess-\ning domain walls in ferromagnetic nanowires. Physical Review B , 84(10):104445,\n2011.\n[20] T Dohnal, J Rademacher, H Uecker, and D Wetzel. pde2path 2.0. ENOC, 2014.\n[21] Jens DM Rademacher and Hannes Uecker. Symmetries, freezin g, and hopf bifurca-\ntions of traveling waves in pde2path, 2017.\n[22] Yuri A Kuznetsov. Elements of applied bifurcation theory , volume 112. Springer\nScience & Business Media, 2013.\n6 Appendix\n6.1 Proof of Theorem 2\nWe use the notation\nu=u(ξ;η,α,β,µ)= (θ(ξ;η,α,β,µ),p(ξ;η,α,β,µ),q(ξ;η,α,β,µ))T\nand bifurcation parameters η= (ccp,s,h)T, wheres0andh0=h∗are defined below (see\n§3.2 for details). The starting point for our perturbation analysis ar e the unperturbed\nparameters and explicit heteroclinic solution in the center case (12) , where the frequency\nis Ω0=s2\n0/2+β/α. These are given by\nη0:=\nccp0\ns0\nh0\n:=\n0\n2√−µ\nαβ\nα−2µ−2µ\nα2\n\n29as well as\nu0=u0(ξ;η0,α,β,µ) :=\nθ0(ξ;η0,α,β,µ)\np0(ξ;η0,α,β,µ)\nq0(ξ;η0,α,β,µ)\n:=\n2arctan(exp(√−µξ))√−µ\n0\n.\nUnless stated otherwise, we suppress the explicit dependence of uonα,β, andµin the\nfollowing discussion. Let us write Zπ:=Zπ\n−with the notation from Remark 3 so that\nthe unperturbed right asymptotic state is given by\nZπ(η0) =/parenleftbigg\nπ,αs0\n2,s0\n2−/radicalbigg\n−µ\nα2/parenrightbiggT\n=/parenleftbig\nπ,√−µ,0/parenrightbigT\nand its derivative with respect to ηis given by\nZπ\nη(η0) =\n0 0 0\n0α\n20\nβ\n2√−µ2+α2\n2−α\n2√−µ\n.\nWe write system (7) for brevity as\nu′=f(u;η), (22)\nsof(u;η) denotes the right side of (7). The linearization w.r.t. ηin the unperturbed\nheteroclinic connection u0, given by (17), is the non-autonomous linear equation\nu′\nη=fu(u0;η0)uη+fη(u0;η0)η, (23)\nwhereuη= (θη,pη,qη)T. Its homogeneous part is\nθ′\nη=√−µcos(θ0)θη+pηsin(θ0)\np′\nη=−(αs0+2√−µcos(θ0))pη+s0qη\nq′\nη=−s0pη−(αs0+2√−µcos(θ0))qη, (24)\nwithθ0(ξ) = 2arctan(exp(√−µξ)) due to (17). We next solve (24) and determine its\nfundamental solution matrix.\nThe first obvious vector-solution of it is U1=u′\n0= (θ′\n0,0,0) since the second and the\nthird equation of (24) do not depend on θη. The other solutions can be obtained from\nU1and the result of Lemma 1. Changing to polar coordinates\npη=rcosϕ, q η=rsinϕ,\nthe equations for pηandqηbecome\nr′=−(αs0+2√−µcos(θ0))r\nϕ′=−s0,\n30whose general solution can be written as\npη=r0r(ξ)cos(−s0ξ+ϕ0)\nqη=r0r(ξ)sin(−s0ξ+ϕ0)\nwhere\nr(ξ) = exp\n−αs0ξ−2√−µ/integraldisplay\nξcos(θ0(τ))dτ\n=/parenleftig\n1+e2√−µξ/parenrightig2\ne(−2√−µ−αs0)ξ,\nandr0,ϕ0arearbitrary integrationconstants corresponding to suitable init ial conditions.\nNote that lim\nξ→±∞r(ξ) =∞for 0≤s0<2√−µ/α.\nNext, the values of the integration constants have to be selected in order for the second\nand the third vector-solutions\nU2=\nθ1\n1\nr1r(ξ)cos(−s0ξ+ϕ1)\nr1r(ξ)sin(−s0ξ+ϕ1)\n, U3=\nθ2\n1\nr2r(ξ)cos(−s0ξ+ϕ2)\nr2r(ξ)sin(−s0ξ+ϕ2)\n(25)\ntobelinearlyindependent. Here θ1\n1,θ2\n1arenotrelevantforwhatfollows. Thedeterminant\nof the fundamental matrix reads\ndetΦ(ξ) = det(U1(ξ),U2(ξ),U3(ξ)) =r1r2r2(ξ)θ′\n0(ξ)sin(ϕ2−ϕ1),\nwhich is non-zero for r1=r2= 1,ϕ1= 0 andϕ2=π/2, i.e. detΦ( ξ) =r2(ξ)θ′\n0(ξ).\nTogether, we get the fundamental solution matrix of the homogen eous part as\nΦ(ξ) =\nθ′\n0(ξ)θ1\n1(ξ) θ2\n1(ξ)\n0r(ξ)cos(−s0ξ)−r(ξ)sin(−s0ξ)\n0r(ξ)sin(−s0ξ)r(ξ)cos(−s0ξ)\n. (26)\nThe derivative of (22) with respect to ηis given by (23) and from the variation of\nconstants formula we get for some ξ0that\nuη(ξ) = Φξ,ξ0uη(ξ0)+ξ/integraldisplay\nξ0Φξ,τfη(u0(τ);η0)dτ,\nwhere Φ ξ,τ= Φ(ξ)·Φ−1(τ) is the evolution operator. Using (26) we find\nΦξ,τ(ξ,τ;η) =\nΘ1 Θ2 Θ3\n0r(ξ)\nr(τ)cos(−s0(ξ−τ))−r(ξ)\nr(τ)sin(−s0(ξ−τ))\n0r(ξ)\nr(τ)sin(−s0(ξ−τ))r(ξ)\nr(τ)cos(−s0(ξ−τ))\n,(27)\nwhere the explicit forms of the functions Θ 1,2,3(ξ) are not relevant for the remainder\nof this proof. Since uη(ξ) tends to∂ηZ0\n−forξ→ −∞the hyperbolicity of Z0\n−(more\n31precisely the resulting exponential dichotomy) implies Φ ξ,ξ0uη(ξ0)→0 asξ0→ −∞and\nso\nuη(ξ) =ξ/integraldisplay\n−∞Φξ,τfη(u(τ;η0);η0)dτ. (28)\nRegarding the limiting behavior as ξ→ ∞, recall that Corollary 1 states that the right\nasymptotic limit of the perturbed heteroclinic orbit is either the pert urbed equilibrium\nZπ(η) or a periodic orbit around it in the blow-up chart at θ=π. The integral (28)\ndistinguishes these case in the sense that either it has a limit as ξ→+∞so the\nheteroclinic orbit connects the two equilibria, or it does not and the h eteroclinic orbit\nconnects to a periodic solution.\nWe next determine uη(ξ) componentwise\nuη(ξ) =v:=\nv11v12v13\nv21v22v23\nv31v32v33\n,\nwherevijare the components of (28) and index i= 1,2,3 relates to θ,p,qas well as\nj= 1,2,3 toccp,s,h.\nTowards this, we compute\nfη(u0(τ),η0) =\n0 0 0\n−β/α−√−µ\nα(2+α2) 1\n2β\n1+e2√−µ τ√−µ0\n,\nand together with (28) and (27) we obtain\nv21=−β\nαIC−2βJS, v22=−√−µ\nα(2+α2)IC−√−µIS, v23=IC,\nv31=−β\nαIS+2βJC, v32=−√−µ\nα(2+α2)IS+√−µICv33=IS,\nwhere\nIC=IC(ξ):=ξ/integraldisplay\n−∞(1+exp( −2√−µξ))2\n(1+exp( −2√−µτ))2cos(−s0(ξ−τ))dτ,\nIS=IS(ξ):=ξ/integraldisplay\n−∞(1+exp( −2√−µξ))2\n(1+exp( −2√−µτ))2sin(−s0(ξ−τ))dτ,\nJC=JC(ξ):=ξ/integraldisplay\n−∞exp(−2√−µτ)(1+exp( −2√−µξ))2\n(1+exp( −2√−µτ))3cos(−s0(ξ−τ))dτ,\n32ξ ξ0iπ\n2√−µ\nI1I2I3\nI4\nFigure 14: Contour CinCfor the integrals IandJ.\nJS=JS(ξ):=ξ/integraldisplay\n−∞exp(−2√−µτ)(1+exp( −2√−µξ))2\n(1+exp( −2√−µτ))3sin(−s0(ξ−τ))dτ.\nNote that we do not provide explicit formulas for v11,v12andv13, because they are\nnot needed for further computations. This is the reason why we ne glected the explicit\nexpressions of Θ 1,Θ2, and Θ 3before. We now introduce the following complex-valued\nintegrals for further computations:\nI(ξ):=IC(ξ)+iIS(ξ) =ξ/integraldisplay\n−∞/parenleftbig\n1+e−2√−µξ/parenrightbig2\n/parenleftbig\n1+e−2√−µτ/parenrightbig2exp/parenleftbigg\n−i2√−µ\nα(ξ−τ)/parenrightbigg\ndτ,\nforICandISas well as\nJ(ξ):=JC(ξ)+iJS(ξ) =ξ/integraldisplay\n−∞e−2√−µτ/parenleftbig\n1+e−2√−µξ/parenrightbig2\n/parenleftbig\n1+e−2√−µτ/parenrightbig2exp/parenleftbigg\n−i2√−µ\nα(ξ−τ)/parenrightbigg\ndτ\nforJCandJS. We extend the above integrals to the complex plane and integrate a long\nthe counter-clockwise oriented rectangular contour Cas illustrated in Figure 14, and\nletξ0→ −∞. We will provide the details of the computation of Ionly, asJcan be\ncalculated in a fully analogous way.\nThe complex integrand of Iis\ng(z;ξ):=/parenleftbig\n1+e−2√−µξ/parenrightbig2\n/parenleftbig\n1+e−2√−µz/parenrightbig2exp/parenleftbigg\n−i2√−µ\nα(ξ−z)/parenrightbigg\n,\nwith singularities in Cat the pointsi(π+2kπ)\n2√−µ,k∈Z, one of which lies in the interior of C,\nnamelyz0:=iπ\n2√−µ. The contour integral Ican now be written via the residue theorem\nas\nI1(ξ)+I2(ξ)+I3(ξ)+I4(ξ) = 2πi/summationdisplay\nintCResg(z;ξ),\nwhereI1,...,I 4are given by\n33I1:z=x,\nI1(ξ0,ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbiggξ/integraldisplay\nξ0exp/parenleftig\ni2√−µ\nαx/parenrightig\n/parenleftbig\n1+e−2√−µx/parenrightbig2dx\nI(ξ) = lim\nξ0→−∞I1(ξ0,ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbiggξ/integraldisplay\n−∞exp/parenleftig\ni2√−µ\nαx/parenrightig\n/parenleftbig\n1+e−2√−µx/parenrightbig2dx,\nI2:z=ξ+iy,\nI2(ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2π√−µ/integraldisplay\n0iexp/parenleftig\n−2√−µ\nαy/parenrightig\n/parenleftbig\n1+e−2√−µ(ξ+iy)/parenrightbig2dy,\nI3:z=x+π√−µi,\nI3(ξ0,ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbigg\ne−2π\nαξ0/integraldisplay\nξexp/parenleftig\ni2√−µ\nαx/parenrightig\n/parenleftbig\n1+e−2√−µξ/parenrightbig2dx\n=−e−2π\nαI1(ξ0,ξ),\nI4:z=ξ0+iy,\nI4(ξ0,ξ) =/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\ni2√−µ\nα(ξ0−ξ)/parenrightbigg0/integraldisplay\nπ√−µiexp/parenleftig\n−2√−µ\nαy/parenrightig\n/parenleftbig\n1+e−2√−µ(ξ0+iy)/parenrightbig2dy,\nlim\nξ0→−∞I4(ξ0,ξ) = 0.\nUtilizing the Laurent series of gwe obtain\nResg(z;ξ)|z=z0=α+i\n2α√−µe−π\nα/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbigg\n,\nwhich leads to\nI(ξ) =/parenleftig\n1−e−2π\nα/parenrightig−1/parenleftbiggπi(α+i)\nα√−µe−π\nα/parenleftig\n1+e−2√−µξ/parenrightig2\nexp/parenleftbigg\n−i2√−µ\nαξ/parenrightbigg\n−I2(ξ)/parenrightbigg\n.\nNow we can write\nIC(ξ) = ReI(ξ)\n=π/parenleftbig\n1+e−2√−µξ/parenrightbig2\nα√−µ/parenleftbig\neπ\nα−e−π\nα/parenrightbig/bracketleftbigg\n−cos/parenleftbigg\n−2√−µ\nαξ/parenrightbigg\n−αsin/parenleftbigg\n−2√−µ\nαξ/parenrightbigg/bracketrightbigg\n−1\n1−e−2π\nαIr\n2(ξ)\n34as well as\nIS(ξ) = ImI(ξ)\n=π/parenleftbig\n1+e−2√−µξ/parenrightbig2\nα√−µ/parenleftbig\neπ\nα−e−π\nα/parenrightbig/bracketleftbigg\nαcos/parenleftbigg\n−2√−µ\nαξ/parenrightbigg\n−sin/parenleftbigg\n−2√−µ\nαξ/parenrightbigg/bracketrightbigg\n−1\n1−e−2π\nαIi\n2(ξ),\nwhereIr\n2(ξ) andIi\n2(ξ) are the real and imaginary part of I2(ξ), respectively.\nStudying the integral Jin a similar fashion, we obtain\nJC(ξ) = ReJ(ξ)\n=π/parenleftbig\n1+e−2√−µξ/parenrightbig2\n2α2√−µ/parenleftbig\neπ\nα−e−π\nα/parenrightbig/bracketleftbigg\nαcos/parenleftbigg\n−2√−µ\nαξ/parenrightbigg\n−sin/parenleftbigg\n−2√−µ\nαξ/parenrightbigg/bracketrightbigg\n−1/parenleftig\n1−e−2π\nα/parenrightigJr\n2(ξ)\nas well as\nJS(ξ) = ImJ(ξ)\n=π/parenleftbig\n1+e−2√−µξ/parenrightbig2\n2α2√−µ/parenleftbig\neπ\nα−e−π\nα/parenrightbig/bracketleftbigg\ncos/parenleftbigg\n−2√−µ\nαξ/parenrightbigg\n+αsin/parenleftbigg\n−2√−µ\nαξ/parenrightbigg/bracketrightbigg\n−1/parenleftig\n1−e−2π\nα/parenrightigJi\n2(ξ),\nwhere also here Jr\n2(ξ) andJi\n2(ξ) are the real and imaginary part of J2(ξ). Direct compu-\ntations show also that\nlim\nξ→+∞Ii\n2(ξ) =α\n2√−µ/parenleftbigg\n1−exp/parenleftbigg\n−2π\nα/parenrightbigg/parenrightbigg\nand\nlim\nξ→+∞Ir\n2(ξ) = lim\nξ→+∞Jr\n2(ξ) = lim\nξ→+∞Ji\n2(ξ) = 0.\nSumming up, the second and third component of uη(ξ)·ηfor sufficiently large ξare\n/parenleftigg\nα\n2s\nβ\n2√−µccp+2+α2\n2s−α\n2√−µh/parenrightigg\n+\nπ\nρ\n/parenleftig\n−1\nα√−µh+2\nα2s/parenrightig\ncos/parenleftig\n−2√−µ\nαξ/parenrightig\n+/parenleftig\n−1√−µh+(3+α2)\nαs/parenrightig\nsin/parenleftig\n−2√−µ\nαξ/parenrightig\n/parenleftig\n1√−µh−(3+α2)\nαs/parenrightig\ncos/parenleftig\n−2√−µ\nαξ/parenrightig\n+/parenleftig\n−1\nα√−µh+2\nα2s/parenrightig\nsin/parenleftig\n−2√−µ\nαξ/parenrightig\n+\n+O(e−2√−µξ),\nwhereρ:= exp(π/α)−exp(−π/α). One readily verifies that the oscillatory part in the\nexpression above vanishes if and only if sandhare zero and thus we infer that the\nheteroclinic connection cannot be between equilibria to first order in the parameters.\nIn order to detect cancellations of these oscillatory parts for high er orders of sandh, we\nnext consider the behavior of the quantity (13) with respect to pa rameter perturbations.\nWith slight abuse of notation, for u= (θ,p,q)Twe writeH(u;η) :=H(p,q) evaluated\n35at parameters η, and other parameters at some fixed value, and we always consider the\nheteroclinic solutions from Corollary 1.\nOur strategy in the following steps is as follows: we utilize the quantity Hbecause\nlimξ→+∞Halways exists along these solutions. In order to distinguish whether this\nlimit is an equilibrium or a periodic orbit, we consider\n/tildewideH(u;η):=H(u;η)−H(Zπ;η),\ni.e., the difference of the H-values of the (parameter dependent) equilibrium Zπand the\nlimit ofuasξtends to infinity. Expanding /tildewideHin the limit ξ→ ∞with respect to the\nparameterηyields conditions for periodic asymptotics. In the following, subindice s of\nHdenote partial derivatives, e.g. Hu=∂uH.\nClearly,H(u0;η0) =H(Zπ(η0);η0), thus/tildewideH0= 0 and, since equilibria are critical points\nofH, we have/tildewideHu(u0;η0) =/tildewideHη(u0;η0) = (0,0,0)T. The second derivative is given by\nd2\ndη2/tildewideH=/an}bracketle{tuη,/tildewideHuuuη/an}bracketri}ht+/an}bracketle{tuη,/tildewideHuη/an}bracketri}ht+/an}bracketle{t/tildewideHηu,uη/an}bracketri}ht, (29)\nsince/tildewideHηηis the zero matrix, /tildewideHuthe zero vector, and /tildewideHuη=/tildewideHT\nηu. Thus\n/tildewideH(u0+uηη;η) =1\n2(uηη)THuu(u0;η0)(uηη)+(uηη)THuη(u0;η0)η\n−1\n2/parenleftbig\nZπ\nη(η0)η/parenrightbigTHuu(Zπ(η0);η0)/parenleftbig\nZπ\nη(η0)η/parenrightbig\n−/parenleftbig\nZπ\nη(η0)η/parenrightbigTHuη(Zπ(η0);η0)η+O/parenleftbig\n/bardblη/bardbl3/parenrightbig\n.(30)\nWith the derivatives Huu,Huηin (30) given by\nHuu(u;η) =\n0 0 0\n02\nq−s/2−2p−αs\n(q−s/2)2\n0−2p−αs\n(q−s/2)22p2−αsp+h−β−/α+µ−s2/4\n(q−s/2)3\n,\nHuη(u;η) =\n0 0 0\n0p−αq\n(q−s/2)2 0\nβ/α\n(1−ccp)2(q−s/2)2−p2−αpq−αs\n2p−s\n2q+h−β−/α+µ\n(q−s/2)3 −1\n(q−s/2)2\n,\nfor the right hand side of (30) in the limit ξ→+∞we obtain\n1\n2(uηη)THuu(u0;η0)(uηη)+(uηη)THuη(u0;η0)η=−αβ2\n4µ√−µc2\ncp\n+(4+5α2+α4)(α2ρ2−4(1+α2)π2)\n4α3ρ2√−µ(s−s0)2−α4ρ2−4(1+α2)π2\n4αρ2µ√−µ(h−h0)2\n−αβ(2+α2)\n2µccp(s−s0)+α2β\n2µ√−µccp(h−h0)\n+(2+α2)(α4ρ2−4(1+α2)π2)\n2α2ρ2µ(s−s0)(h−h0),\n36as well as\n1\n2/parenleftbig\nZπ\nη(η0)η/parenrightbigTHuu(Zπ(η0);η0)/parenleftbig\nZπ\nη(η0)η/parenrightbig\n+/parenleftbig\nZπ\nη(η0)η/parenrightbigTHuη(Zπ(η0);η0)η=\n−αβ2\n4µ√−µc2\ncp+α(4+5α2+α4)\n4√−µ(s−s0)2−α3\n4µ√−µ(h−h0)2\n−αβ(2+α2)\n2µccp(s−s0)+α2β\n2µ√−µccp(h−h0)+α2(2+α2)\n2µ(s−s0)(h−h0).\nTherefore, the expansion in the limit ξ→+∞is independent of ccpand reads\nlim\nξ→∞/tildewideH(u0+uηη;η) =−(1+α2)2(4+α2)π2\nα3ρ2√−µ(s−s0)2\n−2(1+α2)(2+α2)π2\nα2ρ2µ(s−s0)(h−h0)\n+(1+α2)π2\nαρ2µ√−µ(h−h0)2+O/parenleftbig\n/bardblη−η0/bardbl3/parenrightbig\n.(31)\nRecallρ= exp(π/α)−exp(−π/α). One readily verifies that the resulting (binary)\nquadratic form of (31) is negative definite for all α >0 so the only solution to the\nleading order problem\nd2\ndη2/tildewideH(u0;η0) = 0\nis thetrivial one( s,h) = (0,0), and anynon-trivial solution satisfies |s−s0|2+|h−h0|2=\nO(|ccp|3).\nIn particular, for ccp= 0 there is a neighborhood of ( s0,h0) such that the only solution\nis the trivial one, which is therefore also the case in the LLG equation . In caseccp/ne}ationslash= 0,\nhigher orders may lead to a solution with non-zero sand/orh, but there is numerical\nevidence that such solutions do not exist (see §4 for details).\n6.2 Proof of Theorem 3\nThe idea of the proof is to apply Lyapunov-Schmidt reduction, i.e., to determine a bi-\nfurcation equation whose solutions are in one-to-one correspond ence with heteroclinic\nconnections between equilibria (7) near one of the explicit solutions u0from (17) con-\nnecting the equilibria Z0\n−andZπ\n−. In the present context this is known as Melnikov’s\nmethod, see for example [22].\nRecallu0corresponds to a homogeneous DW for ccp= 0 with speed s0and rotation\nfrequency Ω 0given by (19). In the present codim-2 parameter regime we will show\nthat the bifurcation equation defines a codimension two bifurcation curve in the three-\ndimensional parameter space ( ccp,s,Ω), which passes through the point (0 ,s0,Ω0). The\nmain part of the proof is to show the existence of certain integrals f or the considered\nparameter set. These integrals are almost identical to the ones st udied within the proof\nof Theorem 2 and we use the same approach.\n37In this section we denote the parameter vector by η:= (ccp,s,Ω)T∈R3, with initial\nvalueη0= (0,s0,Ω0)Tcorresponding to the unperturbed values. The solutions of the\nperturbedsystemcloseto u0hastheform u(ξ;η) =u0(ξ)+uη(ξ;η0)(η−η0)+O(/bardblη−η0/bardbl2),\nwhereuη= (θη,pη,qη)T=O(/bardblη−η0/bardbl).\nAs discussed in Appendix 6.1, the linearization (23) of system (7) aro und the unper-\nturbed heteroclinic connection u0has the fundamental solution matrix Φ( ξ) as defined\nin(26). Inthepresent codim-2casewithdim( W0\nu) = dim(Wπ\ns) = 1inR3, thebifurcation\nequationM(η) = 0 entails two equations. Here M(η) measures the displacement of the\nmanifoldsW0\nuandWπ\ns, andwe willchoose thistobenearthepoint u0(0) =/parenleftbigπ\n2,√−µ,0/parenrightbigT\nin the directions given by vectors v1(0) andv2(0) from adjoint solutions as detailed be-\nlow. From the Taylor expansion M(η) =Mη(η0)(η−η0)+O(/bardblη−η0/bardbl2) we infer by the\nimplicit function theorem that a full rank of Mη(η0) implies a one-to-one correspondence\nof solutions to the bifurcation equation with elements in the kernel o fM(η0).\nIn order to compute Mη(η0) and its rank, we project onto the transverse directions to\nu0, which means to project the inhomogeneous part of equation (23) onto two linearly\nindependent bounded solutions v1,v2of the adjoint variational equation v′=−AT·v,\nwhere\nAT=\n√−µcos(θ0) 0 0\nsin(θ0)−αs0−2√−µcos(θ0) −s0\n0 s0 −αs0−2√−µcos(θ0)\n.\nThe solutions are given in terms of (25) by\nv1=U1×U2\ndetΦ=\n0\n−1\nr(ξ)sin(−s0ξ)\n1\nr(ξ)cos(−s0ξ)\nandv2=U3×U1\ndetΦ=\n0\n1\nr(ξ)cos(−s0ξ)\n1\nr(ξ)sin(−s0ξ)\n.\nImplementing the projection onto these, we obtain the so-called Melnikov integral\nMη(η0):=+∞/integraldisplay\n−∞(v1,v2)T·fη(u0;η0)dξ\n=/parenleftbigg\nβICCα√−µIS−√−µICIS+αIC\nβICS−α√−µIC−√−µIS−IC+αIS/parenrightbigg\n,(32)\nwhere\nICC:=+∞/integraldisplay\n−∞/parenleftbig\n1−e2√−µξ/parenrightbig\neαs0ξ+2√−µξcos(−s0ξ)\n/parenleftbig\n1+e2√−µξ/parenrightbig3dξ,\nICS:=+∞/integraldisplay\n−∞/parenleftbig\n1−e2√−µξ/parenrightbig\neαs0ξ+2√−µξsin(−s0ξ)\n/parenleftbig\n1+e2√−µξ/parenrightbig3dξ,\n38IC:=+∞/integraldisplay\n−∞eαs0ξcos(−s0ξ)/parenleftbig\n1+e2√−µξ/parenrightbig\n·/parenleftbig\n1+e−2√−µξ/parenrightbigdξ,\nIS:=+∞/integraldisplay\n−∞eαs0ξsin(−s0ξ)/parenleftbig\n1+e2√−µξ/parenrightbig\n·/parenleftbig\n1+e−2√−µξ/parenrightbigdξ.\nWe next show that the second and thirdcolumns in (32) have non-va nishing determinant\nso that the rank is always 2, in particular also for β= 0.\nFor brevity, we present the calculations of ICCandICSonly, which are based on the\nsame idea as the computations in Appendix 6.1. The solutions for ICandIScan be\ncomputed in an analogous way.\nFrom Appendix 6.1 we know that the following integral would not exist in cases0=\n2√−µ/αand one readily verifies the existence for s0= 0. Therefore, we first assume\n0<s0<2√−µ/αfor the moment and discuss the case s0= 0 later. We set\nI:=+∞/integraldisplay\n−∞g(ξ)dξ, g(ξ):=/parenleftbig\n1−e2√−µξ/parenrightbig\ne(αs0+2√−µ)ξe−is0ξ\n/parenleftbig\n1+e2√−µξ/parenrightbig3,\nso thatICC= Re(I) andICS= Im(I). Utilizing the same idea for the contour integral\nas in Appendix 6.1 (cf. Figure 14) and the residue theorem, we obtain\nI−eαs0iπ√−µ·es0π√−µ·I= 2πi/summationdisplay\nRes(g).\nThe function ghas a pole of order three atiπ\n2√−µand theLaurentseries gives\nRes(g) =−(α−i)2s2\n0\n8µ√−µexp/parenleftbigg\n(α−i)iπs0\n2√−µ/parenrightbigg\n,\nand therefore\nI=2πs2\n0eπs0\n2√−µ/bracketleftig\n−2αcos/parenleftig\nπαs0\n2√−µ/parenrightig\n+(α2−1)sin/parenleftig\nπαs0\n2√−µ/parenrightig/bracketrightig\n8µ√−µ/parenleftig\n1−eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig\n−ieπs0√−µsin/parenleftig\nπαs0√−µ/parenrightig/parenrightig\n+i2πs2\n0eπs0\n2√−µ/bracketleftig\n(1−α2)cos/parenleftig\nπαs0\n2√−µ/parenrightig\n−2αsin/parenleftig\nπαs0\n2√−µ/parenrightig/bracketrightig\n8µ√−µ/parenleftig\n1−eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig\n−ieπs0√−µsin/parenleftig\nπαs0√−µ/parenrightig/parenrightig.\nSeparating the real and imaginary parts of Iwe get\nICC=πs2\n0√−µeπs0\n2√−µ\n4µ2·2α/parenleftig\n1−eπs0√−µ/parenrightig\ncos/parenleftig\nπαs0\n2√−µ/parenrightig\n+(1−α2)/parenleftig\n1+eπs0√−µ/parenrightig\nsin/parenleftig\nπαs0\n2√−µ/parenrightig\n1+e2πs0√−µ−2eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig\n39ICS=πs2\n0√−µeπs0\n2√−µ\n4µ2·(1−α2)/parenleftig\n1−eπs0√−µ/parenrightig\ncos/parenleftig\nπαs0\n2√−µ/parenrightig\n+2α/parenleftig\n1+eπs0√−µ/parenrightig\nsin/parenleftig\nπαs0\n2√−µ/parenrightig\n1+e2πs0√−µ−2eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig .\nAs mentioned before, one analogously gets\nIC=πs0eπs0\n2√−µ\n2µ·/parenleftig\n1−eπs0√−µ/parenrightig\ncos/parenleftig\nπαs0\n2√−µ/parenrightig\n−α/parenleftig\n1+eπs0√−µ/parenrightig\nsin/parenleftig\nπαs0\n2√−µ/parenrightig\n1+e2πs0√−µ−2eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig\nIS=πs0eπs0\n2√−µ\n2µ·α/parenleftig\n1−eπs0√−µ/parenrightig\ncos/parenleftig\nπαs0\n2√−µ/parenrightig\n+/parenleftig\n1+eπs0√−µ/parenrightig\nsin/parenleftig\nπαs0\n2√−µ/parenrightig\n1+e2πs0√−µ−2eπs0√−µcos/parenleftig\nπαs0√−µ/parenrightig .\nHaving the explicit expressions for IC,IS,ICC, as well as ICS, we can study the rank\nof (32) in case 0 < s0<2√−µ/α. The determinant of the second and third column\nof (32) simplifies to\n(αIS−IC)2+(IS+αIC)2=/parenleftbig\n1+α2/parenrightbig2π2s2\n0eπs0√−µ/ne}ationslash= 0,∀s0/ne}ationslash= 0.\nTherefore,Mη(η0) has full rank and we obtain\nM(η) =/parenleftbiggβICCα√−µIS−√−µICIS+αIC\nβICS−α√−µIC−√−µIS−IC+αIS/parenrightbigg\n·(η−η0)+O/parenleftbig\n/bardblη−η0/bardbl2/parenrightbig\n.(33)\nIn the remaining case s0= 0, which means h=β/α, we haveICC=ICS=IS= 0,\nIC=1\n2√−µ, and thus\nMη(η0) =/parenleftigg\n0−1\n2α\n2√−µ\n0−α\n2−1\n2√−µ/parenrightigg\n, (34)\nwhose rank is always 2. Thus the splitting directions are independent ofccpin first order\nin caseh=β/α. Moreover, note that the splitting in sis independent of the anisotropy\nµto first order in case h=β/α. This completes the proof of Theorem 2.\n40"    },    {        "title": "1907.08454v2.A_cryogenic_memory_element_based_on_an_anomalous_Josephson_junction.pdf",        "content": "A cryogenic memory element based on an anomalous Josephson junction\nC. Guarcello1and F.S. Bergeret1, 2\n1Centro de Física de Materiales, Centro Mixto CSIC-UPV/EHU,\nPaseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain\n2Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain\n(Dated: September 27, 2019)\nWe propose a non-volatile memory element based on a lateral ferromagnetic Josephson junction with spin-\norbit coupling and out-of-plane magnetization. The interplay between the latter and the intrinsic exchange\nfield of the ferromagnet leads to a magnetoelectric effect that couples the charge current through the junction\nand its magnetization, such that by applying a current pulse the direction of the magnetic moment in F can\nbe switched. The two memory states are encoded in the direction of the out-of-plane magnetization. With\nthe aim to determine the optimal working temperature for the memory element, we explore the noise-induced\neffects on the averaged stationary magnetization by taking into account thermal fluctuations affecting both the\nJosephson phase and the magnetic moment dynamics. We investigate the switching process as a function of\nintrinsic parameters of the ferromagnet, such as the Gilbert damping and strength of the spin-orbit coupling, and\nproposed a non-destructive readout scheme based on a dc-SQUID. Additionally, we analyze a way to protect the\nmemory state from external perturbations by voltage gating in systems with a both linear-in-momentum Rashba\nand Dresselhaus spin-orbit coupling.\nI. INTRODUCTION\nSuperconducting electronics is suggested as playing an im-\nportant role in the search of ultra-low-power computers [1–5].\nOne of the key challenges towards this objective is the fabrica-\ntion of a reliable and scalable cryogenic memory architecture.\nSuperconductor-ferromagnet-superconductor (SFS) junctions\nare promising structures suggested for such memories [6–15].\nIndeed, the interplay between the intrinsic exchange field and\nthe induced superconductivity in the ferromagnet leads to the\nso-called p-junction, i.e., a Josephson junction exhibiting an\nintrinsic p-phase shift in its ground state. Vertical ferromag-\nnetic multilayer structures are being used as Josephson mag-\nnetic memories. The two logic states of these memories usu-\nally correspond to states with different relative orientation of\nmagnetic layers, that in turn determines whether the junction\nis in the 0- or p-state. Readout schemes are commonly based\non distinguish between resistive and non-resistive states.\nHere, we suggest an alternative cryogenic memory element\nbased on a ferromagnetic anomalous Josephson junction, usu-\nally called j0-junction [16]. It consists of a SFS Josephson\njunction with a Rashba-like spin-orbit coupling (SOC). Its\nground state corresponds to a finite phase shift in its current-\nphase-relation 0 <j0<p. Such anomalous phase has been\nrecently detected experimentally in hybrid Josephson junc-\ntions fabricated with the topological insulator Bi 2Se3[17] and\nAl/InAs heterostructures [18]. Both materials have strong\nspin-orbit coupling and in those experiments, time reversal is\nbroken by an external magnetic field that acts as a Zeeman\nfield. In our proposal, the memory element is a Josephson\njunction with a ferromagnetic link, thus in principle time-\nreversal is broken intrinsically by the exchange field. As\ndemonstrated theoretically, in these junctions the magneti-\nzation of F can be controlled by passing an electric current\nthrough the device [19–24]. We propose to use such a junc-\ntion as a memory element with the information encoded in the\nmagnetization direction of the F layer. An important issue for\nthe memory element is the effects stemming from the unavoid-able thermal fluctuations on both phase and magnetization dy-\nnamics. Therefore we present an exhaustive analysis of the\nnoisy dynamics of a current-biased SFS Josephson junction,\nsee Fig. 1, considering the influence of stochastic thermal fluc-\ntuations. Indeed, we explore this current-induced magnetic\nbistability, in order to define two well-distinguishable logic\nstates, and investigate the robustness of such memory against\nnoise-induced effects. We also suggest a suitable non-invasive\nreadout scheme based on a current-controlled superconduct-\ning quantum interference device (SQUID), which does not\nrequire an additional magnetic flux to set the optimal oper-\nating point. Moreover, we discuss the intriguing possibility\nof effectively shielding the memory state by voltage gating,\nin a device formed by a ferromagnetic layer with a linear-in-\nmomentum Dresselhaus spin-orbit coupling term.\nThe work is organized as follows: In Sec. II, we present\nthe theoretical model used to describe the time evolution of\nboth the magnetic moment and the Josephson phase of a\ncurrent-driven SFS junction. In Sec. III we discuss the mag-\nSSFx yzIbiasMzImax\nMz\nFIG. 1. S/F/S Josephson junction driven by a rectangular bias current\npulses, Ibias, with amplitude Imax. The z-component of the magneti-\nzation, Mz, is the observable used to define the logic memory states\n0 and 1.arXiv:1907.08454v2  [cond-mat.supr-con]  26 Sep 20192\nnetic configuration of the junction after applying a current\npulse. Specifically we determine its stationary magnetiza-\ntion, as a function of the Gilbert damping parameter and the\nSOC strength. In Sec. IV, we focus on the effect of stochas-\ntic thermal fluctuations on both the phase and the magnetiza-\ntion dynamics. In Sec. V, we explore how a generic Rashba-\nDresselhaus type SOC can affect the stationary magnetization.\nIn Sec. VI, we discuss a feasible readout scheme based on a\ndc-SQUID magnetometer. In Sec. VII we present our conclu-\nsions.\nII. THE MODEL\nWe consider a similar setup as the one studied in Ref. 20. It\nconsists of a SFS junction in which the ferromagnet is a thin\nfilm with an out-of-plane magnetic anisotropy and a Rashba-\nlike SOC, see Fig. 1.\nBecause of the interplay between the exchange field and the\nSOC, the current-phase relation of the SFS Josephson junc-\ntion reads Ij=Icsin(j\u0000j0), where Icis the critical current\nof the junction, jis the Josephson phase difference, and j0\nis the so-called anomalous phase shift. The latter depends on\nseveral parameters of the system, such us the Rashba coeffi-\ncient a[25, 26], the transparency of S/F interfaces, the spin\nrelaxation, and the disorder degree of the junction. The ex-\nact dependence on these parameters is here not crucial, but\nthe geometric one. If we consider a two-dimensional SOC\nwith momenta in the plane of the F film, and the charge cur-\nrent flows in x-direction, then the phase shift j0is propor-\ntional to the y-component of the magnetic moment according\nto [16, 20, 27, 28]\nj0=rMy\nM; (1)\nwhere M=q\nM2x+M2y+M2zis the modulus of the magneti-\nzation vector, and the parameter rquantifies the SOC strength\nand contains the a-dependence. The specific dependence of\nron the SOC will be discussed in Sec. IV, when we consider\nthe coexistence of both Rashba and Dresselhaus SOC contri-\nbutions and their impact on the magnetization dynamics.\nAs discussed in Refs. 20, 22, 24, and 29, Eq. (1) estab-\nlishes a direct coupling between the magnetic moment and\nthe supercurrent. The time evolution of the magnetization can\nbe described in terms of the Landau–Lifshitz–Gilbert (LLG)\nequation [30, 31]\ndM\ndt=\u0000grM\u0002Heff+g\nM\u0012\nM\u0002dM\ndt\u0013\n; (2)\nwhere grdenotes the gyromagnetic ratio. The first term on\nthe right-hand side describes the precession motion around\nHeff, whereas the second term describes the dissipation, which\nis accounted by the phenomenological dimensionless Gilbert\ndamping parameter g. The i-th component of the effective\nfieldHeff, with i=x;y;z, can be calculated as [32]\nHeff;i=\u00001\nW¶F\n¶Mi; (3)where Wis the volume of the F layer, and Fis the free energy\nof the junction, which can be written as\nF=\u0000EJjIbias+Es(j;j0)+EM: (4)\nHere, EJ=F0Ic=(2p)withF0being the flux quantum, Ibiasis\nthe external current in units of Ic,Es(j;j0) =EJ[1\u0000cos(j\u0000\nj0)], and EM=\u0000KW\n2\u0010\nMz\nM\u00112\nis the magnetic energy depend-\ning on the anisotropy constant K. The ratio between the en-\nergy scales of the system will be indicated by the parameter\ne=EJ=(KW).\nThe effective magnetic field calculated from Eq. (3) reads\nHeff=K\nM[ersin(j\u0000rmy)ˆy+mzˆz]; (5)\nwith mx;y;z=Mx;y;z=M.\nSince the normalized components of the magnetization sat-\nisfy the condition m2\nx+m2\ny+m2\nz=1, it is convenient to\nwrite the LLG equations in spherical coordinates [33], so that\nthe normalized components of the magnetization can be ex-\npressed in terms of the polar and azimuthal angles qandf\nas\nmx(t) =sinq(t)cosf(t)\nmy(t) =sinq(t)sinf(t) (6)\nmz(t) =cosq(t):\nBy normalizing the time to the inverse of the ferromagnetic\nresonance frequency wF=grK=M, that is t=wFt, the LLG\nequations in spherical coordinates reduce to the following two\ncoupled equations [33]\ndq\ndt=1\n1+g2\u0010\neHeff;f+geHeff;q\u0011\n(7)\nsinqdf\ndt=1\n1+g2\u0010\ngeHeff;f\u0000eHeff;q\u0011\n; (8)\nwhere the qandfcomponents of the normalized effective\nfield are defined as\neHeff;q=ersin(j\u0000rmy)cosqsinf\u0000mzsinq (9)\neHeff;f=ersin(j\u0000rmy)cosf: (10)\nNext, we discuss how the Josephson phase j(t)of a mag-\nnetic junction responds to a driving bias current. The dy-\nnamics of an overdamped SFS Josephson junction can be\ndescribed through the modified resistively shunted junction\n(RSJ) model [34, 35] generalized to include the anomalous\nphase shift j0=rmy. The phase dynamics is described by the\nequation:\ndj\ndt=w[Ibias(t)\u0000sin(j\u0000rmy)]+rdmy\ndt; (11)\nwhere the time is still normalized to the inverse of the ferro-\nmagnetic resonance frequency and w=wJ=wF, with wJ=\n2pIcR=F0being the characteristic frequency of the junc-\ntion [34] ( Ris the normal-state resistance of the device).3\nmzst\n-1.0-0.500.51.0\nmzst\n-1.0-0.500.51.0(a)\n(d)(b)\n(e)(c)\n(f)\nFIG. 2. Stationary magnetization, mstz, as a function of randg, at\ndifferent values of the amplitude Imaxof the current pulse, in the\nabsence of noise fluctuations. The other parameters are: e=10,\nw=1,s=5, and mz(t=0) = + 1.\nLast term in the right-hand side of Eq. (11) stems from the\ntime derivative of the anomalous phase, ˙j0,cf.Eq. (1), and\nwas ignored in Refs. 20, 22, and 24, but has to be included in\norder to preserve gauge-invariance [35].\nWe assume that the SFS junction is driven by a rectangular\ncurrent pulse, Ibias, centered at tc:\nIbias(t) =\u001a\nImax;tc\u0000s\u0014t\u0014tc+s\n0; elsewhere :(12)\nHere, sis the width and Imaxis the intensity, in units of Ic,\nof the pulse, so that a value Imax>1 indicates a bias current\nlarger than the critical value.\nBecause of the magnetoelectric effect in a j0-junction, the\ncharge current induces an in-plane magnetic moment [27, 28,\n36–39], which in turn acts as a torque on the out-of plane mag-\nnetization of the F layer and eventually leads to its switch-\ning [20, 22].\nIn the next sections we search for an optimal combination\nof system parameters to induce the magnetization reversal.\nSpecifically, we explore the response of the magnetization by\nvarying gandrin suitable ranges, whereas the energy and\ntimescales ratios eandw, are fixed. The energy ratio eranges\nfrom e\u0018100 [20] in systems with weak magnetic anisotropy,\ntoe\u00181 for stronger anisotropy [40]. In our calculation we\nchoose an intermediate value e=10. The typical ferromagnet\nresonance frequency is wF'10 GHz, while the characteristic\nJosephson frequency, usually of the order of gigahertz, may\nbe tuned experimentally. Therefore we choose w=1. As\nlong as the injected bias current is below the critical value, the\nresults discussed in this work are only weakly affected by the\nvalue of w. In contrast, if Ibias>1 the magnetic switching\nwould become more unlikely as wincreases. In particular,\nforw\u001d1 the torque exerted by the Josephson current oscil-\nlates very fast, in comparison with the timescale of the mag-\nnetization [21]. This means that the magnetization would ex-\nperience an effective torque averaged over many oscillations,\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstr=0.1 -γ= 0.25\nImax=0.8\n●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=0.9\n●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.\n●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.1\n●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●\n●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●\n●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.2\n●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●\n●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.3\n●●●\n●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.4\n●●●\n●●●●●●●●●●●●●\n●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●0 10 20 30 40 50-1.0-0.50.00.51.0\nσmzstImax=1.5FIG. 3. Stationary magnetization, mstz, as a function of the current\npulse width, s, at different values of the amplitude Imax2[0:8\u00001:5],\nin the absence of noise fluctuations. The other parameters are: r=\n0:1,g=0:25,e=10,w=1, and mz(t=0) = + 1.\nwhich results in a small contribution due to a partial cancella-\ntion of the net torque.\nAtt=0 we assume that the F magnetization points towards\nthez-direction, that is M= (0;0;1). With this initial values\nwe solve Eqs. (7)-(8) and (11) self-consistently for different\nvalues of the parameters. From the solution we determine the\nmagnetization direction after the current pulse.\nIII. THE DETERMINISTIC ANALYSIS\nWe first neglect the effect of thermal noise, as in Refs. 22\nand 24, and explore the magnetic switching of the junction.\nThe overall behavior of the stationary magnetization mst\nz,\nnamely, the value of mzat the time t=tmax=100, as a\nfunction of both the Gilbert damping parameter, g, and the\nSOC strength, r, at different current pulse intensities Imaxand\ns=5, is summarized in Fig. 2. We note that each contour plot\nis characterized by a dark fringes pattern, namely, we observe\nregions of the (g;r)parametric space in which the magneti-\nzation reversal systematically occurs, i.e., in which mst\nz=\u00001.\nThis means, for instance, that by increasing rat a fixed gwe\nobserve a sequence of mst\nz= +1 and mst\nz=\u00001 values.\nFor small enough rvalues, the magnetization reversal effect\nis absent. Interestingly, the pattern of dark fringes evidently4\nchanges for Imax>1, so that for Imax=1:6 the dark fringes\nmerge together in large areas of (g;r)values in which magne-\ntization reversal takes place.\nWith the aim of selecting a driving pulse suitable for a\nmemory application, we observe that the magnetization rever-\nsal effect should be sufficiently robust against small changes\nof both the current pulse intensity and the duration of the\npulse. In Fig. 3 we show the stationary magnetization as a\nfunction of the pulse width s, by changing the pulse am-\nplitude Imax. We have chosen the parameters r=0:1 and\ng=0:25. We observe that for a bias current below the crit-\nical value, the need for an accurate current-width regulation is\nsignificantly relaxed, since the magnetization reversal defini-\ntively occurs for any width above a specific value. Instead, for\ncurrent amplitudes higher than the critical value, the station-\nary magnetization versus sis highly scattered between the\ntwo possible values, mst\nz=\u00061, so that even a slight change\nin the pulse width may lead to a non-switching situation. To\nunderstand this behavior, we observe that, for a bias current\nhigher than Ic, a pulse sufficiently long can make the junction\nto switch to the resistive state, so that the Josephson phase\nrapidly evolves and a voltage drop across the device appears,\nsince Vµdj\ndt[34]. In this case, the steady magnetization\nstrongly depends on the dynamical state of the system when\nthe current pulse is switched off. Moreover, the higher the bias\ncurrent is, the faster the Josephson phase evolve and more pro-\nnounced will be the s-dependence of mst\nz. In view of a mem-\nory application, a current pulse smaller than the critical value\nis therefore recommended, in order to make the magnetization\nreversal unrelated on the pulse width s. For these reasons in\nthe subsequent analysis we set Imax=0:9 and s=5 and fo-\ncus on the thermal effect on the phase and the magnetization\ndynamics.\nIV . EFFECTS OF NOISE\nIn this section, we focus on the noisy dynamics of the junc-\ntion, specifically on how it affects the magnetization reversal.\nThe temperature can significantly influence the time evolution\nof the system, eventually inducing unwanted magnetization\nflip or preventing a stable magnetization reversal. Therefore,\nwe consider stochastic thermal fluctuations in both the phase\nand the magnetic moment dynamics. We start discussing the\neffect of a thermal noise source only on the phase within the\nRSJ model. In the second part of this section we include also\nthe thermal noise on the magnetization dynamics.\nA. Thermal current effects on the RSJ model\nThe phase dynamics can be directly perturbed by thermal\nfluctuations accounted by adding to the RSJ model, Eq. (11),\na Langevin Gaussianly distributed, delta correlated stochas-\ntic term, Ith(t). This “thermal current” has the usual white-\nnoise statistical properties that, in normalized units, can be\nFIG. 4. (a) Stationary magnetization, mstz, as a function of rand\ng. (b) Average stationary magnetization, mstz, as a function of rand\ng, atDI=0:01, calculated by averaging over Nexp=100 indepen-\ndent numerical repetitions. (c) Average stationary magnetization,\nmstz, as a function of the thermal current intensity, DI, atg=0:25\nandr=0:1 [namely, the (g;r)-values highlighted with a red circle\nin panel (b)], calculated by averaging over Nexp=1000 independent\nnumerical repetitions. The inset shows the normalized temperatures\ncorresponding to the noise intensities DI. For all panels Imax=0:9\nands=5, whereas the values of other parameters are the same used\nto obtain Fig. 2.\nexpressed as [34, 41, 42]\nhIth(t)i=0 (13)\nIth(t)Ith(t0)\u000b\n=2DId\u0000\nt\u0000t0\u0001\n: (14)\nHere, we introduced the dimensionless amplitude of thermal\ncurrent fluctuations defined as\nDI=kBT\nRwF\nI2c=1\nwkBT\nEJ: (15)\nFor example, if w=1 we obtain DI\u00180:04T\nIcmA\nK, so that, for\ninstance, a junction with Ic=1mA atT=250 mK is affected\nby a thermal fluctuation of intensity DI\u001810\u00002.\nBy taking into account the noise contribution, Eq. (11) be-\ncomes\ndj\ndt=w[Ibias(t)\u0000sin(j\u0000rmy)+Ith(t)]+rdmy\ndt:(16)\nIn Fig. 4 we compare the current-induced magnetization re-\nversal obtained without and with accounting of the noise ef-\nfects, see panel (a) and (b) respectively. We set the intensity\nof the current pulse Imax=0:9 and its width s=5.5\nIn Fig. 4(a) we show the behavior of mst\nzas a function of\nrandgin the deterministic case, namely, in the absence of\nnoise, DI=0. Here, we observe a contour plot composed by\nmany narrow dark fringes in which mst\nz=\u00001, see Fig. 4(a).\nThe situation drastically changes if we include the thermal\nnoise. In this case we focus on the average stationary magne-\ntization, mstz, which is computed by averaging the stationary\nmagnetization over Nexp=100 independent numerical runs.\nThe behavior of mstzas a function of randgforDI=0:01 is\nillustrated in Fig. 4(b). At small values of rthe magnetiza-\ntion reversal is still absent, whereas noise mostly affects the\nregions with large rwhere the averaged value of the magne-\ntization is mainly distributed around zero. Nevertheless, one\ncan still identify dark regions in which magnetization switch-\ning takes place. With a red circle we highlight in Fig. 4(b) the\nregion around the point (g;r) = ( 0:25;0:1)where the mag-\nnetization takes the largest negative average magnetization\nmstz'\u00001. In other words, the region with the most robust\nswitching.\nBy increasing the noise intensity the switching process is\nsuppressed, as shown in Fig. 4(c), at the optimal values r=0:1\nandg=0:25 in Fig. 4(b). In Fig. 4(c) the value of mstzis\nthe average over Nexp=1000 independent numerical repeti-\ntions. In the inset we show the normalized temperatures cor-\nresponding to the noise intensities DI, calculated by assuming\na junction with a temperature-dependent critical current and\nIc=100mA at low temperatures [43]. From this figure, one\nsees that mstz'\u00001 only for DI.0:01, that is for T.0:75Tc.\nFor higher noise intensities both the average magnetization\nand the error bar increase, approaching a zero magnetization\naverage only for DI&0:3.\nIn Fig. 5 we explore the time evolution of the different ob-\nservables with and without thermal noise. Specifically, we\nshow the response of the junction with g=0:25 and r=0:1\nto a current pulse with amplitude Imax=0:9 depicted in panel\n(a). In the absence of noise, DI=0, we plot in Fig. 5(b)\nthe time evolution of the phase and the supercurrent, and in\nFig. 5(c) the different components of the magnetic moment.\nDuring the current pulse, i.e., the yellow shaded region, the\nphase first increases, and then it goes to zero when the pulse\nis turned off, see Fig. 5(b). To understand the phase behav-\nior, we observe that, in the washboard-like picture [34], the\ntilting imposed by the bias current Imax=0:9 is not enough\nfor allowing the “particle” to overcome the nearest potential\nbarrier and switch the system to the finite voltage “running”\nstate. Instead, the phase-particle remains confined within a\npotential minimum, so that when the current is turned off, the\nslope of the washboard potential goes again to zero and the\nphase restores its initial position, i.e.,j!0.\nWe observe that the larger the bias current pulse is, the\nhigher is the washboard potential slope, and therefore for\nImax>1 the greater the speed of the phase particle, so that\nit can take a longer time to restore the initial position after\nthe current pulse is switched off. Moreover, a large bias cur-\nrent pulse may also longer switching times. Hence, a current\nImax<1 is, in general, more advantageous for a memory ap-\nplication.\nIn Fig. 5(c) we show how all components of the magne-\n0 10 20 30 400.00.20.40.60.81.0Ibias\nφ/π\nIφ\n0.00.20.40.60.8\nmx\nmy\nmz\n0 10 20 30 40-1.0-0.50.00.51.0mx,my,mz\n-0.20.00.20.40.60.81.0\nDI=0.05\n0 10 20 30 40-1.0-0.50.00.51.0\ntmx,my,mzDI=0.05(b)\n(c)\n(d)(a)\n(e)FIG. 5. Current pulse (a) and following time evolution of phase and\nJosephson current, see panel (b), and magnetization components, see\npanel (c), in the absence of noise and including a thermal current\ncontribution with amplitude DI=0:05, see panels (e) and (d). The\nvalues of other parameters are Imax=0:9,s=5,r=0:1, and g=\n0:25. The legends in panels (b) and (c) refer also to panels (d) and\n(e), respectively.\ntization are induced by the current pulse. Whereas mxand\nmyare generated during the current pulse, and they undergo a\ndamped oscillations around zero when the current is switched\noff, the z-component, after a transient regime, flips definitively\nto the value mz=\u00001. From this figure we can also estimate\nthe switching time tSW'10, as the time mzroughly takes to\napproach the value \u00001 after switching off the current.\nThe scenario described so far essentially persists also in the\nstochastic case, as shown in Figs. 5(d-e) for DI=0:05. There-\nfore, at the temperature that we are considering, the overall\nbehavior is still quite similar to the one obtained in the ab-\nsence of noise. In fact, the z-component of the magnetization\nflips again to the value \u00001, while the xandycomponents tend\nto oscillate around zero, without, however, vanishing defini-\ntively.\nThe magnetization switch can be achieved in a short time\nscale, by passing through the junction a sequence of current6\nFIG. 6. Time evolution of the magnetization mz, in response to a se-\nquence of three current pulses shown in the top panel, in the presence\nof a thermal current noise with amplitude DI=0:05. The values of\nthe other parameters are: Imax=0:9,s=5,r=0:1, and g=0:25.\npulses, as it is shown in Fig. 6. In the bottom panel we show\nthe time evolution of the magnetization mz, when the junction\nis excited by the three subsequent current pulses presented\nin the top panel, in the presence of a thermal current noise\nwith amplitude DI=0:05. In response to each current pulse,\nmzfollows first a transient regime, and then, as the current is\nswitched off, it approaches the steady value with an opposite\nsign.\nB. Effect of thermal noise on the magnetization dynamics\nThermal noise also affects directly the magnetization dy-\nnamics [44–49] via a stochastic field Hth, a sort of “thermal\nfield”, which is added to the effective magnetic field term in\nEq. (2), as done in Ref. [50]. Inclusion of the thermal noise in\nEq. (2) leads to [33]\ndM\ndt=\u0000grM\u0002(Heff+Hth)+g\nM\u0012\nM\u0002dM\ndt\u0013\n: (17)\nThis stochastic differential equation has to be solved numeri-\ncally by a stochastic integration prescription by keeping the\nmodulus of the magnetic moment constant during the time\nevolution (see Ref. 33 and references therein). For this pur-\npose it is again convenient to write the equations in spherical\ncoordinates, see Eq. (6), so that the stochastic LLG equation\nreads [33, 51]:\ndq\ndt=1\n1+g2h\neHeff;f+eHth;f+g\u0010\neHeff;q+eHth;q\u0011i\n(18)\nsinqdf\ndt=1\n1+g2h\ng\u0010\neHeff;f+eHth;f\u0011\n\u0000eHeff;q\u0000eHth;qi\n;(19)\nwhere\neHth;q=eHth;xcosqcosf+eHth;ycosqsinf\u0000eHth;zsinq(20)\neHth;f=\u0000eHth;xsinf+eHth;ycosf: (21)\nThe normalized field, eHth= (M=K)Hthis assumed to be a\n10-40.001 0.01 0.1 1-1.0-0.50.00.51.0\nDImzstr = 0.1  - γ = 0.25  - Nexp = 1000FIG. 7. Average stationary magnetization, mstz, as a function of the\nnoise intensity, DI, calculated by taking into account both the thermal\ncurrent and the thermal field noise contribution, and by averaging\nover Nexp=1000 independent numerical repetitions. The values of\nother parameters are Imax=0:9,s=5,r=0:1, and g=0:25.\nGaussianly distributed random field with the following statis-\ntical features\nD\neHth;i(t)E\n=0 (22)\nD\neHth;i(t)eHth;i(t0)E\n=2DHd\u0000\nt\u0000t0\u0001\n; (23)\nwhere i=x;y;zand\nDH=\u0012g\nMkBT\njgrjW\u0013\u0012M\nK\u00132\nwF=gkBT\nKW(24)\nis the dimensionless amplitude of thermal field fluctuations.\nIn all previous equations the time is still normalized to the\ninverse of wF.\nInterestingly, by recalling the definition of the parameter\ne=EJ=(KW), from Eqs. (15) and (24) we can easily obtain\nthe following relation between the normalized thermal noise\nintensities\nDH= (g ew)DI: (25)\nThus, by changing the magnetization energy, the Gilbert\ndamping parameter, or the magnetic resonance frequency we\ncan effectively modify the relative strength of the two noise\nmechanisms. This means that one could optimize the system\nparameters in such a way to make, for instance, the impact of\nthe thermal field negligible with respect to the thermal current.\nThis allows us to study the effects produced by these noise\nsources independently. In the following, even if we explicitly\nwrite only the value of DI, we are taking into account both\nthermal current and thermal field independent noise sources,\nwhich amplitudes are related by Eq. (25).\nThe overall effect of both the thermal current and field is\npresented in Fig. 7, where we show the behavior of mstz, calcu-\nlated by averaging over Nexp=1000 independent numerical\nruns, at different values of the noise intensity DI, and by set-\ntingImax=0:9,s=5,g=0:25, and r=0:1. We observe that\nthe average magnetization remains close to the value mstz'\u000017\nFIG. 8. Time evolution of phase and Josephson current (a) and mag-\nnetization components (b) as the system is excited by the current\npulse in Fig. 5. Here we are taking into account both a thermal cur-\nrent and a thermal field contribution, with noise intensity DI=0:005.\nThe values of other parameters are Imax=0:9,s=5,r=0:1, and\ng=0:25.\nonly for DI.0:003, that is for T.0:58Tc, see the inset\nof Fig. 4(c). For larger values of DI,mstzapproaches zero\nand hence the magnetization reversal probability is reduced,\nFig. 4(c). In view of the memory application, one should, in\nprinciple, carefully choose the F layer and its characteristics\n(such as its volume or the Gilbert damping parameter) in or-\nder to make the thermal field effect as small as possible. The\naim is to reduce the thermal field intensity in order to increase\nthe working temperature suitable for a memory application,\ne.g., through a lower Gilbert damping or a larger F volume,\naccording to Eq. (25).\nThe time evolution of j,Ij, and mi(with i=x;y;z), as\nthe junction dynamics if affected by both a thermal current\nand a thermal field, for r=0:1,g=0:25, and DI=0:005, is\nshown in Fig. 8. Here, we consider again the system excited\nby a current pulse with intensity Imax=0:9, as that one shown\nin Fig. 5(a). We observe that all noisy curves still resemble\nin shape the deterministic evolution presented in Figs. 5(b)-\n(c). The value tSW'10 is a quite good estimation for the\nswitching time of the device also in this noisy case.\nV . RASHBA-DRESSELHAUS SOC\nIn all previous analysis it was assumed a pure Rashba SOC.\nHowever, the theory of j0-junctions can be generalized for\nany linear-in-momentum SOC [27, 28], by using the SU(2)-\ncovariant formulation [28], where the SOC is described in\nterms of a SU(2) vector potential A. For a 2D SOC with\nboth Rashba and Dresselhaus contributions one obtains Ax=\n\u0000asy+bsxandAy=asx\u0000bsy(here, aandbare the\nRashba and Dresselhaus coefficients and sxandsyare the\nfirst two Pauli matrices).\nThe appearance of the anomalous phase is related to the\nexistence of a finite Liftshitz invariant term in the free en-\nergy [16, 52–54] which is proportional to Ti¶ij, where Tiis the i-th component of a polar vector which is odd under\ntime reversal, ¶iis the i-the derivative of the superconduct-\ning phase, and the sum over repeated indices is implied here\nand below. For the particular junction geometry sketched in\nFig. 1 the supercurrent, and hence the phase gradient, is fi-\nnite in x-direction. Thus, according to Eq.(5.17) of Ref. 27,\nthe anomalous phase can be written in the following compact\nform [27, 28]:\nj0=reb(ebmx+my): (26)\nHere, we defined the SOC coefficients ratio,eb=b=a, and\nthe parameter reb=r(1\u0000eb2), with rdepending this time on\nbothaandb. In the absence of the Dresselhaus SOC, that\nis wheneb=0 and reb!r, we recover Eq. (1). If both con-\ntributions are similar in magnitude, i.e., wheneb!1, since\nreb!0 the phase shift vanishes, i.e.,j0!0. This is a very\ninteresting situation that we explore in this section. In fact,\nwhereas the Dresselhaus contribution is due to the breaking\nof crystal inversion symmetries, the Rashba SOC stems from\nstructural broken symmetry and therefore can be controlled by\na gate voltage [55, 56]. In other words, a voltage gate can con-\ntrol the ratioebbetween Dresselhaus and Rashba coefficients,\nand hence the phase shift and the supercurrent flow, accord-\ning to Eq. (26). Specifically, by tuning asuch thateb'1 one\ncan fully decouple the phase and magnetic moment dynamics.\nSuch a process can be eventually used to protect the memory\nstate in one of the storage elements of a distributed architec-\nture.\nWe provide next a quantitative analysis of this situation, so\nthat by taking into account the generic j0, Eq. (26), into the\nexpression for the effective field, Eq. (5) becomes\nHeff=K\nM\u001a\neberebsinh\nj\u0000reb\u0010ebmx+my\u0011i\nˆx+\nerebsinh\nj\u0000reb\u0010ebmx+my\u0011i\nˆy+mzˆz\u001b\n:(27)\nTheqandfcomponents of the normalized effective field to\nbe included in LLG Eqs. (7)-(8), read\neHeff;q=eberebsinh\nj\u0000reb\u0010ebmx+my\u0011i\ncosqcosf (28)\n+erebsinh\nj\u0000reb\u0010ebmx+my\u0011i\ncosqsinf\u0000mzsinq\neHeff;f=\u0000eberebsinh\nj\u0000reb\u0010ebmx+my\u0011i\nsinf (29)\n+erebsinh\nj\u0000reb\u0010ebmx+my\u0011i\ncosf;\nwhereas the RSJ equation becomes\ndj\ndt=wn\nIbias(t)\u0000sinh\nj\u0000reb\u0010ebmx+my\u0011io\n(30)\n+reb\u0012\nebdmx\ndt+dmy\ndt\u0013\n:\nThe behavior of the stationary magnetization as a function of\nrandg, at different values ofeb2[0\u00001]is shown in Fig. 9.8\nmzst\n-1.0-0.500.51.0\nmzst\n-1.0-0.500.51.0(a)\n(d)(b)\n(e)(c)\n(f)\nFIG. 9. Stationary magnetization, mstz, as a function of randg, at\ndifferent values of the relative Dresselhaus coefficienteb=b=a, in\nthe absence of noise fluctuations, by imposing Imax=0:9 and s=5.\nThe current pulse intensity and width are chosen equal to\nImax=0:9 and s=5, respectively. As expected from the dis-\ncussion above, the region where no magnetization switching\noccurs, bright color in Fig. 9, increases by increasingebto-\nwards 1. Foreb=1, the j0behavior is fully suppressed, cf.\nEq. (26), and hence no magnetization switching takes place,\ndespite the current pulse flowing through the junction. Inter-\nestingly, we note that for intermediate values ofeb, the area of\nthe switching fringes, i.e., where mst\nz=\u00001, increases consid-\nerably.\nIn summary of this section, by tuning a=bone can decou-\nple the magnetic behavior from the Josephson dynamics and\nfreeze the memory state in order to protect it from external\ncurrent pulses and other perturbations. The major challenge\nin this regard is to find materials with a sizable magnetic mo-\nment and tunable by means of a gate voltage.\nVI. THE MEMORY READ-OUT\nAs discussed in previous section, the writing operation of\nthe proposed memory element can be performed by exciting\nthe junction with controlled current pulses. We could envis-\nage an array of j0-junction-based memory elements, each one\neventually having its own current line so that it can be written\nby sending individual current inputs. Alternatively, exploiting\nthe tuning of the SOC discussed in previous section, one could\ncontrol locally, via individual gates at each junction, several\nmemory elements connected in series to a common current\nline. In this way one could selectively write via a common\ncurrent pulse only a specific set of memory units.\nThe read-out of the memory state can be non-destructively\nperformed by direct measurement of the magnetization state\nthrough a dc-SQUID inductively coupled to the j0-junction.\nA SQUID is essentially a magnetic flux detector [57], which\ncan be employed to measure with a very high sensitivity any\nV V\nIbias MzIreadΦµ\nIbias MzIread\n (a) (b)ΦµSQ\nSQFIG. 10. SQUID-based memory readout and cartoon showing the\ncritical current interference pattern of the SQUID, in the cases of both\npositive and negative orientation along the z-axis of the magnetic\nmoment, see panel (a) and (b), respectively.\nphysical quantity that can be converted in a magnetic flux [58].\nWe suggest a SQUID sensor along the lines of the read-\nout scheme implemented in Ref. 13 for a p-junction mem-\nory. Our scheme is based on an asymmetric inductive dc-\nSQUID, which consists of a superconducting ring with a non-\nnegligible total inductance, L, with two Josephson junctions\nwith different critical currents, i.e.,Ic;16=Ic;2(here, we are ne-\nglecting for simplicity any asymmetry in the ring inductance).\nWith such asymmetric SQUID, one can avoid the use of an ad-\nditional magnetic flux to adjust the working point of the device\nin a high sensitivity position of the ISQ\nc\u0000Fcharacteristics,\nwhere ISQ\ncis the SQUID critical current and Fis the magnetic\nflux threading the loop. In fact, such asymmetric dc-SQUID\nshows non-negligible screening and asymmetry parameters,\nthat is bL=2p\nF0L\n2(Ic;1+Ic;2)6=0 and aI=Ic;1\u0000Ic;2\nIc;1+Ic;26=0, re-\nspectively. In this case, the ISQ\ncmaximum is not centered in\nF=0, but shifted from zero by DF, see Fig. 10, where [57]\n2pDF=F0bLaI. Accordingly, our readout SQUID demon-\nstrates a high sensitivity point of the ISQ\nc\u0000Fcharacteristics\nalso in F=0, that is in the absence of an external magnetic\nflux.\nWe assume that the unbiased critical current, ISQ\nc(F=0),\nlies in the positive branch of the critical current diffraction\npattern of the SQUID, that isdISQ\nc\ndF\f\f\f\nF=0>0, just like in the\ncase shown in Fig. 10. Thus, the magnetic moment mz= +1\ngenerates a positive magnetic flux F= +Fmthrough the loop,\nand gives a critical current higher than the unbiased value, i.e.,\nI+\nc=ISQ\nc(+Fm)>Ic(0), see the red dashed line in Fig. 10(a).\nConversely, if mz=\u00001 the magnetic flux through the loop\nis negative, i.e.,F=\u0000Fm, and the critical current is lower\nthan the unbiased value, I\u0000\nc=ISQ\nc(\u0000Fm)<Ic(0), see the blue\ndashed line in Fig. 10(b).\nThe SQUID readout loop is then sensed by passing a bit-\nread current with intensity Iread=ISQ\nc(0)through the device,\nsee the orange dashed line in Fig. 10, that is a current which\nlays in between I\u0000\ncandI+\nc. In this way, if the magnetic mo-\nment points in the negative z-direction, which encodes the9\n“1” logic state of the memory, the bit-read current makes the\nSQUID to switch to the voltage state, since Iread>I\u0000\nc. Con-\nsequently, a voltage drop appears across the readout SQUID\nin response to the bit-read current. For the opposite magnetic\nmoment orientation, which encodes the “0” logic state, the\nSQUID critical current is larger than the bit-read current, that\nisIread<I+\nc, so that the SQUID remains in the superconduct-\ning, zero-voltage state.\nTo provide an estimate of the dimensions of a SQUID loop\nsuitable to detect the magnetic moment reversal we consider\nthe simple case of a circular readout SQUID with radius a\nand a magnetic moment oriented along the z-axis and placed\non the symmetry axis of the SQUID at a height z0above the\nx\u0000yplane. In this case, the total magnetic flux through the\nloop can be estimated as [58, 59] Fm=m0Mz\n2a2\u000e\u0000\na2+z02\u00013=2.\nThis flux is maximum at z0=0 where it reads Fm=m0Mz\nd,\nwith d=2abeing the loop diameter. If we assume a satura-\ntion magnetization [60] M=8\u0002105A=m and a volume W=\n(10\u0002100\u0002100)nm3=10\u000022m3, we get a total magnetic\nmoment Mz=8\u000210\u000017Am2(that is Mz=8:6\u0002106mB,\nwithmBbeing the Bohr magneton), so that the total magnetic\nflux reads Fm=32p10\u000024d\u00001Wb m. Thus, for a SQUID with\ndiameter'0:5mm we obtain a measurable flux Fm'F0=10,\n(see vertical dot-dashed lines in Fig. 10).VII. CONCLUSIONS\nIn conclusion, we propose a non-volatile superconducting\nmemory based on a bistable magnetic behavior of a current-\nbiased j0-junction, that is a superconductor-ferromagnet-\nsuperconductor Josephson junction with a Rashba-like spin-\norbit coupling (SOC). The memory state is encoded in the\nmagnetization direction of the ferromagnetic layer, which can\nbe switched via controlled current pulses. We explore the ro-\nbustness of the current-induced magnetization reversal against\nthermal fluctuations, in order to find the optimal working tem-\nperature at which the magnetization switching induced by a\ncurrent pulse is stable. We also suggest a way of decoupling\nthe Josephson phase and the magnetization dynamics, by tun-\ning the Rashba SOC strength via a gate voltage.\nFinally, we discuss a suitable non-destructive readout\nscheme based on a dc-SQUID inductively coupled to the j0-\njunction. The suggested readout scheme is exclusively con-\ntrolled by current pulses, and no additional magnetic flux is\nneeded.\nACKNOWLEDGMENTS\nThis work was supported by the Horizon Research and In-\nnovation Program under Grant Agreement No. 800923 (SU-\nPERTED) and the Spanish Ministerio de Economía, Industria\ny Competitividad (MINEICO) under Project No. FIS2017-\n82804-P.\n[1] Q. P. Herr, A. Y . Herr, O. T. Oberg, and A. G. Ioannidis, “Ultra-\nlow-power superconductor logic,” J. Appl. 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Buzdin,\n“Enhancement of the superconducting transition temperature in\nNb=permalloy bilayers by controlling the domain state of the\nferromagnet,” Phys. Rev. Lett. 93, 057002 (2004).\n[61] F. S. Bergeret, A. F. V olkov, and K. B. Efetov, “Odd triplet\nsuperconductivity and related phenomena in superconductor-\nferromagnet structures,” Rev. Mod. Phys. 77, 1321–1373\n(2005)."    },    {        "title": "1907.09039v1.Critical_Thresholds_in_One_Dimensional_Damped_Euler_Poisson_Systems.pdf",        "content": "arXiv:1907.09039v1  [math.AP]  21 Jul 2019CRITICAL THRESHOLDS IN ONE DIMENSIONAL DAMPED\nEULER-POISSON SYSTEMS\nMANAS BHATNAGAR AND HAILIANG LIU\nAbstract. This paper is concerned with the critical threshold phenomenon for one\ndimensional damped, pressureless Euler-Poisson equations with ele ctric force induced by\na constant background, originally studied in [S. Engelberg and H. Liu a nd E. Tadmor,\nIndiana Univ. Math. J., 50:109–157, 2001]. A simple transformation is used to linearize\nthe characteristicsystem of equations, which allowsus to study th e geometricalstructure\nof critical threshold curves for three damping cases: overdampe d, underdamped and\nborderlinedampedthroughphaseplaneanalysis. Wealsoderivethee xplicitformofthese\ncritical curves. These sharp results state that if the initial data is within the threshold\nregion, the solution will remain smooth for all time, otherwise it will hav e a finite time\nbreakdown. Finally, we apply these general results to identify critic al thresholds for a\nnon-local system subjected to initial data on the whole line.\n1.Introduction\nIt is well known that the finite-time breakdown of the systems of Eu ler equations for\ncompressible flows is generic in the sense that finite-time shock form ation occurs for\nall but a “small” set of initial data. For pairs of conservation laws, La x [17] showed\nthat the C1-smoothness of solutions can be lost unless its two Riemann invariant s are\nnondecreasing. On the other hand, with the Poisson forcing the sy stem of Euler–Poisson\nequationsadmitsa“large”setofinitialconfigurationswhichyieldglob alsmoothsolutions,\nsee, e.g. [8, 20, 21]. The Euler-Poisson system of equations is used t o model various\nphenomena, ranging from plasma physics to applications in semicondu ctors. Physically,\nwe would desire to know as to whether the concerned particles aggr egate or the smooth\ndensity profile exists forever. Indeed, for a class of Euler–Poisso n equations, the question\naddressed in [8] is whether there is a critical threshold for the initial data such that the\npersistence ofthe C1solutionregularitydepends onlyoncrossing such acritical thresho ld.\nFor example, for system of Euler–Poisson equations with only electr ic force,\nρt+(ρu)x= 0,\n(ρu)t+(ρu2)x=−kρφx,\n−φxx=ρ−c,\nsubject to initial data ( ρ0>0,u0(x)), [8] has shown that for repulsive force k >0 it\nadmits a global smooth solution if and only if\nu0x(x)>−/radicalbig\n2kρ0(x) ifc= 0,\n1991Mathematics Subject Classification. Primary, 35L65; Secondary, 35L67.\nKey words and phrases. Euler-Poisson system, critical threshold, global regularity, shoc k formation.\n12 MANAS BHATNAGAR AND HAILIANG LIU\nand\n|u0x(x)|</radicalbig\nk(2ρ0(x)−c) ifc >0.\nThese two critical thresholds indicate that with the background ch arge, the solutions of\nthe above system will be oscillatory, hence initial slope cannot be too big either. The zero\nbackground case when augmented with the usual γ-law pressure, is shown by Tadmor\nand Wei [27] to still admit global solutions for a large class of initial dat a identified by\nan intrinsic critical threshold. The non-zero background case with pressure is different.\nUsing special energy techniques with proper normal form of trans formations, the authors\nin [11] have shown that smooth solutions with small amplitude persist f orever with no\nshock formation in the case of cubic law of pressure.\nIn this paper we revisit the onedimensional pressureless, damped E uler–Poisson system\nwith potential induced by a constant background,\nρt+(ρu)x= 0,\nut+uux=−kφx−νu,\n−φxx=ρ−c,(1.1a)\nsubject to initial conditions,\nρ(x,0) =ρ0(x)>0, ρ 0∈C1(R),\nu(x,0) =u0(x), u 0∈C1(R),(1.1b)\nwherec >0 is the constant background, ν >0 is the damping coefficient, and parameter\nksignifies theproperty ofthe underlying forcing, repulsive if k >0 and attractive if k <0.\nWe consider only repulsive force between particles and hence, k >0. Here, we also need\nthe neutrality condition/integraldisplay∞\n−∞(ρ0(ξ)−c)dξ= 0,\nwhich is conserved for all time if ρuvanishes at far fields. Therefore, we have a fixed\nbackground charge density of cand an equal amount of movable charge, ρ(x,t).\nThe main objective of our revisit to this problem is to introduce altern ative tools,\ninsteadoftheuseofflowmaptechniques in[8], toidentifythecritical thresholdsfor(1.1a).\nWe hope these tools can be useful for the study of critical thresh old phenomena in other\nproblems of similar nature. More precisely, we want to get an explicit c haracterization\nof the critical threshold curve as a function of initial density and ve locity slope for three\ndifferent cases:\n(1)ν >2√\nkc, strong damping,\n(2)ν <2√\nkc, weak damping, and\n(3)ν= 2√\nkc, borderline damping.\nWe are able to recover the results in [8] for weak damping case, and o btain a sharp critical\nthreshold for strong damping case, for which only a sufficient condit ion was identified in\n[8], plus a sharp critical threshold for the borderline case.\nWe present two methods in analysis, each gives the critical thresho ld curve for all three\ncases but by different techniques. The initial step in both methods is to transform a non-\nlinear system of equations into a linear system and then analyze the o btained system.\nThe first method is more rudimentary and involves explicit solution tec hniques of linearCRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 3\ndifferential equations with constant coefficients. The second meth od involves vector field\nanalysis.\nOn the solution behavior of Euler–Poisson equations there is a consid erable amount\nof literature available. Consult [9, 28] for nonexistence results and singularity formation;\n[7, 29] for global existence of weak solutions with geometrical symm etry; [10, 12, 13] for\nglobal existence for 3-D irrotational flow, [24] for isentropic case , and [25] for isothermal\ncase. Smoothirrotationalsolutionsforthe two dimensional Euler– Poisson system arecon-\nstructed independently in [14, 23]. See also [15, 16] for related resu lts on two dimensional\ncase. The question of critical thresholds in multi-D Euler-Poisson sy stems remains largely\nopen; we refer to [21] for sharp conditions on global regularity vs finite time breakdown\nfor the 2-D restricted Euler–Poisson system, and [20] for sufficien t conditions on finite\ntime breakdown for the general n-dimensional restricted Euler–P oisson systems. A rel-\native complete analysis of critical thresholds in 3-D restricted Euler –Poisson systems is\ngiven in [19] for both attractive and repulsive forcing.\nAs a direct benefit of our present results, we illustrate how to apply them to an inter-\nesting system in the context of biological aggregation:\nρt+(ρu)x= 0,(x,t)∈R×(0,∞), (1.2a)\nut+uux+u=−∂W ⋆ρ, (1.2b)\nwhere\nW(x) =−|x|+|x|2\n2,\nsubject to initial conditions\nρ(x,0) =ρ0(x)>0, ρ 0∈C1(R),\nu(x,0) =u0(x), u 0∈C1(R).(1.3)\nInstead of electric force governed by the Poisson equation, here non-local interactions\nbetween particles are modeled by Newtonian attractive forces. Sy stem (1.2) has been\nformally derived from interacting particle systems in collective dynam ics; see e.g. [5],\n[6], and kinetic equations for collective behavior can be derived rigoro usly from particle\nsystems via the mean-field limit, see [4, 2], and the references there in.\nWhen initial data is compactly supported, belonging to space ( H2(U),H3(U)), where\nU⊂Rhascompact support, thecritical thresholds forthis problem hav e beenestablished\nin [3] by flow map techniques. We should point out that in [3, Remark 3.1] there is an\nexplanation of what additional assumptions need to be made for the initial data so that\nthe result still holds for U=R. However, the analysis of the local existence result for\nclassical solutions in [3, Appendix A] does not seem to be applicable dire ctly to the setting\nwhen initial data is defined on the whole line. Hence, we present a new lo cal existence\ntheorem of classical solutions and give a self-contained proof using a different approach, in\nwhich some control of solution behavior at far fields is essential. In a ddition, the C1class\nof initial data we consider is larger than the H2×H3class for ( ρ0,u0). With this local\nexistence theory, our results obtained for (1.1a) when applied to ( 1.2) lead to Theorem\n5.2 and Theorem 5.3. To our best knowledge, the geometrical struc ture of the critical\nthreshold curves for(1.2)giveninTheorem 5.3isnew. The explicit thr esholds inTheorem\n5.2 are essentially the same as those in [3], although here the initial da ta is defined on the\nwhole line.4 MANAS BHATNAGAR AND HAILIANG LIU\nA related model is the one-dimensional Euler–alignment system which has a non-local\nvelocity alignment force (such force becomes the linear damping whe n the alignment force\nis localized), for such model thresholds for global regularity vs finit e time breakdown were\nanalyzed in [26]. Such result was further improved in [1] by closing the gap between\nlower and upper thresholds. When both linear damping and nonlocal in teraction forces\nare present, sharp critical thresholds were obtained in [3] for a sp ecial system (1.2) with\nsmooth, compactly supported initial data.\nTherest ofthispaperisorganizedasfollows. InSection2, westate themainresultsand\nintroduce the key transformation as a preparation for the analys is carried out in Sections\n3 and 4. In Section 3, we prove our main results, providing sharp crit ical thresholds for\ninitial configurations which yield either global smooth solution or finite time breakdown.\nIn Section 4, we give dynamic representation of the critical thresh old curve in each case.\nFinally, in Section 5 we apply our obtained results to identify the critica l thresholds for\n(1.2). The proof of the needed local wellposedness result is deferr ed into Appendix A.\n2.Preliminaries and main results\nThe threshold analysis to be carried out is the a priori estimate on sm ooth solutions\nas long as they exist. For the one-dimensional Euler-Poisson proble m, local existence of\nsmooth solutions was long known, it can be justified by using the char acteristic method\nin the pressureless case. We only state the result here.\nTheorem 2.1. (Local existence )Ifρ0∈C1andu0∈C1, then there exists T >0,\ndepending on the initial data, such that the initial value pr oblem (1.1a), (1.1b) admits a\nunique solution (ρ,u)∈C1([0,T)×R).Moreover, if the maximum life span T∗<∞, then\nlim\nt→T∗−∂xu(t,x∗) =−∞\nfor some x∗∈R.\nWeproceedtoderive thecharacteristic system which isessentially u sed toinourcritical\nthreshold analysis. Differentiate the second equation in (1.1a) with r espect to x, and set\nux:=dto obtain:\nρ′+ρd= 0, (2.1a)\nd′+d2+νd=k(ρ−c), (2.1b)\nwhere we have used the Poisson equation in (1.1a) for φ, and\n{}′=∂\n∂t+u∂\n∂x,\ndenotes the differentiation along the particle path,\nΓ ={(x,t)|x′(t) =u(x(t),t),x(0) =α∈R}.\nHere, we employ the method of characteristics to convert the PDE system (1.1a) to ODE\nsystem (2.1) along the particle path which is fixed for a fixed value of t he parameter\nα. Consequently, the initial conditions to the above equations are ρ(0) =ρ0(α) and\nd(0) =d0(α) =u0x(α) for each α∈R.\nThrough analysis of system (2.1) we find the following.CRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 5\nTheorem 2.2. For the given 1D Euler Poisson system (1.1a) with initial dat a (1.1b),\nthere is finite time breakdown iff ∃x∈Rsuch that\n(1)(ν >2√\nkc) Strong damping\nmax{u′\n0(x), u′\n0(x)+λ2(c−ρ0(x))}<0,and\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglecλ2u′\n0(x)+k(c−ρ0(x))\nkρ0(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ1\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsinglecλ1u′\n0(x)+k(c−ρ0(x))\nkρ0(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ2\n,\nwhereλ1=ν−√\nν2−4kc\n2candλ2=ν+√\nν2−4kc\n2c.\n(2)(ν= 2√\nkc) Borderline damping\nmax{u′\n0(x),u′\n0(x)+ν\n2c(c−ρ0(x)}<0,and\nln/parenleftbigg\n−2cu′\n0(x)+ν(c−ρ0(x))\nνρ0(x)/parenrightbigg\n≥2cu′\n0(x)\n2cu′\n0(x)+ν(c−ρ0(x)).\n(3)(ν <2√\nkc) Weak damping\n/parenleftbigg\nu′\n0(x)+ν(c−ρ0(x))\n2c/parenrightbigg2\n≥µ2/bracketleftbiggρ2\n0(x)\nc2/parenleftbiggkceνt∗\nµ2−1/parenrightbigg\n+(2ρ0(x)−c)\nc/bracketrightbigg\n,\nwhereµ=/radicalbig\nkc−ν2/4and\nt∗=1\nµ/bracketleftbigg\nβ+tan−1/parenleftbigg2µu′\n0(x)\nνu′\n0(x)+2k(c−ρ0(x))/parenrightbigg/bracketrightbigg\n,\nβ=\n\n0νu′\n0(x)< min{0,2k(ρ0(x)−c)},\nπ 2k(ρ0(x)−c)< νu′\n0(x),\n2π0< νu′\n0(x)<2k(ρ0(x)−c).\nThis reconfirms the rather remarkable phenomena investigated in [ 8], namely that the\nnon-zero background is able to balance the nonlinear convective eff ects, damping, and the\nrepulsive forces, to yield a global smooth solution if the initial data is w ithin the threshold\nregion.\nThe above result improves and extends the result stated and prov ed in [8].\nIn the next theorem we present critical thresholds in an alternativ e form which can be\ndetermined from the phase plane analysis.\nTheorem 2.3. Consider the 1D Euler Poisson system (1.1a) subject to C1initial data\n(1.1b). There exists a unique solution ρ,u∈C1(R×(0,∞))iff∀x∈R,\n(1)(ν >2√\nkc) Strong damping\n(ρ0(x),u′\n0(x))∈ {(ρ,d) :d >−ρQa(1/ρ), ρ >0},\nwhereQa: [0,∞)−→[0,∞)is a continuous function satisfying\n(2.2)dQa\nds=ν+k\nQa(1−cs), Qa(0) = 0.6 MANAS BHATNAGAR AND HAILIANG LIU\n(2)(ν= 2√\nkc) Borderline damping\n(ρ0(x),u′\n0(x))∈ {(ρ,d) :d >−ρQb(1/ρ), ρ >0},\nwhereQb: [0,∞)−→[0,∞)is a continuous function satisfying\n(2.3)dQb\nds= 2√\nkc+k\nQb(1−cs), Qb(0) = 0.\n(3)(ν <2√\nkc) Weak damping\n(ρ0(x),u′\n0(x))∈ {(ρ,d) :−ρQ1(1/ρ)< d <−ρQ2(s∗−1/ρ), ρ∈(1/s∗,∞)},\nwheres∗>0is uniquely determined, and Q1: [0,s∗]−→R+∪{0}is a continuous\nfunction satisfying\ndQ1\nds=ν+k\nQ1(1−cs), Q1(0) = 0,\nandQ2: [0,s∗]−→R−∪{0}is another continuous function satisfying\ndQ2\nds=−ν+k\nQ2(c(s∗−s)−1), Q2(0) = 0.\nThe details of the proofs of Theorems 2.2 and 2.3 are carried out in Se ction 3 and\nSection 4, respectively.\nThe main tool in our analysis is a transformation of variables with which we can reduce\nthe non-linear system of equations into a linear system. The solution s to the linear system\ncan then be analyzed or analytically found. More precisely, we introd uce\nr=−d\nρ, (2.4a)\ns=1\nρ, (2.4b)\nso that (2.1) reduces to\nr′=−νr−k(1−cs), (2.5a)\ns′=−r. (2.5b)\nClearly, given any initial data, we can find the solution curves r(t) ands(t) which exist for\nallt∈R. In order to return to the original unknowns ( ρ,d), we need to make sure that\nsremains greater than 0 for all t >0. In this way, we avoid the finite time breakdown of\nρ. Consequently, we find dto be bounded from below since\nd=−rρ >−∞ ∀t >0.\nIn all fairness, we should point out that actually s= Γ/ρ0, where Γ is nothing but\nthe ‘indicator’ function introduced in [8] to denote ∂αx(t;α). The variable rhas also\nappeared as β(t) in [8] in the case ν= 0 and c= 0. Here with these two variables\ncombined, we obtain the novel system (2.5) for the first time. The lin earity of (2.5) and\nits special structure allow us to derive explicit solutions as shown in Se ction 3, which is\nessentially the same as those preformed in [8, 3] using the flow map te chniques. However,\nthe geometric structure in terms of phase plane analysis we presen t here has not beenCRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 7\nreported in the literature. We hope this approach to study the geo metric structure of\ncritical threshold curves can be extended to other systems in a dy namic way.\n3.Proof of Theorem 2.2: Analyzing the explicit solution\nDifferentiating (2.5b) and using (2.5a), we obtain the following initial va lue problem\n(IVP) for s,\ns′′+νs′+kcs=k, (3.1a)\ns(0) =1\nρ0, s′(0) =d0\nρ0. (3.1b)\nThe type of damping pertains to the type of solutions to this IVP.\n3.1.Strong damping ( ν >2√\nkc).On solving (3.1), we get,\ns(t) =1\nc/bracketleftbig\n1+Ae−λ1ct+Be−λ2ct/bracketrightbig\n,\nwhere\nA=1\n(λ2−λ1)ρ0[d0+λ2(c−ρ0)],\nB=−1\n(λ2−λ1)ρ0[d0+λ1(c−ρ0)],\nλ1=ν−√\nν2−4kc\n2c, λ2=ν+√\nν2−4kc\n2c.\nAlso notethat if either of AorBis 0, then s(t)>0 forallt >0 is trivially achieved. Since\nour solution comprises negative exponentials, sdecays to 1 /c. Also, such expressions can\nhave at most one extremum. Therefore, on observation we conclu de that if s′(0)≥0,\nthensremains positive; that is, d0≥0 ensures global existence. Next, we differentiate\nthe expression for s,\ns′(t) =−/bracketleftbig\nAλ1e−λ1ct+Bλ2e−λ2ct/bracketrightbig\n,\nand equate it to 0 to obtain an expression for time t=t∗at which the extrema occurs,\ne(λ2−λ1)ct∗=−Bλ2\nAλ1.\nFurthermore, we differentiate s′again to obtain,\ns′′(t) =c/bracketleftbig\nAλ2\n1e−λ1ct+Bλ2\n2e−λ2ct/bracketrightbig\n,\nand write it in the following way,\n=Ace−λ1ctλ1/bracketleftbigg\nλ1+B\nAλ2\n2\nλ1e−(λ2−λ1)ct/bracketrightbigg\n,\nand substitute the expression for e−(λ2−λ1)ct∗to obtain,\ns′′(t∗) =−(λ2−λ1)Ace−λ1ct∗λ1.\nHence, if A >0 then the extremum, if it exists, is a maximum. Furthermore, if A <0, we\nconclude from the expression of t∗thatB >0 becomes a necessary condition for t∗>0 to\nexist. Therefore, A >0 ensures global solution anyways. To obtain the critical threshold\ncurve, we apply the condition that if there exists t=t∗>0, where the minimum of sis8 MANAS BHATNAGAR AND HAILIANG LIU\nachieved, then s(t∗)>0 which, from (3.1a), is equivalent to the condition that s′′(t∗)< k.\nTherefore, we get\n−(λ2−λ1)Ace−λ1ct∗λ1< k.\nAfter substituting the expression for t∗, the above equation can be rewritten as,\n−(λ2−λ1)Acλ1< k/parenleftbigg\n−Bλ2\nAλ1/parenrightbiggλ1\n(λ2−λ1)\n.\nNoting that A <0 andB >0, we get\nc(λ2−λ1)(−Aλ1)λ2\nλ2−λ1< k(Bλ2)λ1\nλ2−λ1.\nOnsubstituting theexpressions for AandBandusing thatthedifference oftheexponents\non either side of the inequality is one, we obtain\n(3.2)/bracketleftbigg\n−cλ1d0+k(c−ρ0)\nkρ0/bracketrightbiggλ2\n</bracketleftbigg\n−cλ2d0+k(c−ρ0)\nkρ0/bracketrightbiggλ1\n.\nFrom the discussion above, we also need d0<0 andd0+λ2(c−ρ0)<0 for uniquely\ndetermining the curve as a critical threshold. These together lead to the conclusion in (1)\nof Theorem 2.2.\n3.2.Weak damping ( ν <2√\nkc).On solving (3.1), we get,\ns(t) =1\nc+e−νt\n2\nc/bracketleftbigg/parenleftbiggc−ρ0\nρ0/parenrightbigg\ncosµt+1\nµ/parenleftbiggcd0\nρ0+ν(c−ρ0)\n2ρ0/parenrightbigg\nsinµt/bracketrightbigg\n,\nwhereµ=√\nkc−0.25ν2. Once again, the idea is that if s= (1/ρ) becomes 0 at some\ntimet=tc, means limt→t−\ncρ(t) =∞, then there is breakdown of the solution. Since, the\nsolutions are decaying with time, we need the first local minimum of sto be greater than\n0. In view of this, we find all the times for which\ns′(t) = 0, s′′(t)>0.\nIn view of this, we calculate\ns′(t) =e−νt\n2/bracketleftbigg/parenleftbiggd0\nρ0/parenrightbigg\ncos(µt)−/parenleftbiggνd0+2k(c−ρ0)\n2µρ0/parenrightbigg\nsin(µt)/bracketrightbigg\n.\nSos′(t) = 0 if\n(3.3) tan( µt) =2µd0\nνd0+2k(c−ρ0).\nWe further perform the second derivative test,\ns′′(t) =ce−νt\n2\nρ0cos(µt)/bracketleftbigg\ntan(µt)/parenleftbigg−4µ2d0+ν2d0+2kν(c−ρ0)\n4µ/parenrightbigg\n−νd0+2k(c−ρ0)\n2−νd0\n2/bracketrightbigg\n,\nwhich in virtue of (3.3) gives\ns′′(t) =−ce−νt\n2cos(µt)\nρ0(νd0+2k(c−ρ0))/bracketleftBigg\n2µ2d2\n0+(νd0+2k(c−ρ0))2\n2/bracketrightBigg\n.CRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 9\nThe condition for a minimum to occur is sgn{cos(µt)}=−sgn{νd0+2k(c−ρ0)}. This\ngives us a sequence of minima at different times. At the first time t∗, we have\nµt∗=β+tan−1/parenleftbigg2µd0\nνd0+2k(c−ρ0)/parenrightbigg\n,\nβ=\n\n0νd0< min{0,2k(ρ0−c)},\nπ 2k(ρ0−c)< νd0,\n2π0< νd0<2k(ρ0−c).\nNote that for\nf(t) =c0+e−γt[c1cos(θt)+c2sin(θt)], γ,θ > 0,\nthe value of the function fat a local minima τis\nc0−e−γτθ/radicalbig\nc2\n1+c2\n2/radicalbig\nθ2+γ2.\nOn comparing swithfabove we have,\nγ=ν\n2, θ=µ, c0= 1/c, c 1=c−ρ0\nρ0, c2=1\nµ/parenleftbiggcd0\nρ0+ν(c−ρ0)\n2ρ0/parenrightbigg\n.\nApplying the above formula on s(t), we have for s(t∗)>0 to hold,\n1−e−νt∗\n2µ/radicalbigg/parenleftBig\nc−ρ0\nρ0/parenrightBig2\n+1\nµ2/parenleftBig\ncd0\nρ0+ν(c−ρ0)\n2ρ0/parenrightBig2\n√\nkc>0.\nThis is equivalent to the following\n(3.4)/parenleftbigg\nd0+ν(c−ρ0)\n2c/parenrightbigg2\n< µ2/bracketleftbiggρ2\n0\nc2/parenleftbiggkceνt∗\nµ2−1/parenrightbigg\n+(2ρ0−c)\nc/bracketrightbigg\n.\nThis proves (2) in Theorem 2.2.\n3.3.Borderline damping ( ν= 2√\nkc).On solving (3.1), we get\ns(t) =1\nc+/bracketleftbigg\nD+/parenleftbiggd0\nρ0+Dν\n2/parenrightbigg\nt/bracketrightbigg\ne−νt\n2,\nwhereD=1\nρ0−1\nc.Settings′(t∗) = 0 to obtain extremum, hence from\ns′(t) =/bracketleftbiggd0\nρ0−/parenleftbiggd0\n2ρ0+Dν\n4/parenrightbigg\nνt/bracketrightbigg\ne−νt\n2,\nwe get\nt∗=4d0\nν(2d0+νDρ0),\nwhich is positive if d0<0 and\n2d0+νDρ0<0.\nThe latter ensures that s′′(t∗)>0 since\ns′′(t) =−ν/bracketleftbiggd0\nρ0+Dν\n4−/parenleftbiggd0\nρ0+Dν\n2/parenrightbiggνt\n4/bracketrightbigg\ne−νt\n2.10 MANAS BHATNAGAR AND HAILIANG LIU\nThe above t∗when inserted into s(t∗)>0 gives us,\n−c/parenleftbigg\nD+2d0\nνρ0/parenrightbigg\n< eνt∗\n2,\nwhich is equivalent to the following\n(3.5) ln/parenleftbigg\n−(2cd0+ν(c−ρ0))\nνρ0/parenrightbigg\n<2cd0\n2cd0+ν(c−ρ0)\nprovided d0<0 and 2cd0+ν(c−ρ0)<0. This proves (3), hence completes the proof of\nTheorem 2.2.\n4.Proof of Theorem 2.3: Critical threshold curve\nWe will look into the geometrical interpretation of the critical thres hold curve for the\n3 cases in this section. First, we note from (2.5) that (0 ,1/c) is the only critical point in\nphase plane, and the vector field for the system (2.5) is shown in Figu re 1.\nrs\n−νr−k(1−cs) = 0\nr=Q(s)\nFigure 1. Vector field for (2.5).\nThe key point is that in the r−splane,s= 0 corresponds to ρ=∞by (2.4b). We\nneed to identify an invariant region Σ in phase plane so that s(t)>0 for all t >0 if\n(r0,s0)∈Σ. Its boundary when transformed onto the ρ−dplane through (2.4) would\ngive us the critical threshold curve. By observation, a trajector y curve starting at the\norigin and moving backwards in time would give us ∂Σ, the boundary of Σ.\nWe proceed to discuss each case as stated in Theorem 2.3.\n4.1.Strong damping ( ν >2√\nkc).We rewrite (2.5) as the following form\nd\ndt/parenleftbigg\nr\ns−1/c/parenrightbigg\n=/parenleftbigg\n−ν ck\n−1 0/parenrightbigg\n·/parenleftbigg\nr\ns−1/c/parenrightbigg\n. (4.1)\nThe coefficient matrix on the right hand side has eigenvalues\nλ±= (−ν±√\nν2−4kc)/2\nwith the corresponding eigenvectors\nv−= (−λ−,1)⊤, v+= (−λ+,1)⊤.CRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 11\nUnder the strong damping condition, the critical point (0 ,1/c) is an asymptotically stable\nnode. In order to determine Σ, we know that the boundary curve ( r(t),s(t)) satisfy the\nfollowing\nr′=−νr+k(cs−1), s′=−r, t < τ\nwith (r,s) = (0,0) at a time t=τ. When the time parameter is eliminated, we obtain\ndr\nds=ν+k\nr(1−cs), r(0) = 0.\nLet such a trajectory be denoted by r=Qa(s), we have\nΣ ={(r,s), s >0,r < Q a(s)},\nwhereQa(s) is as defined in (2.2). We now show such set is well-defined by looking at the\nasymptotic behavior of Qa(s). Since both the eigenvalues are real and negative, then\nlim\nt→−∞r(t)\ns(t)=−λ−,\nthe slope of the eigenvector v−, for any trajectories. Hence\nlim\ns→∞Qa(s)\ns=ν+√\nν2−4kc\n2.\nOne can show that r=−λ−(s−1/c) is a trajectory, so the curve r=Qa(s) always\nremains below it. As a result, r=Qa(s) also remains below the line νr+k(1−cs) = 0.\nHence, we havedQa\nds>0 fors∈(0,∞), Σ is thus well-defined. We can conclude that\ns(t)>0∀t >0 if and only if ( r0,s0)∈Σ.\nWe also need to ascertain the behavior ofQa(s)\nsassgoes to zero to know the behavior\nofdversusρon theρ−dplane as ρ→ ∞. First we know that\nlim\ns→0+Qa(s)\ns= lim\ns→0+Q′\na(s) = lim\ns→0+ν+k\nQa(s)= +∞,\nwhich from (2.4) implies that d→ −∞asρ→ ∞. Transforming the threshold curve\nback onto the ρ−dplane through (2.4), there is global solution if and only if\n(ρ0,d0)∈ {(ρ,d) :d >−ρQa(1/ρ), ρ∈(0,∞)}.\nRemark 4.1.We could also evaluate lim s→∞Qa(s)\nsusing (2.1). Since r=Qa(s) is a\ntrajectory, using (2.4), the above limit is the value of −dasρ→0. Since ρ= 0 is a\nsolution to (2.1a), we thus have\nd′=−(d2+νd+kc),\nwhich gives\nd′=−(d−λ+)(d−λ−).\nFor this Ricatti equation, dbreaks down in finite time if and only if initial data d0< λ−.\nTherefore,\nlim\ns→∞Qa(s)\ns=−λ−=ν+√\nν2−4kc\n2.12 MANAS BHATNAGAR AND HAILIANG LIU\n4.2.Borderline damping ( ν= 2√\nkc).Note that (0 ,1/c) is an asymptotically sta-\nble improper node with eigenvalue λ=−ν/2 and the corresponding eigenvector v=\n[−λ1]T. Similar to the strong damping case we can identify the invariant regio n\nΣ ={(r,s), s >0,r < Q b(s)},\nwhereQbis a monotone function, satisfying\nlim\ns→∞Qb(s)\ns=ν\n2,\nand it can be determined by the ODE (2.3). We also have\nlim\ns→0+Qb(s)\ns= +∞.\nThese enable us to conclude that s(t)>0∀t >0 if and only if ( r0,s0)∈Σ. Transforming\nthe invariant region back onto the ρ-dplane through (2.4), there is global solution if and\nonly if\n(ρ0,d0)∈ {(ρ,d) :d >−ρQb(1/ρ), ρ∈(0,∞)}.\nWe point out that on the ρ-dplane, the critical threshold curve starts at (0 ,−ν/2) and\nmonotonically goes to negative infinity as ρgoes to infinity.\n4.3.Weak damping ( ν <2√\nkc).In such case, the coefficient matrix on the right hand\nside of (4.1) has eigenvalues\nλ±=−ν\n2±i\n2√\n4kc−ν2.\nHence (0 ,1/c) is an asymptotically stable spiral point for system (2.5). Therefor e, tra-\njectories spiral into the critical point as time increases. Conseque ntly, by vector field\ndiagram, a trajectory beginning at the origin and proceeding backw ards in time (into the\nfirst quadrant on r-splane) would spiral outwards and hit the s= 0 line again in the\nsecond quadrant. This segment of the trajectory is the thresho ld curve on the r-splane.\nThis curve partitions the upper half plane into two sections. The clos ed region formed by\nthis curve and s= 0 would then be an invariant region Σ since any trajectory with initial\ndata (r0,s0) in this region would spiral inwards clockwise without touching the s= 0 line.\nAnd any trajectory with initial data outside this region remains outs ide this region.\nWe proceed to describe Σ using two functions.\nrs\n−νr−k(1−cs) = 0\ns∗\nr=Q(s)\nFigure 2. The curve along with vector field for (2.5).CRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 13\nBecause of the outward spiraling nature of the trajectory curve starting from (0 ,0),\nthen there exists a unique s∗>1/cso that we have ( r,s) = (0,s∗) at some t. The first\nsegment of such curve can be defined via a continuous function,\nQ1: [0,s∗]−→R+∪{0},\nsatisfying for 0 < s < s∗,\ndQ1\nds=ν+k\nQ1(1−cs), Q1(0) = 0.\nDefine a continuous function Q2: [0,s∗]−→R−∪{0}as follows.\ndQ2\ndτ=−ν+k\nQ2(τ)(cs∗−1−cτ), Q2(0) = 0.\nThenr=Q2(s∗−s) gives the left segment of the said trajectory curve in the r−splane.\nThat is the invariant region can be defined by\nΣ ={(r,s) :Q2(s∗−s)< r < Q 1(s),s∈(0,s∗)}.\nIn order to transform Σ back to the ρ-dplane, we evaluate the appropriate limits to\nascertain the behavior of the threshold curve as ρ→ ∞As done before, we have\nlim\ns→0+Q1(s)\ns= lim\ns→0+dQ1\nds(s) = +∞.\nNote that Q2(s∗)<0 due to the outward spiraling nature of the threshold trajectory ,\nhence\nlim\ns→0Q2(s∗−s)\ns=−∞.\nWe can now transform back onto the ρ−dplane through (2.4) to conclude that there is\nglobal solution if and only if\n(d0,ρ0)∈ {(d,ρ) :−ρQ1(1/ρ)< d <−ρQ2(s∗−1/ρ), ρ∈(1/s∗,∞)}.\nNote that the shape of the critical threshold curve is similar to that of a parabola opening\ntowards positive ρaxis and vertex at (1 /s∗,0). This completes the proof to Theorem 2.3.\nRemark 4.2.Using (3.4), we can find s∗explicitly. Note that the left-most point of the\nthreshold curve is ( ρ∗,d∗) = (1/s∗,0) withρ∗< c; for which β=π, setting d0=d∗= 0\nin (3.4) and using µ2=kc−ν2/4, we find that\nρ∗=c\neνπ\n2µ+1.\nHence,s∗=1\nρ∗=eνπ\n2µ+1\nc.\n5.Application to an aggregation system\nIn this section, we illustrate that our results can be applied to syste m (1.2), subject to\nC1initial data (1.3).\nForinitial data( ρ0,u0) defined ontheentire R, weshall make thefollowing assumptions\nconcerning their behavior at far fields. There exists δ >0 such that\nu0x(x)∈C0\nb(R),/an}bracketle{tx/an}bracketri}ht2+δρ0(x)∈C0\nb(R), (5.1)14 MANAS BHATNAGAR AND HAILIANG LIU\nwhere/an}bracketle{tx/an}bracketri}ht:=√\n1+x2, andC0\nb(R) denotes the set of bounded continuous functions on R.\nUnder (5.1) we have\n/integraldisplay\nRρ0(x)dx <∞,/integraldisplay\nRρ0(x)|u0(x)|dx <∞,/integraldisplay\nR|x|ρ0(x)dx <∞. (5.2)\nWe state the local wellposedness in the following.\nTheorem 5.1. (Local existence ) Ifρ0, u0∈C1(R), and (5.1) is satisfied, then there\nexistsT >0, depending upon the initial data so that (1.2), (1.3) has a un ique solution\n(ρ,u)∈C1(R×[0,T)), that for t∈[0,T)andi= 1,2\nρ(x,t)|u|i(x,t)→0as|x| → ∞ andρ(x,t)≤C\n|x|2+δ. (5.3)\nMoreover, if the maximum life span T∗<∞, then\nlim\nt→T∗−∂xu(t,x∗) =−∞\nfor some x∗∈R.\nInotherwords, weprovelocalexistenceanduniqueness forsolut ionsinamorerestricted\nfunction space. We would like to point out that the decaying assumpt ions at far fields\nmake sense physically and are reasonable assumptions as we want th e particle density,\nmomentum, and energy to vanish at ±∞.\nUsing (5.2), set\n/integraldisplay\nRρ0(x)dx=:M0,/integraldisplay\nRρ0(x)u0(x)dx=:M1.\nForC1solutions satisfying (5.3) we first derive a local PDE in terms of ( u,E,d,ρ) with\nE:=∂W ⋆ρandd=ux, and reformulate it into a closed ODE system. First, we integrate\n(1.2a) to get/integraldisplay\nRρ(x,t)dx=M0,\nsinceρu→0 as|x| → ∞. Hence we have\nE(x,t) =/integraldisplay\nR[−sgn(x−y)+x−y]ρ(y,t)dy (5.4)\n= (x+1)M0−/integraldisplay\nRyρ(y,t)dy−2/integraldisplayx\n−∞ρ(y,t)dy.\nHere we see that Eis well defined due to (5.3). (1.2) canbe used to obtain( ρu)t+(ρu2)x+\nρu=−Eρ. Upon integrating,\nd\ndt/integraldisplay\nRρu+/integraldisplay\nRρu= 0.\nHere, we used the decay of ρu2and symmetry of Eso that/integraltext\nREρdx= 0. Therefore,\n/integraldisplay\nRρ(y,t)u(y,t)dy=M1e−t.CRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 15\nDifferentiation of (5.4) with respect to tusing (1.2a) yields\nEt= 2ρu−/integraldisplay\nRρ(y,t)u(y,t)dy= 2ρu−M1e−t.\nAlso, we get Ex=M0−2ρ, so that\nEt+uEx=M0u−M1e−t.\nTogether with the equation for d=uxwe obtain an augmented system\nρt+uρx+ρd= 0, (5.5a)\nut+uux+u=−E, (5.5b)\ndt+udx+d2+d=−Ex= 2ρ−M0, (5.5c)\nEt+uEx=−M1e−t+M0u. (5.5d)\nFromthis system we derive the characteristic system, based on wh ich we further construct\nthelocal-in-timesolution. Finally, weshowsuchconstructedsolution indeedsatisfies(5.3).\nFurther details will be deferred to Appendix.\nSince bothequations for uandEarelinear, and decoupled fromthe equations for ρand\nd. It suffices to consider the following system of equations to find the critical threshold:\n(5.6a) ρ′+ρd= 0,\n(5.6b) d′+d2+d= 2/parenleftbigg\nρ−M0\n2/parenrightbigg\n,\nand\n{}′=∂\n∂t+u∂\n∂x,\ndenotes the differentiation along the particle path,\nΓ ={(x,t)|x′(t) =u(x(t),t),x(0) =α∈R}.\nThis is a particular case of system (2.1), with\nν= 1, k= 2, c=M0\n2.\nConsequently, we have the following theorems.\nTheorem 5.2. For the given 1D pressureless damped Euler system of equatio ns (1.2)\nsubject to initial data (1.3), there is finite time breakdown iff∃x∈Rsuch that\n(1)(Subcritical mass M0<1/4)\nmax{u′\n0(x), u′\n0(x)+λ2(0.5M0−ρ0(x))}<0and\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle0.5M0λ2u′\n0(x)+M0−2ρ0(x)\n2ρ0(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ1\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle0.5M0λ1u′\n0(x)+M0−2ρ0(x)\n2ρ0(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ2\n,\nwhereλ1=1−√1−4M0\nM0andλ2=1+√1−4M0\nM0.16 MANAS BHATNAGAR AND HAILIANG LIU\n(2)(Critical mass M0= 1/4)\nmax{u′\n0(x),M0u′\n0(x)+0.5M0−ρ0(x)}<0and\nln/parenleftbigg\n−M0u′\n0(x)+0.5M0−ρ0(x)\nρ0(x)/parenrightbigg\n≥M0u′\n0(x)\nM0u′\n0(x)+0.5M0−ρ0(x).\n(3)(Supercritical mass M0>1/4)\n/parenleftbigg\nu′\n0(x)+0.5M0−ρ0(x)\nM0/parenrightbigg2\n≥/parenleftbigg\nM0−1\n4/parenrightbigg/bracketleftbigg4ρ2\n0(x)\nM2\n0/parenleftbiggM0et∗\nM0−1\n4−1/parenrightbigg\n+4ρ0(x)−M0\nM0/bracketrightbigg\n,\nwhere\nt∗=1/radicalBig/parenleftbig\nM0−1\n4/parenrightbig\nβ+tan−1\n2u′\n0(x)/radicalBig/parenleftbig\nM0−1\n4/parenrightbig\nu′\n0(x)+2M0−4ρ0(x)\n\n,\nβ=\n\n0u′\n0(x)< min{0,4(ρ0(x)−0.5M0)},\nπ 4(ρ0(x)−0.5M0)< u′\n0(x),\n2π0< u′\n0(x)<4(ρ0(x)−0.5M0).\nTheorem 5.3. Considerthe given 1Dpressureless damped Euler system of eq uations (1.2)\nsubject to initial data (1.3). There exists a unique global s olutionρ,u∈C1(R×(0,∞))\niff∀x∈R,\n(1)(Subcritical mass M0<1/4)\n(ρ0(x),u′\n0(x))∈ {(ρ,d) :d >−ρRa(1/ρ), ρ >0},\nwhereRa: [0,∞)−→[0,∞)is a continuous function satisfying\ndRa\nds= 1+1\nRa(2−M0s), Ra(0) = 0.\n(2)(Critical mass M0= 1/4)\n(ρ0(x),u′\n0(x))∈ {(ρ,d) :d >−ρRb(1/ρ), ρ >0},\nwhereRb: [0,∞)−→[0,∞)is a continuous function satisfying\ndRb\nds= 1+1\nRb(2−s/4), Rb(0) = 0.\n(3)(Supercritical mass M0>1/4)\n(ρ0(x),u′\n0(x))∈ {(ρ,d) :−ρR1(1/ρ)< d <−ρR2(γ−1/ρ), ρ∈(1/γ,∞)},\nwhere\nγ=2\nM0/parenleftBig\n1+eπ√\n4M0−1/parenrightBig\n,\nandR1: [0,γ]−→R+∪{0}is a continuous function satisfying\ndR1\nds= 1+1\nR1(2−M0s), R1(0) = 0,\nandR2: [0,γ]−→R−∪{0}is another continuous function satisfying\ndR2\nds=−1+1\nR2(M0(γ−s)−2), R2(0) = 0.CRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 17\nAppendix A.Proof of Theorem 5.1\nDenote′=∂/∂t+u∂/∂xas the derivative along the particle path,\n(A.1) Γ = {(x,t) : ˙x=u(x,t), x(0) =α, α∈R},\nand define\nf(α,t) :=ρ(x(α,t),t),\nQ(α,t) :=E(x(α,t),t),\nd(α,t) :=ux(x(α,t),t),\nz(α,t) :=u(x(α,t),t),\nso to obtain the following closed ODE system\nf′+fd= 0, (A.2a)\nz′+z=−Q, (A.2b)\nd′+d2+d= 2f−M0, (A.2c)\nQ′=−M1e−t+M0z, (A.2d)\nsubject to initial data\nf(α,0) =ρ0(α), (A.3a)\nz(α,0) =u0(α), (A.3b)\nd(α,0) =u0x(α), (A.3c)\nQ(α,0) =E0(α), (A.3d)\nwhereE0(α) is defined by\nE0(α) =/integraldisplay\nR[−sgn(α−β)+(α−β)]ρ0(β)dβ (A.4)\nwhich is well defined since |E0(α)−M0α| ≤M0+/integraltext\nR|x|ρ0(x)dx.\nThisODEproblem, foreachfixed α0∈R,admitsauniquelocal C1solution( f,z,d,Q)(α,t)\nin a neighborhood of ( α0,0). Our aim is to find a T >0, such that f(α,t),z(α,t) are in\nC1(R×[0,T]).\nWe do this by ensuring that the deformation of the path has a strict ly positive uniform\nlower bound. In the neighborhood of any ( α0,0), using (A.2b), (A.2d), (A.3b) and(A.3d),\nwe see that a:=zα(α,t) solves\na′′+a′+M0a= 0,\na(0) =d0(α), a′(0) =−d0(α)+2ρ0(α)−M0.\nSinced0=u0x,ρ0are bounded uniformly in terms of α, we find that for any t >0,\n|a(α,t)| ≤Cuniformly in α. Hence, choosing T <1/(2C), we ensure\n∂x\n∂α= 1+/integraldisplayt\n0zα(α,s)ds >1\n2∀α∈R, t∈[0,T].\nEventually, by the inverse function theorem, for each x∈Randt∈(0,T], we can\nuniquely solve the equation\nx=x(α,t), α=α(x,t),18 MANAS BHATNAGAR AND HAILIANG LIU\nwhere the mapping x→α,tisC2. Finally let us define\nρ(x,t) :=f(α(x,t),t),\nE(x,t) =Q(α(x,t),t),\nu(x,t) =z(α(x,t),t),\np(x,t) =d((α(x,t),t)\nfor eachx∈Randt∈(0,T]. It remains to show ( ρ,u) is indeed a solution to (1.2). One\ncan very that f′=ρt+uρx,so that\nρt+uρx+ρp= 0.\nWe still need to show p=ux, that is to show d=zα·∂α\n∂xalong the particle path Γ. In\nview of this, fix αand set\nΘ(t) :=zα−∂x\n∂αd.\nUpon differentiating,\nΘ′=z′\nα−zαd−∂x\n∂αd′\n=−Qα−(d+1)/parenleftbigg\nzα−d∂x\n∂α/parenrightbigg\n−∂x\n∂α(2f−M0)\n=−Qα−Θ(d+1)+∂x\n∂α(M0−2f).\nHere, we used (A.2b), (A.2c) and (A.1). Differentiating once again an d using the expres-\nsion for Θ along with (A.2d), (A.2a), (A.2c) and (A.1),\nΘ′′=−M0zα−Θ′(d+1)−d′Θ+zα(M0−2f)+∂x\n∂α(−2f′)\n=−Θ′(d+1)+(d2+d−2f+M0)Θ−2f/parenleftbigg\nzα−d∂x\n∂α/parenrightbigg\n=−(d+1)Θ′−(4f−M0−d2−d)Θ.\nNote that Θ(0) = 0 and Θ′(0) = 0, therefore, Θ ≡0. So,d=ux. In a similar manner we\nhave\nut+uux+u=−E(x,t).\nWe need to show E(x,t) =∂W∗ρ. In order to achieve this, it suffices to prove\nq(α,t)≡0,\nwhere\nq(t) =Q(α,t)−/integraldisplay\nR[−sgn(x(α,t)−y(β,t))+(x(α,t)−y(β,t)]ρ0(β)dβ.\nObserve that q(0) = 0. Since the characteristics don’t cross, we have\nsgn(x(α,t)−x(β,t)) = sgn( α−β).\nThis observation along with the characteristic flow equation gives,\nq′=Q′−/integraldisplay\nR(z(α,t)−z(β,t))ρ0(β)dβCRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 19\n=M0z−M1e−t−z/integraldisplay\nRρ0(β)dβ+/integraldisplay\nRz(β,t)ρ0(β)dβ\n=/integraldisplay\nRz(β,t)ρ0(β)dβ−M1e−t.\nWe claim the right hand side is zero. Hence q≡0, we have arrived at equation (1.2b).\nTo prove the above claim, we first show/integraltext\nRQ(α,t)ρ0(α)dα= 0 which will be essential\nlater. To see this, set\na(t) :=/integraldisplay\nRQ(α,t)ρ0(α)dα.\nOne can check that a(0) = 0. From (A.2d),\na′(t) =M0/parenleftbigg/integraldisplay\nRzρ0dα−M1e−t/parenrightbigg\n.\nObserve that a′(0) = 0. Differentiating once again, using the above expression for d eriv-\native and using (A.2b),\na′′+a′+M0a= 0.\nConsequently,/integraldisplay\nRQ(α,t)ρ0(α)dα= 0.\nMultiplying (A.2b) by ρ0, integrating and using the above result, we get\n/integraldisplay\nRz(α,t)ρ0(α)dα=M1e−t.\nThis proves the claim. Finally, we show that the solution obtained for ( 1.2) with the\nabove construction indeed satisfies the designated boundary con ditions. Since f(α,t) =\nρ0(α)/(∂x/∂α), we have ρ→0 as|x| → ∞with the same rate as ρ0since∂x/∂αhas\na strictly positive lower bound ∀t∈[0,T]. And since (A.2b) and (A.2d) form a closed\nlinear system, we can explicitly solve for z(α,t) via\nz′′+z′+M0z=M1e−t,\nz(0) =u0(α), z′(0) =−u0(α)−E0(α).\nSinceu0xis bounded and observing the αM0term inE0(α), bothz(0) andz′(0) may grow\nat most linearly in α, so does z. Hence, by hypothesis,\nlim\n|x|→∞ρu2= lim\n|α|→∞(∂α/∂x)ρ0z2= 0.\nFinally, we show that for a fixed α∈R, for a finite time breakdown to occur, we must\nhave lim t→T∗−d(t) =−∞. Otherwise, we would have lim t→T∗−d(t) =∞. Then∃ǫ >0\nsuch that for t∈I:={t:T∗−ǫ < t < T∗},\nf′=−fd <0.\nTherefore, there exists a constant D >0 such that 2 f−M0< D. Hence, from (A.2c),\nd′< D; Integration from some t0∈Igives\nd(T∗)< d(t0)+D(T∗−t),\nwhich is a contradiction. This completes the proof to Theorem 5.1.20 MANAS BHATNAGAR AND HAILIANG LIU\nAppendix B.Uniqueness of critical threshold curves\nLemma B.1. Qa,Qb,Q1andQ2in Theorem 2.3 are uniquely defined.\nProof.We will prove the existence and uniqueness for Qaonly, and similar analysis can\nbe carried out for Qb,Q1,Q2. Consider an auxiliary problem of the form\ndS\ndr=r\nνr+k(1−cS), S(0) = 0,r >0.\nOne can check the right hand side function is continuous and locally Lip schitz in Saround\nthe origin. Hence, a unique, strictly increasing solution Sexists on [0 ,δ] for some small\nδ >0. We can show that S−1satisfies (2.2). Now, suppose Qand/tildewideQare two solutions to\n(2.2). Note from (2.2) that any solution has to be strictly increasing in a neighbourhood\nfors≥0. Hence, Qand/tildewideQhave inverses, Q−1and/tildewideQ−1. But the inverse being unique,\nwe haveQ−1=/tildewideQ−1=S. Hence, Qais uniquely defined. /square\nAcknowledgement\nThis work was partially supported by the National Science Foundatio n under Grant\nDMS1812666 and by NSF Grant RNMS (Ki-Net) 1107291.\nReferences\n[1] J. A. Carrillo, Y.-P. Choi, E. Tadmor, and C. Tan. Critical thresho lds in 1D Euler equations with\nnon-local forces. Math.Mod.Meth.Appl.Sci., 26:185–206, 2016.\n[2] J. A. Ca¨ nizo, J. A. Carrillo, and J. Rosado. A well-posedness tho ery in measures for some kinetic\nmodels of collective motion. Math.Mod.Meth.Appl.Sci., 21:515–539, 2011.\n[3] J.A. Carrillo, Y-Pil Choi and E. Zatorska. On the pressureless da mped Euler-Poisson equations\nwith quadratic confinement: critical thresholds and large-time beh avior.Mathematical Modelsand\nMethods inApplied Sciences, 26(12): 2311-2340, 2016.\n[4] J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil. Particle, Kine tic, and Hydrodynamic Models\nof Swarming, Mathematical Modeling ofCollective Behavior inSocio-Economic andLifeSciences,\nSeries: Modelling and Simulation in Science and Technology, Birkhauser , 297–336, 2010.\n[5] Y.-L. Chuang, M. R. D’Orsogna, D. Marthaler, A. L. Bertozzi an d L. Chayes. State transitions and\nthe continuum limit for a 2D interacting, self-propelled particle syste m.PhysicaD, 232: 33–47, 2007.\n[6] J. A. Carrillo, M. R. D’Orsogna, and V. Panferov. Double milling in self -propelled swarms from\nkinetic theory. KineticandRelatedModels, 2:363–378, 2009.\n[7] Chen, G.-Q., Wang, D. Convergence of shock capturing scheme f or the compressible Euler–Poisson\nequation. 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Int.Math.Res.Notices, 2013(4):761– 826, 2013.CRITICAL THRESHOLDS IN EULER-POISSON SYSTEMS 21\n[15] Jang, J. The two-dimensional Euler–Poisson system with spher ical symmetry. J.Math.Phys.,\n53:023701, 2012.\n[16] Jang, J., Li, D., Zhang, X. Smooth global solutions for the two-d imensional Euler–Poisson system.\nForumMath., 26:645–701, 2014.\n[17] P.D. Lax. Development of singularities of solutions of nonlinear hy perbolic partial differential equa-\ntions.J.Math.Phys., 5: 611–613, 1964.\n[18] Y. Lee and H. Liu. Thresholds for shock formation in traffic flow m odels with Arrhenius look-ahead\ndynamics. Discrete Contin.Dyn.-A, 35(1): 323–339, 2015.\n[19] Y. Lee and H. Liu. Thresholds in three-dimensional restricted E uler-Poisson equations. PhysicaD,\n262: 59–70, 2013.\n[20] H. Liu, E. Tadmor. Spectral dynamics of the velocity gradient fi eld in restricted fluid flows. Comm.\nMath.Phys., 228: 435–466, 2002.\n[21] H. Liu, E. Tadmor. Critical thresholds in 2-D restricted Euler–P oisson equations. SIAMJ.Appl.\nMath., 63(6): 1889–1910, 2003.\n[22] H. Liu and E. Tadmor. Critical Thresholds in a convolution model f or nonlinear conservation laws.\nSIAMJ.Math.Anal. 33: 930–945, 2002.\n[23] Li, D. and Wu, Y. The Cauchy problem for the two dimensional Eule r–Poissonsystem. J.Eur.Math.\nSoc., 10:2211–2266, 2014.\n[24] Marcati, P., Natalini, R. Weak solutions to a hydrodynamic model f or semiconductorsand relaxation\nto the drift-diffusion equation. Arch.Ration.Mech.Anal., 129:129–145, 1995.\n[25] Poupaud, F., Rascle, M., Vila, J.-P. Global solutions to the isother mal Euler–Poisson system with\narbitrarily large data. J.Differential Equations, 123:93–121,1995.\n[26] Tadmor, E., Tan, C. Critical Thresholds in flocking hydrodynamic s with non-local alignment. Phil.\nTrans.R.Soc.A, 372: 20130401, 2014.\n[27] E. Tadmor, D. Wei. On the global regularity of subcritical Euler- Poisson equations with pressure. J.\nEur.Math.Soc., 10: 757–769, 2008.\n[28] Wang, D., Chen, G.-Q. Formation of singularities in compressible Eu ler–Poisson fluids with heat\ndiffusion and damping relaxation. J.Differential Equations, 144:44–65, 1998.\n[29] Wang, D., Wang, Z. Large BV solutions to the compressible isothe rmal Euler–Poisson equations\nwith spherical symmetry. Nonlinearity, 19:1985–2004, 2006.\nDepartment of Mathematics, Iowa State University, Ames, Io wa 50010\nE-mail address :manasb@iastate.edu\nE-mail address :hliu@iastate.edu"    },    {        "title": "1907.09731v1.Electron_transport_in_high_entropy_alloys__Al___x__CrFeCoNi_as_a_case_study.pdf",        "content": "arXiv:1907.09731v1  [cond-mat.mtrl-sci]  23 Jul 2019Electron transport in high-entropy alloys:\nAlxCrFeCoNi as a case study.\nJ. Kudrnovsk´ y,∗V. Drchal, and F. M´ aca\nInstitute of Physics ASCR, Na Slovance 2, CZ-182 21 Praha 8, Cz ech Republic\nI. Turek\nInstitute of Physics of Materials ASCR, ˇZiˇ zkova 22, CZ-616 62 Brno, Czech Republic\nS. Khmelevskyi\nCenter for Computational Materials Science, Institute for Applied Physics,\nVienna University of Technology, Wiedner Hauptstrasse 8, A -1040 Vienna, Austria\n(Dated: July 24, 2019)\nThe high-entropy alloys Al xCrFeCoNi exist over a broad range of Al concentrations (0 < x <2).\nWith increasing Al content their structure is changed from t he fcc to bcc phase. We investigate\nthe effect of such structural changes on transport propertie s including the residual resistivity and\nthe anomalous Hall resistivity. We have performed a detaile d comparison of the first-principles\nsimulations with available experimental data. We show that the calculated residual resistivities for\nall studied alloy compositions are in a fair agreement with a vailable experimental data as concerns\nboth the resistivity values and concentration trends. We em phasize that a good agreement with\nexperiment was obtained also for the anomalous Hall resisti vity. We have completed study by\nestimation of the anisotropic magnetoresistance, spin-di sorder resistivity, and Gilbert damping. The\nobtained results prove that the main scattering mechanism i s due to the intrinsic chemical disorder\nwhereas the effect of spin polarization on the residual resis tivity is appreciably weaker.\nI. INTRODUCTION\nThe high-entropy alloys (HEA), the multicomponent\ncrystalline alloys, often also called multi-principal ele-\nmentalloyshaveattractedaquitesignificantandgrowing\ninterest in the last decade. Of the vast existing litera-\nture we just mention a recent book,1a critical review,2\nand an overview of possible theoretical approaches.3The\nhigh entropy of mixing of these multicomponent alloys\nsuppresses the formation of ordered intermetallic com-\npounds leading to well disordered phases with simple\nlattice structures such as the face-centered cubic (fcc)\nor the body-centered cubic (bcc) ones. Magnetic HEA’s\nthat consist of magnetic 3 d-elements are particularly in-\nteresting. A typical example is the so-called quinary\nCantor alloy (CrMnFeCoNi) consisting of equiconcentra-\ntion disordered mixture of magnetic Cr, Mn, Fe, Co, and\nNi elements with an fcc structure. Such alloy offers a\nrichness of magnetic properties depending on the sam-\nple preparation.4The structural change from the fcc to\nbcc phase was predicted by ab initio molecular dynamics\nsimulations5for Cantor-like alloys with Cr substituted\nby another metallic element with formula CoFeMnNiX\n(X = Al, Ga, and Sn).\nBy doping with sp-elements, which influences carrier\nconcentration in the conduction band and thus both the\nmagnetic and transport properties, and even the alloy\nstructure one can search for new functional properties.\nA typical example of such alloy is Al xCrFeCoNi alloy6,7\nwithxranging from x=0 tox=2. In particular, alloy-\ning with increasing Al content stabilizes the bcc phase\nfrom the original fcc phase of quaternary CrFeCoNi al-\nloy. Another interesting property, also present in theCantor alloy, is a large residual resistivity of the order\nof 100µΩcm which is in a striking contrast to a much\nsmallerresistivityofthebinaryfccNiFeorfccNiCocoun-\nterparts. The large values of the residual resistivity in\nHEA’s are caused by strong scattering on the intrinsic\nchemical disorder. In the present work we apply the\nalloy-specific first-principles methodology based on the\nKubo-Greenwood formula8which was successfully used\nfor binary alloys9,10also to HEA’s in order to estimate\nthe intrinsic contribution to the resistivity and compare\nthem to available experimental data.\nContrary to the experimental and theoretical studies\nof structural and thermodynamical properties of HEA’s\nthe studies of electronic transport are very rare.2The\ntransport properties, together with the electronic struc-\nture are among the most important material proper-\nties. The theoretical tools for the resistivity study are\nmore complicated and not so broadly available as elec-\ntronic structure codes focused on total energies, electron\ndensities and magnetic moments. Recently, theoretical\ntransport studies of the Cantor fcc CrMnFeCoNi alloy11\nand of a related medium-entropy fcc NiCoMn alloy12ap-\npeared which studied various possible scattering mecha-\nnisms contributing to the residual resistivity. However,\ntheoretical investigation of the effect of Al-doping on the\nresistivity, as well as of the role of different structures\n(bcc, fcc) for electron transport in AlCrFeCoNi, is still\nmissing. Moreover, in addition to residual resistivities\nalsothe anomalousHall resistivity(AHR) for both struc-\nturephaseswasdetermined experimentally.6,7Therefore,\nthese alloys are an obvious choice for ab initio based\nstudies of transport properties in HEA’s containing sp-\nelements. In the present study, also the anisotropic mag-2\nnetoresistance (AMR), the spin-disorder resistivity and\nthe Gilbert damping for both structures and typical Al\nconcentrations are calculated and discussed.\nII. FORMALISM\nThe disordered fcc and bcc phases of Al xCrFeCoNi al-\nloy with xranging from x=0 tox=2 were studied for\nexperimentally observed phases and lattice constants.6\nThe fcc phase exists roughly for x <0.5 while the bcc\nphase is stable for x >1.0, but boundaries are not well\ndefined. Duplex phase (a mixture of fcc and bcc phases)\nexists for the intermediate Al concentrations. We note\nthat sometimes the Al xCrFeCoNi alloy is presented as\nAl1−4yCryFeyCoyNiy, wherey= 1/(4+x) and the sum\nof all component concentrations is one.2\nThe spin-polarized electronic structure calculations\nwere done using the Green function formulation of\nthe tight-binding linear muffin-tin orbital (TB-LMTO)\nmethodintheatomicsphereapproximation(ASA).13We\nemploy the scalar-relativistic version of the TB-LMTO\nmethod and, in order to assess the importance of the rel-\nativistic effects, we also made calculations using the fully\nrelativistic version of the TB-LMTO method. In both\ncases the exchange-correlation potential of Vosko, Wilk\nand Nusair (VWN)14and the spd-basis set were used.\nThe alloy disorder in studied multicomponent alloys is\ndescribed in the framework of the coherent potential ap-\nproximation (CPA).15The use of the CPA allows us to\nwork very efficiently in small fcc or bcc unit cells. On\nthe contrary, large special-quasirandom structure (SQS)\nsupercells are needed in conventional density-functional-\ntheory (DFT) studies.4,5It should be noted that contin-\nuously varying Al content imposes additional non-trivial\nconstraintsonthechoiceofasuitableSQS-supercell. The\ncostwepayforusingtheCPAistheneglectofpossiblelo-\ncal environment and clustering effects in the alloy which,\non the other hand, are captured by the SQS-supercell\napproach. The CPA is a reliable approach in well disor-\ndered alloys, particularly when the concentration trends\nare concerned. Even more important advantage of the\nCPA is the fact that it provides naturally transport re-\nlaxation times which need not be taken from outside like,\ne.g., in the Boltzmann equation approach.\nThe transport properties are described by the conduc-\ntivity tensor σwith components σµν(µ,ν=x,y,z). The\nresistivity tensor ρwith components ρµνis obtained sim-\nply by inversion of the conductivity tensor, ρ=σ−1.\nThe conductivity tensor is determined in the framework\nof the Kubo-Greenwood (K-G) approach (only diagonal\nelements of σµνare non-zero in present cubic systems\nin the scalar-relativistic model).8The off-diagonal com-\nponents of σµνare needed for the AMR/AHR studies\nand they are calculated in the framework of the Kubo-\nBastin (K-B) formulation of the fully-relativistic trans-\nport in disordered magnetic alloys which includes both\nthe Fermi-surface and Fermi-sea terms on equal footing.9The Fermi-surface term contains contribution only from\nthe states at the Fermi energy and includes the most im-\nportant elastic scattering effects due to impurities. The\nFermi-sea term, on the contrary, depends on all occupied\nstates below the Fermi energy; this term contributes only\nto the antisymmetric part of the tensor σµν. Once the\ntransport tensor is determined, the AHR= ρxywhile the\nAMR=(ρzz−ρxx)/ρtot, whereρtotis the average value\nof diagonal components of the resistivity tensor. In rel-\nativistic calculations we assume that the magnetic mo-\nment points in the z-direction. The disorder-induced ver-\ntexcorrections,10whichdescribe thecorrelatedmotionof\ntwo electrons in a random alloy potential, are included.\nThey correspondto the backwardscatteringcontribution\nin the conventional Boltzmann equation approach.\nThe Gilbert damping (GD) constant is an important\nphenomenological parameter describing the magnetiza-\ntion dynamics. It is evaluated here with the help of a\nrecently developed approach using nonlocal torques16as\nanalternativetotheusuallocaltorqueoperatorsentering\nthe torque-correlation formula.17–19This leads to effec-\ntivetorquesthatarerepresentedasnon-site-diagonaland\nspin-independent matrices, which simplifies evaluation of\ndisorder-induced vertex corrections which play essential\nrole in the present formulation since their neglect would\nlead to quantitatively and physically incorrect results.16\nOur formulation gives results that compare well to\nother first-principles studies.17–19In this study we will\nconcentrate on the GD due to chemical disorder, espe-\ncially the effect of Al-doping. It should be noted that\nthere are other sources of damping, e.g., the tempera-\nture effects due to phonons and spin fluctuations which\nare neglected here.\nIII. RESULTS\nA. Electronic structure and magnetic moments\nThe results of electronic structure calculations serve\nas an input for transport calculations of Al xCrFeCoNi\nalloys.\nTo illustrate the underlying electronic structure, we\nshow in Figs. 1 and 2 the total and component-resolved\ndensities of states (DOS) for two typical alloys, namely,\nfcc Al 0.25CrFeCoNi and bcc Al 1.25CrFeCoNi. The fol-\nlowing conclusions can be done: (i) We note a typical\ntwo-peak-like total DOS characteristic of the bcc phase\nas compared to an essentially one-peak-like total DOS\nfor the fcc phase. In both cases the Fermi level is located\ndeep inside the valence band as it is typical for metal-\nlic systems in contrast, e.g., to doped semiconductors in\nwhich the clustering has a non-negligible effect on DOS\nclose to band edges. The CPA is thus a good approxima-\ntionforelectrontransportstudies; (ii)Wealsonotelarger\ntotal DOS at the Fermi energy for fcc phase as compared\nto the bcc phase. This indicates a larger amount of car-\nriers and thus a smaller resistivity/larger conductivity3\n-30-25-20-15-10-5 0 5 10 15 20 25 30\n-0.8-0.6-0.4-0.2  0 0.2Local DOS (states/Ry)\nEnergy (Ry)maj\nminCr\nFe\nCo\nNi\nAl-15-10-5 0 5 10 15 20Total DOS (states/Ry)fcc Al0.25CrFeCoNi maj\nmin\nFIG. 1: Calculated spin-resolved DOS’s for fcc\nAl0.25CrFeCoNi alloy: the total DOS (upper frame) and\natom-resolved DOS’s (lower frame) are shown. The vertical\nlines denote the position of the Fermi level.\nof fcc phases because the amount of disorder is similar\nin both phases (see atom-resolved DOS’s below and dis-\ncussion there); (iii) The Al-resolved DOS is free-electron\nlike with only small modifications in the energy region\nwhere it hybridizes with transition metal states; (iv) The\nmajority Ni-, Co-, and Fe-resolved DOS’s indicate only\nnegligible influence of the disorder: they all have similar\nshapesand centersofgravityand resemblecorresponding\ntotalDOS.Onthecontrary,themajorityCr-DOShasthe\ncenter of gravity shifted to higher energies and its shape\nis different from those of Ni-, Co-, and Fe-states. This is\ndue to a lower atomic number of Cr and thus a weaker\nCoulomb attraction in comparison with Fe, Co, and Ni.\nMajorityCrstatesthusintroduceasignificantdisorderin\nthe majority band. Below we show that in present alloys\nthe resistivity in both majority and minority channels\nis comparable (see also Ref.20); (v) The minority Ni-,-30-25-20-15-10-5 0 5 10 15 20 25 30\n-0.8-0.6-0.4-0.2  0 0.2Local DOS (states/Ry)\nEnergy (Ry)maj\nminCr\nFe\nCo\nNi\nAl-15-10-5 0 5 10 15 20Total DOS (states/Ry)bcc Al1.25CrFeCoNi maj\nmin\nFIG. 2: The same as in Fig. 1 but for bcc Al 1.25CrFeCoNi\nalloy.\nCo-, Fe-, and Cr-resolved DOS’s have centers of gravity\nshifted to different positions thus indicating the presence\nof disorder among all components as contrasted to the\nmajority states. It should be noted that the character of\ndisorder in both fcc and bcc phases is quite similar. The\npresence of non-negligible disorder in both majority and\nminority states is the reason of much larger resistivity of\nAlxCrFeCoNi alloys as compared to the Ni-rich NiFe and\nNiCo alloys in which the disorder effect in majority spin\nchannel is negligible.20The disorder effect is essential for\nboth, minority and majority spin channels, but operates\nin them differently. This observation provides us with a\nmotivation to study the magnetotransport phenomena.\nWe present magnetic properties in Table 1, where we\nshow the total and local magnetic moments for alloys\nwithx= 0.25 andx= 1.25. We can make the follow-\ning conclusions: (i) The induced local Al moments are\nvery small and negative. Also moments on Cr atoms\nare negative and their absolute value is reduced with in-4\nTABLE I: Calculated total magnetic moment ( Mtot) and\nlocal magnetic moments ( mX, X=Al, Cr, Fe, Co, Ni) for\nAlxCrFeCoNi alloys in the fcc ( xAl= 0.25) and bcc ( xAl=\n1.25) phases. Magnetic moments are in µB.\nxAlMtotmAlmCrmFemComNi\n0.25 (fcc) 0.606−0.054−0.6201.9331.0270.253\n1.25 (bcc) 0.646−0.045−0.1032.1171.2430.191\ncreasing Al content. The present alloys are thus ferri-\nmagnets. The values of Ni-local moments are strongly\nreduced as compared to the fcc Ni crystal; (ii) Dominat-\ning moments are those on Co and, first of all, on Fe sites\nwhich have values close to the values in bcc Fe crystal\nwhile moments on Co-sites are smaller as compared to\nfcc-Co crystal. Both moments depend weakly on the Al\ndoping; and (iii) Due to the character of local moments,\nboth alloys have non-zero total magnetization with total\nmoments slightly larger for the bcc phase and relatively\nsmall in their sizes, being of order 0.5 µB. We note a\ngood quantitative agreement of present moments with a\nrecent theoretical study.21\nThe present CPA calculations ignored any spin fluc-\ntuations in the ground state of the alloys. However, ex-\nisting studies of the Cantor CrMnFeCoNi alloy11and of\nthe ternary fcc NiCoMn system12revealed an instability\nof Mn atoms to form more complicated moment distribu-\ntions; we haveverifiedthis featurefor the quinaryCantor\nalloy by using the well-known CPA approach.22In order\ntoexamineasimilarinstabilityofCratomsinthepresent\nAlxCrFeCoNi systems, we have performed the CPA cal-\nculations which started with multiple Cr magnetic mo-\nments. For both structures (fcc and bcc), the iterations\nconverged always to the same single value of Cr moment.\nThis indicates that the present AlCrFeCoNi systems can\nbe reasonably described by assuming the same (average)\nlocal moment attached to each alloy species.\nB. Residual resistivities\nThe theoretical estimate of residual resistivity ρ0and\nits comparison with available experiments6,7,23is the\nmainresultofthepresentpaper. Wetreatthefccandbcc\nphases as disordered alloys described by the CPA. Corre-\nsponding electronic structure provides also naturally the\ntransportrelaxationtimes as used in the K-G formulafor\nestimate of resistivities. We note that the SQS-supercell\ncalculations (see, e.g., Refs.4,5) indicate the presence of\nlocal environment effects, both in atomic structure and\nspins, which are neglected here. On the other hand, local\nenvironment effects appear as fluctuations around aver-\nageatomiclevels. Although such fluctuations arelargein\nsome cases, they usually correspond to states with small\nweights. One can thus say that the intrinsic chemical\ndisorder due to many atomic components will dominate.\nConsidering further the fact that the CPA gives reliablyconcentration trends, we can conclude that present ap-\nproach represents reasonable first approximation to esti-\nmate resistivities in the present HEA’s.\nResults are shown in Fig. 3 for both fcc and bcc\nAlxCrFeCoNi alloys for which experiments6,7are avail-\nable.\nLet us start with experiment Ref.6 (see also Ref.23). It\nshould be noted that experiment was done at the room\ntemperature while calculations relate to T= 0 K. We\nnote an enhancement of the resistivity due to the lattice\nvibrations and spin fluctuations induced by a finite tem-\nperature. It is possibleto include, forsimple systems, the\neffect of temperature in the framework of the alloy anal-\nogymodelasformulatedintheCPA.24Itshouldbenoted\nthat although a success was recently reached by a modi-\nfied approach for fcc alloys with few alloy constituents25\nsuch a detailed study of temperature effects is beyond\nthe scope of the present paper. Under such situation we\ndecided just to scale the zero temperature results by an\nempirical constant to account for the finite temperature\neffects. We have chosen this factor to be 1.2 motivated\nby the experiment7in which resistivity ratio for 300 K\nto 0 K was measured for many samples of different com-\npositions. The purpose is just to see the effect of the\nfinite temperature. Results are shown in Figs. 3a,3b.\nAlready large values of ρ0exist for xAl= 0.0, i.e., for\nthe equiconcentration quaternary CrFeCoNi alloy, which\nagreesvery well with a recent theoretical study.11We ob-\nserve an increase of ρ0with increasing Al content in both\nthe fcc and bcc phases. This result as well as large values\nofρ0are due to the fact that d-states of Al are miss-\ning and thus Al has a low density of states around the\nFermi energy in comparison with other alloy components\nintroducing thus a strong scattering. For example, in\ndisordered bcc Fe 1−xAlxalloys26the experimental ρ0is\nabout 150 µΩcm atT=4 K for xAlaround 0.3. Similarly,\nthe random bcc V 0.75Al0.25alloy exhibits practically the\nsame resistivity.27Comparable values are obtained for\nAlxCrFeCoNi for large xAlin the bcc phase.\nThere is a good quantitative agreement of calculated\nandmeasured ρ0inbothfccandbccregions,inparticular\nfor the scaled model. In agreement with the experiment\nthe slope of the concentration dependence of ρ0is larger\nfor the fcc phase as compared to the bcc one, although\nthe effect is more pronounced in the experiment.\nIt is a well-known fact20that a significant increase of\nthe resistivity occurs in Ni-rich NiFe and NiCo fcc alloys\ndue to the mixing of spin-channels by the spin-orbit cou-\npling. We have therefore performed also fully-relativistic\ncalculations for xAl= 0.25 andxAl= 1.25 alloys with\nfcc and bcc phases, respectively. We have obtained\nonly a small changes of ρ0, namely, ρ0was 86.16 µΩcm\nvs 84.70 µΩcm for xAl= 0.25, and 137.42 µΩcm vs\n135.03µΩcm for xAl= 1.25. In both cases, higher val-\nues correspond to the fully-relativistic model. The ori-\ngin of large enhancement of ρ0by spin-orbit coupling\nin the above-mentioned binary alloys is the existence\nof disorder-free majority bands, which are missing here.5\n 0 25 50 75 100 125 150 175 200 225 250\n 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2ρο (µΩ cm)\nxAlAlxCrFeCoNi alloy (c)\nfcc fcc+bcc bcc 0 25 50 75 100 125 150 175 200 225 250\n         ρο (µΩ cm)(b) AlxCrFeCoNi alloy\nfcc fcc+bcc bcc 0 25 50 75 100 125 150 175 200 225 250\n         ρο (µΩ cm)AlxCrFeCoNi alloy (a)\nfcc fcc+bcc bcc\nFIG. 3: Calculated and experimental residual resistivitie sρ0\nas a function of Al concentration xAl. Dashed vertical lines\ndenote approximate boundaries between the fcc, duplex, and\nbcc phases. Note that the fcc phase extendsinto duplexphase\n(fcc + bcc) and similarly the bcc phase starts in duplexphase .\nFilled circles denote theoretical results while empty symb ols\ncorrespond to experimental data. (a) Experiment Ref.6 at\nroomtemperaturecomparedwiththeoretical resultsfor T= 0\nK. (b) The same as (a) but theoretical results are scaled by\na factor 1.2 to fit experiment done at room temperature (see\ntext). (c) Comparison of experiment, Ref.7, with theoretic al\nresults. Empty circles and empty boxes denote alloys ’as-ca st’\nand ’homogenized’, respectively (see experimental paper f or\ndetails). Both the theory and experiment now correspond to\nT= 0 K.Thecontributionsofthe spin-up( σ↑) andspin-down( σ↓)\nconductivity channels to the total conductivity are com-\nparable. For example, σ↑(σ↓) = 4.77 (7.04) kS/cm\nforxAl= 0.25, and σ↑(σ↓) = 3.85 (3.56) kS/cm for\nxAl= 1.25, respectively. We can thus conclude that rel-\nativistic effects are small in the studied alloy.\nTheoretical description of the duplex region in which\nboth phases co-exist is difficult because of the lack of\nstructural details. We therefore present separately re-\nsults for the fcc phase with Al concentrations extending\ninto the duplex (fcc+bcc) region, and similarly, we start\nthebcc phaseinthe duplexregion. Correspondinglattice\nconstants were taken from Ref.6. We have thus avoided\nany processing of results like, e.g., the serial or parallel\nresistivities, the arithmetic weighting, etc. Inhomogene-\nity of samples in the duplex region is obvious.\nRecently Singh et al.28have shown on the basis of cal-\nculations of chemical interatomic interactions that con-\nsideredAl xCrFeCoNialloyscanexhibitatendency tothe\nclustering in the fcc phase while in the bcc phase can ex-\nist ordering tendency. One can thus speculate that this\ncan be one of possible reasons for the smaller/larger cal-\nculatedresistivitiesforfcc/bccphasesascomparedtothe\nexperimental ones .\nNext we will discuss the experiment of Ref.7. Results\nare presented in Fig. 3c with the following comments: (i)\nContrary to the experiment6there is no clear concentra-\ntion trend. Fluctuating values of ρ0may indicate sample\ninhomogeneity due to its preparation. Largest fluctu-\nations are, as expected, in the duplex region. Larger\nfluctuations are also for ’as-cast’ samples as compared to\n’homogenized’ones; and (ii) Nevertheless, calculated and\nmeasured resistivity values are still in acceptable agree-\nment, as well as larger resistivities for higher xAl(bcc\nphase). Experiment gives no detailed structural data\nconcerning studied samples, just its Al-content so that\nmore detailed discussion of measured resistivity fluctu-\nations and their comparison with the experiment is not\npossible.\nWe note that the performed calculations of residual\nresistivity ignore the effect of local atomic relaxations,\nwhich provide an additional mechanism of electron scat-\ntering. Since we cannot determine the magnitude of the\nlocal relaxations quantitatively, we have performed only\na preliminary study in order to get rough estimation of\ntheir effect on the resistivity by employing the alloy anal-\nogy model in the CPA.24,29As a typical mean value of\nthe atomic displacement, we took ∆ u= 0.05˚A for fcc\nalloys (obtained for the fcc Cantor alloy)11and a slightly\nhigher value ∆ u= 0.075˚A for bcc alloys (because of\nthe more open bcc geometry). The resulting increase of\nthe residual resistivity ofAl xCrFeCoNi wassmall in both\nstructures, being about 2.8 % for x= 0.25 in the fcc case\nand about 1.4 % for x= 1.25 in the bcc case. These\nresults agree qualitatively with those of Ref.11 proving\nthe dominating effect of strong intrinsic chemical disor-\nder on the resistivity. A more systematic study of the\nrole of local atomic relaxations goes beyond the scope of6\nthe present work.\nOne can summarize that the CPA, despite of its sim-\nplicity, isabletoreproducethemainfeaturesofmeasured\nresistivitiesalsoinsuchcomplexalloyslikeAl xCrFeCoNi.\nClearly, the main reasonfor this successis the dominance\nofintrinsic chemical disorderin alloy. On the otherhand,\none should keep in mind that calculated values are influ-\nenced by the neglect of lattice relaxations. In general,\none could say that lattice relaxations roughly represent\nsite-off diagonal disorder which have much smaller effect\nas compared to the dominating chemical disorder related\nto different positions of atomic alloy levels.\nC. Spin-disorder resistivity (SDR)\nThe SDR is the resistivity caused by spin fluctuations\nthat exist at finite temperature in the paramagneticstate\nabove the Curie temperature. The local moments still\nexist but they are oriented randomly in such a way that\nthe total magnetic moment is zero. From the theoret-\nical point of view the SDR can be simulated success-\nfully in the framework of the CPA as the resistivity of\nan equiconcentration alloy of spin moments pointing in\nopposite directions (the disordered local moment (DLM)\nstate).30The fluctuating local moments are then deter-\nmined selfconsistently in the framework of the DFT. In\nthe fcc regiononlylocalmomentsonFe-sitesarenonzero,\nall other collapse to zero. In the bcc region, in addition,\nlocal moments on Co atoms survive. Such result, in gen-\neral, is not correct as, e.g., the local DLM moment in\nfcc Ni collapses to zero but the experiment indicates its\nnonzero value at the Curie temperature. These values\ncan be found theoretically not only for fcc Ni,31but also\nfor binary alloys.32The situation is much more compli-\ncated forthe presentmulticomponent alloy. We therefore\ndetermine just the lower and upper limits of the SDR.\nThe lower limit is the above DLM result, the upper limit\ncorresponds to the DLM state which is constructed on\nthe basis of an FM solution assuming the frozen Fermi\nenergy and frozen potential parameters.31It is denoted\nasρSDR\nmax.\nTABLE II: The spin disorder resistivity (SDR, the resistiv-\nity due to spin fluctuations in the paramagnetic state) of\nAlxCrFeCoNi for two values of Al concentrations, namely,\nxAl= 0.25 (fcc) and xAl= 1.25 (bcc) are shown. We present\nthe SDR results for two models, one in which the SDR is iden-\ntified with the resistivity of the DLM state ( ρSDR\nDLM) and the\nother (ρSDR\nmax) in which the DLM state is constructed from the\ncorresponding FM solutions with frozen Fermi energies and\nfrozen potential parameters. For a comparison we also show\nconventional resistivities ( ρFM, see Fig. 3) and resistivities of\nnon-magnetic phases ( ρNM). All values are in µΩcm.\nxAlρFMρNMρSDR\nDLMρSDR\nmax\n0.25 (fcc) 84.7072.1683.9889.97\n1.25 (bcc) 135.03117.17132.17137.49Results for two Al concentrations are summarized in\nTable 2 in which we have added for a comparison also\nresistivities of the reference FM state and resistivities of\ncorresponding non-magnetic phases. We have following\ncomments: (i) The non-magnetic phases have slightly\nsmaller resistivities as compared not only to the DLM\nphases but also as compared to the reference FM phase.\nThe effect of magnetic scatterings is thus less relevant\nthan the effect of different atom types and their differ-\nent potentials; (ii) Slightly larger values of the reference\nρFMas compared to ρSDR\nDLMare due to the fact that in the\nDLM state in the fcc/bcc phase are non-zero only Fe and\nperhaps also Co moments. Missing magnetic scattering\nthus leads to smaller resistivities (see also discussion in\n(i)); and (iii) On the contrary, the ρSDR\nmaxis slightly larger\ndue to the presence of fluctuating moments on Cr, Fe,\nCo, and Ni atoms. One can thus conclude that due to\nalready large resistivity of the reference FM state, the\nspin disorder influences resistivity only weakly.\nD. Anisotropic magnetoresistance, anomalous Hall\nresistivity, and Gilbert damping\nWe calculate further quantities which are due to the\nspin-orbit coupling, namely, the AMR and the AHR for\ntwo typical Al-concentrations, namely, xAl= 0.25 (fcc\nphase) and xAl= 1.25(bcc phase). The relativistic input\nis needed to solve the K-B transport equation.9While we\nhave found no experimental data for the AMR, the AHR\ndata are available for the above two alloys.6\nTABLE III: Calculated AMR and AHR for Al xCrFeCoNi al-\nloys in the fcc ( xAl= 0.25) and bcc ( xAl= 1.25) phases. The\nAMR values are in % while the AHR values are in µΩcm.\nxAlAMRAHRthAHRexp\n0.25 (fcc) 0.0310.879 0.5\n1.25 (bcc) 0.0441.699 1.5\nCalculated results are summarized in Table 3 with the\nfollowing conclusions: (i) The AMR is positive, but its\nvalues are very small, considering the fact that, e.g., for\nNi-rich NiFe the AMR can be as large as 15%.20,33It was\nshown that large values of the AMR in fcc Ni-rich alloys\nare due to essentially disorder-free majority bands. On\nthe contrary, the Ni-rich NiMn alloy has disorder in both\nthe majority and minority bands and significantly lower\nAMR than Ni-rich NiFe, but still few times larger than\nthe present alloys. In addition to very similar disorder\nin both channels in present alloys, the other reason of\nsuch small AMR can be the ferrimagnetic rather than\nthe FM character of present alloys with the antiparallel\nCr moments. Its role plays also small total moment of\nstudied HEA alloys; and (ii) There is a good agreement\nbetween calculated and measured AHR for the bcc phase\n(xAl= 1.25)while the agreementfor the fcc phase ( xAl=\n0.25) is worse but still reasonable. We note that a good7\nagreement of both calculated resistivities and AHR with\nexperiment is a non-trivial result.\nWehaveestimatedGDparametersforthe sametypical\nconcentrations as above for the AHR. Calculated values\nof the GD parameter for fcc ( xAl= 0.25) and bcc ( xAl=\n1.25) phases are, respectively, 0.00655and 0.00585. Both\nvalues are similar which is compatible with similar values\noftheDOSattheFermilevelandthetotalmagnetization\nwhose ratio is a rough estimate of the GD parameter,\nwhich explains also rather large values due to small total\nspin moments in both alloys. Calculated values of GD\nparameter are comparable to those in Ni-rich fcc NiFe\nalloy but are larger as compared to bcc-FeCo alloy.16\nIV. CONCLUSIONS\nTransport properties of the fcc and bcc phases of the\nhigh-entropy Al xCrFeCoNi alloys were calculated over a\nbroad range of Al concentrations using the DFT-based\nsimulations. The main conclusions from numerical stud-\nies can be summarized as follows: (i) The agreement\nof calculated residual resistivities with available exper-\nimental data is good for both fcc and bcc phase. In\nparticular, the resistivity values as well as larger resis-\ntivity of the bcc-phase as compared to the fcc one agree\nwith both experiments. Calculation even reproduce de-\ntails of concentration trends in one of the experiment.6\n(ii) The major contribution to the residual resistivity is\ndue to the intrinsic chemical disorder while the magneticdisorder has smaller effect. The increase of ρ0with in-\ncreasing Al concentration in both fcc and bcc phases and\nits large values in particular in the latter one are due to\nstrong scatterings on Al atoms; (iii) The calculated val-\nuesofanisotropicmagnetoresistancearepositivebutvery\nsmall being less than 0.05% for both fcc and bcc phases;\n(iv) The spin disorder influences resistivity only weakly\nbecause of already large resistivity of the reference FM\nstate; (v) Estimated values of the Gilbert damping are\ncomparable for chosen typical fcc and bcc phases and are\nrather large (of order 0.006) due to small total spin mo-\nments; and (vi) The estimated anomalous Hall resistivity\nagain agrees well for the bcc phase while agreement with\nthe experiment for the fcc phase is worse though still\nacceptable.\nThe present results thus suggest that the CPA cap-\ntures the main scattering mechanism due to intrinsic\nalloy disorder and gives acceptable description even for\nsuch complex alloys like the studied one.\nAcknowledgments\nThe work of J.K., V.D, F.M., and I.T. was supported\nby a Grant from the Czech Science Foundation (No. 18-\n07172S) and S.K. thanks for support from the Center for\nComputational Materials Science, Vienna University of\nTechnology. 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B 68, 012402\n(2003)."    },    {        "title": "1907.11853v1.Two_improved_Gauss_Seidel_projection_methods_for_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "Two improved Gauss-Seidel projection methods for\nLandau-Lifshitz-Gilbert equation\nPanchi Lia, Changjian Xiea, Rui Dua,b,\u0003, Jingrun Chena,b,\u0003, Xiao-Ping Wangc,\u0003\naSchool of Mathematical Sciences, Soochow University, Suzhou, 215006, China.\nbMathematical Center for Interdisciplinary Research, Soochow University, Suzhou, 215006, China.\ncDepartment of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay,\nKowloon, Hong Kong, China\nA B S T R A C T\nMicromagnetic simulation is an important tool to study various dynamic behaviors of\nmagnetic order in ferromagnetic materials. The underlying model is the Landau-Lifshitz-\nGilbert equation, where the magnetization dynamics is driven by the gyromagnetic torque\nterm and the Gilbert damping term. Numerically, considerable progress has been made in\nthe past decades. One of the most popular methods is the Gauss-Seidel projection method\ndeveloped by Xiao-Ping Wang, Carlos Garc\u0013 \u0010a-Cervera, and Weinan E in 2001. It \frst solves\na set of heat equations with constant coe\u000ecients and updates the gyromagnetic term in the\nGauss-Seidel manner, and then solves another set of heat equations with constant coe\u000ecients\nfor the damping term. Afterwards, a projection step is applied to preserve the length con-\nstraint in the pointwise sense. This method has been veri\fed to be unconditionally stable\nnumerically and successfully applied to study magnetization dynamics under various controls.\nIn this paper, we present two improved Gauss-Seidel projection methods with uncondi-\ntional stability. The \frst method updates the gyromagnetic term and the damping term\nsimultaneously and follows by a projection step. The second method introduces two sets of\napproximate solutions, where we update the gyromagnetic term and the damping term simul-\ntaneously for one set of approximate solutions and apply the projection step to the other set\nof approximate solutions in an alternating manner. Compared to the original Gauss-Seidel\nprojection method which has to solve heat equations 7 times at each time step, the improved\nmethods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time\nand second-order accuracy in space are veri\fed by examples in both 1D and 3D. In addi-\ntion, unconditional stability with respect to both the grid size and the damping parameter is\ncon\frmed numerically. Application of both methods to a realistic material is also presented\nwith hysteresis loops and magnetization pro\fles. Compared with the original method, the\nrecorded running times suggest that savings of both methods are about 2 =7 and 4=7 for the\nsame accuracy requirement, respectively.\nKeywords: Landau-Lifshitz-Gilbert equation, Gauss-Seidel projection method, unconditional\nstability, micromagnetic simulation\n2000 MSC: 35Q99, 65Z05, 65M06\n1. Introduction\nIn ferromagnetic materials, the intrinsic magnetic order, known as magnetization M=\n(M1;M2;M3)T, is modeled by the following Landau-Lifshitz-Gilbert (LLG) equation [1, 2, 3]\n@M\n@t=\u0000\rM\u0002H\u0000\r\u000b\nMsM\u0002(M\u0002H) (1)\n\u0003Corresponding authors\ne-mail: LiPanchi1994@163.com (Panchi Li), 20184007005@stu.suda.edu.cn (Changjian Xie),\ndurui@suda.edu.cn (Rui Du), jingrunchen@suda.edu.cn (Jingrun Chen), mawang@ust.hk (Xiao-Ping Wang)\n1arXiv:1907.11853v1  [math.NA]  27 Jul 2019with\rthe gyromagnetic ratio and jMj=Msthe saturation magnetization. On the right-\nhand side of (1), the \frst term is the gyromagnetic term and the second term is the Gilbert\ndamping term with \u000bthe dimensionless damping coe\u000ecient [2]. Note that the gyromagnetic\nterm is a conservative term, whereas the damping term is a dissipative term. The local \feld\nH=\u0000\u000eF\n\u000eMis computed from the Landau-Lifshitz energy functional\nF[M] =1\n2Z\n\n\u001aA\nM2sjrMj2+ \b\u0012M\nMs\u0013\n\u00002\u00160He\u0001M\u001b\ndx+\u00160\n2Z\nR3jrUj2dx; (2)\nwhereAis the exchange constant,A\nM2sjrMjis the exchange interaction energy; \b\u0010\nM\nMs\u0011\nis the anisotropy energy, and for simplicity the material is assumed to be uniaxial with\n\b\u0010\nM\nMs\u0011\n=Ku\nM2s(M2\n2+M2\n3) withKuthe anisotropy constant; \u00002\u00160He\u0001Mis the Zeeman\nenergy due to the external \feld with \u00160the permeability of vacuum. \n is the volume occupied\nby the material. The last term in (2) is the energy resulting from the \feld induced by the\nmagnetization distribution inside the material. This stray \feld Hs=\u0000rUwhereU(x)\nsatis\fes\nU(x) =Z\n\nrN(x\u0000y)\u0001M(y)dy; (3)\nwhereN(x\u0000y) =\u00001\n4\u00191\njx\u0000yjis the Newtonian potential.\nFor convenience, we rescale the original LLG equation (1) by changes of variables t!\n(\u00160\rMs)\u00001tandx!LxwithLthe diameter of \n. De\fne m=M=Msandh=MsH. The\ndimensionless LLG equation reads as\n@m\n@t=\u0000m\u0002h\u0000\u000bm\u0002(m\u0002h); (4)\nwhere\nh=\u0000Q(m2e2+m3e3) +\u000f\u0001m+he+hs (5)\nwith dimensionless parameters Q=Ku=(\u00160M2\ns) and\u000f=A=(\u00160M2\nsL2). Here e2= (0;1;0),\ne3= (0;0;1). Neumann boundary condition is used\n@m\n@\u0017j@\n= 0; (6)\nwhere\u0017is the outward unit normal vector on @\n.\nThe LLG equation is a weakly nonlinear equation. In the absence of Gilbert damping,\n\u000b= 0, equation (4) is a degenerate equation of parabolic type and is related to the sympletic\n\row of harmonic maps [4]. In the large damping limit, \u000b!1 , equation (4) is related to\nthe heat \row for harmonic maps [5]. It is easy to check that jmj= 1 in the pointwise sense\nin the evolution. All these properties possesses interesting challenges for designing numerical\nmethods to solve the LLG equation. Meanwhile, micromagnetic simulation is an important\ntool to study magnetization dynamics of magnetic materials [3, 6]. Over the past decades,\nthere has been increasing progress on numerical methods for the LLG equation; see [7, 8, 9]\n2for reviews and references therein. Finite di\u000berence method and \fnite element method have\nbeen used for the spatial discretization.\nFor the temporal discretization, there are explicit schemes such as Runge-Kutta methods\n[10, 11]. Their stepsizes are subject to strong stability constraint. Another issue is that the\nlength of magnetization cannot be preserved and thus a projection step is needed. Implicit\nschemes [12, 13, 14] are unconditionally stable and usually can preserve the length of magne-\ntization automatically. The di\u000eculty of implicit schemes is how to solve a nonlinear system\nof equations at each step. Therefore, semi-implicit methods [15, 16, 17, 18, 19] provide a com-\npromise between stability and the di\u000ecult for solving the equation at each step. A projection\nstep is also needed to preserve the length of magnetization.\nAmong the semi-implicit schemes, the most popular one is the Gauss-Seidel projection\nmethod (GSPM) proposed by Wang, Garc\u0013 \u0010a-Cervera, and E [15, 18]. GSPM \frst solves a\nset of heat equations with constant coe\u000ecients and updates the gyromagnetic term in the\nGauss-Seidel manner, and then solves another set of heat equations with constant coe\u000ecients\nfor the damping term. Afterwards, a projection step is applied to preserve the length of mag-\nnetization. GSPM is \frst-order accurate in time and has been veri\fed to be unconditionally\nstable numerically.\nIn this paper, we present two improved Gauss-Seidel projection methods with uncondi-\ntional stability. The \frst method updates the gyromagnetic term and the damping term\nsimultaneously and follows by a projection step. The second method introduces two sets of\napproximate solutions, where we update the gyromagnetic term and the damping term simul-\ntaneously for one set of approximate solutions and apply the projection step to the other set\nof approximate solutions in an alternating manner. Compared to the original Gauss-Seidel\nprojection method, which solves heat equations 7 times at each time step, the improved\nmethods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time\nand second-order accuracy in space are veri\fed by examples in both 1D and 3D. In addi-\ntion, unconditional stability with respect to both the grid size and the damping parameter is\ncon\frmed numerically. Application of both methods to a realistic material is also presented\nwith hysteresis loops and magnetization pro\fles. Compared with the original method, the\nrecorded running times suggest that savings of both methods are about 2 =7 and 4=7 for the\nsame accuracy requirement, respectively.\nThe rest of the paper is organized as follows. For completeness and comparison, we \frst\nintroduce GSPM in Section 2. Two improved GSPMs are presented in Section 3. Detailed\nnumerical tests are given in Section 4, including accuracy check and e\u000eciency check in both\n1D and 3D, unconditional stability with respect to both the grid size and the damping\nparameter, hysteresis loops, and magnetization pro\fles. Conclusions are drawn in Section 5.\n32. Gauss-Seidel projection method for Landau-Lifshitz-Gilbert equation\nBefore the introduction of the GSPM [15, 18], we \frst use the \fnite di\u000berence method\nfor spatial discretization. Figure 1 shows a schematic picture of spatial grids in 1D. Let\ni= 0;1;\u0001\u0001\u0001;M;M + 1,j= 0;1;\u0001\u0001\u0001;N;N + 1, andk= 0;1;\u0001\u0001\u0001;K;K + 1 be the indices of\ngrid points in 3D.\n0 1𝑥−1\n2𝑥1\n2𝑥𝑁−1\n2𝑥𝑁+1\n2𝑥3\n2𝑥𝑁−3\n2\nFig. 1. Spatial grids in 1D. Nodes x\u00001\n2andxN+1\n2are ghost points.\nSecond-order centered di\u000berence for \u0001 mreads as\n\u0001hmi;j;k=mi+1;j;k\u00002mi;j;k+mi\u00001;j;k\n\u0001x2\n+mi;j+1;k\u00002mi;j;k+mi;j\u00001;k\n\u0001y2\n+mi;j;k+1\u00002mi;j;k+mi;j;k\u00001\n\u0001z2; (7)\nwhere mi;j;k=m((i\u00001\n2)\u0001x;(j\u00001\n2)\u0001y;(k\u00001\n2)\u0001z). For the Neumann boundary condition,\na second-order approximation yields\nm0;j;k=m1;j;k;mM;j;k =mM+1;j;k; j = 1;\u0001\u0001\u0001;N;k = 1;\u0001\u0001\u0001;K;\nmi;0;k=mi;1;k;mi;N;k =mi;N+1;k; i= 1;\u0001\u0001\u0001;M;k = 1;\u0001\u0001\u0001;K;\nmi;j;0=mi;j;1;mi;j;K =mi;j;K +1; i= 1;\u0001\u0001\u0001;M;j = 1;\u0001\u0001\u0001;N:\nTo illustrate the main ideas, we \frst consider the following simpli\fed equation\nmt=\u0000m\u0002\u0001m\u0000\u000bm\u0002(m\u0002\u0001m);\nwhich can be rewritten as\nmt=\u0000m\u0002\u0001m\u0000\u000bm(m\u0001\u0001m) +\u000b\u0001m: (8)\nWe split (8) into two equations\nmt=\u0000m\u0002\u0001m; (9)\nmt=\u000b\u0001m: (10)\nHowever, (9) is still nonlinear. Therefore, we consider a fractional step scheme to solve\n(9)\nm\u0003\u0000mn\n\u0001t= \u0001hm\u0003\nmn+1=mn\u0000mn\u0002m\u0003\n4or\nmn+1=mn\u0000mn\u0002(I\u0000\u0001t\u0001h)\u00001mn;\nwhereIis the identity matrix. This scheme is subject to strong stability constraint, and thus\nthe implicit Gauss-Seidel scheme is introduced to overcome this issue. Let\ngn\ni= (I\u0000\u0001t\u0001h)\u00001mn\ni; i= 1;2;3: (11)\nWe then have 0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=0\n@mn\n1+ (gn\n2mn\n3\u0000gn\n3mn\n2)\nmn\n2+ (gn\n3mn+1\n1\u0000gn+1\n1mn\n3)\nmn\n3+ (gn+1\n1mn+1\n2\u0000gn+1\n2mn+1\n1)1\nA: (12)\nThis scheme solve (9) with unconditional stability. (10) is linear heat equation which can be\nsolved easily. However, the splitting scheme (9) - (10) cannot preserve jmj= 1, and thus a\nprojection step needs to be added.\nFor the full LLG equation (4), the GSPM works as follows. De\fne\nh=\u000f\u0001m+^f; (13)\nwhere ^f=\u0000Q(m2e2+m3e3) +he+hs.\nThe original GSPM [15] solves the equation (4) in three steps:\n\u000fImplicit Gauss-Seidel\ngn\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(mn\ni+ \u0001t^fn\ni); i= 2;3;\ng\u0003\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^fn\ni); i= 1;2; (14)\n0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA=0\n@mn\n1+ (gn\n2mn\n3\u0000gn\n3mn\n2)\nmn\n2+ (gn\n3m\u0003\n1\u0000g\u0003\n1mn\n3)\nmn\n3+ (g\u0003\n1m\u0003\n2\u0000g\u0003\n2m\u0003\n1)1\nA: (15)\n\u000fHeat \row without constraints\n^f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +he+hn\ns; (16)\n0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA=0\n@m\u0003\n1+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n1+^f\u0003\n1)\nm\u0003\n2+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n2+^f\u0003\n2)\nm\u0003\n3+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n3+^f\u0003\n3)1\nA: (17)\n\u000fProjection onto S2\n0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003\u0003j0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA: (18)\nHere the numerical stability of the original GSPM [15] was founded to be independent of\ngridsizes but depend on the damping parameter \u000b. This issue was solved in [18] by replacing\n(14) and (16) with\ng\u0003\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^f\u0003\ni); i= 1;2;\n5and\n^f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +he+h\u0003\ns;\nrespectively. Update of the stray \feld is done using fast Fourier transform [15]. It is easy\nto see that the GSPM solves 7 linear systems of equations with constant coe\u000ecients and\nupdates the stray \feld using FFT 6 times at each step.\n3. Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert\nequation\nBased on the description of the original GSPM in Section 2, we introduce two improved\nGSPMs for LLG equation. The \frst improvement updates both the gyromagnetic term and\nthe damping term simultaneously, termed as Scheme A. The second improvement introduces\ntwo sets of approximate solution with one set for implicit Gauss-Seidel step and the other set\nfor projection in an alternating manner, termed as Scheme B. Details are given in below.\n3.1. Scheme A\nThe main improvement of Scheme A over the original GSPM is the combination of (13)\n- (17), or (9) - (10).\n\u000fImplicit-Gauss-Seidel\ngn\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(mn\ni+ \u0001t^fn\ni); i= 1;2;3;\ng\u0003\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^f\u0003\ni); i= 1;2; (19)\n0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA=0\n@mn\n1\u0000(mn\n2gn\n3\u0000mn\n3gn\n2)\u0000\u000b(mn\n1gn\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n1+\u000bgn\n1\nmn\n2\u0000(mn\n3g\u0003\n1\u0000m\u0003\n1gn\n3)\u0000\u000b(m\u0003\n1g\u0003\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n2+\u000bgn\n2\nmn\n3\u0000(m\u0003\n1g\u0003\n2\u0000m\u0003\n2g\u0003\n1)\u0000\u000b(m\u0003\n1g\u0003\n1+m\u0003\n2g\u0003\n2+mn\n3gn\n3)mn\n3+\u000bgn\n31\nA:(20)\n\u000fProjection onto S2\n0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003j0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA: (21)\nIt is easy to see that Scheme A solves 5 linear systems of equations with constant coe\u000ecients\nand uses FFT 5 times at each step.\n3.2. Scheme B\nThe main improvement of Scheme B over Scheme A is the introduction of two sets of\napproximate solutions, one for (19) - (20) and the other for (21) and the update of these two\nsets of solutions in an alternating manner.\nGiven the initialized g0\ng0\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m0\ni+ \u0001t^f0\ni); i= 1;2;3; (22)\nScheme B works as follows\n6\u000fImplicit Gauss-Seidel\ngn+1\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^f\u0003\ni); i= 1;2;3 (23)\nm\u0003\n1=mn\n1\u0000(mn\n2gn\n3\u0000mn\n3gn\n2)\u0000\u000b(mn\n1gn\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n1+\n\u000b((mn\n1)2+ (mn\n2)2+ (mn\n3)2)gn\n1\nm\u0003\n2=mn\n2\u0000(mn\n3gn+1\n1\u0000m\u0003\n1gn\n3)\u0000\u000b(m\u0003\n1gn+1\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n2+\n\u000b((m\u0003\n1)2+ (mn\n2)2+ (mn\n3)2)gn\n2\nm\u0003\n3=mn\n3\u0000(m\u0003\n1gn+1\n2\u0000m\u0003\n2gn+1\n1)\u0000\u000b(m\u0003\n1gn+1\n1+m\u0003\n2gn+1\n2+mn\n3gn\n3)mn\n3+\n\u000b((m\u0003\n1)2+ (m\u0003\n2)2+ (mn\n3)2)gn\n3 (24)\n\u000fProjection onto S2\n0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003j0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA: (25)\nHere one set of approximate solution fm\u0003gis updated in the implicit Gauss-Seidel step and\nthe other set of approximate solution fmn+1gis updated in the projection step. Note that\n(23) is de\fned only for fm\u0003gwhich can be used in two successive temporal steps, and thus\nonly 3 linear systems of equations with constant coe\u000ecients are solved at each step and 3\nFFT executions are used for the stray \feld. The length of magnetization can be preserved\nin the time evolution.\nThe computational cost of GSPM and its improvements comes from solving the linear\nsystems of equations with constant coe\u000ecients. To summarize, we list the number of linear\nsystems of equations to be solved and the number of FFT executions to be used at each step\nfor the original GSPM [18], Scheme A, and Scheme B in Table 1. The savings represent the\nratio between costs of two improved schemes over that of the original GSPM.\nGSPM Scheme Number of linear systems Saving Execution of FFT Saving\nOriginal 7 0 4 0\nScheme A 5 2=7 3 1=4\nScheme B 3 4=7 3 1=4\nTable 1. The number of linear systems of equations to be solved and the number of FFT\nexecutions to be used at each step for the original GSPM [18], Scheme A, and Scheme B. The\nsavings represent the ratio between costs of two improved schemes over that of the original\nGSPM.\n4. Numerical Experiments\nIn this section, we compare the original GSPM [15, 18], Scheme A, and Scheme B via a\nseries of examples in both 1D and 3D, including accuracy check and e\u000eciency check, uncon-\nditional stability with respect to both the grid size and the damping parameter, hysteresis\n7loops, and magnetization pro\fles. For convenience, we de\fne\nratio\u0000i=Time(GSPM)\u0000Time(Scheme i)\nTime(GSPM);\nfori= A and B, which quanti\fes the improved e\u000eciency of Scheme A and Scheme B over\nthe original GSPM [15, 18].\n4.1. Accuracy Test\nExample 4.1 (1D case). In 1D, we choose the exact solution over the unit interval \n =\n[0;1]\nme= (cos(\u0016x) sin(t);sin(\u0016x) sin(t);cos(t));\nwhich satis\fes\nmt=\u0000m\u0002mxx\u0000\u000bm\u0002(m\u0002mxx) +f\nwith \u0016x=x2(1\u0000x)2, and f=met+me\u0002mexx+\u000bme\u0002(me\u0002mexx). Parameters are\n\u000b= 0:00001 andT= 5:0e\u00002.\nWe \frst show the error kme\u0000mhk1withmhbeing the numerical solution with respect\nto the temporal stepsize \u0001tand the spatial stepsize \u0001x. As shown in Figure 2(a) and Fig-\nure 2(c), suggested by the least squares \ftting, both \frst-order accuracy in time and second-\norder accuracy in space are observed. Meanwhile, we record the CPU time as a function\nof accuracy (error) by varying the temporal stepsize and the spatial stepsize in Figure 2(b)\nand Figure 2(d), Table 2 and Table 3, respectively. In addition, from Table 2 and Table 3,\nthe saving of Scheme A over GSPM is about 2=7, which equals 1\u00005=7, and the saving of\nScheme B over GSPM is about 4=7, respectively. This observation is in good agreement with\nthe number of linear systems being solved at each step for these three methods, as shown in\nTable 1.\nXXXXXXXXXXCPU time\u0001tT/1250 T/2500 T/5000 T/10000 Reference\nGSPM 7.7882e-01 1.5445e+00 3.1041e+00 6.2196e+00 -\nScheme A 4.8340e-01 9.9000e-01 2.0527e+00 4.4917e+00 -\nScheme B 3.3010e-01 6.3969e-01 1.2281e+00 2.5510e+00 -\nratio-A 0.38 0.36 0.34 0.28 0.29(2/7)\nratio-B 0.58 0.59 0.60 0.59 0.57(4/7)\nTable 2. Recorded CPU time in 1D with respect to the approximation error when only \u0001tis\nvaried and \u0001x= 1=100.\nExample 4.2 (3D case). In 3D, we choose the exact solution over \n = [0;2]\u0002[0;1]\u0002[0;0:2]\nme= (cos(\u0016x\u0016y\u0016z) sin(t);sin(\u0016x\u0016y\u0016z) sin(t);cos(t));\nwhich satis\fes\nmt=\u0000m\u0002\u0001m\u0000\u000bm\u0002(m\u0002\u0001m) +f\n8log(∆t)-12.5 -12 -11.5 -11 -10.5 -10log(error)\n-12.5-12-11.5-11-10.5-10\nGSPM\nScheme A\nScheme B(a) Temporal accuracy\nlog(error)-12.5 -12 -11.5 -11 -10.5 -10log(time)\n-1.5-1-0.500.511.52\nGSPM\nScheme A\nScheme B (b) CPU time versus approximation error (\u0001 t)\nlog(∆x)-5.1 -5 -4.9 -4.8 -4.7 -4.6log(error)\n-16-15.9-15.8-15.7-15.6-15.5-15.4-15.3-15.2-15.1-15\nGSPM\nScheme A\nScheme B\n(c) Spatial accuracy\nlog(error)-16 -15.8 -15.6 -15.4 -15.2 -15log(time)\n77.588.599.5\nGSPM\nScheme A\nScheme B (d) CPU time versus approximation error (\u0001 x)\nFig. 2. Approximation error and CPU time in 1D. (a) Approximation error as a function of the\ntemporal step size; (b) CPU time as a function of the approximation error when \u0001tis varied\nand \u0001xis \fxed; (c) Approximation error as a function of the spatial step size; (d) CPU time\nas a function of the approximation error when \u0001xis varied and \u0001tis \fxed.\nwith \u0016x=x2(1\u0000x)2,\u0016y=y2(1\u0000y)2,\u0016z=z2(1\u0000z)2andf=met+me\u0002\u0001me+\u000bme\u0002(me\u0002\n\u0001me). Parameters are T= 1:0e\u000005and\u000b= 0:01.\nLike in the 1D case, we \frst show the error kme\u0000mhk1withmhbeing the numerical\nsolution with respect to the temporal stepsize \u0001tand the spatial stepsize \u0001x. As shown in\nFigure 3(a) and Figure 3(c), suggested by the least squares \ftting, both \frst-order accuracy in\ntime and second-order accuracy in space are observed. Meanwhile, we record the CPU time\nas a function of accuracy (error) by varying the temporal stepsize and the spatial stepsize in\nFigure 3(b) and Figure 3(d), Table 4 and Table 5, respectively. In addition, from Table 4 and\nTable 5, the saving of Scheme A over GSPM is about 2=7, and the saving of Scheme B over\nGSPM is about 4=7, respectively. This observation is in good agreement with the number of\nlinear systems being solved at each step for these three methods, as shown in Table 1.\nIt worths mentioning that all these three methods are tested to be unconditionally stable\nwith respect to the spatial gridsize and the temporal stepsize.\n9XXXXXXXXXXCPU time\u0001x1/100 1/120 1/140 1/160 Reference\nGSPM 3.3752e+03 5.2340e+03 9.0334e+03 1.0495e+04 -\nScheme A 2.4391e+03 3.7175e+03 6.5149e+03 8.0429e+03 -\nScheme B 1.4740e+03 2.2448e+03 3.9152e+03 4.8873e+03 -\nratio-A 0.28 0.29 0.28 0.23 0.29(2/7)\nratio-B 0.56 0.57 0.57 0.53 0.57(4/7)\nTable 3. Recorded CPU time in 1D with respect to the approximation error when only \u0001xis\nvaried and \u0001t= 1:0e\u00008.\nXXXXXXXXXXCPU time\u0001tT/10 T/20 T/40 T/80 Reference\nGSPM 3.5188e+01 6.8711e+01 1.4146e+02 2.9769e+02 -\nScheme A 2.3015e+01 4.3920e+01 8.6831e+01 1.7359e+02 -\nScheme B 1.3984e+01 2.6313e+01 5.1928e+01 1.0415e+02 -\nratio-A 0.35 0.36 0.39 0.42 0.29(2/7)\nratio-B 0.60 0.62 0.63 0.65 0.57(4/7)\nTable 4. Recorded CPU time in 3D with respect to the approximation error when only \u0001tis\nvaried and the spatial mesh is 128\u000264\u000210.\n4.2. Micromagnetic Simulations\nTo compare the performance of Scheme A and Scheme B with GSPM, we have carried out\nmicromagnetic simulations of the full LLG equation with realistic material parameters. In\nall our following simulations, we consider a thin \flm ferromagnet of size \n = 1 \u0016m\u00021\u0016m\u0002\n0:02\u0016m with the spatial gridsize 4 nm \u00024 nm\u00024 nm and the temporal stepsize \u0001 t= 1\npicosecond. The demagnetization \feld (stray \feld) is calculated via FFT [15, 18].\n4.2.1. Comparison of hysteresis loops\nThe hysteresis loop is calculated in the following way. First, a positive external \feld\nH0=\u00160His applied and the system is allowed to reach a stable state. Afterwards, the\nexternal \feld is reduced by a certain amount and the system is relaxed to a stable state\nagain. The process continues until the external \feld attains a negative \feld of strength H0.\nThen the external \feld starts to increase and the system relaxes until the initial applied\nexternal \feld H0is approached. In the hysteresis loop, we can monitor the magnetization\ndynamics and plot the average magnetization at the stable state as a function of the strength\nof the external \feld. The stopping criterion for a steady state is that the relative change of\nthe total energy is less than 10\u00007. The applied \feld is parallel to the xaxis. The initial state\nwe take is the uniform state and the damping parameter \u000b= 0:1.\nIn Figure 4, we compare the average magnetization in the hysteresis loop simulated by\nGSPM, Scheme A and Scheme B. Pro\fles of the average magnetization of these three methods\nare in quantitative agreements with approximately the same switch \feld 9 ( \u00060:4) mT.\n4.2.2. Comparison of magnetization pro\fles\nIt is tested that GSPM in [15] was unstable with a very small damping parameter \u000band\nwas resolved in [18]. This section is devoted to the unconditional stability of Scheme A and\n10log(∆t)-14 -13.5 -13 -12.5 -12 -11.5log(error)\n-14-13.5-13-12.5-12-11.5\nGSPM\nScheme A\nScheme B(a) Temporal accuracy\nlog(error)-14 -13.5 -13 -12.5 -12 -11.5log(time)\n2.533.544.555.56\nGSPM\nScheme A\nScheme B (b) CPU time versus approximation error (\u0001 t)\nThe spatial step size log( ∆x)-2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7log(error)\n-32.5-32-31.5-31\nGSPM\nScheme A\nScheme B\n(c) Spatial accuracy\nlog(error)-32.5 -32 -31.5 -31log(time)\n22.533.544.555.56\nGSPM\nScheme A\nScheme B (d) CPU time versus approximation error (\u0001 x)\nFig. 3. Approximation error and CPU time in 3D. (a) Approximation error as a function of the\ntemporal step size; (b) CPU time as a function of the approximation error when \u0001tis varied\nand \u0001x= \u0001y= \u0001zis \fxed; (c) Approximation error as a function of the spatial step size; (d)\nCPU time as a function of the approximation error when space is varied uniformly and \u0001tis\n\fxed.\nScheme B with respect to \u000b. We consider a thin \flm ferromagnet of size 1 \u0016m\u00021\u0016m\u00020:02\u0016m\nwith the spatial gridsize 4 nm \u00024 nm\u00024 nm and the temporal stepsize is 1 picosecond.\nFollowing [18], we consider the full LLG equation with \u000b= 0:1 and\u000b= 0:01 and without\nthe external \feld. The initial state is m0= (0;1;0) ifx2[0;Lx=5][[4Lx=5;Lx] and\nm0= (1;0;0) otherwise. The \fnal time is 10 ns. In Figures 5 to 7, we present a color plot\nof the angle between the in-plane magnetization and the xaxis, and an arrow plot of the\nin-plane magnetization for the original GSPM [15], Scheme A, and Scheme B, respectively.\nIn these \fgures, \u000b= 0:1 is presented in the top row and \u000b= 0:01 is presented in the bottom\nrow; a color plot of the angle between the in-palne magnetization and the xaxis is presented\nin the left column and an arrow plot of the in-plane magnetization is presented in the right\ncolumn.\n5. Conclusion\nIn this paper, based on the original Gauss-Seidel projection methods, we present two\nimproved Gauss-Seidel projection methods with the \frst-order accuracy in time and the\nsecond-order accuracy in space. The \frst method updates the gyromagnetic term and the\n11XXXXXXXXXXCPU time\u0001x1/6 1/8 1/10 1/12 Reference\nGSPM 2.1066e+01 9.2615e+01 1.9879e+02 3.7820e+02 -\nScheme A 1.5278e+01 6.5953e+01 1.4215e+02 2.6725e+02 -\nScheme B 8.9698e+00 3.8684e+01 8.4291e+01 1.5977e+02 -\nratio-A 0.27 0.29 0.28 0.29 0.29(2/7)\nratio-B 0.57 0.58 0.58 0.58 0.57(4/7)\nTable 5. Recorded CPU time in 3D with respect to the approximation error when only the\nspatial gridsize is varied with \u0001x= \u0001y= \u0001zand \u0001t= 1:0e\u000009.\n-50 -40 -30 -20 -10 0 10 20 30 40 50\n0 H (mT)-1-0.8-0.6-0.4-0.200.20.40.60.81M/Ms\n GSPM\n Scheme A\n Scheme B\nFig. 4. Comparison of hysteresis loops for GSPM, Scheme A and Scheme B. Pro\fles of the av-\nerage magnetization of these three methods are in quantitative agreements with approximately\nthe same switch \feld 9 (\u00060:4) mT . The applied \feld is parallel to the xaxis and the initial state\nis the uniform state.\ndamping term simultaneously and follows by a projection step, which requires to solve heat\nequations 5 times at each time step. The second method introduces two sets of approximate\nsolutions, where we update the gyromagnetic term and the damping term simultaneously for\none set of approximate solutions and apply the projection step to the other set of approximate\nsolutions in an alternating manner. Therefore, only 3 heat equations are needed to be solved\nat each step. Compared to the original Gauss-Seidel projection method, which solves heat\nequations 7 times at each step, savings of these two improved methods are about 2 =7 and\n4=7, which is veri\fed by both 1D and 3D examples for the same accuracy requirement. In\naddition, unconditional stability with respect to both the grid size and the damping parameter\nis con\frmed numerically. Application of both methods to a realistic material is also presented\nwith hysteresis loops and magnetization pro\fles.\nAcknowledgments\nThis work is supported in part by the grants NSFC 21602149 (J. Chen), NSFC 11501399\n(R. Du), the Hong Kong Research Grants Council (GRF grants 16302715, 16324416, 16303318\n12(a) Angle pro\fle ( \u000b= 0:1)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (b) Magnetization pro\fle ( \u000b= 0:1)\n(c) Angle pro\fle ( \u000b= 0:01)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (d) Magnetization pro\fle ( \u000b= 0:01)\nFig. 5. Simulation of the full Landau-Lifshitz-Gilbert equation using GSPM without any exter-\nnal \feld. The magnetization on the centered slice of the material in the xyplane is used. Top\nrow:\u000b= 0:1; Bottom row: \u000b= 0:01. Left column: a color plot of the angle between the in-plane\nmagnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization.\n13(a) Angle pro\fle ( \u000b= 0:1)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (b) Magnetization pro\fle ( \u000b= 0:1)\n(c) Angle pro\fle ( \u000b= 0:01)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (d) Magnetization pro\fle ( \u000b= 0:01)\nFig. 6. Simulation of the full Landau-Lifshitz-Gilbert equation using Scheme A without any\nexternal \feld. The magnetization on the centered slice of the material in the xyplane is used.\nTop row: \u000b= 0:1; Bottom row: \u000b= 0:01. Left column: a color plot of the angle between the in-\nplane magnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization.\n14(a) Angle pro\fle ( \u000b= 0:1)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (b) Magnetization pro\fle ( \u000b= 0:1)\n(c) Angle pro\fle ( \u000b= 0:01)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (d) Magnetization pro\fle ( \u000b= 0:01)\nFig. 7. Simulation of the full Landau-Lifshitz-Gilbert equation using Scheme B without any\nexternal \feld. The magnetization on the centered slice of the material in the xyplane is used.\nTop row: \u000b= 0:1; Bottom row: \u000b= 0:01. Left column: a color plot of the angle between the in-\nplane magnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization.\n15and NSFC-RGC joint research grant N-HKUST620/15) (X.-P. Wang), and the Innovation\nProgram for postgraduates in Jiangsu province via grant KYCX19 1947 (C. Xie).\nReferences\n[1] L. Landau, E. Lifshitz, On the theory of the dispersion of magetic permeability in ferromagnetic bodies,\nPhys. Z. Sowjetunion 8 (1935) 153{169.\n[2] T. Gilbert, A lagrangian formulation of gyromagnetic equation of the magnetization \feld, Phys. Rev.\n100 (1955) 1243{1255.\n[3] W. F. B. Jr., Micromagnetics, Interscience Tracts on Physics and Astronomy, 1963.\n[4] P. Sulem, C. Sulem, C. Bardos, On the continuous limit limit for a system of classical spins, Comm.\nMath. Phys. 107 (1986) 431{454.\n[5] M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Di\u000berential Geom. 28 (1988)\n485{502.\n[6] I. \u0014Zuti\u0013 c, J. Fabian, S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76\n(2004) 323{410.\n[7] M. Kruzik, A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism,\nSIAM Rev. 48 (2006) 439{483.\n[8] I. Cimr\u0013 ak, A survey on the numerics and computations for the Landau-Lifshitz equation of micromag-\nnetism, Arch. Comput. Methods Eng. 15 (2008) 277{309.\n[9] C. J. Garc\u0013 \u0010a-Cervera, Numerical micromagnetics: a review, Bol. Soc. Esp. Mat. Apl. 39 (2007) 103{135.\n[10] A. Fran\u0018 cois, J. Pascal, Convergence of a \fnite element discretization for the Landau-Lifshitz equations\nin micromagnetism, Math. Models Methods Appl. Sci. 16 (2006) 299{316.\n[11] A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo, B. Azzerboni, A numerical solution of\nthe magnetization reversal modeling in a permalloy thin \flm using \ffth order runge-kutta method with\nadaptive step size control, Physica B. 403 (2008) 1163{1194.\n[12] Y. H, H. N, Implicit solution of the Landau-Lifshitz-Gilbert equation by the Crank-Nicolson method,\nJ. Magn. Soc. Japan 28 (2004) 924{931.\n[13] S. Bartels, P. Andreas, Convergence of an implicit \fnite element method for the Landau-Lifshitz-Gilbert\nequation, SIAM J. Numer. Anal. 44 (2006) 1405{1419.\n[14] A. Fuwa, T. Ishiwata, M. Tsutsumi, Finite di\u000berence scheme for the Landau-Lifshitz equation, Japan\nJ. Indust. Appl. Math. 29 (2012) 83{110.\n[15] X. Wang, C. J. Garc\u0013 \u0010a-Cervera, W. E, A gauss-seidel projection method for micromagnetics simulations,\nJ. Comput. Phys. 171 (2001) 357{372.\n[16] W. E, X. Wang, Numerical methods for the Landau-Lisfshitz equation, SIAM J. Numer. Anal. 38 (2000)\n1647{1665.\n[17] J. Chen, C. Wang, C. Xie, Convergence analysis of a second-order semi-implicit projection method for\nLandau-Lifshitz equation, arXiv 1902.09740 (2019).\n[18] C. J. Garc\u0013 \u0010a-Cervera, W. E, Improved gauss-seidel projection method for micromagnetics simulations,\nIEEE Trans. Magn. 39 (2003) 1766{1770.\n[19] I. Cimr\u0013 ak, Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation\nwith an exchange \feld, IMA J. Numer. Anal. (2005) 611{634.\n16"    },    {        "title": "1908.08629v2.Damping_enhancement_in_coherent_ferrite_insulating_paramagnet_bilayers.pdf",        "content": "Damping enhancement in coherent ferrite/insulating-paramagnet bilayers\nJacob J. Wisser,1Alexander J. Grutter,2Dustin A. Gilbert,3Alpha T. N'Diaye,4\nChristoph Klewe,4Padraic Shafer,4Elke Arenholz,4, 5Yuri Suzuki,1and Satoru Emori6,\u0003\n1Department of Applied Physics, Stanford University, Stanford, CA, USA\n2NIST Center for Neutron Research, Gaithersburg, MD, USA\n3Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA\n4Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA, USA\n5Cornell High Energy Synchrotron Source, Ithaca, NY, USA\n6Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n(Dated: October 29, 2019)\nHigh-quality epitaxial ferrites, such as low-damping MgAl-ferrite (MAFO), are promising\nnanoscale building blocks for all-oxide heterostructures driven by pure spin current. However, the\nimpact of oxide interfaces on spin dynamics in such heterostructures remains an open question. Here,\nwe investigate the spin dynamics and chemical and magnetic depth pro\fles of 15-nm-thick MAFO\ncoherently interfaced with an isostructural \u00191-8-nm-thick overlayer of paramagnetic CoCr 2O4\n(CCO) as an all-oxide model system. Compared to MAFO without an overlayer, e\u000bective Gilbert\ndamping in MAFO/CCO is enhanced by a factor of >3, irrespective of the CCO overlayer thickness.\nWe attribute this damping enhancement to spin scattering at the \u00181-nm-thick chemically disordered\nlayer at the MAFO/CCO interface, rather than spin pumping or proximity-induced magnetism. Our\nresults indicate that damping in ferrite-based heterostructures is strongly in\ruenced by interfacial\nchemical disorder, even if the thickness of the disordered layer is a small fraction of the ferrite\nthickness.\nI. INTRODUCTION\nEmerging spintronic device schemes leverage magnon\nspin currents in electrically insulating magnetic oxides\n(e.g., ferrites), unaccompanied by dissipative motion\nof electrons, for computing and communications\napplications1,2. Low-dissipation spintronic devices\nbecome particularly attractive if insulating ferrite thin\n\flms with low magnetic damping can serve as sources\nof magnon spin currents. Such low-damping ferrites\ninclude not only epitaxial garnet ferrites (e.g., YIG)3{11\nthat have been widely used in studies of insulating\nspintronics2{4,12{15, but also coherently strained epitaxial\nspinel ferrites16{18with crucial technical advantages over\ngarnets, such as lower thermal budget for crystallization,\nhigher magnon resonance frequencies, and potential to be\nintegrated coherently with other spinels and perovskites\nwith various functionalities19{22.\nIn general, low-damping ferrite thin \flms must be\ninterfaced with other materials to realize spintronic\ndevices. It is therefore essential to understand whether\nand how damping in the ferrite is impacted by the\nproximity to another material. For instance, to convert\nbetween electronic and magnonic signals through direct\nand inverse spin Hall or Rashba-Edelstein e\u000bects23,\nthe low-damping ferrite needs to be interfaced with\na nonmagnetic metal with strong spin-orbit coupling.\nSpin transport and enhanced damping through spin\npumping24in ferrite/spin-orbit-metal structures has\nalready been extensively studied3,4,12{15,25. Moreover,\nthe low-damping ferrite can be interfaced with an\ninsulating antiferromagnetic or paramagnetic oxide, in\nwhich signals can be transmitted as a pure magnon\nspin current26{40. While interfacing low-damping ferriteswith insulating anti/paramagnetic oxides has enabled\nprototypes of magnon spin valves37{39, the fundamental\nimpact of insulating oxide interfaces on spin dynamics\nhas remained mostly unexplored. In particular, it is an\nopen question whether or how damping of the ferrite is\nenhanced from spin dissipation within the bulk of the\nadjacent anti/paramagnetic oxide or from spin scattering\nat the oxide interface.\nHere, we investigate how room-temperature magnetic\ndamping in epitaxial ferrimagnetic spinel MgAl-ferrite\n(MgAl 1=2Fe3=2O4, MAFO) is impacted when interfaced\nwith an overlayer of insulating paramagnetic spinel\nCoCr 2O4(CCO)41,42. This epitaxial MAFO/CCO\nbilayer is an isostructural model system, possessing\na coherent interface with continuous crystal lattices\nbetween the spinel ferrite and paramagnet. We \fnd that\nthe presence of MAFO/CCO interface increases damping\nby more than a factor of >3 compared to MAFO without\nan overlayer. We attribute this damping enhancement {\nwhich is comparable to or greater than spin pumping\ne\u000bects reported for ferrite/spin-orbit-metal bilayers { to\nspin scattering by the ultrathin ( \u00181 nm) chemically\ndisordered layer at the MAFO/CCO interface. Our\n\fndings show that spin scattering at oxide interfaces\nhas a profound in\ruence on damping, even when the\nchemically disordered layer is a small fraction of the total\nmagnetic layer thickness.\nII. FILM GROWTH AND STRUCTURAL\nCHARACTERIZATION\nEpitaxial thin \flms of 15-nm-thick MAFO interfaced\nwith 1.3-8 nm of CCO overlayer were grown on as-arXiv:1908.08629v2  [cond-mat.mtrl-sci]  26 Oct 20192\n40 42 44 46MAO (004)\nMAFO/CCO\n      (004)\nMAFO (004)log10(Intensity) (arb. units)\n2q (deg)CCO (004)\n-0.180 -0.175 -0.1700.570.580.590.600.610.620.63-\n \n   (115)-(a) (c)\nqip(Å-1)qop(Å-1)\nCCO\n(25 nm)MAO(b)\n-0.2-0.1 0.00.10.2MAFOCCOIntensity (arb. units)\nDw004 (deg)MAFO/CCO\nFigure 1. (a) 2 \u0012-!scans of epitaxial MAFO(15 nm), CCO(25 nm), and MAFO(15 nm)/CCO(8 nm). The data are o\u000bset for\nclarity. (b) Rocking curve scans about the (004) \flm peak for the \flms shown in (a). (c) Reciprocal space map of epitaxial\nCCO(25 nm) coherently strained to the MAO substrate.\nreceived single-crystal MgAl 2O4(MAO) substrates via\npulsed laser deposition. A KrF 248 nm laser was\nincident on stoichiometric targets of MAFO and CCO\nwith \ruences of \u00191.5 J/cm2and\u00191.3 J/cm2,\nrespectively. Both \flms were grown in 10 mTorr (1.3\nPa) O 2and were cooled in 100 Torr (13 kPa) O 2.\nMAFO \flms were grown at 450\u000eC, whereas CCO \flms\nwere deposited at 300\u000eC in an attempt to minimize\nintermixing between the MAFO and CCO layers. These\ngrowth temperatures, much lower than >700\u000eC typically\nrequired for epitaxial garnets3{11, are su\u000ecient to fully\ncrystallize MAFO and CCO. The low crystallization\ntemperatures of the spinels o\u000ber an advantage over\nthe oft-studied garnets, with more opportunities for\nisostructural integration with coherent interfaces. The\nMAFO \flms exhibit a room-temperature saturation\nmagnetization of\u0019100 kA/m and a Curie temperature of\n\u0019400 K18. To obtain consistent ferromagnetic resonance\nresults, MAFO \flms were grown and subsequently\ncharacterized by ferromagnetic resonance (FMR) ex-situ;\nafter surface cleaning with ultrasonication in isopropanol,\nCCO overlayers were then deposited as described above.\nGrowth rates were calibrated via X-ray re\rectivity.\nOur structural characterization of MAFO and\nCCO shows high-quality, coherently strained \flms.\nIn symmetric 2 \u0012-!X-ray di\u000braction scans, only\npeaks corresponding to the (00 `) re\rections are\nobserved, indicating that the \flms are highly epitaxial.\nAdditionally, as seen in Fig. 1(a), Laue oscillations\naround the (004) Bragg re\rections in both single-layer\nMAFO and CCO layers as well as MAFO/CCO bilayers\ndenote smooth interfaces. Furthermore, MAFO, CCO,\nand MAFO/CCO samples all exhibit essentially the\nsame \flm-peak rocking curve widths (FWHM) of \u00190.06\u000e\n(Fig. 1(b)). Reciprocal space mapping of the ( \u00161\u001615)\nre\rection in 25-nm-thick single-layer CCO on MAO\n(Fig. 1(c)) reveals that the in-plane lattice parameter of\nthe \flm coincides with that of the substrate, indicating\nCCO is coherently strained to MAO. We note thatdespite the relatively large lattice mismatch between\nCCO and MAO of \u00193 %, coherently strained growth of\nCCO of up to 40 nm has been previously reported on\nMAO substrates41. For our CCO \flm, we calculate an\nout-of-plane lattice constant c\u00198:534\u0017A from the 2 \u0012-!\nscan; taking the in-plane lattice parameter a= 8:083\u0017A of\nthe MAO substrate, the resulting tetragonal distortion of\ncoherently strained CCO is c=a\u00191:055, similar to that\nfor coherently strained MAFO18.\nStructural characterization results underscore the\nquality of these epitaxial \flms grown as single layers and\nbilayers. Considering the comparable high crystalline\nquality for MAFO, CCO, and MAFO/CCO { as\nevidenced by the presence of Laue oscillations and narrow\n\flm-peak rocking curves { we conclude that MAFO/CCO\nbilayers (with the total thickness limited to \u001423 nm) are\ncoherently strained to the substrate. In these samples\nwhere the substrate and \flm layers are isostructural, we\nalso do not expect antiphase boundaries43{46. Indeed,\nwe \fnd no evidence for frustrated magnetism, i.e., high\nsaturation \feld and coercivity, that would arise from\nantiphase boundaries in spinel ferrites43{46; MAFO/CCO\nbilayers studied here instead exhibit soft magnetism, i.e.,\nsquare hysteresis loops with low coercivity <0.5 mT,\nsimilar to our previous report on epitaxial MAFO thin\n\flms18. Thus, MAFO/CCO is a high-quality all-oxide\nmodel system, which permits the evaluation of how spin\ndynamics are impacted by a structurally clean, coherent\ninterface.\nIII. FERROMAGNETIC RESONANCE\nCHARACTERIZATION OF DAMPING\nTo quantify e\u000bective damping in coherently strained\nMAFO(/CCO) thin \flms, we performed broadband\nFMR measurements at room temperature in a coplanar\nwaveguide setup using the same procedure as our prior\nwork16,18. We show FMR results with external bias3\nmagnetic \feld applied in the \flm plane along the [100]\ndirection of MAFO(/CCO); essentially identical damping\nresults were obtained with in-plane \feld applied along\n[110]47. Figure 2(a) shows the frequency fdependence of\nhalf-width-at-half-maximum (HWHM) linewidth \u0001 Hfor\na single-layer MAFO sample and a MAFO/CCO bilayer\nwith a CCO overlayer thickness of just 1.3 nm, i.e., less\nthan 2 unit cells. The linewidth is related to the e\u000bective\nGilbert damping parameter \u000beffvia the linear equation:\n\u0001H= \u0001H0+h\u000beff\ng\u00160\u0016Bf (1)\nwhere \u0001H0is the zero-frequency linewidth, his Planck's\nconstant,g\u00192:05 is the Land\u0013 e g-factor derived from the\nfrequency dependence of resonance \feld HFMR ,\u00160is the\npermeability of free space, and \u0016Bis the Bohr magneton.\nIt is easily seen from Fig. 2(a) that with the addition\nof ultrathin CCO, the damping parameter is drastically\nincreased, i.e., >3 times its value in bare MAFO.\nFigure 2(b) shows that the damping enhancement\nseen in MAFO/CCO is essentially independent of\nthe CCO thickness. This trend suggests that\nthe damping enhancement is purely due to the\nMAFO/CCO interface, rather than spin dissipation in\nthe bulk of CCO akin to the absorption of di\u000busive\nspin current reported in antiferromagnetic NiO26,35,48.\nWe note that other bulk magnetic properties of\nMAFO (e.g., e\u000bective magnetization, Land\u0013 e g-factor,\nmagnetocrystalline anisotropy) are not modi\fed by the\nCCO overlayer in a detectable way. We also rule\nout e\u000bects from solvent cleaning prior to CCO growth\nor thermal cycling in the deposition chamber up to\n300\u000eC, as subjecting bare MAFO to the same ex-\nsitu cleaning and in-situ heating/cooling processes as\ndescribed in Section II, but without CCO deposition,\nresults in no measurable change in damping. The\ndamping enhancement therefore evidently arises from the\nproximity of MAFO to the CCO overlayer.\nWe consider two possible mechanisms at the\nMAFO/CCO interface for the observed damping\nenhancement:\n(1) Spin current excited by FMR in MAFO\nmay be absorbed via spin transfer in an interfacial\nproximity-magnetized layer49of CCO, whose magnetic\nmoments may not be completely aligned with those of\nMAFO. While CCO by itself is paramagnetic at room\ntemperature, prior studies have shown that Co2+and\nCr3+cations in epitaxial CCO interfaced with a spinel\nferrite (e.g., Fe 3O4) can develop measurable magnetic\norder50. Such damping enhancement due to interfacial\nmagnetic layer is analogous to spin dephasing reported\nfor ferromagnets interfaced directly with proximity-\nmagnetized paramagnetic metal (e.g., Pt, Pd)49.\n(2) Even if CCO does not develop proximity-induced\nmagnetism, chemical disorder at the MAFO/CCO\ninterface may enhance spin scattering. For instance,\nchemical disorder may lead to an increase of Fe2+\n0 10 20 300246810HWHM Linewidth (mT)\nFrequency (GHz)(a)\n(b)MAFO/CCO\neff≈ 0.007\nMAFO\neff≈ 0.002\n0 2 4 6 80.0000.0020.0040.0060.0080.0100.012eff\nCCO thickness (nm)Figure 2. (a) HWHM FMR linewidth versus frequency\nfor MAFO(15 nm) and MAFO(15 nm)/CCO(1.3 nm). The\ne\u000bective Gilbert damping parameter \u000beffis derived from\nthe linear \ft. (b) \u000beffplotted against the CCO overlayer\nthickness. The dashed horizontal line indicates the average of\n\u000befffor MAFO without an overlayer.)\ncations at the MAFO surface, thereby increasing\nthe spin-orbit spin scattering contribution to Gilbert\ndamping in MAFO compared to its intrinsic composition\ndominated by Fe3+with weak spin-orbit coupling18,51.\nAnother possibility is that chemical disorder at the\nMAFO/CCO interface introduces magnetic roughness\nthat gives rise to additional spin scattering, perhaps\nsimilar to two-magnon scattering recently reported for\nferromagnet/spin-orbit-metal systems52.\nIn the following section, we directly examine interfacial\nproximity magnetism and chemical disorder to gain\ninsight into the physical origin of the observed damping\nenhancement in MAFO/CCO.\nIV. CHARACTERIZATION OF INTERFACE\nCHEMISTRY AND MAGNETISM\nTo evaluate the potential formation of a magnetized\nlayer in the interfacial CCO through the magnetic\nproximity e\u000bect, we performed depth-resolved\nand element-speci\fc magnetic characterization\nof MAFO/CCO bilayers using polarized neutron\nre\rectometry (PNR) and soft magnetic X-ray\nspectroscopy. PNR measurements were performed\nusing the PBR instrument at the NIST Center for\nNeutron Research on nominally 15-nm-thick MAFO\nlayers capped with either thick (5 nm) or thin (3 nm)4\nCCO overlayers. PNR measurements were performed in\nan in-plane applied \feld of 3 T at temperatures of 300\nK and 115 K, the latter case being slightly above the\nnominal 97 K Curie temperature of CCO41,42. Incident\nneutrons were spin-polarized parallel or anti-parallel to\nthe applied \feld both before and after scattering from\nthe sample, and the re\rected intensity was measured\nas a function of the perpendicular momentum transfer\nvector Q. The incident spin state of measured neutrons\nwere retained after scattering, corresponding to the\ntwo non-spin-\rip re\rectivity cross sections ( \"\"and##).\nSince all layers of the \flm are expected to saturate well\nbelow the applied \feld of 3 T, no spin-\rip re\rectivity is\nexpected and these cross sections were not measured.\nSince PNR is sensitive to the depth pro\fles of the\nnuclear and magnetic scattering length density (SLD),\nthe data can be \ftted to extract the chemical and\nmagnetic depth pro\fles of the heterostructure. In this\ncase, we used the Re\r1D software package for this\npurpose53. Figure 3(a,b) shows the 300 K re\rectivities\nand spin asymmetry curves of a nominal MAFO (15\nnm)/CCO (5 nm) sample alongside the depth pro\fle\n(Fig. 3(c)) used to generate the \fts shown. The\nbest \ft pro\fle (Fig. 3(c)) provides no evidence of a\nlayer with proximity-induced magnetization in the CCO.\nRather, we note that there appears to be a layer of\nmagnetization suppression near both the MAO/MAFO\nand MAFO/CCO interfaces. Further, the interfacial\nroughnesses of both the MAO/MAFO and MAFO/CCO,\n0.9(1) nm and 1.35(5) nm respectively, are signi\fcantly\nlarger than the CCO surface roughness of 0.27(3) nm\nand the bare MAFO surface roughness of <\u00180.5 nm54.\nThe interfacial roughnesses are signatures of chemical\nintermixing at the spinel-spinel interface leading to\ninterfacial suppression of the magnetization and/or Curie\ntemperature. Thus, we \fnd that the MAFO/CCO\ninterface, although structurally coherent, exhibits a\nchemically intermixed region on the order of one spinel\nunit cell thick on either side.\nTo obtain an upper limit of the proximity-induced\ninterfacial magnetization in CCO, we performed Markov-\nchain Monte-carlo simulations as implemented in the\nDREAM algorithm of the BUMPS python package.\nThese simulations suggest an upper limit (95% con\fdence\ninterval of) 7 emu/cc in the 1.5 nm of the CCO closest\nto the interface. In this case, the model evaluated the\nMAFO as a uniform structural slab but allowed for total\nor partial magnetization suppression at both interfaces,\nwhile the CCO layer was treated as a uniform slab with\nan allowed magnetization layer of variable thickness at\nthe interface.\nHowever, we note that equivalently good \fts are\nobtained using simpler models that \ft a single MAFO\nlayer with magnetically dead layers at the interfaces and\na completely nonmagnetic CCO layer. Equivalent results\nwere obtained for the thick CCO sample at 115 K and\nfor the thin CCO sample. We therefore conclude that the\nPNR results strongly favor a physical picture in which the\nFigure 3. (a) Spin-polarized neutron re\rectivity and (b)\nspin asymmetry of a MAFO (15 nm)/CCO (5 nm) bilayer\nalongside theoretical \fts. (c) Nuclear and magnetic scattering\n(scaled \u000210) length density pro\fle used to generate the \fts\nshown. Error bars represent \u00061 standard deviation.\nCCO is notmagnetized through the magnetic proximity\ne\u000bect.\nTo con\frm the PNR results and examine the e\u000bect\nof a CCO overlayer on the local environment of Fe\ncations in MAFO, we performed temperature-dependent\nX-ray absorption (XA) spectroscopy and X-ray magnetic\ncircular dichroism (XMCD) measurements at Beamline\n4.0.2 of the Advanced Light Source at Lawrence Berkeley\nNational Laboratory. We note that the detection\nmode (total electron yield) used here for XA/XMCD\nis sensitive to the top \u00195 nm of the sample, such that\nFe L edge signals from CCO-capped MAFO primarily\ncapture the cation chemistry near the MAFO/CCO\ninterface. Measurements were performed in an applied\n\feld of 400 mT along the circularly polarized X-ray beam,\nincident at 30\u000egrazing from the \flm plane. To minimize\ndrift e\u000bects during the measurement, multiple successive\nenergy scans were taken and averaged, switching both\napplied \feld direction and photon helicity so that all\nfour possible combinations of \feld direction and helicity\nwere captured at least once. XA and XMCD intensities\nwere normalized such that the pre-edge is zero and\nthe maximum value of the average of the (+) and\n(\u0000) intensities is unity. In the case of the Co L-\nedge, measurements were taken with energy sweeps\ncovering both Fe and Co edges, and for consistency\nboth edges were normalized to the highest XAS signal,\ncorresponding to the Fe L 3-edge.\nFigure 4(a) compares the XA of a bare MAFO \flm5\nFigure 4. (a) 300 K X-ray absorption spectra of MAFO and\nMAFO/CCO (3 nm) grown on MAO. (b) Photon helicity-\ndependent XA spectra and XMCD of the Fe L-edge for a\nMAFO/CCO (3 nm) bilayer at 300 K. (c) Co and (d) Cr\nL-edge XA and XMCD of the same bilayer.\nwith one capped by 3 nm of CCO. The two XA lineshapes\nare nearly identical, indicating the same average Fe\noxidation state and site-distribution in CCO-capped\nand uncapped MAFO \flms. It is therefore likely that\nthe reduced interfacial magnetization observed through\nPNR is a result of a defect-induced Curie temperature\nreduction, rather than preferential site-occupation of Co\nand Cr that might increase the Fe2+content in the\nintermixed interfacial region.\nWe further note that although a large XMCD signal\nis observed on the Fe-edge at 300 K (Fig. 4(b)), neither\nthe Co nor Cr L edges exhibit any signi\fcant magnetic\ndichroism, as shown in Figs. 4(c)-(d). Similar results\nare obtained on the Cr L edge at 120 K. Consistent\nwith the PNR results, we thus \fnd no evidence for\na net magnetization induced in the CCO through the\ninterfacial magnetic proximity e\u000bect.\nOur \fnding of suppressed interfacial magnetism\nin MAFO/CCO is reminiscent of earlier reports\nof magnetic dead layers in epitaxially-grown ferrite-\nbased heterostructures55{57. For example, prior\nPNR experiments have revealed magnetic dead layers\nat the interfaces of ferrimagnetic spinel Fe 3O4and\nantiferromagnetic rock-salt NiO or CoO, even when the\ninterfacial roughness is small (e.g., only 0.3 nm)55,56.\nA magnetic dead layer of 1 spinel unit cell has also\nbeen reported at the interface of Fe 3O4and diamagnetic\nrock-salt MgO grown by molecular beam epitaxy57.\nWe note that in these prior studies, the spinel ferrite\flms interfaced with the rock salts (NiO, CoO, MgO)\npossess antiphase boundaries. Suppressed magnetism\nis known to result from antiphase boundaries, as they\nfrustrate the long-range magnetic order and reduce\nthe net magnetization of the ferrite44. By contrast,\nthere is no evidence for antiphase boundaries in all-\nspinel MAFO/CCO grown on spinel MAO; therefore,\nthe suppressed magnetism at the MAFO/CCO interface\ncannot be attributed to antiphase-boundary-induced\nmagnetic frustration.\nAnother possible scenario is that magnetic dead layer\nformation is a fundamental consequence of the charge\nimbalance between di\u000berent lattice planes, as recently\nshown in a recent report of (polar) Fe 3O4undergoing\natomic reconstruction to avoid \\polar catastrophe\" when\ngrown on (nonpolar) MgO58. In our study on all-\nspinel heterostructures, there may also be some degree of\ncharge mismatch depending on the relative populations\nof cations on the tetrahedrally- and octahedrally-\ncoordinated sites at the MAFO/CCO interface, although\nthe charge mismatch is expected to be only \u0019\u00061, i.e.,\na factor of\u00195-6 smaller than that in MgO/Fe 3O458.\nThus, atomic reconstruction driven by charge imbalance\nappears unlikely as a dominant source of the magnetic\ndead layer in MAFO/CCO. We instead tentatively\nattribute the dead layer to atomic intermixing driven by\ndi\u000busion across the MAFO/CCO interface during CCO\noverlayer deposition.\nV. DISCUSSION\nOur PNR and XA/XMCD results (Section IV) indicate\nthat the damping enhancement observed in Section III\narises from chemical disorder, rather than proximity-\ninduced magnetism, at the MAFO/CCO interface.\nWe emphasize that this interfacial disordered layer\nis con\fned to within \u00192 spinel unit cells. We\nalso note that this interfacial disorder is due to\natomic intermixing, but not structural defects (e.g.,\ndislocations, antiphase boundaries), in this coherent\nbilayer system of MAFO/CCO. Nevertheless, this\nultrathin chemically disordered layer alone is evidently\nsu\u000ecient to signi\fcantly increase spin scattering.\nConsidering that the cation chemistry of Fe in MAFO\ndoes not change substantially (Fig. 4(a)), the interfacial\nspin scattering is likely driven by magnetic roughness,\nleading to a mechanism similar to two-magnon scattering\nthat accounts for a large fraction of e\u000bective damping in\nmetallic ferromagnet/Pt bilayers52.\nWe now put in context the magnitude of the damping\nenhancement \u0001 \u000beff, i.e., the di\u000berence in the e\u000bective\nGilbert damping parameter between CCO-capped and\nbare MAFO,\n\u0001\u000beff=\u000bbilayer\neff\u0000\u000bferrite\neff; (2)\nby comparing it with ferrite/spin-orbit-metal systems\nwhere spin pumping is often considered as the source6\n0.0000.0020.0040.0060.008\n MAFO/CCO\n   [this study] MAFO/W\n [Riddiford] MAFO/Pt\n [Riddiford]YIG/Pt\n[Wang]Daeff\nYIG/Pt\n [Sun]\nFigure 5. Comparison of the enhancement of the e\u000bective\nGilbert damping parameter \u0001 \u000befffor MAFO/CCO and\nferrite/spin-orbit-metal bilayers. YIG/Pt [Sun], YIG/Pt\n[Wang], and MAFO/Pt(W) [Riddiford] are adapted from\nRefs.59,60, and61respectively. The values of \u0001 \u000befffrom the\nliterature are normalized for the saturation magnetization\nof 100 kA/m and magnetic thickness of 15 nm for direct\ncomparison with our MAFO/CCO result.\nof damping enhancement. Since damping enhancement\nfrom spin pumping or interfacial scattering scales\ninversely with the product of the saturation of\nmagnetization Msand the magnetic layer thickness tm,\nthe values of \u0001 \u000befftaken from the literature59{61are\nnormalized for direct comparison with the MAFO \flms\nstudied here with Ms= 100 kA/m and tm= 15 nm.\nAs summarized in Fig. 5, \u0001 \u000befffor MAFO/CCO\nis comparable to { or even greater than { \u0001 \u000beff\nfor ferrite/metal bilayers. This \fnding highlights that\nthe strength of increased spin scattering in a ferrite\ndue to interfacial chemical disorder can be on par\nwith spin dissipation due to spin pumping in metallic\nspin sinks. More generally, this \fnding suggests that\nspecial care may be required in directly relating \u0001 \u000beff\nto spin pumping across bilayer interfaces (i.e., spin-\nmixing conductance52), particularly when the FMR-\ndriven magnetic layer is directly interfaced with a spin\nscatterer.\nFurthermore, the strong interfacial spin scattering {\neven when the oxide interface is structurally coherent\nand the chemically disordered layer is kept to just <\u00182\nunit cells { poses a signi\fcant challenge for maintaining\nlow damping in ferrite/insulator heterostructures. This\nchallenge is partially analogous to the problem of reduced\nspin polarization in tunnel junctions consisting of spinelFe3O4and oxide barriers (e.g., MgO)62{65, which is also\nlikely due to interfacial chemical disorder and magnetic\ndead layers. However, we emphasize that the problems of\nantiphase boundaries43{46and charge-imbalance-driven\natomic reconstruction58, which have posed intrinsic\nchallenges for devices with MgO/Fe 3O4interfaces, are\nlikely not applicable to all-spinel MAFO/CCO. It is\ntherefore possible that deposition schemes that yield\nsharper interfaces, e.g., molecular beam epitaxy, can be\nemployed to reduce interfacial imperfections and hence\nspin scattering at MAFO/CCO for low-loss all-oxide\ndevice structures.\nVI. CONCLUSIONS\nWe have shown that e\u000bective damping in epitaxial\nspinel MgAl-ferrite (MAFO) increases more than\nthreefold when interfaced coherently with an insulating\nparamagnetic spinel of CoCr 2O4(CCO). This damping\nenhancement is not due to spin pumping into the\nbulk of CCO. Our depth-resolved characterization of\nMAFO/CCO bilayers also reveals no proximity-induced\nmagnetization in CCO or signi\fcant change in the\ncation chemistry of MAFO. We attribute the giant\ndamping enhancement to spin scattering in an ultrathin\nchemically disordered layer, con\fned to within 2 spinel\nunit cells across the MAFO/CCO interface. Our results\ndemonstrate that spin dynamics in ferrite thin \flms are\nstrongly impacted by interfacial disorder.\nAcknowledgements - This work was supported in\npart by the Vannevar Bush Faculty Fellowship program\nsponsored by the Basic Research O\u000ece of the Assistant\nSecretary of Defense for Research and Engineering and\nfunded by the O\u000ece of Naval Research through grant\nno. N00014-15-1-0045. 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Phys.\nLett.105, 102410 (2014)."    },    {        "title": "1908.11084v1.Enhancement_of_ultrafast_demagnetization_rate_and_Gilbert_damping_driven_by_femtosecond_laser_induced_spin_currents_in_Fe81Ga19_Ir20Mn80_bilayers.pdf",        "content": "1 \n Enhancement of u ltrafast demagnetization  rate and Gilbert damping driven by \nfemtosecond  laser -induced spin currents in Fe81Ga 19/Ir20Mn 80 bilayers  \nWei Zhang1,2, Qian Li u3, Zhe Yuan3, Ke Xia3, Wei He1, Qing -feng Zhan4, Xiang -qun \nZhang1, and Zhao -hua Cheng1,2,5* \n1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed \nMatter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, \nChina  \n2School o f Physical Sciences, University of Chinese Academy of Sciences, Beijing \n100049, China  \n3The Center for Advanced Quantum Studies and Department of Physics, Beijing \nNormal University, 100875 China  \n4State Key Laboratory of Precision Spectroscopy, School of Phy sics and Materials \nScience, East China Normal University, Shanghai 200241, China  \n5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China  \n \n \n \n \n 2 \n Abstract  \nIn spintronics application s, ultrafast spin dynamics have  to be controlled at  \nfemtosecond (fs) timescale s via f s-laser radiation. At such ultrafast timescale s, the \neffect of the Gilbert damping factor α on ultrafast demagnetization time\nM should  be \nconsidered.  In previous explorations for the relationship between these two parameters, \nit was found that the theoretical calculations based on the local spin -flip scattering \nmodel do not agree  with the experimental results.  Here,  we find that in \nFe81Ga19(FeGa) /Ir20Mn 80(IrMn) bilayers, the unconventional IrMn  thickness \ndependence of α results  from  the competition between spin currents pumped from the \nferromagnetic (FM) FeGa layer to the antiferromagnetic (AFM) IrMn layer and those \npumped from the AFM layer to the FM layer. More importantly , we establish a \nproportional relationship between the change of the ultrafast demagnetization rate and \nthe enhancement of Gilbert damping induced by the spin currents via interfacial spin \nchemical potential\ns  . Our work build s a bridge to connect the ul trafast \ndemagnetization time and Gilbert damping in ultrafast photo -induced spin currents  \ndominated systems , which  not only explains the disagreement  between experimental \nand theoretical results in the relation  of 𝜏𝑀 with α, but pr ovides  further insight into \nultrafast spin dynamics  as well .    \nPACS numbers: 75.78.Jp, 75.40.Gb, 76.50.+g, 78.47.+p  \n*To whom all correspondence should be addressed.  zhcheng@iphy.ac.cn  \n 3 \n I. INTRODUCTION  \nThe understanding of spin dynamics from nanosecond  (ns) down to femtosecond  \n(fs) timescales is an essential task toward s the realization of ultrafast spintronic devices \nin the frequency range  from GHz to THz  [1,2] . The study of ultrafast demagnetization \ntime,\nM, is one of the most challenging problem s in laser -induced ultrafast spin \ndynamics . The Gilbert damping factor , α , is of the utmost importance for high \nfrequency switching of spintronic devices.  Since both \nM  and α require a transfer of \nangular momentum from the electronic system to the lattice, the unification of these \ntwo seemingly unrelated parameters can facilitate the exploration of the microscopic \nmechanism of laser -induced ultrafast spin dynamics. An inverse ly proportional \nrelation ship between  \nM and α was predicted  by theoretical calculations  based on the \nlocal phonon -mediated Elliott -Yafet scattering mechanism  [3-5] as well as the \nstochastic Landau -Lifshitz -Bloch  (LLB) model  [6]. However,  the relationship between \nM\nand α has been debated for over one decade  [7]. Until now, all experimental results  \nhave show n that \nM increases with α [8-12].  \nApart from the local spin -flip scattering mechanism  [13], we proposed that the \nnon-local spin current s should be taken into account to coordinate the contradiction in \nthe relationship between 𝜏𝑀 and α. Previous work suggest ed that the superdiffusive \nspin current contribut ed to ultrafast demagnetization  [14], whilst the Gilbert damping \ncould  also be enhanced via non -local spin currents in ferromagnetic  (FM)/nonmagnetic  \n(NM) [15 ] and FM/antiferromagnetic  (AFM) heterostructures  [16]. Femtosecond laser 4 \n irradiation of  ferromagnetic thin films is a  fascinating novel approach to create large \nspin currents  [17,18 ]. Figure 1(a) shows that i n the case of time -resolved magneto -\noptical Kerr effect  (TRMOKE) experiments, hot electrons excited by fs -laser pulses \ncan travel at high velocities and over tens of  nanometers through the films. The \ndifference of mean free path between spin majority and spin minority hot electrons in \nferromagnetic thin film s generates superdiffusive spin currents on fs timescales. Such \nspin current s dissipated at the interface of the heterostructure result in the out -of-\nequilibrium spin accumulation represented by spin chemical potential  𝜇𝑠. Moreover,  \nfigure 1(b) shows the damped magnetization precession around the effective field  could \nbe influenced via spin current.  Tveten et al.  [19] predicted that the ultrafast \ndemagnetization time \nM could be described in the language of spin current -induced \ndamping  𝛼𝑠𝑝 in magnetic heterostructures based on the  electron -magnon scattering \ntheory. However, the experimental evidence on the connection of ultrafast \ndemagnetization  time with damping driven by f s laser-induced spin currents  is not yet \nunderstood.   \n  II. RESULTS  \nA. Sample properties   \n    Ir20Mn 80 (tIrMn)/Fe 81Ga19 (10 nm) bilayers [20 ] were deposited on optically \ntransparent single -crystalline MgO (001) substrates in a magnetron sputtering system \nwith a base pressure below 3×10−7 Torr. The substrates were annealed at 700 °C for 1 h \nin a vacuum chamber and then held  at 250 °C during deposition. FeGa layers were 5 \n obliquely deposited at an incidence angle of 45°. The IrMn layers were deposited while \ncontinuously rotating the substrates. In order to induce an exchange bias (EB) along the \nFeGa [010] direction, a magnetic field of 500 Oe provided by a permanent magnet was \napplied along the MgO [110] axis during growth. After deposition, a 3 nm protective \nTa layer was deposited on the samples to avoid oxidation. The static longitudinal Kerr \nloops  of Fe 81Ga19 (10 nm)/Ir 20Mn 80 (𝑡𝐼𝑟𝑀𝑛) along FeGa  [010] direction with various \nAFM IrMn thicknesses  (𝑡𝐼𝑟𝑀𝑛) at room temperature were acquired using a laser diode \nwith a wavelength of 650 nm.  \n     Figure 2(a) shows the longitudinal Kerr loops  of Fe 81Ga19 (10 nm)/Ir 20Mn 80 (\nIrMnt\nnm) along FeGa  [010] direction with various AFM IrMn thicknesses  (\nIrMnt ) at room \ntemperature, whereas the thickness of FM FeGa layer was fixed at 10 nm. For\nnm2IrMnt\n, the width of the hysteresis loops is enlarged with no obvious shift along  \nthe x-axis, implying that the thickness of IrMn layer is too thin to form an \nantiferromagnetic order for pinning the magnetization reversal of FeGa  [21] (Insert in \nFig. 2(b) (left)). For\nnm2IrMnt , the antiferromagnetic orders are well established, and \nconsequently the antiferromagnetic moments pin FM ones reversal to induce a \nunidirectio nal anisotropy (Insert in Fig. 2 (b) (Right)). The loops therefore exhibit \nevidently exchange bias behavior. The exchange bias field achieves a value of about 60 \nOe when \nnm2IrMnt , whilst the largest value of coercivity (~72 Oe) occurs at \nIrMnt\n2 nm.  \nB. TRMOKE measurements for ultrafast demagnetization and Gilbert damping   6 \n We performed the polar TRMOKE experiment to measure ultrafast \ndemagnetization time under a saturated  applied field of 20 kOe in the normal direction \nof the samples  [22]. The details of the TRMOKE experiment are described in \nAPPENDIX A.  Figure 3(a) shows th e demagnetization curves  for various IrMn \nthicknesses with a maximum magnetization quenching  of ~10%  [23,24 ]. The temporal \nchanges of the Kerr signals ∆𝜃𝑘(𝑡) were normalized by the saturation value 𝜃𝑘 just \nbefore the pump laser excitation. The time evolution of magnetization on sub -\npicosecond timescales can be fitted according to Eq. (1) in terms of the three -\ntemperature model (3TM)  [17]. \n−∆𝑀(𝑡)\n𝑀={{[𝐴1\n(𝑡𝜏0+1⁄)0.5−𝐴2𝜏𝐸−𝐴1𝜏𝑀\n𝜏𝐸−𝜏𝑀𝑒−𝑡\n𝜏𝑀−𝜏𝐸(𝐴1−𝐴2)\n𝜏𝐸−𝜏𝑀𝑒−𝑡\n𝜏𝐸]Θ(𝑡)}∗𝐺(𝑡,𝜏𝐺)}∗𝐺(𝑡,𝜏𝐺)        (1)                                                                        \nwhere \n),(*GtG represents the convolution product with the Gaussian laser pulse \nprofile, \nG  is the full width at half maximum (FWHM) of the laser pulses , \n)(t  is a \nstep function , \n)(t  is the Dirac delta function . 𝐴1 represents the value of ∆𝑀(𝑡)\n𝑀 after \nequilibrium between electrons, spins , and lattices . 𝐴2 is proportional to the initial \nelectrons temperature rise. Here, we used the 780 nm laser as the pump pulse to excite \nthe magnetic system out of equilibrium, while the 390 nm laser pulse was used as a \nprobe beam. Therefore, i n Eq. (1), t he state filling e ffects during pump probe \nexperiment are neglected due to the different wavelength of pump and probe beams \nused in this study. The cooling time by heat diffusion is described by  𝜏0, which should \nbe about one order of magnitude larger than  𝜏𝐸 representing the timescale of electron -\nphonon interactions. The best -fitted value of 𝜏𝐸=500 𝑓𝑠 for all samples is in good 7 \n agreement with that of previous reports  [18]. The fitting parameters in Eq. (1) are shown \nin Table I, from which one notes the pulse  width is 350 fs for all the samples. In our \nexperimental setup, the time -resolution is about 80 fs. In order to obtain a high time \nresolution, we measured the ultrafast demagnetization with very fine step of time delay \n(15 fs). The values of ultrafast dem agnetization time (120 -220 fs) obtained from Eq. (1) \nare defined as the time needed for the magnetization to reach a level of 𝑒−1 of its \nmaximum demagnetization. The time needed for magnetization to reach its maximum \ndemagnetization (>500fs) should be longer than the time extracted from Eq. (1). A \nsimilar result was reported by B. V odungbo et al .[25]. The very large temporal \nstretchi ng of the laser pulse up to 430 fs was attributed to the conversion of the incident \nlaser pulse into a cascade of hot electrons. This could be one of the possible reasons \nresulting in the spread of laser pulse up on the samples in this study. Via changing the \nsingle parameter ， , we can accurately reproduce the experimental results for various \nsamples.  The ultrafast demagnetization time\nM was observed to decrease from  220±\n10 fs for \nIrMnt   = 0 nm to 120±10 fs for \nIrMnt   = 2 nm, then increase with  further \nincreasing 𝑡𝐼𝑟𝑀𝑛  [Fig. 3(b)].  \nThe precessional frequency and damping factor can be derived by means of the \nTRMOKE signal s as well [26, 27]. Figure 4(a) shows the typical time evolution of the \npolar component of magnetization after pump laser excitation at different fields applied \nalong  with the [110] direction of F eGa for 𝑡𝐼𝑟𝑀𝑛=2 𝑛𝑚. It is observed clearly that \nthe spin precession process can be influenced obviously by applied fields. The exact \nM8 \n values for 𝑓 with various applied fields can be obtained using the damped harmonic \nfunction added to an ex ponential -decaying background : \n𝛥𝑀(𝑡)=𝐴+𝐵𝑒𝑥𝑝(−𝑣𝑡)+𝐶𝑒𝑥𝑝(−𝑡\n𝜏)𝑠𝑖𝑛 (2𝜋𝑓𝑡+𝜑)                     (2)                   \nwhere 𝐴 and 𝐵 are the background magnitudes, and 𝑣 is the background recovery \nrate. 𝐶,𝜏,𝑓 and 𝜑are the magnetization precession amplitude, relaxation time, \nfrequency and phase, respectively. The field dependence of frequency 𝑓 extracted \nfrom the fitting procedure is shown in Fig.  4(b). We note that the experimental  f-H \nrelation can be reproduced very well by Kittel equation ( 3) [27 ]. \n             .                                                 (3) \nwith                                                          and             \n                                            . \nAnd γ=𝛾𝑒𝑔2⁄ is the gyromagnetic ratio. 𝜑𝑀 and 𝜑𝐻 are the angles of in -plane \nequilibrium M and H respect to the FeGa [010] easy axis. 𝐾1,𝐾𝑢,𝐾𝑒𝑏 and 𝐾𝑂𝑢𝑡 are \nthe in -plane magneto crystalline, uniaxial , unidirectional and out -of-plane magnetic \nanisotrop y constants of FeGa films,  respectively . The value of magnetocrystalline \nanisotropy  constant  is 𝐾1=4.5×105 𝑒𝑟𝑔/𝑐𝑚3 for the samples with various AFM \nlayer thickness during the fitting procedure and the uniaxial  magnetic anisotropy \nconstant𝐾𝑢=(1.5±0.3)×105𝑒𝑟𝑔/𝑐𝑚3. For 𝑡𝐼𝑟𝑀𝑛=3 𝑛𝑚 and 5 nm, the \nunidirectional magnetic anisotropy constant of  𝐾𝑒𝑏=3×104𝑒𝑟𝑔/𝑐𝑚3 has to be \nincluded for more accurate fitting, although it is one order magnitude smaller than those \n2 1 22 1)2( HHMf\ns\nM eb H M s M M u s O K HM KK K M K H      cos ) cos( 2sin 2 cos2 4 2-2\n1 12 2\nut 1    \nM eb H M s M u M K HM K K H     cos ) cos( 2cos2 4cos21 2    9 \n of magneto crystalline and uniaxial anisotropy .   \nThe effective G ilbert damping factor  𝛼𝑒𝑓𝑓 shown in Fig. 4 (c) is determined \nfrom the relaxation time 𝜏 by Eq. ( 4) [28]: \n \n) (/22 1H Heff                                              (4) \nSince the overall effective damping factor 𝛼𝑒𝑓𝑓 consists of intrinsic damping and \nextrinsic damping whereby the second one arises from  both the two-magnon -scattering \nand the dephasing effect  in the sample s, the overall effective  Gilbert damping factor \ndecreases monotonously to a constant  value  with increasing the applied field (Fig . 4(c)). \nAs one of the main ly extrinsic contributions , the two -magnon -scattering induced \ndamping has been extensively studied in exchange biased heterostructures  [29-34]. The \nmature theory was developed to explain  the two-magnon scattering process due to \nspatial fluctuations of anisotropy and exchange bias field [30,35 ]. The two -magnon \nscattering process comes from  the scatterings of the uniform ( 𝑘=0) precession mode \ninto nonuniform modes ( k≠0 magnons) that  are degenerate in frequency. This \nprocess is described by the Hamiltonian, in which the spatial fluctuation in the exchange \ncoupling caused by interface roughness determines the scattering strength. The \nroughness gives rise to a large fluctuating field because the FM magnetization interacts \nalternatively with one or the other AF sublattice via the atomic exchange coupling. It is \na well -known relaxation mechanism effective in exchange biased heterostructur es due \nto the interface roughness occurring on the short length scales. When a low external \nfield comparable with the exchange bias field was applied, the two-magnon scattering 10 \n effect will result in the increase of Gilbert damping with the exchange bias fi eld \naccording to previous reports  [33, 34 ]. However, as shown in Ref. 36, a strong enough \napplied field can be used to exclude the contributions from the two-magnon -scattering, \nwhere the value of Gilbert dam ping factor keeps as a constant with various two -\nmagnon -scattering strength. Based on this  result , a similar method using strong enough \nexternal fields was applied in this study to exclude the two -magnon -scattering effect. \nMoreover, previous works show that the two -magnon -scattering induced damping \nincreases with precession frequency because of the increased degeneracy of spin waves  \n[37, 38]. Our work demonstrated that the damping factor keeps almost a constant value \nat high enough applied fields, i ndicating the minor contributions from the two-magnon -\nscattering to Gilbert damping. Besides, it has been demonstrated previously that the \ntwo-magnon -scattering contri butions decrease monotonously with increasing the film \nthickness  [33, 34 ]. This again disagrees with the tendency of thickness dependence of \ndamping at high applied field shown in Fig. 5(c). Therefore, in this study, the two -\nmagnon -scattering strength was suppressed effectively by applying a high enough \nexternal field. On the other hand, inhomogeneities in FeGa thin film may cause \nvariations in the local magnetic anisotropy field. It leads to the variations of spin \norientations when the external field is not large enough, and gives rise to t he enhanced \ndamping arising from spin dephasing e ffect [28]. However, an applied field (~ kOe) \nmuch la rger than the anisotropy field makes the spin orientation uniform, as a result, \nthe dephasing effect was suppressed largely.  Based on the above analysis, t he intrinsic \npart of damping is independent of t he external field or precession frequency, while the 11 \n extrinsic part  including both the depha sing effect and the two-magnon -scattering effect  \nare field-dependent.  In order to avoid the effect of the extrinsic damping factor , the \nintrinsic damping factors were obtained by fitting the overall damping factor as the \nfunction of applied fields with the Eq.  (5) [39, 40 ] shown as the red line in Fig. 4(c): \n0/\n1HH\neff e\n                                                   (5) \nwhere α and 𝛼1𝑒−𝐻𝐻0⁄ are the intrinsic and extrinsic parts of the damping factor, \nrespectively.   \nFor the derivation of spin precessional frequency as well as the Gilbert damping, \nthe similar producers as shown above were adapted to various samples. Fig ure 5(a) \nshows the precessional frequency  from oscillation curves with various IrMn thicknesses. \nSince the exchange bias field and coercivity are much weaker tha n applied fields, the  f-\nH curves  of FeGa films are therefore slightly different  with various AFM layer \nthicknesses , which is in contra st to the observation that the enhanced uniaxial \nanisotropy of Fe/CoO bilayers  [28] increases the precessional frequency largely.  More \nimportantly, w e find th e effective damping factor \neff decreases with applied fields \n[Fig. 5(b)]. The solid lines represent the fittin g expression shown as the Eq. (5 ). \nInterestingly, the effective Gilbert damping factors drop to a nearly constant value as \nthe intrinsic damping factor when the applied fields increase strong enough to suppress \nthe extrinsic contributions  as stated above .  \nThe value s of the intrinsic damping factor as a function of the thickness of the \nIrMn layer  are illustrated in Fig. 5(c ). It increases firstly  and reaches the maximum 12 \n value when the thickness of the IrMn layer at \nIrMnt = 2 nm, and finally  decreases with \nfurther increasing the thickness of the IrMn AFM layer. A drastic change of 2.5 times \nfor damping occurs at \nIrMnt = 2 nm . Similarly, S. Azzawi et al. showed around 2 times \nenhancement of damping in NiFe/Pt bilayers when a continuous Pt capping layer is just \nforming at 0.6 nm  by TRMOKE measurements  [41]. Moreover, once a continuous IrMn \nlayer is forming at 2 nm, the accompanied strong intrinsic anisotropy of AFM would \ncontribute partly t o the damping enhancement superimposed to  spin pumping effect. \nThis has been demonstrated previously by W. Zhang et al where the damping of \nPy/IrMn bilayers is 3 or 4 times larger than that in the Py/Cu/IrMn samples  [42]. Based \non the discussions  in Fig. 4 , we can exclude the  extrinsic mechanisms such as the two-\nmagnon -scattering and the dephasing effect as the dominant contributions to the \ndamping process when the external fields are high enough  [43]. Besides,  FeGa alloys \nare particularly interesting because of their magneto -elastic properties  [44]. The \nacoust ic waves are possible to be tri ggered by ultrashort laser and as a result, spin \nprecession would be excited  non-thermally via a magnetoelastic effect  [45]. However, \nthis effect  can be  excluded based on the following reasons: firstly, the external field has \nto be applied along with the hard axis of FeGa, otherwise , the magnetization precession \ncannot be induced. It agrees with the fact that  the canted  magnetization from the easy \naxis is necessary when the spin precession arising  from instantaneous anisotropy \nchange  accompanied by ultrafast demagnetization occurs  [26]. In contrast,  the \noccurrence of spin precession from the magnetoelastic effect is independent of  initial \nmagnetization orientation. Secondly,  in order to check the contribution of resonance 13 \n mode from the magnetoelastic effect, we per formed a fast Fourier transform  in \nAPPENDIX B . Only the uniform  field-dependent  precession mode  was excited at present \nstudy.  This is not the expected behavior for the acoustically induced modulation of the \nmagneto -optical effects.  Therefore, the magnetoelastic effect of FeGa was suppressed \nlargely in this study. It  probably because the laser fluence of around 1 mJ/cm2 is not \nhigh enough to induce a large amplitude of strain pulse. According to Ref. 45 , the \noscillations amplitude of acoustic mode  increases linearly with the laser energy density \nwithin the probed range . Moreover, the FeGa material with a thickness as thic k as 60 \nnm is preferred to induce an obvious magnetoelastic behavior  [46], while 10 nm  at the \npresent experiment is  probably too thin.  As a result, t he intrinsic damping can be \ninfluenced by the following  paramenters : (1) magnetocrystalline anisotropy  of FM [47]; \n(2) exchange  bias field  [30, 31, 36 ], and (3) spin pumping effect  at the interface between \nFM and  the AFM [ 15, 16, 42, 48 ]. In the case of FeGa/IrMn bilayers , the \nmagnetocrystalline anisotropy constant of FeGa\n1K  =\n3 5/ 105.4 cm erg  , which is \nobtained fro m Fig. 4 and Fig. 5 , is invariant  with AFM layer thickness. Moreover , \nreferring to Fig. 2 (b), it seems that there is no direct relationship between  the intrinsic \ndamping factor and the exchange  bias field Heb. When the applied field is far higher \nthan the exchange bias field, both the precessional frequency and the damping factor \nshow independence of excha nge bias field  [36]. Therefore, the IrMn  thickness \ndependence of the intrinsic damping is not attributed to the magnetocrystalline \nanisotropy and the exchange bias field . Due to the strong spin-orbit coupling of the \nheavy metal  (HM) Ir in the IrMn alloy, the contribution of spin pumping to the damping 14 \n factor must be taken into account. It is noteworthy that the IrMn thickness dependence \nof damping in FeGa/IrMn is different from  that in other normal FM/HM bilayers, where \nthe damping factor increases monotonically with the thickness  of HM layer and \napproa ches a saturation valu e [49]. However,  the damping of FeGa ferromagnetic layer \ndecreases again after reaching a peak value at 𝑡𝐼𝑟𝑀𝑛=2 𝑛𝑚. The change of the \ndamping  factor  is always accompanied by the spin currents transfer between FM and \nAFM layers.  More spin currents absorbed by the neighbor ing layer result in larger \ndamping in the FM layer. A n unconventional  decrease of the damping factor implies \nthat not only the effect of heavy metal Ir in IrMn alloy has to be taken into account, but \nalso the ant iferromagnetic magnetization. The heavy metal Ir serves as a perfect spin \nsink to absorb the spin currents , and consequently increases the damping in FeGa, while \nthe antiferromagnetic magnetization in IrMn serves as a new source to compensate the \ndissipati on of magnetization precession  and decrease the damping of FeGa.  \nC. First -principle calculations for IrMn layer thickness dependence of \nGilbert damping  \nTo understand the behavior of the IrMn thickness -dependent damping factor, we \ncalculated the damping factor  using the scattering theory of magnetization dissipation \ncombined with the first -principles electronic structure  [50]. The calculated FM/AFM \nbilayer structure  shown in Fig. 6(a) are the same as that in  the experiment.  Here, the \nmagnetic moments of AFM sublattices serve as not only a spin sink to absorb the spin \ncurrent pumped from the adjacent FM layer, but also a spin current emitter to partly 15 \n cance l the spin pumping effect of the FM. The interfacial exchange coupling force s the \nmagnetic moments of  the IrMn sublattices in a few layers near the interface to preces s \nfollowing the adjacent FM , generat ing spin current s back into the FM layer  [Fig. 6 (b)]. \nBased on this model, t he enhancement of damping due to the spin current  𝛼𝑠𝑝=∆𝛼=\n𝛼𝑡𝐼𝑟𝑀𝑛−𝛼𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚 as a function of IrMn thickness was calculated  and shown as the \nsolid circle in Fig. 6(c). It increases firstly to a peak value at \nIrMnt = 2 nm, and then \ndrops with further increasing  the IrMn layer thickness . When \nIrMnt  2 nm , the \nthickness of the IrMn  layer is too thin to establish the antiferromagnetic order, which \ncan be supported by the negligible exchang e bias as shown in Fig. 2 (b). In this case, the \npumped spin current from the AFM back into the FM to partially cancel the spin \npumping effect by  the FM is largely reduced because of the disorder of the \nantiferromagnetic moments as illust rated on  the left side in Fig. 6 (b). In this re gion, \ntherefore, the magnetic moments in the AFM serve as a perfect spin sink to absorb the \nspin current pumped from the adjacent FM  resulting in a significant enhancement in \nthe damping  factor . For the samples with the thickness of IrMn  𝑡𝐼𝑟𝑀𝑛>2𝑛𝑚, however , \nthe antiferromag netic order is well established and the accompanied exchange bias is \nremarkably large ( See Fig . 2(b) and its insert ). Because of the exchange coupling \nbetween FM and AFM at the interface, the magneti c moments of the AFM sublattices \nin a few layers near the interface is forced to precess following the magnetic moment \nof the FM, while those far away from the interface would keep static. Such an exchange \nspring effect at the interface caused spin precession in the AFM layer,  and consequently , \nspin currents would be transfe rred from AFM to the FM layer. Moreover, these  spin 16 \n current s from the AFM would be enhanced due to the coherent precession of \nmagnetization in different sublattices as illustr ated in the right side of Fig. 6 (b). The \nexchange  spring effect induced  precession of the AFM has two effects: (1) the AFM \nhas intrinsic damping that increases the overall dam ping of the FM/AFM bilayer. (2) \nthe precessional motion of magnetic moments in AFM sublattices pumps spin current s \ninto the FM, which cancels partly the spin pumping by the FM. As a result, the overall \ndamping of the bilayers is reduced. From  the solid circles in Fig. 6(c), one can find that \nthe damping decreases with increasing\nIrMnt  when \nIrMnt 2 nm, indicating that the \nlatter effect of the pumped spin currents is dominant over the intrinsic damping.  Besides, \nby comparing the calculated and experimental values  [Fig. 6(c) and (d)], one can find \nthat the calculated Gilbert damping  is larger than the experiment al one for \nIrMnt 1 nm . \nThe reason for t he deviation is the assumption of a  perfect ly flat  FeGa/ IrMn interface \nin the  calculation, which leads to a larger spin current pumped from the FM . \nUnfortunately, it is almost impossible to fabricate the perfect ly flat film when the \nthickness is less than 1  nm.  \nIn order to separate the contribution of the precession of the magnetic moment of \nthe AFM sublattice to damping, we also calculated the damping by assuming perfectly \nstatic AFM ordered IrMn without precession (solid diamonds  in Fig. 6 (c)) and a \nparamagnetic IrMn layer with vanishing Néel order  (solid triangles  in Fig. 6(c)). The \ncalculated results demonstrate that if the magnetic moments of the AFM sublattice \neither do not precess or align randomly, the IrMn layers serve only as a perfect spin 17 \n sink to absorb the spin current s pumped from the adjacent FM  resulting in a significant \nenhancement of damping . The damping increases monotonically to a saturation value \nwith IrMn thickness, which is similar to that of heavy metals  [49].  \nD. Relationship between ultrafast demagnetization rate and Gilbert \ndamping induced by non -local spin currents  \nThe central strategy of our study is to establish a direct correlation between \nM\nand α. According to Fig. 3(b ) and Fig. 5(c), we find that the femtosecond laser -induced \nultrafast demagnetization time \nM and the Gilbert damping  α show an opposite IrMn \nthickness dependence  in FeGa/IrMn bilayers.  By plotting \nM versus α as shown in \nFig. 7 (a), one can clearly observe that the value of \nM decreases with  α, suggesting \nthat spin transport plays an additional dissipation channel for accelerating the ultrafast \ndemagnetization and enhanc ing the d amping.  The damping  factor  𝛼𝑡𝐼𝑟𝑀𝑛 for \nIrMnt 0 \nnm is ascribed to the spin pumping effect induced by various AFM thicknesses 𝛼𝑠𝑝 \nand the contribution from the FM itself 𝛼𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚.To give further insight into th e \nrelationship, we replo tted Fig.  7(a) by using t he change of the ultrafast demagnetization \nrate ∆1\n𝜏𝑀=1\n𝜏𝑀|𝑡𝐼𝑟𝑀𝑛−1\n𝜏𝑀|𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚  versus the enhancement of Gilbert damping  \n𝛼𝑠𝑝=∆𝛼=𝛼𝑡𝐼𝑟𝑀𝑛−𝛼𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚  induced by the spin current.  An approximately \nlinear relationship is  confirmed and shown in Fig. 7 (b), which can be fitted using Eq. \n(6): (For the de rivation of Eq. (6 ), please see APPENDIX D  for details)  \n∆1\n𝜏𝑀=𝜇𝑠\nℏ∆𝛼,                                                      (6) 18 \n Where ∆1\n𝜏𝑀 , ∆α  represents the enhancement of ultrafast demagnetization rate and \nGilbert damping induced by the  spin current, respectively, 𝜇𝑠 is the spin chemical \npotential,  and ℏ is the Planck constant. A reasonable value of  𝜇𝑠≈1 𝑒𝑉 which is \nsimilar to that of spin splitting in 3d transition metals was obtained  by the linear fitting \nusing Eq. (6).  \n    The spin chemical potential 𝜇𝑠  is proportional to spin accumulation s at the \ninterface between different layers . It contributes largely to ultrafast demagnetization \naccording to the model of laser -induced ultrafast superdiffusive spin transpor t in \nlayered heterostructures  [14, 5 1]. There is a large difference in velocities or lifetimes \nfor spin -dependent hot electrons  [52]. As a result,  the transport properties of ho t \nelectrons are spin -dependent. For instance , the minority -spin electrons excited by \nultrashort laser survive  for a quite short time an d they decay to non -mobile bands \napproximately at the position  they were excited. Instead, majority -spin electrons have \nlonger lifetimes and higher velocities. So they leave  fast from the excitation reg ion after \nbeing created, resulting in  part of the demagnetization process. Because the directions \nof motion for all the electrons are random , they can obtain a velocity directed back \ntowards the ferromagnetic film . A second part of the demagnetization is ascribed to  the \nbackflow of spin -minority elec trons from the substrate or the neighbor layer. Spin -\nmajority electrons entering the ferromagnetic  layer will find good transport properties \nand continu e diffusing without severely decay ing. However , spin-minority electrons \nexperience a considerable worsen ing of the transport properties as soon as they enter 19 \n the ferromagnetic layer. The consequence is that they are trapped at the entrance of the \nferromagnetic layer , giving rise to  the spin accumulations  at the interface . Nevertheless, \nthe quantitative description for spin accumulations during ultrashort laser -induced \ndemagnetization in heterostructures  is still lacking. This work aims at filling this gap \nby relating ultrafast demagnetization time and Gilbert damping. A de tailed calculation \nfor the value of 1 eV for spin chemical potential obtained in this experiment is highly \ndesirable.    \nThe non -local spin current s dissipated at the interface of FeGa/IrMn open an \nadditional channel to accelerate the ultrafast demagnetization and enhance  the Gilbert  \ndamping. However , in the case of the sample with \nIrMnt = 0 nm without the assistant \nAFM layer , both the local spin -flip and non -local spin transport mechanism s probably \ncontribute to the ultrafast demagnetization in the ferromagnetic layer. For instance, \nbased on the breathing Fermi -surface model of the Gilbert damping and the Elliott -\nYafet relation for the spin -relaxation time, a relation shown as Eq. (7 ) is established \nbetween the conductivity -like Gilbert damping \n and ultrafast demagnetization time  \n𝜏𝑀 [10]. \n                                                (7) \nTaking the values of \nnm tMIrMn 0 and \nnm tIrMn 0 are 220  fs and 0.004, respectively , a \nvalue of 𝛼𝜏𝑀⁄=1.8×1010𝑠−1 is derived.  This value is reasonable and agrees well \nwith that  of 3d transition metal  Ni calculated by  the breathing Fermi -surface model [ 53], \n2pbFM\nelM20 \n indicating that  the ultrafast demagnetization of ferromagnetic FeGa film  itself  is mainly \ngoverned by the local spin -flip scattering events . Nonetheless, we have to address that, \nultrafast demagnetization in the ferromagnetic layer was accelerated and the Gilbert \ndamping was enhanced via the interfacial spin  accumulations o nce the IrMn layer was \nattached .  \nIII. CONCLUSIONS  \n The unconventional IrMn thickness dependence of α  is attributed to the \ncancellation of the spin currents pumped from the AFM IrMn layer to the FM FeGa \nlayer.  We establish a  proportional  relationship between the change of ultrafast \ndemagnetization rate and the enhancement of Gilbert damping induced by the spin \ncurrents via interfacial spin chemical potential . This result  can facili tate the utilization \nof ultrafast spintronic devices in the THz region.  \n \n \n \n \n \n \n 21 \n Acknowledgments  \nThis work is supported by the National Key Research Program of China (Grant Nos. \n2015CB921403, 2016YFA0300701,  and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos. 91622126, 51427801, and 51671212) and \nthe Key Research Program of Fro ntier Sciences, CAS (Grant Nos. QYZDJ -SSW -\nJSC023, KJZD -SW-M01 and ZDYZ2012 -2). The work at Beijing Normal University \nis partly supported by the National Natural Sciences Foundation of China (Grant Nos. \n61774017,  61704018, and 11734004), the Recruitment Program of Global Youth \nExperts and the Fundamental Research Funds for the Central Universities (Grant No. \n2018EYT03).  The work at East China Normal University  is partly supported by the \nNational Natural Sciences F oundation of China  (Grant No. 11874150).   \n \n \n \n \n \n \n \n 22 \n APPENDIX A: TRMOKE MEASUREMENTS  \nIn this study, the dynamical process of fast and ultrafast spin dynamics was \nmeasured by TRMOKE. The experiments were carried out using an all -optical pump -\nprobe technique. A  train of optical pulses with a wavelength of 780 nm, 55 fs duration , \nand 100 nJ/pulse is generated at 5.2 MHz repetition rate by a Ti: sapphire oscillator \n(FEMTOLASER, XL -100). A 200  μm thickness BBO crystal was used to double the \nfrequency of femtosecond laser. The laser beam from the source is split into both 780 \nnm and 390 nm beams. We use the 780 nm laser as the pump pulse to excite the \nmagnetic system out of equilibrium, while the 3 90 nm laser pulse was used as a probe \nbeam to measure the subsequent magnetization dynamics with the timescale from sub -\npicosecond to nanosecond. The pump laser beam is much stronger than the probe with \nan intensity ratio of about 100 for all the measureme nts. Both the pump and probe \nbeam s are incident along the normal axis (z -axis) of the sample s. The detection \ngeometry is only sensitive to the out -of-plane component of the magnetization Mz. For \nfast spin dynamics, we applied various external fields along the Fe 81Ga19 [110] direction \nto trigger the spin precession, while a large enough field about 20 kOe  was applied \nalong the Fe 81Ga19 [001] direction to obtain the ultrafast demagnetization curves. W e \nadjusted the pump laser fluence from 1 mJ/cm2 to 1.25 mJ/cm2 to obtain the same \nmaximum quenching for various samples. The pump and probe beams are focused onto \nthe sample s with spot diameters of ~10 μm  and ~5μm  via an objective lens, \nrespectively. For the spin precession measurements, t he scheme  of the TRMOKE 23 \n experiment is illustrated in Fi g. 8. The signal s are sensitive with the polar component \nof magnetization after pump laser excitation at different fields applied along the [110] \ndirection of FeGa .   \n \nAPPENDIX B: FAST FOURIER TRANSFORM ANALYSIS  \n    The ferromagnetic FeGa is a famous material for its magneto -elastic properties. \nAfter femtosecond laser irradiation, an external field -independent resonance mode \nwould be triggered due to the excitation of coherent acoustic phonons. However, only \none f ield-dependent resonance mode was excited in this study according to fast Fourier \ntransform analysis  in Fig.  9.   \n \nAPPENDIX C: FIRST -PRINCIPLE CALCULATIONS  \nThe electronic structure of FeGa /IrMn bilayer is calculated self -consistently using \nthe local density approximation of the density functional theory. The spin -dependent \npotentials, charge and spin densities are obtained with the minimal basis of tight -\nbinding linear muffin -tin orbitals. In the calculation of the total damping, the scattering \nregion  consisting of the repeated FeGa /IrMn bilayers are connected to two semi -infinite \nCu leads. We have introduced the thermal lattice disorder into a 4x4 supercell a nd \ndisplaced the atoms in the scattering region randomly away from their equilibrium \npositions  with a Gaussian distribution. The root -mean -square atomic displacements of \nthe Gaussian distribution are determined using a simple Debye model with the Debye 24 \n temperature of 470 K. The two -dimensional Brillouin zone of the supercell is sampled \nby a 24x24 k -mesh corresponding to the 96x96 mesh for the Brillouin zone for the 1x1 \nunit cell. The effect of magnons in the FM FeGa is neglected in our calculation. This is \nbecause the magnetic damping is dominated by electrons at the Fermi level in metals, \nwhich can  efficiently transfer spin angular momentum into the orbital motion via spin -\norbit interaction. In metals and alloys, the influence of magnon -phonon coupling is \nnegligible except for near the Curie temperature  [54]. \nIf magnetization precession occurs only in the FM FeGa layer, the calculated \ndamping enhancement does not sensitively depend on the specific order of the AFM \nIrMn. Here we take two limits: the perfectly antiferromagnetic ordered IrMn and the \nparamagnetic IrMn (the magnetic moments of Mn are rand omly distributed such that \nboth the Néel order and total magnetization vanish). The damping enhancements \ncalculated for the two cases are nearly identical, where the damping factor is enhanced \nand saturates at the thickness of 2 nm. It indicates that the p umped spin current by the \nprecessional FeGa is immediately absorbed by the IrMn layer. The large moment on the \nMn atom can absorb the pumped transverse spin current efficiently. On the other hand, \nthe AFM IrMn is forced to precess due to the interfac ial exchange coupling,  however,  \nthe efficient of the spin current generation by AFM depends on its specific order . It is \nsuppressed largely in the case of paramagnetic  IrMn because of the cancellation via \nmagnetic moments with various orientations shown on the  left side of Fig. 6(b) in the \nmain text . In contrast, the efficient of the spin current generation by the AFM is \nenhanced remarkably by the coherent precession of the ordered magnetic moments 25 \n shown in the right side of Fig. 6(b) in the main text . And the cone angle of precessional \nIrMn is modeled to exponentially decay from the interface with a typical decay length \nof 2 nm. The precessional AFM has mainly two contributions to the damping \nenhancement of the bilayer. First, the AFM has intrinsic damping that increases the total \nenergy loss during the magnetization dynamics. The second effect is that the \nprecessional AFM pumps spin current into the FM that cancels partly the spin pumping \nby the FM and decreases the damping enhancement.  \n \nAPPENDIX D: DERIV ATION OF EQ. (6 ) IN THE MAIN TEXT  \nIt is well known that the magnetic moment \nsM  is proportional to the spin angular \nmomentum \nS via gyromagnetic ratio \nBg ： \nS Ms\n                                                          (8) \nwhere\ng is Lande factor, \nB is Bohr magneton. Normally, we take \nsMVm as the \ntotal magnetic moments , where \nV is the volume of the atom.  \n    \nM is the ultrafast demagnetization time. Therefore, t he value of \nM1 is taken as \nthe demagnetization rate. The demagnetization is alw ays accompanied by the \ndissipation of spin angular momentum, and hence the rate of spin angular momentum \ndissipation is :  \nMm\n1\n.                                                           (9) \nOn the other hand, the spin current 𝑗𝑠⃗⃗⃗  of per unit area generated by spin pumping effect 26 \n reads:  \nj𝑠⃗⃗ =1\n4𝜋𝑔𝑒𝑓𝑓𝜇𝑠⃗⃗⃗ ,                                                     (10)  \nwhere 𝑔𝑒𝑓𝑓is the effective interfacial spin -mixing conductance including the influence \nof the backflow spin current from the AFM IrMn to FeGa , \ns is the spin accumulation -\ndriven chemical potential. The pumped spin current across the interface is  𝐼𝑠⃗⃗⃗ =𝑗𝑠⃗⃗⃗ 𝐴, \nwhere\nA is the area of the interface.    \n𝑔𝑒𝑓𝑓=4𝜋𝑀𝑠𝑑∆𝛼\n𝑔𝜇𝐵,                                                    (11) \nwhere 𝑑is the thickness of the ferromagnetic layer, \nnm t tIrMn IrMn 0  is the \nenhancement of Gilbert damping induced by the absorption and generation of spin \ncurrent via various IrMn thicknesses.  \nTherefore, if we correlate the spin angular momentum dissipated by the ultrafast \ndemagnetization and that induced by spin pumping, the relationship reads:  \n \ns\nMIm1                                                        (12) \nAnd then we take Eq. (10) into Eq. (12), we can correlate the parameters \nM  and \n\nvia: \n\ns\nM1\n,                                                       (13) \nTo exclude the contributions from local spin -flip scattering mechanisms to the ultrafast \ndemagnetization rate represented by\nnm t\nMIrMn 01\n , the value of \nM1 is replaced by \nnm t\nMt\nM MIrMn IrMn 01 1 1\n\n.  \n 27 \n References  \n[1] T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. Freimuth, \nA. Kronenberg, J. Henrizi, and I. Radu , Nat. Photonics. 10, 483 (2016).  \n[2] E. Beaurepaire, J. C. Merle, A. Daunois, and J.Y . Bigot, Phys. Rev. Lett. 76, 4250 \n(1996).  \n[3] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge, Phys. Rev. \nLett. 95, 267207 (2005).  \n[4] U. Atxitia and O. Chubykalo -Fesenko, Phys. Rev. B. 84, 144414 (2011).  \n[5] U. Atxitia, O. Chubykalo -Fesenko, J. Walowski, A. Mann, and M. M ünzenberg, \nPhys. Rev. B. 81, 174401 (2010).  \n[6] B. Koopmans, G. Malinowski, F. 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Rev. B. 86, 024404 (2012).  \n[52] V . P. Zhukov, E. V . Chulkov, and P. M. Echenique, Phys. Rev. B . 73, 125105 (2006).  32 \n [53] M. Fähnle, J. Seib, and C. Illg, Phys. Rev. B. 82, 144405 (2010).  \n[54] Y . W. Zhao, Y . Liu, H. M. Ta ng, H. H. Jiang, Z. Yuan, and K.  Xia, Phys. Rev. B, \n98, 174412 (2018).   \n \nFigure Captions  \nTable. I . Values of the main fit parameters of ultrafast demagnetizations curves for \nvarious thicknesses of the samples.  \nFIG.1. (color online ) Basic concept of both ultrafast demagnetization and spin \nprecession induced  by spin currents . (a) The excitation of fs laser pulse transforms slow \nmajority -spin d electrons (red) into fast sp electrons, thereby launching a sp in current \ntoward s the AFM layer. The spin current crossing the interface results in the spin \naccumulation at the interface represented by spin chemical potential  𝜇𝑠. (b) The typical \ntime evolu tion of magnetization after femtosecond laser irradiation measured by \nTRMOKE experiment.  \nFIG. 2. ( Color online) Static magnetic properties of of MgO/Fe 81Ga19 (10 nm)/Ir 20Mn 80 \n(t nm) bilayers. (a) Longitudinal -MOKE loops with various thicknesses of IrMn layer  \nIrMnt\n. (b) Coercivity Hc and exchange bias field \nebH as a function of IrMn layer \nthickness \nIrMnt . \nFIG.3. (Color online ) Ultrafast demagnetization. (a) Ultrafast demagnetization curves \nwith various IrMn layer thickness es. The solid lines represent  the fitting results by Eq.  \n(1) in the text.  The insert  shows the configuration of the measurement for ultrafast 33 \n demagnetization. (b)  Ultrafast demagnetization time  as a function of IrMn layer \nthickness.  \nFIG.4. (Color Online)  Spin precession.  (a) TRMOKE signals of FeGa/IrMn bilayers \nwith \nMnIrt = 2 nm in various applied fields.  (b) Precessional frequency as a function of \napplied fields . (c) Effective Gilbert damping constant as a function of applied fields.  \nFIG. 5 . (Color Online)  Frequency and damping of spin precession. (a) Fr equency of \nspin precession as a function of applied fields with various IrMn thickness. The solid \nlines represent the fitting results by Kittle equations.  (b) Effective Gilbert damping \nconstants as a function  of applied fields with various IrMn thicknesses.  (c) Intrinsic \nGilbert damping as a function of IrMn thickness.  \nFIG.6. (Color online) Results of First -principle  calculation s. (a) Illustration of the \nferromagnet (FM)/antiferromagnet  (AFM) structure employe d to investigate the spin \ntransport. (b) The configuration of the IrMn magnetic moments located at the first layer \nnear the interface. (c)  The calculated damping enhancement as a function of the \nthickness of the antiferromagnetic IrMn. The solid circles sh ow the calculated damping \nenhancement with the precession of AFM magnetic moments. The solid diamonds \nshow the calculated damping enhancement with perfectly static AFM ordered IrMn \nwithout precession, while the  solid  triangles correspond to the calculated values using \na static paramagnetic IrMn layer with vanishing Néel order.  (d) The experimental \ndamping enhancement as a function of the thickness of antiferromagnetic IrMn.  34 \n FIG.7. (Color online ) (a) Ultrafast demagnetization time as a function of Gilbert  \ndamping . (b) The variation of ultrafast demagnetization rate as a function of Gilbert \ndamping  enhancement . The red line indicates the fitting via Eq. (6 ) in the text.  \nFIG. 8. (Color online ) Scheme of TRMOKE experiment for spin precession dynamics.  \nFIG. 9. (Color online ) Fourier transform spectra measured between 0.85 kOe and 3.0 \nkOe for 𝑡𝐼𝑟𝑀𝑛=2 𝑛𝑚. \n  35 \n  \n \nTable I  \n \n \n \n \n \n. \n \n \n \n tIrMn (nm)                      A1       A2           \n0  220±10 500    5  350  0.8       2      \n1  160±10 500    6  350  0.8       2      \n2  120±10 500    7  350        0.8       2      \n3  145±10 500    4  350  0.8       2       \n5  200±10 500    5  350  0.8       2       \n)(fsM\n)fsE（\n)(0ps\n)(fsG36 \n  \n  \n \n \n \n \n \n \n \n \n \nFIG.1.  \n \n \n \n  \n37 \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFIG. 2.  \n \n \n  \n38 \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFIG.3. \n \n \n  \n39 \n  \n \n \n \n \n \n \n \n \n \n \nFIG.4. \n \n \n \n \n  \n40 \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFIG. 5 . \n \n \n \n  \n41 \n  \n \n \n \n \n \n \n \n \nFIG.6. \n \n \n \n \n \n \n  \n42 \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFIG.7. \n \n \n  \n43 \n  \n \n \n  \n \n \n \n \nFIG. 8.  \n \n \n \n \n \n \n \n  \n44 \n  \n \n \n \n \n \n \n \n \n \n \n \nFIG.9.  \n \n"    },    {        "title": "1908.11761v1.Magnetization_reversal__damping_properties_and_magnetic_anisotropy_of_L10_ordered_FeNi_thin_films.pdf",        "content": "Magnetization reversal, damping properties and magnetic anisotropy of L 10\nordered FeNi thin \flms\nV. Thiruvengadam,1B. B. Singh,1T. Kojima,2K. Takanashi,2M. Mizuguchi,2,a)and S. Bedanta1,b)\n1)Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences,\nNational Institute of Science Education and Research (NISER), HBNI, P.O.- Bhimpur Padanpur, Via Jatni, 752050,\nIndia\n2)Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577,\nJapan\n(Dated: August 2019)\nL10ordered magnetic alloys such as FePt, FePd, CoPt and FeNi are well known for their large magnetocrys-\ntalline anisotropy. Among these, L10-FeNi alloy is economically viable material for magnetic recording media\nbecause it does not contain rare earth and noble elements. In this work, L10-FeNi \flms with three di\u000berent\nstrengths of anisotropy were fabricated by varying the deposition process in molecular beam epitaxy system.\nWe have investigated the magnetization reversal along with domain imaging via magneto optic Kerr e\u000bect\nbased microscope. It is found that in all three samples, the magnetization reversal is happening via domain\nwall motion. Further ferromagnetic resonance (FMR) spectroscopy was performed to evaluate the damping\nconstant and magnetic anisotropy. It was observed that the FeNi sample with moderate strength of anisotropy\nexhibits low value of damping constant \u00184:9\u000210\u00003. In addition to this, it was found that the \flms possess\na mixture of cubic and uniaxial anisotropies.\nIn order to increase the storage density in magnetic\nrecording media it requires reduction in the bit size1. On\nthe other hand, in ferromagnetic materials, the super-\nparamagnetic (SPM) limit is inevitable at a critical ra-\ndius, below which the magnetic moment is thermally un-\nstable and become incapable of storing the information2.\nTherefore, magnetic material with high magnetocrys-\ntalline anisotropy is essential to overcome the SPM\nlimit3. In this context, the L10ordered magnetic alloys\nsuch as FePt, FePd, CoPt and FeNi have potential for\nultra-high density magnetic recording media because of\ntheir large uniaxial magnetocrystalline anisotropy energy\ndensity \u0018107erg cm\u00003.L10ordered alloy is a binary\nalloy system with face centered tetragonal (FCT) crys-\ntal structure where each constituent atomic layers are\nalternatively laminated along the direction of crystallo-\ngraphic c-axis4,5. In these materials the large anisotropy\nenergy is due to their tetragonal symmetry of L10crystal\nstructure6.\nL10-FeNi possess high values of saturation magnetiza-\ntion (1270 emu cm\u00003), coercivity (4000 Oe), and uniaxial\nanisotropy energy density (1.3 \u0002107erg cm\u00003)7{9. In ad-\ndition its Curie temperature is quite high \u0018550\u000eC and it\nexhibits excellent corrosion resistance7. All these above\nproperties make L10-FeNi a promising material for fab-\nricating information storage media and permanent mag-\nnets. Further, L10ordered FeNi alloy is free of noble\nas well as rare-earth elements. Also the constituent ele-\nments (i.e. Fe and Ni) are relatively inexpensive. There-\nfore L10ordered FeNi alloy is economically viable for\ncommercial applications4,10,11. It should be noted that\nalthough, L10- FeNi possess high uniaxial anisotropy,\na)Electronic mail: mizuguchi@imr.tohoku.ac.jp\nb)Electronic mail: sbedanta@niser.ac.inshape anisotropy becomes dominant in thin \flms11. Pre-\nviously the study of the order parameter (S) and Fe-Ni\ncomposition dependence on Kuhas been reported7. Pre-\nviously, magnetic damping constants for L10-FeNi and\ndisordered FeNi have been studied employing three kinds\nof measurement methods12. However, the e\u000bect of di\u000ber-\nent anisotropy values of L10- FeNi thin \flms on evo-\nlution of their magnetic domain structures and damping\nhas not been studied so far. Therefore, focus of this paper\nis to study the domain structures during magnetization\nreversal, anisotropy strength, damping properties of L10\n- FeNi thin \flms with di\u000berent anisotropy values.\nL10-FeNi thin \flms were prepared in a molecular beam\nepitaxy chamber with base pressure of 10\u00008Pa consist-\ning of e-beam evaporators (for Fe and Ni) and Knud-\nsen cells (for Cu and Au). First a seed layer of Fe (1\nnm) and Au (20 nm) were deposited at 80\u000eC on MgO\n(001) substrate followed by Cu (50 nm) layer deposited\nat 500\u000eC. It has been reported that Cu and Au layers\nare prone to form an alloy of Cu 3Au (001)9. In order\nto fabricate highly ordered L10-FeNi \flms, a bu\u000ber layer\nof Au 0:06Cu0:51Ni0:43(50 nm) was deposited on Cu 3Au\nlayer at 100\u000eC by co-deposition13. On top of this bu\u000ber\nlayer, FeNi layer was grown by alternative layer deposi-\ntion (Fe and Ni one after another) or co-deposition (de-\nposition of Fe and Ni simultaneously). Disordered-FeNi\n\flm (sample A) was obtained by co-deposition process\nwhereas L10-ordered FeNi \flms were fabricated by alter-\nnate deposition at 100\u000eC (sample B) and 190\u000eC (sam-\nple C), respectively. Finally, Au capping layer (3 nm)\nwas deposited on top of FeNi layer at 30-40\u000eC. To study\nmagnetization reversal and magnetic domain structures,\nwe have performed hysteresis measurements along with\nsimultaneous domain imaging using magneto optic Kerr\ne\u000bect (MOKE) based microscope manufactured by Evico\nMagnetics Ltd. Germany14. Kerr microscopy was per-arXiv:1908.11761v1  [physics.app-ph]  30 Aug 20192\nFIG. 1. Hysteresis loops measured at room temperature by longitudinal Kerr microscopy at 0\u000e, 30\u000e, 60\u000eand 90\u000efor samples\nA - C shown in (a) - (c), respectively..\nformed at room temperature in longitudinal geometry.\nAngle dependent hysteresis loops were measured by ap-\nplying the \feld for 0\u000e<\b<360\u000eat interval of 10\u000ewhere\n\b is the angle between the easy axis and the direction\nof applied magnetic \feld. Further, magnetization dy-\nnamics of the L10-FeNi thin \flms has been studied by\nferromagnetic resonance (FMR) spectroscopy technique\nin a \rip-chip manner using NanOsc Instrument Phase\nFMR15. Frequency range of the RF signal used in this\nexperiment was between 5 and 17 GHz. To understand\nthe symmetry of anisotropy and quantify the anisotropy\nenergy density, angle dependent FMR measurement has\nbeen performed by applying the magnetic \feld in the\nsample plane with a \fxed frequency of 7 GHz.\nMagnetic hysteresis loops measured using Kerr mi-\ncroscopy for samples A-C are shown in \fgure 1 for various\nvalues of = 0\u000e, 30\u000e, 60\u000eand 90\u000e. Square shaped hystere-\nsis loops have been observed for all three samples for all\nthe angles (0\u000e<\b<360\u000e). This indicates that the mag-\nnetization reversal occurs via nucleation and subsequent\ndomain wall motion16. It is observed that the coercivity\nvaries signi\fcantly among the samples e.g. HCof samplesA, B and C are 2.2, 1.7 and 25 mT, respectively. Kuof\nthese samples are -1.05 \u0002106erg cm\u00003(A), 3.47 \u0002106erg\ncm\u00003(B), and 4.86 \u0002106erg cm\u00003(C).\nDomain images captured near to coercive \feld HCfor\nall three samples at 0\u000e, 30\u000e, 60\u000eand 90\u000eare displayed in\n\fgure 2. The black and gray contrast in the domain im-\nages represent positive and negative magnetized states,\nrespectively. From the domain images it can be con-\ncluded that magnetization reversal occurs via domain\nwall motion. Sample A (\fgure 2 a to d) exhibits large\nwell-de\fned stripe domains which indicates the presence\nof weak magnetic anisotropy. These stripe domains are\nfound to be tilted for e.g. \b = 30\u000eand 60\u000e(\fgure 2\nb and c). For \b = 90\u000e(\fgure 2d) branched domains\nare observed. In addition to 180\u000edomain wall, sample\nB (\fgure 2 e to h) shows 90\u000edomain walls at certain\nangles of measurement (\fgure S1). Sample C (\fgure 2 i\nto l) shows narrow branched domains and are found to\nbe independent of \b , which indicates that the sample\nC is magnetically isotropic in nature. By comparing the\ndomain images at any particular \b among the samples,\nit is observed that the size (i.e. width) of the domain3\nFIG. 2. Domain images captured for samples A, B and C at angles, 0\u000e, 30\u000e, 60\u000eand 90\u000enear to coercivity are shown. Scale\nbars are shown on the domain images for each sample separately which are valid for the images recorded at other angles for\nthose respective samples.\ndecreases with increasing anisotropy strength.\nIn the following we have investigated the magnetiza-\ntion dynamics of the L10FeNi \flms by FMR. Measured\nFMR spectra (open symbol) for samples A and C at se-\nlective frequencies are shown in \fgure 3 (a) and (b), re-\nspectively. Resonance \feld (H res) and line width (\u0001 H)\nwere extracted by \ftting of Lorentzian shape function\n(equation 1) having anti-symmetric (\frst term) and sym-\nmetric components (second term) to the obtained FMR\nderivative signal17;\nS21=K14\u0001H(H\u0000Hres)\n[4(H\u0000Hres)2+ (\u0001H)2]2\n\u0000K2(\u0001H)2\u00004(H\u0000Hres)2\n[4(H\u0000Hres)2+ (\u0001H)2]2\n+ (slopeH ) +Offset(1)\nwhere S21is transmission signal, K1andK2are co-\ne\u000ecient of anti-symmetric and symmetric component,\nrespectively, slope H is drift value in amplitude of the\nsignal. Frequency ( f) dependence of Hresand \u0001 Hare\nplotted in \fgure 3 (a) and (b), respectively. From the\nfvs.Hresplot, parameters such as Lande g-factor, ef-\nfective demagnetization \feld (4 \u0019Meff), anisotropy \feld\nHKwere extracted by \ftting it to Kittel resonance\ncondition18;\nf=\r\n2\u0019q\n(HK+Hres)(HK+Hres+ 4\u0019Meff) (2)\nwhere\r=g\u0016B=~is gyromagnetic ratio, \u0016Bas Bohr\nmagneton, ~as reduced Plancks constant. The values ofdamping constant \u000bfor all three samples were extracted\nby using the equation17{19;\n\u0001H= \u0001H0+4\u0019\u000bf\n\r(3)\nwhere \u0001H0is called as inhomogeneous line width\nbroadening. Values of all the \ftting parameters obtained\nby using equations (1) and (2) are given in table 1. With\nincrease in anisotropy strength of the samples, 4 \u0019Meff\ndecreases whereas \u000bincreases. Sample A exhibits lowest\nvalue of\u000b, 4:9\u000210\u00003which is the same order with nor-\nmal FeNi alloy thin \flm ( \u000b= 1:2\u000210\u00003)20. The value\nof inhomogeneous line width broadening \u0001 H0, which de-\npends on the quality of the thin \flm17, is highest for\nsample C and lowest for sample A.\nApart from frequency dependent FMR, angle depen-\ndent FMR measurements at a \fxed frequency of 7 GHz\nwere performed to analyze the anisotropy energy density\nas well as symmetry of all three samples. Figure 4 shows\nthe angle dependent Hresplot for all three samples. Sam-\nples A and B clearly show the presence of mixed cubic\nand uniaxial anisotropies with minima in Hresapproxi-\nmately at angles 0\u000e, 90\u000e, 180\u000eand 270\u000e. From the an-\ngle dependent Hresplot, the value of uniaxial and cubic\nanisotropy energy constants have been estimated by \ft-\nting those data to solution of LLG equation that includes\nthe cubic and uniaxial anisotropies and it is written as21;4\nFIG. 3. FMR spectra of (a) sample A and (b) sample C measured at frequency 5 17 GHz. The solid lines in (a) and (b) are\n\ftted with equation 1. (c) fvs.Hresand (d) \u0001Hvs.fplots for samples A C extracted from the FMR frequency dependent\nspectra. The lines in (c) and (d) are the best \fts to the equations (2) and (3), respectively.\nTABLE I. The Parameters extracted from the \ftting of FMR experimental data (Figure 3) of all three samples using the\nequations (2) and (3)\nSample H K(Oe) 4\u0019Meff g-factor \u000b \u0001H0(Oe)\nA 49.41 \u00060.82 18279 \u0006365 1.97 \u00060.01 0.0049 \u00060.0001 42.85 \u00060.65\nB -52.39 \u00064.64 8900 \u0006193 1.93 \u00060.01 0.0277 \u00060.0007 66.89 \u00067.13\nC 680.29 \u0006106.88 3509 \u0006106 1.98 \u00060.01 0.0680 \u00060.0019 143.67 \u000616.20\nf=\r\n2\u0019\u0014\nH+2K2\nMscos2\u001e\u00004K4\nMscos4\u001e\u0015\n\u0002\u0014\nH+ 4\u0019Ms+2K2\nMscos2\u001e\u0000K4\nMs(3 +cos4\u001e)\u00151=2\n(4)\nwhere, K2andK4are cubic and uniaxial anisotropy\nenergy density constants, respectively, MSis saturation\nmagnetization, His the applied magnetic \feld, \b is the\nangle between applied \feld direction and easy axis of\nthe sample. The angle dependence of Hresis \ftted to\nequation 4. From the \ft, value of MS,K2andK4have\nbeen extracted and are shown in table 2. It has been\nfound that both cubic and uniaxial anisotropy energy\nconstants of sample A are greater than that of sample B\nby one order of magnitude. Further in sample A the ratioTABLE II. The parameters extracted from the \ftting of angle\ndependent FMR experimental data of two samples using the\nequation (4)\nSamplesMs(emu cm\u00003)K2(erg cm\u00003)K4(erg cm\u00003)\nA 1645 1.66 \u0002104-2.16\u0002104\nB 1026.8 4.8 \u0002103-1.2\u0002103\nC Magnetically isotropic behaviour\nbetween the cubic to uniaxial anisotropy is about 0.75\nwhereas for sample B the ratio is 4. Therefore for sample\nB the cubic anisotropy is four times stronger than the\nuniaxial anisotropy. This is the reason of occurrence of\n90\u000edomain walls for sample B as shown in Fig. S122. In\ncomparison to samples A and B, sample C shows isotropic\nbehavior which is in consistent with the results obtained5\nFIG. 4. Anisotropy symmetry plot and the \fts for (a) sample A, (b) sample B and (c) sample C measured using FMR by\nkeeping the frequency \fxed at 7 GHz. The red solid lines are the best \fts to equation (4).\nfrom Kerr microscopy.\nL10-FeNi thin \flms (samples A, B and C) show dif-\nferent strength of anisotropy, which were fabricated by\ndi\u000berent deposition processes in a MBE system. Do-\nmain images reveals that magnetization reversal for all\nthree samples occur via nucleation and subsequent do-\nmain wall motion. We have observed lowest values of \u000b\n\u00184:9\u000210\u00003for the sample A which shows moderate\nmagnetic anisotropy. Angle dependent FMR measure-\nment shows the mixed cubic and uniaxial anisotropy in\nsamples A and B, while sample C exhibits magnetically\nisotropic behavior. Our work demonstrates that vari-\nable anisotropic L10-FeNi thin \flms can be fabricated in\nwhich the damping constant and magnetization reversal\ncan be tuned. These results may be useful for future\nspintronics based applications.\nAcknowledgements: We acknowledge the \fnancial sup-\nport by department of atomic energy (DAE), Depart-\nment of Science and Technology (DST-SERB) of Govt.\nof India,DST-Nanomission (SR/NM/NS-1018/2016(G))\nand DST, government of India for INSPIRE fellowship.\nREFERENCES\n1Z. Z. Bandic and R. H. Victora, Proceedings of the IEEE 96,\n1749 (2008).\n2S. Bedanta and W. Kleemann, Journal of Physics D: Applied\nPhysics 42, 013001 (2008).\n3D. Weller and A. Moser, IEEE Transactions on magnetics 35,\n4423 (1999).\n4M. Kotsugi, M. Mizuguchi, S. Sekiya, M. Mizumaki, T. Kojima,\nT. Nakamura, H. Osawa, K. Kodama, T. Ohtsuki, T. Ohkochi,\net al. , Journal of Magnetism and Magnetic Materials 326, 235\n(2013).\n5S. Goto, H. Kura, E. Watanabe, Y. Hayashi, H. Yanagihara,Y. Shimada, M. Mizuguchi, K. Takanashi, and E. Kita, Scienti\fc\nreports 7, 13216 (2017).\n6T. Klemmer, C. Liu, N. Shukla, X. Wu, D. Weller, M. Tanase,\nD. Laughlin, and W. So\u000ba, Journal of magnetism and magnetic\nmaterials 266, 79 (2003).\n7T. Kojima, M. Ogiwara, M. Mizuguchi, M. Kotsugi, T. Ko-\nganezawa, T. Ohtsuki, T.-Y. Tashiro, and K. Takanashi, Journal\nof Physics: Condensed Matter 26, 064207 (2014).\n8K. Takanashi, M. Mizuguchi, T. Kojima, and T. Tashiro, Journal\nof Physics D: Applied Physics 50, 483002 (2017).\n9M. Mizuguchi, S. Sekiya, and K. Takanashi, Journal of Applied\nPhysics 107, 09A716 (2010).\n10M. Kotsugi, M. Mizuguchi, S. Sekiya, T. Ohkouchi, T. Kojima,\nK. Takanashi, and Y. Watanabe, in Journal of Physics: Con-\nference Series , Vol. 266 (IOP Publishing, 2011) p. 012095.\n11K. Mibu, T. Kojima, M. Mizuguchi, and K. Takanashi, Journal\nof Physics D: Applied Physics 48, 205002 (2015).\n12M. Ogiwara, S. Iihama, T. Seki, T. Kojima, S. Mizukami,\nM. Mizuguchi, and K. Takanashi, Applied Physics Letters 103,\n242409 (2013).\n13T. Kojima, M. Mizuguchi, T. Koganezawa, K. Osaka, M. Kot-\nsugi, and K. Takanashi, Japanese Journal of Applied Physics\n51, 010204 (2011).\n14\\EVICOmagnetics,\" http://www.evico-magnetics.de/\nmicroscope.html .\n15\\NanOsc FMR Spectrometers,\" https://www.qdusa.com/\nproducts/nanosc-fmr-spectrometers.html .\n16S. Mallick, S. Bedanta, T. Seki, and K. Takanashi, Journal of\nApplied Physics 116, 133904 (2014).\n17B. B. Singh, S. K. Jena, and S. Bedanta, Journal of Physics D:\nApplied Physics 50, 345001 (2017).\n18C. Kittel, Physical Review 73, 155 (1948).\n19B. Heinrich, J. Cochran, and R. Hasegawa, Journal of Applied\nPhysics 57, 3690 (1985).\n20Z. Zhu, H. Feng, X. Cheng, H. Xie, Q. Liu, and J. Wang, Journal\nof Physics D: Applied Physics 51, 045004 (2018).\n21S. Pan, T. Seki, K. Takanashi, and A. Barman, Physical Review\nApplied 7, 064012 (2017).\n22S. Mallik, N. Chowdhury, and S. Bedanta, AIP Advances 4,\n097118 (2014).\nSUPPLEMENTARY MATERIAL6\nFIG. 5. Magnetic domain images for sample B at various \felds\nclose to the reversal \feld. It shows that the magnetization\nreversal is happening through 90\u000edomain wall in sample B."    },    {        "title": "1909.02738v2.The_interplay_of_large_two_magnon_ferromagnetic_resonance_linewidths_and_low_Gilbert_damping_in_Heusler_thin_films.pdf",        "content": "The interplay of large two-magnon ferromagnetic resonance linewidths and low\nGilbert damping in Heusler thin \flms\nW. K. Peria,1T. A. Peterson,1A. P. McFadden,2T. Qu,3C. Liu,1C. J. Palmstr\u001cm,2;4and P. A. Crowell1\n1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455\n2Department of Electrical & Computer Engineering,\nUniversity of California, Santa Barbara, California 93106\n3Department of Electrical and Computer Engineering,\nUniversity of Minnesota, Minneapolis, Minnesota 55455\n4Department of Materials, University of California, Santa Barbara, California 93106\nWe report on broadband ferromagnetic resonance linewidth measurements performed on epitaxial\nHeusler thin \flms. A large and anisotropic two-magnon scattering linewidth broadening is observed\nfor measurements with the magnetization lying in the \flm plane, while linewidth measurements with\nthe magnetization saturated perpendicular to the sample plane reveal low Gilbert damping constants\nof (1:5\u00060:1)\u000210\u00003, (1:8\u00060:2)\u000210\u00003, and<8\u000210\u00004for Co 2MnSi/MgO, Co 2MnAl/MgO, and\nCo2FeAl/MgO, respectively. The in-plane measurements are \ft to a model combining Gilbert and\ntwo-magnon scattering contributions to the linewidth, revealing a characteristic disorder lengthscale\nof 10-100 nm.\nI. INTRODUCTION\nThe theoretical understanding of the damping mech-\nanism believed to govern longitudinal magnetization re-\nlaxation in metallic ferromagnets, originally due to Kam-\nbersk\u0013 y [1, 2], has in recent years resulted in quantita-\ntive damping estimates for realistic transition metal band\nstructures [3{5]. Although of great interest where engi-\nneering of damping is desired [6], these calculations re-\nmain largely uncompared to experimental data. Kam-\nbersk\u0013 y damping may be characterized by the so-called\nGilbert damping constant \u000bin the Landau-Lifshitz-\nGilbert macrospin torque equation of motion, and for-\nmally describes how the spin-orbit interaction in itinerant\nelectron systems results in damping of magnetization dy-\nnamics [2]. Schoen et al. [7] have reported that \u000bis mini-\nmized for Co-Fe alloy compositions at which the density-\nof-states at the Fermi level is minimized, in reasonable\nagreement with Kambersk\u0013 y model predictions [8]. Fur-\nthermore, half-metallic, or near half-metallic ferromag-\nnets such as full-Heusler compounds have been predicted\nto demonstrate an ultralow Kambersk\u0013 y \u000b(\u001410\u00003) due\nto their spin-resolved band structure near the Fermi level\n[9]. Finally, anisotropy of the Kambersk\u0013 y damping in sin-\ngle crystals has been predicted, which is more robust for\nFermi surfaces with single-band character [5, 10].\nThe Gilbert damping constant is often reported\nthrough measurements of the ferromagnetic resonance\n(FMR) linewidth \u0001 H, which may be expressed as a sum\nof individual contributions\n\u0001H=2\u000bf\n\r+ \u0001H0+ \u0001HTMS; (1)\nwhere the \frst term is the Gilbert damping linewidth\n(fis the FMR frequency, \ris the gyromagnetic ratio),\n\u0001H0is a frequency-independent inhomogeneous broad-\nening, and \u0001 HTMS represents an extrinsic two-magnon\nscattering (TMS) linewidth contribution [11, 12] that is,\nin general, a nonlinear function of frequency. In recentyears it has been realized that TMS linewidths are per-\nvasive for the conventional in-plane geometry of thin \flm\nFMR measurements, requiring either the perpendicular-\nto-plane FMR geometry [13] (for which TMS processes\nare suppressed) or su\u000eciently broadband measurements\n[14] to extract the bare Gilbert \u000b. For instance, recent\nFMR linewidth studies on Heusler compounds have re-\nported distinct TMS linewidths [15, 16], which challenged\nsimple inference of the Gilbert \u000b.\nIn this article, we present FMR linewidth measure-\nments for epitaxial Heusler thin \flms for all principal ori-\nentations of the magnetization with respect to the sym-\nmetry axes. For the in-plane con\fguration, large and\nanisotropic TMS-dominated linewidths are observed. In\nthe perpendicular-to-plane con\fguration, for which the\nTMS process is inactive [11], the Gilbert \u000band inhomo-\ngeneous broadening are measured. We \fnd evidence of a\nlow (\u001810\u00003) Gilbert\u000bin these Heusler thin \flms, accom-\npanied by a large and anisotropic TMS contribution to\nthe linewdith for in-plane magnetization. We conclude\nby discussing the interplay of low Gilbert \u000band large\nTMS, and we emphasize the nature by which the TMS\nmay conceal the presence of anisotropic Kambersk\u0013 y \u000b.\nII. SAMPLES\nThe Heusler alloy \flms used for these measurements\nwere grown by molecular beam epitaxy (MBE) by co-\nevaporation of elemental sources in ultrahigh vacuum\n(UHV). The MgO(001) substrates were annealed at\n700\u000eC in UHV followed by growth of a 20 nm thick MgO\nbu\u000ber layer by e-beam evaporation at a substrate temper-\nature of 630\u000eC. The 10 nm thick Co 2MnAl and Co 2MnSi\n\flms were grown on the MgO bu\u000ber layers at room tem-\nperature and then annealed at 600\u000eC for 15 minutes\nin situ in order to improve crystalline order and surface\nmorphology. The 24 nm thick Co 2FeAl sample was grown\nusing the same MgO substrate and bu\u000ber layer prepa-arXiv:1909.02738v2  [cond-mat.mtrl-sci]  9 Apr 20202\nration, but at a substrate temperature of 250\u000eC with\nno post-growth anneal. Re\rection high energy electron\ndi\u000braction (RHEED) was monitored during and after\ngrowth of all samples and con\frmed the expected epitax-\nial relationship of MgO(001) h110ijjHeusler(001)h100i.\nX-ray di\u000braction (XRD) demonstrated the existence of\na single phase of (001)-oriented Heusler, along with the\npresence of the (002) re\rection, con\frming at least B2\nordering in all cases. In addition, for the Co 2MnSi \flm\nonly, the (111) re\rection was observed, indicating L2 1\nordering [see Fig. 1(a)]. All of the \flms were capped\nwith several nm of e-beam evaporated AlOx for pas-\nsivation prior to atmospheric exposure. The e\u000bective\nmagnetization for the 24 nm thick Co 2FeAl \flm was\ndetermined from anomalous Hall e\u000bect saturation \feld\nto be 1200 emu/cm3, which is consistent with measure-\nments of Ref. [17] for L2 1or B2-ordered \flms, along\nwith 990 emu/cm3and 930 emu/cm3for the Co 2MnSi\nand Co 2MnAl \flms, respectively. Hereafter, we will refer\nto the Co 2MnSi(10 nm)/MgO as the \\CMS\" \flm, the\nCo2MnAl(10 nm)/MgO \flm as the \\CMA\" \flm, and the\nCo2FeAl(24 nm)/MgO \flm as the \\CFA\" \flm.\nIII. EXPERIMENT\nBroadband FMR linewidth measurements were per-\nformed at room temperature with a coplanar waveguide\n(CPW) transmission setup, similar to that discussed in\ndetail in Refs. [18, 19], placed between the pole faces\nof an electromagnet. A cleaved piece of the sample\n(\u00182 mm\u00021 mm) was placed face-down over the center-\nline of the CPW. A rectifying diode was used to detect\nthe transmitted microwave power, and a \u0018100 Hz mag-\nnetic \feld modulation was used for lock-in detection of\nthe transmitted power, resulting in a signal /d\u001f=dH\n(where\u001fis the \flm dynamic magnetic susceptibility).\nThe excitation frequency could be varied from 0-50 GHz,\nand a microwave power near 0 dBm was typically used. It\nwas veri\fed that all measurements discussed in this arti-\ncle were in the small precession cone angle, linear regime.\nThe orientation of the applied magnetic \feld could be\nrotated to arbitrary angle in the \flm plane (IP), or ap-\nplied perpendicular to the \flm plane (PP). We empha-\nsize again that TMS contributions are suppressed in the\nPP con\fguration [12]. The resonance \felds were \ft as a\nfunction of applied frequency in order to extract various\nmagnetic properties of the \flms.\nThe magnetic free energy per unit volume used to gen-\nerate the resonance conditions for these samples is given\nby\nFM=\u0000M\u0001H+K1sin2\u001ecos2\u001e+ 2\u0019M2\neffcos2\u0012;(2)\nwhere His the applied \feld, \u001eand\u0012are the azimuthal\nand polar angles of the magnetization, respectively, K1\nis a \frst order in-plane cubic anisotropy constant, and\n4\u0019Meffis the PP saturation \feld, which includes the\nusual demagnetization energy and a \frst order uniaxial\n-150-75075150-6-30361\n0 GHz20 GHz30 GHzdχ/dH (arb. u.)H\n - HFMR (Oe)40 GHzCMS\n090180270360101102103104 <111> \n<202>Intensity (arb. u.)φ\n (°)CMS\n-5000 5 00-101F\nield (Oe)M/MSH\n || 〈110〉H\n || 〈100〉H\n || 〈110〉CFA\n(d)(b)(\nc)(a)C\nFAFIG. 1. (a) Wide-angle x-ray di\u000braction \u001e-scans ofh202i\n(blue) andh111i(red) peaks for the CMS \flm. (b) Typical\nderivative susceptibility lineshapes for these samples at dif-\nferent microwave excitation frequencies. The \fts are shown\nas solid lines. (c) In-plane hysteresis loops for CFA obtained\nwith a vibrating-sample magnetometer (VSM). (d) Atomic\nforce microscopy (AFM) image of surface topography for\nCFA. RMS roughness is 0.2 nm.\nanisotropy due to interfacial e\u000bects. The parameters ob-\ntained by \ftting to Eq. 2 are shown in Table I. The uncer-\ntainty in these parameters was estimated by measuring a\nrange of di\u000berent sample pieces, and using the standard\ndeviation of the values as the error bar. The long-range\ninhomogeneity characteristic of epitaxial samples makes\nthis a more accurate estimate of the uncertainty than\nthe \ftting error. The magnetic-\feld-swept FMR line-\nshapes were \ft to the derivative of Lorentzian functions\n[19] in order to extract the full-width at half-maximum\nlinewidths \u0001 H[magnetic \feld units, Fig. 1(b)], which\nare the focus of this article. The maximum resonant fre-\nquency was determined by the maximum magnetic \feld\nthat could be applied for both IP and PP electromagnet\ncon\fgurations, which was 10.6 kOe and 29 kOe, respec-\ntively. For the IP measurement, the angle of the applied\n\feld in the plane of the \flm was varied to determine the\nin-plane magnetocrystalline anisotropy of our samples,\nwhich was fourfold-symmetric for the three \flms char-\nacterized in this article. The anisotropy was con\frmed\nusing vibrating-sample magnetometry (VSM) measure-\nments, an example of which is shown in Fig. 1(c), which\nshows IP easy and hard axis hysteresis loops for the\nCFA \flm. For the PP measurement, alignment was ver-\ni\fed to within\u00180.1\u000eto ensure magnetization saturation\njust above the PP anisotropy \feld, thus minimizing \feld-3\nTABLE I. Summary of the magnetic properties extracted from the dependence of the resonance \feld on applied frequency\nfor both \feld in-plane ( jj) and \feld perpendicular-to-plane ( ?) con\fgurations, along with the Gilbert \u000band inhomogeneous\nbroadening from the perpendicular-to-plane con\fguration. 2 K1=Msand 4\u0019Meffare the in-plane and perpendicular-to-plane\nanisotropy \felds, respectively (see Eq. 2), and gis the Land\u0013 e g-factor.\nSample 2 K1=Ms(Oe) 4\u0019Mjj\neff(kOe) 4 \u0019M?\neff(kOe) gjjg?\u000b001(\u000210\u00003) \u0001H0(Oe)\nCMS 280 12.3 13.3 2.04 2.04 1 :5\u00060:1 9\u00061\nCMA 35 11.3 11.7 2.06 2.08 1 :8\u00060:2 12\u00063\nCFA 230 15.1 15.5 2.06 2.07 <0.8 100\u00066\nCFA 500\u000eC anneal N/A N/A 15.1 N/A 2.07 1 :1\u00060:1 45\u00061\n01 02 03 04 05 00306090120α\n001 = 1.1×10-3CFA 500 °C annealCFAC\nMAα001 < 8×10-4α\n001 = 1.8×10-3ΔH (Oe)F\nrequency (GHz)α001 = 1.5×10-3CMS\nFIG. 2. Linewidths as a function of frequency with the \feld\napplied perpendicular to plane, for which two-magnon scat-\ntering is inactive. The black squares are data for the CMS\n\flm, the red circles are for the CMA \flm, and the blue trian-\ngles are for the CFA \flm. In addition, linewidths are shown\nfor a CFA \flm that was annealed at 500\u000eCex situ (magenta\ndiamonds). Corresponding linear \fts are shown along with\nthe extracted Gilbert damping factor \u000b. The blue dashed\nlines indicate an upper bound of \u000b001= 8\u000210\u00004and a lower\nbound of\u000b001= 0 for CFA.\ndragging contributions to the linewidth.\nIV. RESULTS AND ANALYSIS\nA. Perpendicular-to-plane linewidths\nFirst we discuss the results of the PP measurement. As\nstated in Sec. III, the TMS extrinsic broadening mecha-\nnism is suppressed when the magnetization is normal to\nthe plane of the \flm. We can thus \ft our data to Eq. 1\nwith \u0001HTMS = 0, greatly simplifying the extraction of\nthe Gilbert damping constant \u000band the inhomogeneous\nbroadening \u0001 H0. Prior knowledge of \u0001 H0is particu-\nlarly important for constraining the analysis of the IP\nmeasurements, as we shall discuss.\n3035404550C\nMS〈100〉C\nMS〈110〉ΔH (Oe)CMS2\n0 GHz2\n800300032003400H\nFMR(Oe)-\n450 4 5200400600C\nFA〈100〉ΔH (Oe)A\nngle (°)CFA2\n0 GHzCFA〈110〉2\n600280030003200H\nFMR(Oe)50100150200(a)(\nc)C\nMA〈100〉C\nMA〈110〉ΔH (Oe)CMA1\n5 GHz(b)2\n00020502100H\nFMR(Oe)FIG. 3. Azimuthal angular dependence of the linewidths (left\nordinate, blue circles) and resonance \felds (right ordinate,\nblack squares) for (a) CMS, (b) CMA, and (c) CFA. The\nexcitation frequency was 20 GHz for CMS, 15 GHz for CMA,\nand 20 GHz for CFA. The solid lines are sinusoidal \fts.\nThe dependence of \u0001 Hon frequency for the CMS,\nCMA, and CFA \flms in the PP con\fguration is\nsummarized in Fig. 2, in which \fts to Eq. 1 are\nshown with \u0001 HTMS set to zero. For the CMS\n\flm,\u000b001= (1:5\u00060:1)\u000210\u00003and \u0001H0= 9 Oe,\nwhile for the CMA \flm \u000b001= (1:8\u00060:2)\u000210\u000034\nand \u0001H0= 12 Oe. Co 2MnSi 2=3Al1=3/MgO and\nCo2MnSi 1=3Al2=3/MgO \flms (both 10 nm thick) were\nalso measured, with Gilbert damping values of \u000b001=\n(1:8\u00060:2)\u000210\u00003and\u000b001= (1:5\u00060:1)\u000210\u00003, re-\nspectively (not shown). For CFA, we obtained a damp-\ning value of \u000b001= 3\u000210\u00004with an upper bound of\n\u000b001<8\u000210\u00004and \u0001H0= 100 Oe. These \ft param-\neters are also contained in Table I. The source of the\nlarge inhomogeneous broadening for the CFA \flm is un-\nclear: AFM measurements [Fig. 1(d)] along with XRD in-\ndicate that the \flm is both crystalline and smooth. Note\nthat the range of frequencies shown in Fig. 2 are largely\ngoverned by considerations involving the Kittel equation\n[20]: measurements below 10 GHz were not used due to\nthe increasing in\ruence of slight misalignment on \u0001 H\n(through \feld-dragging) for resonant \felds just above\nthe saturation value. A piece of the CFA sample was\nannealed at 500\u000eCex situ , which reduced the inhomoge-\nnoeus broadening to \u001845 Oe (still a relatively large value)\nand increased the Gilbert damping to \u000b001= 1:1\u000210\u00003\n(similar behavior in CFA was seen in Ref. [21]). The\nconstraint of \u000b001<8\u000210\u00004is among the lowest of re-\nported Gilbert damping constants for metallic ferromag-\nnets, but the \u000b\u001810\u00004range is not unexpected based\non Kambersk\u0013 y model calculations performed for similar\nfull-Heusler compounds [9] or other recent experimental\nreports [22, 23]. It should be noted that Schoen et al. [7]\nhave recently reported \u000b= 5\u000210\u00004for Co 25Fe75thin\n\flms, where spin pumping and radiative damping con-\ntributions were subtracted from the raw measurement.\nSpin pumping contributions to the intrinsic damping are\nnot signi\fcant in our \flms, as no heavy-metal seed layers\nhave been used and the \flms have thicknesses of 10 nm\nor greater. For the radiative damping contribution [13]\nin the geometry of our CPW and sample, we calculate\ncontributions \u000brad<\u00181\u000210\u00004, which is below the uncer-\ntainty in our damping \ft parameter.\nB. In-plane linewidths\nWith the intrinsic damping and inhomogeneous broad-\nening characterized by the PP measurement, we turn our\nattention to the IP linewidth measurements, for which\nTMS contributions are present. For hard-axis measure-\nments, frequencies <\u00185 GHz were not used due to the\nin\ruence of slight magnetic \feld misalignment on the\nlinewidths. For easy-axis measurements, the lower limit\nis determined by the zero-\feld FMR frequency. Fig-\nure 3 shows the dependences of the resonance \felds and\nlinewidths on the angle of the in-plane \feld. An im-\nportant observation seen in Fig. 3 is that the linewidth\nextrema are commensurate with those of the resonance\n\felds and therefore the magnetocrystalline anisotropy en-\nergy. This rules out \feld-dragging and mosaicity contri-\nbutions to the linewidth, which can occur when the reso-\nnance \feld depends strongly on angle [24]. We note that\nsimilar IP angular dependence of the FMR linewidth,\n01 02 03 04 05 002004006008000204060801000\n60120180240300C\nFA(c)ΔH (Oe)F\nrequency (GHz)〈100〉ΔH (Oe)(a)C\nMSC\nMA[\n001]〈110〉(b)ΔH (Oe)FIG. 4. Linewidths along all three principal directions for\nCMS (a), CMA (b), and CFA (c). Heusler crystalline axes\nare labeled byh100i(black),h110i(red), and [001] (blue). In\nall three cases,h110iis the in-plane easy axis and h100iis the\nin-plane hard axis. The corresponding \fts are shown as the\nsolid curves, where the in-plane linewidths are \ft using Eq. 3\nand the out-of-plane linewidths are \ft to the Gilbert damping\nmodel. The \ft parameters are given in Table II.\nwhich was attributed to an anisotropic TMS mechanism\ncaused by a rectangular array of mis\ft dislocations, has\nbeen reported by Kurebayashi et al. [25] and Woltersdorf\nand Heinrich [14] for epitaxial Fe/GaAs(001) ultrathin\n\flms.\nTo further study the anisotropy of the IP \u0001 Hin our\n\flms, we have measured \u0001 Hat the angles correspond-\ning to the extrema of HFMR (and \u0001H) in Fig. 3 over\na range of frequencies. These data are shown in Fig. 4,\nalong with the PP ([001]) measurements for each sample.\nA distinguishing feature of the data shown in Fig. 4 is the\nsigni\fcant deviation between IP and PP linewidths in all\nbut one case (CMS h100i). Large and nonlinear frequency\ndependence of the IP linewidths is strongly suggestive of\nan active TMS linewidth broadening mechanism. In the\npresence of TMS, careful analysis is required to separate5\n0°90°1\n80°2\n70°01x10501 02 03 04 00.00.51.01.52\nx1045x1041\n1.411.82\n4 GHz32 GHz1\n6 GHz |q2M| (cm-1)ξ\n-1ξ = 100 nm10-41\n0-35\n×10-3ΔHTMS/H'2 (Oe-1) (×10-4)f\nFMR (GHz)α = 10-2(\nq || M)(q ^ M)(b) |q| (cm-1)q\n ^ Mωm (GHz)q\n || MD\negeneracyH = 1 kOe(a)\nFIG. 5. (a) Two-magnon scattering linewidth contribution\nfor values of Gilbert damping \u000b= 10\u00002;5\u000210\u00003;10\u00003;and\n10\u00004. The inset shows magnon dispersions for an applied\n\feld ofH= 1 kOe. (b) Contours of the degenerate mode\nwavenumber q2Min the \flm plane as a function of wavevector\nangle relative to the magnetization for fFMR = 16, 24, and\n32 GHz. The dashed circle indicates the wavenumber of a\ndefect with size \u0018= 100 nm.\nthe Gilbert damping from the TMS linewidth contribu-\ntions. We therefore describe the TMS mechanism in more\ndetail in the following section in order to analyze the IP\nlinewidths in Fig. 4 and extract the Gilbert damping.\nC. Two-magnon scattering model\nThe TMS mechanism leads to a characteristic nonlin-\near frequency dependence of \u0001 H[11, 12]. In Fig. 4,\nthe IP \u0001His not a linear function of frequency, but\npossesses the \\knee\" behavior characteristic of the fre-\nquency dependence of linewidths dominated by the TMS\nmechanism. We have \ft our data to the TMS model\ndescribed by McMichael and Krivosik [12], in which theTMS linewidth \u0001 HTMS is given by [26, 27]\n\u0001HTMS =\r2\u00182H02\ndf=dHjfFMRZ\n\u00000qCq(\u0018)\u000e\u000b(!\u0000!q)d2q;(3)\nwhere \u0000 0qis the defect-mediated interaction term be-\ntween magnons at wavevector 0 and q,Cq(\u0018) = (1 +\n(q\u0018)2)\u00003=2is the correlation function of the magnetic sys-\ntem with correlation length \u0018, andH0is the magnitude\nof the characteristic inhomogeneity (units of magnetic\n\feld). The \u000e\u000b-function in Eq. 3 selects only the magnon\nscattering channels that conserve energy. In the limit of\nzero intrinsic damping, it is identical to the Dirac delta\nfunction, but for \fnite \u000bit is replaced by a Lorentzian\nfunction of width \u000e!= (2\u000b!=\r )d!=dH . The magnon\ndispersion relation determining !qis the usual Damon-\nEshbach thin \flm result [26, 28] with the addition of mag-\nnetocrystalline anisotropy sti\u000bness \feld terms extracted\nfrom the dependence of the resonance \feld on the applied\nfrequency for the IP con\fguration. The \flm thickness\nda\u000bects the states available for two-magnon scattering\nthrough the dispersion relation, namely, the linear term\nwhich gives rise to negative group velocity for small q\n(/\u0000qd). The IP FMR linewidth data shown in Fig. 4\nwere \ft to Eq. 1 (with Eq. 3 used to evaluate \u0001 HTMS)\nwith\u0018,\u000b, andH0as \ftting parameters (shown in Table\nII). The correlation length \u0018remains approximately con-\nstant for di\u000berent in-plane directions, while the strength\nH0is larger for theh100idirections in the CMA and CFA\nsamples and the h110idirections in the CMS sample.\nSome degree of uncertainty results from this \ftting proce-\ndure, because for linewidth data collected over a limited\nfrequency range, \u0018and\u000bare not completely decoupled\nas \ftting parameters. In absolute terms, however, the\nlargest systematic errors come from the exchange sti\u000b-\nness, which is not well-known. The error bars given in\nTable II were calculated by varying the exchange sti\u000b-\nness over the range 400 meV \u0017A2to 800 meV \u0017A2, and\nrecording the change in the \ft parameters. This range of\nvalues was chosen based on previous Brillouin light scat-\ntering measurements of the exchange sti\u000bness in similar\nHeusler compounds [29, 30]. In addition, we note that\nin Eq. 1 \u0001 H0is taken to be isotropic, with the value\ngiven by the PP linewidth measurements shown in Fig.\n2. Although certain realizations of inhomogeneity may\nresult in an anisotropic \u0001 H0(see Ref. [14] for a good\ndiscussion), doing so here would only serve to create an\nadditional \ftting parameter.\nD. E\u000bect of low intrinsic damping\nThe e\u000bect of low intrinsic damping on the two-magnon\nlinewidth can be seen in Fig. 5(a). As \u000bdecreases, with\nall other parameters \fxed, \u0001 HTMS steadily increases\nand becomes increasingly nonlinear (and eventually non-\nmonotonic) with frequency. In particular, a \\knee\" in\nthe frequency dependence becomes more pronounced for6\nTABLE II. Summary of the \ftting parameters used to \ft the\nin-plane data of Fig. 4 (black squares and red circles) to Eqs.\n1 and 3. CFA refers to the unannealed Co 2FeAl sample.\nSample (Field Direction) \u000b(\u000210\u00003)\u0018(nm)H0(Oe)\nCMSh110i 1:6\u00060:2 40\u000625 55\u000630\nCMSh100i 1:5\u00060:1 40\u000625 30\u000615\nCMAh110i 3:1\u00060:2 70\u000620 30\u00065\nCMAh100i 4:7\u00060:4 55\u000610 90\u00065\nCFAh110i 2:0\u00060:3 20\u000610 175\u000660\nCFAh100i N/A N/A N/A\nlow damping (see e.g. Fig. 5(a) curve for \u000b= 10\u00004). The\nphysics giving rise to the knee behavior is illustrated in\nFig. 5(b). The TMS process scatters magnons from zero\nto non-zero wavevector at small q. There is assuemd\nto be su\u000ecient disorder to allow for the momentum q\nto be transferred to the magnon system. There will al-\nways be, however, a length scale \u0018below which the disor-\nder decreases, so that the \flm becomes e\u000bectively more\nuniform at large wavevectors. The corresponding FMR\nfrequencies are those for which the contours of constant\nfrequency (the \fgure eights in Fig. 5) in q-space have ex-\ntrema atq\u0018\u0018\u00001. The TMS rate is also determined by\nthe interplay of the magnon density of states, the e\u000bec-\ntive area in q-space occupied by the modes that conserve\nenergy, and the Gilbert damping. The knee behavior is\nmore pronounced for low \u000bdue to the increased weight\nof the van Hove singularity coming from the tips of the\n\fgure eights, in the integrand of Eq. 3. Although a larger\nwindow of energies, set by the width of \u000e\u000b, is available for\nlarger\u000b, this smears out the singularity in the magnon\ndensity of states, removing the sharp knee in the TMS\nlinewidth as a function of frequency. The PP measure-\nment con\frms that all of these epitaxial Heusler \flms lie\nwithin the range \u000b<2\u000210\u00003. Ferromagnetic \flms with\nultralow\u000bare therefore increasingly prone to large TMS\nlinewidths (particularly for metals with large Ms). The\nTMS linewidths will also constitute a larger fraction of\nthe total linewidth due to a smaller contribution from the\nGilbert damping. In practice, this is why experimental\nreports [7, 22, 23] of ultralow \u000bhave almost all utilized\nthe PP geometry.\nE. Discussion\nThe results of the IP linewidth \fts to Eqs. 1 and 3 are\nsummarized in Table II. In the case of CMS, the high-\nfrequency slopes in Fig. 4(a) approach the same value\nalong each direction, as would be expected when the fre-\nquency is large enough for the TMS wavevector to exceed\nthe inverse of any defect correlation length. In this limit,\n\u000bis isotropic (within error limits).\nNext, we discuss the CMA IP data shown in Fig. 4(b)\nand Table II. It is clear from this \fgure that a good \ftcan be obtained along both h100iandh110idirections.\nIn Table II it can be seen that the value of the defect cor-\nrelation length \u0018is approximately the same along both\ndirections. However, the values of \u000bwe obtain from \ft-\nting to Eqs. 1 and 3 do not agree well with the PP value\nof\u000b001= 1:8\u000210\u00003(Fig. 2). Anisotropic values of \u000b\nhave been both predicted [5, 10] and observed [31], and\nan anisotropic \u000bis possibly the explanation of our best-\n\ft results. The in-plane h100iand [001] directions are\nequivalent in the bulk, so the anisotropy would neces-\nsarily be due to an interface anisotropy energy [31] or\nperhaps a tetragonal distortion due to strain [32].\nFinally, we discuss the CFA linewidths shown in Fig.\n4(c) and Table II. This sample has by far the largest two-\nmagnon scattering contribution, which is likely related\nto the anomalously large inhomogeneous broadening and\nlow intrinsic damping [see Fig. 5(a)] observed in the PP\nmeasurement. A good \ft of the data was obtained when\nthe \feld was applied along the h110idirection. Notably,\nthe IPh110ibest \ft value of 2 :1\u000210\u00003is nearly a factor\nof 3 larger than the \u000b001upper bound on the same sample\n(Table I), strongly suggesting an anisotropic Gilbert \u000b. A\nstriking anisotropy in the IP linewidth was revealed upon\nrotating the magnetization to the h100iorientation. For\ntheh100icase, which yielded the largest TMS linewidths\nmeasured in this family of \flms, we were not able to \ft\nthe data to Eq. 3 using a set of physically reasonable in-\nput parameters. We believe that this is related to the\nconsideration that higher order terms in the inhomoge-\nneous magnetic energy (see Ref. [26]) need to be taken\ninto account. Another reason why this may be the case is\nthat the model of McMichael and Krivosik [12] assumes\nthe inhomogeneities to be grain-like, whereas the samples\nare epitaxial [see Fig. 1(a)]. Atomic force microscopy im-\nages of these samples [Fig. 1(d)] imply that grains, if they\nexist, are much larger than the defect correlation lengths\nlisted in Table II, which are of order 10's of nm. We also\nnote that there does not appear to be a correlation be-\ntween the strength of two-magnon scattering H0and the\ncubic anisotropy \feld 2 K1=Ms, which would be expected\nfor grain-induced two-magnon scattering.\nV. SUMMARY AND CONCLUSION\nWe conclude by discussing the successes and limita-\ntions of the McMichael and Krivosik [12] model in an-\nalyzing our epitaxial Heusler \flm FMR linewidth data.\nWe have shown that two-magnon scattering is the ex-\ntrinsic linewidth-broadening mechanism in our samples.\nAny model which takes this as its starting point will\npredict much of the qualitative behavior we observe,\nsuch as the knee in the frequency dependence and the\nlarge linewidths IP for low \u000b\flms. The TMS model\nused in this article (for the purpose of separating TMS\nand Gilbert linewidth contributions) is, however, only\nas accurate as its representation of the inhomogeneous\nmagnetic \feld and the underlying assumption for the7\nfunctional form of Cq(\u0018). Grain-like defects are as-\nsumed, which essentially give a random magnetocrys-\ntalline anisotropy \feld. We did not, however, explicitly\nobserve grains in our samples with AFM, at least below\nlengthscales of\u001810\u0016m [Fig. 1(d)]. Mis\ft dislocations, a\nmuch more likely candidate in our opinion, would cause\nan e\u000bective inhomogeneous magnetic \feld which could\nhave a more complicated spatial pro\fle and therefore\nlead to anisotropic two-magnon scattering (see Ref. [14]).\nThe perturbative nature of the model also brings its own\nlimitations, and we believe that the CFA h100idata, for\nwhich we cannot obtain a satisfactory \ft, are exemplary\nof a breakdown in the model for strong TMS. Future\nwork should go into methods of treating the two-magnon\nscattering di\u000berently based on the type of crystalline de-\nfects present, which will in turn allow for a more reli-\nable extraction of the Gilbert damping \u000band facilitate\nthe observation of anisotropic Gilbert damping, enabling\nquantitative comparison to \frst-principles calculations.\nRegardless of the limitations of the model, we empha-\nsize three critical observations drawn from the linewidth\nmeasurements presented in this article. First, in all cases\nwe observe large and anisotropic TMS linewidth contri-\nbutions, which imply inhomogeneity correlation length-\nscales of order tens-to-hundreds of nanometers. The mi-\ncroscopic origin of these inhomogeneities is the subjectof ongoing work, but are likely caused by arrays of mis\ft\ndislocations [14]. The relatively large lengthscale of these\ndefects may cause them to be easily overlooked in epi-\ntaxial \flm characterization techniques such as XRD and\ncross-sectional HAADF-STEM, but they still strongly in-\n\ruence magnetization dynamics. These defects and their\nin\ruence on the FMR linewidth through TMS complicate\ndirect observation of Kambersk\u0013 y's model for anisotropic\nand (in the case of Heusler compounds) ultralow intrinsic\ndamping in metallic ferromagnets. Second, we observed\nlow intrinsic damping through our PP measurement,\nwhich was<2\u000210\u00003for all of our samples. Finally, we\nhave presented the mechanism by which FMR linewidths\nin ultralow damping \flms are particularly likely to be en-\nhanced by TMS, the anisotropy of which may dominate\nany underlying anisotropic Kambersk\u0013 y damping.\nThis work was supported by NSF under DMR-1708287\nand by SMART, a center funded by nCORE, a Semi-\nconductor Research Corporation program sponsored by\nNIST. 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Zhang, Giant Anisotropy\nof Gilbert Damping in Epitaxial CoFe Films, Physical\nReview Letters 122, 117203 (2019)."    },    {        "title": "1909.04362v3.Spin_Pumping_from_Permalloy_into_Uncompensated_Antiferromagnetic_Co_doped_Zinc_Oxide.pdf",        "content": "Spin Pumping from Permalloy into Uncompensated Antiferromagnetic Co doped Zinc\nOxide\nMartin Buchner,1,\u0003Julia Lumetzberger,1Verena Ney,1Tadd aus Scha\u000bers,1,yNi\u0013 eli Da\u000b\u0013 e,2and Andreas Ney1\n1Institut f ur Halbleiter- und Festk orperphysik, Johannes Kepler Universit at, Altenberger Str. 69, 4040 Linz, Austria\n2Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland\n(Dated: August 11, 2021)\nHeterostructures of Co-doped ZnO and Permalloy were investigated for their static and dynamic\nmagnetic interaction. The highly Co-doped ZnO is paramagentic at room temperature and becomes\nan uncompensated antiferromagnet at low temperatures, showing a narrowly opened hysteresis and\na vertical exchange bias shift even in the absence of any ferromagnetic layer. At low temperatures in\ncombination with Permalloy an exchange bias is found causing a horizontal as well as vertical shift\nof the hysteresis of the heterostructure together with an increase in coercive \feld. Furthermore,\nan increase in the Gilbert damping parameter at room temperature was found by multifrequency\nFMR evidencing spin pumping. Temperature dependent FMR shows a maximum in magnetic\ndamping close to the magnetic phase transition. These measurements also evidence the exchange\nbias interaction of Permalloy and long-range ordered Co-O-Co structures in ZnO, that are barely\ndetectable by SQUID due to the shorter probing times in FMR.\nI. Introduction\nIn spintronics a variety of concepts have been devel-\noped over the past years to generate and manipulate spin\ncurrents [1, 2]. Amongst them are the spin Hall e\u000bect\n(SHE), which originates from the spin orbit coupling [3],\nspin caloritronics [4] utilizing the spin seebeck e\u000bect [5]\nor spin transfer torque (current induced torque) due to\nangular momentum conservation [6] as examples. Spin\npumping [7], where a precessing magnetization transfers\nangular momentum to an adjacent layer, proved to be a\nvery versatile method since it has been reported for di\u000ber-\nent types of magnetic orders [8{11] or electrical properties\n[12{14] of materials. Furthermore it could also be veri-\n\fed in trilayer systems where the precessing ferromagnet\nand the spin sink, into which the angular momentum is\ntransferred, are separated by a non-magnetic spacer [15{\n18]. This is strongly dependent on the material, while\nfor Cu [15], Au [16], or Al [17] pumping through a few\nnanometers is possible an MgO barrier of 1 nm is enough\nto completely suppress spin pumping [18].\nSpintronic devices are usually based on a ferromagnet\n(FM) although antiferromagnetic spintronics [19] holds\nthe advantages of faster dynamics, less perturbation by\nexternal magnetic \felds and no stray \felds. The latter\ntwo are caused by the zero net magnetization of an an-\ntiferromagnet (AFM), which on the other hand makes\nthem harder to manipulate. One way to control an\nAFM is by using an adjacent FM layer and exploiting\nthe exchange-bias (EB) e\u000bect [20, 21]. Measuring spin-\ntransfer torque in FM/AFM bilayer structures, is possi-\n\u0003Electronic address: martin.buchner@jku.at; Phone: +43-732-\n2468-9651; FAX: -9696\nyCurrent address: NanoSpin, Department of Applied Physics,\nAalto University School of Science, P.O. Box 15100, FI-00076\nAalto, Finlandble [22, 23], but challenging due to Joule heating [24{26]\nor possible unstable antiferromagnetic orders [27]. Anti-\nferromagnets can be used either as spin source [28] or as\nspin sink [11, 29] in a spin pumping experiment. Thereby\nthe spin mixing conductance, a measure for the absorp-\ntion of angular (spin) momentum at the interface [7],\nis described by intersublattice scattering at an antiferro-\nmagnetic interface [30]. Linear response theory predicted\nan enhancement of spin pumping near magnetic phase\ntransitions [31], which could recently also be veri\fed ex-\nperimentally [29].\nIn this work we investigate the behavior of the uncom-\npensated, antiferromagnetic Co xZn1-xO with x2f0.3,\n0.5, 0.6g(in the following 30 %, 50 % and 60 % Co:ZnO)\nin contact to ferromagnetic permalloy (Py). While\nweakly paramagnetic at room temperature, Co:ZnO\nmakes a phase transition to an antiferromagnetic state at\na N\u0013 eel temperature ( TN) dependent on the Co concentra-\ntion [32]. This resulting antiferromagnetism is not fully\ncompensated which is evidenced by a narrow hysteresis\nand a non saturating magnetization up to 17 T [33]. Fur-\nthermore, Co:ZnO \flms exhibit a vertical EB in complete\nabsence of a FM layer [34]. This vertical exchange shift is\ndependent on the Co concentration [32], temperature and\ncooling \feld [35] and the \feld imprinted magnetization\npredominantly shows orbital character [36]. Note that\nbelow the coalesence limit of 20 % the vertical EB van-\nishes. Co:ZnO therefore o\u000bers to study magnetic inter-\nactions between an uncompensated AFM and a FM Py\nlayer. Static coupling, visible as EB, is investigated using\nsuper conducting quantum interference device (SQUID)\nmagnetometry. The dynamic coupling across the inter-\nface is measured using ferromagnetic resonance (FMR)\nat room temperature and around the magnetic transi-\ntion temperatures determined from M(T) SQUID mea-\nsurements. Element selective XMCD studies are carried\nout to disentangle the individual magnetic contributions.\nFinally heterostructures with an Al spacer were investi-\ngated to rule out intermixing at the interface as sourcearXiv:1909.04362v3  [cond-mat.mtrl-sci]  14 Oct 20192\nfor the coupling e\u000bect.\nII. Experimental Details\nHeterostructures consisting of Co:ZnO, Py and Al, as\nshown in Fig. 1 were fabricated on c-plane sapphire sub-\nstrates using reactive magnetron sputtering (RMS) and\npulsed laser deposition (PLD) at a process pressure of 4\n\u000210-3mbar. The di\u000berent layers of a heterostructure are\nall grown in the same UHV chamber with a base pressure\nof 2\u000210-9mbar in order to ensure an uncontaminated\ninterface. While Py and Co:ZnO are grown by magnetron\nsputtering, the Al spacer and capping layers are grown\nby PLD. Al and Py are fabricated at room temperature\nusing 10 standard cubic centimeters per minute (sccm)\nAr as a process gas.\nFor the heterostructures containing a Co:ZnO layer,\nsamples with three di\u000berent Co concentrations of 30 %,\n50 % and 60 % are grown utilizing preparation conditions\nthat yield the best crystalline quality known for Co:ZnO\nsingle layers [32, 33, 36]. For 30 % and 50 % Co:ZnO\nmetallic sputter targets of Co and Zn are used at an\nAr:O 2ratio of 10 : 1 sccm, while for 60 % Co:ZnO no oxy-\ngen and a ceramic composite target of ZnO and Co 3O4\nwith a 3 : 2 ratio is used. The optimized growth temper-\natures are 450\u000eC, 294\u000eC and 525\u000eC. Between Co:ZnO\ngrowth and the next layer a cool-down period is required,\nto minimize inter-di\u000busion between Py and Co:ZnO.\nThe static magnetic properties are investigated by\nSQUID magnetometry. M(H) curves are recorded at\n300 K and 2 K in in-plane geometry with a maximum\nmagnetic \feld of \u00065 T. During cool-down either a mag-\nnetic \feld of\u00065 T or zero magnetic \feld is applied to dif-\nferentiate between plus-\feld-cooled (pFC), minus-\feld-\ncooled (mFC) or zero-\feld-cooled (ZFC) measurements.\nAll measurements shown in this work have been corrected\nby the diamagnetic background of the sapphire substrate\nand care was taken to avoid well-known artifacts [37, 38].\nFor probing the element selective magnetic properties\nX-ray absorption (XAS) measurements were conducted\nat the XTreme beamline [39] at the Swiss Synchrotron\nLightsource (SLS). From the XAS the X-ray magnetic\ncircular dichroism (XMCD) is obtained by taking the\ndirect di\u000berence between XAS with left and right cir-\ncular polarization. The measurements were conducted\nwith total \ruoresence yield under 20\u000egrazing incidence.\nThereby, the maximum magnetic \feld of 6.8 T was ap-\nplied. Both, external magnetic \feld and photon helic-\nity have been reversed to minimize measurement arte-\nfacts. Again pFC, mFC and ZFC measurements were\nconducted applying either zero or the maximum \feld in\nthe respective direction.\nThe dynamic magnetic properties were measured us-\ning multi-frequency and temperature dependent FMR.\nMulti-frequency FMR is exclusively measured at room\ntemperature from 3 GHz to 10 GHz using a short cir-\ncuited semi-rigid cable [40]. Temperature dependentmeasurements are conducted using an X-band resonator\nat 9.5 GHz. Starting at 4 K the temperature is increased\nto 50 K in order to be above the N\u0013 eel-temperature of the\nCo:ZnO samples [32, 35]. At both FMR setups the mea-\nsurements were done in in-plane direction.\nThe measured raw data for SQUID, FMR, XAS and\nXMCD can be found in a following data repository [41].\nIII. Experimental results & Discussion\nFIG. 1: (a) shows the schematic setup of the samples. For the\nCo:ZnO layer three di\u000berent Co concentrations of 30 %, 50 %\nand 60 % are used. The cross section TEM image of the 60 %\nCo:ZnO/Py sample as well as the electron di\u000braction pattern\nof the Co:ZnO layer (b) and a magni\fcation on the interface\nbetween Co:ZnO and Py (c) are shown.\nFigure 1(a) displays the four di\u000berent types of samples:3\nCo:ZnO layers, with Co concentrations of 30 %, 50 % and\n60 %, are grown with a nominal thickness of 100 nm and\nPy with 10 nm. To prevent surface oxidation a capping\nlayer of 5 nm Al is used. For single 60 % Co:ZnO \flms\nthe vertical-exchange bias e\u000bect was largest compared to\nlower Co concentrations. Therefore, for 60 % Co:ZnO\nsamples with an additional Al layer as spacer between\nCo:ZnO and Py have been fabricated. The thickness of\nthe Al spacer (1 nm, 1.5 nm and 2 nm) is in a range where\nthe Al is reported not to suppress spin pumping e\u000bects\nitself [17].\nTEM\nTo get information about the interface between Py\nand Co:ZnO high resolution cross section transmission\nelectron microscopy (TEM) was done. In Fig. 1(b) the\ncross section TEM image of 60 % Co:ZnO/Py with the\nelectron di\u000braction pattern of the Co:ZnO is shown. A\nmagni\fcation of the interface between Co:ZnO and Py is\nshown in Fig. 1(c). From XRD measurements [32] it is\nobvious that the quality of the wurtzite crystal slightly\ndecreases for higher Co doping in ZnO. A similar be-\nhavior is observed in TEM cross section images. While\n35 % Co:ZnO shows the typical only slightly misoriented\ncolumnar grain growth [32] it is obvious from Fig. 1(b)\nthat the crystalline nanocolumns are less well ordered for\n60 % Co:ZnO. Although the electron di\u000braction pattern\ncon\frms a well ordered wurtzite structure, the misorien-\ntation of lattice plains is stronger than for 35 % Co:ZnO\n[32], even resulting in faint Moir\u0013 e fringes which stem from\ntilted lattice plains along the electron path. This cor-\nroborates previous \fndings of !-rocking curves in XRD\n[32, 36] where the increase in the full width at half maxi-\nmum also evidences a higher tilting of the crystallites, i.e.\nan increased mosaicity. The interface to the Py layer is\nsmooth, although it is not completely free of dislocations.\nAlso the interface seems to be rather abrupt within one\natomic layer, i.e. free of intermixing. A similar behavior\nis found for the interface between 50 % Co:ZnO and Py\n(not shown).\nXAS and XMCD\nFigure 2 shows XAS and XMCD spectra recorded at\n3 K and a magnetic \feld of 6.8 T at the Ni L 3/2and\nCo L 3/2edges of 60 % Co:ZnO/Py after pFC, mFC or\nZFC. For all three cooling conditions the Ni L 3/2edges\n(Fig. 2(a)) show a metallic character of the Ni XAS with-\nout any additional \fne structure characteristics for NiO\nand thus no sign of oxidation of the Py. Further, no dif-\nferences in the XAS or the XMCD of the Ni edges of\ndi\u000berent cooling conditions are found. The same is ob-\nserved for the Fe L 3/2edges, however, they are a\u000bected\ngreatly by self-absorption processes in total \ruorescence\nyield (not shown).\n/s56/s52/s48 /s56/s53/s48 /s56/s54/s48 /s56/s55/s48 /s56/s56/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s40/s97/s41\n/s78/s105/s32/s76/s51/s47/s50/s32/s101/s100/s103/s101/s32/s64/s32/s51/s75/s44/s32/s84/s70/s89/s32\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s32/s105/s110/s99/s105/s100/s101/s110/s99/s101/s44/s32/s66/s32/s61/s32/s54/s46/s56/s84\n/s32/s88/s65/s83/s32/s112/s70/s67\n/s32/s88/s65/s83/s32/s90/s70/s67\n/s32/s88/s65/s83/s32/s109/s70/s67\n/s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s112/s70/s67\n/s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s90/s70/s67\n/s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s109/s70/s67/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s88/s65/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s40/s98/s41/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s88/s77/s67/s68/s32/s40/s37/s41\n/s55/s55/s48 /s55/s56/s48 /s55/s57/s48 /s56/s48/s48 /s56/s49/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s67/s111/s32/s76/s51/s47/s50/s32/s101/s100/s103/s101/s32/s64/s32/s51/s75/s44/s32/s84/s70/s89/s32\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s32/s105/s110/s99/s105/s100/s101/s110/s99/s101/s44/s32/s66/s32/s61/s32/s54/s46/s56/s84/s32/s88/s65/s83/s32/s112/s70/s67\n/s32/s88/s65/s83/s32/s90/s70/s67\n/s32/s88/s65/s83/s32/s109/s70/s67\n/s32/s88/s77/s67/s68/s32/s112/s70/s67\n/s32/s88/s77/s67/s68/s32/s90/s70/s67\n/s32/s88/s77/s67/s68/s32/s109/s70/s67\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s88/s65/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s45/s48/s46/s48/s54/s45/s48/s46/s48/s52/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54\n/s88/s77/s67/s68/s32/s40/s37/s41FIG. 2: In (a) the XMCD at the Ni L 3/2edges after pFC,\nmFC and ZFC for 60 % Co:ZnO/Py are shown. (b) shows\nthe same for the Co L 3/2edges.\nThe Co L 3/2edges in Fig. 2(b) are also greatly af-\nfected by the self absorption of the total \ruorescence\nyield, since it is buried below 10 nm of Py and 5 nm of\nAl. In contrast to Ni the XAS and XMCD at the Co\nL3/2edges (Fig. 2(b)) are not metallic and evidence the\nincorporation of Co as Co2+in the wurtzite structure\nof ZnO [32, 36]. The overall intensity of the Co XMCD\nis strongly reduced indicating a small magnetic moment\nper Co atom well below metallic Co. This small e\u000bective\nCo moment in 60 % Co:ZnO can be understood by the\ndegree of antiferromagnetic compensation that increases\nwith higher Co doping concentrations [32]. Furthermore,\nno indications of metallic Co precipitates are visible in\nthe XAS and XMCD of the heterostructure as it would\nbe expected for a strong intermixing at the interface to\nthe Py.\nNo changes between the pFC, mFC and ZFC measure-\nments are visible also for the Co edges either in XAS or\nXMCD indicating that the spin system of the Co dopants\nis not altered in the exchange bias state. This corrob-\norates measurements conducted at the Co K-edge [36].\nAfter \feld cooling the XMCD at the Co main absorption\nincreased compared to the ZFC conditions. At the Co\nK-edge the main absorption stems from the orbital mo-\nment. The spin system is only measured indirectly at the\npre-edge feature which remained una\u000bected by the cool-\ning \feld conditions. The data of K- and L-edges com-\nbined evidences that the imprinted magnetization after\n\feld cooling is composed predominantly of orbital mo-4\nment, which is in good agreement with other EB systems\n[42, 43]\nSQUID\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s49/s48/s49\n/s45/s49/s48 /s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s49/s48/s49\n/s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s45/s49/s54/s48/s45/s49/s50/s48/s45/s56/s48/s45/s52/s48/s48/s52/s48/s56/s48/s49/s50/s48/s49/s54/s48/s51/s48/s48/s75\n/s40/s98/s41/s32/s77/s47/s77/s91/s49/s48/s109/s84/s93\n/s48/s72/s32/s40/s109/s84/s41/s32/s80/s121/s32\n/s32/s51/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s53/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121/s40/s97/s41\n/s32\n/s32/s51/s48/s48/s75\n/s32/s50/s75/s77/s32/s40 /s101/s109/s117/s41\n/s48/s72/s32/s40/s109/s84/s41/s32/s32/s109/s70/s67\n/s32/s32/s112/s70/s67\n/s32/s32/s90/s70/s67/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s50/s75/s32/s97/s102/s116/s101/s114/s58\nFIG. 3: At 300 K the M(H) curves of the single Py \flm al-\nmost overlaps with the M(H) curves of the heterostructures\nwith all three Co:ZnO concentrations (a). In the inset it can\nbe seen that there is no di\u000berence in coercive \feld for Py at\n300 K and 2 K. Measuring the 60 % Co:ZnO/Py heterostruc-\nture after plus, minus and zero \feld cooling, horizontal and\nvertical exchange bias shifts are visible, as well as an increase\nin the coercive \feld (b).\nThe static coupling in the heterostructures was investi-\ngated by integral SQUID magnetometry. Measurements\ndone at 300 K, as shown in Fig. 3(a), do not reveal a sig-\nni\fcant in\ruence of the Co:ZnO on the M(H) curve of\nPy. Just a slight increase in coercive \feld from 0.1 mT\nto 0.4 mT is determined. Some of the M(H) curves\nin Fig. 3(a) are more rounded than the others. This\ncan be attributed to slight variations in the aspect ra-\ntio of the SQUID pieces and thus variations in the shape\nanisotropy. The inset of Fig. 3(a) shows the hysteresis of\nthe single Py \flm at 300 K and 2 K, where no di\u000berence\nin coercivity is visible. Please note that up to now mea-\nsurements were conducted only in a \feld range of \u000610 mT\nand directly after a magnet reset. This is done to avoid\nin\ruences of the o\u000bset \feld of the SQUID [38]. At lowtemperatures, to determine the full in\ruence of Co:ZnO,\nhigh \felds need to be applied, as it has been shown in [35].\nTherefore, coercive \felds obtained from low temperature\nmeasurements are corrected by the known o\u000bset \feld of\n1.5 mT of the SQUID [38].\nSince the paramagnetic signal of Co:ZnO is close to the\ndetection limit of the SQUID and thus, orders of mag-\nnitude lower than the Py signal it has no in\ruence on\nthe room temperature M(H) curve. However, with an\nadditional Co:ZnO layer a broadening of the hysteresis,\na horizontal and a small vertical shift are measured at\n2 K as can be seen exemplary for 60 % Co:ZnO/Py in\nFig. 3(b). Similar to single Co:ZnO \flms where an open-\ning of theM(H) curve is already visible in ZFC mea-\nsurements [32, 34{36] also in the heterostructure no \feld\ncooling is needed to increase the coercive \feld.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50\n/s32/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41\n/s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41\n/s40/s98/s41\n/s32/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s54/s48/s37 /s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s97/s102/s116/s101/s114/s32/s90/s70/s67/s40/s97/s41\n/s51/s48 /s52/s48 /s53/s48 /s54/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s32/s112/s70/s67\n/s32/s109/s70/s67\n/s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41\n/s45/s49/s50/s45/s49/s48/s45/s56/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s32/s112/s70/s67\n/s32/s109/s70/s67/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s58\n/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s109/s84/s41/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s58\nFIG. 4: (a) At 2 K the coercivity increases with Co concen-\ntration in the heterostructure. In the inset the temperature\ndependence of the coercivity of the 60 % Co:ZnO/Py het-\nerostructure is given. (b) The vertical shift (circles) and the\nhorizontal shift (squares) depend on the Co concentration.\nBoth shifts reverse the direction when the measurement is\nchanged from pFC to mFC.\nEarlier works [32, 34] demonstrated that the hystere-\nsis opening and vertical shift in Co:ZnO are strongly de-\npendent on the Co concentration and increase with in-\ncreasing Co doping level. Furthermore, the EB e\u000bects\nare observed in the in-plane and out-of-plane direction,\nwith a greater vertical shift in the plane. Therefore,\nthe heterostructers with Py are measured with the mag-\nnetic \feld in in-plane direction. Figure 4(a) provides an\noverview of the coercive \feld after ZFC for the di\u000ber-5\nent Co concentrations. The coercive \feld increases from\n0.1 mT for single Py to 20.6 mT for 60 % Co:ZnO/Py.\nAdditionally, in the inset the temperature dependence of\nthe coercive \feld of the 60 % Co:ZnO/Py heterostructure\nis shown, since it shows the strongest increase in coercive\n\feld. From the 20.6 mT at 2 K it \frst increases slightly\nwhen warming up to 5 K. That the maximum coercivity\nis not at 2 K is in good agreement with measurements at\nsingle 60 % Co:ZnO \flms where a maximum hysteresis\nopening at 7 K was determined [35]. Afterwards the co-\nercive \feld decreases. At the N\u0013 eel temperature of 20 K a\ncoercive \feld of 11.6 mT is measured. Above T Nit de-\ncreases even further but the coercivity is still 3.65 mT at\n50 K. A coupling above T Ncould stem from long range\nmagnetic ordered structures in Co:ZnO where \frst in-\ndications are visible already in single Co:ZnO \flms [32].\nHowever, for single layers they are barely detectable with\nthe SQUID.\nThe vertical (circles) and horizontal (squares) hystere-\nsis shifts after pFC and mFC are shown in Fig. 4(b) for\nthe Py samples with Co:ZnO layers. Similar to single\nCo:ZnO \flms the vertical shift increases with rising Co\nconcentration. The shift is given in percent of the magne-\ntization at 5 T to compensate for di\u000berent sample sizes.\nDue to the overall higher magnetization at 5 T in combi-\nnation with Py this percentage for the heterostructures\nis lower than the vertical shift for single Co:ZnO \flms.\nWith increasing Co concentration the degree of antiferro-\nmagnetic compensation increases [32, 35], which in turn\nshould lead to a stronger EB coupling. This can be\nseen in the horizontal shift and thus EB \feld which is\nstrongest for 60 % Co:ZnO/Py and nearly gone for 30 %\nCo:ZnO/Py. For both kinds of shift the pFC and mFC\nmeasurements behave similar, except the change of di-\nrection of the shifts.\nMultifrequency FMR\nThe dynamic coupling between the two layers has been\ninvestigated by multifrequency FMR measured at room\ntemperature. The frequency dependence of the resonance\nposition between 3 GHz and 10 GHz of the heterostruc-\ntures is shown in Fig. 5(a). The resonance position of Py\nyields no change regardless of the Co concentration in\nthe Co:ZnO layer or its complete absence. Also in 2 nm\nAl/Py and 60 % Co:ZnO/2 nm Al/Py the resonance po-\nsition stays unchanged. The resonance position of a thin\n\flm is given by Kittel formula [44]:\nf=\r\n2\u0019p\nBres(Bres+\u00160M) (1)\nwith the gyromagnetic ratio \r=g\u0016B\n\u0016hand magnetiza-\ntionM. However, any additional anisotropy adds to Bres\nand therefore alters eq. (1) [44]. The fact that all samples\nshow the identical frequency dependence of the resonance\nposition evidences that neither the gyromagnetic ratio \rand thus the Py g-factor are in\ruenced nor any addi-\ntional anisotropy BAniso is introduced by the Co:ZnO.\nBy \ftting the frequency dependence of the resonance po-\nsition using the Kittel equation with the g-factor of 2.11\n[45] all the samples are in the range of (700 \u000615) kA/m,\nwhich within error bars is in good agreement with the\nsaturation magnetization of (670 \u000650) kA/m determined\nfrom SQUID.\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48/s50/s52/s54/s56/s49/s48\n/s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s48/s46/s48/s49/s48/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s45/s49/s48/s49\n/s40/s98/s41\n/s32/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s66\n/s114/s101/s115/s32/s40/s109/s84/s41/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121 \n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 \n/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 \n/s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 \n/s32/s65/s108/s47/s80/s121 \n/s32/s80/s121 /s40/s97/s41\n/s32/s32/s66\n/s112/s112/s32/s40/s109/s84/s41\n/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s32/s32\n/s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41\n/s32/s32/s110/s111/s114/s109/s46/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s66/s32/s40/s109/s84/s41/s32/s53/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s32/s32/s32/s32/s102/s114/s101/s113/s46/s32/s61/s32/s54/s46/s53/s56/s71/s72/s122\n/s32/s76/s111/s114/s101/s110/s116/s105/s97/s110/s32/s102/s105/s116\nFIG. 5: The resonance \felds determined at room temperature\nwith multifrequency FMR are seen in (a). In the inset an ex-\nemplary FMR spectrum for of 50 % Co:ZnO/Py at 6.58 GHz is\nshown with the corresponding Lorentian \ft. For the linewidth\n(b) and the associated damping parameter \u000b(inset) an in-\ncrease is visible for the heterostructures with higher Co con-\ncentration in the Co:ZnO. The lines are linear \fts to the data.\nEven though the Co:ZnO layer does not in\ruence the\nresonance position of the FMR measurement the het-\nerostructures exhibit an increase in linewidth. This cor-\nresponds to a change of the damping in the system. The\nfrequency dependence of the linewidth can be used to sep-\narate the inhomogeneous from the homogeneous (Gilbert\nlike) contributions, from which the Gilbert damping pa-\nrameter\u000bcan be determined.\n\u0001B= \u0001Bhom+ \u0001Binhom (2)6\nwhere\n\u0001Bhom=4\u0019\u000b\n\rf (3)\nNo di\u000berence in linewidth between Al/Py (open stars)\nand Py (full stars) is found, as can be seen in Fig. 4(b)\nwhere the peak to peak linewidth B ppis plotted over the\nmeasured frequency range for all the heterostructures.\nWhile the heterostructure with 30 % Co:ZnO/Py (green\ntriangles) lies atop the single Py and the Al/Py \flm,\nthe linewidth increases stronger with frequency for 50 %\nCo:ZnO/Py (blue circles). The broadest FMR lines are\nmeasured for the 60 % Co:ZnO/Py heterostructure (red\nsqaures).\nUsing the Py g-factor of 2.11 [45], \u000bcan be calcu-\nlated from the slopes of the frequency dependence ex-\ntracted from the linewidths seen in Fig. 5(b): the result-\ning\u000bare shown in the inset. For the single Py layer \u000bPy\n= (5.7\u00060.3)\u000210-3which compares well to previously re-\nported values [7]. This increases to \u000b50= (8.0\u00060.3)\u000210-3\nfor 50 % Co:ZnO/Py and even \u000b60= (9.4\u00060.3)\u000210-3for\n60 % Co:ZnO/Py. So the damping increases by a factor\nof 1.64 resulting in a spin pumping contribution \u0001 \u000b=\n(3.7\u00060.5)\u000210-3that stems from the angular momentum\ntransfer at the interface of Py and Co:ZnO. By insertion\nof a 2 nm Al spacer layer \u0001 \u000breduces to (0.8\u00060.5)\u000210-3.\nDependence on the Al spacer thickness\nTo obtain information about the lengthscale of the\nstatic and dynamic coupling, heterostructures with Al\nspacer layers of di\u000berent thickness (1 nm, 1.5 nm and\n2 nm thick) between Py and the material beneath (sap-\nphire substrate or 60 % Co:ZnO) were fabricated. With-\nout a Co:ZnO layer the spacer underlying the Py layer\ndoes not exhibit any changes in either SQUID (not\nshown) or FMR (see Fig 5 (a) and (b)). The results ob-\ntained for the 60 % Co:ZnO/Al/Py heterostructure for\nthe coercive \feld, vertical and horizontal shift extracted\nfromM(H) curves are shown in Fig. 6(a), whereas the\ndamping parameter \u000bfrom room temperature multifre-\nquency FMR measurements, analogues to Fig. 5(b), are\ndepicted in Fig. 6(b).\nThe horizontal shift and the increased coercive \feld\nare caused by the coupling of FM and AFM moments in\nrange of a few \u0017Angstrom to the interface [46{48]. There-\nfore, both e\u000bects show a similar decrease by the insertion\nof an Al spacer. While the horizontal shift and coer-\ncive \feld are reduced signi\fcantly already at a spacer\nthickness of 1 nm, the vertical shift (inset of Fig. 6(a))\nis nearly independent of the Al spacer. Comparing with\nthe XMCD spectra of Fig. 2 it can be concluded that\nthe vertical shift in the uncompensated AFM/FM sys-\ntem Co:ZnO/Py stems solely from the increased orbital\nmoment of pinned uncompensated moments in Co:ZnO\nand is independent of the FM moments at the interface.Furthermore, the FM moments do not exhibit any ver-\ntical shift and the exchange between the two layers only\nresults in the horizontal shift.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s48/s46/s48/s49/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100\n/s32/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s109/s84/s41\n/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41/s32/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116\n/s32/s32/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41\n/s115/s112/s97/s99/s101/s114/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s40/s98/s41\n/s32/s32\n/s115/s112/s97/s99/s101/s114/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s65/s108/s32/s115/s112/s97/s99/s101/s114/s47/s80/s121/s40/s97/s41\n/s65/s108/s47/s80/s121\nFIG. 6: When an Al spacer is inserted between the Py and\nthe Co:ZnO layer horizontal shift and coercive \feld show a\nstrong decrease already at 1 nm spacer thickness (a) while the\nvertical shift (inset) is not dependent on the spacer thickness.\n(b) shows the e\u000bect of the Al spacer on the Gilbert damping\nparameter\u000b, which also decreases if the spacer gets thicker\nthan 1 nm. As shaded region the Gilbert damping parameter\nof a Al/Py \flm is indicated within error bars.\nFor the FMR measurments after inserting an Al spacer\nno e\u000bect on the resonance position is found, as was shown\nalready in Fig. 5(a). For a 1 nm thick Al spacer the damp-\ning results in \u000b= (8.8\u00060.3)\u000210-3, which gives a \u0001 \u000b=\n(3.1\u00060.5)\u000210-3. This is only a slight decrease compared\nto the sample without Al spacer. By increasing the spacer\nthickness\u000breduces to values just above the damping ob-\ntained for pure Py or Al/Py, shown as shaded region in\nFig. 6(b). The 1 nm thick Al layer is thick enough to sup-\npress intermixing between the Co:ZnO and the Py layer\nas can be seen in Fig. 1(b). Together with the unchanged\nbehavior of Al/Py without Co:ZnO damping e\u000bects due\nto intermixing between Al and Py can be excluded. Also,\na change in two magnon scattering can be ruled out, since\nit would account for non-linear e\u000bects on the linewidth\nand contribute to \u0001 Binhom [49]. Therefore, the increase\nin Gilbert damping can be attributed to a dynamic cou-\npling, e.g. spin pumping from Py into Co:ZnO. Further-\nmore, the dynamic coupling mechanism is extends over a\nlonger range than the static coupling. With 1 nm spacer\nthe dynamic coupling is only slightly reduced whereas\nthe static coupling is already completely suppressed.7\nTemperature dependent FMR\nIn vicinity to the magnetic phase transition temper-\nature the spin pumping e\u000eciency should be at a max-\nimum [29, 31]. Therefore, the samples are measured\ninside a resonator based FMR setup, as a function of\ntemperature. During the cooldown no magnetic \feld is\napplied and the results shown in Fig. 7 are ZFC mea-\nsurements. For 50 % Co:ZnO/Py the resonance posi-\ntions shifts of Py to lower magnetic \felds as the tem-\nperature decreases as can be seen in Fig. 7(a). Not only\nthe resonance position is shifting, but also the linewidth\nis changing with temperature as shown in Fig. 7(b). The\nlinewidth has a maximum at a temperature of 15 K which\ncorresponds well to T Ndetermined by M(T) SQUID\nmeasurements for a 50 % Co:ZnO layer [32]. This max-\nium of the linewidth in the vicinity of T Nis also ob-\nserved for 60 % Co:ZnO/Py and even 30 % Co:ZnO/Py,\nas shown in Fig. 7(c). The measured maximum of 30 %\nCo:ZnO/Py and 60 % Co:ZnO/Py are at 10.7 K, 19.7 K\nrespectively and are marked with an open symbol in\nFig. 7(c). For comparison the N\u0013 eel temperatures de-\ntermined from M(T) measurements [32] are plotted as\ndashed line. Py on the other hand shows only a slight\nincrease in linewidth with decreasing temperature. The\nobserved e\u000bects at low temperatures vanish for the 60 %\nCo:ZnO/2 nm Al/Py heterostructure.\nFigure 7(d) shows the temperature dependence of the\nresonance \feld for all samples. For Py Bresonly decreases\nslightly whereas for 50 % and 60 % Co:ZnO a strong shift\nofBrescan be observed. This shift evidences a magnetic\ncoupling between the Py and the Co:ZnO layer. Even\nin the heterostructure with 30 % Co:ZnO/Py a clear de-\ncrease in resonance position below 10 K (the previously\ndetermined T N[32]) is visible. This shift of the resonance\nposition is only observed at low temperatures. At room\ntemperature no shift of the resonance position at 9.5 GHz\nhas been observed as shown in Fig. 5(a). From the low-\ntemperature behavior of the single Py layer and eq. 1 it\nis obvious that the gyromagnetic ratio is not changing\nstrongly with temperature, therefore shift of the reso-\nnance position in the heterostructure can be attributed\nto a change in anisotropy. From the SQUID measure-\nments at 2 K, see Fig. 3(b) and Fig. 4(b) EB between the\ntwo layers has been determined, which acts as additional\nanisotropy [20] and therefore causes the shift of the reso-\nnance position. Both the shift of the resonance position\nand the maximum in FMR linewidth vanish if the Py is\nseparated from 60 % Co:ZnO by a 2 nm Al spacer layer.\nSo, also at low temperatures the static EB coupling and\nthe dynamic coupling can be suppressed by an Al spacer\nlayer.\nM(T) measurements indicated a more robust long-\nrange magnetic order in 60 % Co:ZnO by a weak sepa-\nration of the \feld heated and ZFC curves lasting up to\n200 K [32]. Additionally, the coercive \feld measurements\non the 60 % Co:ZnO/Py hetersotructure revealed a weak\ncoupling above T N. However, this has not been observedfor lower Co concentrations. In the heterostructure with\n30 % Co:ZnO the FMR resonance position and linewidth\nreturn quickly to the room temperature value for temper-\natures above the T Nof 10 K. For both 50 % Co:ZnO/Py\nand 60 % Co:ZnO/Py the resonance positions are still de-\ncreased and the linewidths are increased above their re-\nspective N\u0013 eel temperatures and are only slowly approach-\ning the room temperature value. In the 60 % Co:ZnO/Py\nheterostructure measurements between 100 K and 200 K\nrevealed that a reduced EB is still present. It is known for\nthe blocking temperatures of superparamagnetic struc-\ntures that in FMR a higher blocking temperature com-\npared to SQUID is obtained due to much shorter probing\ntimes in FMR of the order of nanoseconds compared to\nseconds in SQUID [50]. Hence, large dopant con\fgura-\ntions in Co:ZnO still appear to be blocked blocked on\ntimescales of the FMR whereas they already appear un-\nblocked on timescales of the SQUID measurements.\nV. Conclusion\nThe static and dynamic magnetic coupling of Co:ZnO,\nwhich is weakly paramagnetic at room temperature and\nan uncompensated AFM at low temperatures, with ferro-\nmagnetic Py was investigated by means of SQUID mag-\nnetometry and FMR. At room temperature no static in-\nteraction is observed in the M(H) curves. After cooling\nto 2 K an EB between the two layers is found resulting\nin an increase of coercive \feld and a horizontal shift.\nAdditionally, a vertical shift is present caused by the un-\ncompensated moments in the Co:ZnO. While this vertical\nshift is nearly una\u000bected by the insertion of an Al spacer\nlayer between Co:ZnO and Py the EB vanishes already\nat a spacer thickness of 1 nm.\nThe FMR measurements at room temperature re-\nveal an increase of the Gilbert damping parameter for\n50 % Co:ZnO/Py and 60 % Co:ZnO/Py, whereas 30 %\nCo:ZnO/Py is in the range of an individual Py \flm. At\nroom temperature the resonance position is not a\u000bected\nfor all the heterostructures. For the 60 % Co doped sam-\nple \u0001\u000b= 3.7\u000210-3, which is equivalent to an increase\nby a factor of 1.64. In contrast to the static magnetic\ncoupling e\u000bects, an increased linewidth is still observed\nin the heterostructure containing a 1 nm Al spacer layer.\nAt lower temperatures the resonance position shifts\nof the heterostructures to lower resonance \felds, due to\nthe additional EB anisotropy. The temperature depen-\ndence of the linewidth shows a maximum at tempera-\ntures, which by comparison with M(T) measurements\ncorrespond well to T Nof single Co:ZnO layers and thus\ncorroborate the increase of the damping parameter and\nthus spin pumping e\u000eciency in vicinity to the magnetic\nphase transition. Furthermore, the shift of the resonance\nposition has been observed at temperatures well above\nTNfor 50 % Co:ZnO/Py and 60 % Co:ZnO/Py. Up to\nnow only indications for a long range AFM order in 60 %\nCo:ZnO/Py had been found by static M(T) measure-8\n/s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s40/s100/s41/s40/s98/s41\n/s40/s99/s41\n/s32/s32/s110/s111/s114/s109/s46/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s66/s32/s40/s109/s84/s41/s32/s84/s32/s61/s32/s32/s32/s52/s46/s48/s75\n/s32/s84/s32/s61/s32/s49/s52/s46/s57/s75\n/s32/s84/s32/s61/s32/s51/s49/s46/s52/s75\n/s32/s84/s32/s61/s32/s53/s48/s46/s50/s75/s40/s97/s41\n/s32/s32/s66\n/s112/s112/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s84\n/s78/s32/s100/s101/s116/s101/s114/s109/s105/s110/s101/s100/s32\n/s102/s114/s111/s109/s32/s77/s40/s84/s41/s32/s83/s81/s85/s73/s68/s32/s91/s51/s50/s93\n/s32/s32/s66\n/s114/s101/s115/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s80/s121\n/s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121/s84\n/s78/s32/s54/s48/s37/s84\n/s78/s32/s53/s48/s37/s84\n/s78/s32/s51/s48/s37\n/s32/s32/s66\n/s112/s112/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s80/s121\n/s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121\nFIG. 7: By decreasing the temperature the resonance position of 50 % Co:ZnO/Py shifts to lower resonance \felds (a) and\nthe linewidth increases, showing a maxium at the T N(b). A similar behavior is observed for the heterostructures with 30 %\nand 60 % Co doping while a single Py \flm does not exhibit a maximum when cooling (c). The maximum is marked as open\nsymbol in the temperature dependence, while the T Ndetermined from M(T) [32] are shown as dashed lines. Furthermore, the\nresonance position of the heterostructures with Co:ZnO shifts at low temperatures (d).\nments. The dynamic coupling, however, is sensitive to\nthose interactions due to the higher time resolution in\nFMR resulting in a shift of the resonance position above\nthe T Ndetermined from M(T) SQUID.\nAcknowledgment\nThe authors gratefully acknowledge funding by the\nAustrian Science Fund (FWF) - Project No. P26164-N20 and Project No. ORD49-VO. All the mea-\nsured raw data can be found in the repository at\nhttp://doi.org/10.17616/R3C78N. 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Lett.,\n70 (2), 250-256 (2005)."    },    {        "title": "1909.05315v2.Chaos_in_nanomagnet_via_feedback_current.pdf",        "content": "arXiv:1909.05315v2  [cond-mat.mes-hall]  23 Nov 2019Chaos in nanomagnet via feedback current\nTomohiro Taniguchi1, Nozomi Akashi2, Hirofumi Notsu3,4,\nMasato Kimura3, Hiroshi Tsukahara5, and Kohei Nakajima2\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan,\n2Graduate School of Information Science and Technology,\nThe University of Tokyo, Bunkyo-ku, 113-8656 Tokyo, Japan,\n3Faculty of Mathematics and Physics, Kanazawa University, K anazawa, Ishikawa 920-1164, Japan,\n4JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Ja pan,\n5High Energy Accelerator Research Organization (KEK), Tsuk uba, Ibaraki 305-0801, Japan\n(Dated: November 26, 2019)\nNonlinear magnetization dynamics excited by spin-transfe r effect with feedback current is studied\nbothnumericallyandanalytically. Thenumericalsimulati onoftheLandau-Lifshitz-Gilbertequation\nindicates the positive Lyapunov exponent for a certain rang e of the feedback rate, which identifies\nthe existence of chaos in a nanostructured ferromagnet. Tra nsient behavior from chaotic to steady\noscillation is also observed in another range of the feedbac k parameter. An analytical theory is\nalso developed, which indicates the appearance of multiple attractors in a phase space due to the\nfeedback current. An instantaneous imbalance between the s pin-transfer torque and damping torque\ncauses a transition between the attractors, and results in t he complex dynamics.\nPACS numbers:\nI. INTRODUCTION\nNonlinear dynamics can be found in a wide variety of\nphysical, chemical, and biological systems from small to\nlarge scale [1,2]. Recent observations of rich magnetiza-\ntion dynamics, such as switching, auto-oscillation (limit\ncycle), and synchronization, excited in a nanostructured\nferromagnet have also proved the applicability of nonlin-\near science to a fine structure [3–12]. These dynamics are\ndriven by spin current carried by, for example, conduct-\ning electrons in metals [13–15]. Since the spin current\nin metals can survive only within nanometer scale [16],\nthese magnetization dynamics had not been observedun-\ntil the development of fabrication technology of nanos-\ntructure was achieved. A new direction investigating the\napplicability of such nonlinear magnetization dynamics\nto non-von Neumann computing scheme, inspired by hu-\nman brain, has been growing very recently [17–20].\nAn attractive and intriguing phenomenon in nonlinear\nscience is chaos [21,22]. It should be noticed here that\nthe previous works in magnetism and spintronics have\nclarified that the magnetization dynamics in a nanos-\ntructured ferromagnet is sufficiently sufficiently well de-\nscribed by two dynamical variables [23–28]. For exam-\nple, the macrospin model has two dynamical variables\ndescribing the zenith and azimuth angles of the mag-\nnetization. The Thiele equation depicting the magnetic\nvortex or skyrmion dynamics includes two variables cor-\nresponding to the radius and phase of the core, whereas\nthe domain wall motion is represented by the center\nof the wall position and the tilted angle of the magne-\ntization at the center. On the other hand, according\nto the Poincar´ e-Bendixson theorem, chaos is prohibited\nin a two-dimensional system [21]. Therefore, an addi-\ntional degree of freedom is necessary to induce chaosin ferromagnets. In previous works, chaos has been\nstudied for systems with alternative current [29,30] or\nmagnetic and/or electric interaction between two ferro-\nmagnets [31,32]. The former makes the system nonau-\ntonomous, whereas the latter utilizes many-body system.\nAnother possible sourcecausing highly nonlinear dynam-\nics is feedback force with delay because the presence of\nthe delay makes the dimension of the system infinite [33].\nRecently, the modulation of the threshold current by the\nself-injection of the feedback current into the vortex fer-\nromagnet was theoretically predicted [34] and was ex-\nperimentally confirmed [35]. Complex dynamics in an\nin-plane magnetized ferromagnet with feedback current\nwas also found by numerical simulation [36]. However,\nit should be emphasized that the existence of the feed-\nback effect does not necessarily guarantee chaos. There-\nfore, a careful analysis is necessary for the magnetization\ndynamics in the presence of feedback effect in order to\nidentify chaos.\nThe purpose of this work is to develop a theoretical\nanalysis of the nonlinear magnetization dynamics in a\nnanostructured ferromagnet in the presence of feedback\ncurrent. We perform the numerical simulation of the\nLandau-Lifshitz-Gilbert (LLG) equation in spin torque\noscillator (STO), and find that the feedback current\ncauses highly nonlinear dynamics of the magnetization.\nThis work identifies chaos by the positive Lyapunov ex-\nponent, which is found in a certain range of the feedback\nrate, whereas transient behavior is also observed in an-\nother range of the feedback rate. We also develop an\nanalytical theory to reveal the origin of such complex\ndynamics. The bifurcation analysis indicates that the\nfeedback current results in the appearance ofmultiple at-\ntractors in the phase space. An instantaneous imbalance\nbetween the spin-transfer torque and damping torque al-\nlows a transition between these attractors, and induces2\npm\nIz\nxχIm.p\ntime (ns)0 20 40 60 80 1001.0\n0.5\n-0.5\n-1.00mx, mz\nmxmz\n1ns01\n-1(a) (b)\nFIG. 1: (a) Schematic view of the system. The direct current\nIis injected from the reference layer to the positive layer,\nwhereas the current, χIm·p, outputted from the STO is\ninjectedintotheSTOwithtimedelay τ. Thefeedbackcurrent\noscillates when the magnetization min the free layer is in a\ndynamical state. (b) Typical magnetization dynamics in the\nabsence of the feedback current. The inset shows an auto-\noscillation in a steady state.\nthe complex dynamics found in the numerical analysis.\nThe paper is organized as follow. In Sec. II, we de-\nscribe the structure of the STO and show the LLG equa-\ntion including feedback current. In Sec. III, the results\nof the numerical simulation of the LLG equation are pre-\nsented. The Lyapunov exponents and bifurcation dia-\ngrams as functions of the feedback rate and delay time\nare also presented. In Sec. IV, a theoretical analysis on a\nmultiple attractor is discussed. Section IVA summarizes\nthis work.\nII. SYSTEM DESCRIPTION\nIn this section, we describe the system under consider-\nation, and provide the comment on the numerical meth-\nods. The details of the algorithms are also given in the\nSupplemental Material [37] (which includes Ref. [38]).\nA. LLG equation\nThesystemunderconsiderationisschematicallyshown\nin Fig. 1(a). The unit vectors pointing in the magnetiza-\ntion directions in free and reference layers are denoted as\nmandp, respectively. Direct current, I, is injected from\nthe reference to free layer, and excites an auto-oscillation\nof the magnetization mvia spin-transfer effect [13,14].\nHere, we focus on the STO consisting of a perpendicu-\nlarly magnetized free layer and an in-plane magnetized\nreference layer because this type of STO can emit large\nemissionpowerwith narrowlinewidth [10], andtherefore,\nis of great interest from viewpoints of both fundamental\nand applied physics. The magnetization pin the refer-\nence layer points to the positive xdirection, whereas the\nzaxis is perpendicular to the film-plane. The magne-\ntization dynamics in the free layer is described by the\nLandau-Lifshitz-Gilbert (LLG) equation given by\ndm\ndt=−γm×H−γHsm×(p×m)+αm×dm\ndt(1)whereγandαare the gyromagnetic ratio and the\nGilbert damping constant respectively. The magnetic\nfieldH= [Happl+ (HK−4πM)mz]ezconsists of an\napplied field Happl, interfacial magnetic anisotropy field\nHK[39–41], and demagnetization field −4πM. The spin-\ntransfer torque strength, Hsis given by\nHs=/planckover2pi1ηI[1+χm(t−τ)·p]\n2e(1+λm·p)MV, (2)\nwhereMandVarethesaturationmagnetizationandthe\nvolume of the free layer, respectively. The spin-transfer\ntorque strength is characterized by the spin polarization\nηand spin-transfer torque asymmetry λ. The values of\nthe parameters used in this work are derived from the\nexperiment [10], as well as a theoretical analysis [42] as\nM= 1448.3 emu/c.c., HK= 18.616 kOe, Happl= 2.0\nkOe,V=π×602×2 nm3,η= 0.537,λ= 0.288,\nγ= 1.764×107rad/(Oe s), and α= 0.005. The cur-\nrent ofI= 1.0 mA corresponds to the current density of\n8.8 MA/cm2. An auto-oscillation in the absence of the\nfeedback is excited in this type of STO when the cur-\nrent magnitude becomes larger than a threshold value\n[42] (see also Appendix A for derivation),\nIc=4αeMV\n/planckover2pi1ηλ(Happl+HK−4πM),(3)\nwhich is about 1 .6 mA for the present parameters. Fig-\nure 1(b) shows a typical magnetization dynamics in the\nabsence of the feedback current, where the direct current\nisI= 2.5 mA. As shown, an auto-oscillation having a\nperiod of 0 .16 ns is excited after a relaxation time on the\norderof10ns. The inset ofFig. 1(b) showsthe dynamics\nofmx(red) and mz(black) in a steady state. It can be\nseen from the figure that mzis almost temporally con-\nstantbut slightlyoscillatesarounda certainvalue. These\nresults will be used for comparison with the dynamics in\nthe presence of the feedback current, as well as for the\ndevelopment of an analytical theory, below.\nB. Description of feedback effect\nThe strength of the spin-transfer torque, Eq. (2), in-\ncludesthe feedback currentgivenby χIm(t−τ)·p, where\nχis the rate of the feedback current with respect to\nthe direct current I, whereas τis the delay time. Due\nto tunnel magnetoresistance effect, the feedback current\ndepends on the relative direction of the magnetizations,\nm·p[10]. The feedback current brings the past infor-\nmation of the magnetization state, and extends the di-\nmension of the phase space, which presents a possibility\nto excite chaos in STO.\nLet us give brief comments on experiment to measure\nchaos in an STO. An experimental work injecting the\nfeedback current to a vortex STO and measuring the\noutput power was already reported [35]. The feedback\ncurrent can be injected to the STO independently from3\nthe direct currentby usinga bias-Teeanddelayline. The\nnumerical analyses shown below, as well as the analytical\ntheory developed in Sec. IV, predict that chaos appears\nfor a large feedback rate χand/or long delay time τcom-\nparedtotypicaltime scalesofthe STO.The typicalvalue\nofthe delay time possible in experiment is on the orderof\n10 ns [35]. On the other hand, the oscillation period ( ∼3\nns) of the vortex STO used in the previous work [35] is\nonly 10 times shorter than the delay time. This might be\nthe reason why chaos was not confirmed in the previous\nworks. Regarding this point, two approaches are taken\ninto account to observe chaos in STO. The first one is to\nmake the delay time long. A long delay time is realized\nby using a long electric cable. The second approach is\nto use an STO having a short oscillation period. In fact,\nthe STO studied in this work has a short period because\nof macrospin structure of the magnetization. Therefore,\nthe theoretical analyses shown below will possibly be ex-\namined experimentally. A possible remaining issue, how-\never, may be an energy loss in a cable, which should be\noptimized in experiments.\nWe also give a comment on the method to identify\nchaos by experiments. The experimental methods to\nidentifychaosare,forexample, theestimationoftheLya-\npunov exponent from time series of data and/or Fourier\nanalysis. The former method requires to measure the dy-\nnamical trajectory in the system, and estimate the Lya-\npunov exponent from a discrete set of time series data\nby evaluating the principal axis of the expansion [43]. A\npossible problem in applying this method to STO is the\nlimitation of the information on the dynamical trajec-\ntory obtained. The magnetization dynamics in the STO\nis measured through the magnetoresistance effect. Both\ngiant and tunnel magnetoresistances are proportional to\nm·p. Therefore, we can measure only the component\nof the magnetization mprojected to the direction of p.\nThis fact might makeit difficult to reproducethe dynam-\nical trajectory and identify chaos from the time series of\ndata. The Fourier analysis, on the other hand, indicates\nchaos from the shape of the spectrum. The Fourier spec-\ntrum shows a sharp peak for a non-chaotic dynamics,\nwhereas it has a broad structure without a unique peak\nin chaos state; see also Sec. IIIA. Therefore, the Fourier\nspectrum provides an evidence to identify chaos.\nC. Numerical method\nHere, let us provide a brief description of the numer-\nical technique used in the next section. The LLG equa-\ntion, Eq. (1), with the feedback current is solved by\na fourth-order Runge-Kutta scheme accompanied with\ncontinuation method. The details of this algorithm are\nsummarized in the Supplemental Material [37]. We also\nevaluate the bifurcation diagram, which is defined as the\nlocalmaximumof mz(t)afterthemagnetizationmovesto\nan attractor. In this work, chaosis defined as the dynam-\nics with the positive Lyapunov exponent. The Lyapunovexponent in this work is defined as an average of the in-\nstantaneous expansion rate of the dynamical trajectory\nin the phase space with respect to a small perturbation\nǫas\n˜λ= lim\nNL→∞1\nNL∆tNL/summationdisplay\ni=1log/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜ǫi\nǫ/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (4)\nwhere ∆ tis the time step of the LLG simulation. The\nnumber of the perturbation applied to the STO is NL,\nwhereas ˜ ǫiis the expansion of the dynamical trajectory\nwith respect to the ith perturbation. The detail of the\nalgorithmtoevaluatetheLyapunovexponentisalsosum-\nmarized in the Supplemental Material [37].\nIII. NUMERICAL ANALYSIS\nIn this section, we show the results of the numerical\nsimulation of the LLG equation, as well as the Lyapunov\nexponent and bifurcation diagram.\nA. Lyapunov exponent as a function of feedback\nrate\nHere, we show the Lyapunov exponent and the bir-\nfurcation diagram as a function of the feedback rate χ.\nThe value of τin this section is set to be 30 ns. Figures\n2(a)-2(c) show the time evolutions of mz(t) forχ= 0.02,\nχ= 0.50, andχ= 0.89, respectively. Note that the time\nrange of each figure is different to understand the char-\nacteristics of each dynamics. In the presence of a small\nfeedback current shown in Fig. 2(a), although the ampli-\ntude of the oscillation is modulated, the dynamics in the\nsteadystateisstillperiodic. Ontheotherhand, whenthe\nfeedback rate becomes relatively large, chaotic behavior\nappears, as shownin Fig. 2(b). In this case, non-periodic\nand highly nonlinear dynamics appears over a wide time\nrange. The value of mzoscillates almost over its possi-\nble value, |mz| ≤1. A further increase of the feedback\nrate leads to a transition of the magnetization dynam-\nics from chaotic to non-chaotic, as shown in Fig. 2(c).\nThe chaotic dynamics suddenly disappears after a com-\nparatively long period, i.e., longer than the oscillation\nperiod of the limit cycle in the absence of the feedback\ncurrent. As mentioned below, the Lyapunov exponents\nof the dynamics in Figs. 2(a) and 2(c) are zero, whereas\nit is positive for the dynamics in Fig. 2(b).\nNote that evaluating the perpendicular component mz\nin time domain is useful for theoretical analysis because\nit is approximately constant in the absence of the feed-\nback effect, whereas it becomes complex by the feedback\nforce, as mentioned above. On the other hand, evalu-\nating the in-plane component mxwill be useful for ex-\nperiments because it directly relates to the experimen-\ntally observed signal through magnetoresistance effect.\nTherefore, we also show the Fourier spectra of mxfor4\ntime (ms)0 0.5 1.0 1.51.0\n0.5\n0(c)mz\nχ=0.89\ntime (μs)0 0.5 1.0 1.51.0\n0.5\n0(b)mz\nχ=0.50\ntime (ns)0 200 100 300 400 500 6001.0\n0.5\n0(a)\n(d)mz\nχ=0.02|mx(f)| (arb. unit)\n050100150\nfrequency (GHz)6.0 6.1 6.2 6.3 6.4 6.5 6.6(e)\n|mx(f)| (arb. unit)\n050100\nfrequency (GHz)5.6 5.8 6.0 6.2 6.4 6.6(f)\n|mx(f)| (arb. unit)\n0600\n500\n400\n300\n200\n100\nfrequency (GHz)5.6 5.9\nFIG. 2: Time evolutions of the perpendicular component mz(t) for the feedback rates of (a) χ= 0.02, (b) 0 .50, and (c) 0 .89.\nThe current and the delay time are I= 2.5 mA and τ= 30 ns. Note that the time range of each figure is different. Fou rier\nspectra of the in-plane component mx(t) for (d) χ= 0.02, (e) 0 .50, and (f) 0 .89 are also shown.\nχ= 0.02, 0.50, and 0 .89 in Figs. 2(d)-2(f), respectively.\nTheFourierspectrumhasasharppeakwith subpeaksfor\nχ= 0.02, which is a typical spectrum of the oscillation\nwith the amplitude modulation. The Fourier spectrum\nforχ= 0.50, on the other hand, shows a broad structure\nover a relatively wide range of the frequency. A main\npeak is not uniquely determined. The structure implies\nthat the dynamics is chaos. The Fourier spectrum for\nχ= 0.89 shows a sharp peak, corresponding to the os-\ncillation frequency after the transition from chaotic to\nlimit cycle oscillation. The oscillation frequency is dif-\nferent from that in the absence of the feedback because\nthe oscillation amplitude is modified due to the feedback\neffect. Regarding these results, the Fourier analysis will\nbe a possible tool to experimentally identify chaos.\nFigures 3(a) and 3(b) show the Lyapunov exponent\nand the bifurcation diagram as a function of the feed-\nback rate in a small range χ≤0.10. The Lyapunov\nexponent remains zero for χ/lessorsimilar0.024, where the dynam-\nics is a limit cycle, such as shown in Fig. 1(b), or the\noscillation with an amplitude modulation as shown in\nFig. 2(a). In the limit cycle state, the local maximum\nofmzis a single value, whereas it takes several values\nand shows symmetric distributions around its center in\nthe modulated dynamics, as can be seen in Fig. 3(b).\nThe Lyapunov exponent becomes positive for χ/greaterorsimilar0.025,\nwhere the bifurcation diagram shows an inhomogeneous\n(asymmetric) structure. The Lyapunov exponent and\nthe bifurcation diagram for a wide range of the feed-\nback rate, χ≤1.00, are shown in Figs. 3(c) and 3(d),\nrespectively. The positive Lyapunov exponent indicates\nthe existence of chaos in STO. The Lyapunov exponent\nmz\n00.2\n0.10.30.40.50.60.70.80.91.0\n0 0.2 0.1 0.3 0.4 0.5 0.7 0.9 0.6 0.8 1.0 0 0.2 0.1 0.3 0.4 0.5 0.7 0.9 0.6 0.8 1.00.06\n0.05\n0.04\n0.03\n0.02\n0.01\n0\n-0.01(c) (d)Lyapunov exponent (1/ns)\nmz\n0.40.50.60.70.8\nfeedback rate, χ\nfeedback rate, χ0 0.020.01 0.03 0.04 0.05 007 0.09 0.06 0.08 0.1\nfeedback rate, χ\nfeedback rate, χ0 0.020.01 0.03 0.04 0.05 007 0.09 0.06 0.08 0.10\n-0.010.010.020.030.04(a) (b)Lyapunov exponent (1/ns)\nFIG. 3: (a) (Maximum) Lyapunov exponent and (b) bi-\nfurcation cascade (local maximum of mz) as a function of\nthe feedback rate χ≤0.10. The current and delay time are\nI= 2.5 mA and τ= 30 ns, respectively. The range of χis\nextended to χ≤1.00 in (c) and (d).\nbecomes zero again when the feedback rate is further in-\ncreased to χ≃0.87. The magnetization dynamics shown\nin Fig. 2(c), corresponding to this parameter region, can\nbe regarded as transient chaos, which can be found in,\nfor example, a spatially extended turbulence model [44],\nwhere the dynamical system finally arrives at an attrac-\ntor with zero or negative Lyapunov exponent long time\nafter showing chaotic behavior [21]. For example, the\ntransient time observed in Fig. 2(c) is on the order of5\n0.1 ms, which is sufficiently longer than the period of the\nauto-oscillation in the absence of the feedback current\n(0.16 ns) but is measurable because it is shorter than the\nexperimentally available time range for STO dynamics\nreported up to date, 1.6 ms [45].\nB. Lyapunov exponent as a function of delay time\nHere, we show the Lyapunov exponent and the birfur-\ncation diagram as a function of the delay time τ. The\nvalue of χin this section is set to be 0 .20. Figures 4(a)\nand 4(b) show the time evolutions of mzfor short delay\ntimes,τ= 0.03 and 0.3 ns, respectively. For such a suffi-\nciently short delay time, the current necessary to excite\nan auto-oscillation of the magnetization is given by (see\nalso Appendix A for derivation)\n˜Ic=4αeMV\n/planckover2pi1ηλp0(Happl+HK−4πM),(5)\nwherep0=p(χ,τ,θ= 0) is\np0= 1−χ\nλcos2πfFMRτ, (6)\nwherefFMR=γ(Happl+HK−4πM)/(2π) is the ferro-\nmagnetic resonance (FMR) frequency. In the absence of\nthe feedback current ( χ→0), Eq. (5) becomes identical\nto Eq. (3). According to Eqs. (5) and (6), the threshold\ncurrent to move the magnetization from the energetically\nstable state ( θ= 0) is anoscillatingfunction of τ. Forex-\nample,Icgiven by Eq. (5) becomes 1 .9 mA for τ= 0.03\nns, which is smaller than the applied current, I= 2.5\nmA. Therefore, the magnetization can move from the\ninitial state, as shown in Fig. 4(a). On the other hand,\nIcbecomes 4 .4 mA for τ= 0.3 ns, and, therefore, the\nmagnetization stays in the energetically stable state in\nFig. 4(b). Such a modification of the instability thresh-\nold was studied in a vortex oscillator both theoretically\nand experimentally [34,35]. For a sufficiently long delay\ntime, the magnetization dynamics becomes highly com-\nplex, and Eq. (5) does not work. The periodic oscillation\nwith the amplitude modulation is found for τ= 9.3 ns,\nwhereas non-periodic dynamics appears for τ= 9.6 ns,\nas shown in Figs. 4(c) and 4(d).\nFigures 4(e) and 4(f) summarize the Lyapunov expo-\nnent and bifurcation diagram as a function of the delay\ntime, respectively. Note that the magnetization stays in\nthe energetically equilibrium state for 0 .3≤τ <1.2 ns,\nas in the case shown in Fig. 4(b). In such a case, the\nLyapunov exponent is negative, indicating that the mag-\nnetization saturates to a fixed point. On the other hand,\nchaos appears with increasing the delay time, whereas\nthe periodic oscillations with the amplitude modulation\nappear for specific values of τ. The negative Lyapunov\nexponent for a short delay time is approximately esti-\nmated from a linearized LLG equation [46] as\n˜λ≃ −2παfFMR/parenleftbigg\n1−I\n˜Ic/parenrightbigg\n. (7)time ( μs)0 100 50 150 200 250 300\ntime ( μs)0 100 50 150 200 250 3001.0\n0.5\n0(c)mz\nτ=9.3 nsτ=0.03 ns\nτ=9.6 nsτ=0.3 ns\n1.0\n0.5\n0(d)mz mztime (ns)0 0.5 1.0 1.51.0\n0.5\n0(a)mz\ntime (ns)0 40 20 60 80 1001.0\n0.5\n0(b)mz\n00.2\n0.10.30.40.50.60.70.80.91.0\ndelay time, τ (ns)0 5 10 15 20 25 30\ndelay time, τ (ns)0 5 10 15 20 25 300.10\n0.05\n0\n-0.15-0.10-0.05(e) (f)Lyapunov exponent (1/ns)\nFIG. 4: Time evolutions of mz(t) for the delay times of (a)\nτ= 0.03, (b) 0 .3, (c) 9.3 ns, and (d) 9 .6 ns. The current\nand the feedback rate are I= 2.5 mA and χ= 0.20. (e)\nThe Lyapunov exponent and (f) bifurcation cascade (local\nmaximum of mz) as a function of the delay time.\nFor example, for τ= 0.3, Eq. (7) is −0.09 GHz, which\nis close to the numerically estimated value, −0.11 GHz.\nWe simultaneously emphasize that the limit of τ→0\ndoes not correspond to the zero-feedback limit (the zero-\nfeedback limit corresponds to χ→0). Even in the limit\nofτ→0, the feedback current exists and affects the dy-\nnamics. For example, for τ= 0.03, the magnetization\nshows a limit cycle oscillation, and the Lyapunov expo-\nnent is zero. Equation (7) works when the magnetization\nstays at a fixed point, and the delay time τis short.\nIV. THEORETICAL ANALYSIS\nThe above numerical results indicate the existence of\nrich variety of nonlinear dynamics, including chaos, in\nan STO. Although it is difficult to solve the LLG equa-\ntion exactly due to its nonlinearity, let us investigate the\nphysical origin of the complex dynamics with help of an\napproximated theory, which has been known to be useful\nto analyze nonlinear dynamics such as auto-oscillation\n(limit cycle) [28,42] and synchronization [47]. An auto-\noscillation in an STO is excited when the spin-transfer\ntorque balances with the damping torque, and the field\ntorque,−γm×H, remains finite. The field torque leads\ntoasustainableoscillationofthe magnetizationonacon-\nstantenergycurveofthe magneticenergydensitydefined6\nasE=−M/integraltext\ndm·H. In the present system, the con-\nstant energy curve corresponds to the trajectory with\na constant zenith angle θ= cos−1mz, where the oscil-\nlation frequency, f(θ), on the constant energy curve is\nf(θ) =γ[Happl+(HK−4πM)cosθ]/(2π). It should be,\nhowever, emphasized that there is often an instantaneous\nimbalance between the spin-transfer torque and damping\ntorque because of their different angular dependencies.\nTherefore, strictly speaking, θ(ormz) in the present\nsystem is not a constant variable [42]; see also the inset\nof Fig. 1(b). However, for a sufficiently small damp-\ning constant α, the real trajectory of the auto-oscillation\nis practically close to a constant energy curve. In such\na case, it is useful to derive the equation of motion of\nθaveraged over the precession period T(θ) = 1/f(θ)\nasdθ/dt≡(1/T)/contintegraltext\ndt(dθ/dt) (see also Appendix A for\nderivation),\ndθ\ndt=−αγ[Happl+(HK−4πM)cosθ]sinθ\n+γHs0\nλtanθ/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\np(χ,τ,θ),(8)\nwhereHs0=/planckover2pi1ηI/(2eMV), whereas p(χ,τ,θ) is given by\np(χ,τ,θ) = 1−χ\nλcos2πf(θ)τ. (9)\nThe angle θsatisfying dθ/dt= 0 and d(dθ/dt)/dθ <\n(>)0 corresponds to a stable (unstable) fixed point in\nthe reduced phase space [1]. In the absence of feedback\ncurrent, there is only one stable fixed point (attractor),\ncorresponding to auto-oscillation state in real space, in\nthe present STO [42]. On the other hand, Fig. 5(a)\nshows an example of dθ/dtin the presence of the feed-\nback. As shown, several attractors satisfying dθ/dt= 0\nandd(dθ/dt)/dθ <0 appear due to the feedback current.\nFigures 5(b) and 5(c) show the attractors mz= cosθas\na function of the feedback rate χand the delay time τ,\nrespectively. It canbe understoodfromthese figuresthat\nthe number of the attractor increases with increasing the\nfeedback rate and/or delay time. Let us here call such\nstructures as multiple attractors. Although these results\nare obtained with an approximation mentioned above,\nthey are useful to understand the origin of the complex\nmagnetization dynamics found by numerical simulation,\nas discussed below.\nThe multiple attractors originate from the function\np(χ,τ,θ) givenby Eq. (9). In the absenceofthe feedback\ncurrent ( χ= 0), the function p(χ,τ,θ) = 1 is indepen-\ndent of the angle θ. On the other hand, in the presence\nof the feedback current ( χ/negationslash= 0), several values of the\nangleθgive an identical value of p(χ,τ,θ) because the\nfunction includes a periodic (cosine) function depending\nonθ. As a result, several θcan simultaneously satisfy\nthe conditions of the stable fixed point.\nThe origin of the complex dynamics found in the nu-\nmerical simulation is considered to be the existence ofmultipleattractors. Sincetheattractorslocatediscretely,\nas shown in Fig. 5, one might consider that once the\nmagnetization is trapped by one of the attractors, it can-\nnot move to the others. It should be, however, reminded\nthat the assumption of a constant angle θwasused in the\nderivation of Eq. (8). As emphasized above, the real an-\ngleθ= cos−1mzin alimit cycleslightlyoscillatesaround\nthe fixed point estimated analytically by Eq. (8) because\nof the instantaneous imbalance between the spin-transfer\ntorque and damping torque. As a result, the magnetiza-\ntion can move from one attractor to the other when the\ndistance between the attractors is smaller than the oscil-\nlation amplitude of the angle θ. The transition between\nthe attractorscausesthe highlycomplexdynamicsshown\nin Fig. 2, contrary to the system without feedback in\nwhich an auto-oscillation state is uniquely determined.\nIt is considered that the above analytical theory can\nbe applied to any type of STO, although Eq. (8) was\nderived for its specific type. For example, the complex\ndynamics found in an in-plane magnetized STO [36] may\nbe causedby the samemechanism, i.e., the appearanceof\nmultiple attractors due to the existence of feedback cur-\nrent. The periodicity of the multiple attractors in this\ntype of STO is described by elliptic functions in contrast\nwith Eq. (9) where the periodicity is described by a sim-\nple trigonometric function; see Appendix B.\nA. Conclusion\nIn conclusion, the nonlinear magnetization dynamics\nin a spin-torque oscillator was studied by taking into ac-\ncount the effect of spin-transfer torque excited by the\nfeedback current. The numerical simulation reveals rich\nvariety of the nonlinear magnetization dynamics, which\ncan be controlled by the feedback parameter. The posi-\ntive Lyapunov exponent for a certain range of the feed-\nback rate indicated the existence of chaos in the spin-\ntorque oscillator, whereas transient behavior from the\nchaotic to the steady state was also observed in another\nrange of the feedback parameter. The analytical the-\nory based on the averaged equation of motion revealed\nthat the feedback current results in the multiple attrac-\ntors in the phase space. The number of the attractors\nincreased with increasing the feedback rate and/or de-\nlay time. An instantaneous imbalance between the spin-\ntransfer torque and damping torque caused a transition\nbetween the attractors, and induces the complex magne-\ntization dynamics.\nAcknowledgement\nThe authors are thankful to Joo-Von Kim, Take-\nhiko Yorozu, Sumito Tsunegi, and Shinji Miwa for\nvaluable discussion. T. T. is grateful to Satoshi\nIba, Aurelie Spiesser, Hiroki Maehara, and Ai Emura\nfor their support and encouragement. The results7mz\n00.2\n0.10.30.40.50.60.70.80.91.0\nfeedback rate, χ0 0.20.1 0.3 0.4 0.5 0.7 0.9 0.6 0.8 1.0angle, θ (deg)-0.2-0.10.2\n00.1\n0 10 20 30 40 50 60 70 80 90(a) (b)dθ/dt (1/ns)(c)mz\n00.2\n0.10.30.40.50.60.70.80.91.0\ndelay time, τ (ns)0 5 10 15 20 25 30\nFIG. 5: (a) The averaged dθ/dtgiven by Eq. (8) solved in the phase space as a function of θ= cos−1mz. The current,\nfeedback rate, and delay time are I= 2.5 mA,χ= 0.10, and τ= 30 ns. (b), (c) Stable fixed points mz= cosθestimated\nanalytically as a function of (b) the feedback rate χwithτ= 30 ns and (c) the delay time τwithχ= 0.10.\nwere partially obtained from a project (Innovative AI\nChips and Next-Generation Computing Technology De-\nvelopment/(2) Development of next-generation comput-\ning technologies/Exploration of Neuromorphic Dynam-\nics towards Future Symbiotic Society) commissioned by\nNEDO. K. N. is supported by JSPS KAKENHI Grant\nNumbers JP18H05472, and JP16KT0019. H. N. is sup-\nported by JSPS KAKENHI Grant Number JP18H01135,\nand JST PRESTO Grant Number JPMJPR16EA. M.\nK. is supported by JSPS KAKENHI Grant Numbers\nJP16H02155, JP17H02857.\nAppendix A: Averaged LLG equation of\nperpendicularly magnetized STO\nIntroducing the zenith and azimuth angles ( θ,ϕ) as\nm= (sinθcosϕ,sinθsinϕ,cosθ), the LLG equation (1),\nforθis given by\ndθ\ndt=−γ/planckover2pi1ηI[1+χm(t−τ)·p]\n2e(1+λsinθcosϕ)MVcosθcosϕ\n−αγ[Happl+(HK−4πM)cosθ]sinθ,(A1)\nwhere the higher order terms of αare neglected. As\nmentioned in the main text, an auto-oscillationis excited\nwith a trajectory depicting practically on a constant en-\nergy curve of E=−M/integraltext\ndm·H=−MHapplcosθ−\n[M(HK−4πM)/2]cos2θ. The dynamical trajectory\non the constant energy curve, which is the solution of\ndm/dt=−γm×H, is given by mx= sinθcosω(θ)t,\nmy= sinθsinω(θ)t, andmz= cosθ, whereθis constant\nwhereas\nω(θ) =γ[Happl+(HK−4πM)cosθ].(A2)\nThe frequency and period of the auto-oscillation are\nf(θ) =ω(θ)/(2π) andT(θ) = 1/f(θ), respectively. Sub-\nstituting these solutions, mx,my, andmz, into Eq. (A1),we find that\n1\nT(θ)/contintegraldisplay\ndtdθ\ndt\n=−γ/planckover2pi1ηI\n2eMVT(θ)/integraldisplayT(θ)\n0dt[1+χsinθcosω(t−τ)]cosθcosωt\n1+λsinθcosωt\n−αγ\nT(θ)/integraldisplayT(θ)\n0dt[Happl+(HK−4πM)]sinθ.\n(A3)\nUsing the integral formulas, we find that\n1\nT(θ)/contintegraldisplay\ndtdθ\ndt=γ/planckover2pi1ηI\n2eλMVtanθ/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\np(χ,τ,θ)\n−αγ[Happl+(HK−4πM)cosθ]sinθ,\n(A4)\nwherep(χ,τ,θ) isgivenbyEq.(9). Equation(A4)isiden-\ntical to Eq. (8). The threshold current given by Eq. (5)\nis the current satisfying lim θ→0dθ/dt= 0, whereas Eq.\n(3) is Eq. (5) in the limit of χ→0.\nAppendix B: Averaged LLG equation of in-plane\nmagnetized STO\nIn the main text, the multiple attractors are investi-\ngated for an STO consisting of a perpendicularly mag-\nnetized free layer and an in-plane magnetized reference\nlayer. On the other hand, previous works had focused on\nan STO consisting of in-plane magnetized free and refer-\nence layers[29–31,36]. Therefore, let us show that the in-\nplane magnetized STOalso showsthe multiple attractors\nstructurewhen thespin-transfertorqueincludes thefeed-\nback current. In this Appendix, the values of the param-\neters are derived from Refs. [47–49], The magnetic field\nandthestrengthofthespin-transfertorqueofanin-plane\nmagnetized STO are given by H=HKmyey−4πMmzez\nandHs=/planckover2pi1ηJ/(2eMd), respectively, where HK= 200Oe\nis an in-plane anisotropy field along the easy ( y) axis,J\nis the current density, and d= 2.0 nm is the thickness\nof the free layer. The saturation magnetization and the8\nGilbert damping constant are M= 1500 emu/c.c. and\n0.01, respectively. The spin polarization ηis 0.5, whereas\nthe spin-transfer torque asymmetry λis assumed to be\nzero, for simplicity. The spin-polarization direction pis\nparallel to the easy axis, p=ey.\n1. Energy range of in-plane auto-oscillation\nAs mentioned in the main text, the averaged LLG\nequation is derived by assuming an auto-oscillation on\na constant energy curve. The energy density of an in-\nplane magnetized ferromagnet is given by\nE=−MHK\n2m2\ny+4πM2\n2m2\nz. (B1)\nThe minimum, saddle, and maximum energy densities\nareEmin=−MHK/2,Es= 0, and Emax= 4πM2/2,\ncorresponding to the magnetization states of m=±ey,\n±ex, and±ez, respectively. Here, we focus on the auto-\noscillation around the easy axis, where the corresponding\nenergy density Eis in the range of Emin< E < E s. The\nauto-oscillation is excited when the current density is in\nthe range of Jc< J < J∗[47–49], where JcandJ∗are\nthe critical and switching current densities given by\nJc=2αeMd\n/planckover2pi1η(HK+2πM), (B2)\nJ∗=4αeMd\nπ/planckover2pi1η/radicalbig\n4πM(HK+4πM).(B3)\n2. Averaged LLG equation in the absence of\nfeedback current\nThe LLG equation averaged over the constant energy\ncurve of Ein the in-plane magnetized ferromagnet with-\nout the feedback current is given by [47]\n/contintegraldisplay\ndtdE\ndt=Ws+Wα, (B4)\nwhere WsandWαare the work done by the spin-transfer\ntorque and the energy dissipation by the damping torque\nduring a precession on a constant energy curve,\nWs=γM/contintegraldisplay\ndtHs[p·H−(m·p)(m·H)]\n= 2πMHs2E/M+HK/radicalbig\nHK(HK+4πM),(B5)\nWα=−αγM/contintegraldisplay\ndt/bracketleftBig\nH2−(m·H)2/bracketrightBig\n=−4αM/radicalBigg\n4πM−2E/M\nHK/bracketleftbigg2E\nMK(k)+HKE(k)/bracketrightbigg\n,\n(B6)E/(MHK/2)dE/(MHK/2)\n-1.0 -0.9 -0.8 -0.6 -0.7 -0.5 -0.4 -0.3 -0.2 -0.1 080\n40\n0\n-40\n-80\n-120(a)\nE/(MHK/2)dE/(MHK/2)\n-1.0 -0.9 -0.8 -0.6 -0.7 -0.5 -0.4 -0.3 -0.2 -0.1 080\n40\n0\n-40\n-80\n-120(b)\nFIG. 6: The averaged energy change, dE≡/contintegraltext\ndt(dE/dt), an\nin-plane magnetized ferromagnet as a function of the energy\ndensityE. The vertical and horizontal axes are renormalized\nbyMHK/2. The feedback current is (a) zero and (b) χ= 0.10\nwithτ= 3 ns.\nwhereK(k) =/integraltext1\n0dx//radicalbig\n(1−x2)(1−k2x2) andE(k) =/integraltext1\n0dx/radicalbig\n(1−k2x2)/(1−x2) are the first and second kind\nof complete elliptic integral with the modulus k:\nk=/radicalBigg\n4πM(HK+2E/M)\nHK(4πM−2E/M). (B7)\nThe precession period T(E) on a constant energy curve\nofEis\nT(E) =4K(k)\nγ/radicalbig\nHK(4πM−2E/M).(B8)\nFigure 6(a) shows an example of dE≡/contintegraltext\ndt(dE/dt) in\nthe absence of the feedback current, where the current\ndensity is chosen to be J= (Jc+J∗)/2. The energy den-\nsityEsatisfying dE= 0 and d(dE)/dE <0 corresponds\nto a stable attractor. As in the case of the STO in the\nmain text, there is only one attractor in this system.\n3. Work done by feedback current\nNow let us consider the role of the feedback current.\nIn the presence of the feedback current, the spin-transfer\ntorque performs an additional work given by\nWχ\ns≡γM/contintegraldisplay\ndtHsχm(t−τ)·p[p·H−(m·p)(m·H)],\n(B9)\nwhere we assume that the feedback current density is\ngiven by χJm(t−τ)·p. The averaged LLG equation in\nthe presence of the feedback current becomes\n/contintegraldisplay\ndtdE\ndt=Ws+Wχ\ns+Wα. (B10)\nTo evaluate Wχ\ns, it is useful to note that the solution\nof the magnetization oscillating around the easy axis on\na constant energy curve of Eis given by [47]\nmx(t) =/radicalbigg\n1+2E\nMHKsn/bracketleftbigg4K(k)\nT(E)t,k/bracketrightbigg\n,(B11)9\nmy(t) =/radicalBigg\n4πM−2E/M\nHK+4πMdn/bracketleftbigg4K(k)\nT(E)t,k/bracketrightbigg\n,(B12)\nmz(t) =/radicalBigg\nHK+2E/M\nHK+4πMcn/bracketleftbigg4K(k)\nT(E)t,k/bracketrightbigg\n,(B13)where sn( u,k), dn(u,k), and cn( u,k) arethe Jacobiellip-\ntic functions with u= 4K(k)t/T(E). Introducing a new\nvariablex= sn(u,k), Eq. (B9) becomes\nWχ\ns=4χMHs/radicalbig\nHK(4πM−2E/M)/integraldisplay1\n0dx/bracketleftbig\nHKmy−my(HKm2\ny−4πMm2\nz)/bracketrightbig\n/radicalbig\n(1−x2)(1−k2x2)my(t−τ), (B14)\nwhere dn( u,k) and cn( u,k) inmy(t) andmz(t) are re-\nplaced by√\n1−k2x2and√\n1−x2, respectively. On theother hand, my(t−τ) in Eq. (B14) is given by [50]\nmy(t−τ) =/radicalBigg\n4πM−2E/M\nHK+4πMdn(u,k)dn(v,k)+k2sn(u,k)sn(v,k)cn(u,k)cn(v,k)\n1−k2sn2(u,k)sn2(v,k)\n=/radicalBigg\n4πM−2E/M\nHK+4πMdn(v,k)√\n1−k2x2+k2sn(v,k)cn(v,k)x√\n1−x2\n1−k2sn2(v,k)x2,(B15)\nwherev= 4K(k)τ/T(E). Equation (B15) indicates that\nthe multiple attractors originate from the periodicity of\nthe elliptic function. In contrast with Eqs. (B5) and\n(B6), the analytical expression of Eq. (B14) is complex;\nsee next section. Therefore, we evaluate Eq. (B14) nu-\nmerically.\nFigure 6(b) shows/contintegraltext\ndt(dE/dt) in the presence of the\nfeedback current, where χ= 0.10 andτ= 3 ns. As\nshown, the multiple attractors appear, as in the STO\nstudied in the main text. Therefore, we consider that\nthe chaotic dynamics studied in Ref. [36] might be also\nrelated to the multiple attractors.\n4. Analytical expression of Wχ\ns\nSubstituting Eq. (B15) into Eq. (B14), Wχ\nsis rewrit-\nten as\nWχ\ns=4χMHs/radicalbig\nHK(4πM−2E/M)5/summationdisplay\nℓ=1Iℓ,(B16)where we introduce Iℓas\nI1=c2\nyHKdn(v,k)/integraldisplay1\n0dx√\n1−k2x2\n√\n1−x2[1−k2sn2(v,k)x2]\n≡c2\nyHKdn(v,k)˜I1,\n(B17)\nI2=−c4\nyHKdn(v,k)/integraldisplay1\n0dx(1−k2x2)3/2\n√\n1−x2[1−k2sn2(v,k)x2]\n≡ −c4\nyHKdn(v,k)˜I2,\n(B18)\nI3=c2\nyc2\nz4πMdn(v,k)/integraldisplay1\n0dx/radicalbig\n(1−x2)(1−k2x2)\n1−k2sn2(v,k)x2\n≡c2\nyc2\nz4πMdn(v,k)˜I3,\n(B19)\nI4=c2\ny/bracketleftbig/parenleftbig\n1−c2\ny/parenrightbig\nHK+c2\nz4πM/bracketrightbig\nk2sn(v,k)cn(v,k)/integraldisplay1\n0dxx\n1−k2sn2(v,k)x2\n≡c2\ny/bracketleftbig/parenleftbig\n1−c2\ny/parenrightbig\nHK+c2\nz4πM/bracketrightbig\nk2sn(v,k)cn(v,k)˜I4,(B20)10\nI5=c2\ny/parenleftbig\nc2\nyHKk2−c2\nz4πM/parenrightbig\nk2sn(v,k)cn(v,k)/integraldisplay1\n0dxx3\n1−k2sn2(v,k)x2\n≡c2\ny/parenleftbig\nc2\nyHKk2−c2\nz4πM/parenrightbig\nk2sn(v,k)cn(v,k)˜I5.(B21)\nHere, we introducethe followingnotations, forsimplicity.\ncy=/radicalBigg\n4πM−2E/M\nHK+4πM, cz=/radicalBigg\nHK+2E/M\nHK+4πM.(B22)The integrals ˜Iℓ(ℓ= 1−5) can be performed as\n˜I1=/integraldisplay1\n0dx√\n1−k2x2\n√\n1−x2[1−k2sn2(v,k)x2]\n=1\nsn2(v,k)/integraldisplay1\n0dx/radicalbig\n(1−x2)(1−k2x2)−1−sn2(v,k)\nsn2(v,k)/integraldisplay1\n0dx/radicalbig\n(1−x2)(1−k2x2)[1−k2sn2(v,k)x2]\n=K(k)\nsn2(v,k)−cn2(v,k)\nsn2(v,k)Π[k2sn2(v,k),k],(B23)\n˜I2=/integraldisplay1\n0dx(1−k2x2)3/2\n√\n1−x2[1−k2sn2(v,k)x2]\n=1\nsn2(v,k)/integraldisplay1\n0dx/radicalbigg\n1−k2x2\n1−x2−cn2(v,k)\nsn2(v,k)/integraldisplay1\n0dx√\n1−k2x2\n√\n1−x2[1−k2sn2(v,k)x2]\n=E(k)\nsn2(v,k)−cn2(v,k)\nsn2(v,k)˜I1,(B24)\n˜I3=/integraldisplay1\n0dx/radicalbig\n(1−x2)(1−k2x2)\n1−k2sn2(v,k)x2\n=1\nk2sn2(v,k)/integraldisplay1\n0dx/radicalbigg\n1−k2x2\n1−x2−dn2(v,k)\nk2sn2(v,k)/integraldisplay1\n0dx√\n1−k2x2\n√\n1−x2[1−k2sn2(v,k)x2]\n=E(k)\nk2sn2(v,k)−dn2(v,k)\nk2sn2(v,k)˜I1,(B25)\n˜I4=/integraldisplay1\n0dxx\n1−k2sn2(v,k)x2\n=−log[1−k2sn2(v,k)]\n2k2sn2(v,k)\n=−logdn(v,k)\nk2sn2(v,k),(B26)\n˜I5=/integraldisplay1\n0dxx3\n1−k2sn2(v,k)x2\n=−1\nk2sn2(v,k)/integraldisplay1\n0dxx+1\nk2sn2(v,k)/integraldisplay1\n0dxx\n1−k2sn2(v,k)x2\n=−1\n2k2sn2(v,k)+˜I4\nk2sn2(v,k),(B27)11\nwhere Π( a2,k) =/integraltext1\n0dx/[(1−a2x2)/radicalbig\n(1−x2)(1−k2x2)]is the third kind of complete elliptic integral.\n1S. 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The current-voltage character-\nistics of the junction demonstrate a pattern of subharmonic current steps which forms a devil’s\nstaircase structure. We show that a width of the steps become s maximal at ferromagnetic reso-\nnance. Moreover, we demonstrate that the structure of the st eps and their widths can be tuned\nby changing the frequency of the external magnetic field, rat io of Josephson to magnetic energy,\nGilbert damping and the junction size.\nThis paper is submitted to LTP Journal.\nI. INTRODUCTION\nJosephson junction with ferromagnet layer (F) is\nwidely considered to be the place where spintronics and\nsuperconductivity fields interact1. In these junctions\nthe supercurrent induces magnetization dynamics due\nto the coupling between the Josephson and magnetic\nsubsystems. The possibility of achieving electric con-\ntrol over the magnetic properties of the magnet via\nJosephson current and its counterpart, i.e., achieving\nmagnetic control over Josephson current, recently at-\ntracted a lot of attention1–7. The current-phase rela-\ntion in the superconductor-ferromagnet-superconductor\njunction (SFS) junctions is very sensitive to the mutual\norientation of the magnetizations in the F-layer8,9. In\nRef.[10] the authors demonstrate a unique magnetization\ndynamics with a series of specific phase trajectories. The\norigin of these trajectories is related to a direct coupling\nbetween the magnetic moment and the Josephson oscil-\nlations in these junctions.\nExternal electromagnetic field can also provide a cou-\npling between spin wave and Josephson phase in SFS\njunctions11–17. Spin waves are elementary spin excita-\ntions which considered to be as both spatial and time\ndependent variations in the magnetization18,19. The fer-\nromagnetic resonance(FMR) correspondsto the uniform\nprecession of the magnetization around an external ap-\nplied magnetic field18. This mode can be resonantly ex-\ncited by ac magnetic field that couples directly to the\nmagnetization dynamics as described by the Landau-\nLifshitz-Gilbert (LLG) equation18,19.\nIn Ref.[18] the authors show that spin wave resonance\nat frequency ωrin SFS implies a dissipation that is mani-\nfested as adepressionin the IV-characteristicofthe junc-\ntion when /planckover2pi1ωr= 2eV, where/planckover2pi1is the Planck’s constant,\ne is the electron charge and Vis the voltage across the\njunction. The ac Josephson current produces an oscil-\nlating magnetic field and when the Josephson frequencymatches the spin wave frequency, this resonantly excites\nthe magnetization dynamics M(t)18. Due to the non-\nlinearity of the Josephson effect, there is a rectification\nof current across the junction, resulting in a dip in the\naverage dc component of the suppercurrent18.\nIn Ref.[13] the authors neglect the effective field due\nto Josephson energy in LLG equation and the results re-\nveal that even steps appear in the IV-characteristic of\nSFS junction under external magnetic field. The ori-\ngin of these steps is due to the interaction of Cooper\npairs with even number of magnons. Inside the ferro-\nmagnet, if the Cooper pairs scattered by odd number of\nmagnons, no Josephson current flows due to the forma-\ntion of spin triplet state13. However, if the Cooper pairs\ninteract with even number of magnons, the Josephson\ncoupling between the s-wave superconductor is achieved\nand the spin singlet state is formed, resulting in flows of\nJosephsoncurrent13. In Ref.[20]weshowthat takinginto\naccount the effective field due to Josepshon energy and\nat FMR, additional subharmonic current steps appear in\nthe IV-characteristic for overdamped SFS junction with\nspin wave excitations (magnons). It is found that the po-\nsition of the current steps in the IV-characteristics form\ndevil’s staircase structure which follows continued frac-\ntion formula20. The positions of those fractional steps\nare given by\nV=\nN±1\nn±1\nm±1\np±..\nΩ, (1)\nwhere Ω = ω/ωc,ωis the frequency of the external ra-\ndiation, ωcis the is the characteristic frequency of the\nJosephson junction and N,n,m,pare positive integers.\nIn this paper, we present a detailed analysis for the\nIV-characteristics of SFS junction under external mag-\nnetic field, and show how we can control the position\nof the subharmonic steps and alter their widths. The\ncoupling between spin wave and Josephson phase in SFS\njunction is achieved through the Josephson energy and\ngauge invariant phase difference between the S-layers. In\nthe framework of our approach, the dynamics of the SFS2\njunction isfully describedbytheresistivelyshuntedjunc-\ntion (RSJ) model and LLG equation. These equations\nare solved numerically by the 4thorder Runge-Kutta\nmethod. The appearance and position of the observed\ncurrent steps depend directly on the magnetic field and\njunction parameters.\nII. MODEL AND METHODS\nF\nss\nHacxyz\nH0I\nI\nFIG. 1. SFS Josephson junction. The bias current is applied\nin x-direction, an external magnetic field with amplitude Hac\nand frequency ωis applied in xy-plane and an uniaxial con-\nstant magnetic field H0is applied in z-direction.\nIn Fig 1 we consider a current biased SFS junction\nwhere the two superconductors are separated by ferro-\nmagnet layer with thickness d. The area of the junction\nisLyLz. An uniaxial constant magnetic field H0is ap-\nplied in z-direction, while the magnetic field is applied in\nxy-plane Hac= (Haccosωt,Hacsinωt,0)withamplitude\nHacand frequency ω. The magnetic field is induced in\nthe F-layer through B(t) = 4πM(t), and the magnetic\nfluxes in z- and y-direction are Φ z(t) = 4πdLyMz(t),\nΦy(t) = 4πdLzMy(t), respectively. The gauge-invariant\nphase difference in the junction is given by21:\n∇y,zθ(y,z,t) =−2πd\nΦ0B(t)×n, (2)\nwhereθis the phase difference between superconducting\nelectrodes, and Φ 0=h/2eis the magnetic flux quantum\nandnis a unit vector normal to yz-plane. The gauge-\ninvariantphasedifference in terms ofmagnetizationcom-\nponents reads as\nθ(y,z,t) =θ(t)−8π2dMz(t)\nΦ0y+8π2dMy(t)\nΦ0z,(3)\nwhere Φ 0=h/(2e) is the magnetic flux quantum.\nAccordingtoRSJ model, the currentthroughthe junc-\ntion is given by13:\nI\nI0c= sinθ(y,z,t)+Φ0\n2πI0cRdθ(y,z,t)\ndt,(4)\nwhereI0\ncis the critical current, and R is the resistance\nin the Josephson junction. After taking into account thegaugeinvarianceincludingthemagnetizationoftheferro-\nmagnetandintegratingoverthejunction areatheelectric\ncurrent reads13:\nI\nI0c=Φ2\nosin(θ(t))sin/parenleftBig\n4π2dMz(t)Ly\nΦo/parenrightBig\nsin/parenleftBig\n4π2dMy(t)Lz\nΦo/parenrightBig\n16π4d2LzLyMz(t)My(t)\n+Φ0\n2πRI0cdθ(y,z,t)\ndt. (5)\nThe applied magnetic field in the xy-plane causes pre-\ncessionalmotionofthemagnetizationinthe F-layer. The\ndynamics of magnetization Min the F-layer is described\nby LLG equation\n(1+α2)dM\ndt=−γM×Heff−γ α\n|M|[M×(M×Heff)](6)\nThe total energy of junction in the proposed model is\ngivenby E=Es+EM+EacwhereEsistheenergystored\nin Josephson junction, EMis the energy of uniaxial dc\nmagnetic field (Zeeman energy) and Eacis the energy of\nac magnetic field:\nEs=−Φ0\n2πθ(y,z,t)I+EJ[1−cos(y,z,t)],\nEM=−VFH0Mz(t),\nEac=−VFMx(t)Haccos(ωt)−VFMy(t)Hacsin(ωt)(7)\nHere,EJ= Φ0I0\nc/2πis the the Josephson energy, H0=\nω0/γ,ω0is the FMR frequency, and VFis the volume of\nthe ferromagnet. We neglect the anisotropy energy due\nto demagnetizing effect for simplicity. The effective field\nin LLG equation is calculated by\nHeff=−1\nVF∇ME (8)\nThus, the effective field Hmdue to microwave radiation\nHacand uniaxial magnetic field H0is given by\nHm=Haccos(ωt)ˆex+Hacsin(ωt)ˆey+H0ˆez.(9)\nwhile the effective field ( Hs) due to superconducting part\nis found from\nHs=−EJ\nVFsin(θ(y,z,t))∇Mθ(y,z,t).(10)\nOne should take the integration of LLG on coordinates,\nhowever, the superconducting part is the only part which\ndepends on the coordinate so, we can integrate the ef-\nfective field due to the Josephson energy and insert the\nresult into LLG equation. Then, the y- and z-component\nare given by\nHsy=EJcos(θ(t))sin(πΦz(t)/Φ0)\nVFπMy(t)Φz(t)/bracketleftbigg\nΦ0cos(πΦy(t)/Φ0)\n−Φ2\n0sin(πΦy(t)/Φ0)\nπΦy(t)/bracketrightbigg\nˆey, (11)\nHsz=EJcos(θ(t))sin(πΦy(t)/Φ0)\nVFπMz(t)Φy(t)/bracketleftbigg\nΦ0cos(πΦz(t)/Φ0)\n−Φ2\n0sin(πΦz(t)/Φ0)\nπΦz(t)/bracketrightbigg\nˆez. (12)3\nAs a result, the total effective field is Heff=Hm+\nHs. In the dimensionless form we use t→tωc,ωc=\n2πI0\ncR/Φ0is the characteristic frequency, m=M/M0,\nM0=∝ba∇dblM∝ba∇dbl,heff=Heff/H0,ǫJ=EJ/VFM0H0,hac=\nHac/H0, Ω =ω/ωc, Ω0=ω0/ωc,φsy=4π2LydM0/Φo,\nφsz=4π2lzdM0/Φo. Finally, the voltage V(t) =dθ/dtis\nnormalized to /planckover2pi1ωc/(2e). The LLG and the effective field\nequations take the form\ndm\ndt=−Ω0\n(1+α2)/parenleftbigg\nm×heff+α[m×(m×heff)]/parenrightbigg\n(13)\nwith\nheff=haccos(Ωt)ˆex+(hacsin(Ωt)+ΓijǫJcosθ)ˆey\n+ (1+Γ jiǫJcosθ)ˆez, (14)\nΓij=sin(φsimj)\nmi(φsimj)/bracketleftbigg\ncos(φsjmi)−sin(φsjmi)\n(φsjmi)/bracketrightbigg\n,(15)\nwherei=y,j=z. The RSJ in the dimensionless form is\ngiven by\nI/I0\nc=sin(φsymz)sin(φszmy)\n(φsymz)(φszmy)sinθ+dθ\ndt.(16)\nThe magnetization and phase dynamics of the SFS\njunction can be described by solving Eq.(16) together\nwith Eq.(13). To solve this system of equations, we em-\nploy the fourth-order Runge-Kutta scheme. At each cur-\nrent step, we find the temporal dependence of the volt-\nageV(t), phase θ(t), andmi(i=x,y,z) in the (0 ,Tmax)\ninterval. Then the time-average voltage Vis given by\nV=1\nTf−Ti/integraltext\nV(t)dt, whereTiandTfdetermine the in-\nterval for the temporal averaging. The current value is\nincreased or decreased by a small amount of δI (the bias\ncurrent step) to calculate the voltage at the next point\nof the IV-characteristics. The phase, voltage and mag-\nnetization components achieved at the previous current\nstep are used as the initial conditions for the next cur-\nrent step. The one-loop IV-characteristic is obtained by\nsweeping the bias current from I= 0 toI= 3 and back\ndown to I= 0. The initial conditions for the magnetiza-\ntion components are assumed to be mx= 0,my= 0.01\nandmz=/radicalBig\n1−m2x−m2y, while for the voltage and\nphase we have Vini= 0,θini=0. The numerical param-\neters (if not mentioned) are taken as α= 0.1,hac= 1,\nφsy=φsz= 4,ǫJ= 0.2 and Ω 0= 0.5.\nIII. RESULTS AND DISCUSSIONS\nItiswell-knownthatJosephsonoscillationscanbesyn-\nchronized by external microwave radiation which leads\nto Shapiro steps in the IV-characteristic22. The position\nof the Shapiro step is determined by relation V=n\nmΩ,\nwheren,mare integers. The steps at m= 1 are calledharmonics, otherwise we deal with synchronized subhar-\nmonic (fractional) steps. We show below the appearance\nof subharmonics in our case.\nFirst we present the simulated IV-characteristics at\ndifferent frequencies of the magnetic field. The IV-\ncharacteristics at three different values of Ω are shown\nin Fig 2(a).\nFIG. 2. (a) IV-characteristic at three different values of Ω.\nFor clarity, the IV-characteristics for Ω = 0 .5 and Ω = 0 .7\nhave been shifted to the right, by ∆ I= 0.5 and ∆ I= 1,\nrespectively with respect to Ω = 0 .2; (b) An enlarged part\nof the IV-characteristic with Ω = 0 .7. To get step voltage\nmultiply the corresponding fraction with Ω = 0 .7.\nAs we see, the second harmonic has the largest step\nwidth at the ferromagnetic resonance frequency Ω = Ω 0,\ni.e., the FMR is manifested itself by the step’s width.\nThere are also many subharmonic current steps in the\nIV-characteristic. We have analyzed the steps position\nbetween V= 0 and V= 0.7 for Ω = 0 .7 and found dif-\nferent level continued fractions, which follow the formula\ngiven by Eq.(1) and demonstrated in Fig.2(b). We see4\nthe reflection of the second level continued fractions 1 /n\nand 1−1/nwithN= 1. In addition to this, steps with\nthird level continued fractions 1 /(n−1/m) withN= 1\nis manifested. In the inset we demonstrate part of the\nfourth level continued fraction 1 −1/(n+ 1/(m+1/p))\nwithn= 2 and m= 2.\nIn case of external electromagnetic field which leads to\nthe additional electric current Iac=AsinΩt, the width\nof the Shapiro step is proportional to ∝Jn(A/Ω), where\nJnis the Bessel function of first kind. The preliminary\nresults (not presented here) show that the width of the\nShapiro-like steps under external magnetic field has a\nmore complex frequency dependence20. This question\nwill be discussed in detail somewhere else.\nThe coupling between Josephson phase and magneti-\nzation manifests itself in the appearance of the Shapiro\nsteps in the IV-characteristics at fractional and odd mul-\ntiplies of Ω20. In Fig.3 we demonstrate the effect of the\nratio of the Josephson to magnetic energy ǫJon appear-\nance of the steps and their width for Ω = 0 .5 where the\nenlarged parts of the IV-characteristics at three differ-\nent values of ǫJare shown. As it is demonstrated in\nthe figures, at ǫJ= 0.05 only two subharmonic steps\nappear between V= 1 and V= 1.5 (see hollow ar-\nrows). An enhanced staircase structure appears by in-\ncreasing the value of ǫJ, which can be see at ǫJ= 0.3\nand 0.5. Moreover, an intense subharmonic steps appear\nbetween V= 1.75 andV= 2 forǫJ= 0.5. The posi-\ntions for these steps reflect third level continued fraction\n(N−1)+1/(n+1/m)withN=4 andn=1 [see Fig.3(b)].\nLet us now demonstrate the effect of Gilbert damping\non the devil’s staircase structure. The Gilbert damping\nαis introduced into LLG equation23?to describe the\nrelaxation of magnetization dynamics. To reflect effect\nof Gilbert damping, we show an enlarged part of the IV-\ncharacteristic at three different values of αin Fig.4.\nThewidthofcurrentstepat V= 2Ωisalmostthesame\nat different values of α(e.g., see upward inset V= 2Ω).\nThe subharmonic current step width for V= (n/m)Ω (n\nis odd,mis integer) is decreasing with increasing α. In\naddition a horizontal shift for the current steps occurs.\nWe see the intense current steps in the IV-characteristic\nfor small value of α= 0.03 (see black solid arrows). With\nincrease in Gilbert damping (see α= 0.1, 0.16 and 0 .3)\nthe higher level subharmonic steps disappear. It is well-\nknownthatatlargevalueof αtheFMRlinewidthbecome\nmore broadening and the resonance frequency is shifted\nfrom Ω 0. Accordingly, the subharmonic steps disappear\nat large value of α. Furthermore, using the formula pre-\nsented in Ref.[20] the width at Ω = Ω 0for the fractional\nand odd current steps is proportional to (4 α2+α4)−q/2\n×(12+3α2)−k/2, whereqandkare integers.\nFinally, we demonstrate the effect of the junction size\non the devil’s staircase in the IV-characteristic under ex-\nternalmagneticfield. Thejunction sizechangesthe value\nofφsyandφsz. In Fig.5(a) we demonstrate the effect of\nthe junction thickness by changing φsz(φsyis qualita-\nFIG. 3. (a) An enlarged part of the IV-characteristic at\ndifferent values of ǫJin the interval between V= 1 and V=\n1.5; (b)Thesameintheintervalbetween V= 1.75andV= 2.\nFor clarity, the IV-characteristics for ǫJ= 0.3, and 0 .5 have\nbeen shifted to right, by ∆ I= 0.07, and 0 .14, respectively\nwith respect to the case with ǫJ= 0.05.\ntively the same).\nWe observe an enhanced subharmonic structure with\nincrease of junction size or the thickness of the ferro-\nmagnet. In Ref.[13] the authors demonstrated that the\ncritical current and the width of the step at V= 2Ω as a\nfunction of Lz/Lyfollow Bessel function of first kind. In\nFig.5(b), we can see the parts of continued fraction se-\nquences for subharmonic steps between V= 1 andV= 2\natφsz=φsy= 6. Current steps between V= 1 and\nV= 1.5 reflect the two second level continued fractions\n(N−1)+ 1/nandN−1/nwithN= 3 in both cases,\nwhile for the steps between V= 1.5 andV= 2 follow\nthe second level continued fraction ( N−1) + 1/nwith\nN= 4.\nFinally, wediscussthepossibilityofexperimentallyob-5\nFIG. 4. An enlarged part of IV-characteristic for four differ -\nent values of Gilbert damping for Ω = 0 .5. The inset shows an\nenlargedpartofcurrentstepwithconstantvoltage at V= 2Ω.\nserving the effects presented in this paper. For junction\nsized= 5nm, Ly=Lz= 80nm, critical current I0\nc≈\n200µA, saturation magnetization M0≈5×105A/m,\nH0≈40mT and gyromagnetic ratio γ= 3πMHz/T,\nwe find the value of φsy(z)=4π2Ly(z)dM0/Φ0= 4.8 and\nǫJ= 0.1. With the same junction parameters one can\ncontrol the appearance of the subharmonic steps by tun-\ning the strength of the constant magnetic field H0. Esti-\nmations showthat for H0= 10mT, the value of ǫJ= 0.4,\nand the fractional subharmonic steps are enhanced. In\ngeneral, the subharmonic steps are sensitive to junction\nparameters, Gilbert damping and the frequency of the\nexternal magnetic field.\nIV. CONCLUSIONS\nIn this work, we have studied the IV-characteristics\nof superconductor-ferromagnet-superconductor Joseph-\nson junction under external magnetic field. We used a\nmodified RSJ model which hosts magnetization dynam-\nics in F-layer. Due to the external magnetic field, the\ncouplingbetweenmagneticmomentandJosephsonphase\nis achieved through the effective field taking into account\nthe Josephson energy and gauge invariant phase differ-\nence between the superconducting electrodes. We have\nsolvedasystemofequationswhichdescribethe dynamics\nof the Josephson phase by the RSJ equation and magne-\ntization dynamics by Landau-Lifshitz-Gilbert equation.\nThe IV-characteristic demonstrates subharmonic current\nsteps. The pattern of the subharmonic steps can be con-\ntrolled by tuning the frequency of the ac magnetic field.\nWe show that by increasing the ratio of the Josephson to\nmagneticenergyanenhancedstaircasestructureappears.\nFinally, we demonstrate that Gilbert damping and junc-\nFIG. 5. (a) IV-characteristic at three different values of\nφsz= 0.7,3,6 andφsy=φsz. (b) An enlarged part of the IV-\ncharacteristic at φsz=φsy=6. The hollow arrows represent\nthe starting point of the sequences. To get step voltage we\nmultiply the corresponding fraction by Ω = 0 .5.\ntion parameters can change the subharmonic step struc-\nture. The observed features might find an application in\nsuperconducting spintronics.\nV. ACKNOWLEDGMENT\nWe thank Dr. D. V. Kamanin and Egypt JINR col-\nlaboration for support this work. The reported study\nwas partially funded by the RFBR research Projects No.\n18-02-00318 and No. 18-52-45011-IND. Numerical cal-\nculations have been made in the framework of the RSF\nProject No. 18-71-10095.6\nREFERENCES\n∗majed@sci.cu.edu.eg\n†shukrinv@theor.jinr.ru\n1J. Linder and K. Halterman, Phys. Rev. B 90, 104502\n(2014).\n2Yu. M. Shukrinov, A. Mazanik, I. Rahmonov, A. Botha,\nand A. Buzdin, EPL122, 37001 (2018).\n3Yu. M. Shukrinov, I. Rahmonov, K. Sengupta, and A.\nBuzdin, Appl. Phys. Lett. 110, 182407 (2017).\n4A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008).\n5A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n6F. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys.77, 1321 (2005).\n7A. A. Golubov, M. Y. Kupriyanov, and E. IlIchev, Rev.\nMod. Phys. 76, 411 (2004).\n8M. A. Silaev, I. V. Tokatly, and F. S. Bergeret, Phys. Rev.\nB95, 184508 (2017).\n9I. Bobkova, A. Bobkov, and M. Silaev, Phys. Rev. B 96,\n094506 (2017).\n10Y. M. Shukrinov, I. Rahmonov, and K. Sengupta, Phys.\nRev. B99, 224513 (2019).\n11M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D.\nKoelle, R. Kleiner, and E. 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Rev.\nB 97, 224514 (2018).\n21K. K. Likharev, Dynamics of Josephson junctions and cir-\ncuits, Gordon and Breach science publishers -Switzerland\n(1986).\n22S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).\n23T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443-\n3449 (2004).\n24M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102,\n137601(2009)."    },    {        "title": "1909.09085v1.Magnetization_dynamics_of_the_compensated_ferrimagnet__Mn__2_Ru__x_Ga_.pdf",        "content": "Magnetisation dynamics of the compensated ferrimagnet Mn 2RuxGa\nG. Bon\fglio\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands\nK. Rode, K. Siewerska, J. Besbas, G. Y. P. Atcheson, P. Stamenov, and J.M.D. Coey\nCRANN, AMBER and School of Physics, Trinity College Dublin, Ireland\nA.V. Kimel, Th. Rasing, and A. Kirilyuk\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands and\nFELIX Laboratory, Radboud University, Toernooiveld 7c, 6525 ED Nijmegen, The Netherlands\nHere we study both static and time-resolved dynamic magnetic properties of the compensated\nferrimagnet Mn 2RuxGa from room temperature down to 10 K, thus crossing the magnetic compen-\nsation temperature TM. The behaviour is analysed with a model of a simple collinear ferrimagnet\nwith uniaxial anisotropy and site-speci\fc gyromagnetic ratios. We \fnd a maximum zero-applied-\n\feld resonance frequency of \u0018160 GHz and a low intrinsic Gilbert damping \u000b\u00180:02, making it a\nvery attractive candidate for various spintronic applications.\nI. INTRODUCTION\nAntiferromagnets (AFM) and compensated ferrimag-\nnets (FiM) have attracted a lot of attention over the last\ndecade due to their potential use in spin electronics1,2.\nDue to their lack of a net magnetic moment, they are\ninsensitive to external \felds and create no demagnetis-\ning \felds of their own. In addition, their spin dynamics\nreach much higher frequencies than those of their ferro-\nmagnetic (FM) counterparts due to the contribution of\nthe exchange energy in the magnetic free energy3.\nDespite these clear advantages, AFMs are scarcely\nused apart from uni-directional exchange biasing rela-\ntively in spin electronic applications. This is because\nthe lack of net moment also implies that there is no\ndirect way to manipulate their magnetic state. Fur-\nthermore, detecting their magnetic state is also compli-\ncated and is usually possible only by neutron di\u000braction\nmeasurements4, or through interaction with an adjacent\nFM layer5.\nCompensated, metallic FiMs provide an interesting al-\nternative as they combine the high-speed advantages of\nAFMs with those of FMs, namely, the ease to manipu-\nlate their magnetic state. Furthermore, it has been shown\nthat such materials are good candidates for the emerging\n\feld of All-Optical Switching (AOS) in which the mag-\nnetic state is solely controlled by a fast laser pulse6{8.\nA compensated, half-metallic ferrimagnet was \frst en-\nvisaged by van Leuken and de Groot9. In their model\ntwo magnetic ions in crystallographically di\u000berent po-\nsitions couple antiferromagnetically and perfectly com-\npensate each-other, but only one of the two contributes\nto the states at the Fermi energy responsible for elec-\ntronic transport. The \frst experimental realisation of\nthis, Mn 2RuxGa (MRG), was provided by Kurt et al.10.\nMRG crystallises in the XAHeusler structure, space\ngroupF\u001643m, with Mn on the 4 aand 4csites11.\nSubstrate-induced bi-axial strain imposes a slight tetrag-\nonal distortion, which leads to perpendicular magneticanisotropy. Due to the di\u000berent local environment of\nthe two sublattices, the temperature dependence of their\nmagnetic moments di\u000ber, and perfect compensation is\ntherefore obtained at a speci\fc temperature TMthat\ndepends on the Ru concentration xand the degree of\nbiaxial strain. It was previously shown that MRG ex-\nhibits properties usually associated with FMs: a large\nanomalous Hall angle12, that depends only on the mag-\nnetisation of the 4 cmagnetic sublattice13; tunnel magne-\ntoresistance (TMR) of 40 %, a signature of its high spin\npolarisation14, was observed in magnetic tunnel junc-\ntions (MTJs) based on MRG15; and a clear magneto-\noptical Kerr e\u000bect and domain structure, even in the ab-\nsence of a net moment16,17. Strong exchange bias of a\nCoFeB layer by exchange coupling with MRG through\na Hf spacer layer18, as well as single-layer spin-orbit\ntorque19,20showed that MRG combined the qualities of\nFMs and AFMs in spin electronic devices.\nThe spin dynamics in materials where two distinct\nsublattices are subject to di\u000bering internal \felds (ex-\nchange, anisotropy, . . . ) is much richer than that of a\nsimple FM, as previously demonstrated by the obers-\nvation of single-pulse all-optical switching in amorphous\nGdFeCo21,22and very recently in MRG8. Given that the\nmagnetisation of MRG is small, escpecially close to the\ncompensation point, and the related frequency is high,\nnormal ferromagnetic resonance (FMR) spectroscopy is\nunsuited to study their properties. Therefore, we used\nthe all-optical pump-probe technique to characterize the\nresonance frequencies at di\u000berent temperatures in vicin-\nity of the magnetic compensation point. This, together\nwith the simulation of FMR, make it possible to deter-\nmine the e\u000bective g-factors, the anisotropy constants and\ntheir evolution across the compensation point. We found,\nin particular, that our ferrimagnetic half-metallic Heusler\nalloy has resonance frequency up to 160 GHz at zero-\feld\nand a relatively low Gilbert damping.arXiv:1909.09085v1  [cond-mat.mtrl-sci]  19 Sep 20192\nFIG. 1. Net moment measured by magnetometry and coercive\n\feld measured by static Faraday e\u000bect. The upturn of the net\nmoment below T\u001850 K is due to paramagnetic impurities\nin the MgO substrate. TMis indicated by the vertical dotted\nline. As expected the maximum available applied \feld \u00160H=\n7 T is insu\u000ecient to switch the magnetisation close to TM.\nII. EXPERIMENTAL DETAILS\nThin \flm samples of MRG were grown in a `Sham-\nrock' sputter deposition cluster with a base pressure of\n2\u000210\u00008Torr on MgO (001) substrates. Further infor-\nmation on sample deposition can be found elsewhere23.\nThe substrates were kept at 250\u000eC, and a protective\n\u00183 nm layer of aluminium oxide was added at room tem-\nperature. Here we focus on a 53 nm thick sample with\nx= 0:55, leading to TM\u001980 K as determined by SQUID\nmagnetometry using a Quantum Design 5 T MPMS sys-\ntem (see FIG. 1). We are able to study the magneto-\noptical properties both above and below TM.\nThe magnetisation dynamics was investigated using an\nall-optical two-colour pump-probe scheme in a Faraday\ngeometry inside a \u00160Hmax= 7 T superconducting coil-\ncryostat assembly. Both pump and probe were produced\nby a Ti:sapphire femtosecond pulsed laser with a cen-\ntral wavelength of 800 nm, a pulse width of 40 fs and\na repetition rate of 1 kHz. After splitting the beam in\ntwo, the high-intensity one was doubled in frequency by\na BBO crystal (giving \u0015= 400 nm) and then used as\nthe pump while the lower intensity 800 nm beam acted\nas the probe pulse. The time delay between the two was\nadjusted by a mechanical delay stage. The pump was\nthen modulated by a synchronised mechanical chopper\nat 500 Hz to improve the signal to noise ratio by lock-in\ndetection. Both pump and probe beams were linearly\npolarized, and with spot sizes on the sample of 150 µm\nand 70 µm, respectively. The pump pulse hit the sample\nat an incidence angle of \u001910\u000e. After interaction with\nthe sample, we split the probe beam in two orthogonally\npolarized parts using a Wollaston prism and detect the\nchanges in transmission and rotation by calculating the\nFIG. 2. Comparison of hysteresis loops obtained by Faraday,\nAHE, and magnetometry recorded at room temperature. The\ntwo former were recorded with the applied \feld perpendicular\nto the sample surface, while for the latter we show results for\nboth \feld applied parallel and perpendicular to the sample.\nsum and the di\u000berence in intensity of the two signals.\nThe external \feld was applied at 75\u000eto the easy axis of\nmagnetization thus tilting the magnetisation away from\nthe axis. Upon interaction with the pump beam the mag-\nnetisation is momentarily drastically changed24and we\nmonitor its return to the initial con\fguration via remag-\nnetisation and then precession through the time depen-\ndent Faraday e\u000bect on the probe pulse.\nThe static magneto-optical properties were examined\nin the same cryostat/magnet assembly.\nIII. RESULTS & DISCUSSION\nA. Static magnetic properties\nWe \frst focus on the static magnetic properties as\nobserved by the Faraday e\u000bect, and compare them to\nwhat is inferred from magnetometry and the anomalous\nHall e\u000bect. In FIG. 2 we present magnetic hysteresis\nloops as recorded using the three techniques. Due to the\nhalf metallic nature of the sample, the magnetotrans-\nport properties depend only on the 4 csublattice. As the\nmain contribution to the MRG dielectric tensor in the\nvisible and near infrared arises from the Drude tail16,\nboth AHE and Faraday e\u000bect probe essentially the same\nproperties (mainly the spin polarised conduction band of\nMRG), hence we observe overlapping loops for the two\ntechniques. Magnetometry, on the other hand, measures\nthe net moment, or to be precise the small di\u000berence\nbetween two large sublattice moments. The 4 asublat-\ntice, which is insigni\fcant for AHE and Faraday here\ncontributes on equal footing. FIG. 2 shows a clear di\u000ber-\nence in shape between the magnetometry loop and the3\nFIG. 3. Time resolved Faraday e\u000bect recorded at T= 290 K\nin applied \felds ranging from 1 T to 7 T. After the initial\ndemagnetisation seen as a sharp increase in the signal at t\u0018\n0 ps, magnetisation is recovered and followed by precession\naround the e\u000bective \feld until fully damped. The lines are\n\fts to the data. The inset shows the experimental geometry\nfurther detailed in the main text.\nAHE or Faraday loops. We highlight here that the ap-\nparent `soft' contribution that shows switching close to\nzero applied \feld, is not a secondary magnetic phase, but\na signature of the small di\u000berences in the \feld-behaviour\nof the two sublattices. We also note that this behaviour\nis a result of the non-collinear magnetic order of MRG.\nA complete analysis of the dynamic properties therefore\nrequires knowledge of the anisotropy constants on both\nsublattices as well as the (at least) three intra and in-\nter sublattice exchange constants. Such an analysis is\nbeyond the scope of this article, and we limit our anal-\nysis to the simplest model of a single, e\u000bective uniaxial\nanisotropy constant Kuin the exchange approximation\nof the ferrimagnet.\nB. Dynamic properties\nWe now turn to the time-resolved Faraday e\u000bect and\nspin dynamics. Time-resolved Faraday e\u000bect data were\nrecorded at \fve di\u000berent temperatures 10 K, 50 K, 100 K,\n200 K and 290 K, with applied \felds ranging from 1 T to\n7 T.\nFIG. 3 shows the \feld-dependence of the Faraday ef-\nfect as a function of the delay between the pump and\nthe probe pulses, recorded at T= 290 K. Negative de-\nlay indicates the probe is hitting the sample before the\npump. After the initial demagnetisation, the magneti-\nsation recovers and starts precessing around the e\u000bec-\ntive \feld which is determined by the anisotropy and the\napplied \feld. The solid lines in FIG. 3 are \fts to the\ndata to extract the period and the damping of the pre-cession in each case. The \ftting model was an expo-\nnentially damped sinusoid with a phase o\u000bset. We note\nthat the apparent evolution of the amplitude and phase\nwith changing applied magnetic \feld is due to the quasi-\nresonance of the spectrum of the precessional motion\nwith the low-frequency components of the convolution\nbetween the envelope of the probe pulse and the phys-\nical relaxation of the system. The latter include both\nelectron-electron and electron-lattice e\u000bects. A rudimen-\ntary model based on a classical oscillator successfully re-\nproduces the main features of the amplitude and phase\nobserved.\nIn two-sublattice FiMs, the gyromagnetic ratios of the\ntwo sublattices are not necessarily the same. This is par-\nticularly obvious in rare-earth/transition metal alloys,\nand is also the case for MRG despite the two sublat-\ntices being chemically similar; they are both Mn. Due\nto the di\u000berent local environment however, the degree\nof charge transfer for the two di\u000bers. This leads to two\ncharacteristic temperatures, a \frst TMwhere the mag-\nnetic moments compensate, and a second TAwhere the\nangular momenta compensate. It can be shown that for\nthe ferromagnetic mode, the e\u000bective gyromagnetic ratio\n\re\u000bcan then be written25\n\re\u000b=M4c(T)\u0000M4a(T)\nM4c(T)=\r4c\u0000M4a(T)=\r4a(1)\nsubscripti= 4a;4cdenotes sublattice i,Mi(T)\nthe temperature-dependent magnetisation, and \rithe\nsublattice-speci\fc gyromagnetic ratio. \re\u000bis related to\nthe e\u000bective g-factor\nge\u000b=\re\u000bh\n\u0016B(2)\nwherehis the Planck constant and \u0016Bthe Bohr magne-\nton.\nThe frequency of the precession is determined by the\ne\u000bective \feld, which can be inferred from the derivative\nof the magnetic free energy density with respect to M.\nFor an external \feld applied at a given \fxed angle with\nrespect to the easy axis this leads to the Smit-Beljers\nformula26\n!FMR =\re\u000bvuut1\nM2ssin2\u001e\"\n\u000e2E\n\u000e\u00122\u000e2E\n\u000e\u001e2\u0000\u0012\u000e2E\n\u000e\u0012\u000e\u001e\u00132#\n(3)\nwhere\u0012and\u001eare the polar and azimuthal angles of the\nmagnetisation vector, and Ethe magnetic free energy\ndensity\nE=\u0000\u00160H\u0001M+Kusin2\u0012+\u00160M2\nscos2\u0012=2 (4)\nwhere the terms correspond to the Zeeman, anisotropy\nand demagnetising energies, respectively, and Msis the\nnet saturation magnetisation. It should be mentioned\nthat the magnetic anisotropy constant Kuis related to\nM, which is being considered constant in magnitude, via\nKu=\f\u00160M2\ns=2,\fa dimensionless parameter.4\nFIG. 4. Observed precession frequency as a function of the\napplied \feld for various temperatures. The solid lines are \fts\nto the data as described in the main text.\nBased on Eqs. (1) through (4) we \ft our entire data set\nwith\re\u000bandKuas the only free parameters. The exper-\nimental data and the associated \fts are shown as points\nand solid lines in FIG. 4. At all temperatures our simple\nmodel with one e\u000bective gyromagnetic ratio \re\u000band a\nsingle uniaxial anisotropy parameter Kureproduces the\nexperimental data reasonably well. The model systemat-\nically underestimates the resonance frequency for inter-\nmediate \felds, with the point of maximum disagreement\nincreasing with decreasing temperature. We speculate\nthis is due to the use of a simple uniaxial anisotropy in\nthe free energy (see Eq. 4), while the real situation is\nmore likely to be better represented as a sperimagnet. In\nparticular, the non-collinear nature of MRG that leads\nto a deviation from 180\u000eof the angle between the two\nsublattice magnetisations, depending on the applied \feld\nand temperature.\nFrom the \fts in FIG. 4 we infer the values of ge\u000band\nthe anisotropy \feld \u00160Ha=2Ku=Ms. The result is shown\nin FIG. 5. The anisotropy \feld is monotonically increas-\ning with decreasing temperature as the magnetisation\nof the 4csublattice increases in the same temperature\nrange. We highlight here the advantage of determining\nthis \feld through time-resolved magneto-optics as op-\nposed to static magnetometry and optics. Indeed the\nanisotropy \feld as seen by static methods is sensitive to\nthe combination of anisotropy and the netmagnetic mo-\nment, as illustrated in FIG. 1, where the coercive \feld\ndiverges as T!TM. In statics one would expect a di-\nvergence of the anisotropy \feld at the same temperature.\nThe time-resolved methods however distinguish between\nthe net and the sublattice moments, hence better re\rect-\ning the evolution of the intrinsic material properties of\nthe ferrimagnet.\nThe temperature dependence of the anisotropy con-\nstants was a matter for discussion for many years27,28.\nFIG. 5. E\u000bective g-factor,ge\u000b, and the anisotropy \feld\nas determined by time-resolved Faraday e\u000bect. ge\u000b, orange\nsquares, increases from near the free electron value of 2 to 4\njust belowTM, while the anisotropy \feld, blue triangles, in-\ncreases near-linearly with decreasing temperature. A M3\ft,\nred dashes line, of the anisotropy behaviour shows the almost-\nmetallic origin of it, indicating the dominant character of the\n4c sublattice.\nWritten in spherical harmonics the 3 danisotropy can\nbe expressed as, k2Y0\n2(\u0012) +k4Y0\n4(\u0012)29wherek2/\nM(T)3andk4/M(T)10. The experimental measured\nanisotropy is then, K2(T) =ak2(T)+bk4(T), withaand\nbthe contributions of the respective spherical harmonics.\nFIG. 5 shows that a reasonable \ft of our data is ob-\ntained with M(T)3which means, \frst, that the contri-\nbution of the 4thorder harmonic can be neglected, and\nsecond, that the contribution of the TMand 2ndsublat-\ntice is negligible, indicating the dominant character of\nthe 4c sublattice.\nIn addition, we should note here that the high fre-\nquency exchange mode was never observed on our exper-\niments. While far from TMits frequency might be too\nhigh to be observable, in the vicinity of TM, in contrast,\nits frequency is expected to be in the detection range.\nMoreover, given the di\u000berent electronic structure of the\ntwo sublattices, it is expected that the laser pulse should\nselectively excite the sublattice 4c, and therefore lead to\nthe e\u000bective excitation of the exchange mode. We argue\nthat it is the non-collinearity of the sublattices (see sec-\ntion III A) that smears out the coherent precession at\nhigh frequencies.\nThe e\u000bective gyromagnetic ratio, ge\u000b, shows a non-\nmonotonic behaviour. It increases with decreasing Tto-\nwardsTM, reaching a maximum at about 50 K before\ndecreasing again at T= 10 K. We alluded above to\nthe di\u000berence between the magnetic and the angular mo-\nmenta compensation temperatures. We expect that ge\u000b\nreaches a maximum when T=TA30, here between the\nmeasurement at T= 50 K and the magnetic compensa-\ntion temperature TM\u001980 K.5\nFIG. 6. Intrinsic and anisotropic broadening in MRG across\ntheTM. The inset shows the evaluation process of the two\ndamping parameters. A linear \ft is used to evaluate intercept\n(anisotropic broadening) and slope (intrinsic damping) of the\nfrequencies versus the inverse of the decay time. The data\npoint are obtained from the \ft of time-resolved Faraday e\u000bect\nmeasurements (an example is shown in Fig.4).\nFrom XMCD data11, we could estimate spin and or-\nbital moment components of the magnetic moments of\nthe two sublattices, what allowed us to derive the ef-\nfective g-factors for the sublattices as g4a= 2:05 and\ng4c= 2:00. In this case we expect the angular momentum\ncompensation temperature TAto be below TM, opposite\nto what is observed for GdFeCo21. Given this small dif-\nference however, TAandTMare expected to be rather\nclose to each other, consistent with the limited increase\nofge\u000bacross the compensation points.\nWe turn \fnally to the damping of the precessional mo-\ntion of Maround the e\u000bective \feld \u00160He\u000b. Damping is\nusually described via the dimensionless parameter \u000bin\nthe Landau-Lifshiz-Gilbert equation, and it is a measure\nof the dissipation of magnetic energy in the system. In\nthis model, \u000bis a scalar constant and the observed broad-\nening in the time domain is therefore a linear function of\nthe frequency of precession31{33. We infer \u000b0, the total\ndamping, from our \fts of the time-resolved Faraday e\u000bect\nas\u000b0= (\u001cd)\u00001, where\u001cdis the decay time of the \fts. We\nthen, for each temperature, plot \u000b0as a function of the\nobserved frequency and regress the data using a straight\nline \ft. The intrinsic \u000bis the slope of this line, while the\nintercept represents the anisotropic broadening.\nFIG. 6 shows the intrinsic damping \u000band the\nanisotropic broadening as a function of temperature.\nAnisotropic broadening is usually attributed to a vari-\nation of the anisotropy \feld in the region probed by the\nprobe pulse34. For MRG this is due to slight lateral vari-\nations in the Ru content xin the thin \flm sample. Such a\nvariation leads to a variation in e\u000bective TMandTAand\ncan therefore have a large in\ruence on the broadening asa function of temperature. Despite this, the anisotropic\nbroadening is reasonably low in the entire temperature\nrange above TM, and a more likely explanation for its\nrapid increase below TMis that the applied magnetic\n\feld is insu\u000ecient to completely remagnetize the sam-\nple between two pump pulses. As observed in Fig.5, the\nanisotropy \feld reaches almost 4 T at low temperature,\ncomparable to our maximum applied \feld of 7 T. The\nintrinsic damping \u000bis less than 0.02 far from TM, but\nincreases sharply at T= 100 K. We tentatively attribute\nthis to an increasing portion of the available power be-\ning transferred into the high-energy exchange mode, al-\nthough we underline that we have not seen any direct\nevidence of such a mode in any of the experimental data.\nIV. CONCLUSION\nWe have shown that the time-resolved Faraday e\u000bect\nis a powerful tool to determine the spin dynamic proper-\nties in compensated, metallic ferrimagnets. The high spin\npolarisation of MRG enables meaningful Faraday data to\nbe recorded even near TMwhere the net magnetisation\nis vanishingly small, and the dependence of the dynamics\non the sublattice as opposed to the net magnetic prop-\nerties provides a more physical understanding of the ma-\nterial. Furthermore, we \fnd that the ferromagnetic-like\nmode of MRG reaches resonance frequencies as high as\n160 GHz in zero applied \feld, together with a small in-\ntrinsic damping. This value is remarkable if compared\nto well-known materials such as GdFeCo which, at zero\n\feld, resonates at tens of GHz21or [Co/Pt] nmultilay-\ners at 80 GHz35but with higher damping. We should\nhowever stress that, in the presence of strong anisotropy\n\felds, higher frequencies can be reached. Example of that\ncan be found for ferromagnetic Fe/Pt with \u0019280 GHz\n(Ha= 10T)36, and for Heusler-like ferrimagnet (Mn 3Ge\nand Mn 3Ga) with\u0019500 GHz (Ha= 20T)37,38. Never-\ntheless, the examples cited above show a considerably\nhigher intrinsic damping compared to MRG. In addi-\ntion, it was recently shown that MRG exhibits unusu-\nally strong intrinsic spin-orbit torque20. 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Our main findings include:\n(1) in the absence of Gilbert damping, the spin wave propagat es at a constant velocity with the\nincrement proportional tothe strength of the magnetic field ; (2) in the absence of magnetic field, at a\ngiven time the spin wave converges exponentially fast to its initial profile as the damping parameter\ngoes to zero and in the long time the relaxation dynamics of th e spin wave converges exponentially\nfast to the easy-axis direction with the exponent proportio nal to the damping parameter; (3) in\nthe presence of both Gilbert damping and magnetic field, the s pin wave converges to the easy-axis\ndirection exponentially fast at a small timescale while pro pagates at a constant velocity beyond\nthat. These provides a comprehensive understanding of spin waves in ferromagnetic materials.\nPACS numbers: 05.45.Yv, 75.70.-i, 75.78.-n\nI. INTRODUCTION\nA spin wave is the disturbance of intrinsic spin order in magnetic mater ials. It is usually excited using magnetic\nfields and offers unique properties such as charge-less propagatio n and high group velocities, which are important for\nsignal transformations and magnetic logic applications [1–6].\nThepropagationofspinwavesisdescribedbytheLandau-Lifshitz- Gilbert(LLG) equation[7,8]in thedimensionless\nform\nmt=−m×h−αm×(m×h), (1)\nwhere the magnetization m= (m1,m2,m3)Tis a three dimensional vector with unit length, αis the Gilbert damping\nparameter. The effective field hincludes the exchange term, the anisotropy term with easy axis alon g the x-axis and\nthe anisotropy constant q, and the external field\nh= ∆m+qm1e1+hexte1. (2)\nhextis the strength of the external field applied along the x-axis with e1the unit vector. This model is often used to\ndescribe the magnetization dynamics in ferromagnetic thin films.\nFrom a theoretical perspective, a spin wave is known as a solitotary wave, which appears as the solution of a weakly\nnonlinear dispersive partial differential equation. In LLG equation ( 1)-(2), a soliton is caused by the cancellation of\nnonlinear and dispersive effects in the magnetic material. Solitons are of interests for quite a long time [9–13]. Most\nof works consider the one dimensional case and drop the damping te rm [9, 10, 13]. In [11], using the stereographic\nprojection, the authors found that the presence of Gilbert damp ing was merely a rescaling of time by a complex\nconstant. However, this was found to be valid only for a single spin in a constant magnetic field [12].\nIn this work, we give a comprehensive study of an explicit spin wave in t he LLG equation. Our starting point is\nthat the spin wave maintains its shape while it propagates at a consta nt velocity and the derivation is based on the\ngeneralization of the method of characteristics. The main findings a re: (1) in the absence of Gilbert damping, the\nspin wave propagates at a constant velocity with the increment pro portional to the strength of the magnetic field; (2)\nin the absence of magnetic field, at a given time the spin wave converg es exponentially fast to its initial profile as the\ndamping parameter goesto zeroand in the long time the relaxationdy namics of the spin waveconvergesexponentially\n∗Electronic address: jingrunchen@suda.edu.cn2\nfast to the easy-axis direction with the exponent proportional to the damping parameter; (3) in the presence of both\nGilbert damping and magnetic field, the spin wave converges to the ea sy-axis direction exponentially fast at a small\ntimescale while propagates at a constant velocity beyond that.\nII. DERIVATION AND RESULTS\nAs mentioned above, we start with the assumption that a spin wave m aintains its shape while it propagates at a\nconstant velocity. This can be seen from the method characterist ics in simple situations.\nIn 1D when α=q=hext= 0, one can check that\nm(x,t) =\ncosθ0\nsinθ0cos/parenleftig\nc\ncosθ0(x+ct)/parenrightig\nsinθ0sin/parenleftig\nc\ncosθ0(x+ct)/parenrightig\n(3)\nsolvesmt=−m×mxx. Hereθ0is determined by the initial condition and u=x+ctis the characteristic line. (3)\nprovides a solitary solution with the traveling speed c. A detailed derivation of (3) can be found in Chapter 2 of [13].\nA generalization of the method of characteristics yields a spin wave t omt=−m×∆m\nm(x,t) =\ncosθ0\nsinθ0cosv\ncosθ0\nsinθ0sinv\ncosθ0\n, (4)\nwherev=c1x+c2y+c3z+(c2\n1+c2\n2+c2\n3)t=c·x+(c·c)twithc= (c1,c2,c3)T. The speed field is cwith magnitude\n|c|. Actually, both (3) and (4) can be rewritten as\nm(x,t) =\ncosθ0\nsinθ0cos(w0·x+ϕ(t))\nsinθ0sin(w0·x+ϕ(t))\n, (5)\nwherew0=c/cosθ0andϕ(t) =/parenleftbig\n|c|2/cosθ0/parenrightbig\nt.\n(3)-(5) are obtained in the absence of Gilbert damping. In order to take the Gilbert damping and the other terms\nin (2) into account, we make an ansatz for the spin wave profile in the following form\nm(x,t) =\ncosθ(t)\nsinθ(t)cos(w0·x+ϕ(t))\nsinθ(t)sin(w0·x+ϕ(t))\n, (6)\nwhereθandϕare independent of xand only depend on t.\nSubstituting (6) into (2) and (1) and denoting w0·x+ϕ(t) byu(x,t), we have\nh=\n0\n−|w0|2sinθcosu(x,t)\n−|w0|2sinθsinu(x,t)\n+q\ncosθ\n0\n0\n+hext\n1\n0\n0\n,\nm×h=\n0\n|w0|2sinθcosθsinu(x,t)\n−|w0|2sinθcosθcosu(x,t)\n+q\n0\nsinθcosθsinu(x,t)\n−sinθcosθcosu(x,t)\n+hext\n0\nsinθsinu(x,t)\n−sinθcosu(x,t)\n,\nm×(m×h) =\n−|w0|2sin2θcosθ\n|w0|2sinθcos2θcosu(x,t)\n|w0|2sinθcos2θsinu(x,t)\n+q\n−sin2θcosθ\nsinθcos2θcosu(x,t)\nsinθcos2θsinu(x,t)\n+hext\n−sin2θ\nsinθcosθcosu(x,t)\nsinθcosθsinu(x,t)\n,\nand\nmt=\n−θtsinθ\nθtcosθcosu(x,t)−ϕtsinθsinu(x,t)\nθtcosθsinu(x,t)+ϕtsinθcosu(x,t)\n.\nAfter algebraic simplifications, we arrive at\n/braceleftbigg\nθt=−α(|w0|2+q)sinθcosθ−αhextsinθ\nϕt= (|w0|2+q)cosθ+hext. (7)3\nA. The absence of Gilbert damping\nWhenα= 0, we have θ=θ0andϕ=/parenleftbig\n(|w0|2+q)cosθ0+hext/parenrightbig\nt. Therefore we have the solution\nm=\ncosθ0\nsinθ0cos/parenleftbig\nw0·x+t/parenleftbig\n|w0|2cosθ0+qcosθ0+hext/parenrightbig/parenrightbig\nsinθ0sin/parenleftbig\nw0·x+t/parenleftbig\n|w0|2cosθ0+qcosθ0+hext/parenrightbig/parenrightbig\n. (8)\nNote that this recovers (5) when q= 0 and hext= 0. It is easy to see that the spin wave (8) propagates at a consta nt\nvelocity. The increment of the velocity field is qcosθ0w0\n|w0|2with magnitude|qcosθ0|\n|w0|, due to the magnetic anisotropy.\nThe increment of the velocity field is hextw0\n|w0|2with magnitude|hext|\n|w0|, due to the magnetic field.\nB. The absence of magnetic field\nWhenhext= 0, (7) reduces to\n/braceleftbigg\nθt=−α(|w0|2+q)sinθcosθ\nϕt= (|w0|2+q)cosθ. (9)\nFor the first equation in (9), assuming 0 ≤θ0< π/2, by separation of variables, we have\nα(|w0|2+q)t+C1= lncotθ,\nwhereC1is a constant determined by the initial condition.\nDenote˜t=α(|w0|2+q)t+C1. It follows that\ntanθ=e−˜t, (10)\nfrom which one has\ncosθ=1/radicalbig\n1+e−2˜t, (11)\nsinθ=1/radicalbig\n1+e2˜t. (12)\nWhent= 0, (11) turns to\ncosθ0=1√\n1+e−2C1, (13)\nfrom which we can determine C1by the initial condition θ0.\nAs forϕ, from the second equation in (9), one has that\ndϕ\ndθ=dϕ\ndt·dt\ndθ=−1\nαsinθ.\nTherefore\nαϕ=−/integraldisplaydθ\nsinθ=1\n2ln/parenleftbigg1+cosθ\n1−cosθ/parenrightbigg\n+C2= lncot1\n2θ+C2, (14)\nwhere\nC2=−1\n2ln/parenleftbigg1+cosθ0\n1−cosθ0/parenrightbigg\n.\nSubstituting (11) and (13) into (14) yields\nϕ=1\nαln/parenleftigg\ne˜t+/radicalbig\ne2˜t+1\neC1+√\ne2C1+1/parenrightigg\n=1\nα/parenleftbigg\nlncot1\n2θ−lncot1\n2θ0/parenrightbigg\n. (15)4\nIn short summary, the spin wave when α/negationslash= 0 takes the form\nm=1/radicalbig\n1+e2˜t\ne˜t\ncos(w0·x+ϕ)\nsin(w0·x+ϕ)\n. (16)\nThe above derivation is valid when 0 ≤θ0< π/2. Ifπ/2< θ0≤π, we choose the other solution of (11)\ncosθ=−1/radicalbig\n1+e−2˜t, (17)\nand\nϕ=−1\nαln/parenleftigg\ne˜t+/radicalbig\ne2˜t+1\neC1+√\ne2C1+1/parenrightigg\n.\n(16) remains unchanged.\nWhenα→0, we have ˜t→C1and\nlim\nα→0m=1√\n1+e2C1\neC1\ncos/parenleftig\nw0·x+ lim\nα→0ϕ/parenrightig\nsin/parenleftig\nw0·x+ lim\nα→0ϕ/parenrightig\n.\nBy L’Hospital’s rule, one has that\nlim\nα→0ϕ= lim\nα→0d\ndα/parenleftigg\nln/parenleftigg\ne˜t+/radicalbig\ne2˜t+1\neC1+√\ne2C1+1/parenrightigg/parenrightigg\n= lim\nα→0e˜t\n/radicalbig\ne2˜t+1(|w0|2+q)t=eC1\n√\neC1+1(|w0|2+q)t.(18)\nTherefore it follows that\nlim\nα→0m=\ncosθ0\nsinθ0cos/parenleftbig\nw·x+cosθ0/parenleftbig\n|w0|2+q/parenrightbig\nt/parenrightbig\nsinθ0sin/parenleftbig\nw·x+cosθ0/parenleftbig\n|w0|2+q/parenrightbig\nt/parenrightbig\n. (19)\nThis is exactly the solution (8) when hext= 0.\nIn addition, when α→0, bothθandϕconverges exponentially fast to initial conditions; see equations (1 1), (12),\n(13), and (18). Therefore, at a giventime t, (19) convergesexponentiallyfast to the initial spin wave(8) when hext= 0\nwith the exponent proportionalto the damping parameter α. Moreover,in the long time, i.e., when t→+∞,˜t→+∞\nandθ→0, (16) converges to (1 ,0,0)T(the easy-axis direction) exponentially fast with the rate proport ional to the\ndamping parameter α. Whenπ/2< θ0≤π, from (17), we have that (16) converges to ( −1,0,0)T(again the easy-axis\ndirection) exponentially fast with the rate proportional to the dam ping parameter α.\nIt is easy to check that the right-hand side of (19) is the solution of (1) when hext= 0 with the initial condition\nθ0=π/2. Therefore, Gilbert damping does not have any influence on magne tization dynamics in this case.\nIn [11], the authors used the stereographic projection and obser ved that the effect of Gilbert damping was only a\nrescaling of time by a complex constant. However, this was latter fo und to be valid only for a single spin in a constant\nmagnetic field [12]. Our result provides an explicit characterization of magnetization dynamics in the presence of\nGilbert damping.\nC. The presence of both Gilbert damping and magnetic field\nIt is difficult to get the explicit solution of (7) in general. To understan d the magnetization dynamics, we use the\nmethod of asymptotic expansion. For small external magnetic field ,θandϕadmit the following expansions\nθ(t,hext) =θ0(t)+θ1(t)hext+θ2(t)h2\next+···,\nϕ(t,hext) =ϕ0(t)+ϕ1(t)hext+ϕ2(t)h2\next+···.5\nTherefore one has that\nθt(t,hext) =θ0\nt(t)+θ1\nt(t)hext+θ2\nt(t)h2\next+···, (20)\nϕt(t,hext) =ϕ0\nt(t)+ϕ1\nt(t)hext+ϕ2\nt(t)h2\next+···. (21)\nOn the other hand, from (7), it follows that\nθt=−α(|w0|2+q)sinθ0cosθ0−/parenleftbig\nα(|w0|2+q)θ1cos2θ0+αsinθ0/parenrightbig\nhext+···, (22)\nϕt= (|w0|2+q)cosθ0+/parenleftbig\n−(|w0|2+q)θ1sinθ0+1/parenrightbig\nhext+···. (23)\nCombining (20) and (21) with (22) and (23), for the zero-order te rm, one has\n/braceleftbigg\nθ0\nt=−α(|w0|2+q)sinθ0cosθ0\nϕ0\nt= (|w0|2+q)cosθ0 , (24)\nwhich recovers (9) with solution (10) and (15).\nAs for the first-order term, one has that\n/braceleftbigg\nθ1\nt=−α(|w0|2+q)θ1cos2θ0−αsinθ0\nϕ1\nt=−(|w0|2+q)θ1sinθ0+1. (25)\nUsing variation of parameters, one can assume θ1=C(t)\ne˜t+e−˜tand it follows that\nC′(t) =−α(e˜t+e−˜t)sinθ0=−α(tanθ0+tan−1θ0)sinθ0.\nSince\n/integraldisplay\n−αtanθ0sinθ0dt=/integraldisplay\n−α(|w0|2+q)sinθ0cosθ0∗(|w0|2+q)−1sinθ0\ncos2θ0dt\n=/integraldisplay\n(|w0|2+q)−1sinθ0\ncos2θ0dθ0\n=(|w0|2+q)−11\ncosθ0,\nand\n/integraldisplay\n−αtan−1θ0sinθ0dt=/integraldisplay\n−αcosθ0dt\n=−α(|w0|2+q)−1ϕ0,\none can get C(t) = (|w0|2+q)−1(1\ncosθ0−αϕ0), and it follows that\nθ1= (|w0|2+q)−1(sinθ0−αsinθ0cosθ0ϕ0). (26)\nSubstituting the first equation in (24) into the second equation in (2 5), one has\n−/integraldisplay\n(|w0|2+q)θ1sinθ0dt\n=1\nα/integraldisplayθ1\ncosθ0dθ0\n=1\nα(|w0|2+q)/integraldisplay\ntanθ0−sinθ0(lncot1\n2θ0+C2)dθ0(using (26))\n=−t+1\n|w0|2+q(ϕ0cosθ0+α−1C1),\nand thus\nϕ1=1\n|w0|2+q(ϕ0cosθ0+α−1C1). (27)6\nTherefore, when hextis small, it has the approximate solution\n/braceleftbigg\nθ∗=θ0+(|w0|2+q)−1(sinθ0−αsinθ0cosθ0ϕ0)hext\nϕ∗=ϕ0+(|w0|2+q)−1(ϕ0cosθ0+α−1C1)hext(28)\nwithθ0andϕ0satisfying (9).\nFrom (10) and (15), θ0converges exponentially fast to the easy-axis direction, while ϕ0grows linearly. Therefore,\nfrom (26), θ1converges exponentially fast to 0 as well with a larger exponent. Th is relaxation dynamics happens at\na small timescale.\nMeanwhile, from (25), the difference between ϕ∗andϕ0satisfies\nϕ∗\nt−ϕ0\nt=ϕ1\nthext=−(|w0|2+q)θ1sinθ0hext+hext. (29)\nSinceθ1sinθ0convergesto 0 at a small timescale, the dynamics of ϕ∗−ϕ0is determined by the external field at longer\ntimescales. As a consequence, the increment of the velocity field is hextw0\n|w0|2with magnitude|hext|\n|w0|. This validates the\nWalker’s ansatz [14] for a spin wave.\nWhenπ/2< θ0≤πand the magnetic field is applied along the negative x-axis, and if ( |w|2+q)|cosθ0| ≤hext, the\nresult above will be correct.\nNote that θ0=π/2 does not fall into the above two cases since the magnetization dyn amics will change the spin\nwave profile. In fact, as t→+∞,θ→0 if the magnetic field is applied along the positive x-axis direction and θ→π\nif the magnetic field is applied along the negative x-axis direction.\nIII. CONCLUDING REMARKS\nIn this work, we study the magnetization dynamics in Landau-Lifshit z-Gilbert equation. By generalizing the\nmethod of characteristics, we are able to have an explicit characte rization of spin wave dynamics in the presence of\nboth Gilbert damping and magnetic field. Gilbert damping drives the spin wave converge exponentially fast to the\neasy-axis direction with the exponent proportional to the damping parameter at a small timescale and the magnetic\nfield drives the spin wave propagate at a constant velocity at longer timescales.\nIt will be of interests whether the technique developed here applies to the antiferromagnetic case [15, 16] and how\nrigorous the results obtained here can be proved from a mathemat ical perspective.\nIV. ACKNOWLEDGEMENTS\nWe thank Professor Yun Wang for helpful discussions. This work wa s partially supported by National Natural\nScience Foundation of China via grant 21602149 and 11971021.\n[1] M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Applied Physics Letters 87, 153501 (2005).\n[2] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi,\net al., Nature 464, 262 (2010).\n[3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebra nds, Nature Physics 11, 453 (2015).\n[4] S. Woo, T. Delaney, and G. S. D. Beach, Nature Physics (201 7).\n[5] A. V. Chumak and H. Schultheiss, Journal of Physics D: App lied Physics 50, 300201 (2017).\n[6] M. Langer, R. A. Gallardo, T. Schneider, S. Stienen, A. Ro ld´ an-Molina, Y. Yuan, K. Lenz, J. Lindner, P. Landeros, and\nJ. Fassbender, Physical Review B 99(2019).\n[7] L. Landau and E. Lifshitz, Physikalische Zeitschrift de r Sowjetunion 8, 153 (1935).\n[8] T. Gilbert, Physical Review 100, 1243 (1955).\n[9] K. Nakamura and T. Sasada, Physics Letters A 48, 321 (1974).\n[10] H. J. Mikeska, Journal of Physics C: Solid State Physics 11, L29 (1977).\n[11] M. Lakshmanan and K. Nakamura, Physical Review Letters 53, 2497 (1984).\n[12] E. Magyari, H. Thomas, and R. Weber, Physical Review Let ters56, 1756 (1986).\n[13] B. Guo and S. Ding, Landau-Lifshitz Equation (World Scientific, 2007).\n[14] N. L. Schryer and L. R. Walker, Journal of Applied Physic s45, 5406 (1974).\n[15] H. J. Mikeska, Journal of Physics C: Solid State Physics 13, 2913 (1980).\n[16] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Reviews of Modern Physics 90, 015005 (2018)."    },    {        "title": "1910.11692v2.The_lifespan_of_solutions_of_semilinear_wave_equations_with_the_scale_invariant_damping_in_two_space_dimensions.pdf",        "content": "arXiv:1910.11692v2  [math.AP]  17 Apr 2020The lifespan of solutions of semilinear wave\nequations with the scale-invariant damping in\ntwo space dimensions\nTakuto Imai∗, Masakazu Kato†, Hiroyuki Takamura‡\nand Kyouhei Wakasa§\nKeywords: semilinear wave equation, scale-invariant damping, lifespan\nMSC2010: primary 35L71, secondary 35B44\nAbstract\nIn this paper, we study the initial value problem for semilin ear\nwave equations with the time-dependent and scale-invarian t damping\nin two dimensions. Similarly to the one dimensional case by K ato,\nTakamura and Wakasa in 2019, we obtain the lifespan estimate s of\nthe solution for a special constant in the damping term, whic h are\nclassified by total integral of the sum of the initial positio n and speed.\nThe key fact is that, only in two space dimensions, such a spec ial\nconstant in the damping term is a threshold between “wave-li ke” do-\nmain and “heat-like” domain. As a result, we obtain a new type of\nestimate especially for the critical exponent.\n∗Accenture Japan Ltd, Harumi Triton Square Office TowerZ, 1-8-1 2Harumi, Chuo-ku,\nTokyo, 104-0053, Japan. e-mail: takuto.imai@accenture.com.\n†College of Liberal Arts, Mathematical Science Research Unit, Muro ran Insti-\ntute of Technology, 27-1 Mizumoto-cho, Muroran, Hokkaido 050- 8585, Japan. email:\nmkato@mmm.muroran-it.ac.jp.\n‡Mathematical Institute, Tohoku University, Aoba, Sendai 980-8 578, Japan. e-mail:\nhiroyuki.takamura.a1@tohoku.ac.jp.\n§Department of Creative Engineering, National Institute of Techn ology, Kushiro\nCollege, 2-32-1 Otanoshike-Nishi, Kushiro-Shi, Hokkaido 084-0916 , Japan. e-mail:\nwakasa@kushiro-ct.ac.jp.\n11 Introduction\nWe areconcerned withthe following initial valueproblem for semilinear w ave\nequations with the scale-invariant damping:\n/braceleftBigg\nvtt−∆v+µ\n1+tvt=|v|pinRn×[0,∞),\nv(x,0) =εf(x), vt(x,0) =εg(x), x∈Rn,(1.1)\nwherev=v(x,t) is a real valued unknown function, µ >0,p >1,n∈N,\nthe initial data ( f,g)∈H1(Rn)×L2(Rn) has compact support, and ε >0\nis a “small” parameter.\nIt is interesting to look for the critical exponent pc(n) such that\n/braceleftbiggp > pc(n) (and may have an upper bound) = ⇒T(ε) =∞,\n1< p≤pc(n) = ⇒T(ε)<∞,\nwhereT(ε) is the lifespan, the maximal existence time, of the energy solution\nof (1.1) with an arbitrary fixed non-zero data. Then, we have the f ollowing\nconjecture:\n\n\nµ > µ0(n) =⇒pc(n) =pF(n) (heat-like) ,\nµ=µ0(n) =⇒pc(n) =pF(n) =pS(n+µ) (intermediate) ,\n0< µ < µ 0(n) =⇒pc(n) =pS(n+µ) (wave-like) ,\n(1.2)\nwhere\nµ0(n) :=n2+n+2\nn+2. (1.3)\nHere\npF(n) := 1+2\nn(1.4)\nis the so-called Fujita exponent which is the critical exponent of the associ-\nated semilinear heat equations vt−∆v=vp, and\npS(n) :=\n\n∞ (n= 1),\nn+1+√\nn2+10n−7\n2(n−1)(n≥2)(1.5)\nis the so-called Strauss exponent which is the critical exponent of t he associ-\nated semilinear wave equations vtt−∆v=|v|p. We note that pS(n) (n≥2)\nis the positive root of\nγ(p,n) := 2+( n+1)p−(n−1)p2= 0 (1.6)\n2and 0< µ < µ 0(n) is equivalent to pF(n)< pS(n+µ).\nThe conjecture (1.2) shows the critical situation of our problem in t he\nfollowing sense. If one replaces the damping term µvt/(1 +t) in (1.1) by\nµvt/(1+t)β, then one can see that there is no such a pc(n), namely T(ε) =∞\nfor anyp >1 whenβ <−1, the so-called over damping case. Moreover one\nhaspc(n) =pF(n) for any µ >0 when−1≤β <1, the so-called effective\ndamping case, and pc(n) =pS(n) for any µ >0 (it can be any µ∈R) when\nβ >1, the so-called scattering damping case. Therefore one may say t hat\nthe so-called scale-invariant case, β= 1, is an intermediate situation between\nwave-like, in which the critical exponent is related to pS(n), and heat-like, in\nwhich the critical exponent is pF(n). To see all the references above results,\nfor example, see introductions of related papers to the scatterin g damping\ncase, Lai and Takamura [15] (the sub-critical case), Wakasa and Yordanov\n[22] (the critical case), Liu and Wang [17] (partial result of the sup er-critical\ncase).\nFor the conjecture (1.2), D’Abbicco [3] has obtained the heat-like e xis-\ntence partially with\nµ≥\n\n5/3 forn= 1 (cf.µ0(1) = 4/3),\n3 for n= 2 (cf.µ0(2) = 2),\nn+2 for n≥3,\nwhile Wakasugi [25] has obtained the blow-up parts in 1 < p < p F(n) for\nµ≥1 and 1< p < p F(n+µ−1) for 0< µ <1. In the damped case µ >0,\nhis second result is the first blow-up result for super-Fujita expon ents which\nare larger than pF(n).\nIn this paper, we consider a special case of µ= 2. The speciality of this\nvalue is clarified by setting\nu(x,t) := (1+ t)µ/2v(x,t),\nwherevis the solution to (1.1). Then, usatisfies\n\n\nutt−∆u+µ(2−µ)\n4(1+t)2u=|u|p\n(1+t)µ(p−1)/2inRn×[0,∞),\nu(x,0) =εf(x), ut(x,0) =ε{µf(x)/2+g(x)}, x∈Rn,(1.7)\nso that all the technics in the analysis on semilinear wave equations ca n be\nemployed and we may discussed about not only the energy solution bu t also\nthe classical solution. In fact, via this reduced problem (1.7), D’Abb icco,\nLucente and Reissig [5] have proved the intermediate part of the co njecture\n(1.2)forn= 2andthewave-like partfor n= 3when µ= 2. Wenotethatthe\n3assumption of the radial symmetry is considered in [5] for the existe nce part\ninn= 3. Moreover, D’Abbicco and Lucente [4] have obtained the wave-lik e\nexistence part of (1.2) for odd n≥5 whenµ= 2 also with radial symmetry.\nIn the case of µ/ne}ationslash= 2, Lai, Takamura and Wakasa [16] have first studied\nthe wave-like blow-up of the conjecture (1.2) with a loss replacing µbyµ/2\nin the sub-critical case. Initiating this result, Ikeda and Sobajima [8 ] have\nobtained the blow-up part of (1.2).\nFor the lifespan estimate, one may expect that\nT(ε)∼/braceleftbigg\nCε−(p−1)/{2−n(p−1)}for 1< p < p F(n),\nexp/parenleftbig\nCε−(p−1)/parenrightbig\nforp=pF(n)(1.8)\nfor the heat-like domain µ > µ0(n) and\nT(ε)∼/braceleftbigg\nCε−2p(p−1)/γ(p,n+µ)for 1< p < p S(n+µ),\nexp/parenleftbig\nCε−p(p−1)/parenrightbig\nforp=pS(n+µ)(1.9)\nfor the wave-like domain 0 < µ < µ 0(n). Recall the definitions of µ0(n),\npF(n),pS(n) andγ(p,n) in (1.3), (1.4), (1.5) and (1.6). Here T(ε)∼A(ε,C)\nstands for the fact that there are positive constants, C1andC2, independent\nofεsatisfying A(ε,C1)≤T(ε)≤A(ε,C2). Actually, (1.8) for n= 1 and\nµ= 2> µ0(1) = 4/3 is obtained by Wakasa [24], and (1.9) is obtained by\nKato and Sakuraba [12] for n= 3 andµ= 2< µ0(3) = 14/5. One may refer\nLai [14] for the existence part of weaker solution. Moreover, the upper bound\nof (1.8) in the sub-critical case is obtained by Wakasugi [25]. Also the upper\nbound of (1.9) is obtained by Ikeda and Sobajima [8] in the critical cas e,\nlater it is reproved by Tu and Li [21], and Tu and Li [20] in the sub-critic al\ncase.\nIn the non-damped case of µ= 0, it is known that (1.9) is true for n≥3,\norp >2 andn= 2, The open part around this fact is p=pS(n) forn≥9.\nIn other cases, (1.9) is still true if/integraltext\nRng(x)dx= 0. On the other hand, we\nhave\nT(ε)∼\n\nCε−(p−1)/2forn= 1,\nCε−(p−1)/(3−p)forn= 2 and 1 < p <2,\nCa(ε) for n= 2 and p= 2(1.10)\nif/integraltext\nRng(x)dx/ne}ationslash= 0, where a=a(ε) is a positive number satisfying ε2a2log(1+\na) = 1. We note that the bounds in (1.10) are smaller than the one of th e\nfirst line in (1.9) with µ= 0 in each case. For all the references in the case\nofµ= 0, see Introduction of Imai, Kato, Takamura and Wakasa [10].\nThe remarkable fact is that even if µis in the heat-like domain, the\nlifespan estimate for (1.1) is similar to the one for non-damped case. Indeed,\n4forn= 1 and µ= 2> µ0(1) = 4/3, Kato, Takamura and Wakasa [13]\nshow that the result on (1.8) by Wakasa [24] mentioned above is true only if/integraltext\nR{f(x)+g(x)}dx/ne}ationslash= 0. More precisely, they have obtained that\nT(ε)∼\n\nCε−2p(p−1)/γ(p,3)for 1< p <2,\nCb(ε) for p= 2,\nCε−p(p−1)/(3−p)for 2< p <3,\nexp(Cε−p(p−1)) forp=pF(1) = 3,(1.11)\nif/integraltext\nR{f(x) +g(x)}dx= 0, where b=b(ε) is a positive number satisfying\nε2blog(1+b) = 1. We note that the bounds in (1.11) are larger than those\nin (1.8) with n= 1 and µ= 2 in each case.\nOur aim in this paper is to show the lifespan estimates for (1.1) in two\ndimensional case, n= 2, with µ= 2 which is similar to one dimensional case\nas above. We note pc(2) =pF(2) =pS(2 + 2) = 2 and µ0(2) = 2. More\nprecisely, we shall show that\nT(ε)∼/braceleftbiggcε−(p−1)/(4−2p)for 1< p <2,\nexp(cε−1/2) for p= 2(1.12)\nif/integraltext\nR2{f(x)+g(x)}dx/ne}ationslash= 0, and\nT(ε)∼/braceleftbigg\ncε−2p(p−1)/γ(p,4)for 1< p <2,\nexp(cε−2/3) for p= 2(1.13)\nif/integraltext\nR2{f(x)+g(x)}dx= 0. We note that the critical cases in (1.12) and\n(1.13) are new in the sense that they are different from (1.8) and (1 .9).\n(1.12) and (1.13) are announced in Introduction of Lai and Takamu ra [15],\nbut there are typos in the exponents of εin the critical case.\nThestrategyofproofsinthispaperisbasedonpoint-wiseestimate s ofthe\nsolution. Intheexistence part, weemploy theclassical iterationar gument for\nsemilinear wave equations without damping term, which is first introdu ced\nby John [11] in three space dimensions, and its variant, which is develo ped\nby Imai, Kato, Takamura and Wakasa [10] in two space dimensions. In the\nblow-up part, we also employ an improved version of Kato’s lemma on or di-\nnary differential inequality by Takamura [18] for the sub-critical ca ses. We\nnote that, till now, the so-called test function method such as in Ik eda, Soba-\njima and Wakasa [9] cannot be applicable to delicate analysis to catch t he\nlogarithmic growthof thesolution inthecase of p= 2 in(1.10), (1.11), (1.12)\nand (1.13). Therefore we employ the so-called slicing method of the b low-up\ndomain for the critical case, which is introduced by Agemi, Kurokawa and\nTakamura [1] to handle weakly coupled systems of non-damped semilin ear\nwave equations with critical exponents.\n5This paper is organized as follows. In the next section, our goals, (1 .12)\nand (1.13), are described in four theorems, and we introduce the lin ear decay\nestimate and basic lemmas for a-priori estimates. Section 3, or Sec tion 4,\nis devoted to the proof of the lower bound, or upper bound, of the lifespan\nrespectively.\n2 Theorems and preliminaries\nIn this section, we state our results (1.12) and (1.13) in four theor ems. After\nthem, we list useful point-wise estimates of linear wave equations. F or the\nsake of the simplicity, we assume that\nsupp(f,g)⊂ {x∈R2:|x| ≤k}, k≥1 (2.1)\nthroughout this paper.\nThe existence parts of our goals in (1.12) and (1.13) are guarantee d by\nthe following two theorems. Recall the definitions of µ0(n),pF(n),pS(n) and\nγ(p,n) respectively in (1.3), (1.4), (1.5) and (1.6).\nTheorem 2.1 Letn= 2,µ=µ0(2) = 2and1< p≤pF(2) =pS(4) = 2.\nAssume that (f,g)∈C3\n0(R2)×C2\n0(R2)satisfies (2.1). Then, there exists a\npositive constant ε0=ε0(f,g,p,k)such that (1.7) admits a unique solution\nu∈C2(R2×[0,T))ifp= 2, or the integral equation associated with (1.7)\nadmits a unique solution u∈C1(R2×[0,T))otherwise, as far as Tsatisfies\nT≤/braceleftbiggcε−(p−1)/(4−2p)if1< p <2,\nexp(cε−1/2)ifp= 2\nfor0< ε≤ε0, wherecis a positive constant independent of ε.\nTheorem 2.2 Suppose that the assumptions in Theorem 2.1 are fulfilled.\nAssume additionally that\n/integraldisplay\nR2{f(x)+g(x)}dx= 0.\nThen, there exists a positive constant ε0=ε0(f,g,p,k)such that (1.7) admits\na unique solution u∈C2(R2×[0,T))ifp= 2, or the integral equation\nassociated with (1.7) admits a unique solution u∈C1(R2×[0,T))otherwise,\nas far as Tsatisfies\nT≤/braceleftbiggcε−2p(p−1)/γ(p,4)if1< p <2,\nexp(cε−2/3)ifp= 2\nfor0< ε≤ε0, wherecis a positive constant independent of ε.\n6On the other hand, the blow-up parts of our goals in (1.12) and (1.13 )\nare guaranteed by the following two theorems.\nTheorem 2.3 Letn= 2,µ=µ0(2) = 2,1< p≤2 =pF(2) =pS(4).\nAssume that (f,g)∈C2\n0(R2)×C1\n0(R2)satisfies (2.1). Suppose that the\nintegral equation associated with (1.7) has a solution u∈C1(R2×[0,T))with\nsuppu⊂ {(x,t)∈R2×[0,∞) :|x| ≤t+k}. Then, there exists a positive\nconstant ε1=ε1(f,g,p,k)such that the solution cannot exist whenever T\nsatisfies\nT≥\n\ncε−(p−1)/(4−2p)if1< p <2, f(x)≡0andg(x)≥0 (/ne}ationslash≡0),\nexp(cε−1/2)ifp= 2and/integraldisplay\nR2{f(x)+g(x)}dx >0\nfor0< ε≤ε1, wherecis a positive constant independent of ε.\nTheorem 2.4 Suppose that the assumptions in Theorem 2.3 are fulfilled.\nAssume additionally that\nf(x)+g(x)≡0.\nThen, there exists a positive constant ε1=ε1(f,g,p,k)such that the solution\ncannot exist whenever Tsatisfies\nT≥\n\ncε−2p(p−1)/γ(p,4)if1< p <2andf(x)≥0 (/ne}ationslash≡0),\nexp(cε−2/3)ifp= 2and/integraldisplay\nR2f(x)dx <0\nfor0< ε≤ε1, wherecis a positive constant independent of ε.\nFrom now on, we introduce some definitions and useful lemmas. For\n(x,t)∈R2×[0,∞), we set\nuL(x,t) :=∂\n∂tR(f|x,t)+R(f+g|x,t),\nR(φ|x,t) :=1\n2π/integraldisplay\n|x−y|≤tφ(y)/radicalbig\nt2−|x−y|2dy=t\n2π/integraldisplay\n|ξ|≤1φ(x+tξ)/radicalbig\n1−|ξ|2dξ.(2.2)\nWhen (f,g)∈C3\n0(R2)×C2\n0(R2), we note that uLsatisfies that\n/braceleftbigg(uL)tt−∆uL= 0 in R2×[0,∞),\nuL(x,0) =f(x),(uL)t(x,0) =f(x)+g(x), x∈R2\nin the classical sense, and it holds\nsuppuL⊂ {(x,t)∈R2×[0,∞) :|x| ≤t+k}.\nWe introduce the decay estimates for the solutions of (2.2) which will be used\nin the proof of Theorem 2.1 and Theorem 2.2. For the proof, see Lem ma 2.1\nin [10].\n7Lemma 2.1 (Imai, Kato, Takamura and Wakasa [10]) LetuLbe the\none in (2.2). Then, there exist positive constants /tildewiderC0=/tildewiderC0(k)andC0=\nC0(/ba∇dblf/ba∇dblW3,1(R2),/ba∇dblg/ba∇dblW2,1(R2),k)such that uLsatisfies\n/summationdisplay\n|α|≤1|∇α\nxuL(x,t)|\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR2{f(x)+g(x)}dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle·/tildewiderC0\n(t+|x|+2k)1/2(t−|x|+2k)1/2\n+C0\n(t+|x|+2k)1/2(t−|x|+2k)3/2\ninR2×[0,∞).\nNext, we prepare the following decay estimate which will be employed in\nthe proof of Theorem 2.3 and Theorem 2.4.\nLemma 2.2 LetuLbe the one in (2.2). For t−|x| ≥2kandt≥4k, there\nexists a positive constant C=C(f,g)such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleuL(x,t)−/integraldisplay\nR2{f(x)+g(x)}dx\n2π(t+|x|)1/2(t−|x|)1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ck\n(t+|x|)1/2(t−|x|)3/2,(2.3)\nmoreover that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleuL(x,t)+t/integraldisplay\nR2f(x)dx\n2π(t+|x|)3/2(t−|x|)3/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ck\n(t+|x|)1/2(t−|x|)5/2(2.4)\nprovided f(x)+g(x)≡0.\nProof. First we shall prove (2.4). Denote r:=|x|. Fort−r≥2kand\nt≥4k, we shall split the domain into the interior domain t≥2rand the\nexterior domain t≤2r. We set\nDint:={(x,t)∈R2×[0,∞) :t≥2r, t≥4k},\nDext:=/braceleftbig\n(x,t)∈R2×[0,∞) :r+2k≤t≤2r/bracerightbig\n.\nFirst, we prove (2.4) in Dint. Sincef(x)+g(x)≡0, (2.1) and\n|x−y| ≤r+|y| ≤t\n2+k≤tfor (x,t)∈Dintand|y| ≤k,\n8we can rewrite uL(x,t) in (2.2) as\nuL(x,t) =∂\n∂t/braceleftBigg\n1\n2π/integraldisplay\nR2f(y)/radicalbig\nt2−|x−y|2dy/bracerightBigg\n=−t\n2π/integraldisplay\nR2f(y)\n(t2−|x−y|2)3/2dy.\nThis expression gives us\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2π\ntuL(x,t)+1\n(t2−r2)3/2/integraldisplay\nR2f(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤1\n(t2−r2)3/2/integraldisplay\nR2|h1(x,y,t)|\n(t2−|x−y|2)3/2|f(y)|dy, (2.5)\nwhere\nh1(x,y,t) := (t2−r2)3/2−(t2−|x−y|2)3/2.\nUsing the Taylor expansion with respect to yat the origin, we get\nh1(x,y,t) = 3(t2−|x−θy|2)1/2/braceleftbig\n−< x,y > +θ|y|2/bracerightbig\n(2.6)\nwith 0< θ <1. For (x,t)∈Dintand|y| ≤k, we obtain\n(t2−|x−θy|2)1/2≤t+r+|y| ≤3\n2t+k (2.7)\nand\n|−< x,y > +θ|y|2| ≤/parenleftbiggt\n2+k/parenrightbigg\nk. (2.8)\nFrom (2.6), (2.7) and (2.8), it follows that\n|h1(x,y,t)| ≤3/parenleftbigg3\n2t+k/parenrightbigg/parenleftbiggt\n2+k/parenrightbigg\nk≤Ckt2. (2.9)\nTherefore, combining (2.5), (2.9) and\nt+|x−y| ≥t−|x−y| ≥t−r−t\n4≥t−r\n2,\nwe have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2π\ntuL+1\n(t2−r2)3/2/integraldisplay\nR2f(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ckt2\n(t+r)3/2(t−r)9/2.\nSince 3(t−r)≥t+rholds inDint, we obtain (2.4) in Dint.\n9Next, we prove (2.4) in Dext. Here, we employ the following different\nrepresentationformulafrom(2.2)whichisestablishedby(6.24)inH¨ ormander\n[7]:\nuL(x,t) =∂\n∂t/parenleftBigg\n1\n2√\n2π√r/integraldisplay∞\nρ−ρ2z/2I(f)(s,ω,z)/radicalbig\ns−ρ+ρ2z/2ds/parenrightBigg\n,(2.10)\nwhereω=x/r∈S1,ρ=r−t,z= 1/rand\nI(f)(s,ω,z) :=/integraldisplay\ns=<ω,y>−|y|2z/2f(y)dSy.\nFor (x,t)∈Dextand|y| ≤k, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle< ω,y > −|y|2z\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ |y|+|y|2z\n2≤5k\n4. (2.11)\nSince\nρ−ρ2z\n2=−(t+r)(t−r)\n2r≤ −2k,−1+ρz=−t\nrand∂\n∂t=−∂\n∂ρ,\nit follows from (2.10) and (2.11) that\nuL=∂\n∂t/parenleftBigg\n1\n2√\n2π√r/integraldisplay5k/4\n−5k/4I(f)(s,ω,z)/radicalbig\ns−ρ+ρ2z/2ds/parenrightBigg\n=−t\n4√\n2πr3/2/integraldisplay5k/4\n−5k/4I(f)(s,ω,z)\n(s−ρ+ρ2z/2)3/2ds.\nHence, we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle4√\n2πr3/2\ntuL+/braceleftbigg2r\n(t+r)(t−r)/bracerightbigg3/2/integraldisplay\nR2f(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/braceleftbigg2r\n(t+r)(t−r)/bracerightbigg3/2/integraldisplay5k/4\n−5k/4|h2(ρ,s,z)|I(|f|)(s,ω,z)\n(s−ρ+ρ2z/2)3/2ds, (2.12)\nwhere\nh2(ρ,s,z) :=/parenleftbigg\n−ρ+ρ2z\n2/parenrightbigg3/2\n−/parenleftbigg\ns−ρ+ρ2z\n2/parenrightbigg3/2\n.(2.13)\n10Making use of the Taylor expansion with respect to sat the origin, we have\nfrom (2.13)\nh2(ρ,s,z) =−3\n2/parenleftbigg\nθs−ρ+ρ2z\n2/parenrightbigg1/2\nswith 0< θ <1.(2.14)\nSinceρ=r−tandz= 1/r, for|s| ≤5k/4, we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleθs−ρ+ρ2z\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/2\n|s| ≤/braceleftbigg5k\n4+(t+r)(t−r)\n2r/bracerightbigg1/2\n·5\n4k\n≤Ck(t+r)1/2(t−r)1/2\nr1/2(2.15)\nand\ns−ρ+ρ2z\n2≥ −5k\n4+t−r≥3\n8(t−r). (2.16)\nHence, for ( x,t)∈Dext, it follows from (2.12), (2.14), (2.15) and (2.16) that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle4√\n2πr3/2\ntuL+/braceleftbigg2r\n(t+r)(t−r)/bracerightbigg3/2/integraldisplay\nR2f(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤Ckr\n(t+r)(t−r)5/2/integraldisplay\nR2|f(y)|dy.\nTherefore, we obtain (2.4) in Dext.\nFinally, we show (2.3). It follows from the proof of Lemma 2.1 in [10]\nthat\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleR(f+g|x,t)−/integraldisplay\nR2{f(x)+g(x)}dx\n2π(t+r)1/2(t−r)1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ck\n(t+r)1/2(t−r)3/2(2.17)\nfort−r≥2kandt≥4k, whereR(f+g|x,t) is defined in (2.2). From (2.2),\n(2.4) and t−r≥2k, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂\n∂tR(f|x,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ck\n(t+r)1/2(t−r)3/2. (2.18)\nFrom (2.2), (2.17) and (2.18), we obtain (2.3). This completes the pr oof.✷\nIn what follows, we consider the following integral equation:\nu(x,t) =u0(x,t)+L(F)(x,t) for (x,t)∈R2×[0,∞),(2.19)\n11where we set\nu0:=εuL (2.20)\nand\nL(F)(x,t) :=1\n2π/integraldisplayt\n0t−τ\n(1+τ)p−1dτ/integraldisplay\n|ξ|≤1F(x+(t−τ)ξ,τ)/radicalbig\n1−|ξ|2dξ(2.21)\nforF∈C(R2×[0,∞)). We note that (2.21) solves\n/braceleftbigg\nutt−∆u= (1+t)−(p−1)FinR2×[0,∞),\nu(x,0) = 0, ut(x,0) = 0, x∈R2\nwhenF∈C2(R2×[0,∞)). Then, the following lemma is one of the basic\ntools.\nLemma 2.3 (Agemi and Takamura [2]) LetLbe the linear integral op-\nerator defined by (2.21)andΨ = Ψ(|x|,t)∈C(R2×[0,∞)). Then we have\nL(Ψ)(x,t) =L1(Ψ)(r,t)+L2(Ψ)(r,t), r=|x|,\nwhereLj(Ψ) (j= 1,2)are defined by\nL1(Ψ)(r,t)\n:=2\nπ/integraldisplayt\n0(1+τ)−(p−1)dτ/integraldisplayt+r−τ\n|t−r−τ|λΨ(λ,τ)dλ/integraldisplayt−τ\n|λ−r|ρh(λ,ρ;r)/radicalbig\n(t−τ)2−ρ2dρ,\nL2(Ψ)(r,t)\n:=2\nπ/integraldisplay(t−r)+\n0(1+τ)−(p−1)dτ/integraldisplayt−r−τ\n0λΨ(λ,τ)dλ/integraldisplayλ+r\n|λ−r|ρh(λ,ρ;r)/radicalbig\n(t−τ)2−ρ2dρ,\nwherea+:= max{a,0}and\nh(λ,ρ;r) :={(ρ2−(λ−r)2)((λ+r)2−ρ2)}−1\n2.\nMoreover, the following estimates hold in [0,∞)2:\n|L1(Ψ)(r,t)| ≤1√\n2/integraldisplayt\n0(1+τ)−(p−1)dτ/integraldisplayr+t−τ\n|r−t+τ|λ|Ψ(λ,τ)|dλ\n(√ror√\nλ)√\nτ+λ−t+r,\n|L2(Ψ)(r,t)| ≤/integraldisplay(t−r)+\n0(1+τ)−(p−1)dτ\n×/integraldisplayt−r−τ\n0λ|Ψ(λ,τ)|dλ\n(√\n2ror√\nt−r+λ−τ)√\nt−r−τ−λ.\n12In order to construct our solution in the weighted L∞space, we define\nthe following weighted functions:\nw1(r,t) :=τ+(r,t)1/2τ−(r,t)1/2, (2.22)\nw2(r,t) :=τ+(r,t)p1τ−(r,t)p2/parenleftbigg\nlog2τ+(r,t)\nτ−(r,t)/parenrightbigg−p3\n(logτ−(r,t))−p4,(2.23)\nw3(r,t) :=τ+(r,t)1/2τ−(r,t)3/2, (2.24)\nwhere we set\nτ+(r,t) :=t+r+2k\nk, τ−(r,t) :=t−r+2k\nk(2.25)\nand\np1:= min/braceleftbigg3p−4\n2,1\n2/bracerightbigg\n, p2:= max/braceleftbigg\n0,3p−5\n2/bracerightbigg\n,\np3:=/braceleftbigg0 (p/ne}ationslash= 5/3),\n1 (p= 5/3),p4:=/braceleftbigg0 (1< p <2),\n1 (p= 2).(2.26)\nWe remark that w2can be described as\nw2(r,t)−1=\n\nτ+(r,t)(4−3p)/2(1< p <5/3),\nτ+(r,t)−1/2log/parenleftbigg\n2τ+(r,t)\nτ−(r,t)/parenrightbigg\n(p= 5/3),\nτ+(r,t)−1/2τ−(r,t)(5−3p)/2(5/3< p <2),\nτ+(r,t)−1/2τ−(r,t)−1/2logτ−(r,t) (p= 2).(2.27)\nFor these weighted functions, we denote the weighted L∞norms of Vby\n/ba∇dblV/ba∇dbli:= sup\n(x,t)∈R2×[0,T){wi(|x|,t)|V(x,t)|}\nwherei= 1,2,3.\nFinally, we shall introduce some useful representations for L. It is trivial\nthat 1 + τ≥(2k+τ)/(2k) is valid for τ≥0 andk≥1. Setting τ=\n(α+β)/2≥0 withβ≥ −k, we have\n1+τ≥α+2k\n4k. (2.28)\nChanging the variables by\nα=τ+λ, β=τ−λ (2.29)\n13and extending the domain of ( α,β)-integration, we obtain from Lemma 2.3\nand (2.28)\nL1(Ψ)(r,t)≤Ck√r/integraldisplayt−r\n−kdβ/integraldisplayt+r\nt−r{(α+2k)/k}2−p|Ψ∗(α,β)|/radicalbig\nα−(t−r)dα(2.30)\nand\nL1(Ψ)(r,t)≤C√\nk/integraldisplayt−r\n−kdβ/integraldisplayt+r\nt−r{(α+2k)/k}(3−2p)/2|Ψ∗(α,β)|/radicalbig\nα−(t−r)dα,(2.31)\nwhere Ψ∗(α,β) := Ψ((α−β)/2,(α+β)/2).\nSimilarly, we get\nL2(Ψ)(r,t)≤Ck√r/integraldisplayt−r\n−kdβ/integraldisplayt−r\n−k{(α+2k)/k}2−p|Ψ∗(α,β)|√t−r−αdα(2.32)\nand\nL2(Ψ)(r,t)≤Ck/integraldisplayt−r\n−kdβ/integraldisplayt−r\n−k{(α+2k)/k}2−p|Ψ∗(α,β)|√t−r−α√t−r−βdα.(2.33)\n3 Proof of Theorem 2.1 and Theorem 2.2\nIn this section, we prove Theorem 2.1 and Theorem 2.2. The proof is b ased\non the classical iteration method in John [11]. Lemma 3.3 will be used to\nprove Theorem 2.1, whereas we prove Theorem 2.2 by using Lemma 3.4 .\nFirst, we prepare the elementary inequalities in Lemma 3.1 and Lemma 3 .2.\nLemma 3.1 Leta1∈Randk≥1. For0≤r≤t+k, it holds\n/integraldisplayt+r\nt−r{(α+2k)/k}a1\n/radicalbig\nα−(t−r)dα≤C√\nk×\n\nτ+(r,t)a1+1/2(a1>−1/2),\nlog/parenleftbigg\n2τ+(r,t)\nτ−(r,t)/parenrightbigg\n(a1=−1/2),\nτ−(r,t)a1+1/2(a1<−1/2),\nwhereτ+(r,t)andτ−(r,t)are defined in (2.25).\n14Proof.For 0≤r≤t+k, the integration by parts yields\n/integraldisplayt+r\nt−r{(α+2k)/k}a1\n/radicalbig\nα−(t−r)dα\n≤2√\n2r/parenleftbiggt+r+2k\nk/parenrightbigga1\n+2|a1|√\nk/integraldisplayt+r\nt−r/parenleftbiggα+2k\nk/parenrightbigga1−1/2\ndα\n≤2√\n2√\nkτ+(r,t)a1+1/2\n+2|a1|√\nk×\n\n1\na1+1/2/parenleftbiggt+r+2k\nk/parenrightbigga1+1/2\n(a1>−1/2),\nlog/parenleftbiggt+r+2k\nt−r+2k/parenrightbigg\n(a1=−1/2),\n1\n|a1+1/2|/parenleftbiggt−r+2k\nk/parenrightbigga1+1/2\n(a1<−1/2).\nThis completes the proof. ✷\nLemma 3.2 Leta1∈Randk≥1. For0≤r≤t+k, it holds\n/integraldisplayt−r\n−k{(α+2k)/k}a1\n√t−r−αdα≤C√\nk×\n\nτ−(r,t)a1+1/2(a1>−1),\nτ−(r,t)−1/2logτ−(r,t) (a1=−1),\nτ−(r,t)−1/2(a1<−1),\n(3.1)\nwhereτ+(r,t)andτ−(r,t)are defined in (2.25).\nProof.Fora1≥0, we obtain\n/integraldisplayt−r\n−k{(α+2k)/k}a1\n√t−r−αdα≤/parenleftbiggt−r+2k\nk/parenrightbigga1/integraldisplayt−r\n−k1√t−r−αdα\n≤2√\nkτ−(r,t)a1+1/2.\nHence, we obtain (3.1) for a1≥0.\nFora1<0, we show (3.1). Let −k≤t−r≤k, i.e.,k≤t−r+2k≤3k.\nIt follows that\n/integraldisplayt−r\n−k{(α+2k)/k}a1\n√t−r−αdα≤/integraldisplayt−r\n−k1√t−r−αdα\n≤3−a1√\nkτ−(r,t)a1+1/2.\nWe get (3.1) for a1<0 and−k≤t−r≤k.\n15Lett−r≥kwhich implies t−r≥(t−r+2k)/4. Then, breaking the\nintegral up into two pieces, we get\n/integraldisplayt−r\n−k{(α+2k)/k}a1\n√t−r−αdα\n=/integraldisplay(t−r)/2\n−k{(α+2k)/k}a1\n√t−r−αdα+/integraldisplayt−r\n(t−r)/2{(α+2k)/k}a1\n√t−r−αdα\n=:J1+J2. (3.2)\nIt is easy to see that\nJ1≤√\n2(t−r)−1/2/integraldisplay(t−r)/2\n−k/parenleftbiggα+2k\nk/parenrightbigga1\ndα\n≤2√\n2√\nk×\n\n1\na1+1τ−(r,t)a1+1/2(a1>−1),\nτ−(r,t)−1/2logτ−(r,t) (a1=−1),\n1\n|a1+1|τ−(r,t)−1/2(a1<−1).(3.3)\nWe obtain\nJ2≤/braceleftbigg/parenleftbiggt−r\n2+2k/parenrightbigg\n/k/bracerightbigga1/integraldisplayt−r\n(t−r)/21√t−r−αdα\n≤2−a1/parenleftbiggt−r+2k\nk/parenrightbigga1√\n2(t−r)1/2\n≤2−a1+1/2√\nkτ−(r,t)a1+1/2. (3.4)\nBy (3.2), (3.3) and (3.4), we obtain the desired inequality in (3.1) for a1<0\nandt−r≥k. This completes the proof. ✷\nThe following lemma contains one of the most essential estimates.\nLemma 3.3 Let1< p≤2andLbe the linear integral operator defined by\n(2.21). Assume that V∈C(R2×[0,T))withsuppV⊂ {(x,t)∈R2×[0,∞) :\n|x| ≤t+k}. Then, there exists a positive constant C1independent of kand\nTsuch that\n/ba∇dblL(|V|p)/ba∇dbl1≤C1k2/ba∇dblV/ba∇dblp\n1D1(T), (3.5)\nwhereD1(T)is defined by\nD1(T) :=/braceleftbigg\nT4−2p\nk if1< p <2,\n(logTk)2ifp= 2(3.6)\nwithTk:= (T+3k)/k.\n16Proof.In order to show the a-priori estimate (3.5), it is enough to prove\nLj(w−p\n1)≤Ck2w−1\n1D1(T) forj= 1,2, (3.7)\nwhereLjare defined in Lemma 2.3. By (2.22), (2.25) and (2.29), we have\nw1(λ,τ) =/parenleftbiggα+2k\nk/parenrightbigg1/2/parenleftbiggβ+2k\nk/parenrightbigg1/2\n. (3.8)\nWe shall prove (3.7) in the following two cases.\nCase 1: 4 r≥t+r+2k.\nFirst, we evaluate L1. We get from (2.30) and (3.8)\nL1(w−p\n1)≤Ck√r/integraldisplayt+r\nt−r{(α+2k)/k}(4−3p)/2\n/radicalbig\nα−(t−r)dα/integraldisplayt−r\n−k/parenleftbiggβ+2k\nk/parenrightbigg−p/2\ndβ.(3.9)\nFrom Lemma 3.1 and (3.11), we obtain\n/integraldisplayt+r\nt−r{(α+2k)/k}(4−3p)/2\n/radicalbig\nα−(t−r)dα\n≤C√\nk×\n\nτ+(r,t)(5−3p)/2(1< p <5/3),\nτ+(r,t)1/2τ−(r,t)−1/2(p= 5/3),\nτ−(r,t)(5−3p)/2(5/3< p≤2).(3.10)\nHere, for the inequality with p= 5/3, we used the following fact:\nlogs≤sδ\nδfors≥1 andδ >0. (3.11)\nTheβ-integral is estimated by\n/integraldisplayt−r\n−k/parenleftbiggβ+2k\nk/parenrightbigg−p/2\ndβ≤Ck×/braceleftbigg\nτ−(r,t)(2−p)/2(1< p <2),\nlogτ−(r,t) (p= 2).(3.12)\nTherefore, it follows from (3.9), (3.10), (3.12), (2.22) and (3.6) th at\nL1(w−p\n1)≤Ck2τ+(r,t)−1/2τ−(r,t)−1/2×/braceleftbiggτ+(r,t)4−2p(1< p <2),\nlogτ−(r,t) (p= 2)\n≤Ck2w1(r,t)−1D1(T).\nHere, we have used that\nτ+(r,t)≤2t+3k\nk≤2TkandTk≥3.\n17Thus, we have proved (3.7) with j= 1 in Case 1.\nNext, ift > r, we investigate the integral L2. From (2.32) and (3.8), we\nget\nL2(w−p\n1)(r,t)\n≤Ck√r/integraldisplayt−r\n−k{(α+2k)/k}(4−3p)/2\n√t−r−αdα/integraldisplayt−r\n−k/parenleftbiggβ+2k\nk/parenrightbigg−p/2\ndβ.(3.13)\nFrom Lemma 3.2, we obtain\n/integraldisplayt−r\n−k{(α+2k)/k}(4−3p)/2\n√t−r−αdα\n≤C√\nk×/braceleftbigg\nτ−(r,t)(5−3p)/2(1< p <2),\nτ−(r,t)−1/2logτ−(r,t) (p= 2).(3.14)\nFrom (3.13), (3.14), (3.12), (2.22) and (3.6), it follows that\nL2(w−p\n1)≤Ck2τ+(r,t)−1/2×/braceleftbiggτ−(r,t)(7−4p)/2(1< p <2),\nτ−(r,t)−1/2{logτ−(r,t)}2(p= 2)\n≤Ck2w1(r,t)−1D1(T).\nHence, we obtain (3.7) with j= 2 in Case 1.\nCase 2: 4 r≤t+r+2k, i.e.,t+r+2k≤2(t−r+2k).\nFirst, we estimate L1. We have from (2.31) and (3.8)\nL1(w−p\n1)\n≤C√\nk/integraldisplayt+r\nt−r{(α+2k)/k}3(1−p)/2\n/radicalbig\nα−(t−r)dα/integraldisplayt−r\n−k/parenleftbiggβ+2k\nk/parenrightbigg−p/2\ndβ.(3.15)\nWe obtain\n/integraldisplayt+r\nt−r{(α+2k)/k}3(1−p)/2\n/radicalbig\nα−(t−r)dα\n≤/parenleftbiggt−r+2k\nk/parenrightbigg3(1−p)/2/integraldisplayt+r\nt−r1/radicalbig\nα−(t−r)dα\n≤2√\n2rτ−(r,t)3(1−p)/2\n≤C√\nkτ+(r,t)(4−3p)/2. (3.16)\n18Therefore, it follows from (3.15), (3.16), (3.12), (2.22) and (3.6)\nL1(w−p\n1)≤Ck2τ+(r,t)−1/2τ−(r,t)−1/2×/braceleftbiggτ+(r,t)4−2p(1< p <2),\nlogτ−(r,t) (p= 2)\n≤Ck2w1(r,t)−1D1(T).\nThus, we obtain (3.7) with j= 1 in Case 2.\nNext, ift > r, we evaluate L2. From (2.33) and (3.8), we obtain\nL2(w−p\n1)(r,t)\n≤Ck/integraldisplayt−r\n−k{(α+2k)/k}(4−3p)/2\n√t−r−αdα/integraldisplayt−r\n−k{(β+2k)/k}−p/2\n√t−r−βdβ.(3.17)\nFrom Lemma 3.2, we have\n/integraldisplayt−r\n−k{(β+2k)/k}−p/2\n√t−r−βdβ\n≤C√\nk×/braceleftbiggτ−(r,t)−(p−1)/2(1< p <2),\nτ−(r,t)−1/2logτ−(r,t) (p= 2).(3.18)\nFrom (3.17), (3.14), (3.18), (2.22) and (3.6), we have (3.7) with j= 2 in Case\n2. Therefore, the proof of Lemma 3.3 is completed. ✷\nFinally, we state an a-priori estimate of mixed type.\nLemma 3.4 Let1< p≤2andLbe the linear integral operator defined by\n(2.21). Assume that V,V0∈C(R2×[0,T))withsupp (V,V0)⊂ {(x,t)∈R2×\n[0,∞) :|x| ≤t+k}. Then, there exists a positive constant C2independent\nofkandTsuch that\n/ba∇dblL(|V0|p−ν|V|ν)/ba∇dbl2≤C2k2/ba∇dblV0/ba∇dblp−ν\n3/ba∇dblV/ba∇dblν\n2D2,ν(T), (3.19)\nwhereν= 0,p−1,1,pand\nD2,ν(T) :=\n\nTν(5−3p)/2+δνp3\nk (ν= 0,p−1and1< p≤5/3),\nT5−3p+δνp3\nk (ν= 1and1< p≤5/3),\n1 ( ν≤1and5/3< p≤2),\nTγ(p,4)/2\nk (ν=pand1< p <2),\n(logTk)3(ν=pandp= 2),(3.20)\nwhereδstands for any positive constant and p3is defined in (2.26).\n19Proof.In order to show the a-priori estimate (3.19), it is enough to prove\nLj(w−(p−ν)\n3w−ν\n2)≤Ck2w−1\n2D2,ν(T) forj= 1,2,(3.21)\nwhereLjare defined in Lemma 2.3. For δ >0, from (2.23), (2.24), (2.25),\n(2.29) and (3.11), we have\nw3(λ,τ)−(p−ν)w2(λ,τ)−ν\n≤C/parenleftbiggα+2k\nk/parenrightbigg−(p−ν)/2−νp1+δνp3/parenleftbiggβ+2k\nk/parenrightbigg−3(p−ν)/2−νp2−δνp3\n×/parenleftbigg\nlogβ+2k\nk/parenrightbiggνp4\n. (3.22)\nWe shall prove (3.21) in the following two cases.\nCase 1: 4 r≥t+r+2k.\nFirst, we evaluate L1. From (2.30) and (3.22), we get\nL1(w−(p−ν)\n3w−ν\n2)\n≤Ck√r/integraldisplayt+r\nt−r{(α+2k)/k}p5\n/radicalbig\nα−(t−r)dα/integraldisplayt−r\n−k/parenleftbiggβ+2k\nk/parenrightbiggp6/parenleftbigg\nlogβ+2k\nk/parenrightbiggνp4\ndβ,\n(3.23)\nwhere\np5:=4−3p\n2+ν/parenleftbigg1\n2−p1/parenrightbigg\n+δνp3, (3.24)\np6:=−3(p−ν)\n2−νp2−δνp3. (3.25)\nWe have from (3.24) and (2.26)\np5=\n\n4−3p\n2+ν/parenleftbigg5−3p\n2/parenrightbigg\n(1< p <5/3),\n−1\n2+δν (p= 5/3),\n4−3p\n2(5/3< p≤2).(3.26)\n20From Lemma 3.1 and (3.26), we obtain\n/integraldisplayt+r\nt−r{(α+2k)/k}p5\n/radicalbig\nα−(t−r)dα\n≤C√\nk×\n\nτ+(r,t)(1+ν)(5−3p)/2(1< p <5/3),\nlog/parenleftbigg\n2τ+(r,t)\nτ−(r,t)/parenrightbigg\n(ν= 0 and p= 5/3),\nτ+(r,t)δν(ν >0 andp= 5/3),\nτ−(r,t)(5−3p)/2(5/3< p≤2).(3.27)\nWe get from (2.26) and (3.25)\np6=/braceleftbigg−3(p−ν)/2−δνp3 (1< p≤5/3),\n−3(p−ν)/2−ν(3p−5)/2(5/3< p≤2).(3.28)\nFrom (3.28), (2.26) and (1.6), the β-integral is estimated by\n/integraldisplayt−r\n−k/parenleftbiggβ+2k\nk/parenrightbiggp6/parenleftbigg\nlogβ+2k\nk/parenrightbiggνp4\ndβ\n≤Ck×\n\n1 ( ν= 0 orν=p−1),\nτ−(r,t)(5−3p)/2+p2(ν= 1),\nτ−(r,t)1−δνp3(ν=pand 1< p≤5/3),\nτ−(r,t)γ(p,4)/2(ν=pand 5/3< p <2),\n(logτ−(r,t))3(ν=pandp= 2).(3.29)\nHere, for the inequality with ν=pandp= 5/3, we took 0 < δν <1. It\nfollows from (3.23), (3.27), (3.29), (2.26), (2.27) and (3.20) that\nL1(w−(p−ν)\n3w−ν\n2)≤Ck2w2(r,t)−1\n×\n\nTν(5−3p)/2+δνp3\nk (ν= 0,p−1 and 1< p≤5/3),\nT5−3p+δνp3\nk (ν= 1 and 1 < p≤5/3),\n1 ( ν≤1 and 5/3< p≤2),\nTγ(p,4)/2\nk (ν=pand 1< p <2),\n(logTk)2(ν=pandp= 2)\n≤Ck2w2(r,t)−1D2,ν(T).\nWe obtain (3.21) with j= 1 in Case 1.\nNext, ift > r, we investigate L2. From (2.32) and (3.22), we get\nL2(w−(p−ν)\n3w−ν\n2)(r,t)\n≤Ck√r/integraldisplayt−r\n−k{(α+2k)/k}p5\n√t−r−αdα/integraldisplayt−r\n−k/parenleftbiggβ+2k\nk/parenrightbiggp6/parenleftbigg\nlogβ+2k\nk/parenrightbiggνp4\ndβ.\n(3.30)\n21From Lemma 3.2 and (3.26), we have\n/integraldisplayt−r\n−k{(α+2k)/k}p5dα√t−r−α\n≤C√\nk×\n\nτ−(r,t)(ν+1)(5−3p)/2+δνp3(1< p≤5/3),\nτ−(r,t)(5−3p)/2(5/3< p <2),\nτ−(r,t)−1/2logτ−(r,t) (p= 2).(3.31)\nMaking use of (3.30), (3.31), (3.29), (1.6), (2.26), (2.27) and (3.20 ), we get\nL2(w−(p−ν)\n3w−ν\n2)(r,t)\n≤Ck2τ+(r,t)−1/2\n×\n\nτ−(r,t)(ν+1)(5−3p)/2+δνp3(ν= 0,p−1 and 1< p≤5/3),\nτ−(r,t)3(5−3p)/2+δνp3(ν= 1 and 1 < p≤5/3),\nτ−(r,t)(5−3p)/2(ν≤1 and 5/3< p <2),\nτ−(r,t)−1/2logτ−(r,t) (ν≤1 andp= 2),\nτ−(r,t)(5−3p)/2+γ(p,4)/2(ν=pand 1< p <2),\nτ−(r,t)−1/2(logτ−(r,t))4(ν=pandp= 2)\n≤Ck2w2(r,t)−1D2,ν(T).\nHence, we obtain (3.21) with j= 2 in Case 1.\nCase 2: 4 r≤t+r+2k, i.e.,t+r+2k <2(t−r+2k).\nFirst, we evaluate L1. From (2.31) and (3.22), we have\nL1(w−(p−ν)\n3w−ν\n2)\n≤C√\nk/integraldisplayt+r\nt−r{(α+2k)/k}3(1−p)/2+ν(1/2−p1)+δνp3\n/radicalbig\nα−(t−r)dα\n×/integraldisplayt−r\n−k/parenleftbiggβ+2k\nk/parenrightbiggp6/parenleftbigg\nlogβ+2k\nk/parenrightbiggνp4\ndβ. (3.32)\nFrom (3.24), we obtain\n/integraldisplayt+r\nt−r{(α+2k)/k}3(1−p)/2+ν(1/2−p1)+δνp3\n/radicalbig\nα−(t−r)dα\n≤τ−(r,t)3(1−p)/2τ+(r,t)ν(1/2−p1)+δνp3/integraldisplayt+r\nt−r1/radicalbig\nα−(t−r)dα\n≤C√\nkτ+(r,t)p5. (3.33)\n22Therefore, it follows from (3.32), (3.33), (3.26), (3.29), (2.27) an d (3.20) that\nL1(w−(p−ν)\n3w−ν\n2)≤Ck2τ+(r,t)(4−3p)/2\n×\n\nτ+(r,t)ν(5−3p)/2+δνp3(ν= 0,p−1 and 1< p≤5/3),\nτ+(r,t)5−3p+δνp3(ν= 1 and 1 < p≤5/3),\n1 ( ν≤1 and 5/3< p≤2),\nτ−(r,t)γ(p,4)/2(ν=pand 1< p <2),\n(logτ−(r,t))3(ν=pandp= 2)\n≤Ck2w2(r,t)−1D2,ν(T).\nThus, the proof of (3.21) with j= 1 in Case 2 is finished.\nNext, ift > r, we investigate L2. From (2.33), (3.22), (3.24) and (3.25),\nwe get\nL2(w−(p−ν)\n3w−ν\n2)(r,t)≤Ck/integraldisplayt−r\n−k{(α+2k)/k}p5\n√t−r−αdα\n×/integraldisplayt−r\n−k{(β+2k)/k}p6\n√t−r−β/parenleftbigg\nlogβ+2k\nk/parenrightbiggνp4\ndβ.\n(3.34)\nFrom Lemma 3.2 and (3.28), we have\n/integraldisplayt−r\n−k{(β+2k)/k}p6\n√t−r−β/parenleftbigg\nlogβ+2k\nk/parenrightbiggνp4\ndβ\n≤C√\nk×\n\nτ−(r,t)−1/2(ν= 0 orν=p−1),\nτ−(r,t)−p1 (ν= 1),\nτ−(r,t)1/2(ν=pand 1< p <5/3),\nτ−(r,t)−δν+1/2(ν=pandp= 5/3),\nτ−(r,t)−1/2+γ(p,4)(ν=pand 5/3< p <2),\nτ−(r,t)−1/2logτ−(r,t) (ν=pandp= 2).(3.35)\nHere, for the inequality with ν=pandp= 5/3, we took 0 < δν <1. Thus,\nwe obtain (3.21) with j= 2 in Case 2 by (3.34), (3.31), (3.35), (2.27) and\n(3.20). Therefore, the proof of Lemma 3.4 is completed. ✷\nIn the following, we prove Theorem 2.1 and Theorem 2.2. We remark\nthat it is possible to construct a classical solution if p=pS(4) = 2. However,\nits construction is almost the same as for C1solution. Therefore, we shall\nomit the proof.\nProof of Theorem 2.1. We define\nX:=\n\nu(x,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleDα\nxu(x,t)∈C(R2×[0,T)),\n/ba∇dblDα\nxu/ba∇dbl1<∞(|α| ≤1),\nu(x,t) = 0 (|x| ≥t+k)\n\n,\n23whereDα\nx=Dα1\n1Dα2\n2(α= (α1,α2)) andDk=∂/∂xk(k= 1,2). We can\nverify easily that Xis complete with respect to the norm\n/ba∇dblu/ba∇dblX=/summationdisplay\n|α|≤1/ba∇dblDα\nxu/ba∇dbl1.\nUsing the iteration method, we shall construct a solution of (1.7). W e define\nthe sequence of functions {uj}by\nu0=u0, uj+1=u0+L(|uj|p) forj≥0,\nwhereu0is defined in (2.20). It follows from Lemma 1 in [6], p.236 that u0\nsatisfies\n|Dα\nxu0(x,t)| ≤C(f,g)ε(t+r+2k)−1/2(t−r+2k)−1/2\nfor|α| ≤1, where the positive constant C(f,g) depends on Dα\nxgandDβ\nxf\n(|β| ≤2). Hence, we find\n/ba∇dblDα\nxu0/ba∇dbl1≤C(f,g)k−1ε. (3.36)\nAs in [11], p.258, we see from Lemma 3.3 and (3.36) that if εsatisfies\nC1k2D1(T)εp−1/ba∇dbluL/ba∇dblp−1\n1≤1\np2p, (3.37)\nthen{uj}is a Cauchy sequence in X. SinceXis complete, there exists a\nfunction u∈Xsuch that {Dα\nxuj}converges uniformly to Dα\nxuasj→ ∞.\nClearlyusatisfies (2.19) with F(x,t) =|u(x,t)|p. In view of (2.19) and\n(2.21), we note that ∂u/∂tcan be expressed in terms of Dα\nxu(|α| ≤1). From\nthecontinuity of Dα\nxu, thecontinuity of ∂u/∂tisalsovalid. Thus, from(3.36)\nand (3.37), Theorem 2.1 is proved by taking εis small. ✷\nProof of Theorem 2.2. We consider the following integral equation:\nU=L(|u0+U|p), (3.38)\nwhereu0is defined in (2.20). Suppose that we obtain a solution U=U(x,t)\nof (3.38). Then, putting u=U+u0, we get the solution of (2.19) with\nF(x,t) =|u(x,t)|p, and its maximal existence time is the same as that of U.\nThus, we have reduced the problem to the analysis of (3.38). Let Ybe the\nnorm space defined by\nY:=\n\nU(x,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleDα\nxU(x,t)∈C(R2×[0,T)),\n/ba∇dblDα\nxU/ba∇dbl2<∞(|α| ≤1),\nU(x,t) = 0 (|x| ≥t+k)\n\n,\n24which is equipped with the norm\n/ba∇dblU/ba∇dblY=/summationdisplay\n|α|≤1/ba∇dblDα\nxU/ba∇dbl2.\nWe shall construct a solution of the integral equation (3.38) in Y. We define\nthe sequence of functions {Uj}by\nU0= 0, Uj+1=L(|u0+Uj|p) (j= 0,1,2,···).(3.39)\nFrom Lemma 2.1 and the condition/integraltext\nR2(f+g)(x)dx= 0, we see that there\nexists a positive constant C0such that\n/ba∇dblDα\nxu0/ba∇dbl3≤C0ε(|α| ≤1). (3.40)\nWe put\nC3:= (22p+2p)p/(p−1)max/braceleftbig\nC2k2Mp−1\n0,(C2k2Cp−1\n0)p,(C2k2Mp−2\n0C0)p/(p−1)/bracerightbig\n(3.41)\nand\nM0:= 2ppk2Cp\n0C2, (3.42)\nwhereC2is the constant given in Lemma 3.4. We take εandTsuch that\nC3εp(p−1)D2,p(T)≤1. (3.43)\nSimilarly to the proof of Theorem 1 in [10], we shall obtain\n/ba∇dblUj/ba∇dbl2≤2M0εp(3.44)\nby induction. For j= 0, (3.44) holds. Assume that /ba∇dblUj/ba∇dbl2≤2M0εpfor some\nj. From (3.39), (3.19) with ν= 0 andν=p, (3.40), (3.41), (3.42) and (3.44),\nwe have\n/ba∇dblUj+1/ba∇dbl2≤2p−1{/ba∇dblL(|u0|p)/ba∇dbl2+/ba∇dblL(|Uj|p)/ba∇dbl2}\n≤2p−1C2k2{/ba∇dblu0/ba∇dblp\n3D2,0(T)+/ba∇dblUj/ba∇dblp\n2D2,p(T)}\n≤2p−1C2k2{(C0ε)p+(2M0εp)pD2,p(T)}\n≤M0εp+M0C3εp2D2,p(T). (3.45)\nThus, from (3.45), we obtain (3.44) under the conditions (3.43).\nNext, we shall estimate the differences of {Uj}. From (3.39), we obtain\n/ba∇dblUj+1−Uj/ba∇dbl2≤2p−1p{2/ba∇dblL(|u0|p−1|Uj−Uj−1|)/ba∇dbl2\n+/ba∇dblL((|Uj|p−1+|Uj−1|p−1)|Uj−Uj−1|)/ba∇dbl2}.(3.46)\n25Then, in view of the definitions of D2,1(T) andD2,p(T) in(3.20), the estimate\nD2,1(T)≤D2,p(T)1/pholds because of 5 −3p+νp3≤γ(p,4)/2pandTk≥3.\nHere, when p= 5/3, we take δ >0 such that 0 < δ <1/pin Lemma 3.4.\nThus, from (3.19) with ν= 1 and (3.40), we obtain\n/ba∇dblL(|u0|p−1|Uj−Uj−1|)/ba∇dbl2≤C2k2/ba∇dblu0/ba∇dblp−1\n3/ba∇dblUj−Uj−1/ba∇dbl2D2,1(T)\n≤C2k2(C0ε)p−1D2,p(T)1/p/ba∇dblUj−Uj−1/ba∇dbl2.(3.47)\nWe also get from (3.19) with ν=pand (3.44) that\n/ba∇dblL((|Uj|p−1+|Uj−1|p−1)|Uj−Uj−1|)/ba∇dbl2\n≤C2k2(/ba∇dblUj/ba∇dblp−1\n2+/ba∇dblUj−1/ba∇dblp−1\n2)/ba∇dblUj−Uj−1/ba∇dbl2D2,p(T)\n≤2C2k2(2M0εp)p−1D2,p(T)/ba∇dblUj−Uj−1/ba∇dbl2. (3.48)\nHence, we obtain from (3.46), (3.47), (3.48) and (3.41) that\n/ba∇dblUj+1−Uj/ba∇dbl2≤1\n4{(C3εp(p−1)D2,p(T))1/p+C3εp(p−1)D2,p(T)}/ba∇dblUj−Uj−1/ba∇dbl2\n≤1\n2/ba∇dblUj−Uj−1/ba∇dbl2 (3.49)\nprovided (3.43) holds.\nSimilarly to the proof of (3.44) and (3.49), if we assume that (3.43) ho lds,\nthen we obtain the following estimates:\n/ba∇dblDiUj/ba∇dbl2≤2M0εp, (3.50)\n/ba∇dblDi(Uj+1−Uj)/ba∇dbl2≤C4(j+1)2−j(p−1), (3.51)\nwhereC4is a positive constant independent of j. We remark that in order\nto show (3.50) and (3.51), we also use the estimates (3.19) with ν=p−1\nandD2,p−1(T)≤D2,p(T)(p−1)/(p+1). For the actual proof, see the inequalities\n(4.15) and (4.25) in [10] which correspond the estimates (3.50) and ( 3.51)\nrespectively. Then, from (3.49) and (3.51), we see that {Uj}is a Cauchy\nsequence in Yprovided that (3.43) holds. We can verify easily that Yis\ncomplete. Hence, there exists a function Usuch that {Uj}converges to Uin\nY. Therefore, Usatisfies the integral equation (3.38).\nLet us fix ε0as\nC3εp(p−1)\n0≤/braceleftbigg\n6−γ(p,4)/2(1< p <2),\n(log6)−3(p= 2).(3.52)\n26For 0< ε≤ε0, if we assume that\nC3εp(p−1)≤\n\n/parenleftbigg2T\nk/parenrightbigg−γ(p,4)/2\n(1< p <2),\n/parenleftbigg\nlog2T\nk/parenrightbigg−3\n(p= 2),(3.53)\nthen (3.43) holds. In fact, when T≤3k, (3.43) follows from (3.52). When\nT >3k, (3.43) follows from (3.53). Hence, Theorem 2.2 follows immediately\nfrom (3.53). This completes the proof. ✷\n4 Proof of Theorem 2.3 and Theorem 2.4\nIn this section, we prove Theorem 2.3 and Theorem 2.4. For the sub- critical\ncase, we use an improved version of Kato’s lemma on ordinary differen tial\ninequality which was introduced by Takamura [18]. For the critical cas e, we\napplytheslicingiterationmethodwhichwasintroducedbyAgemi, Kuro kawa\nand Takamura [1]. From now on, let u∈C1(R2×[0,T)) be the solution of\nthe integral equation associated with (1.7).\n4.1 Proof of Theorem 2.3\nWe divide the proof of Theorem 2.3 into two cases, 1 < p <2 andp= 2.\nFirst, we shall handle the sub-critical case.\nProof of Theorem 2.3 with 1 < p <2.We shall follow the arguments\nin Section 4 of Takamura [18]. In order to obtain the estimates in Theo rem\n2.3, we shall take a look on the ordinary differential inequality for\nF(t) :=/integraldisplay\nR2u(x,t)dx.\n(1.7) with µ= 2 and (2.1) imply that\nF′′(t) =1\n(1+t)p−1/integraldisplay\nR2|u(x,t)|pdxfort≥0. (4.1)\nHence, the H¨ older’s inequality and (2.1) yield that\nF′′(t)≥π−(p−1)(t+k)−3(p−1)|F(t)|pfort≥0. (4.2)\n27Due to the assumption onthe initial data in Theorem 2.3, f(x)≡0,g(x)≥0\n(/ne}ationslash≡0), we have\nF(0) = 0, F′(0)>0. (4.3)\nIt follows from (4.3) in [18] that\nu(x,t)≥/ba∇dblg/ba∇dblL1(R2)\n2√\n2π√\nt+k/radicalbig\nt−|x|+kεfork≤ |x| ≤t−k.(4.4)\nFrom (4.1), it follows that\nF′′(t)≥1\n(1+t)p−1/integraldisplay\nk≤|x|≤t−k|u(x,t)|pdxfort≥2k.\nPlugging (4.4) into the right-hand side of this inequality, we have that\nF′′(t)≥/parenleftbigg/ba∇dblg/ba∇dblL1(R2)\n2√\n2π(t+k)3/2−1/pε/parenrightbiggp/integraldisplay\nk≤|x|≤t−k1\n(t−|x|+k)p/2dx\n=2π/ba∇dblg/ba∇dblp\nL1(R2)\n(2√\n2π)p(t+k)3p/2−1εp/integraldisplayt−k\nkr\n(t−r+k)p/2dr. (4.5)\nWe evaluate the integral of the last term in (4.5). For t≥3k, we obtain\n/integraldisplayt−k\nkr\n(t−r+k)p/2dr≥1\n2tp/2{(t−k)2−k2}\n≥1\n6t2−p/2. (4.6)\nFrom (4.5) and (4.6), we obtain\nF′′(t)≥/ba∇dblg/ba∇dblp\nL1(R2)\n3·23p−1πp−1εpt3−2pfort≥3k.\nIntegrating this inequality in [3 k,t], we get from (4.3)\nF′(t)>/ba∇dblg/ba∇dblp\nL1(R2)(1−(3/4)4−2p)\n3(4−2p)23p−1πp−1εpt4−2pfort≥4k.\nHence, we obtain from (4.3)\nF(t)> D1εpt5−2pfort≥5k, (4.7)\nwhere\nD1:=/ba∇dblg/ba∇dblp\nL1(R2)(1−(3/4)4−2p)(1−(4/5)5−2p)\n3(4−2p)(5−2p)23p−1πp−1>0.\nIn the sub-critical case, the following basic lemma is useful.\n28Lemma 4.1 (Takamura [18]) Letp >1,a >0andq >0satisfy\nM:=p−1\n2a−q\n2+1>0. (4.8)\nAssume that F∈C2([0,T))satisfies\nF(t)≥Atafort≥T0,\nF′′(t)≥B(t+k)−q|F(t)|pfort≥0,\nF(0)≥0, F′(0)>0, (4.9)\nwhereA,B,k,T 0are positive constants. Then, there exists a positive const ant\nD0=D0(p,a,q,B)such that\nT <22/MT1\nholds provided\nT1:= max/braceleftbigg\nT0,F(0)\nF′(0),k/bracerightbigg\n≥D0A−(p−1)/(2M). (4.10)\nThis is exactly Lemma 2.1 in [18], so that we shall omit the proof here.\nAccording to (4.2), (4.3) and (4.7), we are in a position to apply our\nsituation to Lemma 4.1 with\nA=D1εp, B=π1−p, a= 5−2p, q= 3(p−1)\nwhich imply that (4.8) yields\nM=p−1\n2(5−2p)−3(p−1)\n2+1 =p(2−p)>0.\nIf we set\nT0:=D0A−(p−1)/(2M)=D0D−(p−1)/(p(4−2p))\n1 ε−(p−1)/(4−2p),(4.11)\nwe find that there is an ε0=ε0(g,p,k) such that\nT0≥max/braceleftbiggF(0)\nF′(0),5k/bracerightbigg\n= 5kfor 0< ε≤ε0.\nThis means that T1=T0in (4.10). Therefore, from (4.11), the conclusion of\nLemma 4.1 implies\nT <22/MT1=D2ε−(p−1)/(4−2p),\n29where\nD2:= 22/MD0D−(p−1)/(p(4−2p))\n1 >0.\nThe proof of Theorem 2.3 with 1 < p <2 is now completed. ✷\nProof of Theorem 2.3 with p= 2.\nLet ¯vbe the spherical mean of v∈C0(R2×[0,∞)) with radius r;\n¯v(r,t) :=1\n2π/integraldisplay\n|ω|=1v(rω,t)dSω.\nWe get the following inequality (for the proof, see [2], p.529):\n¯u(r,t)≥u0(r,t)+2\nπ/integraldisplayt−r\n0dτ(1+τ)−1/integraldisplayt−τ−r\n0λ|¯u|2(λ,τ)dλ\n×/integraldisplayλ+r\n|λ−r|ρdρ/radicalbig\nh(λ,ρ;r)((t−τ)2−ρ2), (4.12)\nwhere\nh(λ,ρ;r) := (ρ2−(λ−r)2)((λ+r)2−ρ2).\nSince/integraltext\nR2{f(x)+g(x)}dx >0, fromLemma2.2, thereexist positiveconstants\nE0andKsuch that\nuL(x,t)≥E0/radicalbig\n(t+r)(t−r)\nfort−r≥K≥1. Making use of the positivity of the second term of\nright-hand side in (4.12), we get\n¯u(r,t)≥u0(r,t)≥E0ε/radicalbig\n(t+r)(t−r)fort−r≥K. (4.13)\nWe define\nΣj:={(r,t)|t−r≥Klj},Σ∞:={(r,t)|t−r≥2K},\nwhere\nlj:=j/summationdisplay\nk=02−k(j= 0,1,2,···).\n30For (r,t)∈Σ0, it follows from (4.12) and (4.13) that\n¯u(r,t)≥2\nπ/integraldisplayt−r\n0dτ(1+τ)−1/integraldisplayt−τ−r\n0dλ λ|¯u|2(λ,τ)\n×1/radicalbig\n(t−r−(τ+λ))(t+r−(τ−λ))\n×/integraldisplayλ+r\n|λ−r|ρ/radicalbig\nh(λ,ρ;r)dρ. (4.14)\nFor (r,t)∈Σ0, we introduce\nQj(r,t) :=/braceleftbig\n(λ,τ)∈[0,∞)2|Klj≤τ−λ,0≤λ≤t−r−τ/bracerightbig\n.\nSince\n/integraldisplayλ+r\n|λ−r|ρ/radicalbig\nh(λ,ρ;r)dρ=B/parenleftbigg1\n2,1\n2/parenrightbigg\n=π\n2,\nwe get from (4.14)\n¯u(r,t)≥1/radicalbig\n(t+r)(t−r)/integraldisplay/integraldisplay\nQ0(r,t)(1+τ)−1λ|¯u|2dλdτin Σ0.(4.15)\nFor (r,t)∈Σj, we have\nQj(r,t)⊂Q0(r,t) and Qj(r,t)⊂Σj. (4.16)\nSince Σ j⊂Σ0, it follows from (4.15) and (4.16) that\n¯u(r,t)≥1/radicalbig\n(t+r)(t−r)/integraldisplay/integraldisplay\nQj(r,t)(1+τ)−1λ|¯u|2(λ,τ)dλdτin Σj.(4.17)\nBy using the induction argument, we will show\n¯u(r,t)≥dj/radicalbig\n(t+r)(t−r)logaj/parenleftbiggt−r\nKlj/parenrightbigg\nin Σj, (4.18)\nwhere\na0= 0, aj+1= 2aj+2, (4.19)\nd0=E0ε, dj+1=d2\nj\n23j+9. (4.20)\n31From (4.13), it holds (4.18) with j= 0. We assume that (4.18) holds for\none natural number jand (r,t)∈Σj+1. Substituting (4.18) into (4.17) and\nchanging the variables by (2.29), we get\n/radicalbig\n(t+r)(t−r)¯u(r,t)\n≥d2\nj\n2/integraldisplayt−r\nKljdα/integraldisplayα\nKlj/parenleftbigg\n1+α+β\n2/parenrightbigg−1/parenleftbiggα−β\n2/parenrightbigg\nα−1β−1log2aj/parenleftbiggβ\nKlj/parenrightbigg\ndβ\n≥d2\nj\n8(2aj+1)/integraldisplayt−r\nKljdαα−2/integraldisplayα\nKlj(α−β)d\ndβ/braceleftbigg\nlog2aj+1/parenleftbiggβ\nKlj/parenrightbigg/bracerightbigg\ndβ\n=d2\nj\n8(2aj+1)/integraldisplayt−r\nKljdαα−2log2aj+1/parenleftbiggβ\nKlj/parenrightbigg\ndβ\nand, then,\n/radicalbig\n(t+r)(t−r)¯u(r,t)\n≥d2\nj\n8(2aj+1)/integraldisplayt−r\nKlj+1dαα−2/integraldisplayα\nαlj/lj+1log2aj+1/parenleftbiggβ\nKlj/parenrightbigg\ndβ\n≥d2\nj(1−lj/lj+1)\n8(2aj+1)/integraldisplayt−r\nKlj+1α−1log2aj+1/parenleftbiggα\nKlj+1/parenrightbigg\ndα\n≥d2\nj(1−lj/lj+1)\n8(aj+1)2log2aj+2/parenleftbiggt−r\nKlj+1/parenrightbigg\n. (4.21)\nSolving (4.19) yields\naj= 2j+1−2. (4.22)\nSince (1−lj/lj+1) = 2−(j+1)/lj+1≥2−(j+2), we have\n1−lj/lj+1\n8(aj+1)2≥2−(j+2)\n23·22j+4=1\n23j+9. (4.23)\nTherefore, from(4.21),(4.22)and(4.23),(4.18)holdsforallnat uralnumbers.\nWe get from (4.20)\nlogdj+1= 2j+1logd0−(log2)j/summationdisplay\nk=0/braceleftbig\n(3(j−k)+9)2k/bracerightbig\n.\nWe obtain\ndj= exp/braceleftBigg\n2j/parenleftBigg\nlogd0−(log2)j−1/summationdisplay\nk=0(3(j−k)+9)2k\n2j/parenrightBigg/bracerightBigg\n.(4.24)\n32The sum part in (4.24) converges as j→ ∞by the d’Alembert’s criterion.\nHence, there exists a constant qsuch that it holds\ndj≥exp/braceleftbig\n2jlog(E0eqε)/bracerightbig\n. (4.25)\nSincelj≤2, we get from (4.18), (4.22) and (4.25)\n/radicalbig\n(t+r)(t−r)¯u(r,t)≥exp/braceleftbig\n2jJ(r,t)/bracerightbig\nlog−2/parenleftbiggt−r\n2K/parenrightbigg\nin Σ∞,(4.26)\nwhere\nJ(r,t) := log/parenleftBigg\nε/braceleftbigg\nB−1log/parenleftbiggt−r\n2K/parenrightbigg/bracerightbigg2/parenrightBigg\nandB:=E−1/2\n0e−q/2.(4.27)\nWe take ε0>0 so small that\nBε−1/2\n0≥log(2K). (4.28)\nFor a fixed ε∈[0,ε0), we suppose that Tsatisfies\nT≥exp/parenleftbig\n4Bε−1/2/parenrightbig\n. (4.29)\nNext, we take τ >0 so that\nT > τ > exp/parenleftbig\n2Bε−1/2/parenrightbig\n(>2K).\nFrom (4.28) and (4.29), it follows that\nτ >2Kexp(Bε−1/2). (4.30)\nWe get from (4.27) and (4.30)\nJ(0,τ) = log/parenleftbigg\nε/braceleftBig\nB−1logτ\n2K/bracerightBig2/parenrightbigg\n>0. (4.31)\nSince (0,τ)∈Σ∞, from (4.26) and (4.31), we get u(0,τ)→ ∞(j→ ∞).\nThis completes the proof. ✷\n4.2 Proof of Theorem 2.4\nWe divide the proof of Theorem 2.4 into two cases, 1 < p <2 andp= 2.\nFirst, we shall handle the sub-critical case.\nProof of Theorem 2.4 with 1 < p <2.Due to the assumption on the\ninitial data in Theorem 2.4, f(x)≥0 (/ne}ationslash≡0) andf(x)+g(x)≡0, we have\nF(0)>0, F′(0) = 0.\nFor the key inequality, we employ the following proposition.\n33Proposition 4.1 Let1< p <2. Suppose that the assumptions in Theorem\n2.4 are fulfilled. Then, there exists a positive constant C∗=C∗(f,g,p,k)\nsuch that F(t) =/integraltext\nR2u(x,t)dxsatisfies\nF′′(t)≥C∗εpt2−3p/2fort≥k. (4.32)\nProof.It follows from Lemma 2.2 in [23] that\nF1(t)≥1\n2/parenleftbig\n1−e−2k/parenrightbig/integraldisplay\nR2εf(x)φ1(x)dxfort≥k, (4.33)\nwhere\nφ1(x) =/integraldisplay\nS1ex·ωdωandF1(t) =e−t/integraldisplay\nR2u(x,t)φ1(x)dx.\nFrom (2.4) and (2.5) in [23], we obtain\nF′′(t)≥C(t+k)2−3p/2|F1(t)|pfort≥0. (4.34)\nFrom (4.33) and (4.34), we obtain (4.32). This completes the proof. ✷\nIn the sub-critical case, the following basic lemma is useful.\nLemma 4.2 (Takamura [18]) Assume that (4.9) is replace by\nF(0)>0, F′(0) = 0,\nand additionally that there is a t0>0such that\nF(t0)≥2F(0). (4.35)\nThen, the conclusion of Lemma 4.1 is changed to that there exi sts a positive\nconstant/tildewiderD0=/tildewiderD0(p,a,q,B)such that\nT <22/MT2\nholds provided\nT2:= max{T0,t0,k} ≥/tildewiderD0A−(p−1)/(2M). (4.36)\nThis is exactly Lemma 2.2 in [18], so that we shall omit the proof here.\nIntegrating (4.32) in [ k,t], we have\nF′(t)≥C∗\n3−3p/2εp(t3−3p/2−k3−3p/2)+F′(k)\n34fort≥kbecause of 1 < p <2. Note that F′(k)≥0 follows from F′′(t)≥0\nfort≥0 andF′(0) = 0. Hence we obtain that\nF′(t)≥C∗(1−2−3+3p/2)\n3−3p/2εpt3−3p/2fort≥2k.\nIntegrating this inequality in [2 k,t] together with F(0)>0, we get\nF(t)≥D3εpt4−3p/2fort≥4k, (4.37)\nwhere\nD3:=C∗(1−2−3+3p/2)(1−2−4+3p/2)\n(3−3p/2)(4−3p/2)>0.\nFrom 2F(0) = 2/ba∇dblf/ba∇dblL1(R2)εand (4.37), (4.35) in Lemma 4.2 is fulfilled with\nt0:=D4ε−(p−1)/(4−3p/2),\nift0≥4k, where\nD4:=/braceleftbig\n2/ba∇dblf/ba∇dblL1(R2)D−1\n3/bracerightbig(4−3p/2)−1\n.\nWe are now in a position to apply our result here to Lemma 4.2 with specia l\nchoices on all positive constants except for T0as\nA=D3εp, B=π1−p, a= 4−3\n2p, q= 3(p−1)\nwhich imply that (4.8) yields\nM=p−1\n2a−q\n2+1 =γ(p,4)\n4>0.\nIf we set\nT0:=/tildewiderD0A−(p−1)/(2M)=/tildewiderD0D−2(p−1)/γ(p,4)\n3 ε−2p(p−1)/γ(p,4),\nthen we find that there is an ε0=ε0(f,g,n,p,k )>0 such that\nT0≥max{t0,4k}for 0< ε≤ε0\nbecause of 2 p/γ(p,4)<1/(4−3p/2). This means that T2=T0in (4.36).\nTherefore, the conclusion of Lemma 4.2 implies\nT <22/MT2=D5ε−2p(p−1)/γ(p,4)for 0< ε≤ε0,\n35where\nD5:= 28/γ(p,4)/tildewiderD0D−2(p−1)/γ(p,4)\n3 >0.\nThis completes the proof. ✷\nProof of Theorem 2.4 with p= 2.Since\nf(x)+g(x)≡0 and/integraldisplay\nR2f(x)dx <0,\nby Lemma 2.2, there exist positive constants /tildewiderE0and/tildewideK≥1 such that\nuL(x,t)≥/tildewiderE0\n(t+r)1/2(t−r)3/2fort−r≥/tildewideK.\nFort−r≥/tildewideK, we get from (4.12) that\n¯u(r,t)≥u0(r,t)≥/tildewiderE0ε\n(t+r)1/2(t−r)3/2. (4.38)\nWe define the following domains:\n/tildewideΣj:=/braceleftBig\n(r,t)∈[0,∞)2|t−r≥3/tildewideKlj/bracerightBig\n(j= 0,1,2,···),\n/tildewideΣ∞:=/braceleftBig\n(r,t)∈[0,∞)2|t−r≥6/tildewideK/bracerightBig\n.\nIn the same way as to obtain (4.15), for t−r≥/tildewideK, we get\n¯u(r,t)≥1/radicalbig\n(t+r)(t−r)/integraldisplay /integraldisplay\n/tildewiderQ0(r,t)(1+τ)−1λ|¯u|2dλdτ, (4.39)\nwhere\n/tildewiderQ0(r,t) :=/braceleftBig\n(λ,τ)∈[0,∞)2|/tildewideK≤τ−λ,0≤λ≤t−r−τ/bracerightBig\n.\nFor (r,t)∈/tildewiderΣ0, we set\nS(r,t) :=/braceleftbigg\n(λ,τ)∈[0,∞)2/vextendsingle/vextendsingle/vextendsingle5\n2/tildewideK≤τ+λ≤t−r,/tildewideK≤τ−λ≤5\n4/tildewideK/bracerightbigg\n.\n36For (r,t)∈/tildewiderΣ0, we have S(r,t)⊂/tildewiderQ0(r,t). Substituting (4.38) into (4.39)\nand changing the variables by (2.29), we get\n/radicalbig\n(t+r)(t−r)¯u(r,t)\n≥/integraldisplay /integraldisplay\nS(r,t)(1+τ)−1λ|¯u|2dλdτ\n≥1\n2/integraldisplayt−r\n5/tildewideK/2dα/integraldisplay5/tildewideK/4\n/tildewideK/parenleftbigg\n1+α+β\n2/parenrightbigg−1/parenleftbiggα−β\n2/parenrightbigg/parenleftBigg/tildewiderE0ε\nα1/2β3/2/parenrightBigg2\ndβ.(4.40)\nSinceα+β≤2α,α−β≥α−5k/4≥α/2 and/tildewideK≥1, we get from (4.40)\n/radicalbig\n(t+r)(t−r)¯u(r,t)≥(/tildewiderE0ε)2\n24/integraldisplayt−r\n5/tildewideK/2α−1dα/integraldisplay5/tildewideK/4\n/tildewideKβ−3/2dβ\n≥E∗\n0ε2log/parenleftbiggt−r\n3/tildewideK/parenrightbigg\nin/tildewiderΣ0, (4.41)\nwhereE∗\n0:=/tildewiderE02//parenleftBig\n29/tildewideK1/2/parenrightBig\n. Analogously to the proof of Theorem 2.3 with\np= 2, we obtain from (4.41)\n¯u(r,t)≥/tildewidedj/radicalbig\n(t+r)(t−r)log/tildewideaj/parenleftBigg\nt−r\n3/tildewideKlj/parenrightBigg\nin/tildewiderΣj,\nwhere\n/tildewidea0= 1,/tildewidestaj+1= 2/tildewideaj+2,\n/tildewided0=E∗\n0ε2,/tildewidestdj+1=/tildewidedj2\n3·23j+9.\nThis is the same form as (4.18), (4.19) and (4.20). Hence, we see tha t there\nexists a constant /tildewideqsuch that\n/tildewidedj≥exp/braceleftbig\n2jlog/parenleftbig\nE∗\n0e/tildewideqε2/parenrightbig/bracerightbig\n.\nSincelj≤2, we get\n/radicalbig\n(t+r)(t−r)¯u(r,t)≥exp/braceleftbig\n2jJ(r,t)/bracerightbig\nlog−2/parenleftbiggt−r\n6/tildewideK/parenrightbigg\nin/tildewidestΣ∞,\nwhere\nJ(r,t) := log/braceleftBigg\nε2/braceleftbigg\n/tildewideB−1log/parenleftbiggt−r\n6/tildewideK/parenrightbigg/bracerightbigg3/bracerightBigg\nand/tildewideB:= (E∗\n0)−1/3e−/tildewideq/3.\nIn the same way asin theproof of Theorem 2.3 with p= 2, we get the desired\nestimates. The proof is now completed. ✷\n37Acknowledgement\nThis work started when the first author was a master course stud ent in Fu-\nture University Hakodate, the third author was working in Future U niversity\nHakodate, and the fourth author was working in Muroran Institut e of Tech-\nnology. The third author has been partially supported by Special Re search\nExpenses in FY2017, General Topics (No.B21), Future University H akodate,\nalso by the Grant-in-Aid for Scientific Research (B) (No.18H01132) and (C)\n(No.15K04964), Japan Society for the Promotion of Science. Finally all the\nauthors are grateful to the referee for precise reading and use ful comments.\nReferences\n[1] R.Agemi, Y.Kurokawa and H.Takamura, Critical curve for p-qsystems\nof nonlinear wave equations in three space dimensions , J. Differential\nEquations 167(2000), no. 1, 87-133.\n[2] R.Agemi and H.Takamura, The lifespan of classical solutions to non-\nlinear wave equations in two space dimensions , Hokkaido Math. 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Treboniu Laurean, 400271, Cluj -Napoca, Romania  \n  \n \nAbstract  \nPerpendicular magnetic anisotropy  (PMA)  in ultrathin magnetic structures  is a key ingredient for the \ndevelopment of electrically controlled spintronic devices. Due to their relatively large spin -polarization , \nhigh Curie temperature  and low Gilbert damping t he Co -based full Heusler alloys are of special \nimportance  from  a scientific and application s point of view. Here, we study the mechanisms responsible \nfor the PMA in Pt/Co -based full Heusler alloy /MgO  thin films structures . We show that the  ultrathin  \nHeusler films exhibit  strong PMA even in the absence of magnetic annealing. By means of ferromagne tic \nresonance experiments, we demonstrate that the effective magnetization  shows a two -regime behavior  \ndepending on the thickness of the Heusler layers . Using  Auger spectroscopy measurements,  we evidence \ninterdiffusion at the underlayer/Heusler interface  and the formation of an interfacial CoFe -rich layer which \ncauses  the two -regime behavior. In the case of the ultrathin films, th e interfacial CoFe -rich layer promotes \nthe strong PMA through the electronic hybridization of the metal alloy  and oxygen  orbitals across the \nferromagnet /MgO interface . In addition,  the interfacial CoFe -rich layer  it is also generating  an increase  of \nthe Gilbert damping  for the ultrathin films  beyond the spin -pumping effect . Our results  illustrate  that the \nstrong PMA is not an intrinsic property of the Heusler/MgO interface but  it is actively influenced by the \ninterdiffusion, which can be tuned by a proper choice of the underlayer material, as we show for the case \nof the Pt, Ta and Cr underlayers.  \n \n \n                                                 \na) mihai.gabor@phys.utcluj.ro  2 \n  \n \nIntroduction  \n \nUltrathin films structures showing perpendicular magnetic anisotropy (PMA) are under intensive \nresearch for the development of electrically controlled spintronic devices. Particularly, current induced \nspin–orbit torques (SOTs) in heavy -metal/ferromagnet (FM) heterostructures showing PMA are used to \ntrigger  the magnetization switching1,2. Besides , the antisymmetric  interfacial Dzyaloshinskii -Moriya3,4 \ninteraction ( iDMI)  in similar  PMA  architectures , if strong enough, can lead to the formation of special \nchiral structures like skyrmions5 which are  drivable  by electrical currents6. In the case of  the spin transfer \ntorque magnet ic random -access  memories (STT -MRAMs), considered a s a poten tial replacement for the \nsemiconductor -based  ones, the use of strong PMA materials is required for increased thermal stability7, \nwhile high spin polarization and low Gilbert damping is needed  to obtain large magnetoresistive ratios \nand efficient curre nt induced STT switching8,9.  \nCobalt-based full Heusler alloys are a special class of ferromagnetic materials that attract an increased \nscientific interest , since the ir theoretical  prediction of  half-metallicity10. These compounds are described \nby the formula Co2YZ, where Y is a transition metal,  or a mixture of two trans ition metals , and Z is a main \ngroup , or a mixture of two main group  sp element s. Large magnetoresistive ratios are experimentally \ndemonstrated in certain Co2YZ based in -plane11-18 and out -of-plane19,20 magnetized magnetic tunnel \njunctions  (MTJs)  and r elative ly low Gilbert dampin g parameters were  determined for some compounds21-\n28. Furthermore , PMA was evidenced for  Co2FeAl/ MgO19,29-33, Co 2FeAl 0.5Si0.5/MgO34-37, \nCo2FeSi/MgO38,39 or Co2FexMn 1-xSi/MgO40,41 structures with different  non-magnetic  underla yers. In \nsome of the cases, an annealing stage was necessary to induce PMA, while  for the other the perpendicular \nmagnetiz ation  was achieved even in the as -deposited state. The origin of the strong  PMA in th is type of \nstructures is still under debate . It could be related to both the oxidation at the Heusler/MgO interface42,43 \nand to the spin -orbit interaction effects at the heavy -metal underlayer/Heusler interface44,45. Moreover, it \nwas recently pointed out that in t he case of Co 2FeAl/MgO the diffusion of Al towards the MgO layer \nduring annealing  plays an important role for the  stabiliz ation of  the PMA46,47. The precise knowledge and \ncontrol of the mechanisms responsible for PMA is essential in order to be able to develop viable  spintronic \napplication s. Therefore, i n this paper, we  study  the underlying physics governing the  PMA for Co2FeAl, \nCo2FeAl 0.5Si0.5, Co2FeSi and Co2Fe0.5Mn 0.5Si Heusler alloy thin films  sandwiched between Pt and  MgO \nlayers. We show that b elow  a certain  critical thickness all the Heusler films  show strong PMA even in the 3 \n absence of magnetic annealing . Additionally, using ferromagnetic resonance experiments , we demonstrate \nthat, depending on the thickness  of the Heusler layers,  the effective perpendicular magnetic anisotropy \nshows a  two-regime  behavior.  After excluding other possible mechanism s, we evidence  using Auger \nspectroscopy measu rements, that the diffusion of the lighter elements towards the Pt underlayer and the \nformation of an interfacial CoFe -rich layer causes the  two-regime behavior . In the case of the ultrathin \nfilms , this interfacial CoFe -rich layer  promotes the  strong  PMA through the hybridization of the [Co,Fe] \n3𝑑𝑧2 and O 2𝑝𝑧 orbital s at the interface  and is also responsible  for the increased Gilbert  damping . Our \nstudy reveals that the strong PMA is not intrinsic to the Heusler/MgO interface . It is strongly  influenced \nby the interdiffusion  and can be adjusted  by a proper choice of the underlayer  material , as we show for \nthe case of the Pt, Ta and Cr  underlayers.  \n \nExperimental   \n \nAll the samples studied here were grown at room temperature on thermally oxidized silicon substrates \nin a magnetron sputtering system having a base pressure lower than 2×10-8 Torr.  The main samples have \nthe following structure: Si/SiO 2//Ta (3 nm )/Pt (4 nm)/ FM (0.8-10 nm)/MgO (1 nm)/ Ta (3 nm), where FM \nstands for  Co2FeAl (CFA), Co 2FeAl 0.5Si0.5 (CFAS), Co 2FeSi (CFS), Co 2Fe0.5Mn 0.5Si (CFMS) or CoFeB \n(CFB) , depending on the sample. Additional samples were grown,  and their  structure will be discussed \nlater in the text. The metallic layers were deposited  by dc sputtering  under an argon pressure of 1 mTorr, \nwhile the MgO layer was grown by rf sputtering under an argon pressure of 10 mTorr. The Heusler alloy s \nthin films were sputtered  from stoichiometric targets. The 3 nm thick Ta buffer layer was grown directly \non the substrate to minimize  the roughness and to facilitate the (111) texturing of the  upper Pt  layer. The \n1 nm thick MgO layer was deposited to induce perpendicular m agnetic anisotropy on the Heusler thin \nfilm7. An additional  3 nm th ick Ta capping layer was sputtered  to protect the samples from oxidation due \nto air exposure. The structure of the samples was characterized by x -ray diffraction (XRD) using a four -\ncircle diffractometer. The static magnetic properties have been investigated using a Vibrating Sample \nMagnetometer (VSM) , while  the dynamic magnetic properties by using a TE(011) cavity  Ferromagnetic \nResonance (FMR)  setup working in X -band  (9.79 GHz ). Auger spectra have been recorded  in derivative \nmode, using  a cylindrical mirror analyzer spectrometer working at an electron beam energy of 3keV.  \nDepth profile analysis have been performed by successive  recording of the Auger spectra and Ar ion \nsputter -etching of the surface of the samples by  using a relatively low  ion energy of 600 eV.    \n 4 \n  \n \nResults and discussions  \n \nFigure 1 (a) shows  2θ/ω x-ray diffraction patterns recorded for four representative  Pt (4 nm)/  Co2YZ \n(10 nm)/  MgO (1 nm)  samples . Irrespective of the Heusler composition, the patterns show the (111) and \n(222) peaks  belonging to the Pt  layer , the (022) peak  arising from the Heusler films  and the  (001) peak of \nthe Si substrate . This indicates that the Pt layer has a (111) out -of-plane texture, while the Heusler films \nare (011) out -of-plane textured.  Laue oscillations are observable around the (111) Pt reflection which \nconfirms the good crystalline quality for the Pt films48. Moreover, ϕ -scan measurements (not shown here) \nindicate that both the Pt underlayer and the Heusler fil ms have no in -plane texturing but  show an  in-plane \nisotropic distribution  of the crystallites . No peak belon ging to the Ta capping layer was observed, \nindicating  that the film is in an amorphous or nanocrystalline state.  \nThe static magnetic properties of our films were characterized by VSM measurements . Figure 2 show s \nhysteresis loops measured with the magnetic field applied perpendicular  to the plane of the samples, for \nrepresentative Heusler films thickness es. In order to remove th e substrate diamagnetic contribution,  we \nfitted the large field data with a linear function and extract ed the linear slope  from the raw data.  Regardless  \nof the ir composition , all the Heusler films show a similar behavior. Above a critical spin-reorientation \ntransition thickness  the samples show in-plane magnetic anisotropy . This is indicated by the shape of the \nhysteresis loop s in Fig. 2  (a)–(d), which is typical for a hard axis of magnetization, showing a continuous \nrotation of the magnetization up to saturation.  Below th is critical thickness, the samples show PMA , which \nis attested by  the square shaped hysteresis loops in Fig. 2 (e) –(h). We also determined the saturation \nmagnetization  (𝑀𝑆) and the effective thickness es of the ferr omagnetic layers  using hysteresis loop \nmeasurements  and the procedure described in 31.  The effective thicknesses of the ferromagnetic layers \nare used throughout  the paper and the 𝑀𝑆 is found  to be  790 ± 70 emu/cm3, 660 ± 50 emu/cm3, 935 ± 75 \nemu/cm3 and 895 ± 75 emu/cm3 for CFS, CFMS, CFAS and CFA samples, respectively.   \nIn order to get more insights on the magnetic anisotropy properties of our films, we have performed \nFMR measurements  with the magnetic field applied at diffe rent θH angles  (defined in the inset of Fig. 4) \nwith respect to the normal direction of the layers . Figure 3 shows typical FMR spectra for various fie ld \nangle s recorded  for a 2.4 nm thick Pt/CFAS sample . We define the resonance field HR as the intersection \nof the spectr um with the base line, and the linewidth HPP as the distance between the positive and negative \npeaks of the spectrum. Figure 4 shows the θH dependence of the HR and of the linewidth HPP for the 2.4 5 \n nm thick Pt/CFAS sample. In order to extract the relevant FMR parameters, we analyzed the θH \ndependence of the FMR spectrum using  a model in which the total energy per unit volume is given by  \n𝐸=−𝑀𝑆𝐻cos(𝜃𝐻−𝜃𝑀)+2𝜋𝑀𝑆2cos2𝜃𝑀−𝐾⊥cos2𝜃𝑀,                             (1) \nwhere the first term is the Zeeman energy, the second term is the demagnetizing energy, and the last term \nis the magnetic anisotropy energy.  The 𝑀𝑆 is the saturation magnetization, 𝜃𝐻 and 𝜃𝑀 are the field and \nmagnetization angles defined in the inset of Fig. 4, and the 𝐾⊥ is the effective perpendicular magnetic \nanisotropy constant. From eq. 1 and the Landau -Lifshitz -Gilbert equation, one can der ive the resonance \ncondition as49 \n(𝜔\n𝛾)2\n=𝐻1×𝐻2,                                                                (2)  \nwhere 𝜔 is the angular frequency of the microwave , 𝛾 is the gyromagnetic ratio , given by 𝛾=𝑔𝜇𝐵ℏ where \n𝑔 is the  Landé  g-factor, 𝜇𝐵 is the Bohr magneton and ℏ is the reduced Plan ck constant , and with 𝐻1 and \n𝐻2 given by   \n𝐻1=𝐻𝑅cos(𝜃𝐻−𝜃𝑀)−4𝜋𝑀effcos2𝜃𝑀,                                             (3) \n𝐻2=𝐻𝑅cos(𝜃𝐻−𝜃𝑀)−4𝜋𝑀effcos2𝜃𝑀,                                           (4) \nwhere 4𝜋𝑀eff is the effective magnetization defined as  4𝜋𝑀eff=4𝜋𝑀𝑆−2𝐾⊥𝑀𝑆⁄ and 𝐻𝑅 is the \nresonance field.  For each value of  𝜃𝐻, the 𝜃𝑀 at resonance is calculated from the energy minimum \ncondition 𝜕𝐸 𝜕𝜃𝑀=0 ⁄ . Hence , the 𝐻𝑅 dependence on 𝜃𝐻 can be fitted by Eq. (2)-(4) using 4𝜋𝑀eff and \n𝑔 as adjustable parameters . A typical fit curve is s hown in Fig. 4(a).  \n Figure 5 shows the 𝑔 factor depen dence on the thickness of the Heusler layers for samples with \ndifferent Heusler layer composition s. Depending on the thickness , two regimes are discernable . For \nrelatively  large thickness es, above 2.5-3 nm, the  𝑔 factor shows rather  constant value s between 2.07 and \n2.11, depending on the type of the Heusler layer . For lower thickness es, 𝑔 shows a monotonous decrease, \nregardless  of the Heusler  layer composition. This is an interface effect and it is usually attributed to the \nfact that at the interfaces , due to the symmetry breaking, the orbital motion is n o longer entirely  quenched  \nand will contribute to the gyromagnetic ratio50,51. Another possibility, which cannot be exclude d in our \ncase, is the reduction of the 𝑔 factor due to intermixing between the ferromagnetic  Heusler layer and non -\nmagnetic materials at the interfaces50. \n Figure 6 shows the effective magnetization 4𝜋𝑀eff dependence on the inverse thickness of the    \nferromagnetic layer for samples with different composition s. It is to be mentioned  that the 4𝜋𝑀eff was \ndetermined from FMR experiments only for samples with in-plane magnetic anisotropy (positive 4𝜋𝑀eff). \nIn the case of ultrathin samples showing perpendicular magnetic anisotropy  (negative 4𝜋𝑀eff), due to the 6 \n strong linewidth enhancement , it was not possible to obtain reliable resonance curves.  Therefore, in this \ncase the 4𝜋𝑀eff was estimated from VSM measurements.  Generally, it is considered  that the effective \nperpendicular magnetic anisotropy co nstant  𝐾⊥ can be written as the sum of a volume  (𝐾𝑉), which includes \nmagneto -crystalline and strain related anisotropies, and a surface (𝐾𝑆) contribution : 𝐾⊥=𝐾𝑉+𝐾𝑆𝑡⁄, \nwhere 𝑡 is the thickness of the ferromagnetic layer . Thus, the effective magnetization can be written as  \n \n4𝜋𝑀eff=(4𝜋𝑀𝑆−2𝐾𝑉\n𝑀𝑆)−2𝐾𝑆\n𝑀𝑆1\n𝑡.                                                             (5) \n \nThe above relation implies a linear dependence of  the effective magnetization on the inverse thickness of \nthe ferromagnetic layer.  However, as shown in Fig.6  the Heusler samples do not show a single linear \ndependence for the  entire  thickness range, but two regimes above and below a  certain critical thickness.  \nUsing the 𝑀𝑆 values determined from VSM measurements and b y fitting the experimental data in the large  \nthickness regime to eq. (5) , we extract a surface  anisotropy constant 𝐾𝑆 for the CFA  and CFAS of 0.24  ± \n0.03 erg/ cm2 and 0.22 ± 0.02 erg/cm2 and a volume contribution 𝐾𝑉 of (1.27 ± 0. 69)×106 erg/cm3 and \n(1.51 ± 0.7)×106 erg/cm3, respectively. In the case of the CFMS and CFS, the 𝐾𝑆 was negligible small \nwithin the error bars  and the 𝐾𝑉 was found to be (0.44 ± 0.38)×106 erg/cm3 and ( 0.51 ± 0. 4)×106 erg/cm3, \nrespectively. Using the as extracted values of th e anisotropy constants , we can calculate , for example,  in \nthe case of the CFAS samples a spin-reorientation tra nsition thickness of around 0.55 nm. This is clearly \nnot in agreement with the experimental data , as seen from Fig. 2 and 6 , already  a 1 nm thick CFAS film \nshows strong PMA  and it is spontaneous perpendicular ly magnetized . This is a consequence of the fact \nthat the  1 nm thick CFAS film  falls within  the second anisotropy regime below the critical thickness . The \noccurrence  of this second anisotropy regime with larger effective perpendicular magnetic anisotropy can \nhave several explanations.  For such thin films one must always consider the possible influences of the \nsurface roughness. If the roughness is relatively large , an in -plane demagnetization field will develop at \nthe edges of the terraces  which will reduce the shape anisotropy and fav or perpendicular magneti zation . \nThis is equivalent  to the  emergence  of an additional dipolar surface anisotropy contribution52. The \nroughness is  a parameter which is not easily quantifiable  experimentally  in such thin  multilayer structures. \nHowever, it is reasonable to expect to be comparable  for similar  heterostructures in which the Heusler \nalloy film is replaced with a CFB layer . Atomic force microscopy topography images (not shown here) \nrecorded for heterostructure w ith CFB and CFAS layers are featureless and show a similar RMS \nroughness.  As such, if the low thickness  anisotropy  regime is due to the roughness it must be observable \nalso in the case of CFB samples. However, this is not the case , as shown in Fig. 6, the CFB samples show 7 \n a single linear behavior for the whole range of  thickness . Fitting the data to eq. (5), allowed us to extract  \nfor CFB samples  a surface anisotropy contribution 𝐾𝑆 of 0.79 ± 0.04 erg/cm2 and a  negligible small 𝐾𝑉 \nvolume contribution , in line with previous reports7,53. These findings suggest  that the roughness is not \nresponsible for the two regimes behavior observed in the case of the Heusler samples.  \nAnother possible  physical mechanism  which can explain the presence of the two regimes is  the strain \nvariation  due to coherent –incoherent  growth transition54,55. Within this model, below the critical thickness , \nthe ferromagnetic layer grows uniformly strained in order to account for the lattice  misfit with the adjacent \nlayer s. Above the critical thickness, the strains are partially relaxed through the formation  of misfit \ndislocations.  The changes in the magnetoelastic anisotropy contributions corresponding to this structural \ntransition  can be responsible for t he presence of the two regimes54,55. This scenario is likely in the case of \nthe Heusler samples , since both the bottom Pt layer  the upper Heusler film  grow  out-of-plane textured.  In \norder to test this hypothesis, we have deposited two additional sets of samples. The first set consisted  of \nSi/SiO 2//Ta ( 6 nm)/ CFAS  (tCFAS)/MgO (1 nm)/Ta (3 nm)  samples . The motivation to grow this type of \nsamples was to obtain Heusler films with no out -of-plane texturing. Indeed, x -ray diffra ction measurement  \n[Fig. 1(b)], performed on a Ta/CFAS sample with a Heusler layer thickness of 10 nm , did not indicate the \npresence of any diffraction peak s, except  for the one belonging to the Si substrate. This suggest that both \nthe Ta and the CFAS films are either nanocrystalline or amorphous . Thus , in this type of structure we do \nnot expect the presence of the coherent –incoherent growth transition. The second set of samples consisted \nof epitaxial  MgO  (001)//Cr ( 4 nm)/CFAS  (tCFAS)/MgO (1 nm)/Ta (3 n m) structures . The x -ray diffraction \nmeasurement [Fig. 1 (b)], performed on a Cr/CFAS sample with a Heusler layer thickness of 10 nm, \nindicate s the exclusive presence of the (001) type reflections from the MgO substrate and the Cr and CFAS \nlayers . This confirms the epitaxial growth of the stacks, except for the Ta capping layer, which is \namorphous. Having in view the epitaxial growth we might expect for these samples a possible coherent –\nincoherent growth transition, eventually at higher CFAS thick nesses having in view the relative low \nmismatch between the CFAS lattice and the 45° in-plane rotated Cr lattice  (0.7%). The effective \nmagnetization dependence on the inverse thickness of the ferromagnetic layer for amorphous Ta/CFAS \nand epitaxial Cr/CFAS samples alongside with the Pt/CFAS samples is shown in Fig. 7 . Is to be mentioned \nthat in the for the Ta/CFAS samples the PMA  was obtained for thicknesses  below 1.6 nm , while for the \nCr/CFAS the PMA was not achieved even for thicknesses down to 1 nm. Interestingly, in the case of the \nepitaxial Cr/CFAS samples , for which one might expect possible coherent –incoherent growth transition, \na single linear behavior for the whole thickness range is observ ed. In the case amorphous  Ta/CFAS \nsamples, for which the coherent –incoherent growth transition is not expected, a two-regimes behavior  can 8 \n be distinguished . Although we cannot rule a possible coherent -incoherent growth transition  at larger \nthicknesses, t he results indicate this mechanism is not responsible for the two regimes behavior that we \nobserve  at relatively low thicknesses and other mechanism s must be at play.  \n By fitting the high thickness regime data from Fig.7  to eq. (5)  we extracted for Ta/CFAS  samples a \nsurface anisotropy contribution 𝐾𝑆 of 0.27 ± 0.08 erg/cm2. The volume contribution,  𝐾𝑉, was determine d \nto the be negligible small, as expected for untextured films. Remarkably, the 𝐾𝑆 for the Ta/CFAS samples \nis similar within the error bar s to one obtained for  the Pt/CFAS samples.  Moreover, even in the low \nthickness regime the 𝐾𝑆 might be assumed  similar for the two sets of samples. However, we must  consider  \nthe large uncertainty  having in view the  sparse  data points available for fitting  in the low thickness regime. \nEven so, the clear difference between the two sets of samples is that the Ta/CFAS one shows a  larger \ncritical thickness  (around 2.4 nm ) that separates  the two anisotropy regimes , as compared t o the Pt/CFAS  \none (around 1.5 nm) . This suggests that the possible mechanism responsible for the two -regime behavior \nmight be related to the atomic diffusion at the  Pt/CFAS and  Ta/CFAS interface s. It is well known that Ta \nis prone to diffusion of light elements56. Therefore , a larger critical thickness for the Ta/CFAS samples \nwill imply a larger  atomic  diffusion at the Ta/CFAS interface  compared to the Pt/CFAS one .  \nTo test th e hypothesis  of the interdiffusion , we performed Auger electron spectroscopy (AES) analyses \non th e three set of samples : Pt/CFAS, Cr/CFAS and Ta/CFAS. AES is a surface sensitive technique which \ncan give information about the chemical composition of the surface  with a depth detection limit of 1 -2 \nnm. We started from 10 nm thick CFAS layer samples  and first Ar ion etched the CFAS films down to 4 \nnm thickness and recorded the AES spectra.  Subsequently, t he Ar ion etching and AES spectra recording \nwas repeated  in steps of 1 nm until reaching the underlaye r/CFAS interface.  The etching rate of CFAS \nwas previously calibrated using  ex-situ x-ray reflectometry measurements.  Figure 8 (a) shows two spectra \nrecorded for the Pt/CFAS sample, one after etching the  CFAS layer down to 4 nm (Pt/CFAS 4 nm) and \nthe other  one after etching the CFAS layer down to 1 nm of thickness (Pt/CFAS 1 nm). In the case of the \nPt/CFAS 4 nm spectr um the peaks of Co and Fe are visible alongside with the peaks fro m Al and Si. The \ninset of Fig. 8(a) depicts a n enlargement of the Pt/CF AS 4 nm spectrum around the peaks of Al and Si. \nThe amplitude of the Co and Fe peaks is m uch larger than the amplitude of the of Al and Si ones. This is \ndue to the higher  concentration and higher  Auger relative sensitivity of the  Co and Fe compared to the Al \nand Si. In the case of the Pt/CFAS 1 nm the spectrum  shows the peaks from Co and Fe , with a lower \namplitude,  and the peaks from the Pt underlayer. The presence of the Co and Fe peaks together with the \nPt peaks is not surprising . It is owed to the possible interdiffusion layer at the interface  and to the finite \ndepth resolution  of the  AES which probes both the CFAS layer and the Pt underlayer. The Al and Si peaks 9 \n are not observable , which can be associated  to the relatively low amplitude of the Al and Si falling below \nthe detection limit of the measurement. To test this possibility,  we acquired  Auger  spectra in a narrow \nenergy window around the Al peak , using a longer acquisition time and averaging 10 spectra for e ach \nrecoded  spectrum.  We selected the Al peak and not the Si one because of its larger ampli tude. These \nspectra  recorded for the Pt/CFAS, Ta/CFAS and  Cr/CFAS samples after etching the CFAS layer down to \n4, 3, 2 and 1 nm are shown in Fig.8  (b)-(d). In the case of the Pt/CFAS sample the Al peak is observable \nfor CFAS thickness es down to 1 nm , while in the case of Ta /CFAS  for thicknesses down to 2 nm. \nInterestingly, in the case of the Cr/CFAS sample the Al peak is visible even for a CFAS thickness of 1  \nnm, although with lower amplitude. These findings suggest th at at the underlayer/CFAS interface there is \na diffusion of the light er elements (Al and most likely also Si)  towards the underlayer, with different \ndegree, depe nding  on the nature of the underla yer. As shown schematically in Fig. 8, due to th is lighter \nelements diffusion a CoFe -rich layer form s at the underlayer/CFAS interface. The extent  of the CoFe rich \nlayer depends on the nature of the underlayer . It has the largest thickness for the Ta underlayer (between \n2 and 3 nm) , it is decreasing for the Pt underlayer (between 1 and 2 nm) and it is most likely non -existing \nor extremely thin (below 1 nm) in the case of the Cr underlayer.  \nThe presence of th e CoFe r ich layer agrees  with our findings concerning the occurrence  of the high \nand the low effective PMA  regime s depend ing on the thickness of the Heusler layer. In the case of  the \nPt/CFAS /MgO  samples , the low effective PMA  regime  occurs  for a CFAS layer thickness above  1.6 nm. \nIn this case, the bottom interface consists of Pt/CoFe -rich layer, while the top one of CFAS/MgO . In \nprinciple, both interfaces could contribute to PMA  through Co-O hybridization in the case of the Co-\nterminated CFA S/MgO interface43 or through the d –d hybridization between the  spin-split Co 3d bands \nand the Pt layer 5d bands with large  spin-orbit  coupling44,45. However, their contribution to PMA is small \nand, as we previously mentioned,  would not stabilize perpendicular magnetization except for extremely \nthin CFAS layer s. In the case  of the  high effective  PMA regime (below 1.6 nm) , the bottom interface is \nsimilar consisting of Pt/CoFe -rich layer  and will contribute negligibl y to PMA . However, the top interfa ce \nis now constituted of CoFe -rich layer/MgO  and will induce strong PMA through the hybridization of the \n[Co,Fe] 3𝑑𝑧2 and O 2𝑝𝑧 orbitals42. The premise that the strong PMA is induced by the CoFe -rich \nlayer/MgO interface  is also consistent with our observat ions regarding the dependence of the magnetic \nanisotropy on the nature of the underlayer. As seen in Fig. 7, i n the case of the Cr/CFAS samples, where \nno CoFe -rich layer was evidenced, there is only one anisotropy regime with a relatively low effective \nPMA . In the case of the Ta/CFAS sa mples, the high effective PMA regime is present starting from a larger \nCFAS thickness , as compared to de case of Pt/CFAS samples , which is in agreement  with the thicker 10 \n CoFe -rich layer observed for the Ta/CFAS relative to the Pt/CFAS ones.  It is to be mentioned that in the \ncase of Ru/CFA/MgO and Cr/CFA/Mg O annealed samples  Al diffusion towards the MgO but not towards \nthe underlayer was previously observed46,47. The lack of Al diffusion towards the Cr underlayer is in \nagreement with our findings. In the case of the aforementioned studies, the thermal  annealing of the \nsamples was necessary  to facilitate the Al diffusion  and to achieve strong PMA. In our case, for the Pt and \nTa underlayer , we attain  strong PMA in the low thickness regime without the need of thermal annealing. \nThis indicates that for the Pt and Ta underlayer s the [Al,Si] diffusion takes place during the growth of the \nCFAS film, which results in the formation of the interfacial CoFe -rich layer  directly during deposition . A \nfurther deposition of MgO on this CoFe -rich layer will generate the strong PMA  through the hybridizati on \nof the [Co,Fe] 3𝑑𝑧2 and O 2𝑝𝑧 orbitals42. Having in view the si milar behavior  of the magnetic anisotropy \nfor the CFA, CFAS, CFMS and CFS Heusler alloys thin films that we study here , it is reasonable to \nassume that in all the cases there is  a diffusion of the lighter elements  (Al, Si) towards the Pt underlayer \nand the formation of the CoFe -rich interfacial  layer , which , when MgO is deposited on top,  will give rise \nto the strong PMA in the low thickness regime.  \nWe now discuss the thickness  dependence of the Gilbert damping parameter extracted from  the θH \ndependence of the linewidth HPP. It is known  that generally the linewidth is given by a sum o f extrinsic  \nand intrinsic contr ibution  as49,57-59: \n𝐻PP=𝐻PPint+𝐻PPext,                                                                (5)  \n𝐻PPint=𝛼(𝐻1+𝐻2)|d𝐻𝑅\nd(𝜔𝛾⁄)|,                                                       (6) \n𝐻PPext=|d𝐻𝑅\nd(4𝜋𝑀eff)|Δ(4𝜋𝑀eff)+|d𝐻𝑅\nd𝜃𝐻|Δ𝜃𝐻+Δ𝐻TMS,                            (7) \nwhere, 𝛼 is the intrinsic Gilbert dampi ng parameter and the three  terms in equation (7) are the linewidth \nenhancement due to the anisotropy distribution , due to deviation from planarity of the films  and due to the \ntwo-magnon scattering. In the case of our films, the θH dependence of the linewidth HPP is well fitted \nusing only the intrinsic contribution  and the extrinsic enhancement due to the anisotropy distribution. For \nthis, |d𝐻𝑅d(𝜔𝛾⁄) ⁄ |  and |d𝐻𝑅d(4𝜋𝑀eff) ⁄ |  are numerically calculated using Eqs. (1)-(4) and the HPP vs. \nθH experimental dependence is fi tted to Eq. (5) using 𝛼 and Δ(4𝜋𝑀eff) as adjustable parameters49. An \nexample of a  fit curve is depicted  in Fig. 4(b)  for the case of the 2.4 nm thick Pt/CFAS sample . Figure 9 \nshows the 𝛼 dependence on the inverse ferromagnetic layer  (1/t) thickness for the Pt/CFA, Pt/CFS, \nPt/CFMS , Pt/CF AS and Pt/CFB samples.  We will first discuss the case of CFB, where  a linear dependence \nis observed . The linear increase of the Gilbert damping  parameter with 1/t is expected  and it is due to the \nangular momentum loss due to the spin pumping effect in the Pt layer. In this type of structures it  was 11 \n shown60 that the total damping is given by 𝛼=𝛼0+𝛼SP𝑡⁄, where  𝛼0 is the Gilbert damping of the \nferromagnetic film and 𝛼SP is due to the spin pu mping effect.  By linear fitting the data in Fig. 9  we obtain \na Gilbert damping parameter for the CFB of 0.0028 ± 0.0003 , in agreement with other reports61,62. In the \ncase of the CFAS films, the linear dependence  is observed only for the large thickness region  and by fitting \nthis data we obtain a Gilbert da mping parameter of 0.00 53 ± 0.00 12, consistent wit h previously reported \nvalue s for relative ly thick er films25. The low thickness data deviates from the  linear dependence . This \nbehavior is similar for all the other studied Heusler films, with the low thickness deviation being even \nmore pronounced.  The strong increase of the damping can be related to  the [Al,Si] diffusion and the  \nformation of the interfacial CoFe -rich layer. Since the [Al,Si]  diffusion  is more important for thinner films , \nit will have a stronger impact on the chemical composition relative to the thicker ones. The relatively  small \ndamping of the Co based full Heusl er alloys is a consequence of the ir specific electronic structure21. \nConsequently , deviations from the  correct stoichiometry , which is expected to have a n important  effect \non the electronic structure , will lead to a strong increase of the damping, as shown, for example, by ab-\ninitio calculation in the case of Al deficient CFA films47. Therefore, the increase of the damping beyond \nthe spin pumping effect for the thinner  Heusler films is explained by  the interfacial CoFe -rich layer \nformat ion.  \n \nConclusions  \n \nWe have studied the mechanism s responsible for PMA in the case of  Co2FeAl, Co 2FeAl 0.5Si0.5, Co 2FeSi \nand Co 2Fe0.5Mn 0.5Si Heusler alloy thin films sandwiched between Pt and MgO layers.  We showed that \nthe ultrathin Heusler films exhibit  strong PMA irrespective of their composition. The effective \nmagnetization displays  a two -regime behavior depending on the thickness of the Heusler layers.  The two -\nregime behavior is generated  by the formation of a n CoFe -rich layer at the underlayer/Heusler  interface \ndue to the interdiffusion. The strong PMA observed in the case of the ultrathin films can be explained by \nthe electronic hybridization of the CoFe -rich metal lic layer  and oxygen orbitals across the \nferromagne t/MgO interface . The formation of the  interfacial  CoFe -rich layer  causes the increase of the \nGilbert damping  coefficient  beyond the spin pumping for the ultrathin Heusler films. Our results illustrate \nthat the strong PMA is not an intrinsic property of the  Heusler/MgO interface,  but it is actively influenced \nby the interdiffusion, which can be tuned by a proper choice of the underlayer material.  \n \n  12 \n FIG. 1. (a) 2θ/ω x-ray diffraction patterns recorded for four representative  Pt/Co 2YZ/MgO samples having \na thickness of the Heusler layer of 10 nm.  The patterns show the (111) and (222) peaks belonging to the \nPt layer , the (022) peak from the Heusler films and the (001) peak of the Si substrate.  (b) 2θ/ω x-ray \ndiffraction patterns for the Ta/CFAS (10 nm)/MgO and Cr/CFAS (10 nm)/MgO samples indicating the \namorphous or epitaxial  growth of the CFAS layer , respectively.  \n13 \n FIG. 2. Hysteresis  loops measured with the magnetic field applied perpendicular to the plane of the \nsamples . Depending on the thickness of the Heusler layers, the samples show  in-plane magnetic anisotropy \n(a)-(d) or perpendicular magnetic anisotropy (e) -(h).  \n  \n14 \n FIG. 3. Typical  FMR spectra measured at 9.79 GHz  for different θH field angles for a 2.4 nm thick \nPt/CFAS sample.  \n15 \n FIG. 4. (a) Resonance field HR and (b) linewidth HPP dependence on the θH field angle for a 2.4 nm thick \nPt/CFAS sample . The inset s hows a schematic of the measurement geometry. The points  stand for  \nexperimental data while the lines represent  the result of the theoretical fits, as described in text.  \n \n  \n16 \n FIG. 5. \ng factor dependence on the thickness of the Heusler layers for samples with different Heusler layer \ncomposition.  \n \n  \n  \n17 \n FIG. 6. The effective magnetization \n4effM dependence on the inverse thickness of the  ferromagnetic \nlayer for samples with different composition s. The points are experimental data while the lines are linear \nfits. In the case of the Heusler samples two linear fits correspond to the two anisotropy regimes.   \n \n18 \n FIG. 7. The effective magnetization  \n4effM dependence on the inverse thickness of the ferromagnetic \nlayer for amorphous Ta/CFAS and epitaxial Cr/CFAS samples.  The data for Pt/CFAS is also shown for \ncomparison. The points are experimental data while the lines are linear fits.  \n \n19 \n FIG. 8. (a) AES spectra recoded for the Pt/CFAS sample  after etching the CFAS layer down to 4 and 1 \nnm, respectively . The inset shows a zoom around de Al and Si peaks. AES spectra recorded around the \nAl peak  after etching the CFAS layer down to 4, 3, 2 and 1 nm for the (b) Pt/CFAS, (c) Ta/CFAS and \n(d) Cr/CFAS samples.  Schematic representation of the [Al,Si] diffusion to wards the underlayer and the \ninterfacial CoFe -rich layer formation . \n \n \n20 \n FIG. 9. Gilbert damping parameter ( 𝛼) dependence on the inverse ferromagnetic layer (1/t) thickness for \nthe Pt/CFA, Pt/CFAS , Pt/CFMS, Pt/CFS and Pt/CFB samples.  The points are experim ental data while \nthe lines are linear fits for  Pt/CFB and Pt/CFAS samples. 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Here, we report an approach to manipulate the damping factor of \nFeGa/MgO(001) films by oblique deposition. Owing to  the defects at the surface or \ninterface in thin films , two -magnon scatterin g (TMS)  acts as a non -Gilbert damping \nmechanism in magnetization relaxation.  In this work, the contribution of TMS was \ncharacterized  by in-plane angul ar dependent  ferromagnetic resonance (FMR) . It is \ndemonstrated that the intrinsic Gilbert damping is isotropic and invariant , while the \nextrinsic mechanism  related to TMS is anisotropic  and can be tuned by oblique \ndeposition. Furthermore, the two and fourfold TMS related to  the uniaxial magnetic \nanisotropy  (UMA)  and m agnetocrystalline anisotropy were discussed. Our result s open \nan avenue to manipulate  magnetization relaxation in spintronic devices.  \n1 \n Keywords : Gilbert damping , two -magnon scattering, FMR, oblique deposition, \nmagnetic anisotropy  \n2 \n 1. Introduction  \nIn the past decades, controlling magnetization dynamics in magnetic \nnanostructures has been extensively studied due to its great importance for spintronic \nand spin- torque applications  [1,2] . The magnetic relaxation is described within the \nframework of the Landa u-Lifshitz  Gilbert (LLG)  phenomenology  using the Gilbert \ndamping factor  α [3]. The intrinsic  Gilbert damping depends primarily on the spin- orbit \ncoupling (SOC) [4,5] . It has been demonstrated that alloying or doping with non-\nmagnetic transition metals  provides an opportunity to tune the intrinsic damping  [6,7] . \nUnfortunately, in this way the soft magnetic properties will reduce . In addition to the \nintrinsic damping, the two-magnon scatterin g (TMS) process se rves as a n important  \nextrinsic  mechanism i n magnetization relaxation  in ultrathin films  due to the defects at \nsurface or interface [8,9] . This process  describes the scattering between the uniform \nmagnons and degener ate final -state spin wave modes  [10]. The existence of TMS has \nbeen demonstrated in many systems of ferrites  [11-13]. Since the anisotropic scattering \ncenters , the angular dependence of the extrinsic TMS process  exhibi ts a strong in -plane \nanisotropy  [14], which allows  us to adjust the overall magnetic relaxation , including \nboth the int ensity of relaxation rate and  the anisotropic behavior.  \nHere, we report an approach to engineer  the damping factor of Fe81Ga19 (FeGa ) \nfilms by oblique deposition. The FeGa alloy exhibits large  magnetostriction and narrow \nmicrowave resonance linewidth  [15] , which  could assure  it as a  promising material for \nspintronic devices. For the geometry of off -normal deposition, it has been demonstrated \nto provoke shadow effects and create a periodic stripe defect matrix. This can introduce \na strong uniaxial magnetization anisotropy (UMA) pe rpendicular to the projection of \nthe atom flux  [16-19]. Even though some reports have shown oblique deposition \nprovokes a  twofold TMS channel  [20-22], the oblique angle dependence of the intrinsic \n3 \n Gilbert damping and the TMS  still remain in doubt. For our case, on the basis of the \nfirst-principles calculation and the in -plane angular -dependent FMR measurements, we \nfound that the intrinsic Gilbert damping is isotropic and invariant with varying oblique \ndeposition angles, while the extrinsic mechanism related to the two -magnon -scattering \n(TMS ) is anisotropic and can be tuned by oblique deposition. In addition, importantly \nwe firstly observe a phenomenon that the cubic magnetocrystalline anisotropy \ndetermines the area including degenerate magnon modes, as well as the intensity of fourfold TMS. In general , the strong connection between the extrinsic TMS and the \nmagnetic anisotropy , as well their direct impact on the damping constants , are \nsystem ically investigated, which offer us a useful approach to tailor the damping factor.\n \n2. Experimental details  \nFeGa thin films with a thickness of 20 nm were grown on MgO(001) substrates in \na magnetron sputtering system with a base pressure below 3 × 10−7 Torr. Prior to \ndeposition, t he substrates were annealed at 700 °C for 1 h in a vacuum chamber to \nremove surface contaminations and then held at 250 °C during deposition. The incident \nFeGa beam was at different obl ique angles of ψ =0°, 15°, 30°, and 45°, with respect to \nthe surface normal , and named S1, S2, S3, and S4 in this paper , respectively.  The \nprojection of FeGa beam  on the plane of the substrates  was set perpendicular to the \nMgO[110] direction, which induces a UMA perpendicular to the projection of FeGa \nbeam , i.e., parallel to the MgO[110] direction, due to the we ll-known  self-shadowing \neffect.  Finally , all the samples  were covered with a 5 nm  Ta capping layer to avoid \nsurface oxidation [see figure  1(a)]. The epitaxial relation  of \nFeGa(001)[ 110]||MgO (001)[ 100] was characterized by using t he X -ray in -plane Φ-\nscans , as described elsewhere [23]. Magnetic hy steresis loops were measured  at various \nin-plane magnetic field orientations φ H with respect to the FeGa [100] axis using \n4 \n magneto -optical Kerr effect (MOKE) technique at room temperature . The d ynamic \nmagnetic properties were investigated by broadband FMR measurements  based on a \nbroadband vector network analyzer  (VNA) with a  transmission geometry coplanar \nwaveguide (VNA- FMR ) [24]. This setup allows both frequency and field- sweeps \nmeasurements with external field applied parallel to the sample plane.  During \nmeasurements,  the sampl es were placed face down on the coplanar wavegu ide and the \ntransmission coefficient S 21 was recorded.  \n3. Results and discussion \nFigure  1(b) displays the Kerr  hysteresis loops  of sample  S1 and S4 recorded along \nwith the main crystallographic directions of FeGa [100], [110],  and [010] . The sample \nS1 exhibit s rectangular hysteresis curves with sm all coercivities for the magnetic  field \nalong  [100] and [010]  easy axes. In contrast, the S4 displays a  hysteresis curve with \ntwo step s for the magnetic field along the [010]  axis, which indicates  a UMA along the \nFeGa[100 ] axis superimposed on the four fold magnetocrystalline anisotropy . As a \nresult,  with increasing the oblique angle, the angular dependence of normalized remnant \nmagnetization ( Mr/Ms) gradually reveals a four fold symmetry combined with a uniaxial \nsymmetry, as shown i n the inset of f igure 1(b).  \nSubsequently,  the magnetic anisotropic properties can be further precisely \ncharacterized  by the in -plane angular -dependent FMR measurements.  Figure 1(c) and \n1(d) show  typical  FMR spectra for the real and imaginary part s of coefficient S 21 for \nthe sample S2 . Recorded  FMR spectra contain a symmetric and an antisymmetric \nLorentzian peak , from which the resonant field H r with linewidth ∆𝐻𝐻 can be obtained  \n[24,25] . \nFigure 2(a) shows the in -plane angular dependence of H r measured at 13 .0 GHz  \nand can be fitted  by the following expression [26,27] : \n5 \n 𝑓𝑓=𝛾𝛾𝜇𝜇0\n2𝜋𝜋�𝐻𝐻𝑎𝑎𝐻𝐻𝑏𝑏                                                     (1 ) \nHere,𝐻𝐻𝑎𝑎=𝐻𝐻4(3+𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M)/4+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H)+𝑀𝑀eff and 𝐻𝐻𝑏𝑏=\n𝐻𝐻4𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H), H4 and Hu represent the fourfold \nanisotropy field and the UMA field caused by the self -shadowing effect , respectively. \n𝜑𝜑H(𝜑𝜑M) is the  azimuthal angles of the applied field ( the tipped magnetization ) with \nrespect to the [100] direction , as depicted  in figure 1(a). 𝜇𝜇0𝑀𝑀eff=𝜇𝜇0𝑀𝑀𝑠𝑠−2𝐾𝐾out\n𝑀𝑀𝑠𝑠, Ms is \nthe saturation magnetization  and Kout is the out -of-plane uniaxial anisotropy constant . \n𝑓𝑓 is the resonance frequency , 𝛾𝛾 is the gyromagnetic ratio and here used as the accepted \nvalue for Fe films, 𝛾𝛾=185 rad GHz/T  [28] . \nThe angular dependent  Hr reveals  only a  fourfold symmetry for the none -\nobliquely deposited sample , which indicates  the cubic lattice texture of FeGa on MgO . \nWith increasing the oblique angle , a uniaxial symmetry is found  to be superimposed \non the four fold symmetry, clearly confirming a UMA is produced by the oblique \ngrowth , which agrees with the MOKE’ results.  The fitted  parameter  𝜇𝜇0𝑀𝑀eff=1.90±\n 0.05T is found to be independent on the oblique deposition and close to 𝜇𝜇0𝑀𝑀𝑠𝑠=\n1.89 ± 0.02 T  estimated using VSM, which is almost same as the value  of the  \nliterature  [29] . This indicates negligible out- of-plane ma gnetic anisotropy in the thick \nFeGa films . As shown in figure 2(b), it is observed  that the UMA (Ku=HuMs/2) exhibits \na general increasing  trend with oblique angle , which coincides with the fact  the \nshadowing effect is stronger  at larger  angles of incidence [16-19]. Interestingly the \noblique deposition also affects the cubic anisotropy  K4 (K4=H4Ms/2). Different from  \nthe K4 increases slightly with deposition angle  in Co/Cu system  [16], here the value of \nK4 is the lowest at a n oblique angle of 15°. It is well known th at film stress significantly \ninfluences  the crystallization tendency  [30,31] . FeGa alloy is highly stress ed sensitive \n6 \n due to its larger magnetostriction . Thus, t he change in  K4 of FeGa films  may be \nattributed to the anisotropy dispersion created due to the stress variations  during grain \ngrowth. It should be mentioned that the best way to determine magnetic parameters is \nto measure the out -of-plane  FMR . But the effective saturation magnetization 𝜇𝜇0𝑀𝑀eff=\n1.90T of FeGa alloy leads to the perpendicular applied field beyond our instrument \nlimit. Meanwhile, t he results obtained above are also in accord with those extracted \nby fitting field dependence of the resonance frequency with H//FeGa[100]  shown in \nfigure 2(c).  \nThe effective Gilbert damping 𝛼𝛼eff is extracted by linearly fitting the dependence  \nof linewidth  on frequency : 𝜇𝜇0∆H=𝜇𝜇0∆H0+2𝜋𝜋𝜋𝜋𝛼𝛼𝑒𝑒ff\n𝛾𝛾, where ∆𝐻𝐻0 is the inhomogeneous \nbroadening. For the sake of clarity , figure 3(a) only shows the frequency  dependence \nof linewidth for the  samples S1 and S2 along  [110]  and [100]  axes. It is evident that, \nfor the sample  S1, both linear slopes of two direction s are almost  same. While w ith \nregard to the sample S2 , the slope of the ∆H-f curve along  the easy axis is approximately \na factor of 2 greater than that of the hard axis. The obtained values of  𝛼𝛼eff are shown in \nfigure 3(b). Firstly, the results clearly indicate that the effective damping exhibits \nanisotropy , with  higher value along the easy axis . Secondly, f or the easy  axis, the \noblique angle dependence on the damping parameter indicates an extraordinary trend \nand has a peak at  deposition angle 15°. However, the damping shows an increasing \ntrend  with the oblique angle for the field along the hard axis. In the following part, we \nwill explore the effect of oblique deposition on the mechanism of the anisotropic \ndamping and the magnetic relaxation pr ocess.     \nSo far, convincing experimental evidence is still lacking to prove the existence of \nanisotropic damping in bulk magnets. Chen et al. have shown the emergence of \nanisotropic Gilbert damping in ultrathin Fe (1.3nm)/GaAs and its anisotropy disappears \n7 \n rapidly when the Fe thickness  increases  [32]. We perform  the first-principles \ncalculation of the Gilber t damping of Fe Ga alloy considering  the effect induced by the \nlattice distortion. W e artificially make a tetragonal lattice with varying the lattice \nconstant of the c -axis. The electronic structure of Fe -Ga alloy is calculated self -\nconsistently using the coherent potential approx imation implemented with the tight-\nbinding linear muffin- tin orbitals. Then the atomic potentials of Fe and Ga are randomly \ndistributed in a 5× 5 lateral supercell, which is connected to two semi -infinite Pd leads . \nA thermal lattice disorder is included via  displacing atoms randomly from the perfect \nlattice sites following a Gaussian type of distribution [ 33]. The root -mean -square \ndisplacement at room temperature is determined by the Debye model with the Debye \ntemperature 470 K. The length of the supercell is variable and the calculated total \ndamping is scaled linearly with this length. Thus, a linear least- squares fitting can be \nperformed to extract the bulk damping of the Fe -Ga alloy  [34].  The calculated Gilber t \ndamping is plotted in f igure 3(c) as a funct ion of the lattice distortion (𝑐𝑐−𝑎𝑎)𝑎𝑎⁄.  The \nGilbert damping is nearly independent of the lattice distortion and there is no evidence of anisotropy in t he intrinsic bulk damping of Fe Ga alloy.  \nSo the extrinsic contributions  are responsible for  the anisotropic behavior of \ndamping , which can be separated from the in -plane angular dependent linewidth. The \nrecorded FMR  linewidth  have the following different cont ributions  [11] :  \n     𝜇𝜇0∆𝐻𝐻=𝜇𝜇0∆𝐻𝐻inh+2𝜋𝜋𝛼𝛼𝐺𝐺𝑓𝑓\n𝛾𝛾𝛾𝛾+�𝜕𝜕𝐻𝐻r\n𝜕𝜕𝜑𝜑H∆𝜑𝜑H�+�Γ<𝑥𝑥𝑖𝑖>𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>�\n<𝑥𝑥𝑖𝑖>𝑎𝑎𝑎𝑎𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎 \n                �(�𝜔𝜔2+(𝜔𝜔0\n2)2−𝜔𝜔0\n2)/(�𝜔𝜔2+(𝜔𝜔0\n2)2+𝜔𝜔0\n2)+Γtwofoldmaxcos4(φM- φtwofold)   (2) \n∆Hinh is both frequency and angle independent  term due to the  sample  \ninhomogeneity . The second term is the intrinsic Gilbert damping  (𝛼𝛼𝐺𝐺) contribution. 𝛾𝛾  \n8 \n is a correction factor owing to the field dragging effect caused by magnetic anisotropy  \n[12], 𝛾𝛾 =cos (φM-φH). The 𝜑𝜑M as a function of φH  for the sample S2 at fixed 13 GHz \nis calculated  and show n in figure 4(a).  Note that the draggi ng effect  vanishes  (𝜑𝜑M=\nφH) when the field is along the hard or  easy axes . The third term  describes the mosaicity \ncontribution originating from the angular dispersion of the crystallographic cubic axes \nand yield s a broader linewidth  [35]. The four th term is the TMS contribution. The \nΓ<𝑥𝑥𝑖𝑖> signifies  the intensity  of the TMS along the principal in -plane crystallographic \ndirection  <𝑥𝑥𝑖𝑖>. The 𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>� term indica tes the TMS contribution depending \non the in- plane direction of the field rel ative to <𝑥𝑥𝑖𝑖> and commonly expressed as \ncos2[2(φM-φ<xi>)] [14].  In addition, 𝜔𝜔 is the angular resonant frequency and  𝜔𝜔0=\n𝛾𝛾𝜇𝜇0𝑀𝑀eff. In our case, besides the fourfold TMS caused by expected lattice geometric \ndefects, the other twofold TMS channel is induced by the dipolar fields emerging from \nperiodic stripelike defects  [20,21] . This term is parameterized by its strength Γtwofoldmax \nand the axis of maximal scattering  rate φtwofold.  \nAs an example, t he angle- dependent linewidth measured  at 13 .0 GHz for  the \nsample S2 is  shown in f igure 4(b).  It clearly exhibits a strong in -plane anisotropy, and  \nthe linewidth along the [100] direction is significantly  larger than that along the [110] \ndirection . Taking only  isotropic Gilbert damping  into account , the dragging effect \nvanishes  with field applied  along the hard and easy axes . Meanwhile, the mosaicity \nterm gives an angular variation of the linewidth proportional to |𝜕𝜕𝐻𝐻𝑟𝑟𝜕𝜕𝜑𝜑𝐻𝐻|⁄ , which is \nalso zero along with the principal <100> and < 110>  directions. This gives direct \nevidence that the rel axation is not exclusively governed only considering the intrinsic \nGilbert mechanism and mosaicity term. Because the probability of defect formation \nalong with <100> directions is higher than that  along the <110> directions  [12], the \n9 \n TMS contribution is stronger along the easy axes , which is in accordance with t he fact \nthat the linewidth s along the [100] and [110]  direction s are non -equivalent. Moreover , \nthe linewidth of [010] direction is slightly  larger  than that along the [100] dire ction, \nsuggesting that  another twofold TMS channel is induce d by oblique deposition. As \nindicated by the red solid line  in figure 4(b), the linewidth can be well fitted. D ifferent \nparts making sense to the  linewidth can therefore be sepa rated  and summarized in Tab le \nI. As we know, the TMS predicts the curved non- linear frequency dependence of \nlinewidth, which not appear in a small frequency range for our case (as shown in f igure \n3(a)).  The linewidth as function of frequency was also well fitted including the TMS -\ndamping using the parameters in Table I  (not shown here) . \nThe larger strength of TMS along the easy axis can clearly explain the anisotropic \nbehavior of da mping , with higher value along the easy axis shown in f igure 3(b). The \nobtained Gilbert damping  factor  of ~ 7×10-3 is isotropic and invariant  with different \noblique angle s. The value of damping is slightly larger than the bulk value of 5.5×10-3 \n[29], which may be attributed to spin pumping of the Ta capping layer.   \n The obtained maxi ma of twofold TMS exhibits  an increasing  trend  with the \noblique angle  [shown in f igure 4(c)]. According  to previous works  on the shadowing \neffect [16-19], the larger  deposition angle  makes  the shadow ing effect  stronger , and the \ndipolar fields within stripe like defects  increase just like the UMA. This  can clearly \nexplain that the intensity of two fold TMS  follows exactly the same trend with the \ndeposition angle as the UMA . The axis of the maximal intensity of two fold TMS is \nparal lel to the projection of the  FeGa atom flux  from the fitting data.   As shown in Table  \nI, amazingly the modified growth conditions also influence  the fourfold TMS, \nespecially the strength of TMS along the <100>  axis. Figure 4(c) also presents the \nchanges of the fourfold TMS intensity as the  deposition angle  and shows a peak at 15° , \n10 \n which follows a similar trend as that of 𝛼𝛼eff along [100] axis as shown in f igure 3(b). \nThis indeed confirm s TMS -damping plays an important role in FeGa thin films.    \nFor the dispersion relation ω(k∥) in thin magnetic films , the propagation angle \n𝜑𝜑𝑘𝑘∥����⃗ defined as the angle between k∥���⃗ and the projection of the saturation magnetization \nMs into the sample plane is less than the critical value : 𝜑𝜑max =\n𝑐𝑐𝑎𝑎𝑎𝑎−1�𝜇𝜇0𝐻𝐻r(𝜇𝜇0𝐻𝐻r+𝜇𝜇0𝑀𝑀eff) ⁄  [9,36,37] . This implies  no degenerate modes are \navailable for the angle 𝜑𝜑𝑘𝑘∥����⃗ larger than φmax. Based on this theory, we propose a \nhypothesis that the crystallographic anisotropy  determine s the area including \ndegenerate magnon modes , as well as  the intensity of the fourfold TMS.  The resonance \nfield along <100>  axis change s due to the various crystallographic anisotropy , which \nhas a great effect on the φmax. The values of φmax of samples are shown in f igure 4(d). \nThe data follow the  same trend with the oblique angle as Γ<100>. During the grain \ngrowth,  the cubic anisotropy is  influenced possibly  since the anisotropy dispersion due \nto the stress. For the lower anisotropy of  sample S2 , a relatively larger  amount of stress \nand defects present in the sample and lead to a larger  four fold TMS.  \n4. Conclusions  \nIn conclusion, the effects of oblique deposition on the dynamic properties of FeGa \nthin films have been investigated  systematically . The pronounced TMS  as non-Gilbert \ndamping  results in an anisotropic magnetic relaxation . As the oblique angle increases, \nthe magnitude of the twofold TMS  increases due to the larger shadowing effect . \nFurthermore, the cubic anisotropy dominates  the area including degenerate magnon \nmodes, as well as the intensity of fourfold TMS. The reported results  confirm  that the \nmodified anisotropy can influence the extrinsic relaxation pr ocess  and open a n avenue \nto tailor magnetic  relaxation in spintronic devices. \n11 \n Acknowledgments  \nThis work is supported by the National Key Research Program of China (Grant Nos. \n2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos. 91622126, 51427801, and 51671212)  and \nthe Key Research Program of Frontier Sciences, CAS (Grant Nos. QYZDJ -SSW -\nJSC023, KJZD -SW-M01 and ZDYZ2012 -2). The work at Beijing Norma l University \nis partly supported by the National Natural Sciences Foundation of China (Grant Nos. 61774017, 61704018, and 11734004), the Recruitment Program of Global Youth \nExperts and the Fundamental Research Funds for the Central Universities (Grant No. \n2018EYT03).  \n12 \n References  \n[1] Slonczewski J C 1996 J. Magn. Magn. Mater. 159  L1 \n[2] Žutić  I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. 76 323 \n[3] Gilbert T L2004 IEEE Trans. Magn. 40 3443  \n[4] He P, Ma  X, Zhang  J W, Zhao  H B, Lüpke  G, Shi  Z and Zhou S M 2013 Phys. \nRev. Lett.  110 077203 \n[5] Heinrich B, Meredith D J and Cochran J F 1979 J. Appl. Phys. 50 7726 \n[6] Lee A J, Brangham  J T, Cheng  Y, White  S P, Ruane  W T, Esser  B D, \nMcComb  D W, Hammel  P C and Yang  F Y 2017 Nat. Commun. 8 234 \n[7] Scheck C, Cheng L, Barsukov I, Frait Z and Bailey W E 2007 Phys. Rev. Lett.  \n98 117601 \n[8] Azzawi  S, Hindmarch A and Atkinson D 2017 J. Phys. D: Appl. 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B  80 224421  \n \n \n  \n   \n            \n15 \n Figure Captions  \nFigure  1 (color online) (a) Schematic illustration of the film deposition geometry and \ncoordinate system (b) I n-plane hysteresis loops of samples S1 and S4 with the field \nalong [100], [110], and [010]. The inset shows the polar plot of the normalized \nremanence (M r/Ms) as a functi on of the in- plane angle.  FMR spectrum for the sample \nS2 with H along [100] and [110] axes showing the real (c ) and imaginary (d ) part s of \nthe S 21.  \nFigure  2 (color online) (a) H r vs. φH for FeGa films. (b) The  anisotropy  constants  K4 \nand Ku vs. deposition angle. (c) f  vs. Hr plots measured at H //[100], Symbols are \nexperimental data and the solid lines are the fitted results.  \nFigure  3 (color online) (a) ∆H as a function of f  for samples S1 and S2 with  field along \neasy and hard axis. (b) The dependence of the damping parameter on  the oblique angle \nwith field along [100] and [110] directions. (c) The calculated damping of FeGa alloy \nas a function of lattice distortion.  Figure  4 (color online)  (a) φ\nM and (b) ∆H as a function of φH for the  sample S2  \nmeasured at 13.0 GHz. (c) Oblique angle dependences of Γ<100> and Γtwofoldmax. (d) The \nlargest angle including degenerate magnon modes as a function of the oblique angle \nwith the applied field along <100> direction. \nTable Caption  \nTable I. The magnetic relaxation parameters of the FeGa films prepared via oblique \ndeposition (with experimental errors in parentheses).  \n \n  \n16 \n Figure 1  \n \n \n  \n \n  \n \n  \n \n  \n \n17 \n Figure  2 \n \n \n \n \n \n  \n \n  \n \n  \n \n  \n \n18 \n Figur e 3 \n \n \n  \nFigure  4 \n \n \n \n \n \n \n \n19 \n TableⅠ \nSample  𝜇𝜇0ΔHinh\n(mT) 𝛼𝛼G Δ𝜑𝜑H \n(deg.) Γ<100> \n(107Hz) Γ<110> \n(107Hz) Γtwofoldmax\n(107Hz) 𝜑𝜑twofold  \n(deg. ) \nS1 0 0.007  0.62 17(3)  5.8(1.8)  0(2) 90 \nS2 0.7 0.007  1.2 81.4(3.7)  9.3(1.9)  7.4(3) 90 \nS3 0 0.007  1.0 59.2(4.5)  11.1(2) 13(3.7)  90 \nS4 0 0.007  1.1 33.3(6)  14.8(3.7)  26(4)  90 \n \n20 \n "    },    {        "title": "1911.00744v2.Soft_contribution_to_the_damping_rate_of_a_hard_photon_in_a_weakly_magnetized_hot_medium.pdf",        "content": "Soft contribution to the damping rate of a hard photon in a weakly magnetized\nhot medium\nRitesh Ghosh,1, 2,\u0003Bithika Karmakar,1, 2,yand Munshi G Mustafa1, 2,z\n1Theory Division, Saha Institute of Nuclear Physics,\n1/AF, Bidhannagar, Kolkata 700064, India\n2Homi Bhabha National Institute, Anushaktinagar,\nMumbai, Maharashtra 400094, India\nWe consider weakly magnetized hot QED plasma comprising electrons and positrons.\nThere are three distinct dispersive (longitudinal and two transverse) modes of a photon in a\nthermomagnetic medium. At lowest order in the coupling constant, a photon is damped in\nthis medium via Compton scattering and pair creation process. We evaluate the damping\nrate of hard photon by calculating the imaginary part of the each transverse dispersive modes\nin a thermomagnetic QED medium. We note that one of the fermions in the loop of one-loop\nphoton self-energy is considered as soft and the other one is hard. Considering the resummed\nfermion propagator in a weakly magnetized medium for the soft fermion and the Schwinger\npropagator for hard fermion, we calculate the soft contribution to the damping rate of hard\nphoton. In weak \feld approximation the thermal and thermomagnetic contributions to\ndamping rate get separated out for each transverse dispersive mode. The total damping rate\nfor each dispersive mode in presence of magnetic \feld is found to be reduced than that of\nthe thermal one. This formalism can easily be extended to QCD plasma.\n\u0003ritesh.ghosh@saha.ac.in\nybithika.karmakar@saha.ac.in\nzmunshigolam.mustafa@saha.ac.inarXiv:1911.00744v2  [hep-ph]  7 Mar 20202\nI. INTRODUCTION\nAstrophysical plasma is almost always immersed in magnetic \feld. Extreme, magnetized plasma\nis found in interiors of neutron star, magnetospheres of magnetars and central engines of super-\nnovae and gamma ray bursts [1]. The propagation of photon through the hot magnetized plasma,\nviz., electron-positron plasma (EPP), is of great interest. Because the magnetar phenomena are\nfound by analyzing the high-energy radiation detected at earth. Thus it is very important to have a\ngood understanding of the propagation of photon through the EPP. Furthermore, the phenomenon\nof Faraday rotation i.e., change of polarization of photon while propagating through a medium\nhas been studied in Ref. [2] in a \feld theoretical viewpoint. This has also been detected in several\nastrophysical objects [3]. Also high-intensity laser beams are used to create ultrarelativistic EPP\nof temperature around 10 MeV [4]. This EPP may play an important role in various astrophysical\nsituations. Some properties of such plasma, viz., the equation of state, dispersion relation of collec-\ntive plasma modes of photon and electron, damping rates, mean free paths, transport coe\u000ecients\nand particle production rates, are studied using QED at \fnite temperature [5, 6].\nOn the other hand in noncentral heavy ion collisions, the magnetic \feld as high as (15 \u000020)m2\n\u0019\ncan be generated [7] at LHC energies. After a few fm/ cof the collision, the magnetic \feld strength1\ndecreases to (1\u00002)m2\n\u0019. The e\u000bect of magnetic \feld on the properties of the QCD matter [ viz.\nquark-gluon plasma(QGP)] and on the phase diagram of QCD is of great interest. Recently, several\nstudies have found the e\u000bect of magnetic catalysis [13{16], i.e., the enhancement of phase transition\ntemperature of QCD matter in presence of external magnetic \feld, whereas, some results of inverse\nmagnetic catalysis [17{25] have been reported. Various properties of QCD matter at weak coupling\nin presence of magnetic \feld is being studied including the equation of state [26{28], transport\nproperties [29{31]. Modi\fcation of QCD Debye mass and the two point correlation functions of\nquarks [32] and gluons [33{35] i.e., partons have been analyzed recently. Dilepton production rate\nfrom a hot magnetized QCD plasma [12, 36{42] has been calculated. The photon is also considered\nas a good probe of the QGP medium as photon only interacts electromagnetically. Thus, it comes\nout of the hot QCD system without interacting much. The damping rate of the hard photon is\nassociated with the mean free path of photon [43] and hard photon production rate in QGP [44].\nDamping rate of photon is related to the imaginary part of photon dispersion in the medium [45]\nwhich is again related to the scattering crosssection of the process that we \fnd by cutting the pho-\n1The initial magnitude of this magnetic \feld can be very high at the initial time of the heavy-ion collisions and\nthen it decreases very fast, being inversely proportional to the square of time [8, 9]. However for a di\u000berent point\nof view, see Refs. [10{12], where the time dependence of magnetic \feld is shown to be adiabatic due to the high\nconductivity of the medium.3\nton self-energy diagram [46]. In lowest order coupling constant, photons are damped by Compton\nscattering and pair creation process. In case of low momentum transfer, the damping rate shows\ninfrared singularity. Thus one should consider the e\u000bective resummed propagator instead of bare\npropagator for soft momentum of fermion. We will call this as the soft contribution to the damping\nrate of photon. The hard contribution refers to the case where all the fermions in loop have momen-\ntum order of or much greater than the system temperature T. Both soft and hard contributions\nto the damping rate of hard photon in thermal medium have been calculated in Ref. [45]. The\ndispersion relations of photon are modi\fed for a hot magnetized medium [33]. So the damping rate\nof photon will also get modi\fed in a thermomagnetic medium. In this article we intend to compute\nthe soft contribution to the hard photon damping rate for a weakly magnetized hot medium in\none loop approximation of photon self-energy. In a thermomagnetic medium this would be a good\nindicator as higher loop calculation contributing to higher order would be extremely involved.\nWe consider hard photon of momentum P\u0016= (p0;p) wherep=jpj\u001dTin a relativistic\nhot magnetized QED medium. To \fnd the soft contribution of the damping rate we introduce\na separation scale \u0003 where eT\u001c\u0003\u001cT(gT\u001c\u0003\u001cTin case of QCD). In the soft part of\nthe damping rate, the contribution from soft loop momentum involving a fermion is taken into\naccount up to the separation scale \u0003 . Here we assume that the magnetic \feld strength is weak\ni.e.,p\neB < eT < T (pqfB < gT < T for QCD). We use the recently obtained e\u000bective fermion\npropagator [32] in presence of weak magnetic \feld for the soft fermion and Schwinger propagator\nfor the hard fermion in the loop. The Braaten-Pisarki-Yuan formalism [47] has been used here to\ncalculate the imaginary part of photon self-energy. Extension to the case of damping rate of hard\nphoton in weakly magnetized hot QCD medium is straightforward. We need to consider the loop\nfermions as quark and antiquark in that case.\nIn Sec. II we describe the set up to calculate the photon damping rate associated with imaginary\npart of photon self-energy. In Sec. III the self-energy is obtained in a weak \feld approximation.\nThe imaginary parts of various components of photon self-energy is obtained in Sec. IV. Results\nare given in Sec. V. We conclude in Sec. VI.\nII. SETUP\nWe consider plasma of electrons and positrons at temperature T. Thez-axis of the lab frame\nis oriented along the magnetic \feld. The general structure of the gauge boson self-energy and\ncorresponding e\u000bective propagator have been evaluated in Ref. [33]. The general covariant structure4\nof photon self-energy in a magnetized hot medium can be written as\n\u0005\u0016\u0017=\fB\u0016\u0017+\u001bR\u0016\u0017+\u000eQ\u0016\u0017+\u000bN\u0016\u0017; (1)\nwhere various form factors can be written as\n\f=B\u0016\u0017\u0005\u0016\u0017;\n\u001b=R\u0016\u0017\u0005\u0016\u0017;\n\u000e=Q\u0016\u0017\u0005\u0016\u0017;\n\u000b=1\n2N\u0016\u0017\u0005\u0016\u0017: (2)\nThe general covariant structure of photon propagator can be obtained [33] as\nD\u0016\u0017=\u0018P\u0016P\u0017\nP4+(P2\u0000\u000e)B\u0016\u0017\n(P2\u0000\f)(P2\u0000\u000e)\u0000\u000b2+R\u0016\u0017\nP2\u0000\u001b+(P2\u0000\f)Q\u0016\u0017\n(P2\u0000\f)(P2\u0000\u000e)\u0000\u000b2\n+\u000bN\u0016\u0017\n(P2\u0000\f)(P2\u0000\u000e)\u0000\u000b2: (3)\nWe note that the thermal medium (absence of magnetic \feld) has two dispersive modes of photon\ni.e., one degenerate transverse mode and one medium induced plasmon mode due to breaking of\nboost invariance. Now breaking of rotational invariance in the presence of a magnetic \feld leads\nto three dispersive modes of photon by lifting the degeneracy of the transverse modes. These three\ndispersive modes can be seen from the pole of Eq. (3). Now, the dispersion relations can be written\nas\nP2\u0000\u001b= 0; (4)\n(P2\u0000\u000e)(P2\u0000\f)\u0000\u000b2=\u0012\nP2\u0000\f+\u000e+p\n(\f\u0000\u000e)2+ 4\u000b2\n2\u0013\n\u0002\u0012\nP2\u0000\f+\u000e\u0000p\n(\f\u0000\u000e)2+ 4\u000b2\n2\u0013\n= 0: (5)\nIn weak magnetic \feld approximation \u000bdoes not contribute upto O[(eB)]2, one gets simple form\nof the above dispersive modes [26]\nP2\u0000\u001b= 0;\nP2\u0000\f= 0;\nP2\u0000\u000e= 0: (6)\nDamping rate is de\fned as the imaginary part of photon dispersion relation. The medium induced\nlongitudinal (plasmon) mode does not contribute to the damping rate2and the dispersion relations\n2The longitudinal dispersive mode merges with the light cone at high photon momentum.5\nfor two transverse modes of a photon are given, respectively, as\nP2\u0000\u001b= 0; P2\u0000\u000e= 0; (7)\nDamping rates \r\u000e(p) and\r\u001b(p) (for no overdamping i:e: \ri\u001cp0wherei=\u000e;\u001b) of hard photon\nare given by imaginary part of the form factors as [48]\n\r\u001b(p) =\u00001\n2pIm\u001b(p0=p); (8)\n\r\u000e(p) =\u00001\n2pIm\u000e(p0=p): (9)\nThe tensor structures of R\u0016\u0017andQ\u0016\u0017[33] are given as\nR\u0016\u0017=0\nBBBBBB@0 0 0 0\n0 0 0 0\n0 0\u00001 0\n0 0 0 01\nCCCCCCA; Q\u0016\u0017=0\nBBBBBB@0 0 0 0\n0\u0000cos2\u0012p0 sin\u0012pcos\u0012p\n0 0 0 0\n0 sin\u0012pcos\u0012p0\u0000sin2\u0012p1\nCCCCCCA: (10)\nUsing Eq.(10) in Eq.(2) we can write the form factors \u001band\u000ein weak \feld approximation as\n\u001b=\u0000\u0010\n\u000522\n0+ \u000522\n2\u0011\n; (11)\n\u000e=\u0000cos2\u0012p\u0010\n\u000511\n0+ \u000511\n2\u0011\n\u0000sin2\u0012p\u0010\n\u000533\n0+ \u000533\n2\u0011\n+ 2 sin\u0012pcos\u0012p\u0010\n\u000513\n0+ \u000513\n2\u0011\n: (12)\nCombining Eq.(8) with Eq.(11) and Eq.(9) with Eq.(12), the damping rates become\n\r\u001b(p) =1\n2p\u0010\nIm\u000522\n0+ Im\u000522\n2\u0011\n; (13)\n\r\u000e(p) =1\n2ph\ncos2\u0012p\u0010\nIm\u000511\n0+ Im\u000511\n2\u0011\n+ sin2\u0012p\u0010\nIm \u000533\n0+ Im \u000533\n2\u0011\n\u00002 sin\u0012pcos\u0012p\u0010\nIm \u000513\n0+ Im \u000513\n2\u0011i\n(14)\nThe damping rates in Eqs.(13) and (14) can now be written as\n\r\u001b(p) =\rth(p) +\rB\n\u001b(p); (15)\n\r\u000e(p) =\rth(p) +\rB\n\u000e(p): (16)\nwhere\rthis theO[(eB)0] contribution or thermal contribution is given as\n\rth(p) =1\n2pIm\u000522\n0=1\n2ph\ncos2\u0012pIm\u000511\n0+ sin2\u0012pIm \u000533\n0\u00002 sin\u0012pcos\u0012pIm \u000513\n0i\n: (17)\nThe thermomagnetic corrections of O[(eB)2] are given as\n\rB\n\u001b(p) =1\n2pIm\u000522\n2; (18)\n\rB\n\u000e(p) =1\n2ph\ncos2\u0012pIm\u000511\n2+ sin2\u0012pIm \u000533\n2\u00002 sin\u0012pcos\u0012pIm \u000513\n2i\n: (19)\nWe need to obtain the imaginary parts of 11, 22, 33 and 13 components of the photon self-energy\n\u0005\u0016\u0017which are computed in the following sections.6\nIII. PHOTON SELF-ENERGY IN HOT MAGNETIZED MEDIUM\nThe photon self-energy as shown in Fig. 1 can be written as\n\u0005\u0016\u0017(P) =ie2Zd4K\n(2\u0019)4\u001a\nTr[\r\u0016S\u0003(K)\r\u0017S(Q)] + Tr[\r\u0017S\u0003(K)\r\u0016S(Q)]\u001b\n: (20)\nwhereS\u0003(K) is e\u000bective electron propagator and S(K) is Schwinger propagator for bare electron.\nAs the external photon is hard, we consider one bare and one e\u000bective fermion propagator in the\nloop. In the following we would obtain the propagators for fermion.\nPK\nQ=K−P\nFIG. 1: Photon self-energy where the blob represents the e\u000bective electron propagator in\nmagnetic \feld and double line represents bare electron propagator in magnetic \feld\nA. Fermion propagator in weak \feld approximation\nIn the weak magnetic \feld limit, i.e.,p\neB < m th\u0018eT < T the Schwinger propagator for\nfermion can be written up to O[(eB)2] as [49]\nS(K) == K+mf\nK2\u0000m2\nf+i\r1\r2= Kq+mf\n(K2\u0000m2\nf)2(eB) + 2\"\nf(K\u0001u)= u\u0000(K\u0001n)= ng\u0000= K\n(K2\u0000m2\nf)3\u0000k2\n?(= K+mf)\n(K2\u0000m2\nf)4#\n(eB)2\n+O\u0002\n(eB)3\u0003\n=S0+S1+S2+O[(eB)3]: (21)\nThe general form of fermion self-energy in a weakly magnetized medium can be written as [32]\n\u0006(K) =\u0000a= K\u0000b= u\u0000b0\r5= u\u0000c0\r5= n : (22)\nIn one loop order, the form factors are given as\na(k0;k) =\u0000m2\nth\nk2Q1\u0012k0\nk\u0013\n; (23)\nb(k0;k) =m2\nth\nk\u0014k0\nkQ1\u0012k0\nk\u0013\n\u0000Q0\u0012k0\nk\u0013\u0015\n; (24)\nb0(k0;k) = 4e2M2(T;mf;eB)k3\nk2Q1\u0012k0\nk\u0013\n; (25)7\nc0(k0;k) = 4e2M2(T;mf;eB)1\nkQ0\u0012k0\nk\u0013\n; (26)\nwhere Legendre function of second kind are given as\nQ0(x) =1\n2ln\u0012x+ 1\nx\u00001\u0013\n; (27)\nQ1(x) =xQ0(x)\u00001; (28)\nand the thermomagnetic mass is given as\nM2(T;mf;eB) =eB\n16\u00192\u0014\nln 2\u0000\u0019T\n2mf\u0015\n; (29)\nwhereas thermal mass is given as\nm2\nth=1\n8e2T2: (30)\nThe e\u000bective fermion propagator can be written [32] as\nS\u0003(K) =P\u0000= L(K)\nL2P++P+= R(K)\nR2P\u0000\n=S\u0003\nL(K) +S\u0003\nR(K); (31)\nwhere chirality projection operators are given by\nP\u0006=1\n2(1\u0006\r5); (32)\nandL\u0016andR\u0016are given as\nL\u0016= (1 +a)K\u0016+ (b+b0)u\u0016+c0n\u0016; (33)\nR\u0016= (1 +a)K\u0016+ (b\u0000b0)u\u0016\u0000c0n\u0016: (34)\nFor simplicity of calculation we expand the e\u000bective fermion propagator in Eq. (31) in powers\nofeBand keep terms up to O[(eB)2] as\nS\u0003(K) =S\u0003\n0(K) +S\u0003\n1(K) +S\u0003\n2(K) +O[(eB)3]; (35)\nwhereS\u0003\n0(K) isO[(eB)0] and given as\nS\u0003\n0(K) =(1 +a)= K+b= u\nD2=(1 +a)= K+b= u\nD+D\u0000; (36)\nwhereD\u0006= (1 +a)(k0\u0007k) +b.\nEquation (36) is the e\u000bective HTL fermion propagator [50, 51] in thermal medium. The O[(eB)]\nis obtained as\nS\u0003\n1(K) =1\nD4\u0014\n2(1 +a)= K\r5\u001a\n\u0000(1 +a)k3c0\u0000(1 +a)k0b0\u0000bb0\u001b8\n+= u\r5\u001a\u0010\n(1 +a)2K2\u0000b2\u0011\nb0\u00002(a+ 1)bc0k3\u001b\n+c0= n\r5\u001a\u0010\n2(1 +a)k0+b\u0011\nb+ (a+ 1)2K2\u001b\u0015\n; (37)\nwhereasO[(eB)2] is obtained as\nS\u0003\n2(K) =\u0014\u0010\n2b0\b\n(1 +a)k0+b\t\n+ 2c0k3(1 +a)\u00112\nD6\u0000b02\u0000c02\nD4\u0015\u001a\n(1 +a)= K+b= u\u001b\n\u0000\u0010\n2b0\b\n(1 +a)k0+b\t\n+ 2c0k3(1 +a)\u0011\u0010\nb0= u+c0= n\u0011\nD4\n=\u0012h2(k0;k?;k3)\nD6\u0000h0\nD4\u0013n\n(1 +a)= K+b= uo\n\u0000h(k0;k?;k3)\nD4\u0010\nb0= u+c0= n\u0011\n; (38)\nwhereh= 2b0\b\n(1 +a)k0+b\t\n+ 2c0k3(1 +a) andh0=b02\u0000c02.\nB. Photon self-energy in weak magnetic \feld\nNow theO[(eB)0] contribution of \u0005\u0016\u0017given in Eq. (20) can be written as\n\u0005\u0016\u0017\n0=ie2Zd4K\n(2\u0019)4\u001a\nTr[\r\u0016S\u0003\n0(K)\r\u0017S0(Q)] + Tr[\r\u0017S\u0003\n0(K)\r\u0016S0(Q)]\u001b\n=ie2Zd4K\n(2\u0019)4(\nTr\u0014\n\r\u0016\u0012\r0\u0000~ \r\u0001^k\n2D++\r0+~ \r\u0001^k\n2D\u0000\u0013\n\r\u0017\u0012\nf(1)\n0\r0\u0000f(0)\n0~ \r\u0001~ q\u0013\u0015\n+ Tr\u0014\n\r\u0017\u0012\r0\u0000~ \r\u0001^k\n2D++\r0+~ \r\u0001^k\n2D\u0000\u0013\n\r\u0016\u0012\nf(1)\n0\r0\u0000f(0)\n0~ \r\u0001~ q\u0013\u0015)\n= 8ie2Zd4K\n(2\u0019)4(1 +a)\b\u0000\nK\u0016Q\u0017+K\u0017Q\u0016\u0001\n\u0000g\u0016\u0017K\u0001Q\t\n+b\b\u0000\nQ\u0016u\u0017+Q\u0017u\u0016\u0001\n\u0000g\u0016\u0017Q\u0001u\t\nD+D\u0000Q2;\n(39)\nwhere\nf(1)\n0=q0\nQ2; f(0)\n0=1\nQ2;\nf(1)\n1=q0\nQ4; f(0)\n1=1\nQ4: (40)\nTheO[(eB)] contribution of \u0005\u0016\u0017is given as\n\u0005\u0016\u0017\n1=ie2Zd4K\n(2\u0019)4\u001a\nTr[\r\u0016S\u0003\n0(K)\r\u0017S1(Q)] + Tr[\r\u0017S\u0003\n0(K)\r\u0016S1(Q)]\n+ Tr[\r\u0016S\u0003\n1(K)\r\u0017S0(Q)] + Tr[\r\u0017S\u0003\n1(K)\r\u0016S0(Q)]\u001b\n; (41)\nwhich becomes zero.9\nTheO[(eB)2] contribution of \u0005\u0016\u0017is given as\n\u0005\u0016\u0017\n2=ie2Zd4K\n(2\u0019)4\u001a\nTr[\r\u0016S\u0003\n1(K)\r\u0017S1(Q)] + Tr[\r\u0017S\u0003\n1(K)\r\u0016S1(Q)] + Tr[\r\u0016S\u0003\n0(K)\r\u0017S2(Q)]\n+ Tr[\r\u0017S\u0003\n0(K)\r\u0016S2(Q)] + Tr[\r\u0016S\u0003\n2(K)\r\u0017S0(Q)] + Tr[\r\u0017S\u0003\n2(K)\r\u0016S0(Q)]\u001b\n: (42)\nWe calculate the above mentioned trace as follows. The trace of the \frst and second terms of\nEq. (42) can be calculated as\nTr[\r\u0016S\u0003\n1(K)\r\u0017S1(Q)] + Tr[\r\u0017S\u0003\n1(K)\r\u0016S1(Q)]\n=8 (eB)\nD2\u0010\nQ2\u0000m2\nf\u00112\"\nb0n\n(u\u0016n\u0017+u\u0017n\u0016)(Q\u0001u)\u00002u\u0016u\u0017(Q\u0001n) +g\u0016\u0017(Q\u0001n)o\n+c0n\n2n\u0016n\u0017(Q\u0001u)\u0000(n\u0016u\u0017+n\u0017u\u0016)(Q\u0001n) +g\u0016\u0017(Q\u0001u)o#\n\u00008 (eB)\nD4\u0010\nQ2\u0000m2\nf\u00112\n\u0002\"\nh\u001a\u0000\n1 +a\u0001\u0012\ng\u0016\u0017\u0010\n(K\u0001u)(Q\u0001n)\u0000(K\u0001n)(Q\u0001u)\u0011\n\u0000(K\u0016u\u0017+K\u0017u\u0016)Q\u0001n\n+ (K\u0016n\u0017+K\u0017n\u0016)Q\u0001u\u0013\n+b\u0012\ng\u0016\u0017Q\u0001n+ (u\u0016n\u0017+u\u0017n\u0016)Q\u0001u\u00002u\u0016u\u0017Q\u0001n\u0013\u001b#\n: (43)\nThe trace of third and fourth terms in Eq. (42) can be obtained as\nTr[\r\u0016S\u0003\n0(K)\r\u0017S2(Q)] + Tr[\r\u0017S\u0003\n0(K)\r\u0016S2(Q)]\n=8(eB)2\nD+(Q2\u0000m2\nf)3\u0014\nq0\u0010\n^K\u0016g0\u0017+^K\u0017g0\u0016\u0000g\u0016\u0017\u0011\n\u0000q3\u0010\n^K\u0016g3\u0017+^K\u0017g3\u0016\u0000g\u0016\u0017^k3\u0011\n\u0000\u0010\n^K\u0016Q\u0017+^K\u0017Q\u0016\u0000g\u0016\u0017^K\u0001Q\u0011\u0015\n\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\u0014\n^K\u0016Q\u0017+^K\u0017Q\u0016\u0000g\u0016\u0017^K\u0001Q\u0015\n+8(eB)2\nD\u0000(Q2\u0000m2\nf)3\u0014\nq0\u0010\n^K0\u0016g0\u0017+^K0\u0017g0\u0016\u0000g\u0016\u0017\u0011\n\u0000q3\u0010\n^K0\u0016g3\u0017+^K0\u0017g3\u0016+g\u0016\u0017^k3\u0011\n\u0000\u0010\n^K0\u0016Q\u0017+^K0\u0017Q\u0016\u0000g\u0016\u0017^K0\u0001Q\u0011\u0015\n\u00008(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0014\n^K0\u0016Q\u0017+^K0\u0017Q\u0016\u0000g\u0016\u0017^K0\u0001Q\u0015\n;(44)\nwhere ^K0\u0016= (1;\u0000^k).\nThe trace of \ffth and sixth terms in Eq. (42) are obtained as\nTr[\r\u0016S\u0003\n2(K)\r\u0017S0(Q)] + Tr[\r\u0017S\u0003\n2(K)\r\u0016S0(Q)]\n=8\n(Q2\u0000m2\nf)\"\u0010h2\nD6\u0000h0\nD4\u0011\u0010\n1 +a\u0011\u0010\nK\u0016Q\u0017+K\u0017Q\u0016\u0000g\u0016\u0017K\u0001Q\u0011\n+\u0010\nb\u0010h2\nD6\u0000h0\nD4\u0011\n\u0000b0h\nD4\u0011\n\u0002\n\u0010\ng\u00160Q\u0017+g\u00170Q\u0016\u0000g\u0016\u0017q0\u0011\n\u0000c0h\nD4\u0010\ng\u00163Q\u0017+g\u00173Q\u0016\u0000g\u0016\u0017q3\u0011#\n: (45)10\nThe photon self-energy in weak \feld approximation now can be decomposed using Eqs.(39),(41),(42)\nas\n\u0005\u0016\u0017(P) = \u0005\u0016\u0017\n0(P) + \u0005\u0016\u0017\n2(P); (46)\nwhere the \frst term is a pure thermal( O[(eB)0]) contribution and second term is thermomagnetic\ncorrection ofO[(eB)2].\nNow theO[(eB)0] expression of \u000511, \u000522, \u000533, and \u000513can be written from Eq. (39) as\n\u000511\n0(p0;p) = 8ie2Zd4K\n(2\u0019)4(1 +a)(k0q0+ 2k1q1\u0000~k\u0001~ q) +bq0n\u0000\n(1 +a)k0+b\u00012\u0000(1 +a)2k2o\nQ2\n=\u00004e2XZ\"\u0012f(1)\n0\nD++f(1)\n0\nD\u0000\u0013\n+\u0000\n2^k1q1\u0000^k\u0001q\u0001\u001af(0)\n0\nD+\u0000f(0)\n0\nD\u0000\u001b#\n;\n\u000522\n0(p0;p) = 8ie2Zd4K\n(2\u0019)4(1 +a)(k0q0+ 2k2q2\u0000~k\u0001~ q) +bq0n\u0000\n(1 +a)k0+b\u00012\u0000(1 +a)2k2o\nQ2\n=\u00004e2XZ\"\u0012f(1)\n0\nD++f(1)\n0\nD\u0000\u0013\n+\u0000\n2^k2q2\u0000^k\u0001q\u0001\u001af(0)\n0\nD+\u0000f(0)\n0\nD\u0000\u001b#\n;\n\u000533\n0(p0;p) = 8ie2Zd4K\n(2\u0019)4(1 +a)(k0q0+ 2k3q3\u0000~k\u0001~ q) +bq0n\u0000\n(1 +a)k0+b\u00012\u0000(1 +a)2k2o\nQ2\n=\u00004e2XZ\"\u0012f(1)\n0\nD++f(1)\n0\nD\u0000\u0013\n\u0000\u0000^k\u0001q\u00002^k3q3\u0001\u001af(0)\n0\nD+\u0000f(0)\n0\nD\u0000\u001b#\n;\n\u000513\n0(p0;p) = 8ie2Zd4K\n(2\u0019)4(1 +a)(k1q3+q1k3)n\u0000\n(1 +a)k0+b\u00012\u0000(1 +a)2k2o\nQ2\n=\u00004e2XZ\"\n(^k1q3+q1^k3)\u001af(0)\n0\nD+\u0000f(0)\n0\nD\u0000\u001b#\n: (47)\nUsing Eqs.(42),(43),(44) and (45), one can write the O[(eB)2] expression of \u000511, \u000522, \u000533, and\n\u000513as\n\u000511\n2=\u0000e2XZ\"\n\u00008eB\nD2(Q2\u0000m2\nf)2\u001a\nb0q3+c0q0\u001b\n+8eB\nD4(Q2\u0000m2\nf)2h\u001a\n(1 +a)(k0q3\u0000k3q0) +bq3\u001b\n+8(eB)2\nD+(Q2\u0000m2\nf)3\u0010\n^k2q2\u0000^k1q1\u0011\n\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\u0010\nq0\u0000^k\u0001q+ 2^k1q1\u0011\n\u00008(eB)2\nD\u0000(Q2\u0000m2\nf)3\u0010\n^k2q2\u0000^k1q1\u0011\n\u00008(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0010\nq0+^k\u0001q\u00002^k1q1\u0011\n+8\n(Q2\u0000m2\nf)\u001a\u0010h2\nD6\u0000h0\nD4\u0011\n(1 +a)(2k1q1+K\u0001Q) +\u0012\nb\u0010h2\nD6\u0000h0\nD4\u0011\n\u0000b0h\nD4\u0013\nq0\n\u0000c0h\nD4q3\u001b#\n; (48)11\n\u000522\n2=\u0000e2XZ\"\n\u00008eB\nD2(Q2\u0000m2\nf)2\u001a\nb0q3+c0q0\u001b\n+8eB\nD4(Q2\u0000m2\nf)2h\u001a\n(1 +a)(k0q3\u0000k3q0) +bq3\u001b\n+8(eB)2\nD+(Q2\u0000m2\nf)3\u0010\n^k1q1\u0000^k2q2\u0011\n\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\u0010\nq0\u0000^k\u0001q+ 2^k2q2\u0011\n\u00008(eB)2\nD\u0000(Q2\u0000m2\nf)3\u0010\n^k1q1\u0000^k2q2\u0011\n\u00008(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0010\nq0+^k\u0001q\u00002^k2q2\u0011\n+8\n(Q2\u0000m2\nf)\u001a\u0010h2\nD6\u0000h0\nD4\u0011\n(1 +a)(2k2q2+K\u0001Q) +\u0012\nb\u0010h2\nD6\u0000h0\nD4\u0011\n\u0000b0h\nD4\u0013\nq0\n\u0000c0h\nD4q3\u001b#\n; (49)\n\u000533\n2=\u0000e2XZ\"\n8eB\nD2(Q2\u0000m2\nf)2\u001a\n\u0000b0q3+c0q0\u001b\n+8eB\nD4(Q2\u0000m2\nf)2h\u001a\n(1 +a)(k0q3+k3q0) +bq3\u001b\n+8(eB)2\nD+(Q2\u0000m2\nf)3\u0010\n\u0000q3^k3+^k\u0001q\u0011\n\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\u0010\nq0\u0000^k\u0001q+ 2q3^k3\u0011\n+8(eB)2\nD\u0000(Q2\u0000m2\nf)3\u0010\nq3^k3\u0000^k\u0001q\u0011\n\u00008(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0010\nq0+^k\u0001q\u00002q3^k3\u0011\n+8\n(Q2\u0000m2\nf)\u001a\u0010h2\nD6\u0000h0\nD4\u0011\n(1 +a)(2k3q3+K\u0001Q) +\u0012\nb\u0010h2\nD6\u0000h0\nD4\u0011\n\u0000b0h\nD4\u0013\nq0\n+c0h\nD4q3\u001b#\n; (50)\n\u000513\n2=\u0000e2XZ\"\n8eB\nD4(Q2\u0000m2\nf)2h(1 +a)(k1q0)\u00008(eB)2\nD+(Q2\u0000m2\nf)3^k3q1\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\n\u0002\u0010\n^k1q3+^k3q1\u0011\n+8(eB)2\nD\u0000(Q2\u0000m2\nf)3^k3q1+8(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0010\n^k1q3+^k3q1\u0011\n+8\n(Q2\u0000m2\nf)\n\u0002\u001a\u0010h2\nD6\u0000h0\nD4\u0011\n(1 +a)\u0010\nk1q3+k3q1\u0011\n+c0h\nD4q1\u001b#\n: (51)\nIV. IMAGINARY PARTS OF THE COMPONENTS OF THE PHOTON SELF-ENERGY\nBefore obtaining the imaginary parts, we discuss below the various approximations used in this\ncalculation.\n1. We have considered the momentum of photon as hard ( p\u001dT). The momentum of soft\nfermionk\u001cT. Thus we can take the following approximations:\nnF(!)\u00181; nF(p\u0000!)\u0018e\u0000p=T; e\u0000p=T\u00180: (52)12\n2. An upper cuto\u000b \u0003( < T) of the soft fermion momentum khas been introduced in the inte-\ngrations.\n3. We consider mf=mthfor electron.\nθ\nxyz\nk\np\nφθp\nFIG. 2: Choice of reference frame for computing the various components of photon self-energy.\nThe magnetic \feld is along z-direction and \u0012pis the angle between momentum of photon and the\nexternal magnetic \feld.\n4. To perform the various integrations we choose a frame of reference as shown in Fig. 2 in\nwhich the external momentum of the photon in xzplane with 0 < \u0012p< \u0019= 2. So one can\nwrite\n~ p\u0011(psin\u0012p;0; pcos\u0012p); (53)\nand then the loop momentum as\n~k\u0011(ksin\u0012cos\u001e; ksin\u0012sin\u001e; kcos\u0012): (54)\nIn the following subsection we will obtain imaginary parts of various self-energy components.\nA. Imaginary parts of the magnetic \feld independent part, i:e:O[(eB)0]\nWe evaluate the imaginary parts of \u000511\n0, \u000522\n0, \u000533\n0, and \u000513\n0using the Braaten-Pisarski-Yuan\nmethod [45, 47].13\nIm \u000511\n0=\u00004e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u001a(1)\n0(!0)\u0010\n\u001aD+(!)\n+\u001aD\u0000(!)\u0011\n\u0000\u0000^k\u0001q\u00002^k1q1\u0001\u0010\n\u001aD+(!)\u0000\u001aD\u0000(!)\u0011\n\u001a(0)\n0(!0)\u001b\n\u000e(!+!0\u0000p); (55)\nIm \u000522\n0=\u00004e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u001a(1)\n0(!0)\u0010\n\u001aD+(!)\n+\u001aD\u0000(!)\u0011\n\u0000\u0000^k\u0001q\u00002^k2q2\u0001\u0010\n\u001aD+(!)\u0000\u001aD\u0000(!)\u0011\n\u001a(0)\n0(!0)\u001b\n\u000e(!+!0\u0000p); (56)\nIm \u000533\n0=\u00004e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u001a(1)\n0(!0)\u0010\n\u001aD+(!)\n+\u001aD\u0000(!)\u0011\n\u0000\u0000^k\u0001q\u00002^k3q3\u0001\u0010\n\u001aD+(!)\u0000\u001aD\u0000(!)\u0011\n\u001a(0)\n0(!0)\u001b\n\u000e(!+!0\u0000p); (57)\nIm \u000513\n0=\u00004e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n(^k1q3+q1^k3)\n\u0002\u0010\n\u001aD+(!)\u0000\u001aD\u0000(!)\u0011\n\u001a(0)\n0(!0)\u001b\n\u000e(!+!0\u0000p); (58)\nwhere\u001a(0)\n0;\u001a(1)\n0;\u001a(0)\n1;\u001a(1)\n1;\u001aD+and\u001aD\u0000are spectral representation of f(0)\n0;f(1)\n0;f(0)\n1;f(1)\n1;1=D+and\n1=D\u0000respectively. These spectral functions are obtained in Appendix A. We know that both \u001aD+\nand\u001aD\u0000have pole containing the mass shell \u000e-function + Landau cut part in space like region\nwhereas\u001a(0)\n0;\u001a(1)\n0;\u001a(0)\n1;\u001a(1)\n1have only pole containing the mass shell \u000efunction. Since imaginary\nparts of various components of the self-energy contain the product of two spectral functions, it\nwould then have the pole-pole and the pole-cut contributions.\nThe pole-pole parts of Im\u0005 11, Im\u0005 22, Im\u0005 33and Im\u0005 13contain\u000e(p\u0000!\u0006\u0000q) where!\u0006is the\nenergy of the fermion quasiparticle, ~kand~ q=~k\u0000~ pare the momenta of soft and hard fermion,\nrespectively. Hence !\u0006>k. The\u000e-function yields\np\u0000!\u0006\u0000q= 0\ncos\u001e\u0019!\u0006=k\u0000cos\u0012cos\u0012p\nsin\u0012sin\u0012p:\nThe value of!\u0006=k\u0000cos\u0012cos\u0012p\nsin\u0012sin\u0012pexcludes the range [ \u00001;1] for all values of the parameters \u0012and\u0012p\n. This restriction is valid for both thermal and the magnetic case. Thus pole-pole parts do not\ncontribute in this calculation [44, 45]. In O[(eB)0] the contribution comes only from the pole-cut\npart.14\n1. Pole-cut part of O[(eB)0]\nNow we would \fnd the pole-cut part of the above self-energy components in Eqs.(55), (56), (57)\nand (58) as\nIm \u000511\n0\f\f\f\f\npole\u0000cut\n= 2e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u000e(!0\u0000q)\u0002(k2\u0000!2)\n\u0002\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k\u0001q\u00002^k1q1)\u000e(!0\u0000q)\u0002(k2\u0000!2)\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u0002\u000e(p\u0000!\u0000!0);\n=\u0000e2\u0019Z\u0003\n0k2dk\n2\u00192Z\u0019\n01\n2sin\u0012d\u0012Z2\u0019\n0d\u001e\n2\u0019Zk\n\u0000kd!\u001a\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k\u0001q\u00002^k1q1)\n\u0002\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u000e(p\u0000!\u0000q); (59)\nIm \u000522\n0\f\f\f\f\npole\u0000cut\n= 2e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u000e(!0\u0000q)\u0002(k2\u0000!2)\n\u0002\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k\u0001q\u00002^k2q2)\u000e(!0\u0000q)\u0002(k2\u0000!2)\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u0002\u000e(p\u0000!\u0000!0)\n=\u0000e2\u0019Z\u0003\n0k2dk\n2\u00192Z\u0019\n01\n2sin\u0012d\u0012Z2\u0019\n0d\u001e\n2\u0019Zk\n\u0000kd!\u001a\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k\u0001q\u00002^k2q2)\n\u0002\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u000e(p\u0000!\u0000q); (60)\nIm \u000533\n0\f\f\f\f\npole\u0000cut\n= 2e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u000e(!0\u0000q)\u0002(k2\u0000!2)\n\u0002\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k?q?\u0000^k3q3)\u000e(!0\u0000q)\u0002(k2\u0000!2)\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u0002\u000e(p\u0000!\u0000!0)\n=\u0000e2\u0019Z\u0003\n0k2dk\n2\u00192Z\u0019\n01\n2sin\u0012d\u0012Z2\u0019\n0d\u001e\n2\u0019Zk\n\u0000kd!\u001a\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k?q?\u0000^k3q3)\n\u0002\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u000e(p\u0000!\u0000q); (61)15\nIm \u000513\n0\f\f\f\f\npole\u0000cut\n= 2e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a^k1q3+q1^k3\nq\u000e(!0\u0000q)\n\u0002\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\n\u0002(k2\u0000!2)\u001b\n\u000e(p\u0000!\u0000!0)\n=\u0000e2\u0019Z\u0003\n0k2dk\n2\u00192Z\u0019\n01\n2sin\u0012d\u0012Z2\u0019\n0d\u001e\n2\u0019Zk\n\u0000kd!1\nq\u0010\n^k1q3+q1^k3\u0011\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\n\u0002\u000e(p\u0000!\u0000q):\n(62)\nHere we note that the terms with \u000e(!0+q)\u000e(p\u0000!\u0000!0) \u0002(k2\u0000!2) will not contribute because\nk2\u0000(p+q)2can not be greater than zero. So we have excluded those terms.\nB. Imaginary part of magnetic \feld dependent part of O[(eB)2]\nSimilar toO[(eB)0] case, the imaginary part of \u000511\n2, \u000522\n2, \u000533\n2and \u000513\n2can be written as\nIm \u000511\n2=\u00008e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\n\u0002\u0014\neBn\nq3\u001a(0)\n1\u0010\n\u001a(1)\n9+\u001a10\u0000\u001a7\u0011\n\u0000\u001a(1)\n1\u0010\n\u001a8+k3\u001a(0)\n9\u0011o\n+ (eB)2\u0010\n^k1q1\u0000^k2q2\u0011\n\u001a(0)\n2\n\u0002\u0010\n\u001aD\u0000\u0000\u001aD+\u0011\n\u0000(eB)2q2\n?n\n\u001a(1)\n3\u0010\n\u001aD++\u001aD\u0000\u0011\n+ (2^k1q1\u0000^k\u0001q)\u001a(0)\n3\u0010\n\u001aD+\u0000\u001aD\u0000\u0011o\n+\u001a(1)\n0\u0010\n\u001a(1)\n15\u0000\u001a(1)\n14+\u001a16\u0000\u001a13\u0000\u001a11\u0011\n+ (2k1q1\u0000~k\u0001~ q)\u001a(0)\n0\u0010\n\u001a(0)\n15\u0000\u001a(0)\n14\u0011\n\u0000q3\u001a(0)\n0\u001a12\u0015\n\u0002\u000e(p\u0000!\u0000!0);\n(63)\nIm \u000522\n2=\u00008e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\n\u0002\u0014\neBn\nq3\u001a(0)\n1\u0010\n\u001a(1)\n9+\u001a10\u0000\u001a7\u0011\n\u0000\u001a(1)\n1\u0010\n\u001a8+k3\u001a(0)\n9\u0011o\n+ (eB)2\u0010\n^k2q2\u0000^k1q1\u0011\n\u001a(0)\n2\n\u0002\u0010\n\u001aD\u0000\u0000\u001aD+\u0011\n\u0000(eB)2q2\n?n\n\u001a(1)\n3\u0010\n\u001aD++\u001aD\u0000\u0011\n+ (2^k2q2\u0000^k\u0001q)\u001a(0)\n3\u0010\n\u001aD+\u0000\u001aD\u0000\u0011o\n+\u001a(1)\n0\u0010\n\u001a(1)\n15\u0000\u001a(1)\n14+\u001a16\u0000\u001a13\u0000\u001a11\u0011\n+ (2k2q2\u0000~k\u0001~ q)\u001a(0)\n0\u0010\n\u001a(0)\n15\u0000\u001a(0)\n14\u0011\n\u0000q3\u001a(0)\n0\u001a12\u0015\n\u0002\u000e(p\u0000!\u0000!0);\n(64)\nIm \u000533\n2=\u00008e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)16\n\u0002\u0014\neBn\nq3\u001a(0)\n1\u0010\n\u001a(1)\n9+\u001a10\u0000\u001a7\u0011\n+\u001a(1)\n1\u0010\n\u001a8+k3\u001a(0)\n9\u0011o\n+ (eB)2\u0010\nq3^k3\u0000^k\u0001q\u0011\n\u001a(0)\n2\n\u0002\u0010\n\u001aD\u0000\u0000\u001aD+\u0011\n\u0000(eB)2q2\n?n\n\u001a(1)\n3\u0010\n\u001aD++\u001aD\u0000\u0011\n+ (2q3^k3\u0000^k\u0001q)\u001a(0)\n3\u0010\n\u001aD+\u0000\u001aD\u0000\u0011o\n+\u001a(1)\n0\u0010\n\u001a(1)\n15\u0000\u001a(1)\n14+\u001a16\u0000\u001a13\u0000\u001a11\u0011\n+ (2k3q3\u0000~k\u0001~ q)\u001a(0)\n0\u0010\n\u001a(0)\n15\u0000\u001a(0)\n14\u0011\n+q3\u001a(0)\n0\u001a12\u0015\n\u0002\u000e(p\u0000!\u0000!0);\n(65)\nIm \u000513\n2=\u00008e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\n\u0002\u0014\neBn\n\u001a(1)\n1k3\u001a(0)\n9o\n+ (eB)2^k3q1\u001a(0)\n2n\n\u001aD\u0000\u0000\u001aD+o\n\u0000(eB)2q2\n?n\n(^k1q3+^k3q1)\u001a(0)\n3\n\u0002\u0010\n\u001aD+\u0000\u001aD\u0000\u0011o\n+ (k1q3+k3q1)\u001a(0)\n0\u0010\n\u001a(0)\n15\u0000\u001a(0)\n14\u0011\n+q1\u001a(0)\n0\u001a12\u0015\n\u000e(p\u0000!\u0000!0): (66)\nVarious spectral functions are obtained in Appendix A. As discussed before we also note that the\nimaginary part of various components of the self-energy contain the pole-pole and the pole-cut\ncontributions. As explained earlier the phase space does not allow the pole-pole part to contribute\nin this order. InO[(eB)2] the contribution comes only from the pole-cut part.\n1. Pole-cut part of O[(eB)2]\nNow the expressions of pole-cut parts of Eqs. (63), (64), (65) and (66) after using the approxi-\nmations, are given below:\nIm \u000511\n2\f\f\f\npole\u0000cut\n= 4e2\u0019Z\u0003\n0k2dk\n2\u00192Z1\n2sin\u0012d\u0012Zd\u001e\n2\u0019Zk\n\u0000kd!Z1\n\u00001d!0\u0014\n\u000e000(!0\u0000!q)\u001a\n\u0000(eB)2q2\n?\n96!3q\n\u0002\u0010\n\f++\f\u0000\u0011\n\u0000(eB)2q2\n?(2^k1q1\u0000^k\u0001q)\n96!4q\u0010\n\f+\u0000\f\u0000\u0011\u001b\n\u0000\u000e00(!0\u0000!q)\u001a3(eB)2\n64!3q\n\u0002\u0010q2\n?(^k1q1\u0000^k\u0001q)\n!2q\u0000(^k1q1\u0000^k2q2)\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\n+3(eB)2q2\n?\n128!4q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e0(!0\u0000!q)\u001aeBq 3\n4!2q\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000eB\n4!q\u0010\n\f8+k3\f(0)\n9\u0011\n\u0000(eB)2\n16!4q\u0010\n^k2q2\u0000^k1q1\n+5q2\n?(^k1q1\u0000^k\u0001q)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u0000(eB)2q2\n?\n32!5q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e(!0\u0000!q)\u001aeBq 3\n4!3q\n\u0002\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000(eB)2\n16!5q\u0010\n3^k2q2\u00003^k1q1+5q2\n?(^k1q1\u0000^k\u0001q)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u00001\n2\u0012\n\f(1)\n15+\f16\u0000\u0010\n\f(1)\n14+\f11+\f13\u0011\u0013\n\u0000(2k1q1\u0000~k\u0001~ q)\n2!q\u0012\n\f(0)\n15\u0000\f(0)\n14\u0013\n+q3\n2!q\f12\u001b\u001517\n\u0002\u000e(p\u0000!\u0000!0); (67)\nIm \u000522\n2\f\f\f\npole\u0000cut\n= 4e2\u0019Z\u0003\n0k2dk\n2\u00192Z1\n2sin\u0012d\u0012Zd\u001e\n2\u0019Zk\n\u0000kd!Z1\n\u00001d!0\u0014\n\u000e000(!0\u0000!q)\u001a\n\u0000(eB)2q2\n?\n96!3q\n\u0002\u0010\n\f++\f\u0000\u0011\n\u0000(eB)2q2\n?(2^k2q2\u0000^k\u0001q)\n96!4q\u0010\n\f+\u0000\f\u0000\u0011\u001b\n\u0000\u000e00(!0\u0000!q)\u001a3(eB)2\n64!3q\n\u0002\u0010q2\n?(^k2q2\u0000^k\u0001q)\n!2q\u0000(^k2q2\u0000^k1q1)\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\n+3(eB)2q2\n?\n128!4q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e0(!0\u0000!q)\u001aeBq 3\n4!2q\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000eB\n4!q\u0010\n\f8+k3\f(0)\n9\u0011\n\u0000(eB)2\n16!4q\u0010\n^k1q1\u0000^k2q2\n+5q2\n?(^k2q2\u0000^k\u0001q)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u0000(eB)2q2\n?\n32!5q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e(!0\u0000!q)\u001aeBq 3\n4!3q\n\u0002\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000(eB)2\n16!5q\u0010\n3^k1q1\u00003^k2q2+5q2\n?(^k2q2\u0000^k\u0001q)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u00001\n2\u0012\n\f(1)\n15+\f16\u0000\u0010\n\f(1)\n14+\f11+\f13\u0011\u0013\n\u0000(2k2q2\u0000~k\u0001~ q)\n2!q\u0012\n\f(0)\n15\u0000\f(0)\n14\u0013\n+q3\n2!q\f12\u001b\u0015\n\u0002\u000e(p\u0000!\u0000!0); (68)\nIm \u000533\n2\f\f\f\npole\u0000cut\n= 4e2\u0019Z\u0003\n0k2dk\n2\u00192Z1\n2sin\u0012d\u0012Zd\u001e\n2\u0019Zk\n\u0000kd!Z1\n\u00001d!0\u0014\n\u000e000(!0\u0000!q)\u001a\n\u0000(eB)2q2\n?\n96!3q\n\u0002\u0010\n\f++\f\u0000\u0011\n\u0000(eB)2q2\n?(^k3q3\u0000^k?q?)\n96!4q\u0010\n\f+\u0000\f\u0000\u0011\u001b\n\u0000\u000e00(!0\u0000!q)\u001a3(eB)2\n64!3q\n\u0002\u0010q2\n?(^k3q3\u0000^k?q?)\n!2q+^k?q?\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\n+3(eB)2q2\n?\n128!4q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e0(!0\u0000!q)\u001aeBq 3\n4!2q\u0010\n(\f(1)\n9+\f10\u0000\f7\u0011\n+eB\n4!q\u0010\n\f8+k3\u001a(0)\n9\u0011\n\u0000(eB)2\n16!4q\u0010\n^k?q?\n+5q2\n?(^k3q3\u0000^k?q?)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u0000(eB)2q2\n?\n32!5q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e(!0\u0000!q)\n\u0002\u001aeBq 3\n4!3q\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000(eB)2\n16!5q\u0010\n3^k?q?+5q2\n?(^k3q3\u0000^k?q?)\n2!2q\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\n\u00001\n2\u0012\n\f(1)\n15+\f16\u0000\u0010\n\f(1)\n14+\f13+\f11\u0011\u0013\n\u0000(2k3q3\u0000~k\u0001~ q)\n2!q\n\u0002\u0012\n\f(0)\n15\u0000\f(0)\n14\u0013\n\u0000q3\n2!q\f12\u001b\u0015\n\u000e(p\u0000!\u0000!0); (69)\nIm \u000513\n2\f\f\f\npole\u0000cut18\n= 4e2\u0019Z\u0003\n0k2dk\n2\u00192Z1\n2sin\u0012d\u0012Zd\u001e\n2\u0019Zk\n\u0000kd!Z1\n\u00001d!0\u0014\n\u000e000(!0\u0000!q)\n\u0002\u001a\n\u0000(eB)2q2\n?(^k1q3+^k3q1)\n96!4q\u0010\n\f+\u0000\f\u0000\u0011\u001b\n\u0000\u000e00(!0\u0000!q)\u001a3(eB)2\n64!3q\u0010q2\n?(^k1q3+^k3q1)\n!2q\n\u0000^k3q1\u0011\u0010\n\f+\u0000\f\u0000\u0011\u001b\n+\u000e0(!0\u0000!q)\u001aeB\n4!qk1\f(0)\n9\u0000(eB)2\n16!4q\u0010\n\u0000^k3q1+5q2\n?(^k1q3+^k3q1)\n2!2q\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\u001b\n+\u000e(!0\u0000!q)\u001a\n\u0000(eB)2\n16!5q\u0010\n\u00003^k3q1+5q2\n?(^k1q3+^k3q1)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u0000(k1q3+k3q1)\n2!q\u0012\n\f(0)\n15\u0000\f(0)\n14\u0013\n\u0000q1\n2!q\f12\u001b\u0015\n\u000e(p\u0000!\u0000!0): (70)\nV. RESULTS\nWe perform the integrations in Eq. (59),(60),(61),(62),(67),(68),(69) and (70) numerically. In\nthis calculation we have taken m\u0019= 0:14 GeV. The results are shown for \u0003 = 0 :25 GeV which\nsatis\feseT\u001c\u0003\u001cT.\np=3GeV, T=0.5GeV, eB=m π2/4γδ\nγσ\nπ/8 π/4 3π/8 π/28.2×10-68.3×10-68.4×10-68.5×10-68.6×10-68.7×10-68.8×10-68.9×10-6\nθpγ[GeV]\nFIG. 3: Plot of damping rate of photon with the propagation angle \u0012pforp= 3 GeV,T= 0:5\nGeV andeB=m2\n\u0019=4.\nThe damping rate of photon in presence of magnetic \feld depends on the angle, \u0012p, between the\nmomentum of photon and the magnetic \feld. Figure 3 shows the variation of the damping rate of\na hard photon with the propagation angle. It increases with the increasing propagation angle. One\ncan see that the two transverse modes of a hard photon are damped in a similar fashion. Since the\nmagnetic \feld strength is very weak, this di\u000berence appears to be very small. We note that the19\nmagnetic correction is \u0018O[(eB)2] and switching the magnetic \feld from zto\u0000zdirection would\nnot a\u000bect the result. These two orientations of the magnetic \feld correspond to the propagation\nangle of photon \u0012pand\u0019\u0000\u0012p. These two situations are identical which correspond to the same\ndamping rates of photon at \u0012pand\u0019\u0000\u0012p.\n�=�� ������=� π�/�θ�=π/��θ�=π/�γ��\nγδ\nγσ\n���������������������×��-����×��-����×��-����×��-����×��-�����×��-�����×��-�\n�[���]γ[���]\nFIG. 4: Plot of damping rate of photon with the energy for T= 0:5 GeV and eB=m2\n\u0019=4 at\npropagation angles \u0012p=\u0019=10 and\u0019=2.\nIn Fig. 4 we display the damping rate as a function of photon momentum for two propagation\nangles\u0019=10 and\u0019=2. The soft contribution of the damping rate in a thermal medium agrees well\nwith that obtained in Ref. [45]. In presence of a thermomagnetic medium, the soft contribution\nto the damping rate is found to be reduced than that of the thermal one. For small propagation\nangle, the reduction of the damping rate is more compared to that of thermal medium. For\nhigher momentum the damping rate approaches the thermal value as the temperature becomes the\ndominant scale as compared to the strength of the magnetic \feld considered.\nFigure 5 displays the variation of damping rate with temperature for a speci\fc value of mo-\nmentum and magnetic \feld for two propagation angles \u0019=10 and\u0019=2. It is found that the soft\ncontribution to the damping rate increases with the increase in temperature both in thermal and\nthermomagnetic medium. For small propagation angle the damping rate is more reduced compared\nto that of large propagation angle. This observation is consistent with Fig. 4.\nFigure 6 shows the variation of the damping rate with the magnetic \feld strength for speci\fc\nvalues of photon momentum and temperature for two propagation angles. The thermal damping\nrate (O[(eB)0]) is represented by the black dashed horizontal line. The thermomagnetic damping20\n�=�������=� π�/�θ�=π/��θ�=π/�\nγ��\nγδ\nγσ\n���� ���� ��� ���� ������×��-����×��-����×��-����×��-����×��-�����×��-�\n�[���]γ[���]\nFIG. 5: Plot of damping rate of the hard photon with temperature at p= 3 GeV and eB=m2\n\u0019=4\nfor two propagation angles \u0019=10 and\u0019=2.\n�=������=������θ�=π/��θ�=π/�\nγ��\nγδ\nγσ\n����π�����π�����π�����π����×��-����×��-����×��-����×��-����×��-�\n��[����]γ[���]\nFIG. 6: Plot of damping rate of the hard photon with the magnetic \feld strength at T= 0:5 GeV\nandp= 3 GeV for two propagation angles \u0019=10 and\u0019=2.\nrate decreases with the increasing magnetic \feld. At smaller propagation angles the photons are\nless damped than that of higher propagation angles which are consistent with Fig. 4.\nFig. 7 shows the variation of the photon damping rate with the separation scale \u0003 keeping\nthe scale hierarchy eT\u001c\u0003\u001cT. As the allowed phase space increases with the increase of \u0003,\nthe damping rate is also found to increase with it3. The magnetic correction to the thermal\n3Nevertheless, the damping rate is expected to be \u0003 independent when hard contribution is added.21\np=3GeV, T=0.5GeV, eB=m π2/4θp=π/4 γth\nγδ\nγσ\n0.18 0.21 0.24 0.27 0.304×10-66×10-68×10-610×10-612×10-6\nΛ[GeV]γ[GeV]\nFIG. 7: Plot of damping rate of photon with \u0003 for \u0012p=\u0019=4,p= 3 GeV,T= 0:5 GeV and\neB=m2\n\u0019=4.\ndamping rate is negative. So, the di\u000berence between the thermal and thermomagnetic damping\nrate increases with \u0003.\nVI. CONCLUSION\nWe have calculated the soft contribution to the damping rate of a hard photon in a weakly\nmagnetized QED medium where momentum of one of the fermion in the loop is considered as soft.\nThe two degenerate transverse modes of photon in thermal medium are damped in a similar fashion\nin presence of weak magnetic \feld as shown in Fig. 3. The di\u000berence between two transverse modes\nis very marginal due to weak \feld approximation. The soft contribution to the damping rate in\nthermomagnetic medium is reduced compared to that of thermal medium. When the magnetic\n\feld is switched o\u000b thermomagnetic damping modes reduce to its thermal value. The e\u000bect of\nmagnetic \feld is found to be dominant at low temperature and low photon momentum.\nThe soft contribution to the hard photon damping rate is \u001810\u00006GeV. Thus, a photon of a few\nGeV energy traversing in the QED medium of temperature \u00180:5 GeV and background magnetic\n\feld\u00180:005 GeV2has a mean free path ( \u0015=\r\u00001=2) of a few \u0017A. When the present calculation is\nextended to the case of relativistic heavy ion collisions, the mean free path of photon is found to\nbe of a few hundred fm. This con\frms that the mean free path of photon is larger than the size of\nthe \freball and photon can be treated as a direct probe.22\nThe damping rate is found to be dependent on the separation scale \u0003. One needs to add the\nhard contribution with the soft contribution to cancel the \u0003 dependence of the result. The hard\ncontribution to the photon damping rate comes from two-loop order with hard particles in the loop\nhaving momentum of the order of or higher than the temperature. This itself is a huge calculation\nwhich is in progress.\nAcknowledgement: RG is funded by University Grants Commission (UGC). BK and MGM were\nfunded by Department of Atomic Energy (DAE), India via the project TPAES. RG and BK would\nlike to thank Aritra Das for useful discussions.\nAppendix A: Spectral representation of the propagators\nlim\n\u000f!0Z1\n\u00001dx\u000f\nx2+\u000f2f(x)\u0019f(0)Z1\n\u00001dx\u000f\nx2+\u000f28\n><\n>:signi\fcant contribution comes from\nintegration ;wherex'0\n=f(0)\u000fZ1\n\u00001dx1\nx2+\u000f2=\u0019f(0); (A1)\nwheref(x) is a test function.\nFrom the above equations we can write,\nlim\n\u000f!0\u000f\nx2+\u000f2=\u0019\u000e(x); (A2)\nlim\n\u000f!0Im1\nx+i\u000f=1\n2ilim\n\u000f!0\u00141\nx+i\u000f\u00001\nx\u0000i\u000f\u0015\n=1\n2ilim\n\u000f!0\u00002i\u000f\nx2+\u000f2=\u0000\u0019\u000e(x): (A3)\nlim\n\u000f!0Z1\n\u00001dx2\u000fx\n(x2+\u000f2)2f(x) = lim\n\u000f!0Z1\n\u00001dx\u000ff (x)d\ndx\u0014\n\u00001\n(x2+\u000f2)\u0015\n= lim\n\u000f!0Z1\n\u00001dxf0(x)\u000f\nx2+\u000f2\n=\u0019f0(0) =\u0019Z\ndxf0(x)\u000e(x)\n=\u0000\u0019Z\ndxf(x)\u000e0(x) (A4)\nFrom the above equation we \fnd,\nlim\n\u000f!02\u000fx\n(x2+\u000f2)2=\u0000\u0019\u000e0(x): (A5)23\nNow using Eq. (A5) one can calculate,\nlim\n\u000f!0Im1\n(x+i\u000f)2=1\n2ilim\n\u000f!0\u00141\n(x+i\u000f)2\u00001\n(x\u0000i\u000f)2\u0015\n=1\n2ilim\n\u000f!0\u0000i4\u000fx\n(x2+\u000f2)2=\u0019\u000e0(x): (A6)\nSimilarly,\nlim\n\u000f!0Z1\n\u00001dx\u000f3\u00003x2\u000f\n(x2+\u000f2)3f(x) = lim\n\u000f!0Z1\n\u00001dx\u000f3\n(x2+\u000f2)3f(x)\u0000lim\n\u000f!0Z1\n\u00001dx3x2\u000f\n(x2+\u000f2)3f(x)\n=I1+I2; (A7)\nwhere\nI1= lim\n\u000f!0Z1\n\u00001dx\u000f3f(x)\n(x2+\u000f2)3\u0019lim\n\u000f!0\u000f3f(0)Z1\n\u00001dx1\n(x2+\u000f2)3= lim\n\u000f!0\u000f3f(0)3\n81\n\u000f5= lim\n\u000f!03\n8\u000f2f(0);\nI2=\u00003 lim\n\u000f!0Z1\n\u00001dxx2\u000ff(x)\n(x2+\u000f2)3=\u00003 lim\n\u000f!0\u000fZ\ndxf(x)\u00141\n8d2\ndx2\u00121\nx2+\u000f2\u0013\n+1\n82\n(x2+\u000f2)2\u0015\n=\u0000lim\n\u000f!03\n8\u000f2f(0)\u00003\n8lim\n\u000f!0\u000fZ\ndxf(x)d2\ndx2\u00121\nx2+\u000f2\u0013\n: (A8)\nSo ,\nI1+I2=\u00003\n8lim\n\u000f!0\u000fZ\ndxf(x)d2\ndx2\u00121\nx2+\u000f2\u0013\n=3\n8lim\n\u000f!0Z\ndx\u000ff0(x)d\ndx\u00121\nx2+\u000f2\u0013\n=\u00003\n8lim\n\u000f!0Z\ndx\u000ff00(x)1\nx2+\u000f2=\u00003\n8\u0019Z\ndxf00(x)\u000e(x) =\u00003\n8\u0019Z1\n\u00001dxf(x)\u000e00(x):(A9)\nWe can conclude from the last few steps that,\nlim\n\u000f!0\u000f3\u00003x2\u000f\n(x2+\u000f2)3=\u00003\n8\u0019\u000e00(x): (A10)\nUsing Eq. (A10) we can \fnd\nlim\n\u000f!0Im1\n(x+i\u000f)3=1\n2ilim\n\u000f!0\u00141\n(x+i\u000f)2\u00001\n(x\u0000i\u000f)3\u0015\n= lim\n\u000f!0\u000f3\u00006\u000fx2\n(x2+\u000f2)3=\u00003\n8\u0019\u000e00(x) (A11)\nNow ,\nlim\n\u000f!0Im1\n(x+i\u000f)4=1\n2ilim\n\u000f!0\u00141\n(x+i\u000f)4\u00001\n(x\u0000i\u000f)4\u0015\n= lim\n\u000f!04x\u000f3\u00004x3\u000f\n(x2+\u000f2)4; (A12)\nlim\n\u000f!0Z1\n\u00001dxf(x)4x\u000f3\u00004x3\u000f\n(x2+\u000f2)4= lim\n\u000f!04\u000fZ1\n\u00001dxf(x)1\n24d3\ndx3\u00121\nx2+\u000f2\u0013\n=\u0000lim\n\u000f!0\u000f\n6Z1\n\u00001dxf000(x)1\nx2+\u000f2=\u0000\u0019f000(0)\n6=\u0019\n6Z1\n\u00001dxf(x)\u000e000(x): (A13)\nThus, lim\n\u000f!0Im1\n(x+i\u000f)4=\u0019\n6\u000e000(x): (A14)24\nNow we write the spectral representations for the free propagators as\n\u001a(1)\n0(!0;q) =1\n2\u0019lim\n\u000f!0Im\u00141\n!0+i\u000f+!q+1\n!0+i\u000f\u0000!q\u0015\n=\u00001\n2\u0014\n\u000e(!0+!q) +\u000e(!0\u0000!q)\u0015\n;(A15)\n\u001a(0)\n0(!0;q) =1\n2\u0019!qlim\n\u000f!0Im\u00141\n!0+i\u000f\u0000!q\u00001\n!0+i\u000f+!q\u0015\n=1\n2!q\u0014\n\u000e(!0+!q)\u0000\u000e(!0\u0000!q)\u0015\n;(A16)\n\u001a(1)\n1(!0;q) =1\n4\u0019!qlim\n\u000f!0Im\u00141\n(!0+i\u000f\u0000!q)2\u00001\n(!0+i\u000f+!q)2\u0015\n=\u00001\n4!q\u0014\n\u000e0(!0+!q)\u0000\u000e0(!0\u0000!q)\u0015\n; (A17)\n\u001a(0)\n1(!0;q) =1\n4\u0019!2qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)2+1\n(!0+i\u000f\u0000!q)2\u00001\n!q\u00121\n!0+i\u000f\u0000!q\n\u00001\n!0+i\u000f+!q\u0013\u0015\n=1\n4!2q\u0014\n\u000e0(!0+!q) +\u000e0(!0\u0000!q)\u00001\n!q\u0012\n\u000e(!0+!q)\u0000\u000e(!0\u0000!q)\u0013\u0015\n; (A18)\n\u001a(1)\n2(!0;q) =1\n8\u0019!2qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)3+1\n(!0+i\u000f\u0000!q)3+1\n2!q\u001a1\n(!0+i\u000f+!q)2\n\u00001\n(!0+i\u000f\u0000!q)2\u001b\u0015\n=1\n8!2q\u0014\n\u00003\n8\u0012\n\u000e00(!0+!q) +\u000e00(!0\u0000!q)\u0013\n+1\n2!q\u0012\n\u000e0(!0+!q)\u0000\u000e0(!0\u0000!q)\u0013\u0015\n;(A19)\n\u001a(0)\n2(!0;q) =\u00001\n8\u0019!3qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)3\u00001\n(!0+i\u000f\u0000!q)3+1\n2!q\u001a1\n(!0+i\u000f+!q)2\n+1\n(!0+i\u000f\u0000!q)2+3\n!q\u00121\n!0+i\u000f+!q\u00001\n!0+i\u000f\u0000!q\u0013\u001b\u0015\n=\u00001\n8!3q\u0014\n\u00003\n8\u0012\n\u000e00(!0+!q)\u0000\u000e00(!0\u0000!q)\u0013\n+1\n2!q\u001a\n\u000e0(!0+!q) +\u000e0(!0\u0000!q)\n\u00003\n!q\u0012\n\u000e(!0+!q)\u0000\u000e(!0\u0000!q)\u0013\u001b\u0015\n; (A20)\n\u001a(1)\n3(!0;q) =\u00001\n16\u0019!3qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)4\u00001\n(!0+i\u000f\u0000!q)4+1\n!q\u001a1\n(!0+i\u000f+!q)3\n+1\n(!0+i\u000f\u0000!q)3+1\n2!q\u00121\n(!0+i\u000f+!q)2\u00001\n(!0+i\u000f\u0000!q)2\u0013\u001b\u0015\n(A21)\n=\u00001\n16!3q\u00141\n6\u0012\n\u000e000(!0+!q)\u0000\u000e000(!0\u0000!q)\u0013\n+1\n!q\u001a\n\u00003\n8\u0012\n\u000e00(!0+!q)\n+\u000e00(!0\u0000!q)\u0013\n+1\n2!q\u0012\n\u000e0(!0+!q)\u0000\u000e0(!0\u0000!q)\u0013\u001b\u0015\n; (A22)\n\u001a(0)\n3(!0;q) =1\n16\u0019!4qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)4+1\n(!0+i\u000f\u0000!q)4+1\n2!q\u001a4\n(!0+i\u000f+!q)3\n\u00004\n(!0+i\u000f\u0000!q)3+1\n2!q\u001210\n(!0+i\u000f+!q)2+10\n(!0+i\u000f\u0000!q)2+1\n2!q\n\u0002\u001220\n!0+i\u000f+!q\u000020\n!0+i\u000f\u0000!q\u0013\u0013\u001b\u0015\n(A23)25\n=1\n16!4q\"\n1\n6\u0012\n\u000e000(!0+!q) +\u000e000(!0\u0000!q)\u0013\n+1\n2!q\u001a\n\u00003\n2\u0012\n\u000e00(!0+!q)\n\u0000\u000e00(!0\u0000!q)\u0013\n+5\n!q\u0012\n\u000e0(!0+!q) +\u000e0(!0\u0000!q)\u00001\n!q\u0010\n\u000e(!0+!q)\n\u0000\u000e(!0\u0000!q)\u0011\u0013\u001b#\n: (A24)\nThe e\u000bective propagators are given as,\n1\nD2=1\nD+D\u0000=X\ni\u0012@D2\n@!\u0013\u00001\f\f\f\f\n!=!i1\n(!\u0000!i); (A25)\n1\nD4=1\n(D+D\u0000)2=X\ni\"\u0012@D2\n@!\u0013\u00002\f\f\f\f\n!=!i1\n(!\u0000!i)2+@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i1\n(!\u0000!i)#\n=X\ni\"\u0012@D2\n@!\u0013\u00002\f\f\f\f\n!=!i1\n(!\u0000!i)2\u0000@2D2\n@!2\u00121\n3!@3D6\n@!3\u0013\u00001\f\f\f\f\f\n!=!i1\n(!\u0000!i)#\n;(A26)\n1\nD6=1\n(D+D\u0000)3=X\ni\"\u0012@D2\n@!\u0013\u00003\f\f\f\f\n!=!i1\n(!\u0000!i)3+@\n@!\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i1\n(!\u0000!i)2\n+1\n2@2\n@!2\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i1\n(!\u0000!i)#\n=X\ni\"\u0012@D2\n@!\u0013\u00003\f\f\f\f\n!=!i1\n(!\u0000!i)3\u00003\n2@2D2\n@!2\u00121\n4!@4D8\n@!4\u0013\u00001\f\f\f\f\f\n!=!i1\n(!\u0000!i)2\n\u00003\n5 \n@3D2\n@!3\u001a\n6\u0012@4D8\n@!4\u0013\u00001\n+7\n12\u0012@D2\n@!\u0013\u00004\u001b\n+ 6@2D2\n@!2@\n@!\u0012@4D8\n@!4\u0013\u00001!\f\f\f\f\f\n!=!i\n\u00021\n(!\u0000!i)#\n; (A27)\nwhere!i=\u0006!\u0006are the poles of D+andD\u0000.\nThe spectral functions of the dressed propagators are given as\n\u001aD\u0006=\u0000(!2\u0000k2)\n2m2\nth\u0014\n\u000e(!\u0000!\u0006) +\u000e(!+!\u0007)\u0015\n+\f\u0006\u0002(k2\u0000!2); (A28)\nwhere\n\f\u0006=\u00001\n2(k\u0007!)m2\nth\u0014\nk(!\u0007k)\u0000m2\nth\u001a\nQ0(!\nk)\u0007Q1(!\nk)\u001b\u00152\n+\u0014\n1\n2(1\u0007!\nk)m2\nth\u0019\u00152; (A29)\nwhere we use the Legendre function of second kind\nQ0\u0012!\nk\u0013\n=1\n2ln\f\f\f\f!+k\n!\u0000k\f\f\f\f\u0000i\u0019\n2\u0002(k2\u0000!2): (A30)\n\u001a4(!;k) =1\n\u0019Im\u00101\nD2\u0011\n=\u0000X\ni\u0012@D2\n@!\u0013\u00001\f\f\f\f\n!=!i\u000e(!\u0000!i) +\f4\u0002(k2\u0000!2)26\n=!2\u0000k2\n4m2\nth(k2\u0000!2+m2\nth)\u0014\n(!\u0000k)\u0012\n\u000e(!\u0000!+) +\u000e(!+!\u0000)\u0013\n+ (!+k)\u0012\n\u000e(!\u0000!\u0000)\n+\u000e(!+!+)\u0013\u0015\n+\f4\u0002(k2\u0000!2); (A31)\n\u001a5(!;k) =1\n\u0019Im\u00101\nD4\u0011\n=X\ni\"\u0012@D2\n@!\u0013\u00002\f\f\f\f\n!=!i\u000e0(!\u0000!i) +@2D2\n@!2\u00121\n3!@3D6\n@!3\u0013\u00001\f\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f5\u0002(k2\u0000!2); (A32)\n\u001a6(!;k) =1\n\u0019Im\u00101\nD6\u0011\n=X\ni\"\n\u00003\n8\u0012@D2\n@!\u0013\u00003\f\f\f\f\n!=!i\u000e00(!\u0000!i)\u00003\n2@2D2\n@!2\u00121\n4!@4D8\n@!4\u0013\u00001\f\f\f\f\f\n!=!i\n\u0002\u000e0(!\u0000!i) +3\n5 \n@3D2\n@!3\u001a\n6\u0012@4D8\n@!4\u0013\u00001\n+7\n12\u0012@D2\n@!\u0013\u00004\u001b\n+ 6@2D2\n@!2@\n@!\u0012@4D8\n@!4\u0013\u00001!\f\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f6\u0002(k2\u0000!2); (A33)\n\u001a7(!;k) =1\n\u0019Im\u0012b0\nD2\u0013\n=\u0000b0X\ni\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00001\n\u000e(!\u0000!i) +\f7\u0002(k2\u0000!2); (A34)\n\u001a8(!;k) =1\n\u0019Im\u0012c0\nD2\u0013\n=\u0000c0X\ni\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00001\n\u000e(!\u0000!i) +\f8\u0002(k2\u0000!2); (A35)\n\u001a(0)\n9(!;k) =1\n\u0019Im\u0012h(1 +a)\nD4\u0013\n=h(1 +a)\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\n\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(0)\n9\u0002(k2\u0000!2); (A36)\n\u001a(1)\n9(!;k) =1\n\u0019Im\u0012h(1 +a)!\nD4\u0013\n=!h(1 +a)\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\n\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(1)\n9\u0002(k2\u0000!2); (A37)\n\u001a10(!;k) =1\n\u0019Im\u0012hb\nD4\u0013\n=hb\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f10\u0002(k2\u0000!2); (A38)\n\u001a11(!;k) =1\n\u0019Im\u0014hb0\nD4\u0015\n=hb0\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f11\u0002(k2\u0000!2); (A39)\n\u001a12(!;k) =1\n\u0019Im\u0014hc0\nD4\u0015\n=hc0\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i27\n\u0002\u000e(!\u0000!i)#\n+\f12\u0002(k2\u0000!2); (A40)\n\u001a13(!;k) =1\n\u0019Im\u0014h0b\nD4\u0015\n=h0b\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f13\u0002(k2\u0000!2); (A41)\n\u001a(0)\n14(!;k) =1\n\u0019Im\u0014h0(1 +a)\nD4\u0015\n=h0(1 +a)X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\n\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(0)\n14\u0002(k2\u0000!2); (A42)\n\u001a(1)\n14(!;k) =1\n\u0019Im\u0014k0h0(1 +a)\nD4\u0015\n=!h0(1 +a)X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\n\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(1)\n14\u0002(k2\u0000!2); (A43)\n\u001a(0)\n15(!;k) =1\n\u0019Im\u0014h2(1 +a)\nD6\u0015\n=h2(1 +a)\u0002X\ni\"\n\u00003\n8\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00003\n\u000e00(!\u0000!i)\n+@\n@!\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e0(!\u0000!i)\u0000@2\n@!2\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(0)\n15\u0002(k2\u0000!2); (A44)\n\u001a(1)\n15(!;k) =1\n\u0019Im\u0014h2(1 +a)k0\nD6\u0015\n=!h2(1 +a)\u0002X\ni\"\n\u00003\n8\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00003\n\u000e00(!\u0000!i)\n+@\n@!\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e0(!\u0000!i)\u0000@2\n@!2\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(1)\n15\u0002(k2\u0000!2); (A45)\n\u001a16(!;k) =1\n\u0019Im\u0014h2b\nD6\u0015\n=h2b\u0002X\ni\"\n\u00003\n8\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00003\n\u000e00(!\u0000!i) +@\n@!\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\n\u0002\u000e0(!\u0000!i)\u0000@2\n@!2\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f16\u0002(k2\u0000!2); (A46)\nwhere cut parts of the spectral functions are given as\n\f4=1\n\u0019Im\u00121\nD2\u0013\n=\u00001\n\u0019ImD2\n(ReD2)2+ (ImD2)2; (A47)\n\f5=1\n\u0019Im\u00121\nD4\u0013\n=\u00001\n\u00192 ReD2ImD2\nh\n(ReD2)2+ (ImD2)2i2; (A48)\n\f6=1\n\u0019Im\u00121\nD6\u0013\n=1\n\u0019\u0010\nImD2\u00113\n\u00003\u0010\nReD2\u00112\nImD2\nh\n(ReD2)2+ (ImD2)2i3; (A49)28\n\f7=1\n\u0019Im\u0012b0\nD2\u0013\n=1\n\u0019\u0000Reb0ImD2+ Imb0ReD2\n(ReD2)2+ (ImD2)2; (A50)\n\f8=1\n\u0019Im\u0012c0\nD2\u0013\n=1\n\u0019\u0000Rec0ImD2+ Imc0ReD2\n(ReD2)2+ (ImD2)2; (A51)\n\f(0)\n9=1\n2Im\u0012h(1 +a)\nD4\u0013\n=1\n\u0019Im\u0000\nh(1 +a)\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nh(1 +a)\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A52)\n\f(1)\n9=1\n2Im\u0012h(1 +a)k0\nD4\u0013\n=1\n\u0019Im\u0000\nhk0(1 +a)\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nhk0(1 +a)\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A53)\n\f10=1\n2Im\u0012hb\nD4\u0013\n=1\n\u0019Im\u0000\nhb\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nhb\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A54)\n\f11=1\n2Im\u0012hb0\nD4\u0013\n=1\n\u0019Im\u0000\nhb0\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nhb0\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A55)\n\f12=1\n2Im\u0012hc0\nD4\u0013\n=1\n\u0019Im\u0000\nhc0\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nhc0\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A56)\n\f13=1\n2Im\u0012h0b\nD4\u0013\n=1\n\u0019Im\u0000\nh0b\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nh0b\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A57)\n\f(0)\n14=1\n2Im\u0012h0(1 +a)\nD4\u0013\n=1\n\u0019Im\u0000\nh0(1 +a)\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nh0(1 +a)\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A58)\n\f(1)\n14=1\n2Im\u0012h0(1 +a)k0\nD4\u0013\n=1\n\u0019Im\u0000\nh0(1 +a)k0\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nh0(1 +a)k0\u0001\nh\n(ReD2)2+ (ImD2)2i2;\n(A59)\n\f(0)\n15=1\n\u0019Im\u0012h2(1 +a)\nD6\u001329\n=1\n\u0019Im (h2(1 +a))\u0014\n(ReD2)3\u00003ReD2(ImD2)2\u0015\n+ Re (h2(1 +a))\u0014\n(ImD2)3\u00003ImD2(ReD2)2\u0015\nh\n(ReD2)2+ (ImD2)2i3\n(A60)\n\f(1)\n15=1\n\u0019Im\u0012h2(1 +a)k0\nD6\u0013\n=1\n\u0019Im (h2(1 +a)k0)\u0014\n(ReD2)3\u00003ReD2(ImD2)2\u0015\n+ Re (h2(1 +a)k0)\u0014\n(ImD2)3\u00003ImD2(ReD2)2\u0015\nh\n(ReD2)2+ (ImD2)2i3\n(A61)\n\f16=1\n\u0019Im\u0012h2b\nD6\u0013\n=1\n\u0019Im (h2b)\u0014\n(ReD2)3\u00003ReD2(ImD2)2\u0015\n+ Re (h2b)\u0014\n(ImD2)3\u00003ImD2(ReD2)2\u0015\nh\n(ReD2)2+ (ImD2)2i3\n(A62)\nwhere\nIm(D2) =\u0000\u0019m4\nth\nk2\u0014!\nk+\u0010\n1\u0000!2\nk2\u0011\nQ0\u0010!\nk\u0011\u0015\n; (A63)\nRe(D2) =!2\u0000k2\u00002m2\nth+m4\nth\nk2\u00122!\nkQ0\u0012!\nk\u0013\n\u00001\u0013\n+m4\nth\nk2\u0010\n1\u0000!2\nk2\u0011\u0012\nQ2\n0\u0012!\nk\u0013\n\u0000\u00192\n4\u0013\n; (A64)\nIm(b0) =\u00004e2M2\u0019k3!\n2k3; (A65)\nRe(b0) = 4e2M2k3\nk2\u0014!\nkQ0\u0012!\nk\u0013\n\u00001\u0015\n; (A66)\nIm(c0) =\u00004e2M2\u0019\n2k; (A67)\nRe(c0) = 4e2M21\nkQ0\u0012!\nk\u0013\n; (A68)\nIm\u0010\nh(1 +a)\u0011\n=4\u0019e2M2k3\n2k7\u0012\n\u00002k6\u00002k4\u0000\nk2\n0+ 3m2\nth\u0001\n+ 12k3k0m2\nthQ0\u00004k2m2\nth\u0000\nk2\n0+m2\nth\u0001\n+ 4kk0m2\nthQ0\u0000\nk2\n0+ 4m2\nth\u0001\n+k2\n0m4\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0013\n;\nRe(h(1 +a)) =4e2M2k3\n2k7\u0012\n4k6Q0\u00004k5k0+ 4k4Q0\u0000\nk2\n0+ 3m2\nth\u0001\n+k3k0m2\nth\u0000\n\u000012Q2\n0+ 3\u00192\u00004\u0001\n+ 8k2m2\nthQ0\u0000\nk2\n0+m2\nth\u0001\n+kk0m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 4m2\nth\u0001\n+ 2k2\n0m4\nthQ0\n\u0002\u0000\n4Q2\n0\u00003\u00192\u0001\u0013\n; (A69)\nIm(hb) =4\u0019e2M2k3m2\nth\n2k7\u0012\n4k2k0\u0000\nk2\n0+m2\nth\u0001\n+ 4kQ0\u0000\nk4+ 2m2\nth\u0000\nk2\u00002k2\n0\u0001\n\u0000k4\n0\u0001\n+ 12k0m2\nthQ2\n0(k0\u0000k)(k+k0) +\u00192k0m2\nth(k\u0000k0)(k+k0)\u0013\n; (A70)30\nRe(hb) =4e2M2k3m2\nth\n2k7\u0012\nk5\u0000\n\u00192\u00004Q2\n0\u0001\n+ 2k3\u0000\n2k2\n0+m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0001\n\u00002k2k0Q0\n\u0002\u0010\n4k2\n0+m2\nth\u0000\n\u00004Q2\n0+ 3\u00192+ 4\u0001\u0011\n\u0000kk2\n0\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 4m2\nth\u0001\n+ 2k3\n0m2\nthQ0\n\u0002\u0000\n3\u00192\u00004Q2\n0\u0001\u0013\n; (A71)\nIm(hb0) =16\u0019e4M4k2\n3\n2k7\u0012\n2k4\u00004k3k0Q0+ 4k2\u0000\nk2\n0+m2\nth\u0001\n\u00004kk0Q0\u0000\nk2\n0+ 4m2\nth\u0001\n\u0000k2\n0m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0013\n; (A72)\nRe(hb0) =\u000016e4M4k2\n3\n2k7\u0012\n4k4Q0+k3k0\u0000\n\u00192\u00004\u0000\nQ2\n0+ 1\u0001\u0001\n+ 8k2Q0\u0000\nk2\n0+m2\nth\u0001\n+kk0\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 4m2\nth\u0001\n+ 2k2\n0m2\nthQ0\u0000\n4Q2\n0\u00003\u00192\u0001\u0013\n; (A73)\nIm(hc0) =\u000016\u0019e4M4k3\n2k5\u0000\n4k3Q0\u00002k2k0+ 4kQ0\u0000\nk2\n0+ 2m2\nth\u0001\n+k0m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0001\n; (A74)\nRe(hc0) =\u000016e4M4k3\n2k5\u0012\nk3\u0000\n\u00192\u00004Q2\n0\u0001\n+ 4k2k0Q0+k\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 2m2\nth\u0001\n+ 2k0m2\nthQ0\u0000\n4Q2\n0\u00003\u00192\u0001\u0013\n; (A75)\nIm(h0b) =8\u0019e4M4m2\nth\nk9\u0012\nk6\u0012\u00192\n4\u00003Q2\n0\u0013\n\u00002k5!Q0+k4\u0010\nk2\n3\u0000!2\u0012\u00192\n4\u00003Q2\n0\u0013\u0011\n\u00004k3!k2\n3Q0\u0000k2!2k2\n3\u0012\n\u00003Q2\n0+\u00192\n4+ 3\u0013\n+ 6k!3k2\n3Q0+!4k2\n3\u0012\u00192\n4\u00003Q2\n0\u0013\u0013\n; (A76)\nRe(h0b) =16e4M4m2\nth\nk9 \nk6\u0012\nQ3\n0\u00003\u00192\n4Q0\u0013\n+k5!\u0012\nQ2\n0\u0000\u00192\n4\u0013\n\u0000k4Q0\u0010\n!2\u0010\nQ2\n0\u00003\u00192\n4\u0011\n+k2\n3\u0011\n+k3!k2\n3\u0012\n2Q2\n0\u0000\u00192\n2\u00001\u0013\n+k2!2k2\n3Q0\u0012\n\u0000Q2\n0+3\u00192\n4+ 3\u0013\n+ 3k!3k2\n3\u0010\u00192\n4\u0000Q2\n0\u0011\n+!4k2\n3Q0\u0012\nQ2\n0\u00003\u00192\n4\u0013!\n; (A77)\nIm(h0(1 +a)) =8\u0019e4M4\nk9\u0012\n2k7Q0+ 2k5m2\nthQ0+k4!\u0010\n2k2\n3+m2\nth\u0010\u00192\n4\u00003Q2\n0\u0011\u0011\n\u00002k3!2k2\n3Q0\n+ 3k2!k2\n3m2\nth\u00006k!2k2\n3m2\nthQ0\u0000!3k2\n3m2\nth\u0010\u00192\n4\u00003Q2\n0\u0011\u0013\n; (A78)\nRe(h0(1 +a)) =16e4M4\nk9\u0012\nk7\u0010\u00192\n4\u0000Q2\n0\u0011\n+k5\u0010\nk2\n3+m2\nth\u0010\u00192\n4\u0000Q2\n0\u0011\u0011\n+k4!Q0\n\u0002\u0010\nm2\nth\u0010\nQ2\n0\u00003\u00192\n4\u0011\n\u00002k2\n3\u0011\n+k3k2\n3\u0012\n!2\u0010\nQ2\n0\u0000\u00192\n4\u0011\n+m2\nth\u0013\n\u00003k2!k2\n3m2\nthQ0\n+ 3k!2k2\n3m2\nth\u0012\nQ2\n0\u0000\u00192\n4\u0013\n\u0000!3k2\n3m2\nthQ0\u0010\nQ2\n0\u00003\u00192\n4\u0011\u0013\n; (A79)\nIm\u0010\nh2(1 +a)\u0011\n=16\u0019e4M4k2\n3\n2k11 \n\u00008k9Q0+ 8k8k0\u00008k7Q0\u0000\n2k2\n0+ 5m2\nth\u0001\n+k6k0\u0012\n8k2\n031\n+m2\nth\u0000\n60Q2\n0\u00005\u00192+ 24\u0001\u0013\n\u00008k5Q0\u0000\nk4\n0+ 12k2\n0m2\nth+ 8m4\nth\u0001\n+ 2k4k0m2\nth\n\u0002\u0012\nk2\n0\u0000\n36Q2\n0\u00003\u00192+ 6\u0001\n\u00008m2\nth\u0000\n\u000012Q2\n0+\u00192\u00001\u0001\u0013\n\u00008k3m2\nthQ0\u0012\n3k4\n0\n\u00004k2\n0m2\nth\u0000\n\u00004Q2\n0+\u00192\u00003\u0001\n+ 4m4\nth\u0013\n\u0000k2k0m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0010\nk4\n0+ 12k2\n0m2\nth\n+ 12m4\nth\u0011\n+ 16kk2\n0m4\nthQ0\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 3m2\nth\u0001\n+k3\n0m6\nth\u0000\n80Q4\n0\u000040\u00192Q2\n0+\u00194\u0001!\n; (A80)\nRe\u0010\nh2(1 +a)\u0011\n=\u000016e4M4k2\n3\nk11 \nk9\u0000\n\u00192\u00004Q2\n0\u0001\n+ 8k8k0Q0+k7\u0010\n2k2\n0\u0000\n\u00004Q2\n0+\u00192\u00002\u0001\n+ 5m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0011\n+k6k0Q0\u0010\n8k2\n0+m2\nth\u0000\n20Q2\n0\u000015\u00192+ 24\u0001\u0011\n+k5\u0010\nk4\n0\u0000\n\u00192\u00004Q2\n0\u0001\n+ 4k2\n0m2\nth\u0000\n\u000012Q2\n0+ 3\u00192\u00001\u0001\n+ 8m4\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0011\n+ 2k4k0m2\nthQ0\u0012\nk2\n0\u0010\n12Q2\n0\u00009\u00192+ 6\u0011\n+ 8m2\nth\u0000\n4Q2\n0\u00003\u00192+ 1\u0001\u0013\n+k3m2\nth\u0012\n3k4\n0\u0000\n\u00192\u00004Q2\n0\u0001\n\u00002k2\n0m2\nth\u0010\n8\u0000\n2Q2\n0+ 3\u0001\nQ2\n0\u00006\u00192\u0000\n4Q2\n0+ 1\u0001\n+\u00194\u0011\n+ 4m4\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0013\n+k2k0m2\nthQ0\u0000\n4Q2\n0\u00003\u00192\u0001\u0000\nk4\n0+ 12k2\n0m2\nth+ 12m4\nth\u0001\n\u0000kk2\n0m4\nth\u0000\n16Q4\n0\u000024\u00192Q2\n0+\u00194\u0001\u0000\nk2\n0+ 3m2\nth\u0001\n+k3\n0m6\nthQ0\u0000\n16Q4\n0\u000040\u00192Q2\n0+ 5\u00194\u0001!\n; (A81)\nIm(h2b) =16\u0019e4M4k2\n3m2\nth\n2k11 \nk8\u0000\n\u0000\u0000\n\u00192\u000012Q2\n0\u0001\u0001\n\u00008k7k0Q0\u0000k6\u0010\nk2\n0\u0000\n\u000012Q2\n0+\u00192+ 4\u0001\n+ 4m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0011\n+ 16k5k0Q0\u0000\nk2\n0+m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0001\n+k4\u0012\nk4\n0\u0010\n\u00192\u000012\u0000\nQ2\n0+ 1\u0001\u0011\n\u000016k2\n0m2\nth\u00004m4\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0013\n+ 8k3k0Q0\u0000\n3k4\n0+ 12k2\n0m2\nth+ 4m4\nth\u0000\n\u00004Q2\n0+\u00192+ 1\u0001\u0001\n+k2k2\n0\u0012\nk4\n0\u0000\n\u00192\u000012Q2\n0\u0001\n+ 12k2\n0m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\n+m4\nth\u0010\n80Q4\n0\u00008\u0000\n18 + 5\u00192\u0001\nQ2\n0\n+\u00192\u0000\n12 +\u00192\u0001\u0011\u0013\n\u000016kk3\n0m2\nthQ0\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 3m2\nth\u0001\n\u0000k4\n0m4\nth\u0000\n80Q4\n0\u000040\u00192Q2\n0+\u00194\u0001!\n; (A82)\nRe(h2b) =16e4M4k2\n3m2\nth\nk11 \nk8\u0000\n3\u00192Q0\u00004Q3\n0\u0001\n\u0000k7k0\u0000\n\u00192\u00004Q2\n0\u0001\n+k6Q0\u0010\nk2\n0\u0000\n\u00004Q2\n0+ 3\u00192+ 4\u0001\n+ 4m2\nth\u0000\n3\u00192\u00004Q2\n0\u0001\u0011\n+k5k0\u0000\n2k2\n0\u0000\n\u00004Q2\n0+\u00192\u00002\u0001\n+m2\nth\u0000\n16Q4\n0\u000024\u00192Q2\n0+\u00194\u0001\u000132\n+k4Q0\u0012\nk4\n0\u0000\n4Q2\n0\u00003\u00192+ 12\u0001\n+ 16k2\n0m2\nth+ 4m4\nth\u0000\n3\u00192\u00004Q2\n0\u0001\u0013\n+k3k0\u0012\n3k4\n0\u0000\n\u00192\u00004Q2\n0\u0001\n+ 12k2\n0m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\n+ 2m4\nth\u0000\n16Q4\n0\u00008\u0000\n1 + 3\u00192\u0001\nQ2\n0+\u00192\u0000\n2 +\u00192\u0001\u0001\u0013\n+k2k2\n0Q0\n\u0002\u0012\n4Q2\n0\u0000\nk4\n0+ 12k2\n0m2\nth+ 2\u0000\n6 + 5\u00192\u0001\nm4\nth\u0001\n\u0000\u00192\u0000\n3k4\n0+ 36k2\n0m2\nth+\u0000\n36 + 5\u00192\u0001\nm4\nth\u0001\n\u000016m4\nthQ4\n0\u0013\n\u0000kk3\n0m2\nth\u0000\n16Q4\n0\u000024\u00192Q2\n0+\u00194\u0001\u0000\nk2\n0+ 3m2\nth\u0001\n+k4\n0m4\nthQ0\u0000\n16Q4\n0\n\u000040\u00192Q2\n0+ 5\u00194\u0001!\n; (A83)\nwhereM2is de\fned in Eq. 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The\nmodulation of the Gilbert damping constant is proportional to the product of the spin-up and\nspin-down densities of states of graphene. Consequently, t he Gilbert damping constant in a strong\nmagnetic field oscillates as a function of the external magne tic field that originates from the Landau\nlevel structure of graphene. We find that a measurement of the oscillation period enables the\nstrength of the exchange coupling constant to be determined . The results theoretically demonstrate\nthat the ferromagnetic resonance measurements may be used t o detect the spin resolved electronic\nstructure of the adjacent materials, which is critically im portant for future spin device evaluations.\nIntroduction .—Graphene spintronics is an emergent\nfield aiming at exploiting exotic spin-dependent proper-\ntiesofgrapheneforspintronicsdevices[1]. Althoughpris-\ntinegrapheneisanon-magneticmaterial,therehavebeen\neffortstointroducemagnetismintographenetofindspin-\ndependent phenomena and to exploit its spin degrees of\nfreedom. Placing graphene on a magnetic substrate is\na reasonable way, which leads to magnetic proximity ef-\nfect and lifting of spin degeneracy [2, 3]. Subsequently,\nmagnetization was induced in graphene and spin depen-\ndent phenomena, such as the anomalous Hall effect [4, 5]\nand non-local spin transport [6, 7], were observed. In all\nthese experiments, a spin-dependent current was gener-\nated by an electric field. There is an alternative way to\ngenerate a spin current called spin pumping [8–12]. The\nproximity exchange coupling describes spin transfer at\nthe magnetic interface and a spin current is injected us-\ning ferromagnetic resonance (FMR) from ferromagnetic\nmaterials into the adjacent materials. The generation of\na spin current is experimentally detectable through both\nthe inverse spin Hall effect and modulation of the FMR,\nwhich were experimentally confirmed at magnetic inter-\nfaces between graphene and several magnetic materials\n[13–18].\nThe theory of spin transport phenomena at magnetic\ninterfaces has been formulated based on the Schwinger-\nKeldysh formalism [19], which is applicable to magnetic\ninterfaces composed of a variety of systems, such as a\nparamagnetic metal and a ferromagnetic insulator (FI)\n[20–23], a superconductor and FI [24, 25], and two FIs\n[26, 27]. The modulation of FMR has been investigated\nin several papers. The modulation of Gilbert damping\nwasfound to be proportionalto the imaginarypartofthe\ndynamical spin susceptibility [21, 23–25, 28, 29], which\nmeans that one can detect spin excitations and electronic\nproperties of adjacent materials through the FMR mea-\nsurements. This implies that the FMR measurements\nof FI/graphene bilayer systems allow us to access thespin-dependent properties of graphene in quantum Hall\nregime [30, 31]. However, the modulation of FMR at the\nmagnetic interface between a FI and graphene has not\nbeen investigated and the effect of Landau quantization\non the FMR signal is unclear.\nIn this work, we study the modified magnetization dy-\nnamics of a FI adjacent to graphene. Figure 1 (a) shows\naschematicofthe system. Microwavesareirradiatedand\nthe precession of localized spins is induced. Figure 1 (b)\nand (c) shows the electronic structure of graphene on the\nFI under aperpendicular magneticfield. The spin degen-\neracy is lifted by the exchange coupling at the interface\nand spin-split Landau levels are formed. The densities of\nstates for spin-up and spin-down are shown in the right\npanel; Landau level broadening is included. We find that\nthe modulation of Gilbert damping is proportionalto the\nproduct of the densities of states for spin-up and spin-\ndown, so that the FMR measurements may be used as\na probe of the spin-resolved densities of states. Owing\nto the peak structure of the density of states, the mod-\nulation of Gilbert damping exhibits peak structure and\nan oscillation as a function of Fermi level and magnetic\nfield, which reflects the Landau level structure. One may\ndetermine the exchange coupling constant by analyzing\nthe period of the oscillation.\nModel Hamiltonian .—The totalHamiltonian H(t)con-\nsists of three terms,\nH(t) =HFI(t)+HGr+Hex. (1)\nThe first term HFI(t) describes the bulk FI\nHFI(t) =/summationdisplay\nk/planckover2pi1ωkb†\nkbk−h+\nac(t)b†\nk=0−h−\nac(t)bk=0,(2)\nwhereb†\nkandbkdenote the creation and annihilation\noperators of magnons with momentum k. We assume a\nparabolic dispersion /planckover2pi1ωk=Dk2−/planckover2pi1γB, withγ(<0) the\nelectron gyromagnetic ratio. The coupling between the2\nMicrowaveB(a) System (b) spin splitting \nExchange \ncoupling\nB01230\n-1 \n-2 \n-3 \nDOSE\nup down(c) spin-split Landau level \nkx kyE\nE\nB123\n0-1 \n-2 \n-3 0\nkx kyE\nFIG. 1. (Color online) Schematic picture of the FMR measurem ent and the energy spectrum of graphene in a strong perpen-\ndicular magnetic field. (a) Graphene on a ferromagnetic insu lator substrate. The magnetic field perpendicular to graphe ne\nis applied and the microwave is irradiated to the FI. (b) The s pin degeneracy is lifted by the exchange coupling. (c) The\nperpendicular magnetic field leads to the spin-split Landau level structure. The density of states has a peak structure a nd the\nlevel broadening originating from disorder is included.\nmicrowave and magnons is given by\nh±\nac(t) =/planckover2pi1γhac\n2√\n2SNe∓iΩt, (3)\nwherehacand Ω are the amplitude and frequency of the\nmicrowave radiation, respectively, and Sis the magni-\ntude of the localized spin in the FI. The above Hamilto-\nnian is derived from a ferromagnetic Heisenberg model\nusing the Holstein-Primakoff transformation and the\nspin-wave approximation ( Sz\nk=S−b†\nkbk,S+\nk=√\n2Sbk,\nS−\n−k=√\n2Sb†\nk, whereSkis the Fourier transform of the\nlocalized spin in the FI).\nThe second term HGrdescribes the electronic states\naround the Kpoint in graphene under a perpendicular\nmagnetic field,\nHGr=/summationdisplay\nnXsεnc†\nnXscnXs, (4)\nwherec†\nnXsandcnXsdenote the creation and annihi-\nlation operators of electrons with Landau level index\nn= 0,±1,±2,···, guiding center X, and spin up s= +\nand spin down s=−. The eigenenergy is given by\nεn= sgn(n)√\n2e/planckover2pi1v2/radicalbig\n|n|B, (5)\nwherevis the velocity and the sign function is defined\nas\nsgn(n) :=\n\n1 (n >0)\n0 (n= 0)\n−1 (n <0). (6)\nIn the following, we neglect the Zeeman coupling be-\ntween the electron spin and the magnetic field because\nit is much smaller than the Landau-level separation and\nthe exchange coupling introduced below. In graphene,\nthere are two inequivalent valleys labelled KandK′. Inthis paper, we assume that the intervalley scattering is\nnegligible. This assumption is valid for an atomically flat\ninterface,whichisreasonablegiventherecentexperimen-\ntal setups [4, 17, 18]. Consequently, the valley degree\nof freedom just doubles the modulation of the Gilbert\ndamping.\nThe third term Hexis the exchange coupling at the\ninterface consisting of two terms\nHex=HZ+HT, (7)\nwhereHZdenotes the out-of-plane component of the\nexchange coupling and leads to the spin splitting in\ngraphene,\nHZ=−JS/summationdisplay\nnX/parenleftBig\nc†\nnX+cnX+−c†\nnX−cnX−/parenrightBig\n,(8)\nwithJthe exchange coupling constant. The z-\ncomponent of the localized spin is approximated as\n∝angbracketleftSz\nk∝angbracketright ≈S. The out-of-plane component HZis modeled\nas a uniform Zeeman-like coupling, although in general,\nHZcontains the effect of surface roughness, which gives\noff-diagonal terms. The Hamiltonian HTdenotes the in-\nplane component of the exchange coupling and describes\nspin transfer between the FI and graphene,\nHT=−/summationdisplay\nnX/summationdisplay\nn′X′/summationdisplay\nk/parenleftBig\nJnX,n′X′,ks+\nnX+,n′X′−S−\nk+h.c./parenrightBig\n,\n(9)\nwhereJnX,n′X′,kis the matrix element for the spin trans-\nfer processes and s+\nnX+,n′X′−is the spin-flip operator for\nthe electron spin in graphene.\nModulation of Gilbert Damping .—To discuss the\nGilbert damping, we calculated the time-dependent sta-\ntistical averageof the localized spin under the microwave\nirradiation. The first-order perturbation calculation\ngives the deviation from the thermal average,\nδ∝angbracketleftS+\nk=0(t)∝angbracketright=−h+\nac(t)GR\nk=0(Ω). (10)3\nThe retarded Green’s function is written as\nGR\nk(ω) =2S//planckover2pi1\nω−ωk+iαGω−(2S//planckover2pi1)ΣR\nk(ω),(11)\nwhere we have introduced the phenomenological dimen-\nsionlessdampingparameter αG, calledthe Gilbert damp-\ning constant, which originates from the magnon-phonon\nand magnon-magnon coupling, etc [32–34]. In this pa-\nper, we focus on the modulation of the Gilbert damping\nstemming from the spin transfer processes at the inter-\nface. The self-energy from the spin transfer processes at\nthe interface within second-order perturbation is given\nby\nΣR\nk(ω) =/summationdisplay\nnX/summationdisplay\nn′X′|JnX,n′X′,k|2χR\nn+,n′−(ω).(12)\nThe spin susceptibility is given by\nχR\nn+,n′−(ω) =fn+−fn′−\nεn+−εn′−+/planckover2pi1ω+i0,(13)\nwherefns= 1//parenleftbig\ne(εns−µ)/kBT+1/parenrightbig\nis the Fermi distribu-\ntion function and εns=εn−JSsis the spin-split Landau\nlevel. From the self-energy expression, one sees that the\nmodulation of the Gilbert damping reflects the property\nof the spin susceptibility of graphene. The modulation\nof the Gilbert damping under the microwave irradiation\nis given by [21, 23–25, 28, 29]\nδαK\nG=−2SImΣR\nk=0(ω)\n/planckover2pi1ω, (14)\nwhere the superscript Ksignifies the contribution from\ntheKvalley.\nTo further the calculation, we assume that the ma-\ntrix element JnX,n′X′,k=0is approximated by a constant\nJ0, including detail properties of the interface, that is,\nJnX,n′X′,k=0≈J0. Withthisassumption,theself-energy\nbecomes\nImΣR\nk=0(ω) =−|J0|2π/planckover2pi1ω/integraldisplay\ndε/parenleftbigg\n−∂f(ε)\n∂ε/parenrightbigg\nD+(ε)D−(ε),\n(15)\nwhereDs(ε) is the density of states for spin s=±\nDs(ε) =A\n2πℓ2\nB/summationdisplay\nn1\nπΓ\n(ε−εns)2+Γ2,(16)\nwith magnetic length ℓB=/radicalbig\n/planckover2pi1/(eB) and area of the\ninterface A. Here, we have introduced a constant Γ de-\nscribing level broadening arising from surface roughness\nand impurity scattering. This is the simplest approx-\nimation to include the disorder effect. The density of\nstates shows peaks at the Landau level, which is promi-\nnent when its separation exceeds the level broadening.\nLandau level (JS = 20  meV) δα G [δα 0   10-2 ]\n0.6\nB [T]1.0μ [meV] \n0.4 0.240 \n20 \n-20\n-400\n0.8s=-, n=0\ns=+, n=0123\n-1 \n-2 \n-3 Γ = 1  meV\nkBT = 1  meV (= 11  K)\n6\n03\n0.6\nB [T]1.0μ [meV] \n0.4 0.240 \n20 \n-20\n-400\n0.8\nFIG. 2. (Color online) Modulation of the Gilbert damping\nconstant δαGand spin-split Landau levels as a function of\nthe Fermi level µand the magnetic field B. The spin splitting\nJSis set to 20meV. In the left panel, δαGhas peaks at the\ncrossing points of spin-up and spin-down Landau levels. In\nthe right panel, the blue and red curves identify the spin-up\nand spin-down Landau levels, respectively.\nFinally, the modulation of the Gilbert damping constant\nδαGis derived as\nδαG= 2πgvS|J0|2/integraldisplay\ndε/parenleftbigg\n−∂f(ε)\n∂ε/parenrightbigg\nD+(ε)D−(ε),(17)\nwheregv= 2 denotes the valley degree of freedom.\nFrom this expression, one sees that the modulation of\nthe Gilbert damping is proportional to the product of\nthe densities of states for spin-up and spin-down. There-\nfore, combined with the density of states measurement,\nfor example, a capacitance measurement [35], the FMR\nmeasurement is used to detect the spin-resolved densities\nof states.\nFigure 2 shows the spin-split Landau levels and the\nmodulation of the Gilbert damping δαGas a function of\nthe Fermi level µand the magnetic field B. We use δα0\nas a unit of δαG\nδα0= 2πgvS|J0|2/parenleftbiggA\n2πℓ2\nB1\nmeV/parenrightbigg2\n.(18)\nWe note that δα0(∝B2) depends on the magnetic field.\nBoth the level broadening Γ and the thermal broadening\nkBTare set to 1meV, and JSis set to 20meV [2–4].\nδαGreflects the Landau level structure and has peaks at\ncrossing points of spin-up and spin-down Landau levels.\nThe peakpositions aredetermined bysolving εn+=εn′−\nand the inverse of the magnetic field at the peaks is given\nby\n1\nB=2e/planckover2pi1v2\n(2JS)2/parenleftBig/radicalbig\n|n|−/radicalbig\n|n′|/parenrightBig2\n. (19)\nThe peak structurebecomes prominent when the Landau\nlevel separation exceeds both level and thermal broaden-\ning.4\nΓ = 1 meV\nkBT = 1 meV (= 11 K)\n5 meV (= 57 K)\n10 meV (= 115 K)kBT = 1 meV (= 11 K)\nΓ = 1 meV \n2 meV \n4 meV μ = JS = 20 meV μ = JS = 20 meV (a) (b)\nΔ(1/B)\n3\n1/B [1/T]4δα G [δα 0   10 -2 ]\n2 18\n6\n4\n2\n0\n5 3\n1/B [1/T]4δα G [δα 0   10 -2 ]\n2 18\n6\n4\n2\n0\n5Δ(1/B)\nFIG. 3. (Color online) Quantum oscillation of the modulatio n\nof the Gilbert damping constant δαGas a function of the\ninverse of the magnetic field 1 /B. The Fermi level µand the\nmagnitude of the spin splitting JSare set to 20meV. (a)\nΓ = 1meV and δαGis plotted at several temperatures. (b)\nkBT= 1meV and δαGis plotted for several Γ’s. The period of\nthe oscillation ∆(1 /B) is indicated by double-headed arrows.\nFigure 3 shows the modulation of the Gilbert damping\nδαGas a function of the inverse of the magnetic field\n1/Bwith the Fermi level set to µ= 20meV, where the\nspin-down zeroth Landau level resides. δαGshows peak\nstructure and a periodic oscillatorybehavior. The period\nof the oscillation ∆(1 /B) is derived from Eq. (19) and is\nwritten as\n∆/parenleftbigg1\nB/parenrightbigg\n=2e/planckover2pi1v2\n(2JS)2. (20)\nThe above relation means that the magnitude of the spin\nsplitting JSis detectable by measuring the period of the\noscillation ∆(1 /B). For the peak structure to be clear,\nboth leveland thermalbroadeningmust to be sufficiently\nsmaller than the Landau level separation; otherwise, the\npeak structure smears out.\nDiscussion .—To observe the oscillation of Gilbert\ndamping, at least two conditions must be satisfied. First,\nthe well-separated landau levels have to be realized in\nthe magnetic field where the FMR measurements is fea-\nsible. Second, the FMR modulation caused by the ad-\njacent graphene have to be detectable. The graphene\nLandau levels are observed in recent experiments at 2T\n[36], andrecentbroadbandferromagneticresonancespec-\ntrometer enables the generation of microwaves with fre-\nquencies ≤40GHz and FMR measurements in a mag-\nnetic field ≤2T [37]. The modulation of the FMR\nlinewidth in Permalloy/Graphene [14, 16], yttrium iron\ngarnet/Graphene [17, 18] have been reported by sev-\neral experimental groups, although all of them were per-\nformed at room temperature. Therefore, the above two\nconditionsareexperimentallyfeasibleandourtheoretical\npredictions can be tested in an appropriate experimental\nsetup.Conclusion .—We have studied the modulation of the\nGilbert damping δαGin a ferromagnetic insulator on\nwhich graphene is placed. The exchange coupling at\nthe interface and the perpendicular magnetic field lead\nto the spin-split Landau levels in graphene. We showed\nthatδαGis proportional to the product of the densities\nof states for spin-up and spin-down electrons. Therefore,\nthe spin-resolved densities of states can be detected by\nmeasuring δαGand the total density of states. When the\nFermi level is fixed at a Landau level, δαGoscillates as a\nfunction of the inverse of the magnetic field. The period\nof the oscillation provides information on the magnitude\nof the spin splitting. Our main message is that the FMR\nmeasurement is a probe of spin-resolved electronic struc-\nture. In addition to spin current generation, one may use\ntheFMRmeasurementstodetectthe electronicstructure\nof adjacent materials.\nAcnowledgement WethankJ.Fujimoto, T.Kato,R.\nOhshima, and M. Shiraishi for helpful discussions. 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Titov2, 3\n1Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden\n2Institute for Molecules and Materials, Radboud University Nijmegen, NL-6525 AJ Nijmegen, the Netherlands\n3ITMO University, Saint Petersburg 197101, Russia\n4School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\n5Skolkovo Institute of Science and Technology, Moscow 121205, Russia\n(Dated: November 11, 2019)\nGiant Gilbert damping anisotropy is identi\fed as a signature of strong Rashba spin-orbit coupling\nin a two-dimensional antiferromagnet on a honeycomb lattice. The phenomenon originates in spin-\norbit induced splitting of conduction electron subbands that strongly suppresses certain spin-\rip\nprocesses. As a result, the spin-orbit interaction is shown to support an undamped non-equilibrium\ndynamical mode that corresponds to an ultrafast in-plane N\u0013 eel vector precession and a constant\nperpendicular-to-the-plane magnetization. The phenomenon is illustrated on the basis of a two\ndimensional s-dlike model. Spin-orbit torques and conductivity are also computed microscopically\nfor this model. Unlike Gilbert damping these quantities are shown to reveal only a weak anisotropy\nthat is limited to the semiconductor regime corresponding to the Fermi energy staying in a close\nvicinity of antiferromagnetic gap.\nI. INTRODUCTION\nA gapless character of the spin-wave spectrum in\nisotropic Heisenberg magnets in two dimensions re-\nsults in the homogeneity of magnetic ordering being\ndestroyed by thermal \ructuations at any \fnite tem-\nperatures. In contrast, in van der Waals magnets,\ncharacterized by intrinsic magnetocrystalline anisotropy\nthat stems from spin-orbit coupling1, an ordered mag-\nnetic state can be retained down to a monolayer\nlimit. Two-dimensional (2D) van der Waals magnets\nare currently experiencing a revived attention2{8driven\nby the prospects of gateable magnetism9{12, a con-\ntinuing search for Kitaev materials13,14and Majorana\nfermions15, topologically driven phenomena16as well as\nvarious applications3,4,7. The trade-o\u000b between quan-\ntum con\fnement, nontrivial topology and long-range\nmagnetic correlations determines their unique magneto-\nelectronic properties, in particular a tunable tunneling\nconductance17and magnetoresistance18{20depending on\nthe number of layers in the sample, as well as long-\ndistance magnon transport21.\nFerromagnetic thin \flms have already entered commer-\ncial use in hard drives, magnetic \feld and rotation angle\nsensors and in similar devices7,22,23, while keeping high\npromises for technologically competitive ultrafast mem-\nory elements24and neuromorphic chips25. Moreover, it\nhas recently been suggested that current technology may\nhave a lot to gain from antiferromagnet (AFM) materi-\nals. Indeed, manipulating AFM domains does not induce\nstray \felds and has no fundamental speed limitations\nup to THz frequencies26. Despite their ubiquitousness,\nAFM materials have, however, avoided much attention\nfrom technology due to an apparent lack of control over\nthe AFM order parameter { the N\u0013 eel vector. Switching\nthe N\u0013 eel vector orientation by short electric pulses has\nbeen put forward only recently as the basis for AFM\nspintronics27{29. The proposed phenomenon has beensoon observed in non-centrosymmetric crystals such as\nCuMnAs30{33and Mn 2Au34{36. It should be noted that\nin most cases AFMs are characterized by insulating type\nbehavior37, limiting the range of their potential appli-\ncations, e.g., for spin injection38. Interestingly, antifer-\nromagnetic Mn 2Au possesses a typical metal properties,\ninheriting strong spin-orbit coupling and high conductiv-\nity, and is characterized by collective modes excitations\nin THz range36.\nDespite a lack of clarity concerning the microscopic\nmechanisms of the N\u0013 eel vector switching, these experi-\nments have been widely regarded as a breakthrough in\nthe emerging \feld of THz spintronics26,30,36,39{43. It has\nbeen suggested that current-induced N\u0013 eel vector dynam-\nics in an AFM is driven primarily by the so-called N\u0013 eel\nspin-orbit torques29,32,44{56. The N\u0013 eel spin-orbit torque\noriginates in a non-equilibrium staggered polarization\nof conduction electrons on AFM sublattices29,32,48,50.\nCharacteristic magnitude of the non-equilibrium stag-\ngered polarization and its relevance for the experiments\nwith CuMnAs and Mn 2Au remain, however, debated.\nThe N\u0013 eel vector dynamics in an AFM is also strongly\na\u000bected by an interplay between di\u000berent types of Gilbert\ndampings. Unlike in a simple single-domain ferromagnet\nwith a single sublattice, the Gilbert damping in an AFM\nis generally di\u000berent on di\u000berent sublattices and includes\nspin pumping from one sublattice to another. A proper\nunderstanding of Gilbert damping is of key importance\nfor addressing not only the mechanism of spin pumping\nbut also domain wall motion, magnon lifetime, AFM res-\nonance width and many other related phenomena57{61.\nIt is also worth noting that spin pumping between two\nthin ferromagnetic layers with antiparallel magnetic ori-\nentations share many similarities with Gilbert damping\nin a bipartite AFM62,63.\nA conduction electron mechanism for Gilbert damp-\ning in collinear ferromagnet requires some spin-orbit in-\nteraction to be present. It is, therefore, commonly as-arXiv:1911.03408v1  [cond-mat.str-el]  8 Nov 20192\nFIG. 1. A model of Rashba honeycomb antiferromagnet with\ntwo sublattices, AandB, and on-site exchange interaction\nbetween localized momenta and conduction electrons (see\nEq. 1). The large blue arrow represents the N\u0013 eel vector vector,\nn, that is in general, characterized by non-vanishing in-plane,\nnk, and perpendicular-to-the-plane, n?, components. We re-\nfer to a speci\fc coordinate system with ^xaxis chosen to be\nin the direction of nk.\nsumed that spin-orbit interaction of electrons naturally\nenhances the Gilbert damping. Contrary to this in-\ntuition, we show that Rashba spin-orbit coupling does\ngenerally suppress one of the Gilbert damping coe\u000e-\ncients and leads to the appearance of undamped non-\nequilibrium N\u0013 eel vector precession modes in the AFM.\nSpin dynamics in a bipartite AFM is described in terms\nof two mutually orthogonal vector \felds, namely the vec-\ntorn(t) that is proportional to the N\u0013 eel vector (di\u000berence\nbetween sublattice moments) and the vector m(t) that is\nproportional to the net magnetization (sum of sublattice\nmoments) of a sample. Even though the AFM ground\nstate corresponds to m= 0, it is widely understood that\nno N\u0013 eel dynamics is possible without formation of a small\nbut \fnite nonequilibrium magnetization m. It appears,\nhowever, that Gilbert damping terms associated with the\ntime dynamics of m(t) andn(t) are essentially di\u000berent\nfrom a microscopic point of view.\nIndeed, the Gilbert damping that is proportional to\n@tnis characterized by a coe\u000ecient \u000bn, which is van-\nishing in the absence of spin-orbit interaction, much like\nit is the case in the ferromagnets. This behavior can be\ntraced back to a spin-rotational symmetry of the collinear\nAFM. Indeed, the absolute value of nis conserved up to\nthe orderm2. Thus, the dynamics of the N\u0013 eel vector is\nessentially a rotation that does not change the conduction\nelectron spectrum as far as the spin-rotation invariance\nis present. Breaking the spin-rotation symmetry by spin-\norbit interaction induces, therefore, a \fnite \u000bn, which is\nquadratic with respect to spin-orbit interaction strength.\nIn contrast, the Gilbert damping that is proportional\nto@tmoriginates directly in the conduction electron\nscattering even in the absence of any spin-orbit inter-action. The strength of the damping in a simple sym-\nmetric AFM is characterized by a coe\u000ecient \u000bm, which\nis typically much larger than \u000bn. As a rule, the spin-\norbit interaction tends to suppress the coe\u000ecient \u000bmby\nrestricting the ways in which electrons can damp their\nmagnetic moments. The condition \u000bm\u001d\u000bnhas been\nindeed well documented in a metallic AFM57,59.\nIn this paper, we uncover the microscopic mechanism\nof strong and anisotropic Gilbert damping suppression\ndue to the in\ruence of spin-orbit interaction in a 2D AFM\nmodel on a honeycomb lattice.\nBelow we focus mainly on the AFM in the regime of\ngood metallic behavior, such that the Fermi energy of\nelectrons exceeds by order of magnitude that of an ef-\nfectives-dexchange coupling between electron spins and\nlocalized AFM magnetic momenta. In this case, the tran-\nsition to the highly anisotropic regime takes place pro-\nvided the characteristic spin-orbit energy \u0015exceeds the\nscale ~=\u001c, where\u001cis the electron scattering time. Alter-\nnatively, one may think of characteristic spin-orbit length\nbecoming smaller than the mean free path of conduction\nelectrons. We show here that the splitting of 2D Fermi\nsurfaces by spin-orbit interaction leads to a dramatic sup-\npression of electron spin \rips in certain directions. This\nresults in a strong anisotropy of both Gilbert damping\ntensors ^\u000bnand ^\u000bm, that get some of their principal com-\nponents vanishing. This extreme anisotropy in the damp-\ning leads to essentially undamped N\u0013 eel vector dynamics\nfor certain nonequilibrium modes.\nIn particular, we identify a speci\fc undamped mode\nthat corresponds to perpendicular-to-the-plane magneti-\nzationm/^zand in-plane N\u0013 eel vector n(t)?^z. The\nN\u0013 eel vector corresponding to the mode has a precission\naroundmwith the frequency Jexm=~, whereJexis the\nvalue of the isotropic AFM exchange.\nThe presence of the undamped mode identi\fed here,\nillustrates how lowering the symmetry of the electronic\nbath (by spin-orbit interaction) may induce a conserva-\ntion law in the localized spin subsystem. Based on this\nmicroscopic mechanism we provide qualitative arguments\nin favor of a generality of the giant Gilbert damping\nanisotropy in a 2D metalic AFM with spin-orbit cou-\npling. Even though the undamped mode cannot be asso-\nciated with a single spin-wave or a magnon, its presence\nhas a strong impact on the nonequilibrium N\u0013 eel vector\ndynamics in 2D Rashba AFMs.\nApart from the Gilbert damping our results extend to\ncover conductivity and spin-orbit torques in the Rashba\nhoneycomb AFM model. We also demonstrate how weak\nanisotropy of all these quantities emerge with Fermi en-\nergies approaching the AFM band gap.\nII. PHENOMENOLOGY OF AFM DYNAMICS\nIn this paper, we choose to describe the AFM with a\nclassical Heisenberg model for localized spins SX=SnX\non two sublattices X=A;B. The spins have the same3\nmodulusSand antiparallel directions nA=\u0000nBin the\nground state. The AFM Heisenberg model is coupled to\nan e\u000bective tight-binding model of conduction electrons\n(see Appendix A) by means of exchange interaction,\nHsd=\u0000JX\niX\n\u001b\u001b0Si\u0001\u001b\u001b\u001b0cy\ni\u001bci\u001b0; (1)\nwhereJstands for an s-d-like exchange energy that is the\nsame onAandBsublattices, the operators cy\ni\u001b(ci\u001b) are\nthe standard creation (annihilation) operators for an elec-\ntron on the lattice site iwith the spin index \u001b, and the no-\ntation\u001b= (\u001bx;\u001by;\u001bz) represents the three-dimensional\nvector of Pauli matrices.\nThe real-time dynamics of AFM is, then, de\fned\nby two coupled di\u000berential equations (Landau-Lifshitz-\nGilbert equations) on the unit vectors nAandnB,\n_nA=HA\u0002nA+ (JA=~)nA\u0002sA; (2a)\n_nB=HB\u0002nB+ (JA=~)nB\u0002sB; (2b)\nwhere dot stands for the time derivative, sXis the spin\ndensity of conduction electrons on the sublattice X,\nsA,B(r) =1\n2X\ni\u001b\u001b0D\ncy\ni\u001b\u001b\u001b\u001b0ci\u001b0E2\nA; (3)\nandAis the area of the unit cell in the AFM. The no-\ntationsHA,Brefer to e\u000bective \felds on the sublattices A\nandBthat are de\fned by the Heisenberg model.\nFor an isotropic antiferromagnet, one \fnds an e\u000bec-\ntive \feld28HA+HB=Jexm=~+ 2H, whereHis an\nexternal magnetic \feld in frequency units and Jexis a\ndirect antiferromagnetic exchange energy that is one of\nthe largest energies in the problem. In turn, the combi-\nnationHA\u0000HBis proportional to magnetic anisotropy\nthat we do not specify in this paper.\nMagnetization dynamics in AFM is conveniently for-\nmulated in terms of the N\u0013 eel and magnetization vectors,\nn=\u0000\nnA\u0000nB\u0001\n=2;m=\u0000\nnA+nB\u0001\n=2;(4)\nthat remain mutually perpendicular n\u0001m= 0 and yield\nthe constraint n2+m2= 1. The dynamics necessarily\ninduces a \fnite nonequilibrium magnetization vector m,\nwhile the condition m\u001c1 remains to be ful\flled.\nFrom Eqs. (2) we obtain\n_n=\u0000\nn\u0002m+H\u0002n+n\u0002s++m\u0002s\u0000;(5a)\n_m=H\u0002m+m\u0002s++n\u0002s\u0000; (5b)\nwhere \n = 2 JexS=~ands\u0006=JA(sA\u0006sB)=2~. In\nEqs. (5) we have deliberately skipped terms that are in-\nduced by anisotropy of AFM exchange since the latter\ndepend on particularities of the AFM Heisenberg model\nthat we do not discuss here.\nThe vectors+is proportional to average polariza-\ntion of conduction electrons, while the vector s\u0000is pro-\nportional to the staggered polarization. The quantities\n−2.50.02.5\nvp/∆−2024ε/∆θ=π/2\n−2.50.02.5\nvp/∆θ=π/4K-valley K/prime-valley\n−2.50.02.5\nvp/∆θ= 0FIG. 2. Electronic band structure of the honeycomb AFM\nmodel of Eq. (9) for di\u000berent orientations of the N\u0013 eel vec-\ntor (nz= cos\u0012). Two-dimensional momenta pare measured\nwith respect to the wave-vectors KandK0that specify two\nnonequivalent valleys. Deviation of the N\u0013 eel vector from the\nperpendicular-to-the plane con\fguration ( \u0012= 0) lifts the val-\nley degeneracy. We restrict our analysis to the metallic regime\nwith Fermi energies corresponding to two Fermi surfaces per\nvalley (an example is shown by a black dotted line). The\nenergy scale \u0001 characterizes the strength of s-dexchange in-\nteraction.\ns\u0006=s\u0006\n0+\u000es\u0006contain equilibrium contributions s\u0006\n0that\ncharacterize various interactions induced by conduction\nelectrons. These contributions do renormalize the pa-\nrameters of the AFM Heisenberg model and are not the\nsubject of the present paper.\nThe nonequilibrium contributions \u000es\u0006originate from\nvarious forces applied to conduction electrons. One nat-\nural example is the electric \feld that not only induces an\nelectric current in the sample but also contributes to \u000es\u0006.\nThe electric \feld can be further related to electric current\nby the resistivity tensor. The response of spin densities\nto electric current de\fnes the so-called spin-orbit torques\nin Eqs. (5) that we also compute.\nSimilarly, the response of \u000es\u0006to the time derivatives _n\nand _mdescribe various types of Gilbert damping induced\nby conduction electrons. Quite generally, such a response\ncan be written in the form of a tensor\n\u0012\n\u000es+\n\u000es\u0000\u0013\n=\u0012\n^\u000bm^\u000bmn\n^\u000bnm ^\u000bn\u0013\u0012\n_m\n_n\u0013\n; (6)\nwhere all tensor components may themselves depend on\nthe vectorsnandm.\nGilbert dampings, in their original meaning, corre-\nspond to the contributions to \u000es\u0006that are symmet-\nric under the time reversion. The terms that change\nsign should, more appropriately, be referred to as ef-\nfective spin renormalizations. Both types of terms are,\nhowever, obtained from the microscopic analysis of the\nGilbert damping tensors in Eq. (6) similarly to the case\nof ferromagnets64.4\nTime reversion, mentioned above, applies exclusively\nto the Heisenberg model, while keeping the tight-binding\nmodel (a bath) non-reversed. In other words we do not\nreverse the electron scattering time \u001c. Such a de\fnition\nhelps to identify the dissipative (even with respect to\nthe time reversion) contributions to \u000es\u0006that describe\nGilbert dampings. These contributions must, however,\nchange sign under the transformation \u001c!\u0000\u001c, because\nspin densities s\u0006are always odd with respect to complete\ntime reversion (the one which also includes that of the\nelectron bath). We will see below, indeed, that all Gilbert\ndampings are proportional to the scattering time \u001cin the\nsame way as the longitudinal conductivity does.\nBefore we proceed with the microscopic analysis of \u000es\u0006\nfor a particular model, it is instructive to draw some gen-\neral consequences for Eqs. (5) based on symmetry argu-\nments in the case of collinear AFM with sublattice sym-\nmetry and spin-rotational invariance (i. e. for vanishing\nspin-orbit interaction).\nAssuming that deviations from the AFM ground state\nremain small we shall limit ourselves to the linear order\ninmin Eq. (7a) and to the quadratic order in min\nEq. (7b). Thus, we shall retain terms up to linear order\ninmin the tensors ^ \u000bm, ^\u000bnm, and ^\u000bmnand terms up to\nquadratic order in min ^\u000bn.\nMixing tensors ^ \u000bmnand ^\u000bnmmust be odd in m, which\nimplies, for our precision, a linear in mapproximation.\nAs a result, the sublattice symmetry (the symmetry with\nrespect to renaming AandB) prescribes that the mix-\ning tensors must also be linear in n. In the absence of\nspin-orbit coupling we are also restricted by spin-rotation\ninvariance that (together with the sublattice and time-\nreversion symmetries) dictates the following form of the\nGilbert damping contributions to the non-equilibrium\nspin densities\n\u000es+=\u000bm_m+\u000b0\nmn\u0002(n\u0002_m)+\u000bmnm\u0002(n\u0002_n);(7a)\n\u000es\u0000=\u000bn_n+\u000b0\nnm\u0002(m\u0002_n) +\u000bnmn\u0002(m\u0002_m);(7b)\nwhere all coe\u000ecients are assumed to be constants.\nIt is easy to see that the vector forms n\u0002(m\u0002_n) and\nm\u0002(n\u0002_m), which could have respectively entered the\nspin densities \u000es+and\u000es\u0000, do not contribute to Eqs. (5)\nin the precision explained above. Substitution of Eqs. (7)\ninto Eqs. (5) gives\n_n=\u0000\nn\u0002m+H\u0002n+ \u0016\u000bmn\u0002_m+\u000bnm\u0002_n;(8a)\n_m=H\u0002m+\u000bnn\u0002_n\n+ \u0016\u000bmm\u0002_m+\r(n\u0002m)(n\u0001_m)\u0000\u000b0\nnm2n\u0002_n;(8b)\nwhere \u0016\u000bm=\u000bm\u0000\u000b0\nmand\r=\u000bmn+\u000bnm+\u000b0\nm\u0000\u000b0\nn.\nDiscarding the three last terms in Eq. (8b), which are all\nof the second order in m, we indeed arrive at a set of\nGilbert damping terms that is widely used in the AFM\nliterature57,58,60.\nThe symmetry consideration behind Eqs. (8) has es-\nsentially relied upon the spin-rotation invariance. This\nalso implies \u000bn= 0 as has been pointed out in the in-\ntroductory section. The coe\u000ecient \u000bmcan, in turn, be\fnite and large, even in the absence of spin-orbit inter-\naction. As we will show below, the presence of spin-\norbit interaction does not only provide us with a \fnite\n\u000bnbut also drastically change the symmetry structure of\nEqs. (8). We will demonstrate that the onset of spin-orbit\ninteraction strongly a\u000bects the coupling of the localized\nspin subsystem to the electron bath (described by the\ntight-binding model) resulting in a strong reduction in\nthe ability of conduction electrons to \rip spins in certain\ndirections and, therefore, to impose a friction on magne-\ntization dynamics.\nIn the following, we turn to the microscopic analysis\nof the conductivity (Sec. IV), spin-orbit torques (Sec. V)\nand Gilbert dampings (Sec. VI) in a particular model\nof Rashba honeycomb AFM that has been put forward\nrecently by some of the authors65. Rashba spin-orbit in-\nteraction breaks spin-rotational invariance of the model\nby singling out the direction ^zperpendicular to the 2D\nplane. We, therefore, investigate how such spin-rotation\nbreaking manifests itself in the anisotropy of the above-\nmentioned quantities.\nIII. MICROSCOPIC MODEL\nFor the sake of a microscopic analysis we adopt a sub-\nlattice symmetric s-d-like model of a 2D honeycomb an-\ntiferromagnet with Rashba spin-orbit coupling, that was\nintroduced in Ref. 65. The energy dispersion of this\nmodel is illustrated schematically in Fig. 2. The low en-\nergy model for conduction electrons responsible for the\ndispersion in Fig. 2, is described by an e\u000bective Hamilto-\nnian (see Appendix A) that in a valley-symmetric rep-\nresentation reads\nHe\u000b=vp\u0001\u0006+1\n2\u0015[\u001b\u0002\u0006]^z\u0000\u0001n\u0001\u001b\u0006z\u0003z+V(r):(9)\nHere\u0006,\u0003, and\u001bare the vectors of Pauli matrices in\nsublattice, valley and spin space, respectively, vis the\ncharacteristic Fermi velocity, while \u0015and \u0001 =JSare\nthe energy scales characterizing the strength of Rashba\nspin-orbit coupling and s-d-like exchange energy, corre-\nspondingly.\nThe termV(r) stands for a scalar Gaussian white-noise\ndisorder potential, which is proportional to the unit ma-\ntrix in sublattice, valley and spin space. The potential\nhas a zero mean value hV(r)i= 0 and is fully character-\nized by the pair correlator,\nhV(r)V(r0)i= 2\u0019(~v)2\u000bd\u000e(r\u0000r0); (10)\nwhere the angular brackets denote the averaging over dis-\norder realizations. The dimensionless parameter \u000bd\u001c1\nquanti\fes the disorder strength.\nThe disorder potential is responsible for a momentum\nrelaxation of conduction electrons. Exchange interaction\nand spin-orbit scattering (or the scattering on a non-\ncollinear con\fgurations with m6= 0) enable coupling be-\ntween localized angular momenta and kinetic momenta5\nof electrons. Together these mechanisms form a channel\nto dissipate angular momentum of localized spins into\nthe lattice. Thus, our model provides us with a micro-\nscopic framework to study dissipative quantities such as\nGilbert dampings, anti-damping spin-orbit torques and\nconductivity that we compute below. We also note that\nthe computation of spin-relaxation time can be directly\nrelated to our analysis of Gilbert damping66,67.\nThe spectrum of the model (9) with V(r) = 0 consists\nof two electron and two hole branches for each of the\nvalleys as illustrated in Fig. 2,\n\u000fe\n\u0006;&(p) =p\nv2p2+ \u00012\u0006&\u0015\u0001nz+\u00152=4\u0007\u0015=2;(11a)\n\u000fh\n\u0006;&(p) =\u0000p\nv2p2+ \u00012\u0007&\u0015\u0001nz+\u00152=4\u0006\u0015=2;(11b)\nwhere&=\u0006is the valley index. All spectral branches\nare manifestly isotropic with respect to the direction of\nthe electron momentum pirrespective of the N\u0013 eel vector\norientation (as far as m= 0).\nIn order to limit the complexity of our microscopic\nanalysis we restrict ourselves to the metallic regime that\ncorresponds to the Fermi energy \"F>\u0001 +\u0015above\nthe minimum of the top electron branches, \u000fe\n+;&(p), as\nshown schematically in Fig. 2. Note that the Fermi en-\nergy\"Fis counted in the model from the center of the\nAFM gap. We also focus on the limit of weak disorder\n\"F\u001c=~\u001d1 where\u001c=~=(\u0019\u000bd\"F) stands for the electron\nscattering time. Also, in order to describe spin-orbit in-\nduced anisotropy we \fnd it convenient to decompose the\nN\u0013 eel vector (as well as the magnetization vector) to the\nin-plane and perpendicular-to-the-plane components as\nn=nk+n?, wheren?=nz^z.\nIV. CONDUCTIVITY\nThe electric conductivity in the metallic regime is dom-\ninated by electron di\u000busion. Despite the fact that the\nFermi surface (line) is isotropic and does not depend on\nthe direction of nk, the conductivity appears to be weakly\nanisotropic with respect to in-plane rotations of the N\u0013 eel\nvector due to the onset of spin-orbit interaction. In par-\nticular, fornz= 0 we \fnd the diagonal conductivity com-\nponents\n\u001bxx=4e2\nh\"F\u001c\n~\"2\nF\u0000\u00012\n\"2\nF+ 3\u00012; (12a)\n\u001byy=\u001bxx+4e2\nh\"F\u001c\n~\"2\nF\n\"2\nF+ \u00012\u00152\u00012\n\"4\nF+\"2\nF\u00012+ 2\u00014;\n(12b)\nwhere the principal axes correspond to choosing ^xdirec-\ntion alongnk(see Fig. 1). In the deep metal regime, and\nfor a general direction of n, this anisotropy is evidently\nsmall\n\u001axx\u0000\u001ayy\n\u001axx+\u001ayy=\u00152\u00012\n\"4\nF(1\u0000n2\nz); \"F\u001d\u0015+ \u0001; (13)where\u001aaa= 1=\u001baais the corresponding resistivity tensor\ncomponent. We note that the anomalous Hall conduc-\ntivity is identically vanishing in the model of Eq. (9).\nThe results of Eq. (12) and all subsequent results of our\npaper are technically obtained from linear response Kubo\nformulas evaluated in the di\u000busive approximation (ladder\ndiagram summation). The details of these calculations\ncan be found in Appendixes B, C, and D.\nV. SPIN-ORBIT TORQUE\nBefore proceeding with the microscopic analysis of\nGilbert damping we shall discuss the e\u000bects of spin-orbit\ninduced anisotropy for spin-orbit torques in the model of\nEq. (9). Since this anisotropy appears to be weak in the\nmetal regime, we shall touch on it only brie\ry.\nAs was already mentioned, the spin-orbit torques origi-\nnate in the response of nonequilibrium spin polarizations\n\u000es\u0006to electric current. Technically, we compute \frst\nthe response of \u000es\u0006to electric \feld and, then, express\nthe electric \feld in terms of 2D electric current density j\nusing the conductivity tensor of Eq. (12).\nA straightforward computation of such a response gives\n\u000es\u0000= 0 (see Appendixes B and C for more detail) and\n\u000es+=a(n2\nz)^z\u0002j+b(n2\nz)nk\u0002(nk\u0002(^z\u0002j))\n+c(n2\nz)nk\u0002(n?\u0002(^z\u0002j)); (14)\nwhere the coe\u000ecients a,bandcdo generally depend on\nn2\nz= 1\u0000n2\nx\u0000n2\nyand are shown in Fig. 3. It is appropriate\nto recall here that the computation of the responses from\nthe model of Eq. (9) refers to the case when m= 0. The\nsymmetry form of Eq. (14) in this case has been also\nestablished recently from numerical simulations65.\nImportantly, the \frst term in the right-hand side of\nEq. (14) represents the well-known Rashba-Edelstein\ne\u000bect68, while the other two terms represent higher har-\nmonics of the same \feld-like e\u000bect that arise due to spin-\nrotation symmetry breaking. Anti-damping like torques\n(that are even under time-reversal) are vanishing in the\nmodel due to the valley symmetry constraint. This sym-\nmetry reads \u0003 xH[n]\u0003x=H[\u0000n], from which it follows\nthat the response of \u000es+to charge current must be an\neven function of n.\nThe behavior of the coe\u000ecients a,bandcas a function\nofnzis illustrated in Fig. 3 for two di\u000berent choices of\nthe Fermi energy. For in-plane N\u0013 eel vector orientations\n(nz= 0) we \fnd\na=a01 + 3\u000e2\n1 + 2 \u0016\u00152\u000e2+\u000e4\u00002\u000e6; (15a)\nb= 2a\u000e21\u00002\u0016\u00152\u00004\u000e2+\u000e4\n1 + 2\u000e2\u00003\u000e4; (15b)\nc=\u00002a\u000e21 + 2 \u0016\u00152\u000e2\u00002\u000e2\u00003\u000e4\u00004\u000e6\n1 + 4\u000e2+ 5\u000e4+ 6\u000e6; (15c)6\nFIG. 3. The coe\u000ecients a,b, andcin Eq. (14) as a function of\nthe direction of the N\u0013 eel vector, nz= cos\u0012, for two di\u000berent\nFermi energies: \"F= 4\u0001 (left panel) and \"F= 16\u0001 (right\npanel). We use \u0015= 1:5\u0001. Fornz= 0 the results correspond\nto Eqs. (15).\nwhere\na0=AJ\ne~v\u0015\n\"F; \u0016\u0015=\u0015\n\"F; \u000e =\u0001\n\"F: (16)\nIn the metal regime, \"F\u001d\u0015+\u0001, the results of Eqs. (15)\nare reduced to\na=AJ\ne~v\u0015\n\"F; b =\u0000c= 2AJ\ne~v\u0015\n\"F\u0012\u0001\n\"F\u00132\n:(17)\nOne can, therefore, see that the high harmonics terms\n(proportional to bandc) become irrelevant in the metal\nregime.\nVanishing response of the staggered polarization,\n\u000es\u0000= 0, for the model of Eq. (9) is a simple consequence\nof the sublattice symmetry. As shown below the presence\nof a \fnite, though small, mbreaks such a symmetry and\nleads to a \fnite \u000es\u0000. Taking into account a linear in m\nterm in the Hamiltonian is also necessary to obtain \fnite\nmixed Gilbert damping tensors ^ \u000bnmand ^\u000bmnin Eq. (6).\nA low-energy model that takes into account \fnite mag-\nnetization vector reads (see also Appendix D)\nH=He\u000b\u0000\u0001m\u0001\u001b; (18)\nwhereHe\u000bis given by Eq. (9). The conductivity tensor\ndoes not acquire a linear in mterms in the leading order\nwith respect to the large metal parameter \"F\u001c=~, because\nthe anomalous Hall e\u000bect remains subleading with respet\nto the metal parameter. Similarly, the result of Eq. (14)\nis not a\u000bected by the linear in mcorrections.\nHowever, the direct computation of the staggered po-\nlarization response (in the linear order with respect to m)\ngives rise to a \fnite result. In the limit of large Fermi\nenergy\"F\u001d\u0015+ \u0001, we \fnd\n\u000es\u0000=AJ\ne~v\u0015\n\"F\u0012\u0001\n\"F\u00132h\n2n?\u0002(m?\u0002(^z\u0002j)) (19)\n+ 2mk\u0002(n?\u0002(^z\u0002j))\u00003nk\u0002(m?\u0002(^z\u0002j))i\n;where the overall strength of the e\u000bect is again of the\norder of the coe\u000ecients bandc. This makes the e\u000bects\nof nonequilibrium staggered polarization (including the\ncelebrated N\u0013 eel spin-orbit torque) irrelevant in the metal\nregime. Indeed, staggered polarization can hardly be in-\nduced by electrons with wavelengths that strongly exceed\nthe distance between AandBsublattices.\nThe results of Eqs. (14), (15) clearly suggest that the\nonly torques surviving in the large energy limit are those\nrelated to non-equilibrium polarization \u000es+=a0^z\u0002j,\nwhich is nothing but the standard Rashba-Edelstein\ne\u000bect68. These torques have a form Tn=a0n\u0002(^z\u0002j) in\nthe right-hand side of Eq. (5a) and Tm=a0m\u0002(^z\u0002j)\nin the right-hand side of Eq. (5b). The anisotropy of\ntorques is, however, irrelevant in this limit.\nVI. GILBERT DAMPING\nSurprisingly, the situation is di\u000berent when we consider\nGilbert damping terms. In this case we \fnd that the gi-\nant anisotropy of Gilbert damping persists to arbitrarily\nlarge Fermi energy as soon as spin orbit energy \u0015exceeds\n~=\u001c. The latter condition ensures that the scattering be-\ntween spin-split subbands is suppressed.\nThe direct computation of the Gilbert damping tensors\nfor\u0015\u001d~=\u001cgives\n\u000es+=\u000bk\nm_mk+\r\u0000mm+\r\u0000mn; (20a)\n\u000es\u0000=\u000b?\nn_n?+\r\u0000nm+\r\u0000nn; (20b)\nwhere the terms \u0000 abcontain various vector forms.\nFar in the metal regime, \"F\u001d\u0015+ \u0001, we \fnd\n\u000bk\nm= 2\"F\u001c\n~AJ2S\n\u0019~2v2\u0014\n1\u0000\u00012\n\"2\nF(2 +n2\nz) +:::\u0015\n;(21a)\n\u000b?\nn=\"F\u001c\n~AJ2S\n\u0019~2v2\"\u0012\u0015\n\"F\u00132\n+:::#\n; (21b)\n\r= 2\"F\u001c\n~AJ2S\n\u0019~2v2\"\u0012\u0001\n\"F\u00132\n+:::#\n; (21c)\nwhile the vectors forms \u0000 abcan be written as\n\u0000mm=n\u0002(n\u0002_m) +nk\u0002(nk\u0002_mk)\n\u00002nk\u0002(nk\u0002_m?); (22a)\n\u0000mn=n\u0002(mk\u0002_n?)\u0000m?\u0002(nk\u0002_n?)\n+n?\u0002(m?\u0002_nk)\u0000nk\u0002(mk\u0002_nk)\n\u00003m?\u0002(n?\u0002_nk); (22b)\n\u0000nm= 2nk\u0002(mk\u0002_m?) + 2mk\u0002(n?\u0002_mk)\n\u0000n?\u0002(m?\u0002_m) + 2m?\u0002(n\u0002_m)\n+mk\u0002(n\u0002_m?)\u0000m?\u0002(n?\u0002_mk);(22c)\n\u0000nn=\u0000m\u0002(n\u0002_nk): (22d)\nThus, we see from Eqs. (21) that the coe\u000ecients \u000b?\nnand\n\rare vanishingly small in the metal regime. Moreover,7\nin the limit \"F\u001d\u0001 the only non-vanishing contribu-\ntions to Gilbert dampings are given by the \frst terms\non the right-hand sides of Eqs. (20) that are manifestly\nanisotropic.\nThe onset of spin-orbit interactions therefore makes\nGilbert dampings ultimately anisotropic, also in the deep\nmetal regime. This is in contrast to conductivity and\nspin-orbit torques that are quickly becoming isotropic in\nthe metal limit. For \"F\u001d\u0015+ \u0001, we \fnd the well known\nLandau-Lifshitz-Gilbert equations\n_n=\u0000\nn\u0002m+H\u0002n+ \u0016\u000bk\nmn\u0002_mk+\u000b?\nnm\u0002_n?;\n_m=H\u0002m+\u000b?\nnn\u0002_n?+ \u0016\u000bk\nmm\u0002_mk; (23)\nwhere we again omit terms that originate e. g. from mag-\nnetic anisotropy of the AFM. Eqs. (23) are clearly dif-\nferent from Eqs. (8) derived on the basis of symmetry\nanalysis in the absence of spin-orbit interaction.\nThe very pronounced, highly anisotropic Gilbert\ndamping terms in the Landau-Lifshitz-Gilbert equations\nof Eqs. (23) represent the main result of our paper. The\nphenomenon of the giant Gilbert damping anisotropy in\nthe 2D AFM clearly calls for a qualitative understanding\nthat we provide in Sec. VII.\nVII. QUALITATIVE CONSIDERATION\nThe results of Eqs. (20), (21) suggest that the\nanisotropy of Gilbert damping is most pronounced in the\nmetal limit, \"F\u001d\u0001 +\u0015as far as\u0015\u001c=~\u001d1. In partic-\nular, certain spin density responses are vanishing in this\nlimit. One of them is the response of the average spin\ndensity\u000es+\nzto _mzthat is de\fned by the tensor com-\nponent\u000bzz\nmin Eq. (6). The other four vanishing tensor\ncomponents \u000bxx\nn,\u000bxy\nn,\u000byx\nnand\u000byy\nncorrespond to the re-\nsponses of the in-plane staggered spin densities \u000es\u0000\nxand\n\u000es\u0000\nyto _nxand _ny.\nIt is important to stress that the component \u000bzz\nmis not\nonly \fnite but also quite large in the absence of spin-orbit\ninteraction, i. e. for \u0015= 0. It is, therefore, instructive to\nunderstand how the onset of spin-orbit interaction may\ncancel\u000bzz\nmresponse and lead to the conservation of z\nprojection of magnetization vector.\nSuch a qualitative understanding can be achieved by\nconsidering the Kubo-Greenwood formula for \u000bzz\nmfor the\nmodel of Eq. (18) in the limit \u0001 !0 and\u001c!1 ,\n\u000bzz\nm/X\npX\ns;s0=\u0006jh\tp;sj\u001bzj\tp;s0ij2\u000e(\"F\u0000\u000fe\np;s)\u000e(\"F\u0000\u000fe\np;s0);\n(24)\nwhere\u000fe\np;\u0006=p\nv2p2+\u00152=4\u0007\u0015=2 correspond to the two\nelectronic branches of Eq. (11a) that are evidently valley\ndegenerate in the limit \u0001 !0.\nThe states \t p;sare simply the eigenstates of theHamiltonian H0=vp\u0001\u0006+ (\u0015=2) [\u001b\u0002\u0006]z,\nH0=0\nB@0 0 v(px\u0000ipy) 0\n0 0 \u0000i\u0015 v (px\u0000ipy)\nv(px+ipy)i\u0015 0 0\n0v(px+ipy) 0 01\nCA;\n(25)\nthat can be explicitly written as\n\tp;\u0006=1\n2p\nv2p2\u0007\u0015\u000fe\n\u0006=20\nBB@vpe\u0000i\u001e\n\u0006i\u000fe\n\u0006\n\u000fe\n\u0006\n\u0006ivpei\u001e1\nCCA; (26)\nwhere we have used px=pcos\u001e,py=psin\u001e.\nOne may notice that h\tp;sj\u001bzj\tp;si= 0 for any value\nof\u0015suggesting that the response function \u000bzz\nmin Eq. (24)\nis vanishing. This is, however, not the case for \u0015= 0. In-\ndeed, in the absence of spin-orbit interaction the electron\nbranches become degenerate \u000fe\np;\u0006=vpsuch that the in-\nplane spin-\rip processes contribute to the Kubo formula,\nh\tp;+j\u001bzj\tp;\u0000ij\u0015=0=h\tp;\u0000j\u001bzj\tp;+ij\u0015=0= 1:(27)\nThese processes are exactly the ones responsible for a\n\fnite Gilbert damping component \u000bzz\nmin the absence of\nspin-orbit interaction. The spin-orbit induced splitting\nof the subbands forbids these spin-\rip processes as soon\nas\u0015\u001c=~\u001d1 and leads to a giant anisotropy of Gilbert\ndamping in the metal limit. Indeed, the other elements\nof the Gilbert damping tensor \u000bxx\nmand\u000byy\nmremain \fnite\nirrespective of the subband splitting,\nh\tp;\u0006j(\u001bx+i\u001by)j\tp;\u0006i=\u0006ivpei\u001e\np\nv2p2+\u00152=4: (28)\nOne can further show that for \u0015= 0 the entire Gilbert\ndamping tensor ^ \u000bmbecomes isotropic ^ \u000bxx\nm= ^\u000byy\nm= ^\u000bzz\nm\nas it have been expected on the basis of the symmetry\nanalysis.\nVery similar physics is also responsible for the\nanisotropy of the tensor ^ \u000bn. It is worth noting that\nthe same type of anisotropy is known to take place in\nthe limit of large spin-orbit interaction in 2D Rashba\nferromagnets64. Spin-orbit induced anisotropy of Gilbert\ndamping plays, however, a lesser role in 2D ferromagnets\ndue to the much stricter constraint on the single mag-\nnetization vector. A less restricted dynamics of mand\nnvectors make the Gilbert damping anisotropy play a\nbigger role in 2D AFMs.\nIndeed, it can be directly seen from Eqs. (23) that a\nnonequilibrium state with m=m^zandn=nkbecomes\nundamped in the absence of external \feld H= 0. Such\na state corresponds to the undamped N\u0013 eel vector pre-\ncession around ^zaxis with a frequency given by Jexm.\nThe state clearly survives in the presence of easy plane\nmagnetic anisotropy in the AFM. We believe that such\na phenomenon remains to be rather generic for a vari-\nety of 2D or layered AFM systems with strong spin-orbit\ncoupling of Rashba type.8\nVIII. CONCLUSIONS\nIn this paper, we demonstrate that the presence of suf-\n\fciently strong spin-orbit coupling \u0015\u001c=~\u001d1 results in\nthe ultimate anisotropy of the Gilbert damping tensor\nin the metal regime, \"F\u001d\u0001 +\u0015. One can trace the\nphenomenon to the spin-orbit induced splitting of Fermi\nsurfaces that forbids scattering processes that are respon-\nsible for the relaxation of certain magnetization and N\u0013 eel\nvector components.\nWe also demonstrate that a \fnite in-plane projection\nnkof the N\u0013 eel vector is responsible for a weak anisotropy\nof conductivity and spin-orbit torques for Fermi energies\napproaching the band edge, \"F\u0018\u0001+\u0015. This anisotropy\nis, however, absent in the metallic regime.\nGilbert damping is, however, in the absence of spin-\norbit interaction as it is required by symmetry consider-\nations. Thus, we demonstrate that the onset of Rashba\nspin-orbit interaction in 2D or layered AFM systems\nleads to a giant anisotropy of Gilbert damping in the\nmetallic regime. The physics of this phenomenon origi-\nnates in spin-orbit induced splitting of the electron sub-\nbands that destroys a particular decay channel for mag-\nnetization and leads to undamped precession of the N\u0013 eel\nvector. The phenomenon is based on the assumption that\nother Gilbert damping channels (e. g. due to phonons)\nremain suppressed in the long magnon wavelength limit\nthat we consider. The predicted giant Gilbert damp-\ning anisotropy may have a profound impact on the N\u0013 eel\nvector dynamics in a variety of 2D and layered AFM ma-\nterials.\nACKNOWLEDGMENTS\nWe are thankful to I. Ado, H. Gomonay and J. Sinova\nfor fruitful discussions. This research was supported by\nthe JTC-FLAGERA Project GRANSPORT. D.Y. and\nM.T. acknowledge the support from the Russian Science\nFoundation Project No. 17-12-01359. A.P. acknowledges\nsupport from the Russian Science Foundation Project\n18-72-00058. The work of D.Y. was also supported by\nthe Swedish Research Council (Vetenskapsr\u0017 adet, 2018-\n04383). M.T. is especially thankful to the KITP visitor\nprogram \\Spintronics Meets Topology in Quantum Ma-\nterials\". O.E. acknowledges support from the Swedish\nResearch Council (Vetenskapsr\u0017 adet) and the Knut and\nAlice Wallenberg foundation.\nAppendix A: Model system\nIn this Appendix, we shall brie\ry justify Eqs. (9) and\n(18) of the main text. We start from an s-d-like model\nfor two-dimensional antiferromagnet on a honeycomb\nlattice65. The model includes a local exchange interac-\ntion between localized magnetic moments and conduction\nelectron spins as given by Eq. (1). Itinerant electrons inthe model are, therefore, governed by the tight-binding\nHamiltonian\nH0=\u0000tX\niX\n\u001b\u001b0cy\ni\u001bci\u001b0\u0000JX\niX\n\u001b\u001b0Si\u0001\u001b\u001b\u001b0cy\ni\u001bci\u001b0\n+i\u0015\n3aX\nhi;jiX\n\u001b\u001b0^z\u0001(\u001b\u0002dij)\u001b\u001b0cy\ni\u001bcj\u001b0; (A1)\nwhere we do ignore disorder for a moment. The model\nis characterized by the nearest neighbor hopping energy\ntand the Rashba spin-orbit coupling energy \u0015,z-axis is\naligned perpendicular to the two-dimensional plane, the\nin-plane vectors dijconnect the neighboring sites iandj\non a honeycomb lattice. For any site ion the sublattice\nAwe choose\nd1=a\u0012\n0\n1\u0013\n;d2=a\n2\u0012p\n3\n\u00001\u0013\n;d3=\u0000a\n2\u0012p\n3\n1\u0013\n;\n(A2)\nwhereais the length of the bond between AandB.\nBy projecting the tight-binding model of Eq. (A1) on\nstates in a vicinity of the valley wave-vectors,\nK=4\u0019\n3p\n3a\u0012\n1\n0\u0013\n;andK0=\u0000K; (A3)\nwe \fnd, in the valley symmetric approximation, the ef-\nfective Hamiltonian of Eq. (9) with the assumption that\nSA=\u0000SB, wherev= 3ta=2~. By relaxing the assump-\ntion we obtain the model of Eq. (18).\nAppendix B: Linear Response tensors\nIn order to keep technical expressions compact we let\n~= 1 and\"F=\"below. Our technical analysis is based\non linear response of electron spin density to various per-\nturbations at zero frequency ( dc) limit. In particular, we\nconsider three types of responses: the one with respect\nto electric current (via electric \feld and inverse conduc-\ntivity tensor), the one with respect to the time derivative\nof the N\u0013 eel vector and the other one with respect to the\ntime derivative of magnetization vector. These responses\nare summed up as\n\u000es+=^SSOT\n+j+^SGD\nmn_n+^SGD\nm_m; (B1a)\n\u000es\u0000=^SSOT\n\u0000j+^SGD\nnm_m+^SGD\nn_n; (B1b)\nwhere we de\fne the response tensors ^SSOT\n\u0006 that are\ndescribing spin-orbit torques (both \feld-like and anti-\ndamping) and various ^SGDtensors that are describing\nvarious contributions to Gilbert dampings (and to e\u000bec-\ntive spin renormalizations)64.\nIn order to compute the linear response tensors in\nEqs. (B1) we apply the standard Kubo formula\n\u000es\u0006\n\u000b=J2Sv2A\n2VX\n\fcTrD\n^GR^s\u0006\n\u000b^GA^F\fE@X\f\n@t; (B2)9\nwhereVis the system area, cTr is an operator trace,\n^GR(A)= (\"\u0000H\u0006i0) are retarded (advanced) Green func-\ntion operators, ^ s+\n\u000b=\u001b\u000b, ^s\u0000\n\u000b= \u0003z\u0006z\u001b\u000bare the operators\ncorresponding to the average spin-polarization s+and\nstaggered spin-polarization s\u0000, the product ^F\u0001X(t) rep-\nresents the time-dependent perturbation in the Hamil-\ntonian, while the angular brackets denote the disorder\naveraging that we consider in di\u000busive (ladder) approxi-\nmation.\nThe linear-response formula Eq. (B2) assumes zero\ntemperature and zero frequency limit that corresponds to\ntaking both Green's functions at the same energy \"=\"F.\nWe also neglect the Fermi-sea contribution (also known\nas St\u0014 reda contribution) since such a contribution appears\nto be either zero or subleading in the metal parameter\n\"\u001c\u001d1 with respect to our results.\nThus, in order to compute Gilbert dampings and spin-\norbit torque tensors we consider linear response of \u000es\u0006\nto the three perturbations mentioned above. Each per-\nturbation is parameterized by the term \u000eH=^F\u0001X(t)\nwith\n_X=_n; ^F=\u0000\u0001 \u0003z\u0006z\u001b; (B3a)\n_X=_m; ^F=\u0000\u0001\u001b; (B3b)\n_X= (\u0019v=e )^\u001b\u00001j;^F=\u0006; (B3c)\nwhere ^\u001bis the conductivity tensor (this is computed from\nthe standard Kubo formula which is analogous to the one\nin Eq. (B2) but for the response of current density to\nelectric \feld). The disorder averaging amounts to replac-\ning Green's functions in Eq. (B2) with the corresponding\ndisorder-averaged Green's functions and to replacing one\nof the operators, ^ s\u000bor^F, with the corresponding vertex-\ncorrected operator.\nDisorder-averaged Green's functions become diagonal\nin the momentum space due to restored translational in-\nvariance and take the form GR(A)\np = [\"\u0000H\u0000\u0006R(A)]\u00001,\nwhere the Hamiltonian His de\fned in Eq. (9) of the\nmain text, while the self-energy \u0006R(A)is evaluated in the\nBorn-approximation depicted schematically in Fig. 4a.\nWe \fnd that the real part of the self-energy does renor-\nmalize the Fermi energy \"and thes-dexchange coupling\nstrength \u0001, while the imaginary part reads\nIm \u0006R(A)=\u0007\u0019\u000bd\n2(\"\u0000\u0001 \u0003z\u0006zn\u0001\u001b): (B4)\nIn order to evaluate linear response tensors in the lead-\ning order with respect to the metal parameter \"\u001c\u001d1\none also needs to sum up the ladder diagrams as shown\nin Fig. 4b-c.\nTo do that one de\fnes the vertex corrected operator\n^Fvc=^F+^F(1)+^F(2)+^F(3)+\u0001\u0001\u0001; (B5)\nwhere we denote by ^F(i)the operator ^Fthat is dressed\nby the number of idisorder lines,\n^F(i)= 2\u0019\u000bdZd2p\n(2\u0019)2GR\np^F(i\u00001)GA\np: (B6)\nFIG. 4. Diagrammatic illustration. a) Born-approximation.\nb) Ladder-approximation. c) Disorder-averaged polarization\nbubble. d) Perturbative expansion of the disorder-averaged\npolarization bubble.\nIt appears that the summation in Eq. (B5) can be re-\nduced to geometric series in a \fnite operator space. In-\ndeed, let us de\fne the operator space that is spanned by\n16 operators in each of the valleys\nBi=1\n2\u0003\u0010\u0006\u000b\u001b\f; i=f\u0010;\u000b;\fg; (B7)\nwhereiis a cumulative index with \u0010= 0;za valley parity\nindex and\u000b;\ftaking on the four values f0;x;y;zgeach.\nForB= (B1;B2;:::;B 32) we de\fne the vertex cor-\nrected operator vector as\nBvc=B+FB+F2B+F3B+\u0001\u0001\u0001=1\n1\u0000FB;(B8)\nwhereFstands for a matrix of vertex corrections. Using\nthe normalization condition Tr BiBj= 2\u000eijwe \fnd\nFij=\u0019\u000bdZd2p\n(2\u0019)2Tr\u0002\nGA\npBiGR\npBj\u0003\n; (B9)\nwhere Tr stands for the usual matrix trace in the valley,\nspin and sublattice spaces.\nIt easy to imagine that the matrix inversion in Eq. (B8)\nmight be a daunting analytical task. We note, however,\nthat the matrix Fis evidently diagonal in the valley\nspace, and it can also become block-diagonal in sublattice\nand spin spaces by choosing a more convenient basis.\nA particularly useful choice of basis corresponds to in-\nplane rotation of both spin and sublattice Pauli matrices\nto the frame associated with the in-plane projection nk\nof the N\u0013 eel vector. For spin Pauli matrices this transfor-\nmation is given by\n\u001bx!\u0003znx\u001bx+ny\u001byq\nn2x+n2y; \u001by!\u0003zny\u001bx\u0000nx\u001byq\nn2x+n2y;(B10)\nwhere we took advantage of the fact that the direction of\nnis opposite in the two valleys. The same transformation\n(B10) has to be applied to \u0006 xand \u0006y.\nThe matrixFis instrumental for the analysis of all\nlinear response tensors in Eq. (B1). Indeed, using the10\nde\fnition of Eq. (B9) in Eq. (B2) and summing up the\ndi\u000busion ladders we \fnd\n\u000es\u0006\n\u000b=J2Sv2A\n2\u0019\u000bdX\n\fX\nijTr[^s\u0006\n\u000bBi]RijTr[^F\fBj]@X\f\n@t;\n(B11)\nwhereR=F(1\u0000F)\u00001. Thus, the computation of all\nresponse tensors is reduced in the di\u000busive approximation\nto the computation of the vertex correction matrix Fand\nsubsequent matrix inversion.\nAppendix C: Vertex correction\nStill, \fnding an inverse matrix (1 \u0000F)\u00001is not that\nstraightforward due to a pair of eigenvalues (one per val-\nley) that equal exactly 1. The presence of such eigenval-\nues roots in the particle conservation and is, therefore,\nnot an arti\fcial problem. The unit eigenvalues do evi-\ndently prevent the matrix inversion in Eq. (B8). Nev-\nertheless, it can be shown that the corresponding eigen-\nvectors do not enter the \fnal equations of motion for\nlocalized spins. In the next section, we brie\ry illustrate\nhow one can formally avoid the particle conservation di-\nvergence in the computation of vertex corrections.\nLet us de\fne by a\u0010the eigenvectors of Fthat corre-\nspond to two unit eigenvalues, Fa\u0010=a\u0010, with\u0010= 0;z.\nFor the normalized vector a\u0010we de\fne special operators\n\u0016B\u0010=a\u0010\u0001B=\"\u0000\u0001\u0003\u0010\u0006zn\u0001\u001b\n2p\n\"2+ \u00012; (C1)\nwhich are conserved with respect to impurity dressing\n\u0016B\u0010=\u0016B(i)\n\u0010for any order i. This means that the vertex\ncorrected operator \u0016Bvc\n\u0010is formally diverging in the dc\nlimit. In what follows, we formally write \u0016Bvc\n\u0010=R1\u0016B\u0010,\nwhere the limit R1! 1 is taken at the end of the\ncalculation.\nThe response tensors de\fned by Eqs. (6) consist of\ndi\u000berent correlators of the operators \u0006 \u000b,s+\n\u000b=\u001ba, and\ns\u0000\n\u000b= \u0003z\u0006z\u001b\u000b. It is evident that most of these operators\nare already orthogonal to \u0016B\u0010,\nTr\u0002\n\u0006\u000b\u0016B\u0010\u0003\n= Tr\u0002\ns+\n\u000b\u0016B\u0010\u0003\n= Tr\u0002\ns\u0000\n\u000b\u0016B0\u0003\n= 0; (C2)\nwhile the only dangerous sector is related to the projec-\ntion\nTr\u0002\ns\u0000\n\u000b\u0016Bz\u0003\n=\u00004\u0001n\u000bp\n\"2+ \u00012; (C3)\nwhich is evidently \fnite. The result of Eq. (C3) leads\nto formally diverging contribution \u000es\u0000\ndivthat is generally\npresent in all components of \u000es\u0000,\n\u000es\u0000\ndiv;\u000b/R1X\n\fTr[^s\u0000\n\u000b\u0016Bz] Tr[^F\f\u0016Bz]@n\f\n@t: (C4)One can immediately see, however, that such a diverging\ncontribution corresponds to a particular vector form,\n\u000es\u0000\ndiv;\u000b/R1n\u000bn\u0001@n\n@t= 0; (C5)\nthat manifestly vanishes due to the constraint jnj= 1\nwhich is exact in the limit m= 0. Thus, the diver-\ngency inBvc\ndiv(which originates in the di\u000busion pole of\nthe density-density response) is, in fact, harmless for the\nresponse tensors we are discussing.\nIt is interesting to note that the irrelevance of the di-\nvergency in Bvc\ndivoperator extends to higher orders in m,\neven though it becomes much harder to see. We touch\non this problem in Appendix D.\nAppendix D: Finite magnetization\nThe deviation from a collinear antiferromagnetic order\ncan be accounted by considering a \fnite net magnetiza-\ntion term in the Hamiltonain perturbatively,\nH=He\u000b+U; U =\u0000\u0001m\u0001\u001b: (D1)\nIn the paper, we build the \frst order perturbation theory\nwith respect to U.\nFirst of all, it can be shown that the self-energy ac-\nquires the linear in mcontribution as\nIm \u0006R(A)=\u0007\u0019\u000bd\n2(\"\u0000\u0001 \u0003z\u0006zn\u0001\u001b+ \u0001m\u0001\u001b):(D2)\nSecond, the Dyson expansion of the disorder-averaged\nGreen's functions GR(A)with respect to mreads\nGR(A)!GR(A)+GR(A)UR(A)GR(A); (D3)\nwhereUR(A)=U(1\u0006i\u0019\u000bd=2) and we disregarded terms\nstarting from quadratic order in m. Note, that we\nhave kept the notations GR(A)for the disorder averaged\nGreen's functions of the unperturbed system.\nThe computation of linear response tensors amounts\nto considering an additional contribution to the response\ntensor represented by a complex class of diagrams de-\npicted schematically in Fig. 4d. Before ladder summa-\ntion is applied the diagrams of Fig. 4d correspond to a\ncontribution to the correlator of two operators BiandBj\nof the type\nUij= 2\u0019\u000bdZd2p\n(2\u0019)2Tr\u0002\nGAUAGABiGRBj\n+GABiGRURGRBj\u0003\n; (D4)\nwhich has yet be dressed. The dressing amounts to re-\nplacing both BiandBjoperators with the corresponding\nvertex corrected operators Bvc\niandBvc\nj, respectively.\nThe \fnal result for the response of spin density is still\ngiven by Eq. (B11), where the matrix R=F(1\u0000F)\u00001\nis, however, replaced with\nR=F\n1\u0000F+1\n1\u0000FU1\n1\u0000F; (D5)11\nwhich corresponds to diagrams Fig. 4c-d. 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Technologieentwicklung, Prüssingstr.  27B, 07745 Jena, Germany \n2 Institute of Ion Beam Physics and Materials Resear ch, Helmholtz-Zentrum Dresden-Rossendorf, \nBautzner Landstr. 400, 01328 Dresden, Germany \n3 Institut für Festkörperphysik, Friedrich-Schiller- Universität Jena, Helmholtzweg 3, 07743 Jena, \nGermany \n* Correspondence: cd@innovent-jena.de  \n \nThe field of magnon spintronics is experiencing an increasing interest in the development of \nsolutions for spin-wave-based data transport and pr ocessing technologies that are complementary or \nalternative to modern CMOS architectures. Nanometer -thin yttrium iron garnet (YIG) films have \nbeen the gold standard for insulator-based spintron ics to date, but a potential process technology tha t \ncan deliver perfect, homogeneous large-diameter fil ms is still lacking. We report that liquid phase \nepitaxy (LPE) enables the deposition of nanometer-t hin YIG films with low ferromagnetic \nresonance losses and consistently high magnetic qua lity down to a thickness of 20 nm. The obtained \nepitaxial films are characterized by an ideal stoic hiometry and perfect film lattices, which show \nneither significant compositional strain nor geomet ric mosaicity, but sharp interfaces. Their \nmagneto-static and dynamic behavior is similar to t hat of single crystalline bulk YIG. We found, \nthat the Gilbert damping coefficient α is independent of the film thickness and close to 1 × 10 -4, and \nthat together with an inhomogeneous peak-to-peak li newidth broadening of ∆H0||  = 0.4 G, these \nvalues are among the lowest ever reported for YIG f ilms with a thickness smaller than 40 nm. These \nresults suggest, that nanometer-thin LPE films can be used to fabricate nano- and micro-scaled \ncircuits with the required quality for magnonic dev ices. The LPE technique is easily scalable to YIG \nsample diameters of several inches. \n \n \nI. INTRODUCTION  \n \nYttrium iron garnet (Y 3Fe 5O12 ; YIG) in the micrometer thickness range is the mat erial of choice in \nradio-frequency (RF) engineering for decades (see, e.g., Refs. [1-5]). Especially the lowest spin \nwave loss of all known magnetic materials and the f act, that it is a dielectric are of decisive \nimportance. Since one has learned how to grow YIG f ilms in the nanometer thickness range, there \nhas been a renaissance of this material, as its mag netic and microwave properties are in particular \ndemand in many areas of modern physics.  \nA growing field of application for magnetic garnets  is (i) magnonics, which deals with future \npotential devices for data transfer and processing using spin waves [1,6-9]. The significant thickness  \nreduction achieved today allows reducing the circui t sizes from classical millimeter dimensions [1] \ndown to 50 nm [10-12] . Another important field is (ii) spintronics: By in creasing the YIG surface-\nto-volume ratio as much as possible (while keeping its magnetic properties), physical phenomena, \nsuch as the inverse spin Hall effect [13], spin-tra nsfer torque [14], and the spin Seebeck effect [15]  \n(generated by a spin angular momentum transfer at t he interfaces between YIG and a nonmagnetic \nmetallic conductor layer) become much more efficien t [7,16-29].  Also (iii) the field of terahertz \nphysics, which uses ultrafast spin dynamics to cont rol ultrafast magnetism, for example for potential  2 terahertz spintronic devices [30,31,32], and (iv) t he field of low-temperature physics, which deals \nwith magnetization dynamics at cryogenic temperatur es [33 ] for prospective quantum computer \nsystems, are possible fields of applications for na nometer-thin iron garnet films. \nThere are several different techniques to grow YIG on different substrates. (i) Pulsed laser \ndeposition (PLD) is an excellent technique for fabr icating small samples of nanometer-thin YIG \nfilms with narrow ferromagnetic resonance (FMR) lin ewidths [17,19,21,22,28,34-36] whereas its \nup-scaling to larger sample dimensions of several i nches is challenging. (ii) Magnetron sputtered \nYIG usually yields wider FMR linewidths, and inhomo geneous line broadening is frequently \nobserved [37-40]. (iii) For large-scale, low-cost c hemical solution techniques, such as spin coating, \nstrongly broadened FMR linewidths and increased Gil bert damping parameters were reported \n[41,42]. (iv) Liquid phase epitaxy (LPE) from high- temperature solutions (flux melts), is a well-\nestablished technique. Since nucleation and crystal  growth take place under almost thermodynamic \nequilibrium conditions, this guarantees high qualit y with respect to narrow absolute FMR linewidths \nand a small Gilbert damping coefficient [43-45] at the same time, making LPE comparable or \nsuperior to the other growth techniques. In additio n, LPE allows YIG to be deposited in the required \nquality on 3- or 4-inch wafers [46]. This is import ant for possible applications mentioned above. \nSo far, classical LPE was applied to grow micromete r-thick samples used for magneto-static \nmicrowave devices [47,48] or for magneto-optical im aging systems [49]. The typical shortcomings \nof the LPE technology making thin-film growth so di fficult lie in the fact, that, due to high growth \nrates, nanometer-thin films were technologically di fficult to access. The etch-back processes in high-\ntemperature solutions or interdiffusion processes a t the substrate/film interface at high temperatures  \nusually prevent sharp interfaces. In addition, film  contamination by flux melt constituents (if it is not \na self-flux without foreign components) is unavoida ble in most cases. Nevertheless, it was recently \ndemonstrated, that epitaxial films of 100 nm or thi nner are also accessible with this technique \n[50,51].  \nIn this study, we will show that we are able to dep osit nanometer-thin YIG LPE films with low FMR \nlosses and consistently high magnetic quality down to a thickness of 20 nm. There is no thinnest \n\"ultimate\" thickness for iron garnet LPE films, as it is sometimes claimed. \nIt should be pointed out, that, in addition to the damping properties, magnetic anisotropy \ncontributions as a function of the sample stoichiom etry and film/substrate pairing are also of great \nimportance, since they determine the static and dyn amic magnetization of the epitaxial iron garnet \nfilms and thus their possible applications. For exa mple, large negative uniaxial anisotropy fields \nwere usually observed for garnet films under compre ssion, such as for YIG on gadolinium gallium \ngarnet (Gd 3Ga 5O12 ; GGG) or other suitable substrates with smaller latt ice parameters grown by gas \nphase deposition techniques (see e.g. Refs. [35,36, 52-57]), which favors in-plane magnetization. \nLarge perpendicular magnetic anisotropies, on the o ther hand, can be found for films under tensile \nstrain, e.g. on substrates with larger lattice para meter or for rare earth iron garnet films with smal ler \nlattice parameter than GGG (see e.g. Refs. [58-62] ). Between these two extremes are YIG LPE \nfilms, which are usually grown on standard GGG subs trates and exhibit small tensile strain if no \nlattice misfit compensation, e.g. by La ion substit ution [50,63], has been performed. Such films are \ncharacterized  by a small uniaxial magnetic anisotropy and dominan t shape anisotropy when no \nlarger growth–induced anisotropy contributions due to Pb or Bi substitution occurs [64]. \nHowever, only little information about the structur al properties and the thickness-dependent \nmagnetic anisotropy contributions of nanometer-thin  LPE films has been published so far, which is \nwhy we are concentrating on these properties for YI G films with thicknesses down to 10 nm. This \nallowed us to describe the intrinsic damping behavi or over a wide frequency range and to determine \na set of magnetic anisotropy parameters for all inv estigated films. \n \n  3 II. EXPERIMENTAL DETAILS \n \nNanometer-thin YIG films were deposited on 1-inch ( 111) GGG substrates by LPE from PbO-B 2O3-\nbased high-temperature solutions (HTL) at about 865 °C using the isothermal dipping method (see \ne.g. [65]). Nominally pure Y 3Fe 5O12  films with smooth surfaces were obtained within on e minute \ndeposition time on horizontally rotated substrates with rotation rates of 100 rpm. The only variable \ngrowth parameter for all samples in this study was the degree of undercooling ( ∆T = TL-Tepitaxy ) that \nwas restricted to ∆T ≤ 5 K to obtain films with thicknesses between 10 an d 110 nm. Here TL is the \nliquidus temperature of the high-temperature soluti on and Tepitaxy  is the deposition temperature. After \ndeposition, the samples were pulled out of the solu tion followed by a spin-off of most of the liquid \nmelt remnants at 1000 rpm, pulled out of the furnac e and cooled down to room temperature. \nSubsequently, the sample holder had to be stored wi th the sample in a diluted, hot nitric-acetic-acid \nsolution to remove the rest of the solidified solut ion residues. Finally, the reverse side YIG film of  \nthe doubled-sided grown samples was removed by mech anical polishing and samples were cut into \nchips of different sizes by a diamond wire saw. The  film thicknesses were determined by X-ray \nreflectometry (XRR) and by high-resolution X-ray di ffraction (HR-XRD) analysis, and the latter \ndata were used to calculate anisotropy and magnetiz ation values. \nAtomic force microscopy (AFM) using a Park Scientif ic M5 instrument was carried out for each \nsample at three different regions over 400 µm2 ranges to determine the root-mean-square (RMS) \nsurface roughness. \nThe XRR measurements were carried out using a PANan alytical/X-Pert Pro system. For the HR-\nXRD investigations, a Seifert-GE XRD3003HR diffract ometer using a point focus was equipped \nwith a spherical 2D Göbel mirror and a Bartels mono chromator on the source side. Both systems use \nCu Kα1 radiation. Reciprocal space maps (RSMs) were measu red with the help of a position-sensitive \ndetector (Mythen 1k) at the symmetric (444) and (88 8) as well as the asymmetric (088), (624), and \n(880) reflections. To obtain the highest possible a ngular resolution for symmetric θ−2 θ line scans, a \ntriple-axis analyzer in front of a scintillation co unter was installed on the detector. Using a recurs ive \ndynamical algorithm implemented in the commercial p rogram RC_REF_Sim_Win [66], the vertical \nlattice misfits were calculated.  \nRutherford backscattering spectrometry (RBS) was ap plied to investigate the composition of the \ngrown YIG films using 1.8 MeV He ions and a backsca ttering angle of 168°. Backscattering events \nwere registered with a common Si detector. The ener gy calibration of the multichannel analyzer \nrevealed 3.61 keV per channel. A thin carbon layer was deposited on top of the samples to avoid \ncharging during analysis. The samples were tilted b y 5° with respect to the incoming He ion beam \nand rotated around the axis perpendicular to the sa mple surface in order to obtain reliable random \nspectra. The analysis of the measured spectra was p erformed by a home-made software [67] based \non the computer code NDF [68] and then enables the calculation of the RBS spectra. The measured \ndata were fitted by calculated spectra to extract t he film composition. In this way, the Fe-to-Y ratio  \nof the films was determined. Because of the low mas s of oxygen, the O signal of the deposited films \nis too low for quantitative analysis. \nHigh-resolution transmission electron microscopy (H R-TEM) investigations were performed with \nan image C s-corrected Titan 80-300 microscope (FEI) operated a t an accelerating voltage of 300 kV. \nHigh-angle annular dark-field scanning transmission  electron microscopy (HAADF-STEM) imaging \nand spectrum imaging analysis based on energy-dispe rsive X-ray spectroscopy (EDXS) were done \nat 200 kV with a Talos F200X microscope equipped wi th a Super-X EDXS detector system (FEI). \nPrior to TEM analysis, the specimen mounted in a hi gh-visibility low-background holder was placed \nfor 10 s into a Model 1020 Plasma Cleaner (Fischion e) to remove possible contaminations. Classical \ncross-sectional TEM-lamella preparation was done by  sawing, grinding, polishing, dimpling, and  4 final Ar-ion milling. Quantification of the element  maps including Bremsstrahlung background \ncorrection based on the physical TEM model, series fit peak deconvolution, and application of \ntabulated theoretical Cliff-Lorimer factors as well  as absorption correction was done for the \nelements Y (K α line), Fe (K α line), Gd (L α line), Ga (K α line), O (K line), and C (K line) using the \nESPRIT software version 1.9 (Bruker). \nThe ferromagnetic resonance (FMR) absorption spectr a were taken on two different setups. The \nfrequency-swept measurements were recorded on a Roh de & Schwarz ZVA 67 vector network \nanalyzer attached to a broadband stripline. The YIG /GGG sample was mounted face-down on the \nstripline, and the transmission signals S21  and S12  were recorded using a source power of -10 dBm (= \n0.1 mW). The microwave frequency was swept across t he resonance frequency fres , while the in-\nplane magnetic field H remained constant. Each recorded frequency spectru m was fitted by a \nLorentz function and allowed us to define the reson ance frequency fres  and the frequency linewidth \nΔfFWHM  corresponding to the applied field H = Hres . \nIn addition, field-swept measurements were carried out with another setup using an Agilent E8364B \nvector network analyzer and an 80-µm-wide coplanar waveguide. Again, the microwave \ntransmission parameter S21  was recorded as the FMR signal. This time, the mic rowave frequency \nwas kept constant and the external magnetic field w as swept through resonance. This facilitates \ntracking the FMR signals over large frequency range s. The microwave power was set to 0 dBm (= \n1 mW). In addition, this setup allowed for azimutha l and polar angle-dependent measurements to \ndetermine the anisotropy and damping contributions in detail. The FMR spectra were fitted by a \ncomplex Lorentz function to retrieve the resonance field Hres  and field-swept peak-to-peak linewidth \nΔHpp . By fitting the four sets of resonance field data,  i.e. (i) the in-plane and (ii) the perpendicular-\nto-plane frequency dependence as well as (iii) the azimuthal and (iv) polar angular dependences at f \n= 10 GHz, with the resonance equation for the cubic (111) garnet system, a consistent set of \nanisotropy parameters was determined for each sampl e. In addition, the damping parameters and \ncontributions were determined from the frequency- a nd angle-dependent linewidth data. \nThe vibrating sample magnetometer (VSM, MicroSense LLC, EZ-9) was used to measure the \nmagnetic moments of the YIG/GGG samples magnetized along the YIG film surface. The external \nmagnetic field H was controlled within an error of ≤0.01 Oe. To est imate the volume magnetization \nM of the YIG films, the raw VSM signal was corrected  from background contributions (due to the \nsample holder and the GGG substrate) and normalized  to the YIG volume. The Curie temperatures \nTC for the YIG samples were determined by zero-extrap olation of the temperature dependencies M \n(H=const, T) measured in small in-plane magnetic fields. In or der to verify the Curie temperatures \nmeasured by VSM, a differential thermal analysis of  a 0.55 mm thick YIG single crystal slice was \ncarried out and then used as a reference sample for  the VSM temperature calibration. \n \n \nIII. RESULTS AND DISCUSSION \n \nA. Microstructural properties of nanometer-thin YIG  films \n \nThe thickness values reported in this study are der ived from the Laue oscillations observed in the θ-\n2θ patterns of the high-resolution X-ray diffraction (HR-XRD) measurements and are confirmed by \nX-ray reflectivity (XRR) measurements (see Fig. 1).  The differences between both methods for \ndetermining the film thickness are in the range of ±1 nm. The surface roughness of the films, \nmeasured by atomic force microscopy (AFM) reveals R MS values ranging between 0.2 and 0.4 nm, \nindependent of the film thickness. Sometimes, howev er, partial remnants of dendritic overgrowth  5 increase the surface roughness to RMS values above 0.4 nm for inspection areas larger than 400 µm2 \n(see, e.g., the disturbance in the top right corner  of the AFM image inset in Fig. 1). \n-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 10 010 110 210 310 410 510 610 710 810 910 10 \n  \n t = 10.6 nm Intensity (arb.units) \nIncidence angle [deg] 20 \nµm15 \n10 \n5\n010 15 20 5 0µmnm  1.5 \n 1.0 \n 0.5 \n 0.0 \n-0.5 \n-1.0 \n-1.5 \n t = 21.5 nm  t = 30.9 nm  t = 43.2 nm \n \nFIG. 1. XRR plots of LPE-grown YIG films of differe nt thicknesses. Solid lines correspond to the \nexperimental data, while dashed lines represent the  fitted curves. The spectra shifted vertically for \nease of comparison. The inset shows an AFM image of  the surface topography of the 11 nm YIG \nfilm with a RMS roughness of 0.4 nm. \n \n \n1. Epitaxial perfection studied by high-resolution X-ray diffraction  \n \nCombined high-resolution reciprocal space map (HR-R SM) investigations around asymmetric and \nsymmetric Bragg reflections are useful to evaluate the intergrowth relations of epitaxial films on \nsingle-crystalline substrates as well as to disting uish between lattice strain induced by the film \nlattice distortion or compositional changes due to stoichiometric deviations. \n33.10 33.15 33.20 46.0 46.5 47.0 47.5 48.0 48.5 49.0 \nQx (nm -1 )Qz (nm -1 )\n(088) \nYIG (GGG) \nt = 21 nm 5E0 5E1 5E2 5E3 5E4 5E5 5E6 Intensity (a) \n-0.1 0 0.1 34.5 35.0 35.5 36.0 \nQx (nm -1 )Qz (nm -1 ) (444) \n  6 0 20 40 60 80 100 46810 \n118.5 119.0 119.5 120.0 10 110 210 310 410 510 6\n  t  =     9 nm \n  t   =   11 nm \n  t   =   21 nm \n  t   =   30 nm \n  t   =   42 nm \n  t   = 106 nm     (888) \nYIG / GGG \nScattering angle θ− 2θ (deg) Intensity (counts) (b) \nOut-of-plane misfit \n-δd⊥\nfilm  [10 -4]\nFilm thickness (nm) \n \nFIG. 2. (a) Combined high-resolution reciprocal spa ce maps around the asymmetric YIG/GGG \n(088) Bragg reflection of a 21-nm-thin single-cryst alline YIG LPE film. The inset shows the \ncorresponding symmetric YIG/GGG (444) peak: measure ments were carried out using a position-\nsensitive detector. (b) HR-XRD triple-axis θ–2θ scans around the symmetric YIG/GGG (888) peak \nfor various film thicknesses. The inset shows the ( vertical) out-of-plane misfit vs. film thickness (t he \nsolid line is a guide to the eyes).  \n \nFigure 2(a) shows the HR-RSM of the 21 nm YIG film grown on GGG (111) substrate, measured at \nthe asymmetric (088) reflection in steep incidence,  indicating that both the film and the substrate \nBragg peak positions are almost identical. Besides the nearly symmetrical intensity distribution \nalong the [111]  out-of-plane direction (i.e. the Qz axis), there is only very weak diffuse scattering \nclose to the Bragg peak visible, pointing towards a  nearly perfect crystal lattice without significant  \ncompositional strain or geometric mosaicity. In add ition, no shift of in-plane (the Qx axis) film \nBragg peak position with respect to the substrate i s observed. This behavior indicates a fully straine d \npseudomorphic film growth with a perfect coherent i n-plane lattice match with the GGG substrate. \nThe pattern of the diffuse scattering observed alon g the Qx axis of the symmetric (444) reflection \n(inset in Fig. 2(a)) is very similar to the one fou nd for a comparable GGG substrate (not shown), \nindicating that the defect structure of the system is mainly defined by the substrate and/or substrate  \nsurface. Within the experimental error of ∆Q/Q ∼ 5×10 −6 nm -1 of the high-resolution diffractometer, \nthe same performance was found for all investigated  LPE films with thicknesses below 100 nm, \nclearly demonstrating coherent YIG film growth with out signs of film relaxation. \nHigh-resolution triple-axis coupled θ–2θ scans at the (888) and (444) symmetrical reflectio ns \n(angular accuracy better than 1.5\") were carried ou t to define the strain and film thicknesses of the \nYIG films. Figure 2(b) shows the results obtained a t the (888) reflection. Under these conditions, the  \nBragg reflection of the 106 nm thick YIG layer is c learly visible as a shoulder of the (888) GGG \nsubstrate reflection at higher diffraction angles a nd this indicates a smaller out-of-plane value for the \nlattice parameter d888  than for the GGG substrate. This is characteristic  for tensely stressed “pure” \nYIG LPE films [50,63]. For LPE films with a thickne ss significantly less than 100 nm, however, \nonly simulations can provide the structural paramet ers. For this reason, the diffracted signals shown \nin Fig. 2(b) were simulated and fitted. Using the b est fit of both, the (444) and (888) reflections, t he \nout-of-plane lattice misfit values ( )⊥ ⊥ ⊥ ⊥ ⊥ ⊥∆ − = − = QQ d d d d / /substrate substrate film filmδ  were determined (see \nTable 1). Assuming a fully pseudomorphic [111]-orie nted system, the in-plane stress of the YIG \nfilm can be calculated by σ′|| = -2c 44 δd⊥\nfilm  (see the Supplemental Material [69 ] for a detailed \nderivation and references therein [61,70,71]). The in-plane biaxial ε|| and out-of-plane uniaxial ε⊥  7 strains can be calculated as well using the stiffne ss tensor components c11 , c12 , and c44  for which we \nuse averaged values taken from [72,73] (see also Su pplemental Material [69 ]). The resulting \nparameters are listed in Table I. \nThe inset in Fig. 2(b) shows the out-of-plane misfi t as a function of the film thickness. A weak \nmonotonous increase of ⊥\nfilmdδ with decreasing film thickness is observed between 106 nm and \n21 nm. The same behavior was reported by Ortiz et al . [61] for compressively strained EuIG and \nTbIG PLD-grown films with film thicknesses down to 4 nm and 5 nm, respectively. However, for \nour thinnest LPE films with t ∼ 10 nm, the out-of-plane misfit rapidly drops. Such  a significant \nchange of the misfit with respect to the film thick ness was only mentioned for considerably \ncompressively strained YIG PLD films by O. d’Allivy  Kelly et al.  [17]. They assume, that this \neffect indicated a critical film thickness (below 1 5 nm) for strain relaxation, but did not explain it  in \ntheir letter. \nFor semiconductor LPE films, however, it is known, that interdiffusion processes at the \nfilm/substrate interfaces can generate continuous c omposition profiles in the diffusion zone without \nabrupt changes in the lattice parameters, which lea d to modified stress profiles depending on the \nthickness of the epilayers (see e.g. [74]). A possi ble explanation for the observed behavior could \ntherefore be the presence of a smoothly changing la ttice parameter value in the interface region.  \nSuch composition profiles have recently been discus sed for YIG films grown on GGG substrates by \nhigh-temperature and long-time laser MBE deposition  experiments [75] , and transition layer \nthicknesses have been modeled based on polarized ne utron and X-ray reflectometry techniques. The \nprobability of the existence of such a thin continu ous transition layer and its influence on the \nmagneto-static film properties will be discussed be low. \n \nTABLE I. Structural parameters of the YIG LPE films  grown on GGG (111) substrates: film \nthickness t measured by HR-XRD, RMS roughness obtained by AFM,  vertical lattice misfit ⊥\nfilmdδ  \nobtained by HR-XRD, in-plane strain ε|| and out-of-plane strain ε⊥ and the resulting  in-plane stress σ′||. \n \nt \n(nm)  roughness  \n(nm)  δd⊥\nfilm   \n×10 -4 ε|| \n×10 -4 ε⊥ \n×10 -4 σ′|| \n×10 8 Pa  \n9 - -4.3 2.3 -2.0 0.7 \n11 0.4 -6.1 3.3 -2.8 0.9 \n21  0.2 -10.4 5.6 -4.8 1.6 \n30 0.2 -9.4 5.1 -4.3 1.4 \n42  0.3 -9.2 5.0 -4.2 1.4 \n106  0.4 -8.5 4.6 -3.9 1.3 \n±1 ±0.1 ±0.7 ±0.4 ±0.4 ±0.1 \n \n \n2. Chemical composition studied by Rutherford Backs cattering spectrometry \n \nBesides the epitaxial perfection, the chemical comp osition of the films is of interest to estimate \ndeviations from the ideal Y 3Fe 5O12  stoichiometry and to detect impurity elements. The refore, RBS \nmeasurements were performed for selected LPE films.  As an example, Fig. 3 shows the random \nspectrum of a 30 nm thick YIG film on GGG substrate . The inset presents the main part of the \nspectrum. Applying the NDF software, the computed c urve (solid line) matches perfectly the \nexperimental one (symbols). This enables us to dete rmine the Fe:Y ratio. As for all investigated LPE \nfilms, the Fe:Y ratio was determined to be R = 1.67, which corresponds to the ideal iron garnet   8 stoichiometry with Fe:Y = 5:3. At higher magnificat ions of the backscattering yield in Fig. 3, a very \nlow intensity signal can be observed at ion energie s higher than for backscattering on gadolinium \natoms from the GGG substrate. Although the intensit y is rather low, it can be attributed to heavy \nimpurity elements present to a very low amount over  all in the YIG film. We assign this signal to  \nlead and platinum. These elements may come from the  solvent and the crucible during the \ndeposition of the YIG film. After background correc tion a total quantity of (0.08 ± 0.02) at.% for the \nsum of both elements could be determined. This corr esponds to 0.01 < x + y < 0.02 formula units of \nthe nominal film composition (Y 3-x-yPb xPt y)(Fe 5-x-yPb xPt y)O 12 . In a first approximation, for the \ncalculation of the RBS spectra, it was assumed, tha t both elements contribute in equal parts to the \nhigh-energy signal. So, the calculated spectrum tak es into account the existence of 0.04 at.% lead \nand 0.04 at.% platinum within the YIG layer. This y ields a good representation of the separated \nsignal for these two elements. \n1550 1600 1650 1700 1750 1800 0100 200 300 400 \n  \n measurement       simulation \n separated Pb + Pt signal Backscattering yield  (counts) \nIon energy  (keV) YIG/GGG  \nt = 30 nm \nGd signal \nfrom GGG \nsubstrate 1000 1500 02000 4000 6000 8000 \n   \n Gd Ga YFe \n \nFIG. 3. Energy spectrum of 1.8 MeV He ions backscat tered on the YIG/GGG sample with a YIG \nfilm thickness of t = 30 nm. The inset shows the main part of the spec trum with the edges of the \nsubstrate elements Gd and Ga and the Fe and Y peak from the YIG film. \n \n \n3. Crystalline perfection studied by high-resolutio n transmission electron microscopy \n \nTo analyze the film lattice perfection as well as t he heteroepitaxial intergrowth behavior, HR-TEM \ninvestigations were performed. A cross-sectional im age of an 11 nm thin YIG film on a GGG \nsubstrate makes it possible to visualize both, the entire YIG film volume up to the film surface and \nthe interface in a magnified HR-TEM microscope imag e (see Fig. 4(a)). Besides the perfect \nfilm/substrate interface, neither structural lattic e defects nor significant misalignment could be \nobserved in the coherently strained YIG film lattic e up to the film surface. \nTo prove the homogeneity of the bulk composition an d the performance of the film/substrate \ninterface, HAADF-STEM imaging (Fig. 4(b)) together with element mapping, based on EDXS \nanalysis (Figs. 4(c)-(g)), were performed. The corr esponding HAADF-STEM image in Fig. 4(b) \nallows clearly resolving the film/interface region due to the significant difference of the atomic \nnumber contrast. Because of the uniform spatial dis tribution of both, the film (Y, Fe, O) and the \nsubstrate elements (Gd, Ga, O), which are independe ntly represented by different colors in Figs.  9 4(c)-(g), a homogeneous composition over the entire  YIG film can be confirmed. Small brightness \nvariations within the element maps (on the right ha nd side) result from slight thickness variations of  \nthe classically prepared TEM lamella. Neither an in termixing of the substrate nor of the film \nelements at the YIG/GGG interface is observed in th e element maps within the EDXS detection \nlimit, which is estimated to be slightly below 1 at .-% for the measuring conditions used. For that \nreason, tiny Pb and Pt contributions in the YIG fil m, as shown by RBS (see Fig. 3), where not \ndetected here. \nTo evaluate the lateral element distributions acros s the film near the film/substrate interface, \nquantified line scans were performed as presented i n Fig. 4(h). Using the 10%-to-90% edge \nresponse criterion, it shows a  transition width of (1.9 ± 0.4) nm at the interface . This is lower than \nthe observed 4-6 nm non-magnetic dead layer reporte d for YIG films deposited by RF magnetron \nsputtering [76], and the about 4 nm or the 5–7 nm d eep Ga diffusion observed for PLD [77] or laser \nmolecular beam epitaxy (MBE) [75], respectively . However, at some positions of the sample’s \ncross-section we found a reduced YIG film thickness  on a wavy GGG surface (not shown), which \nwe attribute to a possible etch-back of the substra te at the beginning of film growth or an already \nexisting wavy substrate surface. For further growth  experiments, a careful characterization of the \nsubstrate surfaces by AFM should, therefore, be per formed. The TEM investigations show, that the \nLPE technology is suitable for growing nanometer-th in YIG films without lattice defects and \nwithout significant interdiffusion at the film/subs trate interface, which are necessary preconditions \nfor undisturbed spin-wave propagation and low ferro magnetic damping losses. \n \n \nGGG YIG resist (a) \n  10  0 2 4 6 8 10 12 14 16 18 Composition \n(at.-%) \nDistance (nm) \n(c) Y \n(d) Fe \n(e) Gd \n(f) Ga \n(g) O \n(h) \nGd 3Ga 5O12 Y3Fe 5O12 surface \ninterface \n(b) HAADF Line scan \nY\nFe \nGd \nGa \nO\n \n \nFIG. 4. (a) Cross-sectional high-resolution TEM ima ge of the 11-nm-thin YIG/GGG (111) film. The \narrows mark the YIG/GGG interface. (b) HAADF-STEM i mage highlighting the well-separated \nYIG/GGG interface. (c-g) EDXS element maps of the 1 1-nm-thin YIG/GGG (111) film cross-\nsection. (h) Line scan as marked in (b) of the elem ental concentrations across the film thickness.  \n \n \nB. Static and dynamic magnetization characterizatio n of nanometer-thin YIG films \n \nAfter gaining insight into the YIG film microstruct ure, we want to link these properties to the FMR \nperformance to find out, which of them plays an ess ential role in the observed magneto-static and \ndynamic behavior. Therefore, FMR measurements were carried out within a frequency range of 1 to \n40 GHz, with the external magnetic field either par allel to the surface plane of the sample along the \nH || [11-2] film direction or perpendicular to it ( H || [111 ]). In addition, angle-dependent \nmeasurements, i.e., varying the angle θH of the external magnetic field (polar angular depe ndence, \nwhere θH = 0 is the sample’s normal [111 ] direction) or the azimuth angle φH (in-plane angular \ndependence, where φH = 0 is the sample’s horizontal  [1-10] direction), were performed at \nf = 10 GHz. These four measurement ‘geometries’ allo w to determine Landé’s g-factor, effective \nmagnetization 4π Meff , and anisotropy fields from the resonance field de pendence and to disentangle \nthe damping contributions from the linewidth depend ence [78,79].  \nThe FMR resonance equations to fit the angle- and f requency-dependencies (see eqs. (S22), (S23) in \nthe Supplemental Material [69] for in-plane and out -of-plane bias field conditions after Baselgia et \nal.  [80 ])  are derived from the free energy density of a cubi c (111) system [81]: \n \n ( ) [ ]\n( ) ( )\n\n\n\n− + +− − − ++ − ⋅ −=\n⊥\nϕ θ θ θ θϕϕ θ θ πθ θ ϕϕ θ θ\n3sin cos sin32sin41cos31cos sin cos 2cos cos cos sin sin\n3 4 4\n42 2\n|| 22\n22\nKK K MH M F\nu sH H H s\n, (1) \n  11  where K2⊥, K2|| , and K4 are the uniaxial out-of plane, uniaxial in-plane, and cubic anisotropy \nconstants, respectively.  φ and θ are the angles of the magnetization. Angle φu allows for a rotation of \nthe uniaxial anisotropy direction with respect to t he cubic anisotropy direction.  \n \n \n1. Frequency-dependent FMR linewidth analysis \n \nTo investigate the influence of different contribut ions on the overall magnetic damping, we model \nthe field-swept peak-to-peak linewidth Δ Hpp  of our YIG (111) films as a sum of four contributi ons \n[79,82]:  \n TMS 0 mos G pp H H H H H ∆ + ∆ + ∆ + ∆ = ∆ , (2) \n \nwhere Δ HG is the Gilbert damping, Δ Hmos  the mosaicity, Δ H0 the inhomogeneous broadening, and \nΔHTMS  is the two-magnon scattering contribution, respect ively. Note, that all linewidths in this paper \nare peak-to-peak linewidths, even if not explicitly  stated. \nThe intrinsic Gilbert damping is given by \n  \n f H\nΞ= ∆\nγπα\n34\nG , (3) \n \nwhere γ = gµBħ is the gyromagnetic ratio and Ξ is the dragging function. The dragging function is  a \ncorrection factor to the linewidth needed in field- swept FMR measurements if H and M are not \ncollinear (see e.g . [82]). For H || M follows Ξ = 1. \nThe inhomogeneity term ∆Hmos  accounts for a spread (distribution) of the effect ive magnetization \n4π Meff  [82,83] given by the parameter δ4πMeff : \n \n eff\neffres\nmos 4432MMHH πδπ∂∂= ∆ . (4) \n \n∆H0, i.e. the zero-frequency linewidth, is a general b roadening term accounting for other \ninhomogeneities of the sample, such as the microwav e power dependence of the linewidth in YIG \n(see, e.g., [84]) and systematic fit errors: for ex ample, consistently narrower total full-width at ha lf-\nmaximum linewidths ∆HFWHM  of up to 0.5 Oe were determined by additional freq uency-swept \nmeasurements at a microwave power of -10 dBm compar ed to the field-swept measurements at \n0 dBm discussed here. \nAll kinds of inhomogeneous broadening (including ∆Hmos ) are caused by slightly different \nresonance fields in parts of the sample. These indi vidual resonance lines might be still resolvable at  \nlow frequencies, where Gilbert damping is not large  enough yet–especially for YIG. However, at \nhigher frequencies, these lines become broader and eventually coalesce to a single (apparently \nbroadened) line, which even might exhibit small sho ulders or other kinds of asymmetry. Hence, \nwhat might be nicely fit with a single line at high  frequencies might cause difficulties at low \nfrequencies and sub-mT linewidths. The effect on fi tting the anisotropy constants from the \nresonance fields is not so sensitive. If the resona nce lines cannot be disentangled or the line is not  \nentirely Lorentzian-shaped anymore, the fit might o verestimate the true linewidth resulting in a \nsystematic broader line accounted for by ∆H0.  12  The last term in Eq. (2), ∆HTMS , covers the two-magnon scattering contribution, wh ich is an \nextrinsic damping mechanism due to randomly distrib uted defects. For the in-plane frequency-\ndependence it reads [78,79,82,85-87]:  \n \n \n2 22 2sin\n32\n02\n0 202\n0 2\n1\nTMS\nf fff ff\nH\n+\n\n+−\n\n+\n⋅ Γ\nΞ= ∆−, (5) \n \nwhere f0 = γ4π Meff  and Γ is the two-magnon scattering strength. \nEach of the contributions has a characteristic angl e and frequency dependence. Overall, the \nlinewidth vs. frequency dependencies and the linewi dth vs. angle dependencies can be described \nwith one set of parameters. \nAs we will see, the applied model fits very well to  the experimental results and allows for \ndisentangling the contributions that are responsibl e for the frequency dependence of the linewidth. \nAt first, we discuss the different damping contribu tions. Then, we go into details for the individual \nmagneto-static parameters, the relevant anisotropy contributions mentioned above, which provided \nalso the base input for the fit parameters for the frequency-dependent FMR linewidth of our YIG \nfilms. \nIn Figure 5, the obtained frequency-dependent peak- to-peak linewidths ∆Hpp  (symbols) for the four \nthicknesses 11 nm, 21 nm, 30 nm, and 42 nm are pres ented. The red (solid) curves represent fits \nusing Eq. (2). Figure 5(a) shows data and fits for the out-of-plane bias field configuration ( θH = 0°) \nand Fig. 5(b) for field-in-plane ( θH = 90°), respectively. As mentioned above, due to a  quite complex \nshape of the resonance lines below ~15 GHz for θH = 0° (with more absorption lines needed to \nreflect the shape of the spectrum than for higher f requencies) the linewidths could not anymore be \nevaluated unambiguously with the required precision  for films with thicknesses above 11 nm. \nHowever, for the thinnest film, the evaluation was possible and the overall fit exhibits a linear \nbehavior down to 1 GHz. This means, in the field-ou t-of-plane geometry, the main contribution to \nthe damping is the Gilbert damping α, which can be determined from the linear slope according to \nEq. (3). As it is known from two-magnon scattering (TMS) theory [85,86], there is no TMS \ncontribution if M is perpendicular to the sample plane. The only rem aining contribution is the \ninhomogeneous broadening given by the zero-frequenc y offset \n  13  0369\n036\n036\n0 5 10 15 20 25 30 35 40 036θ H= 0 deg \n 11 nm \n 21 nm ∆Hpp  (Oe) \n 30 nm \n \nf (GHz) 42 nm (a) \n \n \nFIG. 5: Frequency dependence of the linewidth with magnetic field (a) perpendicular-to-plane and \n(b) in-plane. The red (solid) lines are fits to the  data. For the 11-nm sample the individual \ncontributions to the total linewidth are shown in t he top-right panel. Note the different y-axis scaling \nfor the 11 nm sample in the top-left panel. \n \nFrom these out-of-plane measurements, the Gilbert d amping coefficients could be determined, \nranging from α = 0.9 × 10 -4 for the 42-nm-thick sample to α = 2.0 × 10 -4 for the 21 nm sample, \nwhich is about twice the value obtained from in-pla ne measurements (as discussed below). For the \nultrathin 11 nm film, a slightly increased Gilbert damping coefficient of α = 2.7 × 10 -4 and a \nsignificantly enlarged zero-frequency linewidth of 2.8 Oe were found. As mentioned above, the \nreason for the larger offset might be an apparent u nresolvable broadening due to inhomogeneity. For \nthe 21 and 30 nm sample, the zero-frequency interce pt is about ∆H0 = 0.5 Oe, in contrast to ∆H0 = \n1.5 Oe for the 42 nm sample. This indicates, that t he 42 nm sample, in contrast to the thinner \nsamples, seems to have additional microstructural d efects, leading to a superposition of lines. This i s \nvery likely, because the inhomogeneous broadening p reviously reported for 100 nm YIG LPE films \nwas also in the range of ∆H0  = 0.5-0.7 Oe [50].   \nIn Fig. 5(b), the results of the corresponding in-p lane field configuration are given. For the 11 nm \nsample, the four individual fit contributions consi dered in the fit according to Eq. (2) are depicted by \nsolid curves. This sample shows a significant curva ture. The 42 nm sample also shows a small \ncurvature, whereas the other two samples only have a weak curvature at lower frequencies. This \ncurvature usually hints to a contribution from two- magnon scattering, but can also be due to a spread \nof the effective magnetization. Note, the frequency -dependence of the mosaicity and TMS term look \nquite similar at higher frequencies, but show a dif ferent curvature at lower frequencies. Hence, the \nshape of the curve and, thus, the fit reveal, that it is due to a spread of the effective magnetizatio n, \nδ4πMeff  as given by Eq. (4), which lies in the range of 0. 4 to 0.9 G. For the 11 nm sample, this value 036\n036\n036\n0 5 10 15 20 25 30 35 40 03611 nm  Exp.  ∆H  ∆HG ∆Hmos  ∆HTMS  ∆H0\nθ H= 90 deg \n21 nm \n30 nm \nf (GHz) 42 nm ∆Hpp  (Oe) (b) \n 036\n036\n036\n0 5 10 15 20 25 30 35 40 03611 nm  Exp.  ∆H  ∆HG ∆Hmos  ∆HTMS  ∆H0\nθ H= 90 deg \n21 nm \n30 nm \nf (GHz) 42 nm ∆Hpp  (Oe) (b) \n  14  is larger, i.e., δ4π Meff  = 3.2 G, and in addition one needs a small TMS dam ping contribution of Γ = \n1.5×10 7 Hz for a proper fit (see Table II). This is again a distinctive sign, that the 11 nm sample has \nsignificantly different structural and/or magnetic properties, leading to the additional linewidth \ncontributions. The Gilbert damping coefficients of all four samples in in-plane configuration are \nα ≤ 1.3 × 10 -4 and correspond to the best values reported earlier  for 100 nm YIG LPE films [50]. \nThese are also lower than for a recently reported 1 8 nm YIG LPE film [51]. Thus, at room \ntemperature, no significant increase in Gilbert dam ping could be observed for LPE films down to \n10 nm with decreasing thickness. This contrasts wit h various references for PLD and RF-sputtered \nYIG films grown on (111) GGG substrates [88-92].  \n \nTABLE II. Magnetic damping parameters of the LPE (1 11) YIG films: film thickness t, in-plane \nGilbert damping parameter α||, inhomogeneous broadening ∆H0|| , spread of effective magnetization  \nδ4πMeff   and two-magnon scattering contribution Γ. \n \nt \n(nm)  α||  \n (×10 -4) ∆H0||  \n (G)  δ4π Meff   \n(Oe)  Γ \n(10 7 Hz)  \n11 1.2  0.4  3.2  1.5  \n21  1.3  0.6  0.4  0 \n30  1.2  0.4  0.7  0 \n42  1.0  0.4  0.9  0 \naccuracy  ±0.2  ±0.2  ±0.3  ±0.3  \n \n \nAll field-in-plane linewidth parameters of the inve stigated samples are summarized in Table II. It is \nobvious, that inhomogeneous contributions, i.e., th ose originating from magnetic mosaicity δ4πMeff , \nare very small for the samples without two-magnon s cattering. This confirms the high \nmicrostructural perfection and homogeneity of the v olume and interfaces of the LPE-grown films \nwith film thicknesses larger than 11 nm. Contributi ons to two-magnon scattering appear to occur \nonly for LPE films with a thickness of less than 21  nm thick. \n \n \n2. Analysis of magnetic anisotropy contributions \n \nIn the following, we will discuss the anisotropy co ntributions, which provided the base input for the \nfit parameters used for the frequency-dependent FMR  linewidth curves shown above. All curves \nwere fitted iteratively with the respective resonan ce equation (see Eqs. (S22) and (S23) in the \nSupplemental Material [69]) to retrieve a coherent set of fit parameters. The fit parameters are liste d \nin table III. Since the saturation magnetization an d the in-plane stress are known from VSM \nmeasurements and HR-XRD investigations, the anisotr opy constants K can be calculated from the \nanisotropy fields determined by FMR. \n \nTABLE III. Magneto-static parameters of the YIG LPE films of t hickness t: Landé’s g-factor, \neffective magnetization 4 πMeff exp , cubic anisotropy field 2 K4/Ms, and uniaxial in-plane anisotropy \nfield 2 K2||/Ms determined from FMR, saturation magnetization 4 πMs determined from VSM, stress-\ninduced anisotropy field 2 Kσ/Ms calculated from X-ray diffraction data, resulting o ut-of-plane \nuniaxial anisotropy field 2 K2⊥/Ms and effective magnetization 4 πMeff cal , cubic anisotropy constant \nK4, stress-induced anisotropy constant Kσ, and out-of-plane uniaxial anisotropy constant K2⊥. \n  15  t \n(nm) g 4πMeff exp \n(G) 2K4/Ms \n(Oe) 2K2||/Ms \n(Oe) 4πMs \n(G) \n11  2.015 1566 -93 2.0 1494 \n21 2.016 1647 -79 0.8 1819 \n30 2.015 1677 -79 0.6 1830 \n42 2.014 1699 -86 1.1 1860 \naccuracy ±0.002 ±13 ±2 ±3 ±41 \n \nt \n(nm) 2Kσ/Ms \n(G) 2K2⊥/Ms \n(G) 4πMeff cal  \n(G) K4 \n(10 3 erg/cm 3) Kσ \n(10 3 erg/cm 3) K2⊥ \n(10 3 erg/cm 3) \n11 65 127 1368 -5.5 3.9   7.5 \n21 91 143 1676 -5.7 6.6 10.4 \n30 82 135 1696 -5.8 6.0   9.8 \n42 79 136 1724 -6.4 5.8 10.1 \naccuracy ±7 ±4 ±40 ±0.3 ±0.4 ±0.5 \n \n \nThe g-factor of the samples was determined from the freq uency dependencies of the resonance field. \nThere was no significant thickness dependence obser ved yielding a value of g = 2.015(1) for all \nsamples. The cubic anisotropy field 2 K4/Ms was found to be nearly constant, and the average v alue \nis -84(2) Oe, which is in good agreement to reporte d values of -85 Oe for a 120 micrometer thick \nLPE film [81] and of about -80 Oe for a 18 nm thin LPE film [51]. Our calculated anisotropy \nconstants K4 are almost always in the range between -5.7×10 3 and -6.4×10 3 erg/cm 3, which \ncorresponds to YIG single crystal bulk values at 29 5 K [93] . Furthermore, a rather weak in-plane \nuniaxial anisotropy field 2 K2|| /Ms of about 0.6–2 Oe was found, which  had already be en determined \nfor 100 nm YIG LPE films [50].   \nThe stress-induced anisotropy constant Kσ and anisotropy field 2 Kσ/Ms are calculated according to \nRef. [94 ] (for details, see Eqs. (S14), (S15), (S18) in the Supplemental Material [69]). 2 Kσ/Ms is \nsmall and in the same order of magnitude as the cub ic anisotropy field 2 K4/Ms, but with opposite \nsign. Due to the observed monotonous increase of th e out-of-plane lattice misfit (see inset in Fig. \n2(b)), 2 Kσ/Ms grows with decreasing film thickness until it decl ines significantly at a film thickness \nbelow 21 nm. However, the observed stress values ar e almost an order of magnitude smaller than, \ne.g., for as-deposited YIG PLD films on GGG (111) u nder compressive strain (see, e.g., Refs. \n[17,23,35,36]). Only by a complex procedure, applyi ng mid-temperature deposition, cooling, and \npost-annealing treatment, authors of Ref. [95] succ eeded in a change from compressively to tensely \nstrained YIG films. These samples then exhibited th e same stress-induced anisotropy constant as it \nwas observed for our YIG LPE films. \nIn the following, we take a closer look to the cont ributions to the out-of-plane uniaxial anisotropy  \nfield H2⊥ = 2 K2⊥/Ms. A general description for magnetic garnets has been  given for example by \nHansen [94]. Applied to thick [43,64]  as well as to thin epitaxial iron garnet films (see , e.g., \n[37,56,59,60,62]) , the out-of-plane uniaxial anisotropy field H2⊥ is mainly determined by the \nmagnetocrystalline and  uniaxial anisotropy contributions. While the former  refers to the direction of \nmagnetization to preferred crystallographic directi ons in the cubic garnet lattice, the latter origina tes \nfrom lattice strain and growth conditions. Due to t he very low supercooling ( ≤5K), growth-induced \ncontributions, usually observed for micrometer YIG films with larger Pb impurity contents, can be \nneglected in the case of our nanometer-thin YIG LPE  films (see e.g. [64]). Thus,  H 2⊥ can be  16  determined quantitatively by summing the cubic magn etocrystalline anisotropy (first term, \ndetermined by FMR) and the stress-induced anisotrop y (second term, determined by XRD), \n \n \ns sMK\nMKHσ2\n344\n2 + − =⊥ , (6) \n \nor expressed for the (111) substrate orientation (s ee also SM [69 ] and Ref. [93,96]) by:  \n \n \nsMKH39 4111|| 4\n2λσ′ − −=⊥ . (7) \n \nUsing the experimentally determined first-order cub ic anisotropy constant K4 and the in-plane stress \ncomponent ||σ′from Tables I and III along with the room-temperatu re magnetostriction coefficient \nλ111  [94 ], the uniaxial anisotropy field H2⊥ can be calculated, if the saturation magnetization  Ms is \nknown. Ms can be obtained with appropriate accuracy for exam ple from VSM or SQUID \nmeasurements, if the sample volume is exactly known .  \nMagnetic hysteresis loops of YIG LPE films recorded  at room-temperature by VSM measurements \nwith in-plane applied magnetic field are shown in F ig. 6. The paramagnetic contribution of the GGG \nsubstrate was subtracted as described in Ref. [50]. Extremely small coercivity fields with Hc values \nof ∼ 0.2 Oe were obtained for all YIG/GGG samples with the exception of the 21 nm film. These \nvalues are comparable with the best gas phase epita xial films [17,39,76], but the measured saturation \nfields with Hs < 2.0 Oe are significantly smaller. All films exhi bit nearly in-plane magnetization due \nto the dominant contribution of form anisotropy. Ap art from the thinnest sample, the saturation \nmoments determined are not thickness-dependent (see  Table III and Fig. 6) and are very close to \nYIG volume values determined for YIG single crystal s at room temperature (4 πMs ∼ 1800 G) \n[93,97]. However, the observed decrease of the satu ration magnetization in such films with a \nthickness of about 10 nm is significant and will be  discussed below. \n-20 -15 -10 -5 0 5 10 15 20 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 \n10 100 1.4 1.5 1.6 1.7 1.8 1.9 \n  \n H (Oe)  4 πM (10 4 G)  106 nm \n 43 nm \n 30 nm \n 21 nm \n 11 nm \n   4 πM (10 4 G) \n d (nm) \n \nFIG.6: Magnetization loops M(H) of YIG films at room temperature as a function of  the in-plane \nmagnetic field. The inset shows the thickness-depen dent saturation magnetization (the solid line is a \nguide to the eyes). \n  17  However, for nanometer-thin films, it is a big chal lenge to determine Ms precisely enough, because \ntoo large errors can arise from the film’s volume c alculation. While the surface area of the sample \ncan be determined with sufficient precision by opti cal microscopy, thickness measurements with X-\nray or ellipsometry methods can lead to thickness e rrors in the range of ±1 nm due to very small \nmacroscopic morphology or roughness fluctuations. T herefore, for films with thicknesses below \n20 nm, for example, uncertainties up to a maximum o f 10 percent must be considered. This could \nsignificantly affect the effective magnetization 4 πMeff , which can be calculated based on the \nmeasured Ms values by \n ⊥ − =2 eff 4 4 H M Msπ π . (8) \nThis fact can explain the large difference between the calculated 4 πMeff cal  and the measured \n4πMeff exp  values for the 11 nm thin film discussed below, wh ile a much better agreement was \nachieved for the thicker films (see Table III). \nAs expected from micrometer-thick YIG LPE films gro wn on GGG (111) substrates [81], the out-\nof-plane uniaxial anisotropy field H2⊥ and the out-of-plane uniaxial anisotropy constants K2⊥ show, \nthat completely pseudomorphically strained, nanomet er-thin LPE films exhibit no pronounced \nmagnetic anisotropy. Small changes of the in-plane stress σ′|| (see Table I) and thus also in the \nstress-induced anisotropy 2 Kσ/Ms (or Kσ) have no significant influence on the out-of-plane  uniaxial \nanisotropy H2⊥ (see Table III). A comparable H2⊥ value is also expected for films thicker than \n42 nm, since the out-of-plane lattice misfit δd⊥\nfilm  tends to a constant value (see inset in Fig. 2 (b) ). \nThis is in contrast to Ref. [51], where the uniaxia l anisotropy field of YIG LPE films becomes \nnegative above a film thickness of about 50 nm. \n \n \n3. Thickness-dependent analysis of the effective ma gnetization field \n \nTo verify the trend of the calculated 4 πMeff cal  values for decreasing film thicknesses, one can \ncompare the effective magnetization with the experi mentally determined one. This was done for 18 \nYIG films with thicknesses ranging from 10 to 120 n m, including the four samples from above. All \nfilms were grown during the same run under nearly i dentical conditions. Only the growth \ntemperature was varied within a range of 5 K.  This time, the FMR was measured with a constant \nexternal magnetic field applied in-plane and sweepi ng the frequency.  \n0 20 40 60 80 100 120 1500 1550 1600 1650 1700 1750 1800 \n frequency field sweep \n magnetic field sweep Effective Magnetization (G) \nFilm thickness (nm) (a) \n  18  10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 535 540 545 550 555 560 \n fitted VSM values \n Outlier values Curie temperature (K) \nYIG thickness (m) (b) \n \nFig. 7. (a) Thickness dependence of the effective m agnetization 4 πMeff . Blue circles denote \nmeasurements taken by field-sweep and red squares d enote frequency-swept measurements, \nrespectively. (b) Thickness dependence of the Curie  temperature Tc. The dashed line is a guide to the \neyes. \n \n \nIn Fig. 7(a), the obtained thickness dependence of the effective magnetization 4 πMeff  (squares) is \npresented and a monotonous decrease of 4 πMeff  with film thickness reduction can be observed. \nBelow 40 nm, the slope of the curve increases and, for the thinnest films, there is a significant drop  \nof about 100 G. This behavior has been confirmed by  in-plane FMR magnetic field-sweep \nmeasurements for selected samples (circles), as dis cussed before. The values are listed in Table III. \nSimilar results have been reported for YIG PLD film s by Kumar et al. [53]. \nIf we compare the experimental values with the calc ulated ones in Table III, then the same trend of a \nsteady reduction of the effective saturation magnet ization with decreasing film thickness can be \nobserved. The deviation between both 4 πMeff  values is approximately 1–2 %, except for the 11 n m \nfilm. Hence, the saturation magnetization used to c alculate the effective saturation (according to \nequation (8)) does not appear to be as error-prone as it could be due to an inaccuracy in the film \nthickness determination. Therefore, we speculate, t hat the significant drop of 4 πMeff  for the 11 nm \nthin film can be explained by the observed reductio n of 4 πMs (see Table III). \nA similar behavior for 4 πMs was reported for thin PLD or magnetron-sputtered Y IG films, and \ndifferent explanations were given [76,56,91]. One r eason for a reduced saturation magnetization \ncould be an intermixing of substrate and film eleme nts at the GGG/YIG interface, whereby a gradual \nchange of the film composition is assumed [75,77]. In particular, gallium ion diffusion into the first  \nYIG atomic layers will lead to magnetically diluted  ferrimagnetic layers at the interface, due to the \nfact, that magnetic Fe ions are replaced by diamagn etic Ga ions in the various magnetic sublattices. \nThis assumption is supported by recent reports of Y IG films on GGG substrates. One reports about a \n5–7 nm deep Ga penetration found in laser-MBE films  [75 ]. Another group found a Ga penetration \nthroughout a 13-nm-thin PLD film [98]. In these cas es, high-temperature film growth above 850°C \nor prolonged post-annealing at temperatures of 850° C could promote such diffusion processes. In \ncontrast, the deposition time during which the LPE samples were exposed to high temperatures \nabove 860°C was only 5 minutes. Though, the assumed  Ga diffusion depth in our YIG films should \nnot exceed more than 2 nm according to the EDXS ele ment maps in Fig. 4(h). In addition, the Gd  19  diffusion in YIG films, as discussed for RF-magnetr on sputtered [76,99]  or  PLD films [98], could \nlead to the incorporation of paramagnetic ions into  the diamagnetic rare earth sublattice sites, which  \nwould also alter the magnetization [98]. However, n o extended interdiffusion layer was observed at \nthe film/substrate interface for our LPE films, so that the presence of a ‘separate, abrupt’ gadoliniu m \niron garnet interface layer, as reported by Ref. [9 8], is not expected. Therefore, due to possible \ninterdiffusion effects at temperatures of about 860 °C, a gradual reduction of Ms at a postulated \ninterface layer could be the reason for the observe d low saturation value for the thinnest LPE film, \nlisted in Tab. III. \nTo further rule out a discrete magnetic dead layer,  Curie temperature ( Tc) measurements were \nperformed by VSM. It is known from literature, that  Tc remains constant up to a film thickness of \napproximately four YIG unit cells [100], i.e. 2.8 n m, since one YIG unit cell length along the [111 ] \ndirection amounts to d111  ∼ 0.7 nm. Accordingly, the Tc of “pure” YIG films with abrupt interfaces \nand a film thickness of ∼10 nm should be equal to that of bulk material. In order to check this, \ntemperature-dependent VSM measurements (see Fig. 7( b)) were carried out for our LPE films as \nwell as for a bulk YIG single crystal slice, which was used as a reference. We found almost constant \nvalues of Tc = (551±2) K for sample thicknesses between 46 nm ( thin film) and 0.55 mm (bulk). \nThis is in good agreement with the literature, in w hich a Tc of ∼550 K has been reported, e.g. for a \n100-nm-thin sputtered YIG film [76],  while 559 K has been reported for YIG single crysta ls [97]. \nHowever, for our about 10-nm-thin YIG films, Tc decreased significantly to ∼534±1 K (Fig 7(b)), \nwhich is consistent with the observed reduction of 4 πMs listed in Table III. \nHence, the most likely explanation for the observed  reduction of 4 πMs is that the YIG layers at the \nsubstrate/film interface exhibit a reduced saturati on magnetization due to a magnetically diluted iron  \nsublattice, resulting from high-temperature diffusi on of gallium ions from the GGG substrate into \nthe YIG film . While nearly zero gallium content at the film surfa ce leads to a bulk-like value of \n4πMs ∼ 1800 G [93], an increased content of gallium at th e film/substrate interface should, therefore, \nresult in significantly reduced 4 πMs values. In this case, the average saturation magne tization for the \nentire film volume should be reduced and that could  explain the observed decrease in 4 πMs to about \n1500 G for the 11 nm thin LPE film. For thicker fil ms, however, the influence of thin gallium-\nenriched interface layers on the entire film magnet ization decreases, which explains the fast \nachievement of a constant Curie temperature, and th us, a constant Ms with increasing thickness of \nthe YIG volume. In order to confirm our assumptions , additional analyses, such as detailed \nsecondary ion mass spectroscopy (SIMS) investigatio ns, are necessary which, however, go beyond \nthe scope of this report. \n \n \nIV. CONCLUSIONS AND OUTLOOK \n \nIn summary, we have demonstrated that LPE can be us ed to fabricate sub-40 nm YIG films with \nhigh microstructural perfection, smooth surfaces an d sharp interfaces as well as excellent microwave \nproperties down to a minimum film thickness of 11 n m. All LPE films with ≥21 nm thickness \nexhibit extremely narrow FMR linewidths of ∆Hpp  <1.5 Oe at 15 GHz and very low magnetic \ndamping coefficients of α ≤1.3 × 10 -4 which are the lowest values reported within an ext ended \nfrequency range of 1 to 40 GHz. We were able to sho w that LPE-grown YIG films down to a \nthickness of 21 nm have the same magnetization dyna mics influenced by small cubic and stress-\ninduced anisotropy fields. The deviating magnetizat ion dynamics of ultrathin LPE films with \nthicknesses of ∼10 nm are probably caused by an increased inhomogen eous damping and by small \ntwo-magnon scattering contributions, and we specula te that possible inhomogeneities of the \ncomposition in the vicinity of the film/substrate i nterface might be the reason for this. Therefore, i n  20  further studies we will address detailed investigat ions of the composition of the film/substrate \ninterface by high-resolution SIMS measurements and advanced STEM analyses to confirm a gradual \nchange of the LPE film composition at the interface . \nThe results presented here encourage us to take the  next step towards nano- and microscaled \nmagnonic structures, such as directional couplers, logic gates, transistors etc. for a next-generation  \nof computing circuits. The development of nanoscopi c YIG waveguides and nanostructures is \nalready underway and the first circuits are current ly being fabricated [10,12,29]. With its scalabilit y \nto large wafer diameters of up to 3 and 4 inches, L PE technology opens up an alternative way for \nefficient circuit manufacturing for a future YIG pl anar technology on a wafer scale. \n \n \nACKNOWLEDGMENTS \n \nWe thank P. Landeros and R. Gallardo for fruitful d iscussions and A. Khudorozhkov for his help \nduring the measurements. C. D. and O. S. thank R. K öcher for AFM measurements, A. Hartmann \nfor the DSC measurements and R. Meyer and B. Wenzel  for technical support. J. G. thanks A. \nScholz for the support during the XRD measurements.  We would like to thank Romy Aniol for the \nTEM specimen preparation. 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Mater. 170 , L243 (1997).  \n[101 ] Strictly speaking this is valid only for [001] ori ented systems for all others, if the cubic \nlattice constant values of afilm  and asubstrate  are almost identical.  27  Supplemental Material: \nLow damping and microstructural perfection of sub-4 0nm-thin yttrium iron garnet films \ngrown by liquid phase epitaxy  \n \nCarsten Dubs, 1 Oleksii Surzhenko, 1 Ronny Thomas, 2 Julia Osten, 2 Tobias Schneider, 2 Kilian Lenz, 2 \nJörg Grenzer, 2 René Hübner,2 Elke Wendler 3 \n \n1 INNOVENT e.V. Technologieentwicklung, Prüssingstr.  27B, 07745 Jena, Germany \n2 Institute of Ion Beam Physics and Materials Resear ch, Helmholtz-Zentrum Dresden-Rossendorf, \nBautzner Landstr. 400, 01328 Dresden, Germany \n3 Institut für Festkörperphysik, Friedrich-Schiller- Universität Jena, Helmholtzweg 3, 07743 Jena, \nGermany \n \n \nI. STRAIN CALCULATIONS  \n \nIn the following we will derive the in-plane (horiz ontal) stress σ||  as a function of the out-of-plane \n(vertical) lattice misfit ⊥\nfilmdδ  for a [111] oriented cubic system. These calculati ons are based on the \nelasticity theory following mainly Hinckley [70 ], Sander [71 ] and Ortiz et al . [61 ]. Assuming a fully \npseudomorphic system there is only one parameter to  be determined: The out-of-plane lattice misfit \n⊥\nfilmdδ . This value can be directly  obtained from the data of the HR-XRD measurements a nd/or from \nthe corresponding simulations of the symmetrical (4 44) and (888) reflections. \nThe vertical and parallel lattice misfits can be ca lculated by \n , 0||\nsubstrate||\nsubstrate||\nfilm ||\nfilm\nsubstratesubstrate film\nfilm =−=−=⊥⊥ ⊥\n⊥\ndd dddd dd δ δ  (S1) \nwhere dk\ni with i = [substrate, (pseudomorph) film or (cubic) relaxe d film] is the (measured) net \nplane distances for the k = ⊥ (vertical) or || (parallel) direction with respect  to the substrate surface. \nIgnoring any dynamical diffraction effects, the out -of-plane lattice misfit can be directly determined  \nfrom the measurement as follows: \n \nB qq\nqq qq dθθδ δtansubstrate substratesubstrate film\nfilm film∆− =∆− =−− = − =⊥⊥\n⊥⊥ ⊥\n⊥ ⊥, (S2) \nwhere q⊥film  and q⊥substrate are the derived peak positions in the Q-space of th e thin film and the \nsubstrate, respectively. ∆θB is the (kinematical) Bragg angular difference “thi n film – substrate” and \nθB is the Bragg Peak position of the substrate, respe ctively. These formulas follow directly from the \nderivative of Bragg’s law. \nThe in- and out-of-plane strains are given as follo ws: \n .||\nfilm  relaxed||\nfilm  relaxed||\nfilm ||\nfilm  relaxedfilm  relaxed film\ndd d\ndd d −=−=⊥⊥ ⊥\n⊥ε ε  (S3) \nThe general relationship between stress and strain is defined as follows: \n ,kl ijkl ij ε σ C=  (S4) \nwhere σij  are the stress, ε kl  the strain and Cijkl  the second order stiffness tensors and the summati on is \ndone over the repeated indices. The subscripts ij and  kl  refer to the axes of the coordinates system of  28  the unit cell (1,2,3 = x,y,z ). The samples under investigations have a [111] ou t-of-plane orientation; \ntherefore, the corresponding rotation matrices have  to be applied: \n ijkll k j i CUUUU Cδ γ β α αβγδ=′ . (S5) \nFor the [111] oriented surfaces U 111  yields to \n \n\n\n\n −\n=\n310\n3231\n21\n6131\n21\n61\n111U . (S6) \n \n \n \nFigure S1 : Representation of the cubic unprimed and the rota ted, primed coordinate system for an \n<111> oriented thin film; where x´, y´ correspond to the in-plane (||) directions and z´ to the out-of-\nplane ( ⊥) direction; after [71 ]. \n \n \nThe in-plane stress ||σ′ can be expressed in terms of the out-of-plane latt ice misfit ⊥\nfilmdδ obtained by \nXRD measurements.  \n \nFor a cubic pseudomorphic system we can write:  \n 1313 1212 1111 11 ε ε ε σ c c c + + =  (S7) \n ( )⊥′+′+′=′ ε ε σ12 || 12 11 || c c c  (S8) \nwith  \n ⊥′− =ε ν ε||  (S9) \nresulting in: \n ||\n44 12 1112 11\n44 ||4 226 ε σc c cc cc+ ++=′ . (S10) \n \n  29   \nTaking the corresponding rotation matrices and rela tionships into account  [101 ]: \n ⊥ ⊥ ⊥\n+− =+=film 111111\n||\nfilm 1111and11d d δννε δνε , (S11) \nwhere \n .4 4 24 2\n44 12 1144 12 11 111\nc c cc c c\n− ++ +=ν  (S12) \n \nThe in-plane stress can be now expressed in terms o f the out-of-plane lattice misfit by: \n ⊥− =′film 44 || 2 dcδ σ . (S13) \nHere, c44  is the component from the stiffness tensor and ⊥\nfilmdδ is the out-of-plane lattice misfit as \ndefined above. \n \n \nII. ANISOTROPY CALCULATIONS FOR (111) ORIENTED EPIT AXIAL GARNET FILMS  \n \nThe stress-induced anisotropy parameter for the cub ic (111) orientation can be calculated according \nto [94] by \n 111||23λ =Kσ σ′ − , (S14) \nwhere σ´||  is the above calculated in-plane stress for {111} oriented thin films, and λ 111  is the \ncorresponding magnetostriction constant. \nThe stress-induced anisotropy parameter is therefor e given by: \n 111 film 443 λdc=Kσ⊥δ . (S15) \nThe perpendicular magnetic anisotropy field can be calculated according to [43]: \n growth cub 2 H+ H+ H= Hstress ⊥ . (S16) \nAssuming negligible growth-induced contributions Hgrowth  and applying the cubic anisotropy field \nfor (111) film orientation obtained by FMR measurem ents \n \nscubMK= H4\n34− , (S17) \nand taking into account the stress-induced anisotro py field \n \ns sσ\nstressMλ=MK= H111||3 2 σ′\n− , (S18) \nthe effective perpendicular anisotropy field result s in \n \nsMλ +KH39 4111|| 4\n2σ′\n− =⊥ . (S19)  30  From the resonance conditions for the perpendicular  (M ||  [111 ]) magnetized epitaxial thin film, the \neffective saturation magnetization can be obtained by [64 ] \n \n+ − − =⊥\ns ssMK\nMKM Hf2 4\neff2\n344πω,  (S20) \nfrom which the effective saturation magnetization c an be calculated by \n ⊥ −2 eff eff 4 4 H πM= πM=Hs . (S21) \n \nIII. FERROMAGNETIC RESONANCE  \n \nFrom the free energy density given by Eq. (1) of th e main text, the resonance equations have been \ncalculated applying the approach of Baselgia et al.  [80 ]. The resonance conditions for the frequency-\ndependences with field out-of-plane ( f⊥) and in-plane ( f|| ) read: \n \n \n\n\n+ − − \n\n− − =⊥MK\nMKM HMKM H fe e|| 2 4\nff4\nff 23443442π ππγ, (S22)  \n ( ) ( ) ( )\n\n\n\n\n\n− − − − + ×\n\n\n− − = ϕ ϕϕ π ϕϕπγ3 cos 2 cos24 2cos2\n222\n4 2 || 2 4\nff|| 2\n||MK\nMK\nMKM HMKH fu e u\n  (S23). \n \nExamples of angle- and frequency-dependent FMR meas urements with out- and in-plane \nconfiguration of the magnetic bias field are shown in Figure S2.  31  -30° 0° 30° 60° 90° 120° 345f = 10 GHz \n11 nm \n21 nm \n30  nm \n42 nm \n Fits Hres (kOe)\nθH\n0 5 10 15 010 20 30 40 f (GHz)\nH (kOe) \n0 5 10 15 010 20 30 40 \nθH=0° f(GHz)\nH (kOe)  11 nm \n 21 nm \n 30 nm \n 42 nm \n Fit (11 nm) θH=90° \n 11 nm \n 21 nm \n 30 nm \n 42 nm (a) \n(b) \n(c) 2.76 2.77 2.78 2 .7 9 2 .8 0 FMR- S igna l  (arb . units)\nH (kOe) 42 nm \nf = 10 GHz \n4.30 4.32 4.34 4.36 4.38 4 .4 0 FMR-Signa l  (arb . units)\nH (kOe) 11 nm \nf = 8 GHz θH\nφHH→\n[110] _ [112] _M→\nY IG(111) φθ\n \nFigure S2. (a) Polar angular dependencies of the FM R measured at f = 10 GHz. The inset shows the \nFMR coordinate system. Solid lines are fits accordi ng to the resonance equation. (b) Frequency \ndependencies of the resonance field measured with f ield in-plane and (c) out-of-plane. The solid \nblack line is a fit to the 11 nm dataset. Other fit  curves have been omitted for visual clarity. Inset s \nshow FMR spectra and the indicated positions includ ing Lorentzian fits. "    },    {        "title": "1911.12017v2.Ellipticity_and_Dissipation_Effects_in_Magnon_Spin_Valves.pdf",        "content": "Ellipticity and Dissipation E\u000bects in Magnon Spin Valves\nJiansen Zheng,1Andreas R uckriegel,1Scott A. Bender,1and Rembert A. Duine1, 2, 3\n1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n3Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: March 2, 2020)\nWe consider alignment-dependent spin and heat transport across a magnon spin valve in the\ntunneling regime, i.e., a junction consisting of two weakly coupled ferromagnetic insulators. We\ndetermine the di\u000berence in spin and heat conductance between the parallel and antiparallel con\fg-\nuration of the magnetization direction. The dependence of these conductances on both the Gilbert\ndamping and ellipticity is studied. We \fnd that both magnon ellipticity and dissipation open\nchannels for magnons to tunnel through in the antiparallel con\fguration. Our results highlight an\nimportant di\u000berence between electronic and magnon spin transport in spin-valve structures and may\nbe important for the development of devices based on magnetic insulators.\nI. INTRODUCTION\nSpintronics based on spin-polarized charge currents has\nled to a boost in information storage technology with the\ndiscovery of giant magnetoresistance (GMR) in antiferro-\nmagnetically coupled Fe =Cr superlattices1,2. GMR arises\nfrom the spin-dependent transmission of the conduction\nelectron. Magnons, the quanta of collective excitations in\nmagnetically ordered systems, can carry spin current in\nmagnetic insulators in the absence of any charge current,\ne.g., in the spin Seebeck e\u000bect3, where the magnons are\ndriven by a thermal bias, or in nonlocal setups in which\nthe magnons are biased electrically using the spin Hall ef-\nfect in adjacent normal metals4. Magnon spin transport\nis promising, for example, to improve the power e\u000eciency\nof logic devices5{7and for neuromorhpic computing8,9.\nTo \fnd analogies of GMR in magnon spin trans-\nport, spin-valve structures that encompass magnetic in-\nsulators have recently been studied experimentally and\ntheoretically10{12. Wu et al. have observed that the spin\nSeebeck e\u000bect of a heterostructure made of two ferro-\nmagnetic insulators, namely yttrium iron garnet (YIG),\nseparated by a nonmagnetic heavy metal layer, depends\non the relative orientation of the magnetizations of the\ntwo magnetic insulators13. The di\u000berence in spin Seebeck\nsignal between parallel and antiparallel con\fgurations is\nobserved to decrease signi\fcantly as the temperature is\nlowered. In a recent theoretical work, a Green's function\nformalism for magnon tunneling driven by a temperature\nbias across a ferromagnetic junction has been developed\nand applied to compute the diode properties of the tun-\nneling magnon current14. A key aspect of this study is\nthe inclusion of magnon-magnon interactions that are ex-\nploited for the recti\fcation and negative di\u000berential spin\nSeebeck e\u000bects. Furthermore, a tunable spin Seebeck\ndiode based on a magnetic junction structure in which\nthe tunneling spin current can be turned on and o\u000b by\ncontrolling the magnetization orientation has also been\ntheoretically proposed15.In this work, we study the alignment dependence of\nmagnon heat and spin transport across a heterostruc-\nture consisting of two ferromagnetic insulators that are\nweakly exchange coupled, e.g., by a nonmagnetic spacer\nlayer that mediates exchange interactions. The setup we\nconsider is illustrated in Fig. I. The ferromagnetic in-\nsulators act as reservoirs for magnons. The magnons\ncan be coherently driven by ferromagnetic resonance,\nor incoherently generated with an electrical or ther-\nmal bias using adjacent normal metals16. We focus on\nthe e\u000bect of the ellipticity of the magnetization pre-\ncession, which is usually caused by anisotropies, and\nalso on the e\u000bects of dissipation that we parametrize\nwith a Gilbert damping constant. The latter is a phe-\nnomenological parameter that characterizes the decay of\nmagnons. The ellipticity of precession has been shown\nto strongly a\u000bect the parametric excitation of magnons\nin ferromagnetic resonance experiments17, and plays a\nrole in Rayleigh-Jeans condensation of pumped magnons\nin thin-\flm ferromagnets18. Moreover, at the quantum-\nmechanical level, the ellipticity leads to squeezed ground\nstates and, in the case of antiferromagnets, entanglement\nbetween di\u000berent sublattice magnons19,20. Meanwhile, a\nlow Gilbert damping has also been demonstrated to en-\nable long-distance spin transport in magnetic insulators\nsuch as yttrium iron garnet4. Neither the in\ruence of\ndamping nor ellipticity has, however, been considered for\nmagnon tunneling in the insulating spin valve structure\nconsidered here.\nIn our setup, magnons tunnel between the ferromag-\nnets due to the weak exchange coupling and carry heat\nand spin currents in response to applied temperature or\nmagnon chemical potential di\u000berences. For circularly po-\nlarized magnons, conservation of spin forbids magnon\ntunneling in the antiparallel con\fguration. We \fnd that\nanisotropies and dissipation, both of which break spin\nconservation, lead to tunneling currents even in the an-\ntiparallel con\fguration. The di\u000berence between circular\nand elliptical case has no straightforward analog in elec-arXiv:1911.12017v2  [cond-mat.mes-hall]  3 Mar 20202\nFIG. 1. Illustration of a magnetic junction consisting of two\nferromagnetic insulators, which interact with each other with\nweak exchange coupling U. The left (right) insulator has tem-\nperatureTL(TR) and spin accumulation \u0016L(\u0016R). The layer\nin green is a nonmagnetic insulator and is thin enough for the\nmagnons to tunnel across it. The left insulator is the \fxed\nlayer, with the red arrow denoting an up spin, while the right\ninsulator is a free layer in which the magnetization can be\ntuned from a parallel con\fguration (denoted with the red ar-\nrow) to an anti-parallel con\fguration (denoted with the blue\narrow).\ntronic spin valves, as in these latter systems the spin of\nthe electrons at the Fermi level is usually approximately\nconserved, and tunneling current is determined by their\nspin-dependent density of states. Our result therefore\nshows that the di\u000berence between magnon currents in\nthe parallel and antiparallel con\fguration is governed by\ndi\u000berent physics than for metallic structures, which may\nbe useful in designing devices that exploit the tunability\nof the tunneling current.\nThe remainder of the paper is organized as follows:\nIn Sec. II, we introduce the model that we use for our\nsetup. The magnon tunneling currents in the presence of\nboth precession ellipticity and Gilbert damping for both\nparallel and antiparallel con\fgurations are calculated in\nSec. III. In Sec. IV, we numerically calculate the tun-\nneling conductances and discuss their dependencies on\nthe ellipticity and dissipation. Section V summarizes our\nmain \fndings and conclusions. Lastly, the Appendix out-\nlines the derivation of the tunneling currents using rate\nequations.II. MODEL HAMILTONIAN\nA. Lead Hamiltonians\nThe magnetization dynamics in the bulk of each insu-\nlating lead is modeled by the Hamiltonian\nHX=\u00001\n2X\nijJX;ijSX;i\u0001SX;j\u0000~\rX\u00160HXX\niSz\nX;i\n\u00001\n2X\ni\u0014\nKX;x\u0000\nSx\nX;i\u00012+KX;y\u0010\nSy\nX;i\u00112\u0015\n;(1)\nwhereX=L=R denotes the left/right lead and i;jlabel\nthe lattice sites. Here, JX;ij are nearest-neighbor ex-\nchange interactions with strength JX>0, whereasKX;x\nandKX;yare anisotropy constants. Lastly, \u00160HXare the\nmagnetic \felds in the bulk of the leads with gyromagnetic\nratio\rX.\nThe spin operators SX;iare bosonized via a Holstein-\nPrimako\u000b transformation21. ForHX>0 we assume that\nthe magnetic order parameter points in zdirection, so\nthat\nS+\nX;i=Sx\nX;i+iSy\nX;i=p\n2SX\u0002\nbX;i+O(S\u00001\nX)\u0003\n;(2a)\nSz\nX;i=SX\u0000by\nX;ibX;i; (2b)\nwhereSXis the spin quantum number of the magnetic\nmoments in lead X, andbX;iandby\nX;iare the magnon an-\nnihilation and creation operators that satisfy the bosonic\ncommutation relations [ bX;i;by\nX0;j] =\u000eX;X0\u000ei;j. Con-\nversely, for HX<0 we assume that the magnetic order\nparameter points in \u0000zdirection and apply the following\nHolstein-Primako\u000b transformation:\nS+\nX;i=Sx\nX;i+iSy\nX;i=p\n2SXh\nby\nX;i+O(S\u00001\nX)i\n;(3a)\nSz\nX;i=\u0000SX+by\nX;ibX;i: (3b)\nIn the bulk of each lead, we may expand the magnon\noperators in a Fourier series as\nbX;i=1pNXX\nkeik\u0001RibX;k; (4)\nwhereNXdenotes the number of magnetic moments in\nthe leadX. Then the spin Hamiltonian (1) becomes\nHX=X\nk\u0014\nAX;kby\nX;kbX;k+BX\n2\u0010\nby\nX;kby\nX;\u0000k+ h.c.\u0011\u0015\n;\n(5)\nwhere we dropped a constant contribution to the ground\nstate energy as well as O(S\u00001=2\nX) corrections containing\nhigher powers of the magnon operators. The coe\u000ecients\nof the Hamiltonian (5) are given by\nAX;k=~\rX\u00160jHXj+JXSXa2\nXk2\u0000SX\n2(KX;x+KX;y);\n(6a)\nBX=\u0000SX\n2(KX;x\u0000KX;y); (6b)3\nFIG. 2. Semiclassical depiction of a spin Sthat precesses\nelliptically around a magnetic \feld H. The solid blue line is\nthe elliptical precession, whereas the dashed gray line corre-\nsponds to a circular precession. Because the length of the spin\nis conserved, its projection onto the direction of the magnetic\n\feld is not constant during the elliptical precession. There-\nfore, this spin projection is no longer a good quantum number\nfor elliptical magnons.\nregardless of whether we assume HX>0 orHX<0\nand employ the respective Holstein-Primako\u000b transfor-\nmation (2) or (3). In writing down Eq. (6a), we further-\nmore assumed that only long-wavelength magnons with\naXjkj\u001c1 are relevant, where aXis the lattice constant\nof leadX. The quadratic magnon Hamiltonian (5) is\ndiagonalized via a Bogoliubov transformation:\n\u0012bX;k\nby\nX;\u0000k\u0013\n=\u0012\nuX;k\u0000vX;k\n\u0000vX;kuX;k\u0013\u0012\fX;k\n\fy\nX;\u0000k\u0013\n; (7)\nwhere\nuX;k=s\nAX;k+EX;k\n2EX;k; (8a)\nvX;k=BX\njBXjs\nAX;k\u0000EX;k\n2EX;k; (8b)\nwhere\nEX;k=q\n(AX;k+BX) (AX;k\u0000BX): (9)\nThe operators \fX;kand\fy\nX;kcreate and destroy Bogoli-\nubov quasiparticles and obey the bosonic commutation\nrelations [\fX;k;\fy\nX0;k0] =\u000eX;X0\u000ek;k0. Semiclassically, a\nBogoliubov quasiparticle created by \fy\nX;kcorresponds to\nan elliptical spin wave; in contrast, a magnon created\nbyby\nX;kcorresponds to a circular spin wave. The Bogoli-\nubov quasiparticles are often also referred to as magnons,\nor as elliptical or squeezed magnons19,20. For an ellipti-\ncal spin wave, the zcomponent of the spin is not con-\nserved and hence not a good quantum number, unlike\nfor a circular spin wave. This is illustrated semiclassi-\ncally in Fig. 2. With the Bogoliubov transformation (7),the magnon Hamiltonian (5) becomes\nHX=X\nk\u0014\nEX;k\fy\nX;k\fX;k+1\n2(EX;k\u0000AX;k)\u0015\n:(10)\nNote that this Hamiltonian is only valid as long as the\nquasiparticle dispersion (9) is real, i.e., for ~\rX\u00160jHXj>\nSKX;x;SKX;y. If this is not satis\fed, our original as-\nsumption that the magnetic order points in \u0006zdirection\nis not correct and we have to expand around a di\u000berent\nground state. However, for the remainder of this work we\nwill assume that ~\rX\u00160jHXj> SKX;x;SKX;y, so that\nthe quasiparticle Hamiltonian (10) is stable.\nB. Tunneling Hamiltonian\nThe tunneling between the leads is facilitated by a\nlead-lead exchange interaction of the form\nHT=\u0000X\nijUijSL;i\u0001SR;j; (11)\nwhere the exchange coupling Uijis assumed to be small\ncompared to the bulk energy scales and only \fnite for i;j\nclose to the interface. The microscopic origin of such an\ninteraction could either be direct exchange mediated by\nthe conduction electrons in the normal metal or an indi-\nrect superexchange interaction via the ions in the non-\nmagnetic spacer layer.\n1. Parallel Con\fguration\nIn the parallel con\fguration, we take the form (2) for\nboth leads. Then the tunneling Hamiltonian (11) be-\ncomes\nHP\nT=\u0000p\nSLSRX\nijUij\u0010\nby\nL;ibR;j+by\nR;jbL;i\u0011\n;(12)\nwhere we dropped constants and higher order magnon\ncorrections as before, as well as an on-site energy shift\nfor the magnons in each lead. This is justi\fed by our as-\nsumption that the lead-lead exchange is small compared\nto the bulk energy scales. After applying both Fourier\nand Bogoliubov transformations, Eqs. (4) and (7) respec-\ntively, we \fnd\nHP\nT=\u0000r\nSLSR\nNLNRX\nkk0\n\u0002\u0012\nV(n)\nk;k0\fy\nL;k\fR;k0\u0000V(a)\nk;k0\fy\nL;k\fy\nR;\u0000k0+ h.c.\u0013\n;\n(13)\nwhere\nV(n)\nk;k0=Uk;\u0000k0(uL;kuR;k0+vL;kvR;k0); (14a)\nV(a)\nk;k0=Uk;\u0000k0(uL;kvR;k0+vL;kuR;k0) (14b)4\nare the normal ( n) and anomalous ( a) tunneling am-\nplitudes, with the Fourier transform of the lead-\nlead exchange coupling Uk;k0= (U\u0000k;\u0000k0)\u0003=P\ni2LP\nj2Re\u0000ik\u0001Ri\u0000ik0\u0001RjUij. Note that the anomalous\ncoupling (14b) is only \fnite when the magnon ellipticity\nis \fnite, leading to qualitatively new physics in this case.\n2. Antiparallel Con\fguration\nIn the antiparallel con\fguration, we take the Holstein-\nPrimako\u000b transformations (2) for the left and (3) for the\nright lead, yielding\nHAP\nT=\u0000p\nSLSRX\nijUij\u0010\nby\nL;iby\nR;j+bL;ibR;j\u0011\n;(15)\nwithin the same approximations as for the parallel con-\n\fguration considered in the preceding Sec. II B 1. As be-\nfore, we apply the Fourier and Bogoliubov transforma-\ntions given, respectively, in Eqs. (4) and (7) to obtain\nHAP\nT=\u0000r\nSLSR\nNLNRX\nkk0\n\u0002\u0012\nV(n)\nk;k0\fy\nL;k\fy\nR;\u0000k0\u0000V(a)\nk;k0\fy\nL;k\fR;k0+ h.c.\u0013\n(16)\nNote that from the comparison of the magnon tunneling\nHamiltonian between the parallel and antiparallel case,\nEq. (12) and Eq. (15), respectively, it is clear that these\ntwo situations di\u000ber qualitatively. In the parallel case,\none deals with a tunneling Hamiltonian that is also en-\ncountered in the electron transport, whereas in the an-\ntiparallel case, the tunneling corresponds to creation or\ndestruction of a pair of circular magnons. We also stress\nthat the magnon ellipticity, being related to the breaking\nof magnon number, i.e., spin conservation, has no ana-\nlog in electronic systems, where the electron number is\nalways conserved. Therefore, the e\u000bects discussed here\nhave no direct analog in electronic valves. From Eq. (15)\nit is furthermore obvious that there is no spin transport\nin the antiparallel con\fguration without breaking of the\ntotal spin conservation, either by anisotropies or damp-\ning. In this respect, the undamped, circular magnon spin\nvalve resembles a half-metallic system that only trans-\nmits spin when the magnetizations of the two magnets\nare aligned parallel.\nIII. TUNNELING CURRENTS\nThe tunneling currents can be obtained from the rate\nequations for the distribution function of the Bogoliubov\nquasiparticles in each lead,\nnX;k=D\n\fy\nX;k\fX;kE\n=fB\u0012EX;k\u0000\u0016X\nkBTX\u0013\n(17)wherefB(x) = 1=(ex\u00001) is the Bose function, and the\nsecond equality holds in a steady state in which lead X\nis kept at temperature TXand chemical potential \u0016X.\nTo allow for \fnite damping in each lead, we recast the\nsteady state distribution function (17) as\nnX;k=Z1\n\u00001d\u000f\u000e(\u000f\u0000EX;k)fB\u0012\u000f\u0000\u0016X\nkBTX\u0013\n: (18)\nWithin the Gilbert damping phenomenology, we may\nthen add dissipation by broadening the Dirac distribu-\ntions according to14,22\n\u000e(\u000f\u0000EX;k)!A(\u000f\u0000EX;k)\u00111\n\u0019\u000b\u000f\n(\u000f\u0000EX;k)2+ (\u000b\u000f)2;\n(19)\nwhere\u000bis the bulk Gilbert damping parameter.\nDetails of the derivation of the tunneling currents from\nkinetic equations for the quasiparticle distribution func-\ntions can be found in the Appendix; here we only state\nthe results. Labeling the parallel/antiparallel con\fgura-\ntions withY=P=AP , we \fnd the following expressions\nfor the energy current\nIY\nE=2\u0019\n~Z1\n\u00001d\u000f\u000f\n\u0002(\nDY\nE(\u000f)\u0014\nfB\u0012\u000f\u0000\u0016L\nkBTL\u0013\n\u0000fB\u0012\u000f\u0000\u0016R\nkBTR\u0013\u0015\n+~DY\nE(\u000f)\u0014\nfB\u0012\u000f\u0000\u0016L\nkBTL\u0013\n\u0000fB\u0012\u000f+\u0016R\nkBTR\u0013\u0015)\n;\n(20)\nand the spin current\nIY\nS=2\u0019Z1\n\u00001d\u000f\n\u0002(\nDY\nS(\u000f)\u0014\nfB\u0012\u000f\u0000\u0016L\nkBTL\u0013\n\u0000fB\u0012\u000f\u0000\u0016R\nkBTR\u0013\u0015\n+~DY\nS(\u000f)\u0014\nfB\u0012\u000f\u0000\u0016L\nkBTL\u0013\n\u0000fB\u0012\u000f+\u0016R\nkBTR\u0013\u0015)\n;\n(21)\n\rowing from the left to the right lead. Here, DP=AP\nE=S(\u000f)\nare the normal tunneling densities of state, explicitly\ngiven by\n(\nDP=AP\nE (\u000f)\nDP=AP\nS (\u000f))\n=SLSR\nNRNLX\nkk0\f\f\fV(n=a)\nk;k0\f\f\f2(1\u0010\nu2\nR;k0+v2\nR;k0\u0011)\n\u0002A(\u000f\u0000EL;k)A(\u000f\u0000ER;k0): (22)\nNote that their contributions to currents (20) and (21)\nvanish if both leads are at the same temperature and\nchemical potential. On the other hand, ~DP=AP\nE=S(\u000f) are\nanomalous tunneling densities of state that arise because5\nthe Gilbert damping breaks the number conservation of\nthe Bogoliubov quasiparticles. Hence it gives rise to a\nspin current even when both leads are at the same tem-\nperature and chemical potential; it vanishes only if both\nleads are in true thermal equilibrium at the same temper-\nature and vanishing chemical potential. These anomalous\ntunneling densities of state are\n(\n~DP=AP\nE (\u000f)\n~DP=AP\nS (\u000f))\n=SLSR\nNRNLX\nkk0\f\f\fV(a=n)\nk;k0\f\f\f2(1\u0010\nu2\nR;k0+v2\nR;k0\u0011)\n\u0002A(\u000f\u0000EL;k)A(\u000f+ER;k0): (23)\nIt is instructive to consider the limit of conserved quasi-\nparticles (\u000b= 0+) as well as the limit of circular magnons\nin more detail. If there is no dissipation, the anomalous\ncontributions to the currents vanish because energy con-\nservation strictly demands EL;k+ER;k0= 0, which can\nnever be satis\fed since both of these energies are posi-\ntive. Finite damping softens this restriction by allowing\nenergy (and spin) transfer to a thermal bath, thereby\nopening up another channel for energy and spin transfer\nbetween the leads. In the limit of circular magnons, i.e.,\nwhen there are no anisotropies that break rotation sym-\nmetry around the zdirection, we may set uX;k= 1 and\nvX;k= 0; see Eqs. (8). Then ~DP\nE=S(\u000f) = 0 =DAP\nE=S(\u000f).\nThis re\rects the conservation of the total spin Sz\nL+Sz\nR\nin the absence of anisotropies. In the parallel con\fgura-\ntion, tunneling is in this case only allowed for the process\nin which a magnon carrying spin \u0000~is destroyed in one\nlead and another magnon carrying spin \u0000~is created in\nthe other lead. Conversely, in the antiparallel con\fgura-\ntion magnons carry spin \u0000~in the left lead and + ~in the\nright lead. Thus spin conservation only allows anomalous\nprocesses in which magnon pairs in the left and right lead\nare simultaneously destroyed or created. As this process\nviolates energy conservation, it is only possible in the\npresence of dissipation. Therefore, there are no energy\nand spin currents in the antiparallel con\fguration with-\nout either damping (enabling pair creation/annihilation\nprocesses) or breaking of spin conservation (enabling nor-\nmal hopping). This is further illustrated in Fig. 3.\nA. Tunneling Conductances\nIf the biasing is su\u000eciently small, i.e., \u0001 T=TL\u0000TR\u001c\nT, whereT=1\n2(TL+TR) is the average temperature,\nand\u0016L=R\u001cEL=R;k=0, we may linearize the Bose func-\ntions appearing in the currents (20) and (21), yielding\nIY\nE=\u0014Y\u0001T+ \u0005Y(\u0016L\u0000\u0016R) +\rY\nE(\u0016L+\u0016R);(24)\nIY\nS=LY\u0001T+\u001bY(\u0016L\u0000\u0016R) +\rY\nS(\u0016L+\u0016R):(25)\nHere,\n\u0014Y=\u0019\n2~kBT2Z1\n\u00001d\u000f\u000f2\nsinh2\u0010\n\u000f\n2kBT\u0011h\nDY\nE(\u000f) +~DY\nE(\u000f)i\n;\n(26a)\n(a)\nspin current spin current(b)FIG. 3. Tunneling processes allowed by spin conservation for\ncircular magnons. (a) Hopping of a magnon carrying spin \u0000~\nfrom the right to the left lead in the parallel con\fguration.\nThe inverse process of hopping from left to right lead is also\npossible. (b) Pair creation of a magnon carrying spin \u0000~in\nthe left lead and a magnon carrying spin + ~in the right lead\nin the antiparallel con\fguration. The inverse process of pair\nannihilation is also allowed. However, while allowed by spin\nconservation, both pair creation and annihilation processes\nare forbidden by energy conservation if there is no dissipa-\ntion. The spin currents in (a) and (b) are polarized in the\nzdirection. For the inverse processes, the spin currents \row\nin the opposite direction. The blue (red) arrows indicate the\nspin change associated with the creation (annihilation) of a\ncircular magnon.\nis the thermal conductance,\nLY=\u0019\n2kBT2Z1\n\u00001d\u000f\u000f\nsinh2\u0010\n\u000f\n2kBT\u0011h\nDY\nS(\u000f) +~DY\nS(\u000f)i\n;\n(26b)\nis the spin Seebeck conductance,\n\u0005Y=\u0019\n2~kBTZ1\n\u00001d\u000f\u000f\nsinh2\u0010\n\u000f\n2kBT\u0011DY\nE(\u000f); (26c)\nis the spin Peltier conductance, and\n\u001bY=\u0019\n2kBTZ1\n\u00001d\u000f1\nsinh2\u0010\n\u000f\n2kBT\u0011DY\nS(\u000f); (26d)\nis the spin conductance. Lastly,\n\rY\nE=\u0019\n2~kBTZ1\n\u00001d\u000f\u000f\nsinh2\u0010\n\u000f\n2kBT\u0011~DY\nE(\u000f); (26e)\n\rY\nS=\u0019\n2kBTZ1\n\u00001d\u000f1\nsinh2\u0010\n\u000f\n2kBT\u0011~DY\nS(\u000f) (26f)\nare the additional energy and spin loss or gain terms aris-\ning because the \fnite damping breaks the number con-\nservation of Bogoliubov quasiparticles. Since they do not\nvanish when both leads are mutually equilibrated, these\nterms are not part of the transport current and should\nrather be identi\fed with the spin and energy lost to or\ngained from the thermal bath that provides the dissipa-\ntion, which is ultimately the crystal lattice. Microscop-\nically, they correspond to the simultaneous creation or6\nannihilation of a magnon in the left and a magnon in the\nright lead; the required energy and angular momentum\nis provided by the lattice. Therefore, these terms de-\nscribe energy and spin currents \rowing from the lattice\nto both leads, instead of currents \rowing from one lead\nto the other. Because the total angular momentum of\nspins and lattice is conserved, this additional spin trans-\nfer should be experimentally detectable as torques on the\nwhole sample.\nNote also that the spin Seebeck and Peltier conduc-\ntances, Eqs. (26b) and (26c), respectively, are not On-\nsager reciprocals of each other, ~\u0005Y6=TLY. There are\ntwo independent reasons for this: the breaking of time-\nreversal symmetry by the dissipation and the breaking\nof spin conservation by the anisotropies. While the for-\nmer opens up an new channel for bath-assisted energy\ntransfer, namely the pair creation/annihilation processes\ncontained in ~DY\nE=S(\u000f), the latter allows for changes in\nspin without accompanying changes in energy, resulting\ninDY\nE(\u000f)6=DY\nS(\u000f) and ~DY\nE(\u000f)6=~DY\nS(\u000f).\nIV. NUMERICAL RESULTS AND DISCUSSION\nIn realistic systems, the interfaces between layers of\ndi\u000berent materials are usually rough. Such rough inter-\nfaces break the momentum conservation of incident par-\nticles, e\u000bectively randomizing the momenta of the scat-\ntered particles. Therefore, we approximate the interface\ncoupling as Uk;k0\u0019U= const. Furthermore, we work\nin the thermodynamic limit where1\nNLP\nk=\u0000aL\n2\u0019\u00013R\nd3k\nand1\nNRP\nk0=\u0000aR\n2\u0019\u00013R\nd3k0, and take both leads to be\nof the same material, so that we can drop the L=R label.\nThen ~DP=AP\nE (\u000f) = ~DP=AP\nE (\u0000\u000f), resulting in \rP=AP\nE = 0\n[see Eqs. (23) and (26e)]; i.e., there is no additional en-\nergy transfer to the lattice. In keeping with the long-\nwavelength expansion used in Sec. II A, we only con-\nsider low temperatures T\u001cJS=kB. For yttrium iron\ngarnet23, this means T\u001c40 K.\nThe tunneling conductances (26) are displayed in Fig. 4\nas functions of the in-plane anisotropy, i.e., of the spin-\nwave ellipticity. In the parallel con\fguration shown in\nFig. 4(a), all conductances depend only weakly on the\nmagnitude of the anisotropy. With the exception of\nthe dissipation-assisted spin conductance \rP\nS, they de-\ncrease for hard-axis ( Ky<0) and increase for easy-plane\n(Ky>0) anisotropy. This can be attributed to the\nmagnon gap increasing or decreasing, respectively, which\nincreases or decreases the overall magnon population.\nThe strong increase and eventual divergence of the spin\nconductance for Ky!~\r\u00160Hsigni\fes the divergence of\nthe Bose distribution for vanishing spin-wave gap, and is\na precursor to the magnetic reorganization transition in\nwhich the magnetization tilts into the anisotropy plane.\nThe additional dissipation-assisted spin conductance \rP\nS\nmirrors this behavior, but also increases for hard-axis\nanisotropies, in contrast to all other conductances. Thereason for this is that it is an o\u000b-resonance process that\nis less sensitive to the exact value of the gap than the\nresonant ones, whereas its magnitude is determined by\nthe strength of the anisotropies. Because of this, it is\nalso three to four orders of magnitude smaller than the\nother conductances. Also, note that the breaking of the\nOnsager reciprocity of spin Seebeck and Peltier conduc-\ntances by the spin-wave ellipticity and the Gilbert damp-\ning is negligible in the parallel con\fguration.\nIn the antiparallel con\fguration displayed in Fig. 4(b),\non the other hand, the anisotropy dependence of the\nconductances is more pronounced. In agreement with\nthe discussion in Sec. III, spin and spin Peltier conduc-\ntances are in this case both zero if there is no anisotropy.\nHowever, chemical-potential driven spin transfer is still\npossible in this case because the dissipation-assisted spin\nconductance \rAP\nSis \fnite for Ky= 0. Apart from \rAP\nS,\nwhich decreases for Ky<0, all conductances increase\naway from Ky= 0, although they stay small compared\nto the conductances in the parallel con\fguration shown\nin Fig. 4. Note that, as the spin-wave gap closes, the\nspin conductance diverges and the breaking of Onsager\nreciprocity becomes visible.\nTo quantify the e\u000bect of Gilbert damping and spin-\nwave ellipticity on the magnon spin valve, we introduce\nmagnetothermal conductance (MTC), magnetospin con-\nductance (MSC), magneto-Seebeck conductance (MLC),\nand magneto-Peltier conductance (MPC) ratios as fol-\nlows:\nMTC =\u0014P\u0000\u0014AP\n\u0014P; (27a)\nMSC =\u001bP+\rP\nS\u0000\u001bAP\u0000\rAP\nS\n\u001bP+\rP\nS; (27b)\nMLC =LP\u0000LAP\nLP; (27c)\nMPC =\u0005P+\rP\nE\u0000\u0005AP\u0000\rAP\nE\n\u0005P+\rP\nE: (27d)\nIn the absence of dissipation and spin-wave ellipticity\nthere are no currents in the antiparallel con\fguration,\nhence these ratios reduce to 1. Their deviation from\n1 thus measures the magnitude of dissipation and spin-\nwave ellipticity e\u000bects on the magnon spin valve. The\nadditional energy and spin currents \rP=AP\nE and\rP=AP\nS\nare included in the ratios (27) because they a\u000bect the\nconductance ratios that will be measured in an experi-\nment, even though they originate from the lattice and\nnot from the magnons in the other lead.\nAs shown in Fig. 5(a), the in-plane anisotropy a\u000bects\nthe MTC ratio only negligibly. In the presence of large\nGilbert damping, on the other hand, the MTC ratio can\ndeviate from 1 by up to 10%. Responsible for this de-\ncrease are the dissipation-assisted pair creation and an-\nnihilation processes that enable energy transfer in the an-\ntiparallel con\fguration even when there is no spin-wave\nellipticity. The MSC ratio, displayed in Fig. 5(b), be-\nhaves similarly for most values of the anisotropy; when7\nκP/κ0\nσP/σ0\nLP/L0\nΠP/Π0\n103×γSP/γ0\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.0000.0050.0100.0150.0200.0250.0300.035\nKy/ℏγμ0H(a)\nκAP/κ0\nσAP/σ0\nLAP/L0\nΠAP/Π0\nγSAP/γ0\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.00000.00010.00020.00030.00040.00050.00060.0007\nKy/ℏγμ0H(b)\nFIG. 4. Tunneling conductances (26) in the (a) parallel and (b) antiparallel con\fgurations as functions of in-plane anisotropy\nKy, forKx= 0, temperature kBT= 10 \u0002~\r\u00160H, and Gilbert damping parameter \u000b= 10\u00002. The conductances are rescaled\nby the dimensionfull prefactors \u00140=U2p\nSk3\nBT=~2J3,\u001b0=\r0=~\u00140=k2\nBT,L0=~\u00140=kbT, and \u0005 0=TL0=~. With this\nrescaling, spin Seebeck and Peltier conductances, L=L 0and \u0005=\u00050, respectively, lie almost perfectly on top of each for most\nvalues of anisotropy Ky, re\recting Onsager reciprocity.\nα=5×10-3\nα=10-2\nα=5×10-2\nα=10-1\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.900.920.940.960.981.00\nKy/ℏγμ0HMTC(a)\nα=5×10-3\nα=10-2\nα=5×10-2\nα=10-1\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.650.700.750.800.850.900.951.00\nKy/ℏγμ0HMSC(b)\nα=5×10-3\nα=10-2\nα=5×10-2\nα=10-1\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.9850.9900.9951.000\nKy/ℏγμ0HMLC(c)\nα=5×10-3\nα=10-2\nα=5×10-2\nα=10-1\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.9900.9920.9940.9960.9981.000\nKy/ℏγμ0HMPC(d)\nFIG. 5. (a) Magnetothermal, (b) magnetospin, (c) magneto-Seebeck, and (d) magneto-Peltier conductance ratios as de\fned\nin Eqs. (27) as functions of in-plane anisotropy Ky, forKx= 0, temperature kBT= 10 \u0002~\r\u00160H, and various values of the\nGilbert damping parameter \u000b.8\nthe magnon gap closes for Ky!~\r\u00160H, however, it\nrapidly decreases because of the divergence of the spin\nconductance.\nThe magneto-Seebeck and Peltier conductance ratios,\nshown in Figs. 5(c) and 5(d), respectively, display an\nopposite behavior: they are sensitive to the anisotropy,\nbut only slightly a\u000bected by the Gilbert damping. They\nshow a decrease of the order of 1% when the magnon gap\ncloses forKy!~\r\u00160H, and also decrease, albeit more\nslowly, for increasing hard axis anisotropy; this re\rects\nthe increasing strength of the spin-conservation break-\ning in both cases. Unlike the MTC and MSC ratios,\nthe MLC and MPC ratios are actually increased by the\nGilbert damping for larger values of the anisotropy. Note\nalso that the Seebeck and Peltier ratios are both qualita-\ntively and quantitatively almost identical; the breaking of\nthe Onsager reciprocity is only apparent in the stronger\ndecrease of the MLC ratio for Ky!~\r\u00160H.\nV. CONCLUSIONS\nWe have studied the tunneling current in a magnon\nspin valve device. By applying the Holstein-Primako\u000b\ntransformation to the Heisenberg Hamiltonian, we de-\nrived the magnon Hamiltonian, in which transverse\nanisotropies introduce ellipticity of the magnons. We\nhave also added Gilbert damping to the magnon spectral\nfunction to study the e\u000bects of dissipation on the magnon\ntunneling. Both precession ellipticity and Gilbert damp-\ning are found to open new, Onsager-reciprocity breaking\nchannels for heat and spin transport across the junction,\nresulting in \fnite currents even when the magnetizations\nof both leads are aligned antiparallel. We have not only\nfound that dissipation and spin-wave ellipticity decrease\nthe spin and heat conductance ratios, but have also re-\nvealed a clear di\u000berence in the sensitivity of heat and\nspin currents to these two quantities. We hope that our\nresults provide useful guidance for the design and under-\nstanding of magnon spin valve devices.ACKNOWLEDGMENTS\nThis work is supported by the European Research\nCouncil via Consolidator Grant No. 725509 SPINBE-\nYOND. R.D. is a member of the D-ITP consortium, a\nprogram of the Netherlands Organisation for Scienti\fc\nResearch (NWO) that is funded by the Dutch Ministry\nof Education, Culture and Science (OCW). J.Z. would\nlike to thank the China Scholarship Council. This re-\nsearch was supported in part by the National Science\nFoundation under Grant No. NSF PHY-1748958.\nAPPENDIX: DERIVATION OF THE\nTUNNELING CURRENTS\nIn this Appendix, we outline the derivation of the tun-\nneling currents given in Sec. III. The total energy current\nfrom the left to the right lead is given by\nIP=AP\nE =@thHRi (28)\n=X\nk0ER;k0@tnR;k0; (29)\nwhereas the spin current from the left to right lead is\nIP\nS=\u0000@tX\nj~\nSz\nR;j\u000b\n(30)\n=~X\nk0\u0000\nu2\nR;k0+v2\nR;k0\u0001\n@tnR;k0; (31)\nin the parallel con\fguration, or\nIAP\nS=@tX\nj~\nSz\nR;j\u000b\n(32)\n=~X\nk0\u0000\nu2\nR;k0+v2\nR;k0\u0001\n@tnR;k0: (33)\nin the antiparallel con\fguration. Using Fermi's golden\nrule24, we \fnd the following kinetic equations for the\nquasiparticle distribution functions in the parallel con-\n\fguration:\n@tnL;k=2\u0019SLSR\n~NLNRX\nk0h\njUk;\u0000k0j2(uL;kuR;k0+vL;kvR;k0)2\u000e(EL;k\u0000ER;k0) (nR;k0\u0000nL;k)\n+jUk;k0j2(uL;kvR;k0+vL;kuR;k0)2\u000e(EL;k+ER;k0) (1 +nL;k+nR;k0)i\n; (34a)\n@tnR;k0=2\u0019SLSR\n~NLNRX\nkh\njUk;\u0000k0j2(uL;kuR;k0+vL;kvR;k0)2\u000e(EL;k\u0000ER;k0) (nL;k\u0000nR;k0)\n+jUk;k0j2(uL;kvR;k0+vL;kuR;k0)2\u000e(EL;k+ER;k0) (1 +nL;k+nR;k0)i\n: (34b)\nThe corresponding expressions in the antiparallel con- \fguration can be obtained from Eq. (34) by ex-9\nchanging the Bogoliubov-coe\u000ecient prefactors according\nto (uL;kuR;k0+vL;kvR;k0)2$(uL;kvR;k0+vL;kuR;k0)2.\nThe energy and spin currents (20) and (21) are obtained\nby inserting the kinetic equations (34) into their respec-tive de\fnitions (29) and (31) or (33), assuming a steady\nstate as in Eq. 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Flensberg, Many-body quantum theory in\ncondensed matter physics: an introduction (Oxford univer-\nsity press, 2004)."    },    {        "title": "1911.12786v1.Transport_properties_of_spin_superfluids__comparing_easy_plane_ferro__and_antiferromagnets.pdf",        "content": "Transport properties of spin superfluids—comparing easy-plane ferro- and\nantiferromagnets\nMartin Evers and Ulrich Nowak\nFachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany\n(Dated: December 2, 2019)\nWe present a study on spin-superfluid transport based on an atomistic, classical spin model. Easy-\nplane ferro- as well as antiferromagnets are considered, which allows for a direct comparison of these\ntwo material classes based on the same model assumptions. We find a spin-superfluid transport\nwhich is robust against variations of the boundary conditions, thermal fluctuations, and dissipation\nmodeled via Gilbert damping. Though the spin accumulations is smaller for antiferromagnets the\nrange of the spin-superfluid transport turns out to be identical for ferro- and antiferromagnets. Fi-\nnally, we calculate and explore the role of the driving frequency and especially the critical frequency,\nwhere phase slips occur and the spin accumulation breaks down.\nI. INTRODUCTION\nSpin transport in magnetic insulators [1, 2] has been\nintensively studied beacause of the fundamental interest\nin the various physical phenomena that occur in these\nmaterials and because of their potential for future appli-\ncations. Magnetic insulators do not exhibit Joule heat-\ning [3] as no electron transport is involved and many of\nthese are oxides with exceptionally low magnetic damp-\ning [4], which hopefully allows for energy efficient trans-\nport properties. It has even been shown that the realiza-\ntion of logic elements is possible [5], such that devices are\ncompatible and integratable with CMOS technology [6].\nStudies on transport in this material class focuses mostly\non transport of magnons [7], i.e. quanta of spin waves—\nthe elementary excitations of the magnetic ground state.\nAs magnons are quasi particles, their number is not con-\nserved and each magnon mode shows an exponential de-\ncay upon transport through the system on a length scale\nξcalled magnon propagation length [8–13]. This is even\ntrue at zero temperature and in a clean system without\nany disorder due to the coupling of the magnons to elec-\ntronic and phononic degrees of freedom, a fact which is\ndescribed phenomenologically via Gilbert damping in the\nequation of motion as will be explained below.\nIn contrast to this damped magnonic transport, a pro-\nposal for spin transport was made that carries the name\nspin superfluidity. The original idea is in fact quite old\n[14, 15] and rests on a similarity of the magnetic or-\nder parameter—either the magnetization of a ferromag-\nnet or the Néel vector of an antiferromagnet—compared\nto the order parameter of superfluidity—the macroscopic\nwave function—as it occurs for He-4 below the lambda\ntransition. For instance, in easy-plane ferromagnets the\nmagnetizationfeaturesaspontaneouslybrokenrotational\nsymmetry in the easy plane ( SO(2)symmetry) that is\nequivalent to the spontaneously broken gauge invariance\nof the macroscopic wave function ( U(1)symmetry). This\nsymmetry leads in both cases to currents that are sta-\nble against small deviations—the supercurrents. [16] One\nstriking difference of spin-superfluid transport to spin-\nwave transport is its distance dependence: for spin su-perfluidity it is expected to be non-exponential, pushing\nthe limit of the range of magnonic transport.\nThe first experimental realizations of a spin superfluid\nwas achieved in a system of nuclear spins of He-3 atoms\n[17]—a model system which is not in a solid state. Only\nrecently the physics of spin superfluidity has drawn again\nattention for the case of solid magnets [18–23], including\na proposed dissipationless transport in metallic magnets\n[18]. However, König et al. neglected spin-orbit inter-\naction in their model for the electrons, which is one of\nthe reasons for Gilbert damping in magnets [24]. But ev-\nery known material exhibits spin-orbit interaction—since\nspinandangularmomentumofanatomareneverexactly\nzero—and therefore also magnetic damping, even if it is\nsmall. Consequently, spinsuperfluidsdoalwaysshowdis-\nsipation in contrast to their conventional counterparts.\nRecent theoretical work has focused on insulators\nrather than metals, usually based of phenomenological\nmodelsincluding theLandau-Lifshitz-Gilbertequationof\nmotion for both ferro- and antiferromagnets. [16, 19, 20]\nThe experimental detection of spin superfluidity in solid-\nstate magnets has been reported for magnon condensates\n[25], where the origin of the spin-superfluid order param-\neter is different to the cases described above, and also\nin antiferromagnetic solids [23]. However, the interpre-\ntation of the experimental findings is still controversially\ndiscussed [16, 26–28].\nIn the following, we will investigate and compare spin\nsuperfluidityinferro-andantiferromagneticmodels. The\ngeometry of our model resembles that of an experimen-\ntal non-local spin-transport investigation as sketched in\nfig. 1. In the corresponding experiments [29] at one side\n(here on the left) a spin current is injected into the mag-\nnet viathe spin-Halleffect causedby an electricalcurrent\nthrough an attached heavy-metal stripe. The resulting\nspin current is detected using the inverse spin-Hall ef-\nfect at another position (here the right-hand side). In\nour model we avoid the details of the excitation mech-\nanism and model the effect of the injected spin current\nby an appropriate boundary condition that triggers the\ndynamics of the spin systems that we investigate. This\nis done from the perspective of an atomistic, classicalarXiv:1911.12786v1  [cond-mat.mes-hall]  28 Nov 20192\nFigure 1. Basic concept of non-local spin transport as in an\nexperimental setup: heavy metal stripes are attached to the\nmagnet to inject a spin current via the spin-Hall effect (here\non the left hand side). The spin current in a certain distance\n(here at the right end) is detected via inverse spin-Hall effect.\nspin model, which has some advantages: the approach is\nnot restricted to small deviations from the ground state,\nfinite temperatures can be investigated and our calcu-\nlations are not limited to the steady state only. Fur-\nthermore, we are able to compare ferro- and antiferro-\nmagnetic systems. Their behavior turns out to be very\nsimilar, except for the resulting spin accumulation that is\nmuchlowerforthelatter. However,fromanexperimental\npoint of view antiferromagnets are much more promising,\nsince these are not prone to a breakdown of spin super-\nfluidity as a consequence of dipolar interactions, which is\nhard to avoid in ferromagnets. [22]\nII. ATOMISTIC SPIN MODEL\nWe consider the following classical, atomistic spin\nmodel of Heisenberg type [30], comprising Nnormal-\nized magnetic moments Sl=µl/µSon regular lattice\nsitesrl. We assume a simple cubic lattice with lattice\nconstanta. The Hamiltonian for these moments, in the\nfollowing called “spins”, is given by\nH=−J\n2/summationdisplay\n/angbracketleftn,m/angbracketrightSn·Sm−dz/summationdisplay\nn(Sn\nz)2,(1)\ntaking into account Heisenberg exchange interaction of\nnearest neighbors quantified by the exchange constant\nJ, where each spin has Nnbnearest neighbors. Further-\nmore, a uniaxial anisotropy with respect to the zdirec-\ntion with anisotropy constant dzis included. In this work\nwe consider the easy-plane case dz<0, where the mag-\nnets ground state readsgSl=±(cos(gϕ),sin(gϕ),0)with\nsome arbitrary, but uniform anglegϕ∈[0,2π](SO(2)\nsymmetry) and an alternating sign ±in case of antifer-\nromagnetic order ( J <0).\nThe time evolution of the spins Slis governed by\nthe stochastic Landau-Lifshitz-Gilbert (LLG) equationof motion [31–33]\ndSl\ndt=−γ\nµS(1 +α2)/bracketleftbig\nSl×/parenleftbig\nHl+αSl×Hl/parenrightbig/bracketrightbig\n(2)\nHl=−∂H\n∂Sl+ξl\n/angbracketleftbig\nξl\nβ(t)/angbracketrightbig\n= 0,/angbracketleftBig\nξl\nβ(t)ξl/prime\nη(t/prime)/angbracketrightBig\n=δll/primeδβηδ(t−t/prime)2µSαkBT\nγ\ndescribing the motion of a spin in its effective field Hl,\nwhereγisthegyromagneticratio, αtheGilbertdamping\nconstant,kBthe Boltzmann constant and Tthe absolute\ntemperature. The properties of the thermal noise ξlare\nchosen such that the dissipation-fluctuation theorem is\nsatisfied [34]. The material parameters define our sys-\ntem of units,|J|for the energy, tJ:=µS/γ|J|for the\ntime,afor the distance. Numerically the LLG equation\nis solved either by the classical Runge-Kutta method in\ncase of zero temperature, or at finite temperature using\nstochastic Heun’s method. At zero temperature the dis-\nsipated power per spin due to Gilbert damping follows\ndirectly from the time evolution of the spins Sl(t)[35]:\nPdiss=1\nNdH\ndt=1\nN/summationdisplay\nn∂H\n∂Sn/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\neff.field·∂Sn\n∂t/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nLLG.(3)\nWe study a magnetic wire extended along xdirection.\nThe system size for our numerical simulations is given\nbyN=Nx×Ny×Nzspins along x-,y- andzdirec-\ntion, where Nx/greatermuchNy,Nz. For transverse directions we\nuse periodic boundary conditions if not noted otherwise.\nBoundary spins at x=Nxa(the right-hand side) are\ndenotedSl/vextendsingle/vextendsingle\nrightand at this side an open boundary con-\ndition is applied, Sl/vextendsingle/vextendsingle\nright= 0. At the opposite side, at\nx= 0, we use a time-dependent boundary condition,\nSl/vextendsingle/vextendsingle\nleft=±(cos(ω0t),sin(ω0t),0), (4)\nin form of an in-plane precession with frequency ω0that\ninjects a spin current from this side. The alternating sign\n(±) is used only for antiferromagnetic systems, according\nto the sublattices with antiparallel spin orientation.\nThe use of this boundary condition creates an ex-\ncitation with well-defined frequency ω0. Alternatively,\nwe also assumed an externally given spin accumulation\nµ=µezat the left-hand side that causes additional\ntorques on the spins and drives them out of equilibrium,\nwhich directly maps an experimental implementation us-\ning a spin-Hall-generated spin accumulation to the model\nutilized here. This method has been used for instance in\n[22]. In appendix B we calculate how this spin accumula-\ntion maps to the excitation frequency ω0and we further-\nmore confirmed numerically that both mechanisms lead\nto the same response for ferro- and antiferromagnets.\nAlthough an atomistic picture—comprising discrete\ndegrees of freedom—is studied numerically, the micro-\nmagnetic approximation is of particular value for analyt-\nical considerations of ferromagnets. This approximation3\nassumes that spatial variations of magnetic structures\nare small compared to the atomic distance a. In this\ncase differences can be approximated as derivatives and\nthe spins form a continuous field S(r,t). It is handy to\nuse cylindrical coordinates\nS=/parenleftBig/radicalbig\n1−S2zcosϕ,/radicalbig\n1−S2zsinϕ, Sz/parenrightBig\n,where definitions Sz(rl) :=Sl\nzandϕ(rl) :=ϕllink the\natomistic picture to the micromagnetics. Note that for\na spin superfluid Szis considered as the spin-superfluid\ndensity and ϕits phase. The use of the micromagnetic\napproximationforferromagnetsallowstoreformulatethe\nLLGequationintermsofdifferentialequationsfor Szand\nϕthat read\nµS\nγ˙ϕ=Ja2/bracketleftBigg\n1\n1−S2z∆Sz+Sz|∇Sz|2\n(1−S2z)2+Sz|∇ϕ|2/bracketrightBigg\n+ 2dzSz−αµS\nγ˙Sz\n1−S2z(5)\nµS\nγ˙Sz=−Ja2/bracketleftbig/parenleftbig\n1−S2\nz/parenrightbig\n∆ϕ−2Sz∇Sz·∇ϕ/bracketrightbig\n+α/parenleftbig\n1−S2\nz/parenrightbigµS\nγ˙ϕ. (6)\nThese two equations are strictly equivalent to the LLG\nequation eq. (2) for zero temperature with the only as-\nsumption of the micromagnetic approximation. If one\nexpands these equations in lowest order in ∇ϕ,∆ϕ,∇Sz,\nand∆Szfor an easy-plane magnet, which implies espe-\ncially assuming|Sz|/lessmuch1, but keeping|∇ϕ|2, one ends up\nwith\nµS\nγ˙ϕ=Ja2∆Sz+Ja2Sz|∇ϕ|2+ 2dzSz−αµS\nγ˙Sz(7)\nµS\nγ˙Sz=−Ja2∆ϕ+αµS\nγ˙ϕ. (8)\nImportantly, keeping the |∇ϕ|2term is actually required\nif the damping takes relatively high values, a fact which\nwe checked numerically. Furthermore, these equations\nare very similar to others already reported in [19, 21], but\nnot exactly equivalent. Ref. [19] uses more approxima-\ntions, especially neglecting the |∇ϕ|2-term, and ref. [21]\nconsiders a different starting point, namely a quantum\ntheory at low temperatures, where this term has a dif-\nferentSz-dependence. Because of this difference, the re-\nsult from [21] does not exactly match our numerical re-\nsults of the atomistic spin model, nor does it match the\nclassical micromagnetic theory. Hence, we use eqs. (7)\nand (8) that do describe the atomistic spin simulations\nwell. However, eqs. (7) and (8) can be solved in steady\nstate for a special case: a ferromagnet that is of length L\nalongxdirection and exhibits translational invariance in\ny- andzdirection as carried out in appendix A. Steady\nstate means a coherent precession of all spins with a fre-\nquency ˙ϕ=ω0and a stationary profile Sz(x). This so-\nlution of eqs. (7) and (8) reads:\nsϕ(x,t) =α\n2µSω0\nγJ(x−L)2\na2+ω0t+ϕ0(9)\nsSz(x) =sSz(L)\n1 +µ2\nSω2\n0\n2γ2Jdzα2/parenleftbigx−L\na/parenrightbig2, (10)\nwith a spin accumulation at the right end of the sys-\ntem (atx=Nxa=:L) ofsSz(L) =µSω0/2γdz, avalue which is independent of L—one of the striking fea-\ntures of spin superfluidity. Another feature is the mono-\ntone increase of ϕwhich implies the formation of an in-\nplane spin spiral with winding number Nw, which reads\n2πNw=/integraltext\ndϕ=ϕ(L)−ϕ(0). Note furthermore, that an\nopen boundary condition at the right end is an assump-\ntion that leads to solutions eqs. (9) and (10), correspond-\ning to a Neumann condition ∇ϕ/vextendsingle/vextendsingle\nright= 0, which must be\njustified as a realistic choice.\nFor the numerical study of eq. (2) we assume an open\nboundary at the right end. Equation (10) assumes the\nsame and results in a finite Szatx= 0, which contra-\ndicts the numerical driving boundary at this side, eq. (4),\nthat forces Sz(x=0) = 0. Furthermore, in an experiment\nan open boundary at the right end might not be feasi-\nble because of outflowing spin currents, for example into\nan attached heavy metal. Thus, the real behavior at\nthe boundaries for sure deviates from the ideal solution\neq. (10) and raises the question how strong that devia-\ntion is and in how far the boundary conditions influence\nthe overall bulk behavior of the spin transport. This\nis examined numerically from the full model eq. (2) by\nvarying the boundary conditions on the left and right.\nOne example of the variations we tested is an absorbing\nboundary condition on the right, modeling an outflow-\ning spin current by an enhanced damping. As result we\nobserve the profile Sl\nzto show only little change in that\ncase compared to an open boundary and also that in all\ncases the numerical profiles well follow eq. (10) (see in\nthe following fig. 2 a) as example). Other variations of\nthe boundary condition which we tested have also hardly\nany impact on the magnets overall response.\nIII. EASY-PLANE FERROMAGNET\nIn a first step of the numerical investigation, we con-\nsider a collinear ferromagnet as most simple case, with\nparameters J > 0for the ferromagnetic state and dz=\n−0.01Jas in-plane anisotropy. Let us describe the phe-4\n0 1000 2000 3000 4000 5000012345610-3\n010203040506070\n0 0.5 1 1.5 2 2.5\n10-300.010.020.03\n5 6\n10-40.0240.026\nFigure 2. Spin superfluidity in a 1D ferromagnet at T= 0in the steady state: a)depicts the spin accumulation Szand the\nin-plane angle ϕforω0tJ=−2×10−4; numerical data (blue and red symbols) follow perfectly the theoretical curve eqs. (9)\nand (10) (black, dashed lines), except for the vicinity of the left boundary. This is an artifact of the boundary condition, eq. (4),\nused for the numerics. b)shows the spin accumulation at the right end of the system SN\nzversus driving frequency ω0; for small\ndriving frequencies up to a critical value ωcritthe numerical data follow the analytical curvesSz(L); for larger frequencies the\nspin accumulation breaks down, deviating form the theoretical curve, due to phase slips and spin wave excitations.\nnomenology of the spin superfluid in a 1D system of size\nNx×Ny×Nz= 5000×1×1at temperature T= 0.\nThis model is equivalent to a 3D system with transla-\ntional invariance in y- andzdirection. Furthermore, we\nsetα= 0.05andω0tJ=−2×10−4.\nStarting from a uniform ferromagnet as initial condi-\ntion, the boundary spin starts to rotate and due to ex-\nchange the next spin will follow this rotation and ac-\ncordingly drive its neighbor and so on. But since a spin\ncannot immediately follow the dynamics of its neighbor,\nthere is a certain phase difference Dϕbetween the spins,\ni.e., the neighbor to the right is lagging behind. In the\nmicromagnetic approximation this effect is described by\na phase gradient ∇ϕ≈Dϕ/a. The rotation of the spins\nspeeds up, until it reaches the final precession frequency,\ngiven by the driving frequency ω0. At the same time\ntheout-of-planecomponent Sz—thespinaccumulation—\nincreases until it reaches a steady state profile. The time\nscale of this transient phase for reaching a steady state\ncan be quantified: ˙ϕ(t)andSz(t)follow a limited expo-\nnential growth on a characteristic time τt≈5×105tJ\nfor the parameters used here. τtscales positively with\nsystem size Nxand damping α.\nEventually, the numerical time evolution reaches a\nsteady state as shown in fig. 2 a). This steady state\nverifies the analytical solution eqs. (9) and (10) in bulk\nwith a deviation only at the left boundary as anticipated\nand described above. Note that the finite spin accumu-\nlationSzas a consequence of this type of dynamics has\nimportantfeatures: itisalong-rangespintransportsince\nit decays non-exponential and it would allow to measure\nspin transport by means of the inverse spin-Hall effect.\nFurthermore, it could also be addressed, for instance, by\nmagneto-optical measurements—if sensitive to the out-\nof-plane magnetization for a geometry as studied here.\nFor a further investigation, we vary the frequency ω0\nand find two different regimes, one for sufficiently smallω0, where the system is able to follow the excitation\nwithout disturbance, and one for large ω0where the sys-\ntems response deviates from the theoretical expectation.\nThesetworegimes, whichwewillcalllinearandnonlinear\nregime in the following, are sharply separated by a crit-\nical frequency ωcrit. The existence of these two regimes\ncan be seen from the data depicted in fig. 2 b). Here, as\na measure, we consider the spin accumulation of the last\nspinSN\nzat the right end of the system. Below ωcritwe\nfind just the analytical valuesSz(L), see eq. (10), which\nscales linearly with ω0. Atωcritthis behavior breaks\ndown and the spin accumulation SN\nzdecreases with in-\ncreasing pumping frequency. This breakdown can be un-\nderstood in terms of the phase gradient ∇ϕwhich scales\nlinearly with the driving frequency ω0, see eq. (9). How-\never, one can expect a maximum phase gradient ∇ϕfor a\nspin-superfluid state given by the Landau criterion [36]:\nif the phase gradient exceeds locally a critical value, it\nis energetically favorable for the spins at this position to\nrotate out of the x-yplane and return to the plane by\nunwinding the spiral. Hence, the winding number Nw\ndecreases by one—an effect which is called a phase slip.\nThe Landau criterion for the stability of a spin superfluid\nwith respect to phase slips reads [36]\n|∇ϕ|</radicalbigg\n−2dz\nJa2. (11)\nNote that this relation is not exact as a uniform Szis\nassumed for its derivation. Nevertheless do we observe\nthese phase slips numerically. In the linear regime the\nwinding number is constant in the steady state, whereas\nin the nonlinear regime it relaxes by one at a regular\nrate Γpsas shown in fig. 3 at the example of ω0tJ=\n−6.5×10−4.\nTheω0dependence of the phase-slip rate Γpsis de-\npicted in fig. 4. Each phase slip is accompanied by the\nexcitation of a broad spin-wave spectrum on top of the5\n4 4.005 4.01 4.015 4.02\n10748.54949.550\n00.511.510-7\nFigure 3. Winding number Nwand dissipated power Pdiss\nin the steady state of a driven ferromagnet for ω0tJ=\n−6.5×10−4, well in the nonlinear regime. At a rate of Γps\nphase slips relax the winding number of the in-plane spiral.\nFor each such event the dissipated power spikes.\nspin superfluid. These spin waves are visible as oscil-\nlations ofSzaround the spin-superfluid magnitude and,\nhence, there is strictly speaking no steady state any more\nas the phase slips and the spin-wave excitation are def-\ninitely time dependent. In particular, for systems with\nlow Gilbert damping this dynamics leads to determinis-\nticchaos, thoughthe spin-superfluidbackground remains\nvisible. These findings have some severe implications as\nthere is a maximum spin accumulation, which is achieved\nright at the edge between the linear and the nonlinear\nregime. Furthermore, driving the system in the nonlin-\near regime means also to waste energy to the phase slips\nand the excitation of incoherent spin waves.\nFor the parameters here the critical frequency takes\nthe valueωcrittJ≈−5.15×10−4, which is determined\nfrom fig. 2 b). We also tested different parameters, vary-\ningαandL(data not shown), and find that ωcritscales\nnegatively with αandL. Our numerical result can be\ncompared to the analytical prediction above, eq. (11).\nFrom eq. (9) follows that the maximum phase gradient\nis given by∇sϕ(0) =αµSω0L/γJa2, which, inserted into\neq. (11), implies\n|ωcrit|=γJa\nαµSL/radicalbigg\n−2dz\nJ. (12)\nFor our parameter set this takes value 4×10−41/tJ,\nwhich is slightly lower compared to the numerical value\nabove. This discrepancy is probably due to the fact that\neq. (11) ignores the spatial dependence of Sz. Further-\nmore, a test for very low damping α= 10−4showed that\neq. (11) is even more inaccurate in that case.\nAnother important quantity is the dissipated power\ngiven by eq. (3), which takes negative sign as it lowers\nthe total energy. Figure 4 depicts its dependence on ω0.\nBelowthecriticalfrequency, inthesteadystate, itistime\nindependent as the dynamics is completely stationary.\nIn this regime it scales quadratically with the excitation\nfrequency,Pdiss∝ω2\n0, a result which has already been\n10-410-310-210-1010-910-810-7\n10-610-410-2Figure 4. Time-averaged dissipated power /angbracketleftPdiss/angbracketrightand phase-\nslip rate Γpsin the steady state versus driving frequency\nω0comparing ferro- (FM) and antiferromagnets (AFM). The\nperpendicular dash-dotted lines mark ωcritfor the FM and\nthe AFM, where the latter takes on the higher value. For\nω0< ω critthe dissipated power scales as /angbracketleftPdiss/angbracketright∝ω2\n0and is\nidentical for FMs and AFMs. Above ωcritthe increase slows\ndown and the curve flattens for very high ω0. In this regime,\n/angbracketleftPdiss/angbracketrightis higher for AFMs as compared to FMs. For the\nphase-slip rate we find Γps= 0for|ω0|<|ωcrit|and similar,\nincreasing values for the FM and the AFM above ωcrit. The\ndeviation between FM and AFM near ωcritis due to the dif-\nferent critical frequency, i.e. the data almost coincide when\nplotted versus ω0−ωcrit.\nreported before [37]. This behavior changes above ωcrit.\nThetimeevolutionofthedissipationinthisregimeshows\nthat the phase slips notably contribute to the dissipated\npower, i.e. for each phase slip Pdissspikes as shown in\nfig. 3. Because of this time-dependence of Pdiss, we have\nto consider an average over time for the evaluation of the\ndissipated power. Still, the dissipated power increases\nfurtherwith ω0butlessthanlinearandthecurvenotably\nflattens.\nSo far, our results were obtained from zero-\ntemperature simulations. In the following we address\nthe robustness of spin-superfluid transport at finite tem-\nperature. For this, we consider a finite cross section\nNy×Nz>1andNx= 2000 and vary the tempera-\nture. An average over Navrealizations of thermal noise\nis carried out and, furthermore, data are averaged over\nthecrosssectioninordertoreducethenoise. Thespecific\nchoice of parameters in provided is table I.\nFigure 5 presents the numerical results for the exam-\nple ofkBT/J = 10−2forSzandϕ. The spin-superfluid\ntransport remains in tact but, in particular, the spin ac-\ncumulation Szshows strong thermal fluctuations despite\nthe averages taken over the cross section and the Navre-\nalizations. However, on average the spin accumulation\nclearly deviates from its equilibrium value, which is zero.\nTo quantify the influence of the temperature we calculate\nthe spatial average over the xdirection/angbracketleftSz/angbracketrightxand com-\nparethistothezero-temperaturevalue, givenbyeq.(10).\nThe results are included in Table I. Furthermore, the in-\nplane angle/angbracketleftϕ/angbracketrightNavshows only little fluctuations and its6\n00.020.04\n0 500 1000 1500 200001020\nFigure 5. Spin superfluidity in a ferromagnet at finite tem-\nperaturekBT/J = 10−2and forω0tJ=−2×10−4: shown is\nthe spin accumulation Szand the in-plane angle ϕ. Blue lines\nrepresent the numerical data, black dash-dotted lines the an-\nalytical solution at zero-temperature. The spin accumulation\nis subjected to strong thermal fluctuations but still has a fi-\nniteaveragevalue /angbracketleftSz/angbracketrightx=/summationtext\nnSn\nz/Nx, depictedasreddashed\nline. Its value is only slightly lower than the zero-temperature\nvalue. Thermal fluctuations are much less pronounced for the\nin-plane angle.\nTable I. Averaged spin accumulation of a ferromagnet driven\nwithω0tJ=−2×10−4for different temperatures. The cor-\nresponding zero-temperature value is /angbracketleftSz/angbracketright= 0.01, from which\nno significant deviation is observed.\nkBT/JNx×NyNav/angbracketleftSz/angbracketrightx\n10−44×4 38 0.010\n10−24×4 15 0.009\n0.05 8×8 5 0.010\n0.10 8×8 4 0.011\n0.20 14×145 0.012\nspatial profile shows hardly any deviation from the zero-\ntemperature behavior, given by eq. (9). Overall, we find\nno significant difference to the zero temperature case.\nWe also checked whether phase slips due to thermal ac-\ntivation can be observed, but from the available data\nwe could not observe a single one with the conclusion\nthat ΓpstJ<4×10−5. Hence, spin superfluidity is very\nrobust against thermal fluctuations, even though these\nfluctuations are a problem in our simulations in terms of\nthe signal-to-noise ratio.\nIV. EASY-PLANE ANTIFERROMAGNETS\nFor antiferromagnets, the magnetic unit cells comprise\ntwoatoms—denotedAandBinthefollowing—thatform\ntwo sublattices. We write all properties using this label-\ning so thatASlandBSlare spins of the corresponding\nsublattices. In the ground state both sublattices have\nopposite orientation,ASl=−BSl. The field equations,\neqs. (5) and (6), do not hold as these require a small\nin-plane angle difference between two neighboring spins\nDϕ, which is obviously not true in this case. However,it is reasonable to define phase differences and gradients\nwithin each sublattice, i.e.ADϕas phase difference be-\ntween a spin of sublattice A and its next-nearest neigh-\nbor, which is the nearest neighbor within sublattice A.\nAccordingly,BDϕdefines the phase difference of sublat-\nticeB. Assuming sufficiently weak excitation, spatial\nvariationswithineachsublatticearesmallsuchthatami-\ncromagnetic approximation inside the sublattices reads\n∇A,Bϕ≈A,BDϕ/2a. Interestingly, numerical results re-\nveal that the antiferromagnetic system in bulk fulfills\nfield equation (8), applied separately to each sublattice.\nThe other eq. (7) is not valid, as has been reported before\n[20] for a phenomenological model for antiferromagnets.\nConsequently, the antiferromagnet is expected to exhibit\nthe same in-plane angleA,Bϕ(up to phase difference of π\nbetweensublattices)asaferromagnetwithcorresponding\nparameters, but not the same spin accumulationA,BSz.\nBefore we discuss the numerical results in detail, let\nus first introduce two differences to the ferromagnet that\nare essential for understanding the following results: the\nroleofexchangeand(interlinkedwiththis)thetransverse\ngeometry. Just as in a ferromagnet, a spin-superfluid dy-\nnamics imposes a finite spin accumulationA,BSzwhich,\nremarkably, carries the same sign for both sublattices\nleading to a small out-of-plane magnetization. But this is\nof course antagonized by the antiferromagnetic exchange\nthat favors antiparallel orientation of all components be-\ntween sublattices. Consequently, the exchange interac-\ntions must lower the spin accumulation Sztremendously\nas compared to the ferromagnet (compare fig. 6 a) and\nfig. 2 a)). This also implies that the behavior of Szis\ndetermined by the number of nearest neighbors Nnbof\na spin as more neighbors imply stronger exchange cou-\npling. Consequently, a 1D spin chain is less prone to this\nexchange reduction than a 3D system. We checked this\nnumerically by comparing 1D, 2D and 3D models and,\nindeed, the spin accumulation of the spin superfluid Sz\nscales linearly with Nnb.\nThere is another implication: at a boundary the num-\nberofneighborsislocallyreduced—andthereforetheim-\nportance of the exchange—, resulting in deviations of the\nsublattice componentsA,BSz, see fig. 6 a) for a 1D setup\n(the effect is less pronounced in 3D). This 1D setup owns\nonly boundaries along the xdirection and the question\nwhether for finite cross section Ny×Nz>1these devi-\nations aty- andzboundaries significantly influence the\nbulk behavior has also been tested numerically. Fortu-\nnately, deviations at transversal boundaries quickly fall\noff with distance to the boundary over a few lattice con-\nstants. The bulk then behaves qualitatively and quan-\ntitatively just as a 1D system, except for the reduced\nspin accumulation due to the number of neighbors as\ndiscussed above. The study of 1D systems is preferable\nto keep computational costs feasible.\nWe turn now to the presentation of the numerical data\nfor a 1D system. The model parameters are the same as\ngiven above for the ferromagnet, except for the exchange\nconstant which is now negative. Similarly to the ferro-7\n050100024610-5\n10002000300040004900 5000\n0 0.5 1 1.5 2 2.5\n10-300.511.510-4\n5.566.5\n10-41.31.41.510-4\nFigure 6. Spin superfluidity in antiferromagnetic spin chains: a)the spin accumulation in the stead state resolved for the two\nsublattices A and B. In the bulk both take the same value, leading to a finite total spin accumulation, which is two orders of\nmagnitude lower as compared to the ferromagnet. At the boundaries the profiles show deviations from bulk behavior because\nof the broken exchange right at the boundary. b) the spin accumulation at the right end of the system as function of driving\nfrequencyω0; as for the ferromagnet there are two regimes separated by a critical frequency ωcrit.\nmagnet, the system reaches a steady state after a tran-\nsientphasecharacterizedbyalimitedexponentialgrowth\non a time scale τt, which is roughly the same as for the\nferromagnet. In the steady state the sublattice-resolved\nin-plane anglesA,Bϕboth follow exactly the same profile\nas the ferromagnet, i.e. eq. (9), but with a phase differ-\nence ofπbetween the two sublattices because of the an-\ntiferromagnetic order (data for the antiferromagnet not\nshown).\nThe spin accumulation deviates from the behavior of\na ferromagnet as depicted in fig. 6 a). The bulk profiles\n(away from boundaries at x= 0andx=Nxa) are iden-\ntical in the two sublattices,ASz=BSz. Hence, a measur-\nable spin accumulation is present, but it is two orders\nof magnitude lower than in comparable ferromagnetic\ncases. This is the aforementioned exchange reduction.\nIf we consider the spin accumulation Szin bulk, in the\ndata in fig. 6 a) hardly a space dependence is observed in\ncontrast to the ferromagnet, where Sl\nzhas a finite slope.\nThe antiferromagnet exhibits this in the same way, but\nit is also much smaller and the profile becomes roughly\nconstant. Contrary to the ferromagnet, there are distur-\nbances at the boundaries in the profile of Szwhich we\nalready discussed before.\nDriving the antiferromagnet with the time-dependent\nboundary condition eq. (4) at frequency ω0leads to the\nvery same two different regimes as for ferromagnets, a\nlinear regime up to a critical frequency ωcritand above—\nin the nonlinear regime—phase slips occur. These phase\nslips reduce the winding number, lead to the excitation\nof spin waves, and a further increase of the spin accu-\nmulation is not possible. We quantify this behavior in a\nsimilar way as for the ferromagnet. It is, however, not\npossible to use the spin accumulation of the last spin\nSN\nzas a measure because of the deviating profile at the\nboundary. Instead, we take the spin accumulation at the\nend of the bulk in form of a spatial average over the spins\nin the range xl/a∈[4900,4920],Send\nz:=/angbracketleftbig\nSl\nz/angbracketrightbig\n[4900,4920].This range is chosen such that it is sufficiently separated\nfrom the boundary. The data for the ω0dependence of\nthespinaccumulationareshowninfig.6, panelb): These\nshow that critical frequencies takes roughly same values\nfor ferro- and antiferromagnets, a result which has been\ntested and confirmed for another parameter set with dif-\nferentNx,α, anddz. For the data set shown here the\nvalue isωcrittJ≈−5.75×10−4. However, the decrease\nof the spin accumulation Send\nzwith increasing driving fre-\nquencyω0in the nonlinear regime is less pronounced for\nantiferromagnets. We also calculated the ω0dependence\nof the time-averaged dissipated power /angbracketleftPdiss/angbracketrightand of the\nphase-slip rate Γps, both shown in fig. 4. Similar to other\nfeatures these properties behave for the antiferromagnets\nvery much as for ferromagnets: below ωcritthe dissipated\npower shows exactly the same dependence and above it\nis dominated by phase slips. However, a difference is that\naboveωcritthe dissipated power increases faster with ω0.\nOne reason for this might be the dynamics of spin waves\nthat very much differ between ferro- and antiferromag-\nnets. The phase-slip rate differs slightly, however, this\nseems to be solely due to the fact that ωcritdiffers for\nferro- and antiferromagnets. When Γpsis plotted versus\nω0−ωcrit, both curves match almost.\nThe next step is to consider finite temperature. Again\nthis requires a finite cross section for which we use\nNx×Ny×Nz= 2000×4×4and we test two temper-\natures,kBT/J = 10−2andkBT/J = 10−4. As before,\nthe magnetic response is very similar to that of a ferro-\nmagnet: the in-plane angles follow the zero-temperature\nprofiles, as well as does the average spin accumulation\nfor the lower of the two temperatures. The only major\ndifference is the ratio of the spin-superfluid spin accumu-\nlation to the thermal fluctuations, which is two orders of\nmagnitude smaller as a result of the lower spin-superfluid\nsignal and an equal strength of the fluctuations. For the\nhigher temperature, this even leads to an average Szthat\nis essentially zero. This does not mean that there is no8\nspin-superfluid spin accumulation, but rather that the\navailable numerical data are not sufficient to resolve it\nand more averaging is needed. Note that the in-plane\nangle is not affected by this—it is as robust against the\nfluctuations just as for the ferromagnet.\nV. DISCUSSION AND CONCLUSION\nOur comparative study addresses spin superfluidity in\nferro- and antiferromagnets, where one should bear in\nmind that the former are less promising for spin super-\nfluidity as the latter because of the negative influence of\nthe stray field [22]. Nevertheless, the former can help to\nunderstand the behavior of the latter, which we utilize\nin this work. One of the striking features of spin super-\nfluidity is the transport range that leads to a spin ac-\ncumulation at the end of the system Sz(L)(see eqs. (9)\nand (10)) that does depend on the system length L—\na completely different situation compared to spin-wave\ntransport where the intensity decays exponentially with\nthe distance. However, this non-exponential decay does\nnot imply the possibility of an infinite transport range\nsince with increasing system size the critical frequency\nlowers until no undisturbed spin superfluid is possible\nanymore.\nWe present a full analytical solution for the steady\nstate of the ferromagnet, which slightly deviates from the\nanalytical theory reported before [19, 21]. This theory\nis tested numerically by the full atomistic model, which\nallows to test the robustness of the spin-superfluid trans-\nport against varying boundary conditions, against high\nexcitation frequencies and finite temperature. We show\nthat this kind of transport is remarkably robust: bound-\nary conditions and also elevated temperature hardly\nhamper the magnets spin-superfluid response.\nFurthermore, we identify the critical frequency ωcrit—\na manifestation of the Landau criterion—as the limiting\nfactor for the range of this transport. Above this critical\nfrequency phase slips occur, which also sets a limit to\nthe spin accumulation that can be achieved. In ref. [38]\nanother limitation on the spin current of such a spin su-\nperfluid is discussed, which rests on the fact that |Sz|\nis bounded above. But the estimated values would re-\nquire an out-of-plane component that takes quite large\nvalues|Sz|>0.1, which our simulations reveal to be\nhardly possible even for low damping. This is in particu-lar true for the case for antiferromagnets and, therefore,\nwe conclude that the critical frequency—and therefore\nthe phase slips—is a more relevant limitation on spin su-\nperfluid transport.\nThe direct comparison of antiferromagnets to ferro-\nmagnets shows that both exhibit the very same behavior:\nDriven by an in-plane rotation, both form an in-plane\nspin spiral that exhibits exactly the same behavior, in-\ncluding a spin accumulation in form of an out-of-plane\nmagnetization. Antiferromagnets show in principle the\nsame transport range as ferromagnets with a spin accu-\nmulation at the end of the system independent of the\nsystem length, provided the excitation frequency ω0is\nkept constant ( ω0itself depends on the magnets geome-\ntry in experimental setups, see eq. (B12)). Furthermore,\nthe critical frequency takes very similar value for the two\ntypes of magnets. This general accordance of spin super-\nfluidity for both types of magnets is in contrast to spin-\nwavetransportthatisknowntobedifferentforferro-and\nantiferromagnets[39]. Yetthereisamajordeviation: the\nantiferromagnetic exchange lowers tremendously the spin\naccumulation.\nOurstudyalsocoversanexaminationofthedissipation\nofaspinsuperfluidandoftheeffectoffinitetemperature.\nWe proof the principle robustness of spin superfluidity\nagainst thermal fluctuations, i.e. that quite high temper-\natures are required before thermal phase slips start to\nhamper the transport. But the fluctuations are a prob-\nlem from the numerical side as these require integration\noveralargeamountofdatainordertoidentifyanon-zero\nmean spin accumulation. The signal-to-noise ratio might\nbe a problem in experimental setups as well and it could\nbe more promising to measure rather the in-plane an-\ngleϕ, which is more robust against thermal fluctuations\nand which always delivers a clear signal in the cases we\ninvestigated here. A measurement of ϕcan be done in\ntwo ways: either by its time evolution, i.e. the preces-\nsion frequency ω0, or spatially resolved by measuring the\nformation of the in-plane spin spiral.\nACKNOWLEDGMENTS\nFinancial support by the Deutsche Forschungsgemein-\nschaft (DFG) via the SFB 767 “Controlled Nanosystems:\nInteraction and Interfacing to the Macroscale” and the\nprogram “Hematite: A new paradigm for antiferromag-\nnetic spin transport” is gratefully acknowledged.\n[1] M. Wu and A. 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Sonin, Advances in Physics 59, 181 (2010),\nhttps://doi.org/10.1080/00018731003739943.\n[37] H. Ochoa, R. Zarzuela, and Y. Tserkovnyak, Phys. Rev.\nB98, 054424 (2018).\n[38] Y. Tserkovnyak and M. Kläui, Phys. Rev. Lett. 119,\n187705 (2017).\n[39] F. Keffer, H. Kaplan, and Y. Yafet, Amer-\nican Journal of Physics 21, 250 (1953),\nhttps://doi.org/10.1119/1.1933416.\n[40] A. Manchon, J. Železný, I. M. Miron, T. Jungwirth,\nJ. Sinova, A. Thiaville, K. Garello, and P. Gambardella,\nRev. Mod. Phys. 91, 035004 (2019).\nAppendix A: Analytical theory for a 1D ferromagnet\nThe ferromagnet in the micromagnetic approximation\nunder the assumption of small out-of-plane component,\n|Sz|/lessmuch1, is described by the LLG equation in cylindrical\ncoordinates, eqs. (7) and (8). Assuming translational\ninvariancealong y-andzdirectionleadstoa1Dproblem:\nµS\nγ˙ϕ=Ja2∂2\nxSz+Ja2Sz(∂xϕ)2+ 2dzSz−αµS\nγ˙Sz\n(A1)\nµS\nγ˙Sz=−Ja2∂2\nxϕ+αµS\nγ˙ϕ. (A2)\nSteady state means ˙ϕ=ω0and ˙Sz= 0. This allows to\nintegrate the latter equation,\nsϕ(x,t) =α\n2µSω0\nγJ/parenleftbiggx−L\na/parenrightbigg2\n+ω0t+ϕ0,(A3)\nwhere the first integration constant follows from the Neu-\nmann boundary condition at the right end, ∂xϕ(L) = 0\n(no outflow of spin current), and the second one satisfies\nthe condition ˙ϕ=ω0and allows for an arbitrary phase\nϕ0. This is inserted in the first equation, which then\nreads\n−Ja2∂2\nxSz=−µSω0\nγ+µ2\nSω2\n0\nγ2J/parenleftbigg\nαx−L\na/parenrightbigg2\nSz+ 2dzSz.\n(A4)\nWe argue that the second-derivative term can be ne-\nglected−Ja2∂2\nxSz≈0. This is justified in a twofold\nmanner: first we compared the relevance of all terms\nin that equation numerically by calculating those three\nterms from simulations of the full atomistic spin model,\neq. (2). Indeed the result is that in steady state the\nsecond-derivative term is several orders of magnitude\nsmaller compared to the other two. The second reason\nfollows a-posteriori from the calculated solution and is10\nspin injector (using SHE)\nspins not subjected\nto SHEspins driven\nby SHEspins not subjected\nto SHE\nFigure 7. 1D setup for calculation of the excitation frequency\nω0of a magnet driven by a spin injector utilizing the spin-Hall\neffecttoexertexternaltorquesonthespins. Thesetorquesare\napplied in the region [l1,l2]and vanish outside. Furthermore,\nthe Gilbert damping in [l1,l2]is enhanced by αd. The ground\nstateSis in-plane, the spin accumulation µperpendicular.\nexplainedbelow. From −Ja2∂2\nxSz≈0followsthesteady-\nstate solution for Sz:\nsSz=µSω0\n2γdz\n1 +µ2\nSω2\n0\n2γ2Jdz/parenleftbig\nαx−L\na/parenrightbig2. (A5)\nThis solution does not fulfill eq. (A4), however, we can\ninsert it and calculate the deviation by calculating\n∂2\nxsSz=−2µSω0\nγJα2\na2sS2\nz+ 4/parenleftbiggµSω0\nγJ/parenrightbigg2α4(x−L)2\na4sS3\nz\n=O/parenleftbig\nS2\nz/parenrightbig\n.\nThis allows the conclusion that the correction by taking\nthe second derivative into account is of higher order in\nSzand neglecting this is consistent with the original as-\nsumption|Sz|/lessmuch1. Hence, eqs. (A3) and (A5) form the\nanalytical solution for a 1D setup.\nAppendix B: Frequency of a spin superfluid\nThe usual excitation of a spin current in a magnet\nrests on a spin accumulation µat an interface between\nthe magnet and a heavy metal, which is created by an\nelectrical current. Normally for that the spin-Hall effect\nis utilized. The aim of this appendix is to calculate the\nresulting excitation frequency ω0of a spin superfluid.\nWe assume here that the spin accumulation is per-\npendicular to the magnets ground state, i.e. µ∝ez.\nConsequently, there is an additional damping-like torque\n[22, 40] in the LLG equation (here written as viscousdamping):\n˙Sl=−γ\nµSSl×Hl+αlSl×˙Sl+α/prime\nlSl×/parenleftbigg\nSl×µl\n~/parenrightbigg\n.\n(B1)\nA subsetVdof the total volume of the magnet is driven,\ni.e. subjected to the additional torques and the driving\nalso creates an enhanced damping α/prime\nlwithinVd:\nµl=/braceleftbigg\nµdezforrl∈Vd\n0else(B2)\nαl=α0+α/prime\nlwithα/prime\nl=/braceleftbigg\nαdforrl∈Vd\n0else.(B3)\nα0is the intrinsic Gilbert damping of the magnet.\nTo proceed we consider the LLG equation in the fol-\nlowing form, resolved for the time derivative:\n˙Sl=−γ\nµS(1 +α2\nl)Sl×/parenleftbig\nHl+αlSl×Hl/parenrightbig\n+Tl\n1Sl×Al+Tl\n2Sl×/parenleftbig\nSl×Al/parenrightbig\n.(B4)\nTl\n1andTl\n2parameterize arbitrary additional torques with\nrespecttoanaxis Alandforthespecificchoice Al=µl/~,\nTl\n1=αlα/prime\nl/(1+α2\nl)andTl\n2=−α/prime\nl/(1+α2\nl)eq. (B4) is equiva-\nlent to eq. (B1). However, for the sake of generality we\nconsider for the calculation eq. (B4). Assuming Al∝ez\nand using cylindrical coordinates and again the micro-\nmagnetic approximation, this form of the LLG reads\nµS\nγ˙ϕ=Ja2Sz|∇ϕ|2+ 2dzSz−αµS\nγ˙Sz\n−µS\nγAz(T1+αT2) (B5)\nµS\nγ˙Sz=−Ja2∆ϕ+αµS\nγ˙ϕ+µS\nγAz(αT1−T2),(B6)\nan extension of eqs. (7) and (8). In the same spirit as\nin appendix A we can solve these equations in one di-\nmension in steady-state (assuming ˙Sz= 0and ˙ϕ=ω0),\nwhere the geometry depicted in fig. 7 is assumed. We ap-\nply the external spin accumulation in the interval [l1,l2],\nwhereas the total magnet expands over [0,L]. Therefore,\nT1,2(x) =/braceleftbigg\nTd\n1,2forx∈[l1,l2]\n0else\nA(x) =/braceleftbigg\nAd\nzezforx∈[l1,l2]\n0else.11\nIn the 1D setup eq. (B6) reads\n∂2\nxϕ=α(x)µS\nγJa2ω0+µS\nγJa2Az(x) [α(x)T1(x)−T2(x)]\n=\n\n=:¯ω0/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nα0µS\nγJa2ω0 forx∈[0,l1]\n(α0+αd)µS\nγJa2ω0\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:¯ω/prime\n0+µS\nγJa2Ad\nz/bracketleftbig\n(α0+αd)Td\n1−Td\n2/bracketrightbig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:tforx∈[l1,l2]\nα0µS\nγJa2ω0 forx∈[l2,L], (B7)\nwhich can be integrated. There are six boundary conditions to consider, each one at the left and right end of the\nmagnet, where we assume a Neumann condition ∂xϕ(0) =∂xϕ(L) = 0, i.e. no outflow of spin currents. Furthermore, ϕ\nand∂xϕmust be continuous at l1andl2, delivering four internal boundary conditions. But there is another condition,\na gauge condition for ϕ, which allows to add an arbitrary constant phase to ϕ(x)without altering the physics. (In\npractice this gauge phase depends on the prehistory of the magnet, i.e. on how it had reached its steady state, and also\nwhich exact instant in time is considered.) As gauge we use ϕ(0) = 0. Altogether there are 6 integration constants\nand the unknown frequency ω0in combination with 6 boundary conditions and a gauge, such that the problem has a\nunique solution.\nAs result we obtain\nϕ=\n\n1\n2¯ω0x2forx∈[0,l1]\n1\n2(¯ω/prime\n0+t)x2+ (¯ω0−¯ω/prime\n0−t)l1x+1\n2(¯ω/prime\n0−¯ω0+t)l2\n1forx∈[l1,l2]\n1\n2¯ω0x2+ (¯ω/prime\n0−¯ω0+t)/bracketleftbig\n(l2−l1)x+1\n2(l2\n1−l2\n2)/bracketrightbig\nforx∈[l2,L](B8)\nSz=µS\nγω0+Az(x) [T1(x) +α(x)T2(x)]\nJa2(∂xϕ)2+ 2dz(B9)\nand, importantly, we also gain\nω0=−Ad\nz/bracketleftbig\n(α0+αd)Td\n1−Td\n2/bracketrightbig\n(l2−l1)\nα0L+αd(l2−l1). (B10)\nThis holds true for arbitrary torques taking form\neq. (B4). If the specific case of the spin injector utiliz-\ning the spin-Hall effect is considered, then inserting the\nparameters T1,T2andAreads\nω0=−µd\n~αd\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:τ·l2−l1\nα0L+αd(l2−l1).(B11)\nThe former factor τis the strength of the spin-Hall effect\non the magnet [40]:\nτ=γ\nMs~\n2eηϑSHjel1\nd,\nwith spin transparency of the interface η, spin-Hall an-\ngleϑSH, saturation magnetization Msand thickness d\nof the magnet. jelis the electric current density. The\nlatter factor in eq. (B11) is a geometric factor that is ba-\nsically the ratio between the driven volume l2−l1and the\ntotal volume L, weighted with the total damping of the\nmagnet, where the Gilbert-damping enhancement can beexpressed as [22]\nαd=g⊥~2\n2e2γ\nMsd,\nwith transverse spin mixing conductance g⊥of the in-\nterface. This rigorous derivation holds only true for 1D\nferromagnets, however, the natural extension to 2D and\n3D is given by\nω0=−τ·Vd\nα0V+αdVd, (B12)\nwhereVis the magnets total and Vdthe driven volume.\nThe validity of this expression has been checked numeri-\ncally for 1D and 2D systems using various geometries by\ninvestigating the full atomistic LLG eq. (B4). As a result\nwe obtain very good agreement with the analytical calcu-\nlation except for two cases. First, when the assumption\n|Sz|/lessmuch 1is violated and second if the setup is not ef-\nfectively one dimensional, i.e. if the system is not driven\nover the entire transverse width. However, such a mis-\nmatch in usually small for realistic experimental setups.\nWe furthermore did not only simulate ferromagnets, but12\nalso antiferromagnets with same parameters except for\nthe sign of J. These simulations result in exactly the\nsame frequencies ω0as the corresponding ferromagnetsand thus eqs. (B10) to (B12) are also valid for antiferro-\nmagnets, even though note that the resulting spin accu-\nmulation deviates."    },    {        "title": "1912.00310v3.Coarse_graining_in_micromagnetic_simulations_of_dynamic_hysteresis_loops.pdf",        "content": "Coarse-graining in micromagnetic simulations of\ndynamic hysteresis loops\nR Behbahani1;2, M L Plumer1and I Saika-Voivod1;2\n1 Department of Physics and Physical Oceanography, Memorial University of\nNewfoundland, Canada\n2 Department of Applied Mathematics, University of Western Ontario, London,\nOntario, Canada, N6A 3K7\nE-mail: saika@mun.ca\nAbstract. We use micromagnetic simulations based on the stochastic Landau-\nLifshitz-Gilbert equation to calculate dynamic magnetic hysteresis loops at \fnite\ntemperature that are invariant with simulation cell size. As a test case, we simulate\na magnetite nanorod, the building block of magnetic nanoparticles that have been\nemployed in preclinical studies of hyperthermia. With the goal to e\u000bectively simulate\nloops for large iron-oxide-based systems at relatively slow sweep rates on the order\nof 1 Oe/ns or less, we modify and employ a previously derived renormalization group\napproach for coarse-graining (Grinstein and Koch, Phys. Rev. Lett. 20, 207201, 2003).\nThe scaling algorithm is shown to produce nearly identical loops over several decades\nin the model cell volume. We also demonstrate sweep-rate scaling involving the Gilbert\ndamping parameter that allows orders of magnitude speed-up of the loop calculations.\nKeywords : Landau-Lifshitz-Gilbert equation, micromagnetics, coarse-graining, mag-\nnetic hyperthermia, nanorods\nThe fundamental premise of micromagnetics is that the physics of interest can\nbe modeled by a macrospin representing a collection of atomic spins within a small\n\fnite volume, or cell. The approximation that all spins within a cell point in the same\ndirection is valid at temperature T= 0, so long as cells remain smaller than the exchange\nlength [1]. A limiting factor for micromagnetic computer simulations is the number of\ncells used to model the system; using larger cells is computationally advantageous.\nAt \fniteT, a few schemes have been proposed to account for how parameters\nused for modelling the magnetic properties of the material must vary with cell size\nin order to keep system properties invariant with cell size. For example, Kirschner\net al. [2, 3] suggested an approximate scaling of saturation magnetization Msbased\non the average magnetization of blocks of spins in atomistic Monte Carlo simulations,\nand subsequently scaling the exchange and uniaxial anisotropy constants AandKto\npreserve the exchange length and anisotropy \feld. Feng and Visscher [4] proposed that\nthe damping parameter \u000b, which models the dynamics of magnetic energy loss [5], should\nscale with cell size, arguing that using larger cells is analogous to having more degrees of\nfreedom for energy absorption; see also [6] for e\u000borts related to \u000b. The renormalizationarXiv:1912.00310v3  [cond-mat.mtrl-sci]  8 Nov 2021Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 2\ngroup (RG) approach of Grinstein and Koch [7], based on mapping a Fourier space\nanalysis of the non-linear sigma model to ferromagnets in order to scale A,K, \feldH\nand magnetization M, has garnered signi\fcant attention. However, to the best of our\nknowledge, no scaling theory has been applied to the calculation of magnetization-\feld\n(MH) hysteresis loops [8], which are the foundation of experimental characterization of\nmagnetic systems.\nIn this Letter, we modify and employ the approach proposed by Grinstein and\nKoch [7] to the test case of calculating MH loops for magnetite nanorods at sweep rates\nrelevant to magnetic hyperthermia, allowing us to make estimates of speci\fc loss power\nthat would otherwise be computationally impractical.\nThe magnetite nanorods we simulate are the building-blocks of the nanoparticles\nthat were shown by Dennis et al to successfully treat cancerous tumours in mice via\nhyperthermia [9]. It is reasonable to choose the smallest micromagnetic cell to be\nthe cubic unit cell, which is of length a0= 0:839 nm and contains 24 magnetic\nFe ions. We set the exchange sti\u000bness constant to A0= 0:98\u000210\u000011J/m, which\nfor cell length a0yields an e\u000bective exchange constant between neighbouring cells of\nJe\u000b=a0A0= 8:222\u000210\u000021J, which in turn yields a bulk critical temperature of\nTc= 1:44Je\u000b=kB= 858 K for the bulk 3D-Heisenberg-model version of our system.\nThis value of A0is close to what can be theoretically determined by considering the\natomic-level exchange interactions across the faces of neighbouring unit cells [10], and\nis in reasonable agreement with experimental values [11, 12, 13, 14, 15, 16, 17]. The\nnanorod dimensions are approximately 6.7 nm \u000220 nm\u000247 nm (8a0\u000224a0\u000256a0),\nwith its length along the z-axis. We set Ms= 480 kA/m [11, 18, 19], the bulk value\nfor magnetite. We do not consider magnetostatic interactions explicitly, but rather\nimplicitly through an e\u000bective uniaxial anisotropy. For the purposes of this study, we\nchoose a strength of K0= 10 kJ/m3, which is consistent with other studies of iron\noxide nanoparticles [20, 21], and for which a more precise estimate can be obtained by\nconsidering the nanorod's demagnetization tensor [19, 22, 23, 24, 25, 26], maghemite\ncontent [9], and the e\u000bect of neighbouring nanorods within a nanoparticle. We\nomit cubic crystalline anisotropy as it has negligible e\u000bects on the hysteresis loops of\nmagnetite nanoparticles with even modest aspect ratios, as discussed in Refs. [19, 26]\n(we have also veri\fed that adding cubic anisotropy of strength 10 kJ/m3has no impact\non the loops presented here). Anisotropy is set along the z-axis with a 5\u000edispersion to\nmimic lattice disorder [21]. For convenience we set \u000b= 0:1, a choice consistent with\nprevious studies [21, 27] and with magnetite thin \flms [28].\nWhile hysteretic heating is at the heart of magnetic nanoparticle hyperthermia,\npreventing eddy current heating of healthy tissue limits the frequency fand amplitude\nHmaxof the external \feld such that the sweep rate SR = 4 Hmaxfis less than a target\nvalue of 0:25 Oe/ns [29, 18]. For our simulation, we set Hmax= 500 Oe, which for\nthe target SR implies a target value of f= 125 kHz, a value large enough to restrict\nunwanted Brownian relaxation [18].\nTo model the dynamics of the magnetization of a cell Mof \fxed magnitude Ms,Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 3\n𝑎\"𝝁𝑎\"\nNanoparticle\nSingleBlock𝑎$=2𝑎\"(b=2)1344 Cells 𝑎=𝑎\"(b=1) 10752 cells𝑎'=4𝑎\"(b=4)168 Cells 𝑎)=8𝑎\"(b=8)21 Cells \nFigure 1. Coarse-grained modelling of a magnetite nanorod. The smallest\nmicromagnetic cell models the atomic spins within a cubic unit cell of length a0=\n0:839 nm with a single magnetic moment. Our goal is to model the system using\na smaller number of larger cells (of length ab=ba0forb > 1) with appropriately\nscaled parameters. The number of cells drawn and their sizes are only approximate.\nIllustrative spins for half of the tetrahedral Fe3+sites (FCC sites) are drawn over a\nspinel unit cell taken from Ref. [30].\nwe solve the Landau-Lifshitz-Gilbert (LLG) equation [22, 5, 31],\ndM\ndt=\u0000\r1M\u0002He\u000b\u0000\u000b\r1\nMsM\u0002(M\u0002He\u000b) (1)\nwheretis time,\r1=\u00160\re=(1 +\u000b2),\re= 1:76\u00021011rad/(s.T) is the gyromagnetic\nratio for an electron, \u00160is the vacuum permeability, and He\u000bis due to the combination\nof an external \feld, uniaxial anisotropy, exchange interactions and a thermal \feld. We\nperform our simulations using OOMMF (Object Oriented Micromagnetic Framework)\nsoftware [32]. In particular, we include the Theta Evolve module [33] used for simulations\nat \fniteTvia a stochastic thermal \feld [31].\nWe simulate the rod using cubic cells of length ba0, withbtaking on values 1,\n2, 4 and 8. See Fig. 1. For b= 1, 10752 cells make up the rod. For b= 2,\nthere are 10752 =23= 1344 cells. The volume of the rod is \fxed for all simulations\nat 10752a3\n0\u0019(22a0)3. Additionally, we simulate the rod as a single cell { a single\nrectangular prism, or block. While there is some ambiguity in assigning a single length\nscale to represent a rectangular prism, we choose b= 22 from the geometrical mean,\ni.e., the side length of the cube of the same volume as the rod.Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 4\nThe goal of coarse-graining is to determine A(b) andK(b), i.e., how the exchange\nand anisotropy parameters should change with bto keep system properties invariant with\nb. Theb= 22 case is a practical limit where all the atomic spins are represented by a\nsingle macrospin, where exchange interactions are no longer required in the simulations,\nand which provides for an interesting test of a coarse-graining procedure in predicting\nK(b). In calculating hysteresis loops for a system with cell length ba0, we apply an\nexternal \feld along the zaxis ofH(b) =Hmaxsin (2\u0019ft), and report the z-component\nof the average (over cells) magnetization unit vector mH=\u0016Mz=Ms, averaged over 88 to\n100 independent simulations for b>1. Forb= 1 we use 250 simulations.\nIn Fig. 2a we plot hysteresis loops at T= 310 K using di\u000berent cell sizes (varying\nb) while keeping the exchange and anisotropy parameters \fxed at A0andK0. A value\nof SR = 2:5 Oe/ns is chosen to make the simulations computationally feasible at b= 1.\nBoth the coercivity Hcand the remanence increase with increasing b. The increasing\nloop area is consistent with the stronger exchange coupling ( Je\u000b=ba0A0) between\nmagnetization vectors of adjacent cells. For b\u00154, it appears that the exchange is\nstrong enough for the system to be nearly uniformly magnetized, and so Hcremains\nlargely unchanged for b\u00154 sinceKis constant. This means that for b= 1, at this T\nand for our rod size, exchange is not strong enough to be able to treat the nanorod as\na single macrospin in a trivial way. Clearly, varying cell size changes the loops and a\ncoarse-graining procedure is required.\nIn their coarse-graining procedure, Grinstein and Koch introduced a reduced\ntemperature T\u0003, which for a three dimensional system is given by,\nT\u0003=kBT\u0003\nA: (2)\nwhere \u0003 = 2 \u0019=ba 0is a high wave-number cut-o\u000b that re\rects the level of coarse-graining.\nSimilarly, the reduced parameters for \feld and anisotropy constants are de\fned as,\nh=\u00160MsH\nA\u000321000\n4\u0019; g =K\nA\u00032; (3)\nwithHgiven in Oe. Introducing the parameter l= ln(b), they gave the following set of\nequations for calculating the reduced parameters as functions of cell size,\ndT\u0003(l)\ndl= [\u00001 +F(T\u0003(l);h(l);g(l))]T\u0003(l)\ndh(l)\ndl= 2h(l)\ndg(l)\ndl= [2\u00002F(T\u0003(l);h(l);g(l))]g(l)(4)\nwhere\nF(T\u0003;h;g) =T\u0003\n2\u0019(1 +h+g): (5)\nAdditionally, the magnetization of the coarse-grained system is scaled via,\nM(T\u0003;h) =\u0010(l)\u0002M(T\u0003(l);h(l)) (6)Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 5\nwhere\n\u0010(l) =e\u0000Rl\n0F(T\u0003(l0);h(l0);g(l0))dl0: (7)\nFor our system parameters and range of H, bothg\u001c1 andh\u001c1, and so\nF'T=2\u0019, which makes the numerical solution of Eq. 4 practically indistinguishable\nfrom the approximate analytic solution, which we \fnd to be,\nA(b) =\u0010(b)\u0002A0 (8)\nK(b) =\u0010(b)3\u0002K0 (9)\nH(b) =\u0010(b)\u0002H0 (10)\nM0=\u0010(b)\u0002M(b) (11)\nwheret=T=Tcand\u0010(b) =t=b+ 1\u0000t. AtT= 310 K,t= 0:3613,\u0010(2) = 0:8193,\n\u0010(4) = 0:7290,\u0010(8) = 0:6839, and\u0010(22) = 0:6551.\nEqs. 8 and 9 provide a prescription for changing material parameters with b, while\nEqs. 10 and 11 provide the prescription for scaling HandMafter a loop calculation.\nHowever, we \fnd that the prescription does not yield loops that are invariant with b,\non account of Eq. 11; the correction of the coarse-grained values of Mback to those\ncorresponding to the unscaled system is too large (the corrected remanance is too small),\nas we show in Fig. 2b. In Fig. 2c, we apply a correction to Eq. 11 and obtain good\nagreement between the reference ( b= 1) and coarse-grained ( b>1) loops.\nTo motivate our correction to the rescaling of the magnetization, we begin by\nnoting that the same value of T\u0003in Eq. 2 can be achieved by either having a rescaled\ntemperature T(b) or having a rescaled A(b). Combining this idea with Eq. 8 yields,\nT(b) =T0\nb\u0010(b;T0); (12)\nwhich together with Eq. 11 [after solving for M(b)] predicts an overly simple dependence\nofMonT, parametrically through b: a line passing through M0andT0atb= 1 and\nthroughM= 0 andT=Tcasb!0.\nTo obtain a model that better matches the data, we introduce a phenomenolgical\ncorrection to Eq. 11, one in which M0is a weighted average of M(b) and the RG\nexpression for M0,\nM0=\u000e\u0010(b;T0)M(b) + (1\u0000\u000e)M(b): (13)\nWe use\u000eas a free parameter to \ft the M(T) data for the nanorod. This yields a value\nof\u000e= 0:511, which we use in rescaling mHin Fig. 2c. The \ft reasonably recovers M(T)\nin theTrange corresponding to values of bbetween 1 and 22, as shown in Fig. 3.\nThe collapse of the data in Fig. 2c is remarkable, with the biggest discrepancy\narising between b= 1, corresponding to the most \fne-grained simulation, and b= 2, the\n\frst step in coarse-graining. The di\u000berence lies most noticeably in the shoulder region\nwhere magnetization begins to change, where the microscopic details likely matter most.\nLoss of some detail is expected with coarse-graining and consistent with previous studies\ninvolving atomic-level magnetization switching in a grain [34]. The magnetization inCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 6\n400\n 200\n 0 200 400\nH(b) (Oe)1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00\nmH(a)\nb=1\nb=2\nb=4\nb=8\nblock\n400\n 200\n 0 200 400\nH(b)/ (Oe)\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00×mH\n(b)\nb=1\nb=2\nb=4\nb=8\nblock\n400\n 200\n 0 200 400\nH(b)/ (Oe)\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00((1 )+)×mH\n(c)\nb=1\nb=2\nb=4\nb=8\nblock\nFigure 2. Application of RG coarse graining to nanorod MH loops at T= 310 K\nand SR= 2 :5 Oe/ns. (a) Changing cell length ( a=ba0) without changing magnetic\nparameters. (b) AandKare scaled according to Eqs. 8 and 9, respectively, and\nmHandHare scaled according to Eqs. 11 and 10, respectively. (c) As in panel (b),\nexceptmHis scaled according to Eq. 13 with \u000e=0.511. \u0001t= 1 fs for all simulations.\nHorizontal error bars shown for Hcrepresent one standard error and are vertically\ndisplaced to avoid overlap. Uncertainty in Hcis approximately 7 to 13%.\nthe shoulder areas appears to diminish with increasing b. The behavior of b= 22 runs\ncounter to this trend, but at this level of coarse-graining, there is only a single cell. It is\nsigni\fcant, however, that scaling seems to hold even in this limit. (We note that in this\nlimit, even though there are no exchange interactions in the simulations, the value of\nthe e\u000bective anisotropy still depends on exchange through the dependence of TconA0.)\nThe loop areas for b= 1, 2 , 4, 8 and 22 are 495, 488, 443, 432 and 472 Oe, respectively.\nThe smallest loop area (for b= 8) is 13% smaller than the area for b= 1.\nWe note that the unrenormalized exchange length for our simulated material is\nlex;0=q\n2A0\n\u00160M2s= 8:23 nm, which is longer than a8= 6:712 nm, and so only our b= 22\nsingle block simulations scale the cell size beyond lex;0. Under renormalization, however,\nthe exchange length becomes lex;b=q\n2\u0010(b)A0\n\u00160M2s, which decreases with increasing b, and\ntakes on values 7.45, 7.02, 6.80 and 6.66 nm for b= 2;4;8;and 22, respectively. Thus\nforb= 8, the cell length and the exchange length are approximately the same.\nWe now turn our attention to speeding up simulations by considering the\nrelationship between SR and \u000b. A larger value of \u000bsigni\fes a faster loss of energy\nand a shorter relaxation time for alignment of the magnetic moments to the \feld, and\nresults in a smaller hysteresis loop. Likewise, a slower SR is equivalent to a longer\nmeasurement time and consequently a smaller hysteresis loop. To build on these ideas,\nwe recall Sharrock's equation for Hcas a function of T[35],\nHc=HK\"\n1\u0000s\nkBT\nKVln\u0012f0\u001c\nln 2\u0013#\n: (14)\nSharrock derived this equation by calculating the time required for half of the\nmagnetization vectors in the system, which are initially anti-aligned with the \feld,\nto overcome an energy barrier that grows with KV and align with a \feld of strength\nHc. In this context, \u001cis the relaxation time. In the context of hysteresis loops, Hc\nis the \feld required to \rip half of the magentization vectors in an observation time \u001c,Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 7\n0.0 0.2 0.4 0.6 0.8 1.0\nT/Tc0.00.20.40.60.81.0m\nb=1b=2b=4b=8b=22Simulations\nm(b), =0.511\nFigure 3. Determining a scaling function for M(b) from the Tdependence of the\nnanorod magnetization. \u000eis used as a \ftting parameter to match nanorod data,\nyielding a value of 0.511. Vertical dot-dash lines indicate reduced temperatures\ncorresponding to di\u000berent values of b.\nwhich is related to SR via \u001c/1=SR.f0is the so-called attempt frequency, for which\nBrown [31, 36, 37, 38, 39] derived an expression in the high-barrier limit. At small \u000b,\nf0/\u000b, and so the product f0\u001c/\u000b=SR, implying that so long as SR =\u000b= constant, Hc\nshould remain the same.\nIn Fig. 4 we show loops calculated for SR =\u000b= 2:5 (Hmax = 500 Oe, and\nf= 125 kHz), the ratio obtained using a clinically relevant SR = 0 :25 Oe/ns and\nthe estimate of \u000b= 0:1. Data for b= 4 and 8 and for various SR- \u000bpairs show good\nagreement. At 0 :25 Oe/ns, simulations using b= 1 are prohibitively long, taking several\nmonths on available computing resources. The results shown here combine the RG\napproach to reduce the number of cells, the ability to use a larger time step \u0001 tfor\nlarger cells in solving the LLG equation [6], and the SR =\u000bscaling to employ a faster\nSR, all to dramatically reduce simulation time { by a factor of 43to 83for reducing\nthe number of cells, a factor of at least 5 for the time step, and a factor of up to 1000\nwhen using the fastest SR. The average area of the \fve loops for b= 4 in Fig. 4 is\nS= 171:3\u00062:8 Oe, translating to a speci\fc loss power of f\u001601000\n4\u0019MsS=\u001a= 207 W/g\n\u000610% (using \u001a= 5:17 g/cm3), which is consistent with clinical expectations [40]. The\nloop area for b= 8 is 13% lower at 149 :4 Oe.\nIn summary, we show that our modi\fcation to the RG approach of Grinstein and\nKoch [7] yields a scaling of exchange and anisotropy parameters and \fnite temperature\nnanorod hysteresis loops that are, to approximately 10-15%, invariant with cell size. WeCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 8\n400\n 200\n 0 200 400\nH(b)/ (Oe)\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00((1 )+)×mH\nSR=0.25,=0.1,b=8\nSR=0.25,=0.1\nSR=2.5  ,=1\nSR=25   ,=10\nSR=50   ,=20\nSR=250 ,=100\nFigure 4. Invariance of MH loops. We combine RG scaling of magnetic quantities,\nlarger time step with block size, and SR =\u000bscaling to predict the behaviour of\nprohibitively long \fne-grain ( b= 1) simulations. b= 4 unless otherwise noted.\nnote that the coarse-graining of magnetostatic interactions is beyond the framework of\nRef. [7]. We are currently investigating magnetostatic scaling, and intend to report on\nit in future work.\nScaling results hold even to the point where the nanorod is represented by a single\nmagnetization vector that experiences anisotropy only. Whether this limit holds for\nsystems with weaker exchange remains to be studied. This reduction to an e\u000bective\nStoner-Wohlfarth (SW) model [41] should facilitate comparison with experiments on\nnanorods, since an analytic solution to the SW model at \fnite Tand SR exists [27]. It\nshould also simplify computational studies of nanoparticles (nanorod composites) and\ncollections of nanoparticles used in a wide variety of applications and hence facilitate\ncomparison with experimental MH loops and quanti\fcation of system properties through\nsimulations.\nIn addition to the computational speedup resulting from the use of fewer\nmicromagnetic cells, the invariance of loops when SR =\u000bis \fxed provides another avenue\nfor computational speedup by allowing one to use a larger SR than the target value.\nWe caution, however, that the theoretical motivation for this invariance stems from\nconsidering the Sharrock equation for only small \u000b. While both SR and \u000bset time\nscales, we have not provided any reasoning for why the invariance should hold as well\nas it does for larger \u000b.\nThe data that support the \fndings of this study are available from theCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 9\ncorresponding author upon reasonable request.\nAcknowledgments\nWe thank Johan van Lierop, Rachel Nickel and Mikko Karttunen for enlightening\ndiscussions, and Martin D. Leblanc for guidance in using OOMMF. R.B. and I.S.-V.\nthank Mikko Karttunen and Styliani Consta for hosting our stay at Western University.\nWe acknowledge the \fnancial support from the Natural Sciences and Engineering\nResearch Council (Canada). Computational resources were provided by ACENET and\nCompute Canada.\nReferences\n[1] Abo G S, Hong Y K, Park J, Lee J, Lee W and Choi B C 2013 IEEE Trans. Magn. 494937{4939\n[2] Kirschner M, Schre\r T, Hrkac G, Dorfbauer F, Suess D and Fidler J 2006 Physica B 372277{281\n[3] Kirschner M, Schre\r T, Dorfbauer F, Hrkac G, Suess D and Fidler J 2005 J. Appl. Phys. 9710E301\n[4] Feng X and Visscher P B 2001 J. Appl. Phys. 896988{6990\n[5] Gilbert T L 2004 IEEE Trans. Magn. 403443{3449\n[6] Wang X, Gao K and Seigler M 2011 IEEE Transactions on Magnetics 472676{2679\n[7] Grinstein G and Koch R H 2003 Phys. Rev. Lett. 90207201\n[8] Westmoreland S, Evans R, Hrkac G, Schre\r T, Zimanyi G, Winklhofer M, Sakuma N, Yano M,\nKato A, Shoji T, Manabe A, Ito M and Chantrell R 2018 Scripta Materialia 154266{272\n[9] Dennis C, Jackson A, Borchers J, Hoopes P, Strawbridge R, Foreman A, Van Lierop J, Gr uttner\nC and Ivkov R 2009 Nanotechnology 20395103\n[10] Victora R, Willoughby S, MacLaren J and Xue J 2003 IEEE Trans. Magn. 39710{715\n[11] Heider F and Williams W 1988 Geophys. Res. Lett. 15184{187\n[12] Kouvel J 1956 Phys. Rev. 1021489\n[13] Moskowitz B M and Halgedahl S L 1987 J. Geophys. Res. Solid Earth 9210667{10682\n[14] Glasser M L and Milford F J 1963 Phys. Rev. 1301783\n[15] Srivastava C M, Srinivasan G and Nanadikar N G 1979 Phys. Rev. B 19499\n[16] Srivastava C and Aiyar R 1987 J. Phys. C: Solid St. Phys. 201119\n[17] Uhl M and Siberchicot B 1995 J. Phys. Condens. Matter 74227\n[18] Dutz S and Hergt R 2013 Int. J. Hyperth. 29790{800\n[19] Usov N, Gudoshnikov S, Serebryakova O, Fdez-Gubieda M, Muela A and Barandiar\u0013 an J 2013\nJ. Supercond. Nov. Magn. 261079{1083\n[20] Usov N, Serebryakova O and Tarasov V 2017 Nanoscale research letters 121{8\n[21] Plumer M, van Lierop J, Southern B and Whitehead J 2010 J. Phys. Condens. Matter 22296007\n[22] Cullity B D and Graham C D 2011 Introduction to magnetic materials (John Wiley & Sons)\n[23] Newell A J, Williams W and Dunlop D J 1993 Journal of Geophysical Research: Solid Earth 98\n9551{9555\n[24] Aharoni A 1998 Journal of Applied Physics 833432{3434\n[25] Fukushima H, Nakatani Y and Hayashi N 1998 IEEE Trans. Magn. 34193{198\n[26] Usov N and Barandiar\u0013 an J 2012 Journal of Applied Physics 112053915\n[27] Usov N 2010 J. Appl. Phys. 107123909\n[28] Serrano-Guisan S, Wu H C, Boothman C, Abid M, Chun B, Shvets I and Schumacher H 2011\nJ. Appl. Phys. 109013907\n[29] Hergt R and Dutz S 2007 J. Magn. Magn. Mater. 311187{192\n[30] 2019 Introduction to Inorganic Chemistry (Wikibooks) chap 8.6, see Creative CommonsCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 10\nlicence http://creativecommons.org/licenses/by-nc-sa/3.0/us/ URL https://en.wikibooks.\norg/wiki/Introduction_to_Inorganic_Chemistry\n[31] Brown Jr W F 1963 Phys. Rev. 1301677\n[32] Donahue M J and Porter D G 1999 OOMMF User's Guide, Version 1.0, Interagency Report\nNISTIR 6376 National Institute of Standards and Technology Gaithersburg, MD URL https:\n//math.nist.gov/oommf/\n[33] Lemcke O 2004 ThetaEvolve for OOMMF releases: 1.2a3, see\nhttps://math.nist.gov/oommf/contrib/oxsext/oxsext.html URL http://www.nanoscience.\nde/group_r/stm-spstm/projects/temperature/download.shtml\n[34] Mercer J, Plumer M, Whitehead J and Van Ek J 2011 Appl. Phys. Lett. 98192508\n[35] Sharrock M and McKinney J 1981 IEEE Trans. Magn. 173020{3022 ISSN 0018-9464\n[36] Garc\u0013 \u0010a-Palacios J L and L\u0013 azaro F J 1998 Phys. Rev. B 5814937\n[37] Breth L, Suess D, Vogler C, Bergmair B, Fuger M, Heer R and Brueckl H 2012 J. Appl. Phys. 112\n023903\n[38] Taniguchi T and Imamura H 2012 Phys. Rev. B 85184403\n[39] Leliaert J, Vansteenkiste A, Coene A, Dupr\u0013 e L and Van Waeyenberge B 2015 Med. Biol. Eng. Com-\nput.53309{317\n[40] Das P, Colombo M and Prosperi D 2019 Colloids and Surfaces B: Biointerfaces 17442{55\n[41] Stoner E C and Wohlfarth E 1948 Philos. Trans. Royal Soc. A 240599{642"    },    {        "title": "1912.04083v4.Analytical_solution_of_linearized_equations_of_the_Morris_Lecar_neuron_model_at_large_constant_stimulation.pdf",        "content": "arXiv:1912.04083v4  [q-bio.NC]  1 Apr 2021Analytical solution of linearized equations of the Morris- Lecar\nneuron model at large constant stimulation\nA.V. Paraskevov1,2, T.S. Zemskova3,4\n1Institute for Information Transmission Problems, 127051 M oscow, Russia\n2National Research Centre \"Kurchatov Institute\", 123182 Mo scow, Russia\n3Ecole Polytechnique, 91128 Palaiseau, France\n4Moscow Institute of Physics and Technology (National\nResearch University), 141700 Dolgoprudny, Russia\nAbstract\nThe classical biophysical Morris-Lecar model of neuronal e xcitability predicts that\nupon stimulation of the neuron with a sufficiently large const ant depolarizing cur-\nrent there exists a finite interval of the current values wher e periodic spike generation\noccurs. Above the upper boundary of this interval, there is f our-stage damping of\nthe spike amplitude: 1) minor primary damping, which reflect s a typical transient\nto stationary dynamic state, 2) plateau of nearly undamped p eriodic oscillations, 3)\nstrong damping, and 4) reaching a constant asymptotic value of the neuron poten-\ntial. We have shown that in the vicinity of the asymptote the M orris-Lecar equations\ncan be reduced to the standard equation for exponentially da mped harmonic os-\ncillations. Importantly, all coefficients of this equation c an be explicitly expressed\nthrough parameters of the original Morris-Lecar model, ena bling direct comparison\nof the numerical and analytical solutions for the neuron pot ential dynamics at later\nstages of the spike amplitude damping.\nKeywords: neuronal dynamics, Morris-Lecar model, constant current s timulation,\nperiodic spiking, damped oscillations2\n1. Introduction\nThe Morris-Lecar (ML) model [1, 2] is a classical biophysica l model of spike generation\nby the neuron, which takes into account the dynamics of volta ge-dependent ion channels\nand realistically describes the spike waveform. The model p redicts that upon stimulation\nof the neuron with sufficiently large constant depolarizing c urrentIstim, there exists a finite\ninterval of Istimvalues where periodic spike generation occurs [2–7]. Numer ical simulations\nhave shown that in the ML model the cessation of periodic gene ration of spikes above the\nupper boundary of this interval (i.e. at Istim>Imaxin Fig. 1) occurs through a damping\nof the spike amplitude, arising with a delay inversely propo rtional to the value of Istim[8].\nIn particular, the damped dynamics can be divided into four s uccessive stages: 1) minor\nprimary damping, which reflects a typical transient to stati onary dynamic state, 2) plateau\nof nearly undamped periodic oscillations, which determine s the aforementioned delay, 3)\nstrong damping, and 4) reaching constant stationary asympt otic value Vstof the neuron\npotential. This dynamic behavior of the ML model is qualitat ively the same for the 1st and\n2nd types of neuronal excitability.\nIn this paper, we have found a way to linearizing the ML model e quations in the vicinity\nof the asymptote Vst. The resulting equations have been then reduced to an inhomo geneous\nVolterra integral equation of the second kind. In turn, the l atter has been transformed into\nan ordinary differential equation of the second order with a t ime-dependent coefficient at\nthe first-order derivative. As this time dependence was just an exponential decay with the\nsmall pre-exponential factor, we considered its asymptoti c value and analytically solved the\nfinal equation. In order to verify the analytical solution fo und, we have compared it with\nthe numerical solution obtained using the standard MATLAB t ools for systems of ordinary\ndifferential equations (see the Supplementary Material, wh ich contains the MATLAB scripts\nand generated data used for the Figures).\nAs the result, we have accurately shown that the linearized s ystem of equations of the\nML model can be reduced to the standard equation of exponenti ally damped harmonic\noscillations for the neuron potential. Since all coefficient s of this equation are explicitly\nexpressed through parameters of the original ML model, one c an directly (i.e. without\nany fitting) compare the numerical and analytical solutions for dynamics of the neuron\npotential at last two stages of the spike amplitude damping ( left graphs in Fig. 2 and3\nFig. 3). The results allow a quantitative study of the applic ability boundary of ordinary\nbifurcation analysis that implies exponential dynamics.\nFinally, it should be noted that a similar effect of delayed da mped oscillations of the\nneuronal potential has been previously reported for the ML m odel with the 2nd excitability\ntype and the stimulating current value, which is just below t he lower boundary of sustained\nspiking [9, 10]. Emphasize that these findings are only appli cable, first, for small stimulating\ncurrent values lying before the region of sustained oscilla tions (below Iminin Fig. 1). In\nturn, we consider the case of a large stimulating current abo ve that region, i.e., at a different\nstationary point of the ML model. Second, the results [9, 10] are only valid for the excitability\ntype 2 defined by an abrupt occurrence of high-frequency sust ained oscillations at finite\nminimal value Iminof constant stimulating current (see Fig. S1 in the Suppleme ntary\nMaterial). For the type 1, where sustained oscillation freq uency starts as a continuous\nfunction of the stimulating current (Fig. 1), the damped osc illations below Imindo not arise\nin principle, and the results [9, 10] are not applicable. On t he contrary, in the case of large\nstimulating current considered in this paper, the model beh avior is universal for both the\n1st and 2nd types of excitability so that the results are also universal.\n2. Standard Morris-Lecar model\nAs phase plane analysis of the ML model is extensively descri bed in textbooks (e.g.,\n[11–16]), we omit it and provide only basic facts on the model , and its formal description.\nQualitatively, the classical two-dimensional ML model [1, 2] (cf. [17]) couples dynam-\nics of the transmembrane potential Vof the neuron with dynamics of the transmembrane\nconductance wof potassium ions. Spikes represent characteristic pulses ofV(see the gray\ninset in Fig. 1, in the range from ImintoImax). In the ML model the rate of change of V\ndepends on the current value of win such a way that dynamics of wprovides a negative\nfeedback with respect to the dynamics of V. In turn, the rate of change of wis proportional\nto the difference between the current value of wand some \"asymptotic\" value w∞, which\nnonlinearly depends on V. As a result, wtends to reach w∞, which is changed itself in time\ndue to dynamics of V. If one neglects the relaxation of wtow∞, i.e., assumes that this\noccurs instantly, then the generation of spikes in the ML mod el does not happen. The upper\nvalue of the stimulating current, Imax, above which the continuous periodic generation of\nspikes stops, corresponds to the onset of a relatively fast r elaxation of wtow∞.4\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s49/s48/s32/s109/s86/s49/s48/s48/s32/s109/s115/s49/s48/s32/s109/s86\n/s49/s48/s48/s32/s109/s115/s49/s48/s32/s109/s86\n/s49/s48/s48/s32/s109/s115\n/s73\n/s109/s97/s120\n/s32/s32/s83/s112/s105/s107/s105/s110/s103/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s91/s72/s122/s93\n/s73\n/s115/s116/s105/s109/s32/s91 /s65/s47/s99/s109/s50\n/s93/s68/s97/s109/s112/s101/s100/s32/s111/s115/s99/s105/s108/s108/s97/s116/s105/s111/s110/s115\n/s73\n/s109/s105/s110\nFigure 1. Dependence of spike generation frequency (determ ined as the number of spikes divided by\nthe time interval of 20000 ms) on constant stimulating curre ntIstimand, on the gray inset, typical\nexamples of dynamics of the neuron potential in the correspo nding ranges of Istimvalues for the\nMorris-Lecar model with the 1st excitability type [2]. Spik es are characteristic pulses of the neuron\npotential (see the gray inset in the range from Imin= 40µA/cm2toImax= 116.1µA/cm2).\nQuantitatively, the standard ML model equations for dynami cs of the neuronal potential\nVand for relaxation dynamics of the normalized conductance wof potassium ions are given\nby\n\n\nCmdV/dt=−Iion(V,w)+Istim,\ndw/dt= (w∞(V)−w)/τ(V),(1)\nwhere the total sum of ion currents\nIion(V,w) =gCam∞(V)(V−VCa)+gKw(V−VK)+gL(V−VL), (2)5\nIstimis an external stimulating current, and the constituent fun ctions\nm∞(V) =1\n2[1+tanh(( V−V1)/V2)], (3)\nw∞(V) =1\n2[1+tanh(( V−V3)/V4)], (4)\nτ(V) =τmax/cosh((V−V3)/(2V4)). (5)\nFor numerical simulations shown in Figures 1-3 we have used t he following values of\nthe ML model parameters corresponding the 1st neuronal exci tability type [2]: Cm= 20\nµF/cm2,gCa= 4mS/cm2,gK= 8mS/cm2,gL= 2mS/cm2,VCa= 120 mV,VK=−84\nmV,VL=−60mV,V1=−1.2mV,V2= 18mV,V3= 12mV,V4= 17.4mV,τmax= 14.925\nms. These parameters result in the resting potential value Vrest=−59.47mV, which is\nthe solution of equation Iion(V,w∞(V)) = 0 and is very close to VLvalue. Supplementary\nFigures S1-S2 show results for the ML model of the 2nd excitab ility type [2], for which\ngCa= 4.4mS/cm2,V3= 2mV,V4= 30mV,τmax= 25ms, and all the rest parameters are\nthe same as those for the 1st type. In turn, these parameters r esult inVrest=−60.85mV.\nThe initial conditions for all numerical simulations of the ML model in this paper were\nas follows: V(t= 0) =Vrest,w(t= 0) =w∞(Vrest).\n3. Linearization of the Morris-Lecar equations at large con stant stimulation\nIn what follows, we consider the case Istim> Imaxand seek a solution for the potential\nin the form V(t) =Vst+U(t), whereVstis the stationary potential value determined from\nequation Iion(Vst,w∞(Vst)) =IstimandU(t→+∞) = 0. In addition, we assume that for\nany moment of time tthe condition |U(t)| ≪ |Vst|holds. Given this, we expand w∞(V)and\nm∞(V)into a Taylor series up to the linear term with respect to U:\nw∞(Vst+U)≈w∞(Vst)+dw∞(Vst)\ndVU=a+bU,\nm∞(Vst+U)≈m∞(Vst)+dm∞(Vst)\ndVU=p+qU,\nwherea=w∞(Vst),b=dw∞(Vst)/dV,p=m∞(Vst),q=dm∞(Vst)/dV.\nNext, we assume that τ(V)≈τ(Vst)≈τmax≡τ. The assumption is quite important for\nthe linearizing and is based on preliminary numerical simul ations showing that the value of\nImaxdoes not change substantially (at maximum, for a few percent s) with this assumption,\nregardless to the neuronal excitability type (see Fig. S1).6\nAfter that, we exactly solve the linearized ML equation on w,\n\n\nτdw/dt=a+bU(t)−w,\nw(t=t0) =w0.(6)\nIts general solution has form\nw(t) =a+W0(t)+bexp(−t\nτ)t/integraldisplay\nt0exp(t′\nτ)U(t′)\nτdt′, (7)\nwhereW0(t) = (w0−a)exp(−(t−t0)/τ).\nWe find value w(t0) =w0in a local extremum point of the potential V(t=t0) =V0=\nVst+U0, which is determined by conditiondV\ndt(t=t0) = 0. One gets\nw0=Istim−gCam∞(V0)(V0−VCa)−gL(V0−VL)\ngK(V0−VK). (8)\nFurther, writing explicitly the equation on Uand neglecting nonlinear terms, we obtain a\nlinear integro-differential equation for the potential U,\ndU\ndt=−G(t)−A(t)U−Bexp(−t\nτ)t/integraldisplay\nt0exp(t′\nτ)U(t′)\nτdt′, (9)\nwhere coefficients A(t),B, andG(t)are as follows (cf. [10]):\nA(t) = [gCa(p+q(Vst−VCa))+gK(a+W0(t))+gL]/Cm≡A+A0(t),\nA= [gCa(p+q(Vst−VCa))+gKa+gL]/Cm,\nA0(t) =gKW0(t)/Cm≡AKexp(−(t−t0)/τ), AK=gK(w0−a)/Cm,\nB=gKb(Vst−VK)/Cm,\nG(t) =A0(t)(Vst−VK) =B((w0−a)/b)exp(−(t−t0)/τ).\nIntegrating by parts, we obtain\ndU\ndt=G1(t)−A1(t)U(t)+Bexp(−t\nτ)t/integraldisplay\nt0exp(t′\nτ)dU\ndt′dt′, (10)\nwhereG1(t) =−G(t)+BU(t0)exp(−(t−t0)/τ)andA1(t) =A(t)+B.\nFurther, given that U(t) =U(t0)+t/integraldisplay\nt0(dU/dt′)dt′, one can reduce the previous equation\nonUto an integral equation for its derivative f(t) =dU/dt ,\nf(t) =G2(t)+t/integraldisplay\nt0K(t,t′)f(t′)dt′, (11)7\nwhereG2(t) =G1(t)−A1(t)U(t0) =G3+G4exp(−(t−t0)/τ),G3=−(A+B)U(t0),\nG4=U(t0)(−AK+B)−B(w0−a)/b, and\nK(t,t′) =−A1(t)+Bexp(−(t−t′)\nτ) =−(A+B)−AKexp(−(t−t0)\nτ)+Bexp(−(t−t′)\nτ).\n(12)\nThe resulting equation (11) for f(t)is an inhomogeneous Volterra integral equation of the\nsecond kind. Twice differentiating both sides of Eq. (11) wit h respect to t, we obtain that\nthe integral equation (11) is equivalent to ordinary differe ntial equation of the second order\nd2f\ndt2+(2γ+AKexp(−(t−t0)/τ))df\ndt+/parenleftbigg\n−AK\nτexp(−(t−t0)/τ)+ω2\n0/parenrightbigg\nf(t) = 0,(13)\nwhere constants 2γ=A+1/τandω2\n0= (A+B)/τhave been introduced.\nReturning to potential Uand allocating the full derivative, we have\nd2U\ndt2+(2γ+AKexp(−(t−t0)/τ))dU\ndt+ω2\n0U=const. (14)\nAssuming that potential U(t)and all its derivatives tend to zero at t→+∞, one gets\nconst= 0. Finally, we obtain\nd2U\ndt2+(2γ+AKexp(−(t−t0)/τ))dU\ndt+ω2\n0U= 0, (15)\nwith initial conditions\nU(t=t0) =U0=V0−Vst,dU\ndt(t=t0) = 0. (16)\n4. Analytical solution of the linearized equations\nAssuming that w0≈aand neglecting the time-dependent parameter in Eq. (15), we\narrive at\nd2U\ndt2+2γdU\ndt+ω2\n0U= 0. (17)\nGiven the initial conditions (16), the solution of Eq. (17) h as form\nU(t) =U0exp(−γ(t−t0))/bracketleftBig\ncos(ω(t−t0))+γ\nωsin(ω(t−t0))/bracketrightBig\n, (18)\nwith angular frequency ω=/radicalbig\nω2\n0−γ2and oscillation period T= 2π/ω. This solution de-\nscribes exponentially-damped harmonic oscillations and c orresponds well with the numerical\nresult (left graph in Fig. 2 and two left graphs in Fig. 3, see a lso Fig. S2 for the excitability8\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48/s48/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51/s48/s44/s52/s48/s44/s53\n/s97/s110/s97/s108/s121 /s116/s105/s99/s97/s108/s32/s115/s111/s108/s117/s116/s105/s111/s110/s119\n/s116/s32/s91/s109/s115/s93/s110/s117/s109/s101/s114/s105/s99/s97/s108/s32/s115/s111/s108/s117/s116/s105/s111/s110\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48\n/s97/s110/s97/s108/s121 /s116/s105/s99/s97/s108/s32/s115/s111/s108/s117/s116/s105/s111/s110/s86/s32/s91/s109/s86/s93\n/s116/s32/s91/s109/s115/s93/s110/s117/s109/s101/s114/s105/s99/s97/s108/s32/s115/s111/s108/s117/s116/s105/s111/s110\nFigure 2. Left graph: The gray curve is a numerical solution f or dynamics of neuron potential\nV(t)in the Morris-Lecar model with the 1st excitability type at Istim= 116.3µA/cm2>Imax=\n116.1µA/cm2. The red curve is an analytical solution of the linearized sy stem of equations of the\nMorris-Lecar model with initial conditions taken at the poi nt of a local maximum of the potential\n(t0= 693.3 ms, V0= 16.35 mV). Right graph: The corresponding numerical (gray ) and analytical\n(red) solutions for w(t). Parameters of the analytical formulas for this example are as follows:\nVst= 9.28 mV, a= 0.42,ω0= 262.1 Hz, γ= 21.3 Hz, ω= 261.2 Hz, 1/τ= 67.2 Hz, η= 0.08,\nχ= 0.17,2γ/|AK|= 6.78,U0= 7.07 mV, a/w0= 1.04,Wa= 0.05, andWc=−0.02, where the\nlast five parameters depend on t0andV0values.\ntype 2). It is worth noting that ω0andγare independent of t0andV0. Therefore the\ndependencies ω0(Istim),γ(Istim), andω(Istim)are relatively universal and, moreover, these\ncan be continued in the range Istim< Imax(Fig. 3, right graph), though in this case there\nis no correspondence between the numerical and analytical s olutions.\nOne can also obtain an explicit solution for w(t)by substituting U(t)into Eq. (7):\nw(t) =a+(w0−a+w1)exp(−(t−t0)/τ)+w1exp(−γ(t−t0))[−cos(ω(t−t0))+Hsin(ω(t−t0))],\n(19)\nwherew1=bU0A/BandH=ω2\n0/(ωA)−γ/ω.\nUsing auxiliary trigonometric transformations, one can wr ite functions U(t)andw(t)in\na more compact form. Denoting\ncos(η) =1/radicalbig\n1+(γ/ω)2=ω/ω0,sin(η) =γ/ω/radicalbig\n1+(γ/ω)2=γ/ω0, (20)9\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48 /s50/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s73/s109/s40 /s41\n/s32/s32/s91/s72 /s122/s93\n/s73\n/s115/s116/s105/s109/s32/s91 /s65 /s47/s99/s109/s50\n/s93/s48\n/s50 /s40/s83/s112/s105/s107/s105/s110/s103/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s41\n/s48/s44/s48 /s48/s44/s53 /s49/s44/s48 /s49/s44/s53 /s50/s44/s48/s48/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51/s48/s44/s52\n/s40/s86\n/s48/s45/s86\n/s115/s116/s41/s47/s86\n/s115/s116/s83/s47/s40/s86\n/s48 /s45/s86\n/s115 /s116/s41\n/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48\n/s82/s50/s53/s48/s48 /s53/s53/s48 /s54/s48/s48 /s54/s53/s48 /s55/s48/s48 /s55/s53/s48 /s56/s48/s48 /s56/s53/s48/s48/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51/s48/s44/s52\n/s116\n/s48/s32/s91/s109/s115/s93/s83/s47/s40/s86\n/s48 /s45/s86\n/s115 /s116/s41\n/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48\n/s82/s50\nFigure 3. Left graph: Quantitative evaluation of the corres pondence between the numerical and\nanalytical solutions at Istim= 116.3 µA/cm2(see left graph in Fig. 2): dependencies for S=\n1\nn/summationtextn\ni=1|Vst+U(ti)−V(ti)|(blue circles, left scale) and R2=/summationtextn\ni=1(Vst+U(ti)−Vmean)2\n/summationtextn\ni=1(V(ti)−Vmean)2(green squares,\nright scale) on different values of t0andV0. HereVmean=1\nn/summationtextn\ni=1V(ti), and{ti}n\ni=1is the set of\ntime moments ti> t0, for which numerical solution V(ti)is known. As one can see from the lower\ngraph, an approximate empirical condition of the good corre spondence is V0<2Vst. Right graph:\nAnalytical dependencies of ω0,γ, andωon the value of constant stimulating current Istim, with\nsuperimposed spiking frequency from Fig. 1.\nwe get\nU(t) =U0exp(−γ(t−t0))cos(ω(t−t0)−η)\ncos(η), (21)\nwhereη= arctan( γ/ω)is the inverse function of tan(η) =γ/ω. This expression for U(t)is\ncompletely equivalent to the previous solution (18).\nIn turn, introducing notations\ns= (1−γτ)/(ωτ),cos(χ) = 1/√\n1+s2,sin(χ) =s/√\n1+s2, (22)\nwe obtain a compact solution for w(t),\nw(t) =a+Wcexp(−(t−t0)/τ)+Waexp(−γ(t−t0))sin(ω(t−t0)+χ−η), (23)\nwhereχ= arctan( s), quantity Wa= (bU0/(ωτ))/radicalbig\n1+A/Bdetermines the amplitude of the\ndamped oscillations of w(t)(see Fig. 2, right graph), and Wc= (w0−a)−Wasin(χ−η).10\n5. Conclusion\nWe have shown analytically, and confirmed numerically, that for the Morris-Lecar neuron\nmodel upon stimulation by large constant depolarizing curr ent the later stages of spike am-\nplitude damping can be accurately reduced to exponentially damped harmonic oscillations,\nwith the frequency and damping coefficient completely determ ined by the original model\nparameters. Importantly, the obtained analytical formula s converge equally well (near the\nasymptote) with the numerical calculation for both the 1st a nd 2nd types of neuronal ex-\ncitability. These formulas can be directly used to quantify deviations from the harmonic\noscillations when moving away from the asymptote, i.e. with an increase in the oscilla-\ntion amplitude. In other words, the results define quantitat ively the border between truly\nnonlinear and quasi-linear dynamic behavior.\nA particular property of the Morris-Lecar model is the delay in damping occurrence\n(similar to the so-called delayed loss of stability [13]), which is especially pronounced when\nthe stimulating current Istimis just slightly above Imax. During the delay, which can last,\nfor example, 30 oscillation periods, it is practically impo ssible to distinguish the system\ndynamics from the case of periodic spike generation occurri ng atImin< Istim< Imax.\nIn turn, the ordinary bifurcation analysis, which implies e xponential dynamics, does not\ncapture such a delay. The obtained formulas can be helpful fo r studying the limits of\napplicability of the bifurcation analysis in the considere d case.\nAcknowledgments\nThis work was partially funded by the Russian Foundation for Basic Research according\nto the research project # 17-29-07093.\n[1] C. Morris, H. Lecar, Voltage oscillations in the barnacl e giant muscle fiber, Biophys. J. 35,\n193-213 (1981). https://doi.org/10.1016/S0006-3495(81)84782-0\n[2] J. Rinzel, B. Ermentrout, Analysis of neural excitabilit y and oscillations, In: C. Koch, I. Segev\n(Eds.), Methods in neuronal modeling: From synapses to netw orks (2nd Ed., MIT Press, 1998),\npp. 251-291. https://www.researchgate.net/publication/237128320\n[3] T. Tateno, A. Harsch, H. Robinson, Threshold firing frequ ency-current relationships of neurons11\nin rat somatosensory cortex: type 1 and type 2 dynamics, J. Ne urophysiol. 92, 2283-2294\n(2004).https://doi.org/10.1152/jn.00109.2004\n[4] A. Borisyuk, J. Rinzel, Understanding neuronal dynamics by geometrical dissection of minimal\nmodels, In: C. Chow et al. (Eds.), Methods and Models in Neuro physics (1st Ed., Elsevier,\n2005), pp. 17-72. https://doi.org/10.1016/S0924-8099(05)80008-3\n[5] K. Tsumoto et al., Bifurcations in Morris-Lecar neuron mo del, Neurocomputing 69, 293-316\n(2006).https://doi.org/10.1016/j.neucom.2005.03.006\n[6] L.H. Nguyen, K.S. Hong, S. Park, Bifurcation control of th e Morris-Lecar neuron model via a\ndynamic state-feedback control, Biol. Cybern. 106, 587-594 (2012).\nhttps://doi.org/10.1007/s00422-012-0508-4\n[7] C. Liu, X. Liu, S. Liu, Bifurcation analysis of a Morris–Le car neuron model, Biol. Cybern.\n108, 75–84 (2014). https://doi.org/10.1007/s00422-013-0580-4\n[8] T. Zemskova, A. Paraskevov, On damping mechanism of peri odic spiking in Morris-Lecar\nmodel at large values of constant stimulating current, Berns tein Conference (2018).\nhttps://doi.org/10.6084/m9.figshare.11301776\n[9] M.A.D. Roa et al., Scaling law for the transient behavior of type-II neuron models, Phys. Rev.\nE75, 021911 (2007). https://doi.org/10.1103/PhysRevE.75.021911\n[10] S. Ditlevsen, P. Greenwood, The Morris–Lecar neuron mo del embeds a leaky integrate-and-fire\nmodel, J. Math. Biol. 67, 239–259 (2013). https://doi.org/10.1007/s00285-012-0552-7\n[11] C. Koch, Biophysics of Computation: Information Proces sing in Single Neurons (Oxford\nUniversity Press, 1999).\nhttps://global.oup.com/academic/product/biophysics- of-computation-9780195181999\n[12] C.P. Fall et al. (Eds.), Computational Cell Biology (Spr inger-Verlag New York, 2002).\nhttps://doi.org/10.1007/b97701\n[13] E.M. Izhikevich, Dynamical Systems in Neuroscience: T he Geometry of Excitability and Burst-\ning (1st Ed., MIT Press, 2007).\nhttps://mitpress.mit.edu/books/dynamical-systems-ne uroscience\n[14] G.B. Ermentrout, D.H. Terman, Mathematical Foundation s of Neuroscience (Springer-Verlag\nNew York, 2010). https://doi.org/10.1007/978-0-387-87708-2\n[15] D. Sterratt, B. Graham, A. Gillies, D. Willshaw, Princip les of Computational Modelling in Neu-\nroscience (Cambridge University Press, 2011). https://doi.org/10.1017/CBO978051197589912\n[16] W. Gerstner, W.M. Kistler, R. Naud, L. Paninski, Neuron al Dynamics: From Single Neurons\nto Networks and Models of Cognition (Cambridge University P ress, 2014).\nhttps://doi.org/10.1017/CBO9781107447615\n[17] J.M. Gonzalez-Miranda, Pacemaker dynamics in the full Morris-Lecar model, Commun. Non-\nlinear Sci. Numer. Simulat. 19, 3229-3241 (2014).\nhttps://doi.org/10.1016/j.cnsns.2014.02.020/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s50/s49/s48 /s50/s50/s48 /s50/s51/s48 /s50/s52/s48 /s50/s53/s48/s48/s44/s57/s56/s57/s48/s44/s57/s57/s48/s48/s44/s57/s57/s49/s48/s44/s57/s57/s50/s48/s44/s57/s57/s51/s48/s44/s57/s57/s52/s48/s44/s57/s57/s53/s48/s44/s57/s57/s54\n/s32/s32/s40/s86\n/s115/s116/s41/s47\n/s109 /s97/s120\n/s73\n/s115/s116/s105/s109/s32/s91 /s65/s47/s99/s109/s50\n/s93/s83/s112/s105/s107/s105/s110/s103/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s91/s72/s122/s93\n/s73\n/s115/s116/s105/s109/s32/s91 /s65 /s47/s99/s109/s50\n/s93/s77/s76/s32/s110/s101/s117/s114/s111/s110/s32/s40/s84/s121/s112/s101/s32/s50/s41\n/s40/s86/s41/s32/s61/s32\n/s109/s97/s120\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s49/s48/s50/s48/s51/s48\n/s49/s49/s48 /s49/s50/s48 /s49/s51/s48 /s49/s52/s48 /s49/s53/s48/s48/s44/s57/s57/s54/s53/s48/s44/s57/s57/s55/s48/s48/s44/s57/s57/s55/s53/s48/s44/s57/s57/s56/s48/s48/s44/s57/s57/s56/s53/s48/s44/s57/s57/s57/s48/s48/s44/s57/s57/s57/s53\n/s32/s32/s40/s86\n/s115/s116/s41/s47\n/s109 /s97/s120\n/s73\n/s115/s116/s105/s109/s32/s91 /s65/s47/s99/s109/s50\n/s93/s40/s86/s41/s32/s61/s32\n/s109/s97/s120/s83/s112/s105/s107/s105/s110/s103/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s91/s72/s122/s93\n/s73\n/s115/s116/s105/s109/s32/s91 /s65 /s47/s99/s109/s50\n/s93/s77/s76/s32/s110/s101/s117/s114/s111/s110/s32/s40/s84/s121/s112/s101/s32/s49/s41\nFigure S1 . Dependence of spike generation frequency (determined as t he number of spikes\ndivided by the time interval of 20000 ms) on constant stimula ting current Istimfor the\nMorris-Lecar (ML) model with the 1st (left graph) and 2nd (ri ght graph) excitability types.\nThe blue curves correspond to the standard ML model describe d in Sec. 2 of the main text.\nThe orange curves correspond to a simplified version of the ML model with function τ(V),\nsee Eq. (5), taken as a constant equal to its maximal value τmax. In turn, the red curves\nnear the upper boundary of the sustained spiking interval, w hich are virtually superimposed\non the orange ones, correspond to the similar case where func tionτ(V)is also taken as a\nconstant equal to τ(Vst). The value Vstis determined from equation Iion(Vst,w∞(Vst)) =Istim\nand, above the upper boundary, Vstcorresponds to the stationary asymptotic value of the\nneuron potential. Finally, the inset in each graph shows the dependence of τ(Vst)onIstim\nthat is practically negligible (nevertheless, note that it is opposite for type 1 and type 2) so\nthat one can safely use universal approximation τ(V) =τmax.\nThe parameters of the ML model with the excitability type 1 we re as follows: Cm= 20\nµF/cm2,gCa= 4mS/cm2,gK= 8mS/cm2,gL= 2mS/cm2,VCa= 120 mV,VK=−84mV,\nVL=−60mV,V1=−1.2mV,V2= 18mV,V3= 12mV,V4= 17.4mV,τmax= 14.925ms.\nThese parameters result in the following values for the lowe r and upper boundaries of the\nsustained spiking interval of Istim:Imin= 40µA/cm2andImax= 116.1µA/cm2.\nIn turn, the ML model of the excitability type 2 had the follow ing parameters: gCa= 4.4\nmS/cm2,V3= 2mV,V4= 30 mV,τmax= 25 ms, with all the rest parameters being the\nsame as those for the type 1. The corresponding values for the lower and upper boundaries\nof the sustained spiking interval are Imin= 88.3µA/cm2andImax= 216.9µA/cm2./s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48 /s49/s54/s48/s48 /s49/s56/s48/s48/s48/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51/s48/s44/s52/s48/s44/s53/s48/s44/s54/s48/s44/s55\n/s97/s110/s97/s108/s121 /s116/s105/s99/s97/s108/s32/s115/s111/s108/s117/s116/s105/s111/s110/s110/s117/s109/s101/s114/s105/s99/s97/s108/s32/s115/s111/s108/s117/s116/s105/s111/s110/s119\n/s116/s32/s91/s109/s115/s93/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48 /s49/s54/s48/s48 /s49/s56/s48/s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s86/s32/s91/s109/s86/s93\n/s116/s32/s91/s109/s115/s93/s97/s110/s97/s108/s121 /s116/s105/s99/s97/s108/s32/s115/s111/s108/s117/s116/s105/s111/s110/s110/s117/s109/s101/s114/s105/s99/s97/s108/s32/s115/s111/s108/s117/s116/s105/s111/s110\nFigure S2 . Left graph: The gray curve is a numerical solution for dynam ics of neuron\npotential V(t)in the Morris-Lecar (ML) model with the 2nd excitability typ e atIstim=\n216.995µA/cm2> Imax= 216.9µA/cm2, whereImaxis the upper boundary of the sustained\nspiking interval (see the right graph in Fig. S1). The red cur ve is the analytical solution\nof the linearized system of equations of the ML model with ini tial conditions taken at the\npoint of a local maximum of the potential ( t0= 1156 ms, V0= 11.49 mV). Right graph:\nThe corresponding numerical (gray) and analytical (red) so lutions for w(t). Parameters of\nthe analytical formulas for this example are as follows: Vst= 8.25 mV, a= 0.6,ω0= 151.2\nHz,γ= 9.76 Hz, ω= 150.9 Hz, 1/τ= 40.2 Hz, η= 0.065,χ= 0.2,2γ/|AK|= 14.43,U0=\n3.24 mV, a/w0= 1.0,Wa= 0.014, andWc=−0.005, where the last five parameters depend\nont0andV0values.\nThe ML model parameters for the 2nd neuronal excitability ty pe were as follows: Cm= 20\nµF/cm2,gCa= 4.4mS/cm2,gK= 8mS/cm2,gL= 2mS/cm2,VCa= 120 mV,VK=−84\nmV,VL=−60mV,V1=−1.2mV,V2= 18mV,V3= 2mV,V4= 30mV,τmax= 25ms."    },    {        "title": "1912.07728v1.Spin_current_manipulation_of_photoinduced_magnetization_dynamics_in_heavy_metal___ferromagnet_double_layer_based_nanostructures.pdf",        "content": "IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 2017 1\nSpin-current manipulation of photoinduced magnetization dynamics\nin heavy metal / ferromagnet double layer based nanostructures\nSteffen Wittrock1, Dennis Meyer2, Markus M ¨uller2, Henning Ulrichs2, Jakob Walowski3, Maria Mansurova3,\nUlrike Martens3, and Markus M ¨unzenberg3\n1Unit´e Mixte de Physique CNRS/Thales, Universit ´e Paris Sud, 1 Avenue Augustin Fresnel, 91767 Palaiseau, France\n2I. Physikalisches Institut, Georg-August-Universit ¨at G ¨ottingen, Friedrich-Hund-Platz 1, 37077 G ¨ottingen, Germany\n3Institut f ¨ur Physik, Ernst-Moritz-Arndt-Universit ¨at Greifswald, Felix-Hausdorff-Str. 6, 17489 Greifswald, Germany\nSpin currents offer a way to control static and dynamic magnetic properties, and therefore they are crucial for next-generation\nMRAM devices or spin-torque oscillators. Manipulating the dynamics is especially interesting within the context of photo-magnonics.\nIn typical 3dtransition metal ferromagnets like CoFeB, the lifetime of light-induced magnetization dynamics is restricted to about 1\nns, which e.g. strongly limits the opportunities to exploit the wave nature in a magnonic crystal filtering device. Here, we investigate\nthe potential of spin-currents to increase the spin wave lifetime in a functional bilayer system, consisting of a heavy metal (8 nm\nof\f-Tantalum (Platinum)) and 5 nm CoFeB. Due to the spin Hall effect, the heavy metal layer generates a transverse spin current\nwhen a lateral charge current passes through the strip. Using time-resolved all-optical pump-probe spectroscopy, we investigate how\nthis spin current affects the magnetization dynamics in the adjacent CoFeB layer. We observed a linear spin current manipulation\nof the effective Gilbert damping parameter for the Kittel mode from which we were able to determine the system’s spin Hall angles.\nFurthermore, we measured a strong influence of the spin current on a high-frequency mode. We interpret this mode an an exchange\ndominated higher order spin-wave resonance. Thus we infer a strong dependence of the exchange constant on the spin current.\nIndex Terms —Spin Hall effect, spin current, magnetization dynamics, magnetooptical Kerr-effect.\nI. I NTRODUCTION\nTHE spin-transfer effect describes the transfer of spin an-\ngular momentum to a ferromagnet’s magnetization from\nan injected spin polarized current. Since its prediction by L.\nBerger [1], this effect has experienced a high research interest\nas it permits to manipulate and control the magnetization of a\nthin ferromagnetic (FM) layer. Especially when the resulting\nspin transfer torque is collinear with the damping torque,\nmagnetic dissipation can be controlled. Thereby the life time\nof spin wave dynamics can be drastically enhanced.\nAmong the possible methods of creating the necessary spin\ncurrent, the exploitation of the spin Hall effect has become a\npowerful mean since its first observation only a decade ago\n[2], [3], [4], [5]. Governed by spin-orbit coupling phenomena,\nit generates a spin current jsfrom a transverse charge current\njewithout any need for neither a ferromagnet nor an external\nmagnetic field. The efficiency of the conversion process can\nbe described by the spin Hall angle (SHA) \u0002SH=js=je.\nHere, we investigate the photo-induced magnetization dy-\nnamics in a few nanometer thin soft magnetic layer consisting\nof amorphous metallic cobalt iron boron alloy (Co 20Fe60B20)\nunder influence of a strong spin current generated by the SHE\nin an adjacent heavy metal film made of platinum (Pt) or\ntantalum (Ta). The sputter conditions for Ta were chosen such\nthat the film has grown in the high resistive \f-phase for which\na high SHA has recently been reported [6].\nII. E XPERIMENTAL\nThe samples consist of two functional thin layers (fig. 1a):\n8nm Pt or\f-Ta as a SHE-material generating the spin current\nManuscript received April 21, 2017; revised April 21, 2017. Corresponding\nauthor: S. Wittrock (email: steffen.wittrock@u-psud.fr).and5nm of ferromagnetic amorphous CoFeB, into which\nthe spin current is being injected in order to manipulate its\nmagnetization dynamics. The layer stack is complemented by\n3nm of Ru as a capping layer. CoFeB and Ta are deposited by\nargon ion sputtering and Pt and Ru by e-beam evaporation. All\npreparation steps are conducted in situ in ultra-high vacuum.\nSubsequent structuring of the samples by e-beam lithography\nenables the electrical contacting and generation of a high\ncharge current density (fig. 1b).\n(a)\n(b)\n (c)\nFig. 1: Experimental characteristics. (a) Schematic processes\nin the functional layer stack of a SHE material (blue) and the\nferromagnetic CoFeB (yellow). (b) Patterned sample structure,\nthe widthxwas12\u0016m for the results shown here. The red\nmarked area was excited by the laser spot. (c) Pump-probe\nsetup with ratio of powers Ppump :Pprobe = 95 : 5 , time\nresolution is realized by a delay stage, a double modulation\ntechnique of photoelastic modulator (PEM) and chopper fre-\nquency was used [7].\nWe used time resolved pump-probe spectroscopy exploitingIEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 2017 2\nthe magnetooptical Kerr effect to excite and measure magne-\ntization dynamics (schematical setup in 1c). The laser pulses\nwith central wave length \u0015= 800 nm have an autocorrelation\nlength of \u0001\u001c\u001980fs and a repetition rate of 250kHz. The\npump spot size was \u001960\u0016m providing an optical fluence of\nF= 15 mJ=cm2.\nThe incoming pump pulse induces the dynamics at \u001c= 0\nby firstly generating hot electrons, which thermalize on a\ntimescale of \u001c\u0018100fs due to electron-electron-scattering.\nFurther scattering events with phonons and spins lead to\nenergy transfer into the phonon and spin system giving rise to\nultrafast demagnetization [8]. Caused by the high change in\ntemperature on the time scale of \u00181ps, the local anisotropy\nconstant of the ferromagnetic material and thus the effective\nmagnetic field ~Heffis changed. With ~Heffrereaching its\nequilibrium position after a few picoseconds, the magnetiza-\ntion dynamics corresponding to the Landau-Lifshitz-Gilbert-\nequation (LLG) is excited. In order to subsequently trigger\na coherent precession of the magnetization, an external field\n(145mT) was applied at an out-of-plane angle of '= 35\u000e,\ntransversal to ~je.\nIII. R ESULTS -NANOSECOND TIMESCALE\n-0.5-0.4-0.3-0.2-0.10.00.10\n2 004 006 008 001 000-0.04-0.020.000.020.040.06T\nime delay τ [ps]TRMOKE signal [a.u.] \nData \nBackroundµ0H = 145 mT ;  ϕ = 35° ;  j = 6.2e10 A/m2 \nbackround-cleaned, fitted data \nDamping ∼ exp(- τ/τα)\n(a)\n-10-8-6-4-2024681012.012.112.212.312.412.512.612.712.812.9Kittel-Frequency [GHz]C\nurrent density in Ta-layer [1010 A/m2]µ0H = 145 mT ;  ϕ = 35° \n(b)\n-10-8-6-4-202468101.301.351.401.451.501.551.601.65µ0MS [T]C\nurrent density in Ta-layer [1010 A/m2]µ0H = 145 mT ;  ϕ = 35° (c)\nFig. 2: (a) Typical measurement data and Kittel mode after\nsubstraction of exponential background. The TRMOKE-signal\nis proportional to the magnetization. Dependence of (b) Kittel\nfrequency and (c) saturation magnetization on the current\ndensity in the Ta-layer.\nBesides the dominating coherent, in spatially homogeneous\ngeometries called Kittel mode oscillations, also incoherent\nphonons and magnons contribute to the signal and give rise\nto a certain background [9], [8], which can be modelled by\nexponential functions. Substracting it from the raw data (fig.\n2) highlights the magnetic oscillations. These are analysedwith respect to frequency and damping for different current\ndensities passing through the SHE-material.\nA. Kittel frequencies, magnetization, and Oersted field\nThe dependence of the Kittel frequency on jis shown in\nfig. 2b. The mainly parabolic behaviour can be attributed to\nthe reduction of the saturation magnetization due to Joule-\nheating. The observable asymmetries for opposite current\ndirections can be related to the Oersted field produced by the\ncurrent, or to the presence of a field-like torque. Analysing\nthe asymmetries, an Oersted field of HOe=ajwitha=\n(1:23\u00060:04)\u000110\u00008m is determined which agrees well with\ntheoretical estimations. Even if present, the field-like torque\nmust be much smaller than the effect arising from the Oersted\nfield. Taking also into account the in-plane applied magnetic\nfield component Hx, the saturation magnetization can be deter-\nmined from the Kittel formula !=\r\u00160p\nHx(Hx+MS)(fig.\n2c). Note that for the given field geometry, the magnetization’s\nout-of-plane component is only around 2%, as is estimated\nby micromagnetic simulations; therefore this approximation\nis valid. Besides Joule-heating, also the energy deposition of\nthe pump pulse heats up the sample locally to around 400-\n450K at a time scale of up to 1ns. Thus the saturation\nmagnetization at j= 0 A/m2is slightly lower than the\nroom temperature value of \u00160MS= 1:63T, determined by a\nvibrating sample magnetometer [10]. The Joule-heating effect\nleads to a temperature increase especially in Ta of up to 200K\nat a maximum current density of jTa= 9:3\u00011010A/m2.\nB. Effective Gilbert damping parameter of the Kittel mode\nFrom the exponential decay time \u001c\u000bof the Kittel mode (fig.\n2a) and taking into account the full in-plane magnetic field\nHx=Hextcos(') +HOe, and the current dependence of\nthe magnetization, we calculate the effective Gilbert damping\nparameter using:\n\u000bKittel =\u0012\n\u001c\u000b\r\u00160\u0012\nHx+MS\n2\u0013\u0013\u00001\n: (1)\n-10-8-6-4-202468100.0050.0060.0070.0080.0090.0100.011Effective Gilbert dampingC\nurrent density in Ta-layer [1010 A/m2] Calculated data \nLinear fit\n(a)\n-30- 20- 100 1 02 00.000.010.020.030.04C\nurrent density in Pt-layer [1010 A/m2]Effective Gilbert damping Calculated data \nLinear fit (b)\nFig. 3: Current dependence of the effective Gilbert damping\nparameter for (a) the Ta based sample and (b) the Pt based\nsample. From the linear fit, the SHE efficiency can be ex-\ntracted.\nResults for the two SHE-materials Ta and Pt are shown in\nfig. 3. We determined the SHA \u0002SHby fitting the formula:\n~\u000b=\u000b+j\u0001\u0002SH\u0001~\n2ed\u00160MSHx: (2)IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 2017 3\nNote that this expression was derived from linearizing the\nLandau-Lifshitz-Gilbert-Slonczewski-equation (LLGS).\nAveraging a few datasets gives the SHAs of:\n\u0002Ta\nSH=\u00000:043\u00060:011;\nand\n\u0002Pt\nSH= 0:086\u00060:012:\nThese results lie within the values obtained by other groups\n[11] and in particular confirm the different sign to be expected\nfor Pt [12], [13] and Ta [14], [6].\nC. Spin-pumping effect\nThe spin mixing conductance g\"#of a layer combination\ncan be estimated through comparison of the effective damping\nparameters ~\u000bof two layer stacks involving the same ferromag-\nnetic material. Using Ta and Pt as SHE-materials and CoFeB\nas the ferromagnet, the spin mixing conductance of Ta can be\ncalculated using:\ng\"#\nTa=\u00004\u0019MSd\u0001\u0001~\u000b\n~\r+g\"#\nPt (3)\n\u0001~\u000b= ~\u000bPt\u0000~\u000bTadenotes the difference of the effective\nmagnetic damping constants of the d= 5 nm thick CoFeB\nlayer for the two different adjacent SHE-materials. Average\nvalues of a few measurements give ~\u000bPt= (1:20\u00060:15)\u000110\u00002\nand~\u000bTa= (0:69\u00060:05)\u000110\u00002forj= 0. The Pt/CoFeB spin\nmixing conductance of g\"#\nPt= (4\u00061)\u00011019m\u00002[18] was used\nas a reference value.\nWe evaluate the spin mixing conductance of Ta/CoFeB to\ng\"#\nTa= (1:9\u00061:2)\u00011019m\u00002. The smaller value for Ta/CoFeB\ncompared to Pt/CoFeB is expected as similar values were\nalready obtained in Ta/NiFe [15] resp. Pt/NiFe [16] bilayers.\nIn conclusion, this analysis shows that TRMOKE is also a\nsuitable method to determine the spin mixing conductance\nwhich is an important parameter for heavy metal/FM based\nbilayer systems.\nIV. R ESULTS -PICOSECOND TIMESCALE\nOn the timescale of a few picoseconds, when the magneti-\nzation starts relaxating again, we observed a strongly damped,\nultrafast oscillation in the terahertz regime (fig. 4c), usually\nexistent for one or two periods (fig. 4). We identified this os-\ncillation as the well-known perpendicular standing spin-wave\n(PSSW) mode of first order n= 1. Analysing its frequencies\n(fig. 4c) and taking into account the saturation magnetization\ngathered on the nanosecond timescale, especially the exchange\nstiffnessA(fig. 4d) can be obtained from:\n!=\r\u00160s\u0012\nHx+2A\n\u00160MSk2\u0013\u0012\nHx+2A\n\u00160MSk2+MS\u0013\n(4)\nwithk2=k2\nz= (n\u0019=d)2,n2Nthe quantized wavevector\nnormal to the plane. The exchange stiffness is shown in fig.\n4d and found to depend on the spin current.Note that pinning at interfaces can alter the effective wave\nlength of the exchange mode. Without further experimental\nevidence, assuming zero pinning is the simplest model. Since\nthe exchange constant fits to known values determined from\nTRMOKE experiments on thicker films [17], we stick to this\nmodel.\n-2-10123456789-0.5-0.4-0.3-0.2-0.10.00.1 \nsmoothed data \n    data pointsTRMOKE signal [a.u.]T\nime delay τ [ps]Demagnetizationµ0H = 145 mT ;  ϕ = 35° ;  j = 6.2e10 A/m2\n(a)\n34 5 6 7 8 9 -0.06-0.04-0.020.000.020.040.06 original data \nsmoothed data \ndamped sinus fitTRMOKE signal [a.u.]T\nime delay τ [ps]\n(b)\n-10-8-6-4-20246810520540560580600620640Frequency [GHz]C\nurrent density in Ta-layer [1010 A/m2] (c)\n-8-6-4-2024682.32.42.52.62.72.82.93.03.13.23.3Exchange stiffness A [10-11 J/m]C\nurrent density in Ta-layer [1010 A/m2]\n(d)\n-8-6-4-202 4 6 8 0.020.040.060.080.100.120.140.16C\nurrent density in Ta-layer [1010 A/m2]αPSSW (e)\nFig. 4: (a) The first ten picoseconds of the excited dynamics\nincluding an ultrafast, highly damped oscillation during re-\nmagnetization (b). (c-e) Current dependence of certain derived\nparameters.\nThe Gilbert damping parameter of the PSSW mode (fig.\n4e) can again be determined from the oscillation’s exponential\ndecay time\u001c\u000b:\n\u000bPSSW =\u0012\n\u001c\u000b\r\u00160\u0012\nHx+2Ak2\n\u00160MS+MS\n2\u0013\u0013\u00001\n:(5)\nEspecially the exchange term \u0018Ak2=MSproves to be dom-\ninant due to the small CoFeB thickness. We attribute the (at\nleast for positive jTa) linear behaviour of \u000bPSSW to the SHE.\nThe curve’s slope (fig. 4e, jTa\u00150) possesses the same sign\nas the Kittel mode damping \u000bKittel . Furthermore, \u000bPSSW\nis found to be one order of magnitude higher than \u000bKittel .\nThe relative change from \u000bPSSW\u00190:04for a high positive\ncurrent density up to \u000bPSSW\u00190:13forj= 0shows a strong\ndependence on the spin current which is around three times\nhigher than for the Kittel mode.\nV. D ISCUSSION\nAccording to our experiments, the SHA for a Pt-based layer\nstructure is almost double the SHA of the Ta-system. A strongIEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 2017 4\nJoule-heating effect occurred especially in the Ta based struc-\nture because of its high resistivity. The latter also contributes\nto an eventual systematic reduction of the tantalum’s SHA by\nup to 10% assuming the capped Ru displays a small negative\nSHA of \u0002SH\u0019\u00000:001[19]. This would lead to a spin current\nwith opposing sign flowing into the CoFeB from above.\nThe strong spin pumping mechanism and thus increase of\nthe magnetic damping constant especially in the Pt sample\nis theoretically expected due to platinum’s high spin flip\nprobability. Adding an interlayer with high spin diffusion\nlength (such as Cu [20], [21], [22]) could solve this issue.\nThe interesting discovery of the fast, highly damped mag-\nnetic oscillation within the first 10ps of the remagnetization\nprocess is identified as the PSSW mode of first order. It\nshows a strong dependence on the injected spin current which\ninfluences its frequency, the exchange stiffness and the mode’s\ndamping. Note that usually, the PSSW mode is optically\nexcitable and observable in samples, whose layer thickness\nis higher than the optical penetration depth which is \u001830nm\nin our case. To obtain clear evidence of this mode’s properties,\nfurther investigations with enhanced signal to noise ratio are\nnecessary.\nVI. C ONCLUSION AND OUTLOOK\nIn this article we discussed the photoinduced magnetization\ndynamics under influence of an injected spin current, generated\nby the SHE. The powerful tool of time resolved magnetoop-\ntical Kerr effect, compared to more established methods like\nBLS or ST-FMR, allowed to investigate nanosecond as well\nas (sub-)picosecond dynamics and to get insights into high-\nfrequency and non-equilibrium dynamics so far not in the\nfocus within this context.\nWe discussed the influence of Joule-heating on our samples\nand described the linear manipulation of the Kittel mode’s\nGilbert damping through a spin current. The spin Hall angles\nwhich could be determined for the two different, commonly\nused SHE materials Ta and Pt, lie within other reported values\n[11].\nWe reported a magnetic oscillation at the picosecond\ntimescale which is found to be highly dependent on the spin\ncurrent. The exact behaviour of this mode has to be further\ninvestigated in future experiments with enhanced signal to\nnoise ratio. Furthermore, the interesting timescales of the de-\nmagnetization process will be adressed in future publications.\nStandard methods such as ST-FMR do not allow to address\nsuch high-frequency dynamics easily. Our approach enables\nus to enter this temporal regime, which provides new insights\non the action of spin torques on picosecond time scales.\nACKNOWLEDGMENT\nThe authors acknowledge financial support by the DFG,\nwithin the CRC 1073 ’Atomic scale control of energy con-\nversion’.\nREFERENCES\n[1] B ERGER , L.: Exchange interaction between ferromagnetic domain wall\nand electric current in very thin metallic films. Journal of Applied Physics\n55, 6, p. 1954-1956, 1984.[2] K ATO, Y.K.; M YERS , R.C.; G OSSARD , A.C.; A WSCHALOM , D.D.:\nObservation of the Spin Hall Effect in Semiconductors. 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B\n94, 054416, 2016.\n[16] C ZESCHKA , F.D.; D REHER , L.; B RANDT , M.S.; W EILER , M.; A L-\nTHAMMER , M.; I MORT , I.-M.; R EISS, G.; T HOMAS , A.; S CHOCH , W.;\nLIMMER , W.; H UEBL , H.; G ROSS , R.; G OENNENWEIN , S.T.B.: Scaling\nbehavior of the spin pumping effect in ferromagnet/platinum bilayers.\narXiv:1012.3017 , 2011.\n[17] U LRICHS , H; L ENK, B.; M ¨UNZENBERG , M.: Magnonic spin-wave\nmodes in CoFeB antidot lattices. Appl. Phys. Lett. 97, 092506, 2010.\n[18] R UIZ-CALAFORRA , A.; B R¨ACHER , T.; L AUER , V.; P IRRO , P.; H EINZ ,\nB.; G EILEN , M.; C HUMAK , A.V.; C ONCA , A.; L EVEN , B.; H ILLE -\nBRANDS , B.: The role of the non-magnetic material in spin pumping and\nmagnetization dynamics in NiFe and CoFeB multilayer systems. Journal\nof Applied Physics 117, 16, 2015.\n[19] K AMPFRATH , T.; B ATTIATO , M.; M ALDONADO , P.; E ILERS , G.;\nN¨OTZOLD , J.; M ¨AHRLEIN , S.; Z BARSKY , V.; F REIMUTH , F.;\nMOKROUSOV , Y. ; B L¨UGEL , S.; W OLF, M.; R ADU , I.; O PPENEER ,\nP.M. ; M ¨UNZENBERG , M.: Terahertz spin current pulses controlled by\nmagnetic heterostructures. Nature nanotechnology 8, 4, p. 256260, 2013.\n[20] M IZUKAMI , S.; A NDO , Y.; M IYAZAKI , T.: Effect of spin diffusion on\nGilbert damping for a very thin permalloy layer in Cu/permalloy/Cu/Pt\nfilms. Phys. Rev. B 66, 104413, 2002.\n[21] S UN, Y.; C HANG , H.; K ABATEK , M.; S ONG , Y.-Y.; W ANG , Z.;\nJANTZ , M.; S CHNEIDER , W.; W U, M.; M ONTOYA , E.; K ARDASZ , B.;\nHEINRICH , B.; V ELTHUIS , S.G.E.; S CHULTHEISS , H.; H OFFMANN , A.:\nDamping in Yttrium Iron Garnet Nanoscale Films Capped by Platinum.\nPhys. Rev. Lett. 111, 106601, 2013.\n[22] R OJAS -S´ANCHEZ , J.-C.; R EYREN , N.; L ACZKOWSKI , P.; S AVERO ,\nW.; A TTAN ´E, J.-P.; D ERANLOT , C.; J AMET , M.; G EORGE , J.-M.; V ILA,\nL.; J AFFR `ES, H.: Spin Pumping and Inverse Spin Hall Effect in Platinum:\nThe Essential Role of Spin-Memory Loss at Metallic Interfaces. Phys. Rev.\nLett. 112, 106602, 2014."    },    {        "title": "2001.06217v1.Fermi_Level_Controlled_Ultrafast_Demagnetization_Mechanism_in_Half_Metallic_Heusler_Alloy.pdf",        "content": " Fermi Level Controlled Ultrafast Demagnetization Mechanism in Half -Metallic Heusler \nAlloy  \nSantanu Pan1, Takeshi Seki2,3, Koki Takanashi2,3,4, and Anjan Barman1,* \n1Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for \nBasic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India.  \n2Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan.  \n3Center for Spintronics Research Network, Tohoku University, Sendai 980 -8577, Japan.  \n4Center f or Science and Innovation in Spintronics, Core Research Cluster, Tohoku University, Sendai \n980-8577, Japan . \n*E-mail: abarman@bose.res.in  \n \n \n \nThe electronic band structure -controlled ultrafast demagnetization mechanism in Co2FexMn 1-\nxSi Heusler alloy is underpinned by systematic variation of composition. We find the spin-flip \nscattering rate controlled by spin density of states at Fermi level  is responsible for non-\nmonotonic variation  of ultrafast demagnetization time (τ M) with x with a maximum  at x = 0.4 . \nFurthermore, Gilbert damping constant exhibits an inverse relationship with  τM due to the \ndominance of inter -band scattering mechanism. This establishes a unified mechanism of \nultrafast spin dynamics based on Fermi level position.  \n \n \n \n \n \n \n \n \n \n \n  The tremendous application potential of spin -polarized Heusler alloys in advanced spintronic s \ndevices  ignites immense interest to investigate the degree and sustainability of their spin-\npolarization under various conditions [1-4]. However, interpreting spin -polarization from the \nconventional methods  such as photoemission, spin transport measurement, point contact \nAndreev reflection and spin-resolved positron annihilation are non -trivial [5-7]. In the quest of \ndeveloping alternative methods, Zhang et al . demonstrated that all -optical ultrafast \ndemagne tization measurement is a reliable technique for probing spin -polarization [8]. They \nobserved a very large ultrafast demagnetization time as a signature of high spin -polarization in \nhalf-metallic CrO 2. However, Co -based half -metallic Heusler alloys exhibit  a comparatively \nsmaller ultrafast demagnetization time (~ 0.3 ps) which raised a serious debate on the \nperception of ultrafast demagnetization mechanism in Heusler alloys [9-11]. A smaller \ndemagnetization time in Heusler alloys than in CrO 2 is explained d ue to the smaller effective \nband gap in the minority spin band  and enhanced spin-flip scattering  (SFS) rate [9]. However, \nfurther experimental evidence shows that the amount of band gap  in minority spin band  cannot \nbe the only deciding factors for SFS medi ated ultrafast demagnetization efficiency [10]. Rather, \none also has to consider the efficiency of optical excitation for majority and minority spin bands \nas well as the optical pump -induced hole dynamics below  Fermi energy  (EF). Consequently, a \nclear interpretation of spin -polarization from ultrafast demagnetization measurement requires \na clear and thorough understanding of its underlying mechanism. Since its inception in 1996 \n[12], several theoretical models and experimental evi dences based on different microscopic \nmechanisms, e.g. spin -flip scattering (SFS) and super -diffusive spin current have been put \nforward to interpret ultrafast demagnetization [13-20]. However, the preceding proposals are \ncomplex and deterring to each othe r. This complexity increases even more in case of special \nclass of material such as the Heusler alloys. The electronic band structure and the associated \nposition of Fermi level can be greatly tuned by tuning the alloy composition of Heusler alloy \n[21,22]. By utilizing this tunability, h ere, we experimentally demonstrate that the ultrafast \ndemagnetization mechanism relies on the spin density of states at Fermi level in case of half -\nmetallic Heusler alloy system. We extracted the value of ultrafast demagnetiz ation time using \nthree temperature modelling [23] and found its non -monotonic dependency on alloy \ncomposition ( x). We have further showed that the Gilbert damping and ultrafast \ndemagnetization time are inversely proportional in CFMS Heusler alloys suggesti ng the inter -\nband scattering as the primary mechanism behind the Gilbert damping in CFMS Heusler alloys . \nOur work has established a unified theory of ultrafast spin dynamics.   A series of Co 2FexMn 1-xSi (CFMS) thin films have been deposited using magnetron co -\nsputtering system for our investigation with  x = 0.00, 0.25, 0.40, 0.50, 0.60, 0.75 and 1.00 . The \nthickness of the CFMS layer was fixed at 30 nm. It is imperative to study the crystalline phase \nwhich is the most crucial parameter that determines other magnetic properties of Heusler alloy. \nPrior to the magnetization dynamics measurement, we invest igate both the crystalline phase as \nwell as growth quality of all the samples. Fig. 1A shows the ex-situ x-ray diffraction (XRD) \npattern for all the samples. The well -defined diffraction peak of CFMS (400) at 2θ = 66.50º \nindicates that the samples are well  crystalline having cubic symmetry. The intense superlattice \npeak at 2θ = 31.90º represents the formation of B2 phase. The presence of other crucial planes \nare investigated by tilting the sample x = 0.4 by 54.5º and 45.2º from the film plane to the \nnormal direction, respectively  and observed the presence of (111) superlattice peak along with \nthe (220) fundamental peak as shown in Fig. 1B and 1C. The presence of (111) superlattice \npeak confirms the best atomic site ordering in the desired L2 1 ordered phase, whereas the (220) \nfundamental peak results from the cubic symmetry.  The intensity ratios of the XRD peaks are \nanalysed to obtain the microscopic atomic site ordering which remain same for the whole range \nof x (given in Supplemental Materials). The epitaxia l growth of the thin films is ensured by \nobserving the in-situ reflection high -energy electron diffraction (RHEED) images. The square \nshaped hysteresis loops obtained using in -plane bias magnetic field shows the samples have in -\nplane magnetization. The nearly increasing trend of saturation magnetization with alloy \ncomposition ( x) follow the Slater -Pauling curve. In -depth details of sample deposition \nprocedure, RHEED pattern and the hysteresis loops are provided in the Supplemental Materials \n[24]. The ultrafast demagnetization dynamics measurements using  time-resolved magneto -\noptical Kerr effect (TRMOKE) magnetometer have been performed  at a fixed probe fluence of \n0.5 mJ/cm2, while the pump fluence have been varied over a large range . Details of the \nTRMOKE technique is provided in Supplemental Materials [24]. The experi mental data of \nvariation of Kerr rotation corresponding to the ultrafast demagnetization  measured for pump \nfluence = 9.5 mJ/cm2 is plotted in Fig. 2A for different values of x. The data points are then \nfitted with a phenomenological expression derived from  the three temperature  model -based \ncoupled rate equations in order to extract the ultrafast demagnetization time (\nMτ) and fast \nrelaxation (\nEτ) time [23], which is given below:  \n  \nME/τ - /τ- 1 2 E 1 M E 1 2\nk3 1/2\n0 E M E MA (A τ -A τ ) τ (A -A )-Δ {[ - e - e ]H( ) A δ( )} G( )( / t 1) ( τ -τ ) (τ -τ )ttθ t t tt= + +       (1)    where A1 represents the magnetization amplitude after equilibrium between electron, spin and \nlattice is restored,  A2 is proportional to the maximum rise in the electron temperature and  A3 \nrepresents the state filling effects during pump -probe temporal overlap described by a Dirac \ndelta function.  H(t) and δ(t) are the Heaviside step and Dirac delta functions , and G(t) is a \nGaussian function which corresponds to the laser pulse.  \nThe \nMτ  extracted from the fit s are plotted as a function of x in Fig. 2B, which shows  a slight \ninitial increment followed by a sharp  decrement with x. In addition, the ultrafast \ndemagnetization rate is found to be slower in the present Heusler alloys than in the 3d metals \n[9]. The theoretical calculation of electronic band structure of CFM S showed no discernible \nchange in the amount of energy gap  in minority spin band  but a change in position of EF with \nx, which lies at the two extreme ends of the gap for x = 0 and x = 1. Thus, the variation of \nMτ  \nwith x clearly indicates  that the composition dependent EF position is somehow  responsible for \nthe variation in  \nMτ . This warrants the investigation of ultrafast demagnetization with \ncontinuously varying x values between 0 and 1. However, a majority of earlier investigations \n[10,11,2 5], being focused on exploring the ultrafast demagnetization only of Co 2MnSi ( x = 0) \nand Co 2FeSi ( x = 1), lack a convincing conclusion about the role of  electronic band structure \non ultrafast demagnetization  mechanism .  \nIn case of 3d transition metal ferromagnets, Elliott -Yafet (EY) -based SFS mechanism is \nbelieved to be responsible for rapid rise in the spin temperature and ultrafast demagnetization \n[15]. In this theory it has been shown that a  scattering event of an excited electron with a \nphonon changes the probability to find that electron in one of the spin states, namely the \nmajority spin -up (\n ) or minority spin -down (\n ) state, thereby delivering angular momentum \nto the lattice from the electronic system. It arises from the band mixing of majority and minority \nspin states with similar energy value near the Fermi surface owing to the spin -orbit coupling \n(SOC). The spin mixing para meter (b2) from the EY theory [26,27] is given by:  \n                             \n2\nk k k k b min ( ψ ψ , ψ ψ )=                                                               (2) \nwhere \nkψ  represent the eigen -state of a single electron and the bar denotes a defined average \nover all electronic states involved in the EY scattering processes. This equation represents that \nthe spin-mixing due to SFS between spin -up and spin -down states depend o n the  number of  \nspin-up (\n ) and spin -down (\n ) states  at the Fermi level, which is already represented by D F.  A compact differential equation regarding rate of ultrafast demagnetization dynamics as \nderived by Koopmans et al. [27], is given below:  \n                                                  \np C\nCeT TR (1 coth( ))TTm dmmdt=−                                                         (3) \nwhere m = M/MS, and Tp, TC, and Te denote the phonon temperature, Curie temperature and \nelectronic temperature, respectively. R is a material specific scaling factor [28], which is \ncalculated to be:  \n                                                               \n2\nsf C ep\n2\nB D S8a T gRk T D=   ,                                                                  (4) \nwhere asf, gep, DS represent the SFS probability, coupling between electron and phonon sub -\nsystem and magnetic moment divided by the  Bohr -magneton (\nB ), whereas TD is the Debye \ntemperature and kB represents the Boltzmann constant. Further, the expression for gep is: \n22\nF P B D ep\nep3πD D k T λg2=\n, where  DP, and λep denote the number of polarization states of spins and \nelectron -phonon coupling constant, respectively , and ℏ is the reduced Planck’s constant. \nMoreover, the ultrafast demagnetization time at low fluence limit can be derived under various \napproximations as:  \n                                                         \n0C\nM 22\nF si B CC F( / T )τπD λ k TT=\n ,                                                               (5)  \nwhere C0 = 1/4, \nsiλ  is a factor scaling with impurity concentration, and F(T/TC) is a function \nsolely dependent on ( T/TC) [29].  \nEarlier, it has been shown that a negligible DF in CrO 2 is responsible  for large ultrafast \ndemagnetization time. The theoretical calculation  for CFMS  by Oogane et al.  shows that DF \ninitially decreases and then increases with x [30] having  a minima at x = 0.4. As DF decreases, \nthe number of effective minority spin states become less, reducing both SOC strength, as shown \nby Mavropoulos et al. [31], and the effective spin -mixing paramet er is given by Eq. (2), and \nvice versa. This will result in a reduced SFS probability  and rate of demagnetization. In \naddition, the decrease in DF makes gep weaker, which, in turn, reduces the value of R as evident \nfrom Eq. (4). As the value of R diminishes, it will slow down the rate of ultrafast \ndemagnetization which is clear from Eq. (3). In essence , a lower value of DF indicates a lower  value of R, i.e. slower demagnetization rate and larger  ultrafast  demagnetization time. Thus, \ndemagnetization time is highest for x = 0.4. O n both sides of x = 0.4, the value of R will increase \nand ultrafast demagnetization time will decline continuously.  Our experimental results, \nsupported by the existing theoretical re sults for  the CFMS samples with varying alloy \ncomposition, clearly show that the position of Fermi level is a crucial decisive factor for the \nrate of ultrafast demagnetization. This happens due to the continuous tunability of DF with x, \nwhich causes an ensuing variation in the number of scattering channels available for SFS.  To \ncapture the effect of pump fluence on the variation of  \nMτ, we have measured the ultrafast \ndemagnetization curves for various applied pump fluences. All the flu ence dependent ultrafast \ndemagnetization curves are fitted with Eq. (1) and the values of corresponding  \nMτ are \nextracted. The change in \nMτ  with fluence is shown in Fig. 2C. A slight change in \nMτ with \nfluence is observed which is negligible in comparison to the change of \nMτ  with x. However, \nthis increment can be explained using the enhanced spin fluctuations at much higher elevated \ntemperature of the spin sy stem [28].  \nAs the primary microscopic channel for spin angular momentum transfer is the same for both \nultrafast demagnetization and magnetic damping, it is expected to find a correlation between \nthem. We have measured the time -resolved Kerr rotation data corresponding to the \nmagnetization precession at an applied in -plane bias magnetic field  (Hb) of 3.5 kOe  as shown \nin Fig. 3A. The macrospin modelling is employed to analyse the time dependent precessional  \ndata obtained by solving the Landau -Lifshitz -Gilbert equation [32] which is given below:  \n                                                 \neffˆˆˆˆγ( ) α( )dm dmm H mdt dt=−  +                                                    (6) \nwhere \nγ  is the gyromagnetic ratio and is related to Lande g factor by  \n/μg=γB . Heff is the \ntotal effective magnetic field consisting of Hb, exchange field ( Hex), dipolar field ( Hdip) and \nanisotropy field (\nKH ). The experimental variation of precession frequency ( f) against Hb is \nfitted with the Kittel formula for uniform precession to extract HK values. The details of the fit \nare discussed in the Supplementa l Materials [24] . \nFor evaluation of  \nα, all the measured data representing single frequency oscillation are fitted \nwith a general damped sine -wave equation superimposed on a bi -exponential decay function, \nwhich is given as:                                  \nfast slow/τ /τ /τ\n12 ( ) A B e B e (0)e sin( ω ζ)tt tM t M t−− −= + + + −  ,                (7) \nwhere \nζ is the initial phase of oscillation and  \nτ is the precessional relaxation time . \nfastτ  and \nslowτ\n are the fast and slow relaxation times, representing the rate of energy transfer in between \ndifferent energy baths (electron, spin and lattice) following the ultrafast demagnetization and \nthe energy transfer rate between the lattice and surrounding, respec tively. A, B1 and B2 are \nconstant coefficients. The value of \nα  is extracted by further analysing \nτ  using  \n                                        \n( )122α[γτ 2 cos( H H ]=− + +bHδφ                                                         (8)  \nwhere \n22\n12\n1S\nS S S2K 2K sin K (2 sin (2 ))4πMM M MφφH⊥ −= + − +  and \n12\n2\nSS2K cos(2 ) 2K cos(4 )\nMMφφH=+ . Here \n\nand \n  represent the angles of Hb and in -plane equilibrium M with respect to the CFMS [110] \naxis [33]. The uniaxial, biaxial and out -of-plane magnetic anisotropies are denoted as K1, K2 \nand \nK⊥, respectively. In our case K2 has a reasonably large value while K1 and \nK⊥  are \nnegligibly small. Plugging in all parameters including the magnetic anisotropy constant K2 in \nEq. (8), we have obtained the values of \nα  to be 0.0041, 0.0035, 0.0046, 0.0055, 0.0061, and \n0.0075 for x = 0.00, 0.40, 0.50, 0.60, 0.75, and 1.00, respectively. Figure 3B shows the variation \nof \nα  with frequency for all the samples. For each sample, \nα  remains constant with frequency, \nwhich rules out the presence of extrinsic mechanisms contributing to the  \nα. Next, we focus on \nthe variation of \nα  with x. Our experimental results show a non -monotonic variation of  \nα with \nx with a minima at x = 0.4 , which  is exactly opposite to the variation of \nMτ  with x. On the basis \nof Kambersky’s SFS model [34], \nα is governed by the spin -orbit interaction and can be \nexpressed as:  \n                                                      \n22\nF\nSγ (δg)αD4ΓM=\n                                                           (9) \nwhere \ngδ and \n1− represent the deviation of g factor from free electron value (~2.0) and \nordinary electron -phonon collision frequency. Eq. (9) suggests that \nα  is directly proportional \nto DF and thus it become s minimum when DF is minimum [3 0]. This leads to the non -monotonic \nvariation of\nα , which agrees well with earlier observation [30]. To eliminate the possible effects  of \nγ  and \nSM , we have plotted the variation of relaxation frequency, \nSMαγ=G  with x which \nalso exhibits similar variation as  \nα (see the supplementary materials [24] ).  \nFinally , to explore the correlation between\nα , \nMτ and alloy composition, we have plotted these \nquantities  against x as shown in Fig. 4A. We observe that \nMτ  and \nα  varies in exactly opposite \nmanner with x, having their respective maxima and minima at x = 0.4. Although \nMτ  and \nα  \nrefer to two different time scales, both of them follow the trend of variation of  DF with x. This \nshows  that the alloy composition -controlled Fermi level tunability and the ensuing SFS is \nresponsible for both ultrafast demagnetization and Gilbert damping . Figure 4B represents the \nvariation of \nMτ  with inverse of  \nα, which establishes an inversely proportional relation between \nthem . Initially under the assumption of two different magnetic fields, i.e. exchange field and \ntotal effective magnetic field, Koopmans et al.  theoretically proposed that Gilbert damping \nparame ter and ultrafast demagnetization time are inversely proportional [29]. However, that \nraised intense debate and in 2010, Fahnle et al.  showed that \nα  can either be proportional or \ninversely proportional to \nMτ  depending upon the dominating microscopic contribution to the \nmagnetic damping [32]. The linear relation sustains when the damping is dominated by \nconductivity -like contribution, whereas the resistivity -like contribution leads to an inverse \nrelation. The basic difference between the conductivity -like and the resistivity -like \ncontribution s lies in the angular momentum transfer mechanism via electron -hole ( e-h) pair \ngeneration. The generation of e-h pair in the same band, i.e. intra -band mechanism leads to t he \nconductivity -like contribution. On the contrary, when e-h pair is generated in different bands \n(inter -band mechanism), the contribution is dominated by resistivity. Our observation of the \ninversely proportional relation between \nα  and \nMτ  clearly indicates that in case of the CFMS \nHeusler alloy  systems, the damping is dominated by resistivity -like contribution arising from \ninter-band e-h pair generation. This is in contrast to the case of Co, Fe and Ni, where the \nconductivity contribution dominates [35]. Typical resistivity (\nρ ) values for Co 2MnSi ( x = 0) \nare 5\ncm− at 5 K and 20 \ncm− at 300 K [36]. The room temperature value of \nρ\ncorresponds to an order of magnitude larger contribution of the inter -band e-h pair generation \nthan the intra -band generation [36]. This is in strong agreement with our experimental results \nand its conclusion. This firmly establishes that unlike convention al transition metal \nferromagnets, damping in CFMS Heusler alloys  is dominated by resistivity -like contribution , \nwhich results in an inversely proportional relation between \nα  and\nMτ .  In summary, we have investigated the ultrafast demagnetization and magnetic Gilbert damping \nin the CFMS Heusler alloy systems with varying alloy composition ( x), ranging from x = 0 \n(CMS) to x = 1 (CFS) and identified a strong correlation between \nMτ  and x, the latter \ncontrolling the position of Fermi level in the electronic band structure of the system. We have \nfound that \nMτ  varies non -monotonically with x, having a maximum value of ~  350 fs for x = \n0.4 corresponding to the lowest DF and highest degree of spin -polarization. In -depth \ninvestigation has revealed that the ultrafast demagnetization process in CFMS is primarily \ngoverned by the composition -controlled variation in spin -flip scattering rate  due to variable DF. \nFurthermore, we have systematically investigated the precessional dynamics with variation in \nx and extracted the value of \nα  from there. Our results have led to a systematic correlation in \nbetween\nMτ ,\nα and x and we have found an inversely proportional relationship between \nMτ  and \nα\n. Our thorough investigation across the alloy composition ranging from CMS to CFS have \nfirmly establishe d the fact that both ultrafast demagnetization and magnetic  Gilbert  damping \nin CFMS are strongly controlled by the spin density of states at Fermi level. 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Lett.  95, 022509 \n(2009).    \n \n \n \n \n \n \n \n \n \n \n    \n \nFig. 1. (A)  X-ray diffraction (XRD) patterns of Co 2FexMn 1-xSi (CFMS) thin films for different alloy \ncomposition ( x) measured in conventional θ-2θ geometry. Both CFMS (200) superlattice and CFMS \n(400) fundamental peaks are marked along with Cr (200) peak. (B) The tilted XRD patterns reveal the \nCFMS (111) superlattice peak for L2 1 structure. (C) CFMS (220) fundamental peak together with Cr \n(110) peak.  \n \n \n \n \n \n \n \n \n  \nFig. 2. (A) Ultrafast demagnetization curves for the samples with different alloy composition ( x) \nmeasured using TRMOKE. Scattered symbols  are the experimental data and solid lines are fit using Eq. \n3. (B) Evolution of \nMτ  with x at pump fluence of 9.5 mJ/cm2. Symbols are experimental results and \ndashed line is guide to eye.  (C) Variation in \nMτ  with pump fluence.  \n \n \n \n \n  \nFig. 3. (A) Time -resolved Kerr rotation data  showing precessional dynamics  for samples with different \nx values . Symbols  are the experimental data and solid lines are fit with damped sine wave equation ( Eq. \n6). The extracted \nα  values are given  below  every curve. (B) Variation of \nα  with precession frequency \n(f) for all samples as shown by symbols, while  solid lines are linear fit.  \n \n \n \n \n \n \n \n \n \n  \n \nFig. 4. (A) Variation of \nMτ  and \nα  with x. Square and circular symbols  denote the experimental results , \nand dashed , dotted  lines are guide to eye.  (B) Variation of \nMτ  with \n1α− . Symbols represent the \nexperimentally obtained values and solid line refers to linear fit.  \n \n \n \n \n \n \n \n \n \n \n \n \n Supplementa l Material s \n \nI. Sample preparation method  \nA series of MgO Substrate /Cr (20 nm)/ Co 2FexMn 1-xSi (30 nm)/Al -O (3 nm) sample stacks \nwere deposited using an ultrahigh vacuum magnetron co -sputtering system. First a 20 -nm-thick \nCr layer was deposited on top of a single crystal MgO (100) substrate at room temperature \n(RT) followed by annealing it at 600 ºC for 1 h. Next, a Co 2FexMn 1-xSi layer of 30 nm thickness \nwas deposited on the Cr layer followed by an in -situ annealing process at 500 ºC for 1 h. \nFinally, each sample stack was capped with a 3 -nm-thick Al -O protective layer. A wide range \nof values of x is chosen, namely, x = 0.00, 0.25, 0.40, 0.50, 0.60, 0.75 and 1.00. To achieve the \ndesired composition of Fe and Mn precisely, the samples were deposited using well controlled \nco-sputtering of Co 2FeSi and Co 2MnSi. Direct deposition of Co 2FexMn 1-xSi on top of MgO \nproduces strain  due to lattice mismatch in the Co 2FexMn 1-xSi layer which alters its intrinsic \nproperties [1S]. Thus, Cr was used as a buffer layer to protect the intrinsic Co 2FexMn 1-xSi layer \nproperties [2S].  \nII. Details of measurement techniques  \nUsing ex-situ x-ray diffraction ( XRD ) measurement we investigated the crystal structure and \ncrystalline phase of the samples. The in-situ reflection high -energy electron diffraction  \n(RHEED ) images were observed after the layer deposition without breaking the vacuum \ncondition in order to investigate the epitaxial relation and surface morphology of Co 2FexMn 1-\nxSi layer. To quantify the values of M S and H C of the samples, we measured the  magnetization \nvs. in -plane magnetic field  (M-H) loops using a vibrating sample magnetometer ( VSM) at room \ntemperature with H directed along the [110] direction of Co 2FexMn 1-xSi. The ultrafast \nmagnetization dynamics for all the samples were measured by using a time-resolved magneto -\noptical Kerr effect ( TRMOKE ) magnetometer [ 3S]. This is a two -colour  pump -probe \nexperiment in non -collinear arrangement. The fundamental output (wavelength, λ = 800 nm, \npulse -width, \ntσ ~ 40 fs) from an amplified laser system (LIBRA, Coherent) acts as probe and \nits second harmonic signal (λ = 400 nm, \ntσ ~ 50 fs) acts as pump beam. For investigating both \nultrafast demagnetization within few hundreds of femtosecond s and p recessional \nmagnetization dynamics in few hundreds of picosecond time scale, we collected the time -\nresolved Kerr signal in two different time regimes. The time resolution during the  measurements was fixed at 50 fs in -0.5 To 3.5 ps and 5 ps in -0.1 ns to 1 .5 ns to trace both the \nphenomena precisely. The pump and probe beams were focused using suitable lenses on the \nsample surface with spot diameters of ~250 µm and ~100 µm, respectively. The reflected signal \nfrom the sample surface was collected and analysed  using a polarized beam splitter and dual \nphoto detector assembly to extract the Kerr rotation and reflectivity signals separately. A fixed \nin-plane external bias magnetic field ( Hb) of 1 kOe was applied to saturate the magnetization \nfor measurement of ult rafast demagnetization dynamics, while it was varied over a wide range \nduring precessional dynamics measurement.  \nIII. Analysis of XRD peaks  \nTo estimate the degree of Co atomic site ordering, one has to calculate the ratio of integrated \nintensity of (200) and (400) peak. Here, we fit the peaks with Lorentzian profile  as shown in \ninset of Fig.  1S and extracted the integrated intensities as a parameter  from the fit . The \ncalculated ratio of I(200) and I(400) with re spect to alloy composition ( x) is sho wn in Fi g. 1S. \nWe note that there is no significant change in the I(200)/I(400) ratio. This result indicates an \noverall good quality atomic site ordering in the broad range of samples used  in our study . \n \nFig. 1S. Variation of integrated intensity ratio I(200)/I(40 0) with x, obtained from XRD patterns. Inset \nshows the fit to the peaks with Lorentzian profile.  \n \nIV. Analysis of RHEED pattern  \n The growth quality of the CFMS thin films was experimentally investigated using in-situ \nRHEED technique. Figure 2S shows the RHEED images captured  along the MgO [100] \ndirection for all the samples. All the images contain main  thick  streak lines in between the thin \nstreak lines , which are marked by the white arrows, suggesting the formation o f ordered phases. \nThe presence of regularly -aligned streak lines confirms the epitaxial growth in all the films.     \n \nFig. 2S. In-situ RHEED images for all the Co 2FexMn 1-xSi films taken along the MgO [100] direction. \nWhite arrows mark the presence of thin streak lines originating from the L2 1 ordered phase.  \n \nV. Analysis of magnetic hysteresis loops  \nFigure 3SA represents the M-H loops measured at room temperature  using VSM for all the \nsamples. All the loops are square in nature, which indicates a very small saturation magnetic \nfield. We have estimated the values of saturation magnetization ( MS) and coercive field ( HC) \nfrom the M-H loops.  Figure 3SB represents MS as a function of x showing a nearly monotonic \nincreasing trend, which is consistent with the Slater -Pauling rule for Heusler alloys [4S], i.e. \nthe increment in MS due to the increase in the number of valence electrons. However, it deviates \nremarkably at x = 1.0. This deviation towards the Fe -rich region is probably due to the slight \ndegradation in the film quality. Figure 3SC shows that HC remains almost constant with \nvariation of x. \n   \nFig. 3S. (A) Variation of M with H for all the samples. (B) Variation of MS as a function of x. \nSymbols are experimentally obtained values and dashed line is a linear fit. (C) Variation of HC \nwith x.    \n \nVI. Anal ysis of frequency ( f) versus bias magnetic field ( Hb) from TRMOKE \nmeasurements  \nWe have experimentally investigated the precessional dynamics of all the samples using \nTRMOKE technique. By varying the external bias magnetic field ( Hb), various precessional \ndynamics have been measured. The post -processing of these data foll owed by fast Fourier \ntransform (FFT) provides the precessional frequency (f) and this is plotted against Hb as shown \nin Fig. 4S .  \nTo determine the value of in-plane magnetic anisotropy  constant , obtained f-Hb curves have \nbeen analysed  with Kittel formula  which is given below:  \n                                  \n2 1 2\nS\nS S S2K 2K 2K γ(4πM )( )2π M M Mbb f H H= + + + +\n                                      (1S) \n  where MS is saturation magnetization and \nγ  denote the gyromagnetic ratio given by\nBgμγ=\nwhile K1 and K2 represent the two -fold uniaxial and four -fold biaxial magnetic anisotropy \nconstant, respectively.  \n \nFig. 4S. Variation of f as a function of Hb. Circular filled symbols represent the experimental data and \nsolid lines are Kittel fit.  \n \nWe have found the values of several parameters from the fit including K1 and K2. K1 has a \nnegligible value while K2 has reasonably large value in our samples. The e xtracted values of \nthe parameters from the fit are tabulated as follows  in Table 1S : \nTable 1S: The extracted values of Lande g factor and the four -fold biaxial magnetic anisotropy \nconstant K 2 for different values of x. \nx g K2 (erg/cm3) \n0.00 2.20 3.1×104 \n0.40 2.20 2.6×104 \n0.50 2.20 3.0×104 \n0.60 2.20 2.5×104 \n0.75 2.20 2.6×104 \n 1.00 2.20 3.4×104 \n \nVII. Variation of r elaxation frequency  with alloy composition   \nWe have estimated the damping coefficient (α) and presented its variation with alloy \ncomposition ( x) in the main manuscript. According to the Slater -Pauling rule, M S increases \nwhen the valence electron number systematically increases. As in our case the valence electron \nnumber changes with x, one may expect a marginal effect of M S on the estimation of damping. \nThus, to rule out any such possibilit ies, we have calculated the  variation of  relaxation \nfrequency ,\nS GαγM= with x, which is represented in Fig. 5S. It can be clearly observed from \nFig. 5S that relaxation frequen cy exactly follows the trend of\nα . This rules out any possible  \nspurious contribution of M S in magnetic damping.   \n \nFig. 5S. Non-monotonic v ariation of G with x for all the samples.  \n \nReferences:  \n[1S] S. Pan, S. Mondal,  T. Seki,  K. Takanashi, , and A. Barman, Influence of the thickness -dependent \nstructural evolution on ultrafast magnetization dynamics in Co 2Fe0.4Mn 0.6Si Heusler alloy thin films.  \nPhys. Rev. B  94, 184417 (2016).  \n[2S] S. Pan, T. Seki, K. Takanashi, and A. Barman, Role of the Cr buffer layer in the thickness -\ndependent ultrafast magnetization dynamics of Co 2Fe0.4Mn 0.6Si Heusler alloy thin films.  Phys. Rev. \nAppl.  7, 064012 (2017).  \n[3S] S. Panda,  S. Mondal, J. Sinha,  S. Choudhury, and A. Barman, All-optical det ection of interfacial \nspin transparency from spin pumping in β -Ta/CoFeB thin films.  Science Adv.  5, eaav7200 (2019).  \n[4S] I. Galanakis,  P. H.  Dederichs, and  N. Papanikolaou, Slater -Pauling behavior and origin of half -\nmetallicity of the full Hesuler alloys.  Phys. Rev. B  66, 174429 (2002).  \n  "    },    {        "title": "2002.02686v1.Engineering_Co__2_MnAl__x_Si___1_x___Heusler_compounds_as_a_model_system_to_correlate_spin_polarization__intrinsic_Gilbert_damping_and_ultrafast_demagnetization.pdf",        "content": "1 \n Engineering Co 2MnAl xSi1-x Heusler compounds as a model system to \ncorrelate spin polarization, intrinsic Gilbert damping and ultrafast \ndemagnetization  \nC. Guillemard1,2, W. Zhang1*, G. Malinowski1, C. de Melo1, J. Gorchon1, S. Petit -\nWatelot1, J. Ghan baja1, S. Mangin1, P. Le Fèvre2, F. Bertran2, S. Andrieu1*  \n1 Institut Jean Lamour, UMR CNRS 7198, Université de Lorraine, 54500 Nancy France  \n2 Synchrotron SOLEIL -CNRS, Saint -Aubin, 91192 Gif -sur-Yvette, France  \nAbstract:  \nEngineering of magnetic materials f or developing better spintronic  applications relies on \nthe control of two key parameters: the spin polarization and the Gilbert damping \nresponsible for the spin angular momentum dissipation. Both of them are expected to \naffect the ultrafast magnetization dyna mics occurring on the femtosecond time scale. \nHere, we use engineered Co2MnAl xSi1-x Heusler  compounds to adjust the degree of spin \npolarization P from 60 to 100% and investigate how it correlates with the damping. We \ndemonstrate  experimentally that the damping decreases when increasing the spin \npolarization from 1.1  10-3 for Co 2MnAl with 63% spin polarization  to an ultra -low value \nof 4.10-4 for the half -metal magnet Co 2MnSi. This allows us investigating the relation \nbetween these two parameters and the ultrafast demagnetization time characterizing the \nloss of magnetization occurring after femtosecond laser pulse excitation. The \ndemagnetization time is observed to be inversely proportional to 1 -P and as a consequence \nto the magnetic damping, which can  be attributed to the similarity of the spin angular \nmomentum dissipation processes  responsible for these two effects. Altogether, our high \nquality Heusler  compounds allow controlling the band structure and therefore the channel \nfor spin angular momentum dissipation.  \n \n * corresponding authors :  \nwei.z hang @univ -lorraine.fr  \nstephane.andrieu@univ -lorraine.fr   2 \n I - INTRODUCTION  \nDuring the last decades, extensive magnetic materials research has strived to \nengineer denser, faster and more energy efficient processing and data storage devices. On \nthe one hand, a high spin polarization has been one of the most important ingredients th at \nhave  been seek  [1]. For example, the spin polarization is responsible for a high readout \nsignal in magnetic tunnel junction based devices  [2,3] . Additionally, a high spin \npolarization results in a decrease of the threshold current for magnetization reve rsal by \nspin torques  [4] required for the development of spin-transfer -torque magnetic random \naccess memory devices [5] , for gyrotropic dynamics in spin -torque nano -oscillators [6] \nand for magnetic domain wall motion  [7]. On the other hand, the intrinsic m agnetic \nenergy dissipation during magnetization dynamics, which is determined by the Gilbert \ndamping constant, needs to be low in order to build an energy efficient device. Fortunately, \nspin polarization and damping are usually closely  related  in magnetic materials.  \nNowadays, manipulation of the magnetization on the femtosecond timescale has \nbecome an outstanding challenge since the demonstration of subpicosecond \nmagnetization quenching [8] and magnetization reversal on the picosecond timescale  [9]. \nDespite the theoretical and experimental work that has been reported up to now , the \nrelationship between  the polarization at the Fermi level or the magnetic damping and the \nultrafast demagnetization  excited by femtosecond lasers, remains  unclear  [10-15]. Indeed,  \nnumerous mech anisms have been proposed  but no consensus has yet been reached. In \nparticular, efforts have been undertaken to unify the magnetization dynamics on the \nnanosecond timescale and the ultrafast demagnetization considering that the sp in-flip \nmechanisms involved in both phenomena could be the same  [10-11,16] . Regarding the \ninfluence of the damping on the demagnetization time, different predictions have been \nreported  both experimentally and theoretically . In this situation, the need for engineered \nsamples in which the spin -polarization and magnetic damping are well controlled is of \nutmost importance to unveil their role on the ultrafast magnetization dynamics.  \nHeusler compounds are a notable class of magnetic materials allowing for tunabl e \nspin-polarization and magnetic damping [ 17]. The absence of available electronic states \nin the minority band at the Fermi level leads to very high spin polarization and ultra -low \ndamping due to a strong reduction of spin scattering [ 18-23]. Recently,  ultra-low damping 3 \n coefficient associated with full spin polarization at the Fermi energy  was reported in \nCo2Mn-based Heusler compounds , [22-23]. Among those alloys, Co 2MnSi has the \nsmallest damping down to 4.1 x 10-4 with 100% spin -polarization while Co 2MnAl , which \nis not predicted to be a half -meta llic magnet, has a damping of 1.1 x 10-3 and a spin -\npolarization of 60 %.  \nIn the present work, we used Co 2MnAl xSi1-x quaternary Heusler  compounds \ngrown by Molecular Beam Epitaxy (MBE) to tune the spin -polarization at the Fermi \nenergy . Controlling the amount of Al within the alloys allows tuni ng the spin -polarization \nfrom 60  to 100 % as measured by spin resolved photoemission. We show that  the \nmagnetic  damping parameter for these alloys is among the lowest reported in the literature \nand decreases when the spin -polarization increases. Ultrafast magnetization dynamics \nexperiments were thus performed on these prototype samples. This complete \nexperimental characterization allows us to directly correlat e the ultrafast magnetization \ndynamics to these parameters and comparing our results to the different theory discussed \nabove.  \nThe Co 2MnSi compound grows in the L2 1 structure whereas the Co 2MnAl com pound \ngrows in the B2 phase as shown by STEM -HAADF analysis [22]. Such different \nstructures are directly observable during the growth by Reflexion High Energy Electron \nDiffraction (RHEED ) since the surface lattice is different for bot h compounds. Ind eed, \nhalf streaks are observed along Co 2MnSi [110] azimuth due to the L2 1 chemical ordering \n[24] which is not the case for  Co2MnAl  [22]. The RHEED analysis on Co2MnAl xSi1-x \nfilms with x= 0, ¼ ,½ , ¾ ,1 reveals a regular decrease of these half -streaks intensity with \nx (Figure 1 a). This information that concerns only the surface is confirmed in the entire \nthickness of the films  by using x -ray diffraction. Indeed, the (111) peak typical of the \nchemical ordering in the L2 1 structure clearly decreases and disappears with x ( Figure \n1b).  4 \n  \nFigure 1 :  a) RHEED patterns along [110] showing the progressive vanishing of the half -streaks \n(observed on Co 2MnSi, x=0) at the surface with x. b) Confirmation of the transition from L2 1 to \nB2 chemical ordering in the entire film by the vanishing of the (111) peak and displacement of \n(220) peak with x  as shown by x -ray diffraction. c) Spatial  distribution of both chemical ordering \nin the films deduced from STEM -HAADF experiments: as the L 21 structure is observed in the \nentire Co 2MnSi film (x=0 ), and the B2 one i n Co 2MnAl (x=1 ), a mixing of both structure is clearly \nobserved for x=0.5 . \n \nIn addition, the displacement of the (220) peak with x allows us to extract a linear \nvariation of the lattice constant  (Figure 1b ), as observed in the case of a solid solution. \nThis is an indication that the L2 1 chemical ordering progressively vanishes when \nincreasing the Al substitution rate 𝑥. However, the chemical disorder distribution in the \nfilms cannot be easily determined by using the electron and x -ray diffraction analyses. To \naddress this point, a STEM HAADF analysis has been carried  on the Co 2MnAl ½Si½ films \nwith a comparison with Co 2MnSi and Co 2MnAl. A clear mixing of both structures is \n5 \n observ ed for x=½ where around 50% is L2 1 chemically ordered and 50%, B2, with typical \ndomains size around 10nm along the growth axis (001) and a few nm in the plane of the \nfilm ( Figure 1c). \nThe electronic properties of the Co 2MnAl xSi1-x(001) series were studied using spin -\nresolved photoemission (SR -PES) and  ferromagnetic resonance (FMR). The SR -PES \nspectra were obtained by using the largest slit acceptance of the detector (+/ - 8°) at an \nangle of 8° of the normal axis of the surface. Such geometry allows us to analyze all the \nreciprocal space  as confirmed by similar experiments but performed on similar \npolycrystalline films [23]. Getting the spin-polarization dependence with x using raw SR -\nPES spectra  is however not obvious due to the existence of surface states systematically \nobserved on Co 2MnSi  but also on other Co 2Mn-based Heusler compounds [19, 22-23]. \nTo get the bulk spin polarization, we thus used the S polarization of the photon beam. \nIndeed , we have shown that the surface states are no more detected  due to their symmetry \n[19] without any loss of information on the bulk band structure [ 23]. The corresponding \nSR-PES spectra are shown in figure 2 . As expec ted, we thus obtain a tunable spin \npolarization  at EF from 100% to 63% by substituting Si by Al, as shown in figure 3 . \n \nFigure 2 : spin -resolved photoemission spectra using P photon polarization (left), S photon \npolarization (middle) and resulting spin polarization curves (right) for the Co 2MnAl xSi1-x series,  \n6 \n  The radiofrequency magnetic dynamics of the films were thus studied using \nferromagnetic resonance (FMR) . The magnetic damping coefficient  , the effective \nmagnetic moment Ms (close to the true moment in our films due to very small anisotropy \n– see [ 22]), and the inhomogeneous linewidth f0 were thus extracted from the \nmeasurements performed on the Co 2MnAl xSi1-x(001) series. The results obtained on the \nsame series used for photoemission experiments are shown in table I . As shown in figure \n3, a clear correlation is observed between the spin polarization at EF and the magnetic \ndamping  coefficient , as theoretically expected. An ultra -low  value was obtained for \nCo2MnSi (x= 0) due to the large spin gap [ 22]. By substituting Al by Si, the magnetic \ndamping increase is explained by the decrease of the spin polarization.  \nCo2MnAl xSi1-x Spin polarization  \n(%) Ms  \n(µB/f.u.)    \n(x 10-3) f0  \n(MHz)  g factor  \n(0.01)  \nx = 0  973 5.08 0.460.05 14.3 2.01 \nx = 0.25  903 4.85 0.730.15 21.7 1.99 \nx = 0.5  833 4.85 0.680.15 9 2.01 \nx = 0.75  703 4.8 1.000.05 81.5 2.00 \nx = 1  633 4.32 1.100.05 22 2.01 \nTable 1: data extracted from spin -resolved photoemission and ferromagnetic resonance \nexperiments performed on the Co 2MnAl xSi1-x series.  \n \nFigure 3: -top- spin polarization and magnetic damping dependence with Al content for the \nCo2MnAl xSi1-x series and –bottom - magnetic damping versus spin polarization . The lines are \nguide to the eyes.  \n7 \n In addition, t he magnetization is also observed to decrease with x in agreement with the \nSlater -Pauling description of the valence band electrons in Heusler compounds  [25]. \nIndeed, as a 5 µ B magnetic moment per cell is expected for Co2MnSi (type IV valence \nelectrons), it should decrease to  4 when replacing Si by Al (type III)  as actually observed \n(Table I ). Finally, the FMR susceptibilities reach extremely small inhomogeneous \nlinewidth f0, a proof of the excellent homogeneity of the magnetic properties (hence a \nhigh crystal quality) in our films.  \nFigure 4 (a) shows the ultrafast demagnetization curves measured  on the same  \nCo2MnAl xSi1-x series with a maximum magnetization quenching ~1 5%. The temporal \nchanges of the Kerr signals ∆𝜃𝑘(𝑡) were normalized by the saturation value 𝜃𝑘 just before \nthe pump laser excitation. The time evolution of magnetization on sub -picosecond \ntimescales c an be fitted according to Eq. (2 ) in terms of the three -Temperature M odel \n(3TM)  [26], which describes the energy distribution among electrons, phonons, and spins \nafter laser excitation.  \n−∆𝑀(𝑡)\n𝑀={[𝐴1\n(𝑡𝜏0+1 ⁄ )0.5−𝐴2𝜏𝐸−𝐴1𝜏𝑀\n𝜏𝐸−𝜏𝑀𝑒−𝑡\n𝜏𝑀−𝜏𝐸(𝐴1−𝐴2)\n𝜏𝐸−𝜏𝑀𝑒−𝑡\n𝜏𝐸]Θ(𝑡)}∗𝐺(𝑡,𝜏𝐺)          (2) \nwhere  𝐺(𝑡,𝜏𝐺) represents the convolution product with the Gaussian laser pulse profile, \nG\n is the full width at half maximum (FWHM) of the laser pulses. Θ(𝑡) is the Heavyside  \nfunction . The constant  A1 represents the amplitude of demagnetization obtained after \nequilibrium between the electrons, spins, and phonons is reestablished while A 2 is \nproportional to the initial electron temperature raise .  The two critical time parameters  \n𝜏𝑀,𝜏𝐸 are the ultrafast demagnetization time and magnetization recovery time, \nrespectively. In the low fluence regime, which corresponds to our measurements, 𝜏𝐸 \nbecomes close to  the electron -phonon relaxation time . A unique value of  𝜏𝐸=550 ±\n20 𝑓𝑠 was used for fitt ing the demagnetization curves for all samples. T he ultrafast \ndemagnetization time  𝜏𝑀 decrease s from 380 ±10 fs for Co 2MnSi to 165 ±10 fs for \nCo2MnAl (Figure 4b). The evolution of the demagnetization time with both spin \npolarization P and  Gilbert damping 𝛼 is presented in figure 4c and 4d . A clear linear \nvariation between 1𝜏𝑀⁄ and 1−𝑃 is observed in this series . As the magnetic damping 𝛼 \nis proportional to P here, this means that 1𝜏𝑀⁄ is proportional to 𝛼 too. A similar relation 8 \n between these two par ameters was proposed by  Koopmans  et al.  [10]. However, they also \npredicted an influence of the  Curie temperature . As the Curie temperature in Heusler \ncompounds changes with the number of valence electron s and because the Co 2MnAl xSi1-\nx behave as solid solutions  as indicated by  the lattice spacing variation ( Figure 1b), we \nthus consider a linear decrease of  𝑇𝑐 with x going from 985 K to 697 K as exper imentally \nmeasured for x =0 and x=1, respectively. To test this possi ble influence of  the Curie \ntemperature  on the ultrafast magnetization dynamics , we plot in figure 4d  first the product \n𝜏𝑀.𝛼 and second the product   𝜏𝑀.𝛼.𝑇𝑐(𝑥)𝑇𝑐(𝐶𝑜2𝑀𝑛𝑆𝑖 ) ⁄ . These results demonstrate that \nthe Curie temperature does not influence the ultrafas t demagnetization  in our samples . \n \nFigure 4 : (a) Ultrafast demagnetization curves obtained for different Al  concentration x . The \ncurves have been shifted vertically for sake of clarity. The solid lines represent fitted  curves  \nobtained using Eq. ( 2). (b) Ultrafast demagnetization time as a function of Al content  x, (c) the \ninverse of 𝜏𝑀 as a function of  1-P, P being the spin polarization at E F, and d) test of Koopmans \nmodel with and without taking into account the Curie temperature of the films  (see text).   \n9 \n  One can now compare our experimental results with existing theoretica l models. \nWe first discuss  the dependence between the magnetic damping and the spin polarization. \nUltra -low magnetic damping values are predicted in Half -Metal Magnet (HMM) Heusler \ncompounds and explained by the lack of density of state at the Fermi energy for minority \nspin, or in other words by the full spin polarization  [18,27,28] . Consequently, the \nmagnetic damping is expected to increase when creating some states in the m inority band \nstructure around the Fermi energy that is when decreasing the spin polarization  [28]. If \nwe confirmed in previous experimental works that ultra -low magnetic damping \ncoefficients are actually observed especially on HMM Co2MnSi  and Co 2MnGe [19,2 2-\n23], we could not state any quantitative dependence between the damping values and the \nspin polarization. As prospected, the Co 2MnAl xSi1-x alloys are shown here to be ideal \ncandidates to address this point . This allows us getting  a clear experimental demonstration \nof these theoretical expectations. Furthermore, a linear dependence between the magnetic \ndamping and the spin polarization is obtained. This behavior may be explained by the \nmixing of both L2 1 and B2 phases in the films.  To the best of our knowledge, this \nexperimental result is the first quantitative demonstration of the link between the \nmagnetic damping and spin polarization.  \n Second , the dependence between the magnetic damping and the demagnetization \ntime observed here is a clear opportunity to test the different theoretical explanations \nproposed in the literature  to explain ultrafast dynamics . In the last 15 years, t he influence \nof the damping on the ultrafast dynamics has been explored, both theoreticall y and \nexperimentally. The first type of prediction we want to address is the link between the \ndemagnetization time and the electronic structure via the spin polarization  P. Using a \nbasic approach considering  the Fermi golden rule, several groups [12,13]  proposed that \nthe demagnetization process is linked to the population of minority and majority spin \nstates at E F, leading to a dependence of the spin-scattering rate proportional to 1 -P [13]. \nAs this spin scattering rate is linked to the inverse of the dem agnetization delay time , the  \n𝜏𝑀~(1−𝑃)−1 law was proposed . This law is clearly verified i n our samples series. One \nshould note that this is a strong experimental demonstration since we compare samples \ngrown  in the same conditions , so with the same control of the stoichiometry  and structural \nproperties . 10 \n  However, one point is still not clear since much larger demagnetization times in \nthe picosecond timescale would be expected for large band gap and full spin -polarization. \nIn the case of small band ga p of the order of 0.1 eV, Mann et al [13] showed that  thermal \neffects from the heated electron system lead to  a decrease  of 𝜏𝑀. They calculated a \nreduction of the spin -flip suppression factor from 104 for a gap of 1 eV to 40 for a gap of \n0.3 eV. However,  the band gap of our Co 2MnSi was calculated to be around 0.8 eV with \na Fermi energy  in the middle of the gap  [27,28] . This was corroborated by direct \nmeasurement using SR -PES [19, 22 ]. Therefore, according to their model, we should \nexpect a much longer demagnetization time for Co 2MnSi. However, the largest values \nreported by several groups  [13, 29] all on HMM materials are of the same order of \nmagnitude, i.e. around 350 to 400fs . This probably means that a limitation exists due to \nanother physical reason . One hypothesis should be to consider the 1.5eV photon energy \nwhich is much larger than the spin gap. During the excitation, the electrons occupying the \ntop minority spin valence band can be directly excited into the conduction band. In a \nsimilar way, maj ority spin electrons are excited at energies higher than the spin band gap. \nBoth of these effects may allow for spin flips scattering and only the majority electrons \nexcited within the spin band gap energy range cannot flip their spins. Even if such photon  \nenergy influence is not considered based on the argument that the timescale for photon \nabsorption followed by electronic relaxation is very fast compared to the  magnetic \nrelaxation process [16 ], performing experiments by changing the excitation wavelength  \nto energies below the spin band gap would be very interesting to better understand \nultrafast magnetization dynamics.  \n Concerning the dependence between the demag netization  time and the magnetic \ndamping , different theoretical models have been proposed and two opposite trends were \nobtained; 𝛼 and 𝜏𝑀 being either directly  [15] or inversely [10 ] proportional . From the \nexperimental side, the inverse proportionality between 𝜏𝑀 and 𝛼 proposed by Koopmans \net al.  [10] could not be reproduced by doping a thin Permalloy film with rare -earth atoms \n[14]. However, the introduction of these rare -earth elements strongly modifies the \nmagnetic relaxation properties and could induce different relaxation channels for   𝜏𝑀 and \n𝛼 [30]. Zhang  et al.  performed a similar st udy using thin Co/Ni multilayers  and observed  \na direct proportionality between 𝜏𝑀 and 𝛼 [15]. However, the damping extracted in their 11 \n study should be strongly influenced by the heavy metal Pt capping and seed layers which \nmay induce  strong spin pumping effect during the magnetization precession  [30]. \nFurthermore, they did not take into account the influence of the Curie temperature.  \nTherefore, in these studies, extrinsic effects might influence the magnetization dynamics \nin a different way on both time scales  which makes more complex the comparison \nbetween theory and experiments. Therefore, o ur results offer a nice opportunity to \ndisentangle the se different  effects. According to different studies , the ultrafast \ndemagnetization slows down when approaching  the Curie temperature [ 10,16, 32,33]. In \nother words, a larger difference between the initial temperature and 𝑇𝑐 would lead to a \nfaster demag netization . In our samples,  𝑇𝑐 goes up from Co2MnAl to Co2MnSi, whereas \nthe demagnetization process becomes slower . Therefore,  we conclude that, in the  present  \ncase,  the Curie temperatures of our samples are too high to affect  𝜏𝑀 which only depends \non the intrinsic propertie s of the films,  i.e. Gilbert damping and  spin polarization.  This \nalso clarifies some points reported by  Müller et al. work [ 12]. In their paper, they first \nreported a very fast demagnetization process in Co 2MnSi(110)  and second a slow one in  \nCrO 2 and LaSrMnO 3 films with   𝑇𝑐 values close to room temperature (390 K 360 K  \nrespectively). Therefore,  it is not possible to state whether the very slow demag netization \nprocess  in these compounds is due to a low 𝑇𝑐 or a large spin polarization. Furthermore, \nrecent experimental results demonstrated a large decrease in the spin polarization at the \nFermi level in CrO 2 as function of the temperature, resulting in less than 50% at 300 K \n[34]. In our samples  we disentangle these two effects and the longest demagnetization \ntime is found for Co 2MnSi  (𝜏𝑀=380 𝑓𝑠), a true half -metal magnet with a 0.8 eV spin \ngap and a large 𝑇𝑐. \n In summary, we first demonstrate experimentally that substituting Si by Al in \nCo2MnAl xSi1-x Heusler compounds allows us to get a tunable spin polarization  at E F from \n60% in Co 2MnAl to 100% in Co 2MnSi, indicati ng the transition from metallic  to half \nmetallic behaviors.  Second, a strong correlation between the spin polarization and the \nGilbert magnetic damping  is established in these films . This confirms the theoretical \njustification of ultra -low magnetic damping in Ha lf-Metal -Magnet s as a consequence of \nthe spin gap. Third , the ultrafast spin dynamics  results  also nicely confirm that the spin \ngap is at the origin of the increase of the relaxation time. Our experiments allow us to go 12 \n further by establishing clear relati onships between the spin polarization, the magnetic \ndamping and the demagnetization time. A n inverse relationship between demagnetization \ntime and Gil bert damping is established  in these alloys , which  agrees well with the model \nproposed by Mann et al. [13] and with Koopmans et al. [10] but without considering any  \ninfluence of Curie temperature much larger than room temperature in these films . \nExperimental section  \n Co2MnAl xSi1-x(001) quaternary Heusler  compounds are grown by Molecular \nBeam Epitaxy using an MBE machine e quipped with 24 materials. The s toichiometry is \naccurately controlled during the growth by calibration of the Co, Mn, Si and Al atomic \nfluxes using a quartz microbalance located at the pl ace of the sample. The error on each \nelem ent concentration is less than 1 % [23]. The films are grown directly on MgO(001) \nsubstrates, with the epitaxial relationship [100] (001) MgO // [110] (001) Heusler \ncompound. The thickness is fixed to 20nm.  \n The phot oemission experiments were done at the CASSIOPEE beamline at \nSOLEIL synchrotron source. The films were grown in a MBE connected to the beamline \n(see [19,22,35 ] for details). The SR -PES spectra were obtained by using the largest slit \nacceptance of the detec tor (+/ - 8°) at an angle of 8° of the normal axis of the surface. Such \ngeometry allows us to analyze all the reciprocal space  on similar polycrystalline films \n[23]. \n The radiofrequency magnetic dynamics of the films were thus studied  using \nferromagnetic resonance (FMR). A Vectorial Network Analyzor FMR set -up was used in \nthe perpendicular geometry (see [ 22] for experimental details) where the static magnetic \nfield is applied out of the plane of the film in order to avoid extrinsic bro adening of the \nlinewidth due to the 2 -magnons scattering [ 36,37]. \n Ultrafast magnetization dynamics were  investigat ed using polar time -resolved \nmagneto -optical Kerr (TR -MOKE) experiments. An amplified Ti -sapphire laser \nproducing 35 fs pulses at 800 nm with  a repetition rate of 5 KHz  is used . The pump beam \nis kept at the fundamental mode and is focused down to spot size of ~260  𝜇𝑚 while the \nprobe is frequency doubled to 400 nm and focused to a spot size of ~60 𝜇𝑚. 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Rev. B, 90, 214406 (2014)  \n[36]  K. Lenz, H. Wende, W. Kuch, K. Baberscke, K. Nagy, and A. Jánossy,  \n Phys. Rev. B 73,  144424 (2006)  \n[37]  Kh. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M.Farle, U. von Hörsten, H.Wende, \nW. Keune, J. Rocker, S. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, Z. Frait,  \n Phys. Rev.  B 76, 104416 (2007)  "    },    {        "title": "2002.06858v2.Self_similar_shrinkers_of_the_one_dimensional_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "Self-similar shrinkers of the one-dimensional\nLandau–Lifshitz–Gilbert equation\nSusana Gutiérrez1and André de Laire2\nAbstract\nThe main purpose of this paper is the analytical study of self-shrinker solutions of the\none-dimensional Landau–Lifshitz–Gilbert equation (LLG), a model describing the dynamics\nfor the spin in ferromagnetic materials. We show that there is a unique smooth family of\nbackward self-similar solutions to the LLG equation, up to symmetries, and we establish\ntheir asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of\nthe self-similar profiles converge to great circles on the sphere S2, at an exponential rate.\nIn particular, the results presented in this paper provide examples of blow-up in finite\ntime, where the singularity develops due to rapid oscillations forming limit circles.\nKeywords and phrases: Landau–Lifshitz–Gilbert equation, self-similar expanders, backward\nself-similar solutions, blow up, asymptotics, ferromagnetic spin chain, heat flow for harmonic\nmaps, quasi-harmonic sphere.\n2010Mathematics Subject Classification: 82D40; 35C06; 35B44; 35C20; 53C44; 35Q55;\n58E20; 35K55.\n82D40;35C06;35B44; 35C20;53C44;35Q55;58E20;35K55\n1 Introduction\n1.1 The Landau–Lifshitz–Gilbert equation: self-similar solutions\nIn this paper we continue the investigation started in [32, 33] concerning the existence and prop-\nerties of self-similar solutions for the Landau–Lifshitz–Gilbert equation (LLG). This equation\ndescribes the dynamics for the magnetization or spin in ferromagnetic materials [43, 27] and is\ngiven by the system of nonlinear equations\n∂tm=βm×∆m−αm×(m×∆m), (LLG)\nwherem= (m 1,m2,m3) :RN×I−→S2is the spin vector, I⊂R,β≥0,α≥0,×denotes\nthe usual cross-product in R3, and S2is the unit sphere in R3. This model for ferromagnetic\nmaterials constitutes a fundamental equation in the magnetic recording industry [53]. The\nparameters β≥0andα≥0are, respectively, the so-called exchange constant and Gilbert\ndamping, and take into account the exchange of energy in the system and the effect of damping\non the spin chain. By considering a time-scaling, one can assume without loss of generality that\nthe parameters αandβsatisfy\nα∈[0,1]andβ=/radicalbig\n1−α2.\n1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom.\nE-mail: s.gutierrez@bham.ac.uk\n2Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille, France.\nE-mail: andre.de-laire@univ-lille.fr\n1arXiv:2002.06858v2  [math.AP]  21 May 2020From a purely mathematical point of view, the LLG equation is extremely interesting since\nit interpolates between two fundamental geometric evolution equations, the Schrödinger map\nequation and the heat flow for harmonic maps, via specific choices of the parameters involved.\nPrecisely, we recall that in the limit case α= 1(and, consequently, β= 0), (LLG) reduces to\nthe heat flow for harmonic maps onto S2,\n∂tm−∆m=|∇m|2m, (HFHM)\nand, ifα= 0(no damping), it reduces to the Schrödinger map equation\n∂tm=m×∆m. (SM)\nWhen 0<α< 1, (LLG) is of parabolic type. We refer the reader to [40, 29, 32, 33, 12, 14, 15, 13]\nand the references therein for more details and surveys on these equations.\nA natural question, that has proved relevant to the understanding of the global behavior\nof solutions and formation of singularities, is whether or not there exist solutions which are\ninvariant under scalings of the equation. In the case of the LLG equation it is straightforward\nto see that the equation is invariant under the following scaling: If mis a solution of (LLG),\nthenmλ(t,x) =m(λx,λ2t), for any positive number λ, is also a solution. Associated with this\ninvariance, a solution mof (LLG) defined on I=R+orI=R−is called self-similar if it is\ninvariant under rescaling, that is\nm(x,t) =m(λx,λ2t),∀λ>0,∀x∈RN,∀t∈I.\nFixingT∈Rand performing a translation in time, this definition leads to two types of self-\nsimilar solutions: A forward self-similar solution or expander is a solution of the form\nm(x,t) =f/parenleftbiggx√\nt−T/parenrightbigg\n,for (x,t)∈RN×(T,∞), (1.1)\nand a backward self-similar solution or shrinker is a solution of the form\nm(x,t) =f/parenleftbiggx√\nT−t/parenrightbigg\n,for (x,t)∈RN×(−∞,T), (1.2)\nfor some profile f:RN−→S2. In this manner, expanders evolve from a singular value at time\nT, while shrinkers evolve towards a singular value at time T.\nSelf-similar solutions have received a lot of attention in the study of nonlinear PDEs because\nthey can provide important information about the dynamics of the equations. While expanders\nare related to non-uniqueness phenomena, resolution of singularities and long time description\nof solutions, shrinkers are often related to phenomena of singularity formation (see e.g. [26, 18]).\nOn the other hand, the construction and understanding of the dynamics and properties of self-\nsimilar solutions also provide an idea of which are the natural spaces to develop a well-posedness\ntheory that captures these often very physically relevant structures. Examples of equations for\nwhichself-similarsolutionshavebeenstudiedinclude,amongothers,theNavier–Stokesequation,\nsemilinear parabolic equations, and geometric flows such as Yang–Mills, mean curvature flow\nand harmonic map flow. We refer to [37, 48, 36, 51, 5] and the references therein for more\ndetails.\nAlthough the results that will be presented in this paper relate to self-similar shrinkers of\nthe one-dimensional LLG equation (that is, to solutions m:R×I−→S2of LLG), for the sake\nof context we describe some of the most relevant results concerning maps from RN×IintoSd,\nwithN≥2andd≥2. In this setting one should point out that the majority of the works in the\n2literature concerning the study of self-similar solutions of the LLG equation are confined to the\nheat flow for harmonic maps equation, i.e. α= 1. In the case when α= 1, the main works on the\nsubject restrict the analysis to corotational maps taking values in Sd, which reduces the analysis\nof (HFHM) to the study of a second order real-valued ODE. Then tools such as the maximum\nprinciple or the shooting method can be used to show the existence of solutions. We refer to\n[19, 21, 23, 7, 8, 6, 22] and the references therein for more details on such results for maps taking\nvalues in Sd, withd≥3.Recently, Deruelle and Lamm [17] have studied the Cauchy problem\nfor the harmonic map heat flow with initial data m0:RN→Sd, withN≥3andd≥2, where\nm0is a Lipschitz 0-homogeneous function, homotopic to a constant, which implies the existence\nof expanders coming out of m0.\nWhen 0<α≤1, the existence of self-similar expanders for the LLG equation was recently\nestablished by the authors in [33]. This result is a consequence of a well-posedness theorem for\nthe LLG equation considering an initial data m0:RN→S2in the space BMO of functions of\nbounded mean oscillation. Notice that this result includes in particular the case of the harmonic\nmap heat flow.\nAs mentioned before, in the absence of damping ( α= 0), (LLG) reduces to the Schrödinger\nmap equation (SM), which is reversible in time, so that the notions of expanders and shrinkers\ncoincide. For this equation, Germain, Shatah and Zeng [24] established the existence of ( k-\nequivariant) self-similar profiles f:R2→S2.\n1.2 Goals and statements of main results\nThe results of this paper aim to advance our understanding of self-similar solutions of the one-\ndimensional LLG equation. In order to contextualize and motivate our results, we continue to\nprovide further details of what is known about self-similar solutions in this context.\nIn the 1d-case, when α= 0, (SM) is closely related to the Localized Induction Approximation\n(LIA), and self-similar profiles f:R→S2were obtained and analyzed in [34, 35, 41, 10]. In the\ncontext of LIA, self-similar solutions constitute a uniparametric family of smooth solutions that\ndevelop a singularity in the shape of a corner in finite time. For further work related to these\nsolutions, including the study of their continuation after the blow-up time and their stability,\nwe refer to the reader to [4, 3]. At the level of the Schrödinger map equation, these self-similar\nsolutions provide examples of smooth solutions that develop a jump singularity in finite time.\nIn the general case α∈[0,1], the analytical study of self-similar expanders of the one-\ndimensional (LLG) was carried out in [32]. Here, it was shown that these solutions are given by\na family of smooth profiles {fc,α}c,α, and that the corresponding expanders are associated with\na discontinuous (jump) singular initial data. We refer to [32, 33] for the precise statement of this\nresult, and the stability of these solutions, as well as the qualitative and quantitative analysis\nof their dynamics with respect to the parameters candα.\nIt is important to notice that in the presence of damping ( α > 0), since the LLG equation\nis not time-reversible, the notion of expander is different from that of shrinker. It is therefore\nnatural to ask the following question: What can be said about shrinker solutions for the one-\ndimensional LLG equation?\nAnswering this question constitutes the main purpose of this paper. Precisely, our main goals\nare to establish the classification of self-similar shrinkers of the one-dimensional LLG equation\nof the form (1.2) for some profile f:R→S2, and the analytical study of their properties. In\nparticular, we will be especially interested in studying the dynamics of these solutions as ttends\nto the time of singularity T, and understanding how the dynamical behavior of these solutions\nis affected by the presence of damping. Since, as it has been already mentioned, the case α= 0\n3has been previously considered in the literature (see [4, 31]), in what follows we will assume that\nα∈(0,1].\nIn order to state our first result, we observe that if mis a solution to (LLG) of the form (1.2)\nfor some smooth profile f, thenfsolves the following system of ODEs\nxf/prime\n2=βf×f/prime/prime−αf×(f×f/prime/prime),onR, (1.3)\nwhich recasts as\nαf/prime/prime+α|f/prime|2f+β(f×f/prime)/prime−xf/prime\n2= 0,onR, (1.4)\ndue to the fact that ftakes values in S2.\nIn the case α∈(0,1), it seems unlikely to be able to find explicit solutions to (1.4), and even\ntheir existence is not clear (see also equation (1.16)). Nevertheless, surprisingly we can establish\nthe following rigidity result concerning the possible weak solutions to (1.4) (see Section 2 for the\ndefinition of weak solution).\nTheorem 1.1. Letα∈(0,1]. Assume that fis a weak solution to (1.4). Thenfbelongs to\nC∞(R;S2)and there exists c≥0such that|f/prime(x)|=ceαx2/4, for allx∈R.\nTheorem 1.1 provides a necessary condition on the possible (weak) solutions of (1.4): namely\nthe modulus of the gradient of any solution mustbeceαx2/4, for somec≥0. We proceed now to\nestablish the existence of solutions satisfying this condition for any c>0(notice that the case\nwhenc= 0is trivial).\nTo this end, we will follow a geometric approach that was proven to be very fruitful in similar\ncontexts (see e.g. [41, 46, 42, 34, 16]), including the work of the authors in the study of expanders\n[32]. As explained in Subsection 3.1, this approach relies on identifying fas the unit tangent\nvectorm:=fof a curveXminR3parametrized by arclength. Thus, assuming that fis\na solution to (1.4) and using the Serret–Frenet system associated with the curve Xm, we can\ndeduce that the curvature and the torsion are explicitly given by\nk(x) =ceαx2/4,andτ(x) =−βx\n2, (1.5)\nrespectively, for some c≥0(see Subsection 3.1 for further details). In particular, we have\n|m/prime(x)|=k(x) =ceαx2/4, in agreement with Theorem 1.1. Conversely, given c≥0and denoting\nmc,αthe solution of the Serret–Frenet system\n\n\nm/prime(x) =k(x)n(x),\nn/prime(x) =−k(x)m(x) +τ(x)b(x),\nb/prime(x) =−τ(x)n(x),(1.6)\nwith curvature and torsion as in (1.5), and initial conditions (w.l.o.g.)\nm(0) = (1,0,0),n(0) = (0,1,0),b(0) = (0,0,1), (1.7)\nwe obtain a solution to (1.4). Moreover, we can show that the solutions constructed in this\nmanner provide, up to symmetries, all the solutions to (1.3). The precise statement is the\nfollowing.\nProposition 1.2. The set of nonconstant solutions to (1.3)is{Rmc,α:c >0,R∈SO(3)},\nwhereSO(3)is the group of rotations about the origin preserving orientation.\n4The above proposition reduces the study of self-similar shrinkers to the understanding of the\nfamilyofself-similarshrinkersassociatedwiththeprofiles {mc,α}c,α. Thenextresultsummarizes\nthe properties of these solutions.\nTheorem 1.3. Letα∈(0,1],c > 0,T∈Randmc,αbe the solution of the Serret–Frenet\nsystem(1.6)with initial conditions (1.7),\nk(x) =ceαx2/4andτ(x) =−/radicalbig\n1−α2x\n2.\nDefine\nmc,α(x,t) =mc,α/parenleftbiggx√\nT−t/parenrightbigg\n, t<T. (1.8)\nThen we have the following statements.\n(i) The function mc,αbelongs toC∞(R×(−∞,T);S2), solves(LLG)fort∈(−∞,T), and\n|∂xmc,α(x,t)|=c√\nT−teαx2\n4(T−t),\nfor all (x,t)∈R×(−∞,T).\n(ii) The components of the profile mc,α= (m1,c,α,m2,c,α,m3,c,α)satisfy that m1,c,αis even,\nwhilem2,c,αandm3,c,αare odd.\n(iii) There exist constants ρj,c,α∈[0,1], Bj,c,α∈[−1,1],andφj,c,α∈[0,2π),forj∈{1,2,3},\nsuch that we have the following asymptotics for the profile mc,α= (m1,c,α,m2,c,α,m3,c,α)\nand its derivative:\nmj,c,α(x) =ρj,c,αcos(cΦα(x)−φj,c,α)−βBj,c,α\n2cxe−αx2/4\n+β2ρj,c,α\n8csin(cΦα(x)−φj,c,α)/integraldisplay∞\nxs2e−αs2/4ds+β\nα5c2O(x2e−αx2/2),(1.9)\nand\nm/prime\nj,c,α(x) =−cρj,c,αsin(cΦα(x)−φj,c,α)eαx2/4\n+β2ρj,c,α\n8cos(cΦα(x)−φj,c,α)eαx2/4/integraldisplay∞\nxs2e−αs2/4ds+β\nα5cO(x2e−αx2/4),(1.10)\nfor allx≥1, where\nΦα(x) =/integraldisplayx\n0eαs2\n4ds.\nMoreover, the constants ρj,c,α,Bj,c,α, andφj,c,αsatisfy the following identities\nρ2\n1,c,α+ρ2\n2,c,α+ρ2\n3,c,α= 2, B2\n1,c,α+B2\n2,c,α+B2\n3,c,α= 1andρ2\nj,c,α+B2\nj,c,α= 1, j∈{1,2,3}.\n(iv) The solution mc,α= (m 1,c,α,m2,c,α,m3,c,α)satisfies the following pointwise convergences\nlim\nt→T−(mj,c,α(x,t)−ρj,c,αcos/parenleftbigcΦα/parenleftbigx√\nT−t/parenrightbig−φj,c,α/parenrightbig= 0,ifx>0,\nlim\nt→T−(mj,c,α(x,t)−ρ−\nj,c,αcos/parenleftbigcΦα/parenleftbig−x√\nT−t/parenrightbig−φj,c,α/parenrightbig= 0,ifx<0,(1.11)\nforj∈{1,2,3}, whereρ−\n1,c,α=ρ1,c,α,ρ−\n2,c,α=−ρ2,c,αandρ−\n3,c,α=−ρ3,c,α.\n5(v) For anyϕ∈W1,∞(R;R3), we have\nlim\nt→T−/integraldisplay\nRmc,α(x,t)·ϕ(x)dx= 0.\nIn particular, mc,α(·,t)→0ast→T−, as a tempered distribution.\nIt is important to remark that Theorem 1.3 provides examples of (smooth) solutions to the 1d-\nLLG equation that blow up in finite time. In order to see this, let us first recall that the existence\nof smooth solutions to (LLG) on short times can be established as in the case of the heat flow\nfor harmonic maps [45], using that (LLG) is a strongly parabolic system [30, 2]. In particular, in\nthe one-dimensional case, for any initial condition m0∈C∞(R,S2), there exists a maximal time\n0< T max≤∞such that (LLG) admits a unique, smooth solution m∈C∞(R×[0,Tmax);S2).\nMoreover, if Tmax<∞, then\nlim\nt→T−\nmax/bardbl∂xm(·,t)/bardblL∞(R)=∞.\nNext, observe that for any c>0andT∈R, the solution of the initial value problem associated\nwith (LLG) and with initial condition mc,α(·)at timeT−1is given by mc,αin Theorem 1.3,\nfort∈[T−1,T), and blows up at time T. Indeed, from (i)in Theorem 1.3 , we have that\nlim\nt→T−|∂xmc,α(x,t)|= lim\nt→T−c√\nT−teαx2\n4(T−t)=∞,\nforc>0and for allx∈R.\nNotice also that from the asymptotics in part (iii)and the symmetries of the profile estab-\nlished in part (ii), we obtain a precise description of the fast oscillating nature of the blow up of\nthe solution (1.8) given in Theorem 1.3. In this setting, we observe that part (iii)of the above\ntheorem provides the asymptotics of the profile mc,αat infinity, in terms of a fast oscillating\nprincipal part, plus some exponentially decaying terms. Notice that for the integral term in\n(1.9), we have (see e.g. [1])\n/integraldisplay∞\nxs2e−αs2/4ds=2xe−αx2/4\nα/parenleftBig\n1 +2\nαx2−4\nα2x4+···/parenrightBig\n,asx→∞,\nand that using the asymptotics for the Dawson’s integral [1], we also get\nΦα(x) =2eαx2/4\nαx/parenleftBig\n1 +2\nαx2+12\nα2x4+···/parenrightBig\n,asx→∞.\nIt is also important to mention that the big- Oin the asymptotics (1.9) does not depend on the\nparameters, i.e. there exists a universal constant C > 0, such that the big- Oin (1.9) satisfies\n|O(x2e−αx2/2)|≤Cx2e−αx2/2,for allx≥1.\nIn this manner, the constants multiplying the big- Oare meaningful and in particular, big- O\nvanishes when β= 0(i.e.α= 1).\nIn Figure 1 we have depicted the profile mc,αforα= 0.5andc= 0.5, where we can see their\noscillating behavior. Moreover, the plots in Figure 1 suggest that the limit sets of the trajectories\nare great circles on the sphere S2whenx→±∞. This is indeed the case. In our last result we\nestablish analytically that mc,αoscillates in a plane passing through the origin whose normal\nvector is given by B+\nc,α= (B1,c,α,B2,c,α,B3,c,α), andB−\nc,α= (−B1,c,α,B2,c,α,B3,c,α)asx→+∞\nandx→−∞, respectively.\n6m1m2m3\nB+\nc,α\nm1m2m3\n-1.0-0.50.00.51.0-1.0-0.50.00.51.0\nm1m2\nFigure 1: Profile mc,αforc= 0.5andα= 0.5. The figure on the left depicts profile for x∈R+\nand the normal vector B+\nc,α≈(−0.72,−0.3,0.63). The figure on the center shows the profile for\nx∈R; the angle between the circles C±\nc,αisϑc,α≈1.5951. The figure on the right represents the\nprojection of limit cycles C±\nc,αon the plane.\nTheorem 1.4. Using the constants given in Theorem 1.3, let P±\nc,αbe the planes passing through\nthe origin with normal vectors B+\nc,αandB−\nc,α= (−B1,c,α,B2,c,α,B3,c,α), respectively. Let C±\nc,αbe\nthe circles in R3given by the intersection of these planes with the sphere, i.e. C±\nc,α=P±\nc,α∩S2.\nThen the following statements hold.\n(i) For all|x|≥1, we have\ndist(mc,α(x),C±\nc,α)≤30√\n2β\ncα2|x|e−αx2/4. (1.12)\nIn particular\nlim\nt→T−dist(mc,α(x,t),C+\nc,α) = 0,ifx>0,\nlim\nt→T−dist(mc,α(x,t),C−\nc,α) = 0,ifx<0.(1.13)\n(ii) Letϑc,α= arccos(1−2B2\n1,c,α)be the angle between the circles C±\nc,α. Forc≥β√π/√α, we\nhave\nϑc,α≥arccos/parenleftBigg\n−1 +2πβ2\nc2α/parenrightBigg\n. (1.14)\nIn particular\nlimc→∞ϑc,α=π,for allα∈(0,1], and lim\nα→1ϑc,α=π,for allc>0.(1.15)\nThe above theorem above establishes the convergence of the limit sets of the trajectories\nof the profile mc,αto the great circles C±\nc,αas shown in Figure 1. Moreover, (1.12) gives us\nan exponential rate for this convergence. In terms of the solution mc,αto the LLG equation,\nTheorem 1.4 provides a more precise geometric information about the way that the solution\nblows up at time T, as seen in (1.13). The existence of limit circles for related ferromagnetic\nmodels have been investigated for instance in [52, 9] but to the best of our knowledge, this is the\nfirst time that this type of phenomenon has been observed for the LLG equation. In Figure 1\ncan see that ϑc,α≈1.5951forα= 0.5andc= 0.5, where we have chosen the value of csuch\nthat the angle is close to π/2.\nFinally, (1.14) and (1.15) in Theorem 1.4 provide some geometric information about behavior\nof the limit circles with respect to the parameters candα. In particular, formulae (1.15) states\n7that the angle between the limiting circles C+\nc,αandC−\nc,αisπasc→∞, for fixedα∈(0,1], and\nthe same happens as α→1, for fixedc>0. In other words, in these two cases the circles C±\nc,α\nare the same (but differently oriented).\n1.3 Comparison with the limit cases α= 0andα= 1\nIt is well known that the Serret–Fenet system can be written as a second-order differential\nequation. Forinstance, if (m,n,b) = (mj,nj,bj)3\nj=1isasolutionof (1.5)–(1.6), usingLemma3.1\nin [32], we have that new variable\ngj(s) =e1\n2/integraltexts\n0k(σ)ηj(σ)dσ,withηj(x) =nj(x) +ibj(x)\n1 +mj(x),\nsatisfies the equation, for j∈{1,2,3},\ng/prime/prime\nj(x)−x\n2(α+iβ)g/prime\nj(x) +c2\n4eαx2/2gj(x) = 0. (1.16)\nThen, in the case α= 1, it easy to check (see also Remark 3.3) that the profile is explicitly given\nby the plane curve\nmc,1(x) = (cos(cΦ1(x)),sin(cΦ1(x)),0). (1.17)\nIn particular, we see that the asymptotics in Theorem 1.3 are satisfied with\nρ1,c,1= 1, ρ 2,c,1= 1, ρ 3,c,1= 0, φ 1,c,1= 0, φ 2,c,1= 3π/2, φ 3,c,1∈[0,2π).\nThe caseα= 0is more involved, but using (1.16), the solution {mc,0,nc,0,bc,0}of the system\n(1.6)canstillbeexplicitlydeterminedintermsofconfluenthypergeometricfunctions. Thisleads\nto the asymptotics [34, 32, 20]\nmc,0(x) =Ac−2c\nxBccos/parenleftBigg\nx2\n4+c2ln(x) +π\n2/parenrightBigg\n+O/parenleftbigg1\nx2/parenrightbigg\n, (1.18)\nasx→∞, for some vectors Ac∈S2andBc∈R3. In particular, we see that mc,0(x)converges\nto some vector Ac, asx→∞. Hence, there is a drastic change in the behavior of the profile\nin the cases α= 0andα > 0: In the first case mc,0converges to a point at infinity, while in\nthe second case (1.12) tells us that mc,αconverges to a great circle. In this sense, there is a\ndiscontinuity in the behavior of mc,αatα= 0.\nAlso, from equation (1.16), we can formally deduce that the difference between the expanders\nand shrinkers corresponds to flipping the sign in the parameters α→−αandβ→−β. Notice\nthat the exponential coefficient in (1.16) is proportional to the square of the curvature, given by\nce−αx2/4for the skrinkers, and ceαx2/4for the expanders. We used equation (1.16) (with flipped\nsigns) to obtain the asymptotics of the expanders in [32], relying on the fact the exponential\nterm in equation vanishes as x→∞. However, the exponential grow in the case of skrinkers in\n(1.16) changes the behavior of the solution and we cannot use the methods introduced in [32].\nGoing back to Theorem 1.3, it is seems very difficult to get asymptotics for the constants\nin (1.9). Our strategy for the constants appearing in the asymptotics for the expanders in [32]\nrelied on obtaining uniform estimates and using continuity arguments. In particular, using the\nfact that the constants in (1.18) are explicit, we were able to get a good information about the\nconstants in the asymptotics when αwas close to 0. Due to the above mentioned discontinuity\n8ofmc,αatα= 0, it seems unlikely that the use of continuity arguments will provide information\nfor the constants in the asymptotics for the shrinkers.\nFinally, let us also remark that we cannot use continuation arguments to find the behavior\nof the circles for csmall. This is expected since m0,α(x) = (1,0,0)for allx∈R, whenc= 0\n(see (4.6)). In Section 4 we give some numerical simulations for csmall.\nStructure of the paper. The outline of this paper is the following. In Section 2, we study\n(1.4) as an elliptic quasilinear system and prove the rigidity result Theorem 1.1. By using the\nSerret–Frenet system, we prove there existence and uniqueness of solution, up to a rotation, in\nSection 3. We also use this system to obtain the asymptotics of the self-similar profiles. Finally,\nSection 4 is devoted to the proof of Theorem 1.4.\n2 Rigidity result. Theorem 1.1\nThe purpose of this section is to prove the rigidity result stated in Theorem 1.1 concerning\n(weak) solutions of the system\nxf/prime\n2=βf×f/prime/prime−αf×(f×f/prime/prime),onR. (2.1)\nWe start by introducing the notion of weak solution of the above system. To this end, we first\nobserve that the system (2.1) recasts as\nαf/prime/prime+α|f/prime|2f+β(f×f/prime)/prime−xf/prime\n2= 0, (2.2)\nusing the following vector identities for a (smooth) function fwith|f|= 1:\nf×f/prime/prime= (f×f/prime)/prime,\n−f×(f×f/prime/prime) =f/prime/prime+|f/prime|2f.(2.3)\nWe prefer to use the formulation (2.2) since it is simpler to handle in weak sense. Indeed, we\nsay thatf= (f1,f2,f3)∈H1\nloc(R,S2)is aweak solution to the system (2.2) if\n/integraldisplay\nR/parenleftBig\n−αf/prime·ϕ/prime+α|f/prime|2f·ϕ−β(f×f/prime)·ϕ/prime−x\n2f/primeϕ/parenrightBig\ndx= 0, (2.4)\nfor allϕ= (ϕ1,ϕ3,ϕ3)∈C∞\n0(R).\nUsing (2.3), we can recast (2.2) as,\nαf/prime/prime\n1+α|f/prime|2f1+β(f2f/prime/prime\n3−f3f/prime/prime\n2)−x\n2f/prime\n1= 0, (2.5a)\nαf/prime/prime\n2+α|f/prime|2f2+β(f3f/prime/prime\n1−f1f/prime/prime\n3)−x\n2f/prime\n2= 0, (2.5b)\nαf/prime/prime\n3+α|f/prime|2f3+β(f1f/prime/prime\n2−f2f/prime/prime\n1)−x\n2f/prime\n3= 0. (2.5c)\nThus we see that the weak formulation (2.4) can be written as\n/integraldisplay\nRA(f(x))f/prime(x)·ϕ/prime(x) =/integraldisplay\nRG(x,f,f/prime)ϕ(x),for allϕ∈C∞\n0(R), (2.6)\nwith\nA(u) =\nα−βu3βu2\nβu3α−βu1\n−βu2βu1α,\nandG(x,u,p) =\nαu1|p|2−xp1\n2\nαu2|p|2−xp2\n2\nαu3|p|2−xp3\n2\n,\n9whereu= (u1,u2,u3)andp= (p1,p2,p3). We want now to invoke the regularity theory for\nquasilinear elliptic system (see [39, 25]). To verify that the system is indeed uniformly elliptic,\nwe can easily check that\nA(u)ξ·ξ=α|ξ|2,for allξ,u∈R3.\nIn addition, Ghas quadratic growth on bounded domains, i.e.\n|G(x,u,p)|≤√\n3(M|p|2+R|p|),\nfor all|u|≤Mand|x|≤R. Since a weak solution fto (2.6) belongs by definition to H1\nloc(R;S2),\nwe have by the Sobolev embedding theorem that fis Hölder continuous with |f(x)|= 1.\nTherefore we can apply the results in Theorem 1.2 in [25] (see also Lemma 8.6.3 in [38] or\nTheorem 2.4.3 in [49] for detailed proofs), to conclude that f∈H2\nloc(R)∩W1,4\nloc(R), and so\nthatf∈C1,γ\nloc(R), for someγ∈(0,1). We get that G(x,f(x),f/prime(x))belongs toC0,γ\nloc(R), which\nallows us to invoke the Schauder regularity theory (see e.g. Theorem A.2.3 in [38]) to infer that\nf∈C2,γ\nloc(R). This implies that G(x,f(x),f/prime(x))belongs toC1,γ\nloc(R), as well as the coefficients of\nA(u), so the Schauder estimates yield that f∈C3,γ\nloc(R). By induction, we this argument shows\nthatf∈C∞(R).\nWe are now in position to complete the proof of Theorem 1.1. Indeed, let first remark that\ndifferentiating the relation |f|2= 1, we have the identities\nf·f/prime= 0, (2.7)\nf·f/prime/prime=−|f/prime|2. (2.8)\nBy taking the cross product of fand (2.2), and using (2.3), we have\nβf/prime/prime+β|f/prime|2f−α(f×f/prime)/prime+x\n2f×f/prime= 0. (2.9)\nThus, by multiplying (2.2) by α, (2.9) byβ, and recalling that α2+β2= 1, we get\nf/prime/prime+|f/prime|2f−x\n2(αf/prime−βf×f/prime) = 0.\nTaking the scalar product of this equation and f/prime, the identity (2.7) allow us to conclude that\n1\n2(|f/prime|2)/prime−αx\n2|f/prime|2= 0. (2.10)\nIntegrating, we deduce that there is a constant C≥0such that|f/prime|2=Ceαx2/2. This completes\nthe proof of Theorem 1.1.\nWe conclude this section with some remarks.\nRemark 2.1. A similar result to the one stated in Theorem 1.1 also holds for the expanders\nsolutions. Precisely, any weak solution to (2.1), withxf/prime/2replaced by−xf/prime/2in the l.h.s., is\nsmooth and there exists c≥0such that|f/prime(x)|=ce−αx2/4, for allx∈R.\nRemark 2.2. Let us mention that in the case α= 1, a nonconstant solution u:RN→Sdto\nequation\n∆u+|∇u|2u−x·∇u\n2= 0,onRN, (2.11)\n10is usually called quasi-harmonic sphere , since it corresponds to the Euler–Lagrange equations of\na critical point of the (so-called) quasi-energy [44]\nEquasi(u) =/integraldisplay\nRN|∇u(y)|2e−|y2|/4dy.\nIt has been proved in [19] the existence of a (real-valued) function hsuch that\nu(x) =/parenleftBigx\n|x|sin(h(|x|)),cos(h(|x|)/parenrightBig\nis a solution to (2.11)with finite quasi-energy for 3≤N=d≤6. In addition, there is no\nsolution of this form if d≥7[8]. Both results are based on the analysis of the second-order\nODE associated with h. We refer also to [21] for a generalization of the existence result for\nN≥3of other equivariant solutions to (2.11). In the case N= 1andd= 2, the solution to\n(2.11)is explicitly given by (1.17), and its associated quasi-energy is infinity, as remarked in\n[54].\n3 Existence, uniqueness and properties\n3.1 Existence and uniqueness of the self-similar profile. Proposition 1.2\nIn the previous section we have shown that any solution to the profile equation\nαm/prime/prime+α|m/prime|2m+β(m×m/prime)/prime−xm/prime\n2= 0, (3.1)\nis smooth and that there is c≥0such that\n|m/prime(x)|=ceαx2/4,for allx∈R. (3.2)\nWe want to give now the details about how to construct such a solution by using the Serret–\nFrenet frame, which will correspond to the profile mc,αin Theorem 1.3. The idea is to identify\nmas the tangent vector to a curve in R3, so we first recall some facts about curves in the space.\nGivenm:R→S2a smooth function, we can define the curve\nXm(x) =/integraldisplayx\n0m(s)ds, (3.3)\nso thatXmis smooth, parametrized by arclenght, and its tangent vector is m. In addition,\nif|m/prime|does not vanish on R, we can define the normal vector n(x) =m/prime(x)/|m/prime(x)|and the\nbinormal vector b(x) =m(x)×n(x). Moreover, we can define the curvature and torsion of Xm\nask(x) =|m/prime(x)|andτ(x) =−b/prime(x)·n(x). Since|m(x)|2= 1,for allx∈R, we have that\nm(x)·n(x) = 0, for allx∈R, that the vectors {m,n,b}are orthonormal and it is standard to\ncheck that they satisfy the Serret–Frenet system\n\n\nm/prime=kn,\nn/prime=−km+τb,\nb/prime=−τn.(3.4)\nLet us apply this construction to find a solution to (3.1). We define curve Xmas in (3.3), and\nremark that equation (3.1) rewrites in terms of {m,n,b}as\nx\n2kn=β(k/primeb−τkn)−α(−k/primen−kτb).\n11Therefore, from the orthogonality of the vectors nandb, we conclude that the curvature and\ntorsion ofXmare solutions of the equations\nx\n2k=αk/prime−βτkandβk/prime+αkτ= 0,\nthat is\nk(x) =ceαx2\n4andτ(x) =−βx\n2, (3.5)\nfor somec≥0. Of course, the fact that k(x) =ceαx2/4is in agreement with the fact that we\nmust have|m/prime(x)|=ceαx2/4.\nNow, given α∈[0,1]andc>0, consider the Serret–Frenet system (3.4) with curvature and\ntorsion function given by (3.5) and initial conditions\nm(0) = (1,0,0),n(0) = (0,1,0),b(0) = (0,0,1). (3.6)\nThen, by standard ODE theory, there exists a unique global solution {mc,α,nc,α,bc,α}in\n(C∞(R;S2))3, and these vectors are orthonormal. Also, it is straightforward to verify that\nmc,αis a solution to (3.1) satisfying (3.2).\nThe above argument provides the existence of solutions in the statement of Proposition 1.2.\nWe will now complete the proof of Proposition 1.2 showing the uniqueness of such solutions, up\nto rotations.\nTo this end, assume that ˜mis a weak nontrivial solution to (3.1). By Theorem 1.1, ˜mis\ninC∞(R,S2)and there exists c >0such that|˜m/prime(x)|=ceαx2/4, for allx∈R. Following the\nabove argument, the curve X˜m(defined in (3.3)), has curvature ceαx2/4and torsion−βx/2.\nSince the curve Xmc,αassociated with mc,α, andX˜mhave the same curvature and torsion,\nusing fundamental theorem of the local theory of space curves (see e.g. Theorem 1.3.5 in [47]),\nwe conclude that both curves are equal up to direct rigid motion, i.e. there exist p∈R3and\nR∈SO(3)such thatX˜m(x) =R(Xmc,α(x))+p, for allx∈R3. By differentiating this identity,\nwe finally get that ˜m=Rmc,α, which proves the uniqueness of solution, up to a rotation, as\nstated in Proposition 1.2.\n3.2 Asymptotics of the self-similar profile\nThe rest of this section is devoted to establish properties of the family of solutions {mc,α}c,α,\nfor fixedα∈(0,1]andc >0. Due to the self-similar nature of these solutions, this analysis\nreduces to study the properties of the associated profile mc,α, or equivalently, of the solution\n{mc,α,nc,α,bc,α}of the Serret–Frenet system (3.4) with curvature and torsion given in (3.5),\nand initial conditions (3.6).\nIt is important to mention that the recovery of the properties of the trihedron {m,n,b},\nand in particular of the profile m, from the knowledge of its curvature and torsion is a difficult\nquestion. This can be seen from the equivalent formulations of the Serret–Frenet equation in\nterms of a second-order complex-valued highly non-linear EDO, or in terms of a complex-valued\nRiccati equation (see e.g. [11, 50, 42, 32]). For this reason, the integration of the trihedron can\noften only be done numerically, rather than analytically.\nSince the Serret–Frenet equations are decoupled, we start by analyzing the system for the\n12scalarfunctionsmc,α,nc,αandbc,α\n\n\nm/prime\nc,α(x) =ceαx2\n4nc,α(x),\nn/prime\nc,α(x) =−ceαx2\n4mc,α(x)−βx\n2bc,α(x),\nb/prime\nc,α(x) =βx\n2nc,α(x),(3.7)\nwithinitialconditions (mc,α,bc,α,nc,α)(0),thatwesupposeindependentof candα,andsatisfying\nmc,α(0)2+bc,α(0)2+nc,α(0)2= 1.\nThen by ODE theory, the solution is smooth, global and satisfies\nmc,α(x)2+bc,α(x)2+nc,α(x)2= 1,for allx∈R. (3.8)\nMoreover, the solution depends continuously on the parameters c>0andα∈(0,1].\nTo study the behavior of the solution of the system (3.7), we need some elementary bounds\nfor the non-normalized complementary error function.\nLemma 3.1. Letγ∈(0,1]. The following upper bounds hold for x>0\n/integraldisplay∞\nxe−γs2ds≤1\n2γxe−γx2and/integraldisplay∞\nxse−γs2ds=1\n2γe−γx2. (3.9)\nAlso, forγ∈(0,1]andx≥1,\n/integraldisplay∞\nxs2e−γs2ds≤x\nγ2e−γx2,and/integraldisplay∞\nxs3e−γs2ds≤x2\nγ2e−γx2. (3.10)\nProof.We start recalling some standard bounds the complementary error function (see e.g. [1,\n28])\nxe−x2\n2x2+ 1≤/integraldisplay∞\nxe−s2ds≤e−x2\n2x,forx>0. (3.11)\nThe first formula in (3.9) follows by scaling this inequality. The second formula in (3.9) follows\nby integration by parts.\nTo prove the first estimate in (3.10), we use integration by parts and (3.9) to show that\n/integraldisplay∞\nxs2e−γs2ds=xe−γx2\n2γ+1\n2γ/integraldisplay∞\nxe−γs2ds≤e−γx2/parenleftbiggx\n2γ+1\n4γ2x/parenrightbigg\n≤xe−γx2/parenleftbigg1\n2γ+1\n4γ2/parenrightbigg\n,∀x≥1.\nSinceγ∈(0,1], we haveγ2≤γand thus we conclude the estimate for the desired integral. The\nsecond inequality in (3.10) easily follows from the identity\n/integraldisplay∞\nxs3e−γs2ds=1 +γx2\n2γ2e−γx2,∀x∈R,\nnoticing that 1 +γx2≤x2(1 +γ)≤2x2, sincex≥1andγ∈(0,1].\nNow we can state a first result on the behavior of {mc,α,nc,α,bc,α}.\n13Proposition 3.2. Letα∈(0,1]andc>0, and define\nΦα(x) =/integraldisplayx\n0eαs2\n4ds.\nThen the following statements hold.\ni) For allx∈R,\nbc,α(x) =Bc,α+βx\n2ce−αx2/4mc,α(x) +β\n2c/integraldisplay∞\nx/parenleftBigg\n1−αs2\n2/parenrightBigg\ne−αs2/4mc,α(s)ds,(3.12)\nwhere\nBc,α=bc,α(0)−β\n2c/integraldisplay∞\n0/parenleftBigg\n1−αs2\n2/parenrightBigg\ne−αs2/4mc,α(s)ds. (3.13)\nIn particular, for all x≥1\n|bc,α(x)−Bc,α|≤6β\ncαxe−αx2/4. (3.14)\nii) Setting wc,α=mc,α+inc,α, for allx∈R, we have\nwc,α(x) =e−icΦα(x)/parenleftBig\nWc,α−βx\n2ceicΦα(x)−αx2/4bc,α(x)\n−β\n2c/integraldisplay∞\nxeicΦα(s)−αs2/4/parenleftbigβs2\n2nc,α(s) +/parenleftbig1−αs2\n2/parenrightbigbc,α(s)/parenrightbigds/parenrightBig\n,(3.15)\nwhere\nWc,α=wc,α(0) +β\n2c/integraldisplay∞\n0eicΦα(s)−αs2/4/parenleftbigβs2\n2nc,α(s) +/parenleftbig1−αs2\n2/parenrightbigbc,α(s)/parenrightbigds.(3.16)\nIn particular, for all x≥1,\n|wc,α(x)−e−icΦα(x)Wc,α|≤10β\ncα2xe−αx2/4. (3.17)\nFurthermore, the limiting values Bc,αandWc,αare separately continuous functions of (c,α)for\n(c,α)∈(0,∞)×(0,1].\nProof.For simplicity, we will drop the subscripts candαif there is no possible confusion. From\n(3.7), we get\nb(x)−b(0) =/integraldisplayx\n0b/prime(s)ds=β\n2c/integraldisplayx\n0se−αs2\n4m/prime(s)ds\n=β\n2c/parenleftBig\nxe−αx2\n4m(x)−/integraldisplayx\n0/parenleftbig1−αs2\n2/parenrightbige−αs2\n4m(s)ds/parenrightBig\n,(3.18)\nwhere we have used integration by parts. Notice that/integraltext∞\n0(1−αs2/2)e−αs2/4m(s)dsis well-\ndefined, since α∈(0,1]andmis bounded. Therefore, the existence of B:= limx→∞b(x)follows\nfrom (3.18). Moreover,\nB:=b(0)−β\n2c/integraldisplay∞\n0/parenleftBigg\n1−αs2\n2/parenrightBigg\ne−αs2/4m(s)ds.\nFormula (3.12) easily follows from integrating b/primefromx∈Rto∞and arguing as above.\n14To prove (3.14), it is enough to observe that by Lemma 3.1, for x≥1and0<α≤1,\n/integraldisplay∞\nxe−αs2/4ds≤2\nαxe−αx2\n4≤2\nαxe−αx2\n4,and/integraldisplay∞\nxs2e−αs2/4ds≤16\nα2xe−αx2\n4.(3.19)\nSettingw=m+inand using (3.7), we obtain that wsatisfies the ODE\nw/prime+iceαx2/4w=−iβx\n2b(x), (3.20)\nor, equivalently,/parenleftBig\neicΦα(x)w/parenrightBig/prime\n=−iβx\n2b(x)eicΦα(x). (3.21)\nIntegrating (3.21) from 0tox>0, and writing\neicΦα(x)=−i\nc/parenleftBig\neicΦα(x)/parenrightBig/prime\ne−αx2/4,\nintegrating by parts, and using once again (3.7), we get\neicΦα(x)w(x) =w(0)−β\n2cxb(x)eicΦα(x)−αx2/4\n+β\n2c/integraldisplayx\n0eicΦα(s)−αs2/4/parenleftBigβ\n2s2n(s) + (1−αs2\n2)b(s)/parenrightBig\nds.\nSinceα∈(0,1], from the above identity it follows the existence of\nW:= limx→∞eicΦα(x)w(x),\nand formula (3.16) for W.\nFormula (3.15) now follows from integrating (3.21) from x >0to∞and arguing as in the\nprevious lines. The estimate in (3.17) can be deduced as before, since the bounds in (3.19) imply\nthat\n|wc,α(x)−e−icΦα(x)Wc,α|≤β\n2cxe−αx2/4/parenleftbigg\n1 +16(α+β)\n2α2+2\nα/parenrightbigg\n≤10β\ncα2xe−αx2/4,\nwhere we used that α+β≤2andα≤1.\nTo see that the limiting values Bc,αandWc,αgiven by (3.13) and (3.16) are continuous\nfunctions of (c,α), for (c,α)∈(0,∞)×(0,1], we recall that by standard ODE theory, the func-\ntionsmc,α(x),nc,α(x)andbc,α(x)are continuous functions of x,candα. Then, the dominated\nconvergence theorem applied to the formulae (3.13) and (3.16) yield the desired continuity.\nRemark 3.3. As mentioned before, the shrinkers of the 1d-harmonic heat flow can be computed\nexplicitly, because if α= 1, the system (1.5)-(1.6)-(1.7)can be solved easily. Indeed, in this case\nβ= 0, so that we obtain\nmc,1(x) = (cos(cΦ1(x)),sin(cΦ1(x)),0),\nnc,1(x) = (−sin(cΦ1(x)),cos(cΦ1(x)),0),\nbc,1(x) = (0,0,1),\nfor allx∈R.\nIn order to obtain a better understanding of the asymptotic behavior of {mc,α,nc,α,bc,α},\nwe need to exploit the oscillatory character of the function eicΦα(s)in the integrals (3.12) and\n(3.15). In our arguments we will use the following two lemmas.\n15Lemma 3.4. Let0<α≤1. Forσ∈R\\{0}andx∈R, the limit\n/integraldisplay∞\nxseiσΦα(s)ds:= limy→∞/integraldisplayy\nxseiσΦα(s)ds\nexists. Moreover, for all x≥1,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nxseiσΦα(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤11x\n|σ|αe−αx2/4, (3.22)\nand/integraldisplay∞\nxseiσΦα(s)ds=ix\nσeiσΦα(x)−αx2/4+O/parenleftBigg\nx2\nσ2e−αx2/2/parenrightBigg\n. (3.23)\nProof.Letx∈Rand takey≥x. Then, integrating by parts,\n/integraldisplayy\nxseiσΦα(s)ds=1\niσ/integraldisplayy\nxs(eiσΦα(s))/primee−αs2/4ds\n=s\niσeiσΦα(s)−αs2/4/vextendsingle/vextendsingle/vextendsingle/vextendsingley\nx−1\niσ/integraldisplayy\nxeiσΦα(s)−αs2/4/parenleftbig1−αs2\n2/parenrightbigds. (3.24)\nThe existence of the improper integral/integraltext∞\nxseiσΦα(s)dsfollows taking the limit as ygoes to∞\nin the above formula, and bearing in mind that α>0. The estimate (3.22) follows from (3.19)\nand the fact that x≥1and0<α≤1. Finally, integrating by parts once more, we have\niσ/integraldisplay∞\nxeiσΦα(s)−αs2/4/parenleftbig1−αs2\n2/parenrightbigds=−eiσΦα(x)−αx2/2/parenleftbig1−αx2\n2/parenrightbig−/integraldisplay∞\nxeiσΦα(s)−αs2/2/parenleftbigα2s3\n2−2αs/parenrightbigds.\nHence, using Lemma 3.1 and (3.24), we obtain (3.23).\nLemma 3.5. Letσ∈R\\{0},γ∈R,α>0and set ˜γ=γ+α/4. If0<˜γ≤1, then forx≥1,\n/integraldisplay∞\nxeiσΦα(s)−γs2ds=O/parenleftBigg\ne−˜γx2\n|σ|/parenrightBigg\n,/integraldisplay∞\nxseiσΦα(s)−γs2ds=O/parenleftBigg\nxe−˜γx2\n|σ|˜γ)/parenrightBigg\n,\n/integraldisplay∞\nxs2eiσΦα(s)−γs2ds=O/parenleftBigg\nx2e−˜γx2\n|σ|˜γ/parenrightBigg\n.(3.25)\nProof.Forn∈{0,1,2}, we set\nIn=/integraldisplay∞\nxsneiσΦα(s)−γs2ds.\nIn=1\niσ/parenleftbigg\n−xneiσΦα(x)−˜γx2−/integraldisplay∞\nxeiσΦα(s)−˜γs2/parenleftBig\nnsn−1−2˜γsn+1/parenrightBig\nds/parenrightbigg\n.\nThen the desired asymptotics follow from Lemma 3.1.\nUsing previous lemmas, we can now improve the asymptotics in Proposition 3.2 and obtain\nexplicitly the term decaying as e−αx2/4(multiplied by a polynomial).\nCorollary 3.6. With the same notation as in Proposition 3.2, the following asymptotics hold\nforx≥1\nbc,α(x) =Bc,α+βx\n2ce−αx2/4Re(e−icΦα(x)Wc,α) +β\nc2α3O(x2e−αx2/2), (3.26)\nwc,α(x) =e−icΦα(x)/parenleftBig\nWc,α−βBc,α\n2cxeicΦα(x)−αx2/4+iβ2Wc,α\n8c/integraldisplay∞\nxs2e−αs2/4ds/parenrightBig\n(3.27)\n+β\nc2α5O(x2e−αx2/2).\n16Proof.As usual, we drop the subscripts candαin the rest of the proof. Recalling that w=\nm+in, we have from (3.17),\nm= Re(e−icΦα(x)W) +β\ncα2O(xe−αx2/4).\nThus, replacing in (3.12),\nb(x) =B+βx\n2ce−αx2/4Re(e−icΦα(x)W) +β2\nc2α2O(x2e−αx2/2) +Rb(x),(3.28)\nwith\nRb(x) =β\n2cRe/parenleftBigg\nW/integraldisplay∞\nx/parenleftbig1−αs2\n2/parenrightbige−icΦα(s)−αs2/4ds+/integraldisplay∞\nx/parenleftbig1−αs2\n2/parenrightbigO/parenleftbigse−αs2/2\ncα2/parenrightbigds/parenrightBigg\n.\nByusingLemmas 3.1and3.5toestimatethefirstandsecondintegrals, respectively, weconclude\nthat\nRb(x) =β\nc2α3O/parenleftbigx2e−αx2/2/parenrightbig. (3.29)\nBy putting together (3.28) and (3.29), we obtain (3.26). To establish (3.27) we integrate (3.21)\nfromx≥1and∞, and use (3.26) and Lemma 3.1 to get\neicΦw(x)−W=I1(x) +I2(x) +I3(x) +β2\nc2α5O(xe−αx2/2), (3.30)\nwith\nI1(x) =iβB\n2/integraldisplay∞\nxseicΦα(s)ds, I 2(x) =iβ2W\n8c/integraldisplay∞\nxs2e−αs2/4ds,and\nI3(x) =iβ2¯W\n8c/integraldisplay∞\nxs2e2icΦα(s)−αs2/4ds,\nwhere we have used that Re(z) = (z+ ¯z)/2. The conclusion follows invoking again Lemmas 3.1,\n3.4 and 3.5.\nIn Figure 2 we depict the first components of the trihedron {mc,α,nc,α,bc,α}forc= 0.5and\nα= 0.5, andx>0. As described in Corollary 3.6 (recall that wc,α=mc,α+inc,α), in the plots\nin Figure 2 one can observe that, while both m1,c,αandb1,c,αoscillate highly for large values of\nx>0, the component b1,c,αconverges to a limit B1,c,α≈−0.72asx→+∞.\n246810\n-1.0-0.50.51.0\n(i)m1,c,α\n (ii)n1,c,α\n246810\n-1.0-0.8-0.6-0.4-0.2 (iii)b1,c,α\nFigure 2: Functions m1,c,α,n1,c,αandb1,c,αforc= 0.5andα= 0.5onR+. The limit at infinity\nin (iii) isB1,c,α≈−0.72.\n173.3 Proof of Theorem 1.3\nFor simplicity, we will drop the subscripts candαin the proof of Theorem 1.3.\nProof of Theorem 1.3. Let{m,n,b}be the solution of the Serret–Frenet system (3.4)–(3.5)\nwith initial condition (3.6). By ODE theory, we have that the solution {m,n,b}is smooth, is\nglobal, and satisfies\n|m(x)|=|n(x)|=|b(x)|= 1,for allx∈R, (3.31)\nand the orthogonality relations\nm(x)·n(x) =m(x)·b(x) =n(x)·b(x) = 0,for allx∈R. (3.32)\nDefine\nm(x,t) =m/parenleftbiggx√\nT−t/parenrightbigg\n, t<T.\nThen, it is straightforward to check that mis a smooth solution of the profile equation (3.1),\nand consequently msolves (LLG) for t∈(−∞,T). Moreover, using the Serret–Frenet system\n(3.4)–(3.5), we have\n|∂xm(x,t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1√\nT−tm/prime/parenleftbiggx√\nT−t/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=c√\nT−teαx2\n4(T−t).\nThis shows part (i).\nNotice also that, since {mj,nj,bj}forj∈{1,2,3}solves (3.7) with initial condition (1,0,0),\n(0,1,0)and(0,0,1)respectively, the relation (3.8) is satisfied, i.e.\nm2\nj(x) +n2\nj(x) +b2\nj(x) = 1,for allx∈R, j = 1,2,3. (3.33)\nPart(ii)follows from the uniqueness given by the Cauchy–Lipschitz theorem for the solution\nof (3.1)–(3.5) with initial condition (3.6), and the invariance of (3.1)–(3.5)–(3.6) under the\ntransformations\nm(x) = (m1(x),m2(x),m3(x))→(m1(−x),−m2(−x),−m3(−x)),\nn(x) = (n1(x),n2(x),n3(x))→(−n1(−x),n2(−x),n3(−x)),\nb(x) = (b1(x),b2(x),b3(x))→(−b1(−x),b2(−x),b3(−x)),\nSettingWj=ρjeiφj, withρj≥0andφj∈[0,2π], the asymptotics for mjand it derivative\nm/prime\nj, forj∈{1,2,3}in part(iii)are a direct consequence of the asymptotic behavior of the\nprofile established in Proposition 3.2 and Corollary 3.6 (recall also that m/prime(x) =ceαx2\n4n(x)and\nw=m+in). In particular the following limits exist:\nWj:= limx→∞eicΦα(x)(mj+inj)(x), Bj:= limx→∞bj(x).\nNotice also that the relations in (3.33) implies that ρ2\nj+B2\nj= 1, and the fact that bis unitary\nimplies that B2\n1+B2\n2+B2\n3= 1, and thatρ2\n1+ρ2\n2+ρ2\n3= 2.\nThe convergence in part (iv)is an immediate consequence of the definition of min terms of\nthe profilem, and the asymptotics established in part (iii). The relations between ρ−\njandρj,\nforj∈{1,2,3}follow from the parity relations for the profile mestablished in part (ii).\nIt remains to prove part (v). Forϕ∈W1,∞(R), bearing in mind (3.15) and (3.17), it suffices\nto show that\nlim\nt→0+/integraldisplay\nRe−icΦα(x/√\nt)ϕ(x)dx= 0,and lim\nt→0+/integraldisplay\nR/parenleftbig1 +|x|√\nt/parenrightbige−αx2\n4t|ϕ(x)|dx= 0.(3.34)\n18The second limit is a direct consequence of following the explicit computations:\n/integraldisplay\nRe−αx2\n4tdx=/radicalbiggπt\nα,and/integraldisplay\nR|x|e−αx2\n4tdx=4t\nα. (3.35)\nFor the first limit in (3.34), we integrate by parts to obtain\nic/integraldisplay\nRe−icΦα(x/√\nt)ϕ(x)dx=−√\nt/integraldisplay\nR/parenleftbige−icΦα(x/√\nt))/primee−αx2\n4tϕ(x)dx\n=/integraldisplay\nRe−icΦα(x/√\nt)−αx2\n4t/parenleftbigg√\ntϕ/prime(x)−αx\n2√\ntϕ(x)/parenrightbigg\ndx.\nSince|ϕ|and|ϕ/prime|are bounded on R, the conclusion follows again by using (3.35).\n4 Limiting behaviour of the trajectories of the profiles\nIn this section we prove Theorem 1.4. In what follows we denote C±\nc,αthe great circleCc,α=\nP±\nc,α∩S2, withP±\nc,αbeing the planes passing through the origin with (unitary) normal vectors\nB+\nc,α= (B1,c,α,B2,c,α,B3,c,α)andB−\nc,α= (−B1,c,α,B2,c,α,B3,c,α),\ngiven by Theorem 1.3.\nWe will show that the trajectories of the profiles mc,αconverge to the great circles C±\nc,αas\nx→±∞, and study the behavior of the limit circles C±\nc,αwith respect to the parameters cand\nα, analyzing the angle ϑc,α∈[0,π]between their normal vectors B+\nc,αandB−\nc,α.\nWe start this section stating a corollary of Theorem 1.3 that will be used in what follows.\nPrecisely, recalling that w=m+in, from (3.17) and using the constants defined in Theorem 1.3,\nwe have the following\nCorollary 4.1. Forx≥1andj∈{1,2,3}, we have\nmj,c,α(x) =ρj,c,αcos(cΦα(x)−φj) +Rj(x), nj,c,α(x) =−ρj,c,αsin(cΦα(x)−φj) +˜Rj(x),\nfor some functions Rjand ˜Rjsatisfying the bounds |Rj(x)|,|˜Rj(x)|≤10β/(cα2)xe−αx2/4.\nWe will also use the two lemmas below in the proof of Theorem 1.4. The first one establishes\nsome relations between the constants appearing in the asymptotics of the profile mc,αthat can\nbe deduced by using some geometric properties of the Serret–Frenet system.\nLemma 4.2. The constants given by Theorem 1.3 satisfy the following identities\nB1,c,α=ρ2,c,αρ3,c,αsin(φ3,c,α−φ2,c,α), B2,c,α=ρ1,c,αρ3,c,αsin(φ1,c,α−φ3,c,α),\nB3,c,α=ρ1,c,αρ2,c,αsin(φ2,c,α−φ1,c,α),(4.1)\nand\nB1,c,αρ1,c,αeiφ1,c,α+B2,c,αρ2,c,αeiφ2,c,α+B3,c,∞ρ3,c,αeiφ3,c,α= 0, (4.2)\nρ2\n1,c,αe2iφ1,c,α+ρ2\n2,c,αe2iφ2,c,α+ρ2\n3,c,αe2iφ3,c,α= 0. (4.3)\n19Proof.Dropping the subscripts candα, using the relation b=m×nand the asymptotics in\nCorollary 4.1, we get for x≥1,\nb1(x) =−ρ2ρ3/parenleftbigcos(cΦα(x)−φ2) sin(cΦα(x)−φ3)−cos(cΦα(x)−φ3) sin(cΦα(x)−φ2)/parenrightbig+o(1),\nb2(x) =−ρ3ρ1/parenleftbigcos(cΦα(x)−φ3) sin(cΦα(x)−φ1)−cos(cΦα(x)−φ1) sin(cΦα(x)−φ3)/parenrightbig+o(1),\nb3(x) =−ρ1ρ2/parenleftbigcos(cΦα(x)−φ1) sin(cΦα(x)−φ2)−cos(cΦα(x)−φ2) sin(cΦα(x)−φ1),/parenrightbig+o(1),\nwhereo(1)is a function of x, which depends on the parameters αandc, that converges to 0as\nx→∞. Noticing that, for k,j∈{1,2,3},\n−cos(cΦα(x)−φj) sin(cΦα(x)−φk) + cos(cΦα(x)−φk) sin(cΦα(x)−φj) = sin(φk−φj),\nwe obtain\nb1(x) =ρ2ρ3sin(φ3−φ2) +o(1),\nb2(x) =ρ1ρ3sin(φ1−φ3) +o(1),\nb3(x) =ρ1ρ2sin(φ2−φ1) +o(1).\nLettingx→∞, in the above identities we obtain (4.1).\nTo establish the other identities, we first recall that the vectors m,nandbsatisfy the\nfollowing relations\nm·n=n·b=b·m= 0and|m|=|n|=|b|= 1.\nNow, from the asymptotics in Corollary 4.1 and the identity\n(m+in)·b= 0,\nwe have\nb1(x)ρ1e−i(cΦα(x)−φ1)+b2(x)ρ2e−i(cΦα(x)−φ2)+b3(x)ρ3e−i(cΦα(x)−φ3)=o(1),\nfori∈{1,2,3}. Dividing by e−icΦα(x)and letting x→∞, we obtain formula (4.2). Finally,\nformula (4.3) follows easily using a similar argument, bearing in mind the orthogonality relation\nm·n= 0,and that 2 cos(y) sin(y) = Im(e2iy), fory∈R.\nRemark 4.3. Although we do not use (4.3)in this work, this relation could be helpful in estab-\nlishing further properties of the solutions.\nNext we study the angle between the normal vectors to the great circles C±\nc,α, given by\nϑc,α= arccos(2B2\n1,c,α−1),withϑc,α∈[0,π]. We have the following.\nLemma 4.4. Forc≥β√π/√α, we have\nϑc,α≥arccos/parenleftBigg\n−1 +2πβ2\nc2α/parenrightBigg\n. (4.4)\nProof.Using the formula (3.12) in Proposition 3.2 for b1,c,αwithx= 0, we get\n|b1,c,α(0)−B1,c,α|≤β\n2c/integraldisplay∞\n0/parenleftBigg\n1 +αs2\n2/parenrightBigg\ne−αs2/4.\n20Noticing that b1,c,α(0) = 0, and that\n/integraldisplay∞\n0/parenleftBigg\n1 +αs2\n2/parenrightBigg\ne−αs2/4=2√π√α,\nwe conclude that\n2B2\n1,c,α≤2πβ2\nc2α.\nSincec≥β√π/√α, we have−1 + 2πβ2/(c2α)∈[−1,1], so we can use that the function arccos\nis decreasing to obtain (4.4).\nWe continue to prove Theorem 1.4.\nProof of Theorem 1.4. As usual, we omit the subscripts candαwhen there is no confusion.\nIn view of the symmetries established in Theorem 1.3, it is enough to prove the theorem for\nx→∞. We start noticing that |Bc,α|= 1, so that the distance between mandP+is given by\ndist(m(x),P+) =|m1(x)B1+m2(x)B2+m3(x)B3|.\nTo compute the leading term, we notice that using (4.2), we have\n3/summationdisplay\nj=1ρjBjcos(cΦα(x)−φj) = Re/parenleftBigg\neicΦα(x)/parenleftBigg3/summationdisplay\nj=1ρjBje−iφj/parenrightBigg/parenrightBigg\n= 0.\nThus the estimates in Corollary 4.1 give us\ndist(m(x),P+))≤30β\ncα2xe−αx2/4,forx≥1, (4.5)\nLet us fix x≥1and letQ(x)be the orthogonal projection of m(x)on the planeP+, so\nthat dist(m(x),P+) = dist(m(x),Q(x)). We also set C(x)∈C+such that dist(m(x),C+) =\ndist(m(x),C(x)). By the Pythagorean theorem,\n|Q(x)|2= 1−dist(m(x),Q(x))2\nand\ndist(m(x),C(x))2= dist(m(x),Q(x))2+ (1−|Q(x)|)2.\nTherefore\ndist(m(x),C(x))2= 2−2(1−dist(m(x),Q(x))2)1/2,\nand using the elementary inequality√1−y≥1−y, fory∈[0,1], we get\ndist(m(x),C(x))≤√\n2 dist(m(x),Q(x)).\nCombing this estimate with (4.5), we conclude that\ndist(m(x),C+)) = dist(m(x),C(x)))≤√\n2 dist(m(x),P+))≤30√\n2β\ncαxe−αx2/4.\nThe limits in (1.13) follow at once using the definition of mc,αin (1.8) (recall that ϑc,α∈[0,π]).\nThe statement in (ii)is an immediate consequence of Lemma 4.4. This finishes the proof of\nTheorem 1.4.\n21The limit limc→∞ϑc,α=πin part (ii) of Theorem 1.4 helps us to understand the angle\nbetween the great circles as c→∞. However, the behavior of the limit circles C±\nc,αforcsmall\nis much more involved. We conclude this section with some reflections on the behavior of the\nlimit circlesC±\nc,αwhencis small.\nLet us remark that when c= 0, the explicit solution to (1.6)–(1.7) is given by\nm0,α(x) = (1,0,0),\nn0,α(x) = (0,cos(βx2/4),−sin(βx2/4)),\nb0,α(x) = (0,sin(βx2/4),cos(βx2/4)).(4.6)\nThus we see that in this limit case, there is a change in the behavior of the solution: There is\nno limit circle and the vector b0,αdoes not have a limit at infinity. On the other hand, we know\nthat (mc,α(x),nc,α(x),bc,α(x))are continuous with respect to the to c,αandx. Therefore,\nlim\nc→0b1,c,α(x) =b1,0,α(x) = 0,for allx∈R. (4.7)\nOf course, we cannot conclude from (4.7) estimates for B1,c,α, ascgoes to 0. For this reason,\nwe performed some numerical simulations for different values of csmall. In Figures 3 and 4, we\nshow one of these simulations, in the case α= 0.5andc= 0.01, where we see that m1,c,α≈1\nandb1,c,α≈0on[0,2]in agreement with (4.6) and the continuous dependence on c. On the\nother hand, for x≥8.5, we haveb1,c,α(x)≈−1.\nHowever, it seems difficult to infer from our simulations the behavior of Bc,αascgoes to zero.\nFor instance, we have computed numerically Bc,α, and it is not clear that this quantity converges\nforcsmall. Forinstance, wehaveobtained B1,c,α=−0.99215, forc= 10−12,B1,c,α=−0.992045,\nforc= 10−14, andB1,c,α=−0.991965, forc= 10−16.\n2468100.20.40.60.81.0\n(i)m1,c,α\n246810\n-1.0-0.8-0.6-0.4-0.2 (ii)b1,c,α\nFigure 3: Functions m1,c,αandb1,c,αforc= 0.01andα= 0.5. The limit at infinity in (ii) is\nB1,c,α≈−0.996417.\nAcknowledgements. S. Gutiérrez was partially supported by ERCEA Advanced Grant 2014\n669689 - HADE. The Université de Lille also supported S. Gutiérrez’s research visit during July\n2018 through their Invited Research Speaker Scheme. 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Chinese\nUniv. Ser. B , 17(2):164–170, 2002.\n26"    },    {        "title": "2002.08723v2.Stoner_Wohlfarth_switching_of_the_condensate_magnetization_in_a_dipolar_spinor_gas_and_the_metrology_of_excitation_damping.pdf",        "content": "Stoner-Wohlfarth switching of the condensate magnetization in a dipolar spinor gas\nand the metrology of excitation damping\nSeong-Ho Shinn,1Daniel Braun,2and Uwe R. Fischer1\n1Department of Physics and Astronomy, Seoul National University, 08826 Seoul, Korea\n2Eberhard-Karls-Universit at T ubingen, Institut f ur Theoretische Physik, 72076 T ubingen, Germany\n(Dated: May 13, 2020)\nWe consider quasi-one-dimensional dipolar spinor Bose-Einstein condensates in the homogeneous-\nlocal-spin-orientation approximation, that is with unidirectional local magnetization. By analyti-\ncally calculating the exact e\u000bective dipole-dipole interaction, we derive a Landau-Lifshitz-Gilbert\nequation for the dissipative condensate magnetization dynamics, and show how it leads to the Stoner-\nWohlfarth model of a uni-axial ferro-magnetic particle, where the latter model determines the stable\nmagnetization patterns and hysteresis curves for switching between them. For an external magnetic\n\feld pointing along the axial, long direction, we analytically solve the Landau-Lifshitz-Gilbert equa-\ntion. The solution explicitly demonstrates that the magnetic dipole-dipole interaction accelerates\nthe dissipative dynamics of the magnetic moment distribution and the associated dephasing of the\nmagnetic moment direction. Under suitable conditions, dephasing of the magnetization direction\ndue to dipole-dipole interactions occurs within time scales up to two orders of magnitude smaller\nthan the lifetime of currently experimentally realized dipolar spinor condensates, e.g., produced with\nthe large magnetic-dipole-moment atoms166Er. This enables experimental access to the dissipation\nparameter \u0000 in the Gross-Pitaevski\u0014 \u0010 mean-\feld equation, for a system currently lacking a complete\nquantum kinetic treatment of dissipative processes and, in particular, an experimental check of the\ncommonly used assumption that \u0000 is a single scalar independent of spin indices.\nI. INTRODUCTION\nEver since a phenomenological theory to describe the\nbehavior of super\ruid helium II near the \u0015point has\nbeen developed by Pitaevski\u0014 \u0010 [1], the dynamics of Bose-\nEinstein condensates (BEC) under dissipation has been\nintensely studied, see, e.g., [2{8]. Experimentally, the im-\npact of Bose-Einstein condensation on excitation damp-\ning and its temperature dependence has for example been\ndemonstrated in [9{12].\nDissipation in the form of condensate loss is de\fned\nby a dimensionless damping rate \u0000 entering the left-\nhand side of the Gross-Pitaevski\u0014 \u0010 equation, replacing the\ntime derivative as i@t!(i\u0000\u0000)@t. While a micro-\nscopic theory of condensate damping is comparatively\nwell established in the contact-interaction case, using var-\nious approaches, cf., e.g., [5, 13{15], we emphasize the\nabsence of a microscopic theory of damping in dipolar\nspinor gases. While for scalar dipolar condensates, par-\ntial answers as to the degree and origin of condensate-\nexcitation damping have been found see, e.g., Refs. [16{\n19], in spinor or multicomponent gases the interplay of\nanisotropic long-range interactions and internal spinor or\nmulticomponent degrees of freedom leads to a highly in-\ntricate and di\u000ecult-to-disentangle many-body behavior\nof condensate-excitation damping.\nIn this paper, we propose a method to experimen-\ntally access \u0000 in a dipolar spinor condensate by using\nthe dynamics of the unidirectional local magnetization in\na quasi-one-dimensional (quasi-1D) dipolar spinor BEC\nin the presence of an external magnetic \feld. To this\nend, we \frst derive an equation of motion for the mag-\nnetization of the BEC that has the form of a Landau-\nLifshitz-Gilbert (LLG) equation [20{22], with an addi-tional term due to the dipole-dipole interaction between\nthe atoms. The LLG equation is ubiquitous in nano-\nmagnetism, where it describes the creation and dynam-\nics of magnetization. The static limit of this equation\nis, in the limit of homogeneous local spin-orientation, de-\nscribed by the well-known Stoner-Wolfarth (SW) model\n[23{25] of a small magnetic particle with an easy axis of\nmagnetization. We then investigate the magnetization\nswitching after \ripping the sign of the external magnetic\n\feld, and demonstrate the detailed dependence of the\nswitching dynamics on the dissipative parameter \u0000.\nFor a quasi-2D spinor BEC with inhomogeneous local\nmagnetization, Ref. [26] has studied the magnetic domain\nwall formation process by deriving a LLG type equa-\ntion. Here, we derive the LLG equation in a quasi-1D\nspinor BEC with unidirectional local magnetization, in\norder to establish a most direct connection to the orig-\ninal SW model. In distinction to [27], which studied\nthe e\u000bective quasi-1D dipole-dipole interaction resulting\nfrom integrating out the two transverse directions within\na simple approximation, we employ below an exact ana-\nlytic form of the dipole-dipole interaction. In Section II,\nwe establish the quasi-1D spinor Gross-Pitaevski\u0014 \u0010 (GP)\nequation with dissipation, and equations of motion for\nthe magnetization direction (unit vector) M. Section V\nshows how the LLG equation and the SW model result,\nand Section VI derives analytical solutions to the equa-\ntions of motion for Mwhen the external magnetic \feld\npoints along the long, zaxis. We summarize our results\nin section VII.\nWe defer two longer derivations to Appendices. The\nanalytical form of the e\u000bective dipole-dipole interaction\nenergy is deduced in Appendix A, and the quasi-1D GP\nmean-\feld equation with dissipation is described in detailarXiv:2002.08723v2  [cond-mat.quant-gas]  12 May 20202\nin Appendix B. Finally, in Appendix C, we brie\ry discuss\nto which extent relaxing the usual simplifying assumption\nthat dissipation even in the spinor case is described by\na single scalar changes the LLG equation, and whether\nthis a\u000bects the SW model and its predictions.\nII. GENERAL DESCRIPTION OF DAMPING IN\nBECS\nThe standard derivation of the quantum kinetics of\nBose-Einstein condensate damping [5] starts from the\nmicroscopic Heisenberg equation of motion for the quan-\ntum \feld operator ^ (r;t), for a scalar (single compo-\nnent) BEC in the s-wave scattering limit. Using their\nresults, [28] obtained a mean-\feld equation to describe\nthe dissipation of scalar BEC, whose form is\n(i\u0000\u0000)~@ \n@t=H (1)\nwhere is the (in the large Nlimit) dominant mean-\feld\npart upon expanding the full bosonic \feld operator ^ .\nIn Ref. [1], Pitaevski\u0014 \u0010 obtained a similar but slightly\ndi\u000berent form of the dissipative mean-\feld equation\nbased on phenomenological considerations, i~@ \n@t=\n(1\u0000i\u0000)H , by parametrizing the deviation from exact\ncontinuity for the condensate fraction while minimizing\nthe energy [1]. The latter deviation is assumed to be\nsmall, which is equivalent to assuming that \u0000 remains\nsmall. This provides a clear physical interpretation of the\ndamping mechanism, namely one based on particle loss\nfrom the condensate fraction. The version of Pitaevski\u0014 \u0010\ncan be written as\n(i\u0000\u0000)~@ \n@t=\u0000\n1 + \u00002\u0001\nH : (2)\nIt can thus be simply obtained by rescaling time with a\nfactor 1 + \u00002compared to (1). Hence, as long as one\ndoes not predict precisely \u0000, the two dissipative equa-\ntions (1) and (2) cannot be distinguished experimentally\nfrom the dynamics they induce. From the data of [11],\n[4] estimated typical values of \u0000 '0:03 for a scalar BEC\nof23Na atoms (see also [12]), which shows that to distin-\nguish between (1) and (2) experimentally the theoretical\npredictions of \u0000 would need to be precise to the order of\n10\u00004.\nHow eqs.(1) and (2) can be generalized to the dipolar\nspinor gases is comparatively little investigated. Using a\nsymmetry-breaking mean-\feld approach by writing the\nquantum \feld operator as ^ (r;t) as ^ (r;t) = (r;t) +\n\u000e^ (r;t), with (r;t) =h^ (r;t)iandh\u000e^ (r;t)i= 0, [5]\nand [28] showed that \u0000 is derived from the three-\feld\ncorrelation function h\u000e^ y(r;t)\u000e^ (r;t)\u000e^ (r;t)iin a ba-\nsis whereh\u000e^ (r;t)\u000e^ (r;t)i= 0. From this microscopicorigin, based on correlation functions, it is clear that in\nprinciple \u0000 might depend on the spin indices in a spinor\nBECs and hence become a tensor (see Appendix C for\na corresponding phenomenological generalization). Nev-\nertheless, it is commonly assumed cf.,e.g., [26, 29], that\n\u0000 does not depend on spin indices, and the scalar value\nfound speci\fcally in [4] for a scalar BEC of23Na atoms\nis commonly used, while a clear justi\fcation of this as-\nsumption is missing.\nExtending the microscopic derivations in [5] and [28] to\nthe spinor case would be theoretically interesting, but is\nbeyond the scope of the present paper. Here, we instead\nfocus on the question whether the standard assumption\nthat the damping of each spinor component can be de-\nscribed by the mean-\feld equation [28] leads to exper-\nimentally falsi\fable dynamical signatures. It will turn\nout that this assumption introduces an additional strong\ndephasing in the spin-degrees of freedom, ampli\fed by\nthe dipolar interaction. Hence, even on time scales on\nwhich the decay of the condensate fraction according to\n(1) can be neglected, the relaxation of the magnetization\nof the BEC potentially o\u000bers valuable insights whether\nthe scalar-\u0000 assumption is justi\fed. Indeed, in [30] it\nwas shown experimentally that on the time scale of the\nswitching dynamics of the magnetization the number of\nparticles in the condensates remains approximately con-\nstant. One might wonder, then, which dissipative mech-\nanism is left. However, as we will show, by assuming the\nsame GP equation for each component of the spinor as\nfor scalar bosons, additional dephasing occurs that is in\nfact much more rapid than the decay of condensate den-\nsity due to dephasing accelerated by the dipole-dipole\ninteraction.\nIII. MEAN-FIELD DYNAMICS OF DAMPING\nIN DIPOLAR SPINOR BECS\nFor a spinor BEC, linear and quadratic Zeeman inter-\nactions are commonly included in the Hamiltonian. The\nquadratic Zeeman interaction is related to a second-order\nperturbation term in the total energy that can be induced\nby the interaction with an external magnetic \feld ( qB)\nas well as with the interaction with a microwave \feld\n(qMW) [31]. Speci\fcally, by applying a linearly polarized\nmicrowave \feld, one can change qMWwithout changing\nqB[32, 33]. Hence, we will assume that the quadratic\nZeeman term can be rendered zero by suitably changing\nqMW.\nFollowing [26], we thus assert that for a dipolar spinor\nBEC without quadratic Zeeman term, the mean-\feld\nequation can be written as3\n(i\u0000\u0000)~@ (r;t)\n@t=\u0014\n\u0000~2\n2mr2+Vtr(r) +c0j (r;t)j2\u0000~fb\u0000bdd(r;t)g\u0001^f\u0015\n (r;t)\n+SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zF\u00171;\u00172;\u0001\u0001\u0001;\u0017k(r;t)^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k (r;t): (3)\nwhere (r;t) is a vector quantity whose \u000b-th component\nin the spinor basis is  \u000b(r;t) (spin-space indices from\nthe beginning of the Greek alphabet such as \u000b;\f;\r;:::\nare integers running from \u0000StoS). In this expres-\nsion, ~^fis the spin- Soperator where the spin ladder\nis de\fned by ^fzj\u000bi=\u000bj\u000biandh\u000bj\fi=\u000e\u000b;\f, while\nF\u00171;\u00172;\u0001\u0001\u0001;\u0017k(r;t):= y(r;t)^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k (r;t) are the\ncomponents of the expectation value of ^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k.\nThe Larmor frequency vector reads b=gF\u0016BB=~\n(with Land\u0013 e g-factor gF, Bohr magneton \u0016B, and\nthe external magnetic induction B),~bdd(r;t)\u0001e\u0017=\ncddR\nd3r0P\n\u00170=x;y;zQ\u0017;\u00170(r\u0000r0)F\u00170(r0;t):Here,cdd=\n\u00160(gF\u0016B)2=(4\u0019) ande\u0017is a unit vector along the \u0017\naxis [31] (by convention, indices from the middle of the\nGreek alphabet such as \u0014;\u0015;\u0016;\u0017;::: =x;y;z denote spa-\ntial indices), and Q\u0017;\u00170is the spin-space tensor de\fned in\nEq. (A2) of Appendix A. Finally, mis the boson mass,\nc0the density-density interaction coe\u000ecient, and c2kthe\ninteraction coe\u000ecient parametrizing the spin-spin inter-\nactions, where kis an positive integer running from 1\ntoS[26]. For example, c2is the spin-spin interaction\ncoe\u000ecient of a spin-1 gas ( S= 1).\nTo develop a simple and intuitive physical approach,\nwe consider a quasi-1D gas for which one can perform\nanalytical calculations. We set the trap potential as\nVtr(x;y;z ) =1\n2m!2\n?\u0000\nx2+y2\u0001\n+V(z); (4)\nso that the long axis of our gas is directed along the z\naxis and the gas is strongly con\fned perpendicularly.\nFor a harmonic trap along all directions, i.e. when\nV(z) =m!2\nzz2=2, we set!?\u001d!z. For a box trap along\nz, i.e. when V(z) = 0 forjzj \u0014LzandV(z) =1\nFIG. 1. Schematic of the considered geometry in a quasi-1D\ngas (shaded ellipsoid). The length of the red magnetization\narrows, all pointing in the same direction (homogeneous local-\nspin-orientation limit), represents jd(z;t)j.forjzj> Lz, our gas will be strongly con\fned along z\nas long as the quasi-1D condition is satis\fed, where we\nwill discuss below whether the condition is satis\fed, in\nsection VI A.\nSingle-domain spinor BECs have been already real-\nized, for example, using spin-187Rb [34]. This single-\ndomain approximation is common in nanomagnetism, see\nfor example [24], by assuming magnetic particles much\nsmaller than the typical width of a domain wall. The\nlocal magnetization is related to the expectation value\n~F(r;t)\u0011~ y(r;t)^f (r;t) of the spatial spin density\noperator byd(r;t) =gF\u0016BF(r;t). An unidirectional\nlocal magnetization d(z;t) is then given by\ndx(z;t) =d(z;t) sin\u0012(t) cos\u001e(t);\ndy(z;t) =d(z;t) sin\u0012(t) sin\u001e(t); (5)\ndz(z;t) =d(z;t) cos\u0012(t);\nwhered\u0017(z;t) =d(z;t)\u0001e\u0017is the\u0017-th component of\nd(z;t),d(z;t) =jd(z;t)j,\u0012(t) is polar angle of d(z;t),\nand\u001e(t) is azimuthal angle of d(z;t). For an illustra-\ntion of the geometry considered, see Fig. 1. For a single\ncomponent dipolar BEC, F(r;t) has a \fxed direction.\nTo study the relation of the Stoner-Wohlfarth model, in\nwhichF(r;t) changes its direction, with a dipolar BEC,\na multi-component dipolar BEC should therefore be em-\nployed.\nIn the quasi-1D approximation, the order parameter\n \u000b(r;t) is commonly assumed to be of the form\n \u000b(r;t) =e\u0000\u001a2=(2l2\n?)\nl?p\u0019\t\u000b(z;t): (6)\nwherel?is the harmonic oscillator length in the x\u0000\u0000y\nplane and\u001a=p\nx2+y2. Assuming our gas is in the\nhomogeneous local spin-orientation limit, we may also\napply a single mode approximation in space so that\n\t\u000b(z;t) = \t uni(z;t)\u0010\u000b(t). The time-dependent spinor\npart is\n\u0010\u000b(t) =h\u000bje\u0000i^fz\u001e(t)e\u0000i^fy\u0012(t)jSi; (7)\nfor spin-Sparticles [26, 31] and the normalization reads\nj\u0010(t)j2:=\u0010y(t)\u0010(t) = 1. Finally, due to the ( i\u0000\u0000)\nfactor on the left-hand side of Eq. (3), for the ease of\ncalculation, we may make the following ansatz for the\n \u000b(r;t), cf. Ref. [35],\n \u000b(r;t) =e\u0000\u001a2=(2l2\n?)\nl?p\u0019\t (z;t)\u0010\u000b(t)e\u0000(i+\u0000)!?t=(1+\u00002):\n(8)4\nFrom our ans atze in Eq. (7) and (8), one concludes that\nthe expectation value of the (spatial) spin-density oper-\nator is\n~Fx(r;t) =~Se\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002)\n\u0002sin\u0012(t) cos\u001e(t);\n~Fy(r;t) =~Se\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002)\n\u0002sin\u0012(t) sin\u001e(t);\n~Fz(r;t) =~Se\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002)\n\u0002cos\u0012(t): (9)\nThe above equations lead to unidirectional local magneti-\nzation, which has been assumed in Eqs. (5), in the quasi-\n1D limit (after integrating out the strongly con\fning xandyaxes). Note however that our ansatz in Eq. (8) is\nsu\u000ecient, but not necessary for the homogeneous-local-\nspin-orientation limit, and the homogeneous-local-spin-\norientation ansatz is thus designed to render our ap-\nproach as simple as possible.\nBecause we are not assuming any speci\fc form of\n\t (z;t) in our ansatz in Eq. (8),we cover every possible\ntime behavior of j (r;t)j2:= y(t) (t):\nj (r;t)j2=e\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002):(10)\nEq. (10) explicitly shows that Eq. (8) does not imply\nan exponentially decaying wavefunction with time since\nj\t (z;t)j2can be any physical function of time t. How-\never, the ansatz (8) simpli\fes the resulting equation for\n\t(z;t), Eq.(11) below.\nBy integrating out the xandydirections, the GP equa-\ntion for a quasi-1D spin- SBEC can be written as (see\nfor a detailed derivation Appendix B)\n(i\u0000\u0000)~@f\t (z;t)\u0010\u000b(t)g\n@t=\u001a\n\u0000~2\n2m@2\n@z2+V(z) +c0\n2\u0019l2\n?n(z;t)\u001b\n\t (z;t)\u0010\u000b(t)\n+~[\u0000b+SfM(t)\u00003Mz(t)ezgPdd(z;t)]\u00018\n<\n:SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\t (z;t)\u0010\f(t)9\n=\n;\n+SX\nk=1c2k\n2\u0019l2\n?n(z;t)X\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t)8\n<\n:SX\n\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\t (z;t)\u0010\f(t)9\n=\n;;\n(11)\nwhere we de\fned the two functions\nM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t):=1\nSSX\n\u000b;\f=\u0000S\u0010y\n\u000b(t)\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f(t); (12)\nPdd(z;t):=cdd\n2~l3\n?Z1\n\u00001dz0n(z0;t)\u001a\nG\u0012jz\u0000z0j\nl?\u0013\n\u00004\n3\u000e\u0012z\u0000z0\nl?\u0013\u001b\n; (13)\nwith the axial density n(z;t):=R\nd2\u001aj (r;t)j2=\nj\t (z;t)j2e\u00002\u0000!?t=(1+\u00002), whereR\nd2\u001a :=R1\n\u00001dxR1\n\u00001dy. Finally, the function Gappearing\ninPddis de\fned as\nG(\u0015):=r\u0019\n2\u0000\n\u00152+ 1\u0001\ne\u00152=2Erfc\u0012\u0015p\n2\u0013\n\u0000\u0015:(14)\nWe plotG(\u0015) as a function of \u0015in Fig. 2. Eq.(11) rep-\nresents our starting point for analyzing the dynamics of\nmagnetization. We will now proceed to show how it leads\nto the LLG equation and the Stoner-Wolfarth model.IV. EFFECTIVE LANGRANGIAN\nDESCRIPTION\nTo provide a concise phase space picture of the conden-\nsate magnetization dynamics, we discuss in this section a\ncollective coordinate Lagrangian appropriate to our sys-\ntem.\nLetM(t):=d(z;t)=d(z;t) where the magne-\ntizationd(z;t) is de\fned in Eq. (5). Explicitly,\nthe local magnetization direction reads M(t) =\n(sin\u0012(t) cos\u001e(t);sin\u0012(t) sin\u001e(t);cos\u0012(t)). Then, from\nEqs. (9) and (10), F(r;t) =SM(t)j (r;t)j2and one5\n1 2 3 4 50.20.40.60.81.01.2\nFIG. 2. The function G(\u0015) de\fned in Eq. (14). Note that\nG(\u0015)'2=\u00153+O\u0000\n\u0015\u00005\u0001\nfor\u0015\u001d1, soG(\u0015) is always positive\nfor\u0015\u00150.\nobtains (see for a detailed derivation Appendix B)\n@M\n@t=M\u0002fb+S\u00030\ndd(t)Mzezg\u0000\u0000M\u0002@M\n@t;(15)\nwhere the renormalized interaction function \u00030\ndd(t) reads\n\u00030\ndd(t) =3\nN(t)Z1\n\u00001dzn(z;t)Pdd(z;t);(16)\nandN(t):=R\nd3rj (r;t)j2=R1\n\u00001dz n (z;t). From\nEqs. (A9), (A12), and (13), \u00030\ndd(t) is connected to the\ndipole-dipole interaction contribution Vdd(t) by\nVdd(t) =3\n2~S2\u001a\nsin2\u0012(t)\u00002\n3\u001bZ1\n\u00001dzn(z;t)Pdd(z;t)\n=~\n2S2N(t) \u00030\ndd(t)\u001a1\n3\u0000cos2\u0012(t)\u001b\n: (17)\nWe note that in order to obtain the e\u000bective quasi-1D\ndipolar interaction (17), we did not use, in distinction\nto Ref. [27], any simplifying approximation. A detailed\nderivation is provided in Appendix A.\nEq.(15) is the LLG equation with the external mag-\nnetic \feld in z-direction modi\fed by the magnetiza-\ntion inz-direction due to the dipole-dipole interac-\ntion. The corresponding term in units of magnetic \feld,\n~S\u00030\ndd(t)Mzez=(gF\u0016B), can be seen as an additional\nmagnetic \feld that is itself proportional to the magne-\ntization inz-direction, and which leads to an additional\nnonlinearity in the LLG equation.\nFrom Eqs. (13) and (16), to get how \u00030\ndd(t) depends\non timet, one has to calculate the double integral\nZ\ndzZ\ndz0n(z;t)n(z0;t)\u001a\nG\u0012jz\u0000z0j\nl?\u0013\n\u00004\n3\u000e\u0012z\u0000z0\nl?\u0013\u001b\n:\n(18)\nTo achieve a simple physical picture, we assume that\nn(z;t) does not depend on time twithin the time range\nwe are interested in. Then we may write \u00030\ndd(t) = \u00030\ndd.\nThe lifetime of a typical dipolar BEC with large atomicmagnetic dipole moments such as164Dy [36],162Dy and\n160Dy [37], or166Er [30] is of the order of seconds. Since\ntaking into account the time dependence of n(z;t) gen-\nerally requires a numerical solution of Eq. (11), we here\nconsider the case where n(z;t) is constant in time tas\nin [26], to predominantly extract the e\u000bect of magnetic\ndipole-dipole interaction per se .\nWe also neglect the possible e\u000bect of magnetostriction.\nThe latter e\u000bect, amounting to a distortion of the aspect\nratio of the condensate in a harmonic trap as a func-\ntion of the angle of the external magnetic \feld with the\nsymmetry axis of the trap, was measured in a conden-\nsate of Chromium atoms [38] (with a magnetic moment\nof 6\u0016B). The magnetostriction e\u000bect in that experiment\nwas of the order of 10%. For alkali atoms with spin-1\nthe e\u000bect should be a factor 62smaller. In addition, the-\noretical analyses in the Thomas-Fermi limit show that\nmagnetostriction in harmonic traps becomes particularly\nsmall for very small or very large asymmetries of the trap\n[39, 40].\nMore speci\fcally, Ref. [41] has shown that magne-\ntostriction is due to the force induced by the dipole-dipole\nmean-\feld potential \b dd(r;t). In Appendix D, we ap-\nply the approach of [41] to a dipolar spinor BEC. From\nEqs. (16), (17), (A1), and (D5), \u00030\ndd(t) contains \b dd(z;t)\n[the quasi-1D form of \b dd(r;t) de\fned in Eq. (D5)] by\nS2\b\n1\u00003M2\nz(t)\t\nN(t)~\u00030\ndd(t)\n= 3Z1\n\u00001dzn(z;t) \bdd(z;t):(19)\nHence, our LLG-type equation in Eq. (15) e\u000bectively con-\ntains the dipole-dipole mean-\feld potential which causes\nmagnetostriction and the form of Eq. (15) itself will not\nbe changed whether the e\u000bect of magnetostriction is large\nor not. Only the value of \u00030\ndd(t) will be changed because\nmagnetostriction changes the integration domain. Fur-\nthermore, we show in Appendix D that for our quasi-1D\nsystem, the e\u000bect of magnetostriction is smaller in a box\ntrap than in harmonic trap. In fact, for the box trap,\nthis e\u000bect can be neglected if Lz=l?is su\u000eciently large.\nThus, we may neglect the e\u000bect of magnetostriction un-\nder suitable limits for both box and harmonic traps.\nTo get a simple physical idea of the dynamical behavior\nof our system, let us, for now, assume that there is no\ndamping, \u0000 = 0. When the external magnetic \feld is\nchosen to lie in the x\u0000zplane,B= (Bx;0;Bz), Eq. (15)\nbecomes\nd\u0012\ndt=bxsin\u001e;\nd\u001e\ndt=bxcot\u0012cos\u001e\u0000bz\u0000S\u00030\nddcos\u0012: (20)\nwhere we already de\fned the Larmor frequency vector\nb=gF\u0016BB=~below Eq. (3).\nBy using the Lagrangian formalism introduced in [42],6\nthe Lagrangian Lof this system then ful\flls\nL\n~=_\u001ecos\u0012+bxsin\u0012cos\u001e+bzcos\u0012+S\n4\u00030\nddcos (2\u0012);\n(21)\nwhere _\u001e=d\u001e=dt . The equations of motion are\n1\n~@L\n@\u0012=\u0000_\u001esin\u0012+bxcos\u0012cos\u001e\u0000bzsin\u0012\u0000S\n2\u00030\nddsin (2\u0012);\n@L\n@_\u0012= 0;1\n~@L\n@\u001e=\u0000bxsin\u0012sin\u001e;1\n~@L\n@_\u001e= cos\u0012:(22)\nOne easily veri\fes that Eq. (21) is indeed the Lagrangian\nwhich gives Eqs. (20). Let p\u0018be the conjugate momen-\ntum of the coordinate \u0018. Sincep\u0012= 0 andp\u001e=~cos\u0012\n(~times thezcomponent of M), the Hamiltonian His\ngiven by\nH=\u0000bxq\n~2\u0000p2\n\u001ecos\u001e\u0000bzp\u001e+~2\u00002p2\n\u001e\n4~S\u00030\ndd:(23)\nNote that the energy ~E:=H\u0000~S\u00030\ndd=4 is conserved.\nHence, if we put p\u001e= (p\u001e)inand\u001e=\u0019=2 at some time\nt=t0,~E=\u0000bz(p\u001e)in\u0000S\u00030\ndd(p\u001e)2\nin=2~. We can then\nexpress\u001eas a function of p\u001eas\ncos\u001e=\u0000~E+bzp\u001e+1\n2~S\u00030\nddp2\n\u001e\nbxq\n~2\u0000p2\n\u001e\n=\b\n(p\u001e)in\u0000p\u001e\tbz+S\u00030\ndd(p\u001e)in+p\u001e\n2~\nbxq\n~2\u0000p2\n\u001e: (24)\nThe canonical momentum p\u001eremains the initial ( p\u001e)in\nwhenbx= 0, implying that \u0012does not change when\nbx= 0, consistent with Eqs. (20). If jbxjis larger than\njbz\u0006S\u00030\nddj, we can have p\u001e6= (p\u001e)inwithjcos\u001ej\u00141,\nwhich allows for the switching process of the magneti-\nzation. Below a threshold value of jbxjthat depends on\nbzandS\u00030\ndd,p\u001ehas to remain constant for Eq. (24) to\nbe satis\fed, which corresponds to simple magnetization\nprecession about the zaxis.\nWhenp\u001eis a function of time, there are two important\ncases:\n(a)jbzj\u001dS\u00030\ndd: cos\u001e=bz\nbx(p\u001e)in\u0000p\u001eq\n~2\u0000p2\n\u001e;\n(b)jbzj\u001cS\u00030\ndd: cos\u001e=S\u00030\ndd\n2bx(p\u001e)2\nin\u0000p2\n\u001e\n~q\n~2\u0000p2\n\u001e:(25)\nWe plot the corresponding phase diagrams ( \u0012vs\u001e) in\nFig. 3.\nLet (p\u001e)in=~cos\u0012in,bx=bsin\u00120, andbz=bcos\u00120.\nWhen case (a) holds jbzj\u001dS\u00030\ndd, one concludes that\ncos\u00120cos\u0012+ sin\u00120sin\u0012cos\u001e= cos\u00120cos\u0012in, which is\nconstant. Since d\u0001b=db(cos\u00120cos\u0012+ sin\u00120sin\u0012cos\u001e),\nin case (a) the magnetization dprecesses around the ex-\nternal magnetic \feld B, as expected. When (b) holds,\nSW switching can occur, to the description of which we\nproceed in the following.\n0.5 1.0 1.5 2.0\n-1.0-0.50.51.0(p\u001e)in=~=2 and\u001ein=\u0019=2\n0.5 1.0 1.5 2.0\n-1.0-0.50.51.0\n(p\u001e)in=\u0000~=2and\u001ein=\u0019=2\nFIG. 3.p\u001e=~vs\u001e=\u0019when \u0000 = 0 (no dissipation), with initial\nvalues (p\u001e)inand\u001ein(initial value of \u001e) as shown. (1) Dashed\nblue:bz=bx= 0:2 andjbzj\u001dS\u00030\ndd. (2) Black line: bz=bx=\n0:2 andS\u00030\ndd=bx= 0:6. (3) Dash-Dotted red: S\u00030\ndd=bx= 0:6\nandjbzj\u001cS\u00030\ndd. (4) Dotted orange horizontal line: bx= 0.\nV. CONNECTION TO STONER-WOHLFARTH\nMODEL\nThe phenomenological SW model can be directly read\no\u000b from the equations in the preceding section. From\nEq. (23), ~H:=H+~S\u00030\ndd=4 is given by\n~H\n~=\u0000bxsin\u0012cos\u001e\u0000bzcos\u0012+S\u00030\ndd\n2sin2\u0012:\n(26)\nLet (b\u0017)crbe the value of b\u0017at the stability limit where\n@~H=@\u0012 = 0 and@2~H=@\u00122= 0. Then one obtains the\ncritical magnetic \felds\n(bx)crcos\u001e=S\u00030\nddsin3\u0012;(bz)cr=\u0000S\u00030\nddcos3\u0012:\n(27)\nwhich satisfy the equation\nf(bx)crcos\u001eg2=3+ (bz)2=3\ncr=fS\u00030\nddg2=3: (28)\nWe coin the curve in the ( bx;bz)-plane described by\nEq. (28) the switching curve, in accordance with the ter-\nminology established in [43]. Because \u001echanges in time\n[see Eqs. (20) and Fig. 3], the switching curve depends\nin general on the timing of the applied external magnetic\n\felds. We note that, for \u001e= 0, Eqs. (26) and (28) are\nidentical to the SW energy functional\nHSW\n~=\u0000bxsin\u0012\u0000bzcos\u0012+Ksin2\u0012 (29)7\nand the SW astroid [43], respectively, if we identify K=\nS\u00030\ndd=2.\nThe LLG equation in Eq. (15) has stationary solu-\ntions withMparallel to the e\u000bective magnetic \feld\n~fb+S\u00030\ndd(t)Mzezg=(gF\u0016B). Since we set bto lie\nin thexzplane,\u001ewill go to zero for su\u000eciently large\ntimes. Thus Eq. (26) leads to the SW model (29) due\nto the damping term in (15) if \u0000 >0. In Appendix C,\nwe demonstrate that a more general tensorial damping\ncoe\u000ecient \u0000 introduces additional terms on the right-\nhand side of the LLG equation (15), which involve time\nderivatives . While these will thus not a\u000bect the SW phe-\nnomenology, which results from the steady states as func-\ntion of the applied magnetic \felds, and which is thus gov-\nerned by the vanishing (in the stationary limit) of the \frst\nterm on the right-hand side of the LLG equation, they\na\u000bect the detailed relaxation dynamics of the magneti-\nzation and its time scales. These deviations can hence\ncan be used to probe deviations from assuming a single\nscalar \u0000.\nBefore we move on to the next section, we show the\ncharacteristic behavior of \u00030\nddde\fned in Eq. (16), for\na box-trap scenario de\fned by n(z;t) =N=(2Lz) for\n\u0000Lz\u0014z\u0014Lzandn(z;t) = 0 otherwise ( Nis number\nof particles).\nWe stress that due to the \fnite size of the trap along\nthe \\long\" zdirection, in variance with the Hohenberg-\nMermin-Wagner theorem holding for in\fnitely extended\nsystems in the thermodynamic limit, a quasi-1D BEC can\nexist also at \fnite temperatures [44]. This remains true\nup to a ratio of its proper length to the de-Broglie wave-\nlength [45], beyond which strong phase \ructuations set\nin [46]. In fact, these strongly elongated quasi-1D BECs\nat \fnite temperature have been \frst realized already long\nago, cf., e.g. [47].\nFor the box trap, \u00030\ndd= \u0003dd(Lz=l?) where\n\u0003dd(\u0015) =3Ncdd\n2~l3\n?1\n\u0015(Z2\u0015\n0dv\u0010\n1\u0000v\n2\u0015\u0011\nG(v)\u00002\n3)\n:\n(30)\nFrom Eq. (14), G(v)'2=v3+O\u0000\nv\u00005\u0001\nforv\u001d1, so that\n\u0003dd(\u0015)'Ncdd\n2~l3\n?1\n\u0015for\u0015=Lz\nl?\u001d1: (31)\nHence \u0003dd(\u0015) is a slowly decreasing function of the\ncigar's aspect ratio \u0015(keeping everything else \fxed). We\nwill see below that for the parameters of experiments\nsuch as [30], the e\u000bective magnetic \feld due to dipolar\ninteractions greatly exceeds the externally applied mag-\nnetic \felds (in the range relevant for SW switching to be\nobserved) [48].\nVI. ANALYTICAL RESULTS FOR AXIALLY\nDIRECTED EXTERNAL MAGNETIC FIELD\nWithout dissipation, when bx= 0,p\u001e=~cos\u0012=~Mz\nis rendered constant; see Eq. (20). However, in the pres-ence of dissipation, Mzchanges in time even if bx= 0.\nBy employing this change, we propose an experimental\nmethod to measure \u0000.\nFor simplicity, we will assume that the number density\nis constant in time (also see section IV) and the external\nmagnetic \feld points along the zdirection,B=Bzez.\nLet a critical (see for a detailed discussion below) value\nof the magnetization be\n(Mz)cr:=\u0000bz\nS\u00030\ndd: (32)\nThen Eq. (15) can be written as\n@M\n@t=S\u00030\nddM\u0002ezfMz\u0000(Mz)crg\u0000\u0000M\u0002@M\n@t\n=M\u0002ez(bz+S\u00030\nddMz)\u0000\u0000M\u0002@M\n@t:(33)\nSinceM\u0001@M\n@t= 0, by taking the cross product with M\non both sides of Eq. (15), one can derive an expression\nforM\u0002@M\n@t:\n@Mz\n@t=\u0000\u0000S\u00030\ndd\n1 + \u00002fMz\u0000(Mz)crg\u0000\nM2\nz\u00001\u0001\n=\u0000\u0000\n1 + \u00002(bz+S\u00030\nddMz)\u0000\nM2\nz\u00001\u0001\n:(34)\nSinceMis the scaled magnetization, jMj= 1 with a\ncondensate. Hence, \u00001\u0014Mz\u00141. Also, according to\nthe discussion below Eq. (26), the generally positive SW\ncoe\u000ecient (with units of frequency) KisS\u00030\ndd=2.\nFrom Eq. (34), for time-independent \u00030\ndd, one con-\ncludes that there are three time-independent solutions,\nMz= (Mz)crandMz=\u00061. For a box-trapped BEC\nand constant number density, \u00030\ndd= \u0003ddwhich is always\npositive in the quasi-1D limit (cf. Eq. (30) and the discus-\nsion following it). For some arbitrary physical quasi-1D\ntrap potential, in which the number density is not con-\nstant in space, from Eqs. (13), (16), and Fig. 2, one can\ninfer that \u00030\ndd>0, due to the fact that the quasi-1D num-\nber density n(z;t)>0,n(z;t) has its maximum value\nnearz= 0 for a symmetric trap centered there, and then\nG(\u0015) also has its maximum value near \u0015= 0. Then, if\nj(Mz)crj<1,Mz= (Mz)cris an unstable solution and\nMz=\u00061 are stable solutions. When j(Mz)crj<1 and\n\u00001< Mz<(Mz)cr,Mzgoes to\u00001. Likewise, Mzgoes\nto 1 when ( Mz)cr< Mz<1. This bifurcation does not\noccur ifj(Mz)crj>1. For simplicity, we assume that\nj(Mz)crj<1. This is the more interesting case due to\nthe possibility of a bifurcation of stable solutions leading\nto SW switching.\nLet (Mz)inbe the value of Mzatt= 0. The analytic8\nsolution of Eq. (34) satis\fes\nt=1 + \u00002\n\u0000S\u00030\ndd\"\n1\nf(Mz)crg2\u00001ln\u001a(Mz)in\u0000(Mz)cr\nMz\u0000(Mz)cr\u001b\n\u00001\n2f1\u0000(Mz)crgln\u001a1\u0000Mz\n1\u0000(Mz)in\u001b\n+1\n2f1 + (Mz)crgln\u001a1 + (Mz)in\n1 +Mz\u001b\u0015\n=1 + \u00002\n\u0000\"\nS\u00030\ndd\nb2z\u0000(S\u00030\ndd)2ln\u001abz+S\u00030\ndd(Mz)in\nbz+S\u00030\nddMz\u001b\n\u00001\n2 (bz+S\u00030\ndd)ln\u001a1\u0000Mz\n1\u0000(Mz)in\u001b\n\u00001\n2 (bz\u0000S\u00030\ndd)ln\u001a1 + (Mz)in\n1 +Mz\u001b\u0015\n: (35)\nThe above equation tells us that, if ( Mz)in6= (Mz)crand\n(Mz)in6=\u00061,Mzgoes to its stable time-independent\nsolution (jMzj= 1) at time t=1. Thus, we de\fne\nacritical switching time tcrto be the time when jMzj=\n0:99. Also, note that the form of LLG equation (Eq. (33))\ndoes not change whether BEC is con\fned in a quasi-\n1D, quasi-2D, or a three-dimensional geometry. This is\nbecause one can \fnd a connection between \u00030\nddand the\ne\u000bective dipole-dipole-interaction potential Ve\u000b, so one\ncan measure \u0000 even if the BEC is e\u000bectively con\fned in\na space with dimension higher than one, using Eq. (35).\nWe point out, in particular, that tcris inversely pro-\nportional to \u00030\ndd. Hence, for a constant density quasi-\n1D BEC con\fned between \u0000Lz\u0014z\u0014Lz, \u00030\ndd=\n\u0003dd(Lz=l?), and thus tcris also inversely proportional\nto the linear number density along z. This follows from\nthe relation between \u0003 dd(Lz=l?) and the linear numberdensity along zdisplayed in Eq. (30).\nFor large dipolar interaction, the asymptotic expres-\nsion fortcris, assuming \u0000\u001c1\ntcr'1\n\u0000S\u00030\nddln\"\n5p\n2(1\u0000(Mz)2\nin)\nj(Mz)in\u0000(Mz)crj#\n(36)\nprovidedS\u00030\ndd\u001djbzj () j (Mz)crj\u001c1:\nThe above tcrdiverges at ( Mz)in= (Mz)cror\u00061, as\nexpected, since Mz= (Mz)crandMz=\u00061 are time-\nindependent solutions of the LLG equation. We stress\nthat Eq. (36) clearly shows that the magnetic dipole-\ndipole interaction accelerates the decay of Mz. Hence, by\nusing a dipolar spinor BEC with large magnetic dipole\nmoment such as produced from164Dy or166Er one may\nobserve the relaxation of Mzto the stable state within\nthe BEC lifetime, enabling the measurement of \u0000.\nBefore we show how the critical switching time tcr\ndepends on ( Mz)inand \u0000, we will qualitatively discuss\nwhen our quasi-1D assumption and homogeneous-local-\nspin-orientation assumption are valid. Typically, spin-\nspin-interaction couplings are much smaller than their\ndensity-density-interaction counterparts, by two orders\nof magnitude. For spin 123Na BEC or spin 187Rb BEC,\nc0'100jc2j[31, 34]. Thus we may neglect to a \frst ap-\nproximation the S2timesc2kterms in Eq. (11) (see the\ndiscussion at the end of Appendix D). We also require\nj(Mz)crj<1. Thus, we may additionally neglect the b\nterm compared to the Pdd(z;t) term since, for b=bzez,\nS\u00030\ndd>jbjshould be satis\fed to make j(Mz)crj<1 (see\nEq. (32)) and \u00030\nddis related to Pdd(z;t) by Eq. (16).\nWhen \u0000 = 0, using our ansatz in Eq. (8) and integrating\nout thexandydirections, Eq. (D4) can be approximated\nby the expression\n\u0016(t) \t (z;t) =\u001a\n\u0000~2\n2m@2\n@z2+V(z) +c0\n2\u0019l2\n?j\t (z;t)j2+ \bdd(z;t)\u001b\n\t (z;t); (37)\nwhere, from Eqs. (D5), (A1), and (17), the dipole-dipole interaction mean-\feld potential reads\n\bdd(z;t) =~S2\b\n1\u00003M2\nz(t)\t\nPdd(z;t)\n=cdd\n2l3\n?S2\b\n1\u00003M2\nz(t)\tZ1\n\u00001dz0j\t (z0;t)j2\u001a\nG\u0012jz0\u0000zj\nl?\u0013\n\u00004\n3\u000e\u0012z0\u0000z\nl?\u0013\u001b\n=cdd\n2\u0019l2\n?\u0019S2\b\n1\u00003M2\nz(t)\t\u001aZ1\n\u00001d\u0016zj\t (z+ \u0016zl?;t)j2G(j\u0016zj)\u00004\n3j\t (z;t)j2\u001b\n: (38)\nFrom Fig. 2, the function G(\u0015) is positive and decreases\nexponentially as \u0015increases. Thus, if l?is small enough\nsuch thatj\t (z+ \u0016zl?;t)j2does not change within the\nrangej\u0016zj\u00145, one may conclude that\n\bdd(z;t)'2\u0019\n3S2\b\n1\u00003M2\nz(t)\tcdd\n2\u0019l2\n?j\t (z;t)j2;(39)due to the propertyR1\n0d\u0015G (\u0015) = 1.\nA spinor (S= 6) dipolar BEC has been realized using\n166Er [30]. For this BEC, c0= 4\u0019~2a=m wherea'67aB\n(aBis Bohr radius) and 2 \u0019S2cdd=3 = 0:4911c0. Due to\njMz(t)j\u00141 from the de\fnition of M(t), the maximum\nvalue of the chemical potential \u0016(t) is achieved when9\nMz(t) = 0, where\n\u0016(t)'V(z) +\u0012\nc0+2\u0019\n3S2cdd\u0013n(z;t)\n2\u0019l2\n?: (40)\nFrom above Eq. (40), we may regard the 3D number\ndensity as n(z;t)=\u0000\n2\u0019l2\n?\u0001\n. In [30], N= 1:2\u0002105,\n!?=(2\u0019) =p156\u0002198 Hz = 175 :75 Hz,!z=(2\u0019) =\n17:2 Hz,l?= 0:589\u0016m, and the measured peak number\ndensity \u0016npeakis 6:2\u00021020m\u00003. Using Eq. (37) and (39),\nby denoting Lzas the Thomas-Fermi radius along z,\n(\u0000Lz\u0014z\u0014Lz) withV(z) =m!2\nzz2=2, one derives\nLz=(\n3\u0000\nc0+ 2\u0019S2cdd=3\u0001\nN\n4\u0019m!2zl2\n?)1=3\n; (41)\nand the mean number density \u0016 n= (N=2Lz)=\u0000\n2\u0019l2\n?\u0001\n=\n6:721\u00021020m\u00003as well as chemical potential \u0016=(~!?) =\nm!2\nzL2\nz=(2~!?) = 23:22. Note that \u0016 n'1:1 \u0016npeak. Be-\ncause\u0016is not less than ~!?, the experiment [30] is not\nconducted within the quasi-1D limit.\nThe homogeneous-local-spin-orientation approxima-\ntion is valid when the system size is on the order of the\nspin healing length \u0018sor less, which has been experimen-\ntally veri\fed in in [34]. Using c0'100jc2j,\u0018s'10\u0018d\nwhere\u0018d=p\n~2=(2mc0\u0016n) is the density healing length\nand\u0018s=p\n~2=(2mjc2j\u0016n) is the spin healing length.\nThus, ifLzis on the order of 10 \u0018d, the homogeneous-\nlocal-spin-orientation approximation is justi\fed.\nUsing the S= 6 element166Er, we can provide\nnumerical values which satisfy both the quasi-1D and\nhomogeneous-local-spin-orientation limits, as well as\nthey enable us to explicitly show how tcrdepends on\n(Mz)inin a concretely realizable setup. We consider be-\nlow two cases: (A) box trap along z[49] and (B) harmonic\ntrap alongz.\nA. Box traps\nWe setV(z) = 0 forjzj< Lzand1other-\nwise. Then n(z;t) =N=(2Lz) and we estimate \u0016'\u0000\nc0+ 2\u0019S2cdd=3\u0001\nN=\u0000\n4\u0019l2\n?Lz\u0001\nfrom Eq. (40). In this\ncase, \u00030\ndd= \u0003dd(Lz=l?) as is calculated in Eq. (30).\nFixingBz=\u00000:03 mG and N= 100, we con-\nsider the following two cases: (1) !?=(2\u0019) = 2:4\u0002\n104Hz andLz= 3:125\u0016m. Then Lz=l?= 62:03,\n\u0016=(~!?) = 0:1692, and Lz=\u0018d= 29:55. Thus,\nthe system is in both the quasi-1D and homogeneous-\nlocal-spin-orientation limit. S\u0003dd(Lz=l?) = 4:074\u0002\n103Hz,~S\u0003dd(Lz=l?)=(gF\u0016B) = 0:3969 mG, and \u0012cr:=\ncos\u00001(Mz)cris 85:67\u000e.\n(2)!?=(2\u0019) = 1:2\u0002104Hz andLz= 6:250\u0016m.\nThenLz=l?= 87:72,\u0016=(~!?) = 0:0846, and\nLz=\u0018d= 29:55. Thus, again the system is\nin both the quasi-1D and homogeneous-local-spin-\norientation limits. S\u0003dd(Lz=l?) = 1:028\u0002103Hz,\n~S\u0003dd(Lz=l?)=(gF\u0016B) = 0:1002 mG, and \u0012cr= 72:57\u000e.\nFig. 4 shows the relation between tcrand (Mz)in.\n-1.0 -0.5 0.5 1.00.050.100.15!?=(2\u0019) = 2:4\u0002104Hz,Lz= 3:125\u0016m, andl?= 0:0504\u0016m\nwhereN=\u0000\n4\u0019Lzl2\n?\u0001\n= 10:03\u00021020m\u00003((Mz)cr= 0:0756).\n-1.0 -0.5 0.5 1.00.20.40.6\n!?=(2\u0019) = 1:2\u0002104Hz,Lz= 6:250\u0016m, andl?= 0:0712\u0016m\nwhereN=\u0000\n4\u0019Lzl2\n?\u0001\n= 2:508\u00021020m\u00003((Mz)cr= 0:2995).\nFIG. 4.tcras a function of ( Mz)inwhenB=Bzezwhere\nBz=\u00000:03 mG and particle number N= 100. From top to\nbottom: Red for \u0000 = 0 :01, black for \u0000 = 0 :03, and blue for\n\u0000 = 0:09. Lines are from exact analytic formula in Eq. (35),\nand dot-dashed are from asymptotic expression in Eq. (36).\nGenerally,tcrdecreases as \u0000 increases. Also, note that tcr\ndiverges as ( Mz)in!(Mz)cr. For larger mean number den-\nsityN=\u0000\n4\u0019Lzl2\n?\u0001\n(top), the asymptotic expression of tcris\nessentially indistinguishable from the exact analytic formula\noftcr.\nB. Harmonic traps\nWe setV(z) =m!2\nzz2=2. Using the Thomas-Fermi\napproximation, from Eq. (40), \u0016=m!2\nzL2\nz=2 whereLz\nis given by Eq. (41).\u0000\nc0+ 2\u0019S2cdd=3\u0001\nn(z;t)=\u0000\n\u0019l2\n?\u0001\n=\nm!2\nz\u0000\nL2\nz\u0000z2\u0001\nforjzj\u0014Lzandn(z;t) = 0 forjzj>Lz.\nFrom thisn(z;t), we performed a numerical integration\nto calculate \u00030\nddin Eq. (16). Fixing Bz=\u00000:03 mG, we\nconsider the following two cases:\n(1)N= 240,!?=(2\u0019) = 2000 Hz, and !z=(2\u0019) =\n50 Hz, for which Lz= 5:703\u0016m andLz=l?= 32:68.\nWe obtain again the quasi-1D and homogeneous-local-\nspin-orientation limits since \u0016=(~!?) = 0:3337 and\nLz=\u0018d= 17:85. Furthermore, S\u00030\ndd= 1:644\u0002103Hz,\n~S\u00030\ndd=(gF\u0016B) = 1:602\u000210\u00001mG, and\u0012cr= 79:21\u000e.\n(2)N= 340,!?=(2\u0019) = 1000 Hz, and !z=(2\u0019) =\n25 Hz, where Lz= 8:070\u0016m andLz=l?= 32:70. Again,\nwe have the quasi-1D and with homogeneous-local-spin-\norientation limits ful\flled due to \u0016=(~!?) = 0:3341 and10\n-1.0 -0.5 0.5 1.00.10.20.30.4\nN= 240,!?=(2\u0019) = 2000 Hz, and !z=(2\u0019) = 50 Hz.\nLz= 5:703\u0016m andl?= 0:1745\u0016m where\nN=\u0000\n4\u0019Lzl2\n?\u0001\n= 1:010\u00021020m\u00003((Mz)cr= 0:1873).\n-1.0 -0.5 0.5 1.00.20.40.60.8\nN= 340,!?=(2\u0019) = 1000 Hz, and !z=(2\u0019) = 25 Hz.\nLz= 8:070\u0016m andl?= 0:2468\u0016m where\nN=\u0000\n4\u0019Lzl2\n?\u0001\n= 0:550\u00021020m\u00003((Mz)cr= 0:3741).\nFIG. 5.tcras a function of ( Mz)inwhenB=Bzezwhere\nBz=\u00000:03 mG, for two particle numbers Nas shown. From\ntop to bottom: Red for \u0000 = 0 :01, black for \u0000 = 0 :03, and\nblue for \u0000 = 0 :09. Lines are from exact analytic formula in\nEq. (35), and dot-dashed are from asymptotic expression in\nEq. (36). Generally, tcrdecreases as \u0000 increases. Also, note\nthattcrdiverges as ( Mz)in!(Mz)cr. For larger mean number\ndensityN=\u0000\n4\u0019Lzl2\n?\u0001\n(top), the asymptotic expression of tcris\nessentially indistinguishable from the exact analytic formula\noftcr.\nLz=\u0018d= 17:87. In addition, S\u00030\ndd= 8:230\u0002102Hz,\n~S\u00030\ndd=(gF\u0016B) = 8:019\u000210\u00002mG, and\u0012cr= 68:03\u000e.\nFig. 5 shows for the harmonic traps the relation be-\ntweentcrand (Mz)in.\nC. Measurability of critical switching time\nFigs. 4 and 5 demonstrate that the critical switching\ntimetcris much smaller than the lifetime of BEC (sev-\neral seconds [30]) and thus, by measuring tcrby varying\n(Mz)in, one will be able to obtain the value of \u0000, pro-\nvided \u0000 indeed does not depend on spin indices as for\nexample Refs. [26, 29] have assumed. Conversely, if one\nobtains from the measurements a di\u000berent functional re-\nlation which does not follow Eq. (35), this implies that \u0000\nmay depend on spin indices.\nNote that both \fgures, Figs. 4 and 5, show that tcrisinversely proportional to the mean number density\nN=\u0000\n4\u0019Lzl2\n?\u0001\n. Eq. (36) states that tcris inversely pro-\nportional to \u00030\ndd, but except for the box trap case, in\nwhich one can analytically calculate \u00030\ndd= \u0003dd(Lz=l?)\nin Eq. (30), the dependence of \u00030\nddand the mean number\ndensityN=\u0000\n4\u0019Lzl2\n?\u0001\nis not immediately apparent. Thus,\nat least for harmonic traps, and in the Thomas-Fermi ap-\nproximation, one may use the box trap results of Eq. (30)\nfor provide an approximate estimate of the behavior of\ntcr.\nVII. CONCLUSION\nFor a quasi-1D dipolar spinor condensate with\nunidirectional local magnetization (that is in the\nhomogeneous-local-spin-orientation limit), we provided\nan analytical derivation of the Landau-Lifshitz-Gilbert\nequation and the Stoner-Wohlfarth model. For an exter-\nnal magnetic \feld along the long axis, we obtained an\nexact solution of the quasi-1D Landau-Lifshitz-Gilbert\nequation. Our analytical solution demonstrates that the\nmagnetic dipole-dipole interaction accelerates the relax-\nation of the magnetization to stable states and hence\nstrongly facilitates observation of this process within the\nlifetime of typical dipolar spinor BECs. Employing this\nsolution, we hence propose a method to experimentally\naccess the dissipative parameter(s) \u0000.\nWe expect, in particular, that our proposal provides a\nviable tool to verify in experiment whether \u0000 is indeed\nindependent of spin indices, as commonly assumed, and\ndoes not have to be replaced by a tensorial quantity for\nspinor gases. We hope that this will stimulate further\nmore detailed investigations of the dissipative mechanism\nin dipolar BECs with internal degrees of freedom.\nWe considered that the magnetization along z,Mz, has\ncontributions solely from the atoms residing in the con-\ndensate, an approximation valid at su\u000eciently low tem-\nperatures. When the magnetization from noncondensed\natoms is not negligible, as considered by Ref. [5] for a\ncontact interacting scalar BEC, correlation terms mix-\ning the noncondensed part and the mean \feld, such asPS\n\f=\u0000S \u0003\n\f(r;t)h\u000e^ \u000b(r;t)\u000e^ \f(r;t)iwill appear on the\nright-hand side of Eq. (3). Here, \u000e^ \u000b(r;t) is the\u000b-th\ncomponent of quantum \feld excitations above the mean-\n\feld ground state in the spinor basis. Considering the\ne\u000bect of these terms is a subject of future studies.\nACKNOWLEDGMENTS\nThe work of SHS was supported by the National\nResearch Foundation of Korea (NRF), Grant No.\nNRF-2015-033908 (Global PhD Fellowship Program).\nSHS also acknowledges the hospitality of the Uni-\nversity of T ubingen during his stay in the summer\nof 2019. URF has been supported by the NRF11\nunder Grant No. 2017R1A2A2A05001422 and Grant No. 2020R1A2C2008103.\nAppendix A: Derivation of the e\u000bective potential Ve\u000b\nThe dipole-dipole interaction term Vdd(t) in the total energy is given by [31]\nVdd(t) =cdd\n2Z\nd3rZ\nd3r0X\n\u0017;\u00170=x;y;zF\u0017(r;t)Q\u0017;\u00170(r\u0000r0)F\u00170(r0;t); (A1)\nwherecddis dipole-dipole interaction coe\u000ecient, F\u0017(r;t) = y(r;t)^f\u0017 (r;t), andQ\u0017;\u00170(r) is de\fned as the tensor\nQ\u0017;\u00170(r):=r2\u000e\u0017;\u00170\u00003r\u0017r\u00170\nr5(A2)\nin spin space, where r=jrjandr\u0017=r\u0001e\u0017, withe\u0017being the unit vector along the \u0017axis. From now on, we de\fne\n\u001a= (x;y) such that dxdy =d2\u001a=d'd\u001a\u001a where tan'=y=x.\nUsing the convolution theorem, the dipole-dipole interaction term Vdd(t) can be expressed by\nVdd(t) =cdd\n2(2\u0019)D=2Z\nd3k~n(k;t) ~n(\u0000k;t)~Udd(k;t) (A3)\nwith the Fourier transform\nUdd(\u0011;t) =1\nn(r;t)n(r0;t)X\n\u0017;\u00170=x;y;zF\u0017(r;t)Q\u0017;\u00170(\u0011)F\u00170(r0;t); (A4)\nwhere ~g(k;t) = (2\u0019)\u0000D=2R\ndrg(r;t)eik\u0001ris the Fourier transform of the function g(r;t) inD-dimensional space r\n(in our case, D= 3),\u0011=r\u0000r0, andn(r;t) =j (r;t)j2.\nBy denoting k= (k\u001a;kz), wherek\u001a= (kx;ky) withk\u001a=q\nk2x+k2yand tan'k\u001a=ky=kx, with our mean-\feld\nwavefunction in Eq. (8), one derives\n~n(k;t) =1\n\u0019l2\n?1\n(2\u0019)3=2Z\nd2\u001aZ1\n\u00001dze\u0000(\u001a=l?)2n(z;t)ei\u001a\u0001k\u001aeikzz=1\n2\u0019~n(kz;t)e\u0000k2\n\u001al2\n?=4; (A5)\nwheren(z;t):=j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002). Note the factor of (2 \u0019)\u00001appearing, when compared to Eq. (12) in\nRef. [27], which is stemming from our de\fnition of Fourier transform.\nDenoting\u0011=j\u0011j, by writinge\u0011for the unit vector along \u0011, we obtain\nUdd(\u0011;t) =\u00001\n\u00113r\n6\u0019\n5hn\nY2\n2(e\u0011)e\u00002i\u001e(t)+Y\u00002\n2(e\u0011)e2i\u001e(t)o\nS2sin2\u0012(t)\n\u0000n\nY1\n2(e\u0011)e\u0000i\u001e(t)\u0000Y\u00001\n2(e\u0011)ei\u001e(t)o\nS2sinf2\u0012(t)gi\n+1\n\u00113r\n6\u0019\n5Y0\n2(e\u0011)r\n2\n3S2\b\n3 sin2\u0012(t)\u00002\t\n; (A6)\nwhereYm\nl(e\u0011) are the usual spherical harmonics. Its Fourier transform ~Udd(k;t) is\n~Udd(k;t) =1\n(2\u0019)3=24\u0019\n3S2\u001a\n1\u00003\n2sin2\u0012(t)\u001b\u0012\n3k2\nz\nk2\u001a+k2z\u00001\u0013\n+1p\n2\u0019k2\n\u001a\nk2\u001a+k2zS2sin2\u0012(t) cos\b\n2'k\u001a\u00002\u001e(t)\t\n+r\n2\n\u0019k\u001akz\nk2\u001a+k2zS2sinf2\u0012(t)gcos\b\n'k\u001a\u0000\u001e(t)\t\n:(A7)\nBy plugging Eq. (A5) and Eq. (A7) into Eq. (A3), we \fnally obtain Vdd(t) as\nVdd(t) =cdd\n2p\n2\u0019Z1\n\u00001dkz~n(kz;t) ~n(\u0000kz;t)2S2\nl2\n?p\n2\u0019\u001a\n1\u00003\n2sin2\u0012(t)\u001b\u001a\u0000\nk2\nzl2\n?=2\u0001\nek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n\u00001\n3\u001b\n;(A8)12\nwhereE1(x) =R1\nxdue\u0000u=uis exponential integral.\nNote that Eq. (A8) can be also written as\nVdd(t) =cdd\n2p\n2\u0019Z1\n\u00001dkz~n(kz;t) ~n(\u0000kz;t)~Ve\u000b(kz;t) =cdd\n2Z1\n\u00001dzZ1\n\u00001dz0n(z;t)n(z0;t)Ve\u000b(z\u0000z0;t):(A9)\nDue to the fact that ~Ve\u000b(kz;t) can be obtained by Eq. (A8), we can get Ve\u000b(z;t) by inverse Fourier transform. As a\npreliminary step, we \frst write down some integrals of E1(x) as follows:\nZ1\n\u00001dxex2E1\u0000\nx2\u0001\ne\u0000ikx=Z1\n\u00001dxe\u0000ikxZ1\nx2dte\u0000(t\u0000x2)\nt= (\u0019)3=2ek2=4Erfc (jkj=2): (A10)\nDi\u000berentiating Eq. (A10) with respect to ktwo times results in\nZ1\n\u00001dxx2ex2E1\u0000\nx2\u0001\ne\u0000ikx=\u0000(\u0019)3=2\u001a1\n2\u0012k2\n2+ 1\u0013\nek2=4Erfc (jkj=2)\u0000jkj\n2p\u0019\u00002p\u0019\u000e(k)\u001b\n: (A11)\nTherefore,Ve\u000b(z;t) can be calculated as\nVe\u000b(z;t) =1p\n2\u0019Z1\n\u00001dkz2S2\nl2\n?p\n2\u0019\u001a\n1\u00003\n2sin2\u0012(t)\u001b\u001a\u0000\nk2\nzl2\n?=2\u0001\nek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n\u00001\n3\u001b\ne\u0000ikzz\n=S2\nl3\n?\u001a3\n2sin2\u0012(t)\u00001\u001b\u001a\nG(jzj=l?)\u00004\n3\u000e(z=l?)\u001b\n; (A12)\nwhereG(x) is de\fned in Eq. (14), and \u000e(x) is the Dirac delta function.\nThe Fourier transform of Eq. (A12) acquires the form\n~Ve\u000b(kz;t) =1p\n2\u0019Z1\n\u00001dzV e\u000b(z;t)eikzz=r\n2\n\u0019S2\nl2\n?\u001a3\n2sin2\u0012(t)\u00001\u001b\u001aZ1\n0dvG (v) cos (kzl?v)\u00002\n3\u001b\n=r\n2\n\u0019S2\nl2\n?\u001a3\n2sin2\u0012(t)\u00001\u001b\u0014Z1\n0dunp\u0019\u0000\n2u2+ 1\u0001\neu2Erfc (u)\u00002uo\ncos\u0010p\n2kzl?u\u0011\n\u00002\n3\u0015\n:(A13)\nFrom [50], the following integral involving the complementary error function is\nZ1\n0dueu2Erfc (u) cos (bu) =1\n2p\u0019eb2=4E1\u0000\nb2=4\u0001\n: (A14)\nBy di\u000berentiating Eq. (A14) two times with respect to b, we get\nZ1\n0duu2eu2Erfc (u) cos (bu) =\u00001\n2p\u0019\u001a1\n2\u0012b2\n2+ 1\u0013\neb2=4E1\u0000\nb2=4\u0001\n\u00001 +2\nb2\u001b\n: (A15)\nHence, Eq. (A13) becomes\n~Ve\u000b(kz;t) =r\n2\n\u0019S2\nl2\n?\u001a3\n2sin2\u0012(t)\u00001\u001b\u0014\n\u0000\u001a1\n2\u0000\nk2\nzl2\n?+ 1\u0001\nek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n\u00001 +1\nk2zl2\n?\u001b\n+1\n2ek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n+1\nk2zl2\n?\u00002\n3\u0015\n=2S2\nl2\n?p\n2\u0019\u001a\n1\u00003\n2sin2\u0012(t)\u001b\u001a\u0000\nk2\nzl2\n?=2\u0001\nek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n\u00001\n3\u001b\n: (A16)\nComparing Eq. (A8) with Eq. (A16), one veri\fes that Eq. (A12) is the correct result for the e\u000bective interaction of\nthe quasi-1D dipolar spinor gas.13\nAppendix B: Quasi-1D Gross-Pitaevski\u0014 \u0010 equation with dissipation\nBy introducing an identical damping coe\u000ecient for each component of the spinor, cf., e.g. Refs. [26, 29] (i.e. as if\neach component e\u000bectively behaves as a scalar BEC [28]), and neglecting a possible quadratic Zeeman term, the GP\nequation for a spin- SBEC can be written as [26]\n(i\u0000\u0000)~@ \u000b(r;t)\n@t=\u001a\n\u0000~2\n2mr2+Vtr(r) +c0j (r;t)j2\u001b\n \u000b(r;t)\u0000~SX\n\f=\u0000Sfb\u0000bdd(r;t)g\u0001\u0010\n^f\u0011\n\u000b;\f \f(r;t)\n+SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zF\u00171;\u00172;\u0001\u0001\u0001;\u0017k(r;t)SX\n\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f \f(r;t);\n(B1)\nwhere \u000b(r;t) is the\u000b-th component of the mean-\feld wavefunction  (r;t) (the spin-space index \u000bis an integer\ntaking 2S+ 1 values running from \u0000SandS),F\u00171;\u00172;\u0001\u0001\u0001;\u0017k(r;t):= y(r;t)^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k (r;t),~^fis the spin- S\noperator,b=gF\u0016BB=~(gFis the Land\u0013 e g-factor, \u0016Bis the Bohr magneton, and Bthe external magnetic \feld).\nFinally, ~bdd(r;t)\u0001e\u0017=cddR\nd3r0P\n\u00170=x;y;zQ\u0017;\u00170(r\u0000r0)F\u00170(r0;t), wheree\u0017is the unit vector along the \u0017axis\n(\u0017=x;y;z ) [31]. Applying the formalism of Ref. [1] to a spinor BEC assuming that \u0000 does not depend on spin\nindices, one just needs to transform t!\u0000\n1 + \u00002\u0001\ntin Eq. (B1) and (8). We then integrate out the xandydirections\nin Eq. (B1) to obtain the quasi-1D GP equation.\nFrom Eq. (8) in the main text, we have\nZ\nd2\u001aSX\n\f=\u0000Se\u0000\u001a2=(2l2\n?)\nl?p\u0019\u001a\n~bdd(r)\u0001\u0010\n^f\u0011\n\u000b;\f\u001b\n \f(r;t)\n=cdd\n2l3\n?Z1\n\u00001dz0n(z0;t)\u001a\nG\u0012jz\u0000z0j\nl?\u0013\n\u00004\n3\u000e\u0012z\u0000z0\nl?\u0013\u001b\n\t (z;t)e\u0000i+\u0000\n1+\u00002!?tSfM(t)\u00003Mz(t)ezg\u0001SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\u0010\f(t);\n(B2)\nwhereR\nd2\u001a:=R1\n\u00001dxR1\n\u00001dyandn(z;t):=R\nd2\u001aj (r;t)j2=j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002).\nFor a spin-SBEC, from Eq. (B1), for the trap potential given in Eq. (4) and if we use Eq. (8), by integrating out\nthexandydirections, one acquires the expression\n(i\u0000\u0000)~@f\t (z;t)\u0010\u000b(t)g\n@t=\u001a\n\u0000~2\n2m@2\n@z2+V(z) +c0\n2\u0019l2\n?n(z;t)\u001b\n\t (z;t)\u0010\u000b(t)\n+ [\u0000~b+~SfM(t)\u00003Mz(t)ezgPdd(z;t)]\u00018\n<\n:SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\t (z;t)\u0010\f(t)9\n=\n;\n+SX\nk=1c2k\n2\u0019l2\n?n(z;t)X\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t)8\n<\n:SX\n\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\t (z;t)\u0010\f(t)9\n=\n;;\n(B3)\nwhereM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t) is de\fned in Eq. (12) and\nPdd(z;t) =cdd\n2~l3\n?Z1\n\u00001dz0n(z0;t)\u001a\nG\u0012jz\u0000z0j\nl?\u0013\n\u00004\n3\u000e\u0012z\u0000z0\nl?\u0013\u001b\n=cdd\n~S2\b\n3 sin2\u0012(t)\u00002\tZ1\n\u00001dz0n(z0;t)Ve\u000b(z\u0000z0;t);\n(B4)\nwithVe\u000bde\fned in (A12). It is already clear from Eq. (B3) that, besides particle loss from the condensate encoded\nin a decayingj\t(z;t)j, dissipation also leads to a dephasing , i.e. the decay of \u0010(t) due to the term \u0000\u0000@\u0010(t)=@t.\nFrom now on, if there is no ambiguity, and for brevity, we drop the arguments such as x;y;z;t from the functions.14\nFrom Eq. (B3), we then get\n~@\u0010\u000b\n@t=\u0000~\n\t@\t\n@t\u0010\u000b\u0000\u0000 +i\n1 + \u00002\u0012\n\u0000~2\n2m1\n\t@2\t\n@z2+V+c0\n2\u0019l2\n?n\u0013\n\u0010\u000b+\u0000 +i\n1 + \u00002f~b\u0000S(M\u00003Mzez)~Pddg\u00018\n<\n:SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\u0010\f9\n=\n;\n\u0000\u0000 +i\n1 + \u00002SX\nk=1c2k\n2\u0019l2\n?nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSM\u00171;\u00172;\u0001\u0001\u0001;\u0017k8\n<\n:SX\n\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f9\n=\n;; (B5)\nSince@j\u0010j2\n@t= 0 due to the normalization j\u0010j2= 1, we then have\n0 = 2Re\u001a\n\u0000~\n\t@\t\n@t\u0000\u0000\n1 + \u00002\u0012\n\u0000~2\n2m1\n\t@2\t\n@z2+V+c0\n2\u0019l2\n?n\u0013\u001b\n+i\n1 + \u00002~2\n2m\u00121\n\t@2\t\n@z2\u00001\n\t\u0003@2\t\u0003\n@z2\u0013\n+2\u0000\n1 + \u00002f~b\u0000S(M\u00003Mzez)~Pddg\u0001SM\u00002\u0000\n1 + \u00002SX\nk=1c2k\n2\u0019l2\n?nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zS2M2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k: (B6)\nHence the dynamics of the magnetization direction follows the equation\n~S@M\u0017\n@t= 2Re8\n<\n:SX\n\u000b;\f=\u0000S\u0010y\n\u000b\u0010\n^f\u0017\u0011\n\u000b;\f\u0012\n~@\u0010\f\n@t\u00139\n=\n;\n=\u00002\u0000\n1 + \u00002S2M\u0017f~b\u0000S(M\u00003Mzez)~Pddg\u0001M+2\u0000\n1 + \u00002M\u0017SX\nk=1c2k\n2\u0019l2\n?nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zS3M2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k\n+\u0000\n1 + \u00002X\n\u0016=x;y;zf~b\u0016\u0000S(M\u0016\u00003Mz\u000e\u0016;z)~PddgSf\u000e\u0016;\u0017+ (2S\u00001)M\u0016M\u0017g\n\u00001\n1 + \u00002X\n\u0016;\u0014=x;y;zf~b\u0016\u0000S(M\u0016\u00003Mz\u000e\u0016;z)~Pddg\u000f\u0017;\u0016;\u0014SM\u0014\n\u00002Re8\n<\n:\u0000 +i\n1 + \u00002SX\nk=1c2k\n2\u0019l2\n?nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSM\u00171;\u00172;\u0001\u0001\u0001;\u0017kSX\n\u000b;\f=\u0000S\u0010y\n\u000b\u0010\n^f\u0017^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f9\n=\n;; (B7)\nsince the scalar product \u0010y\u0010\n^f\u000b^f\f+^f\f^f\u000b\u0011\n\u0010=Sf\u000e\u000b;\f+ (2S\u00001)M\u000bM\fg[26].\nBy direct comparison, we can identify Eq. (B8) below as being identical to Eq. (B21) in [26], the only di\u000berence\nconsisting in the de\fnition of M\u00171;\u00172;\u0001\u0001\u0001;\u0017k: We employ a scaled version of M\u00171;\u00172;\u0001\u0001\u0001;\u0017k, which is normalized to Sin\n[26]. From Eq. (7) in the main text,\nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM\u00171;\u00172;\u0001\u0001\u0001;\u0017kSX\n\u000b;\f=\u0000S\u0010y\n\u000b\u0010\n^f\u0017^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f=X\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017kS2M\u0017; (B8)\nwhich is real. Therefore, Eq. (B7) can be written in the following form\n@M\n@t=\u0000\u0000\n1 + \u00002M\u0002[M\u0002fb\u0000S(M\u00003Mzez)Pddg] +1\n1 + \u00002M\u0002fb\u0000S(M\u00003Mzez)Pddg\n=1\n1 + \u00002M\u0002(b+ 3SPddMzez)\u0000\u0000\n1 + \u00002M\u0002[M\u0002(b+ 3SPddMzez)]\n=M\u0002(b+ 3SPddMzez)\u0000\u0000M\u0002@M\n@t; (B9)\nsinceM\u0001@M\n@t= 0 holds.\nAsPis a function of zandt, butMis independent of z[Mis the scaled local magnetization and our aim is to\nstudy a dipolar spinor BEC with unidirectional local magnetization (the homogeneous-local-spin-orientation limit)],\nby multiplying with n(z;t) both sides of Eq. (B9) and integrating along z, we \fnally get the LLG equation\n@M\n@t=M\u0002(b+S\u00030\nddMzez)\u0000\u0000M\u0002@M\n@t; (B10)\nwhere \u00030\nddis de\fned in Eq. (16). Note here that \u00030\nddbecomes \u0003 dd(Lz=l?) de\fned in Eq. (30) when n(z;t) =N=(2Lz)\nfor\u0000Lz\u0014z\u0014Lzandn(z;t) = 0 otherwise.15\nAppendix C: Modi\fcation of the LLG equation for \u0000a spin-space tensor\nWhen \u0000 depends on spin indices, i.e. is a tensor, Eq. (B3) can be generalized to read\nSX\n\f=\u0000S(i\u000e\u000b;\f\u0000\u0000\u000b;\f)~@f\t (z;t)\u0010\f(t)g\n@t=SX\n\f=\u0000SH\u000b;\f\t (z;t)\u0010\f(t): (C1)\nThe spinor part of the wavefunction is normalized to unity, j\u0010j2= 1. Hence, we know that@j\u0010j2\n@t= 0. Therefore, from\nEq. (C1), we derive the expression\nSX\n\u000b;\f=\u0000SRe\u0014\n\u0000i\u0010\u0003\n\u000b\u0000\u000b;\f@\u0010\f\n@t\u0000i\u0010\u0003\n\u000b\u0000\u000b;\f\u0010\f1\n\t@\t\n@t\u0000i1\n~\t\u0010\u0003\n\u000bH\u000b;\f\t\u0010\f\u0015\n\u0000Re\u00141\n\t@\t\n@t\u0015\n= 0: (C2)\nThis then leads us to\n@M\u0017\n@t=2\nSSX\n\u000b;\f;\r =\u0000SRe\u0014\n\u0000i\u0010\u0003\n\u000b\u0010\n^f\u0017\u0011\n\u000b;\f\u0000\f;\r@\u0010\r\n@t\u0000i\u0010\u0003\n\u000b\u0010\n^f\u0017\u0011\n\u000b;\f\u0000\f;\r\u0010\r1\n\t@\t\n@t\u0000i1\n~\t\u0010\u0003\n\u000b\u0010\n^f\u0017\u0011\n\u000b;\fH\f;\r\t\u0010\r\u0015\n\u00002Re\u0014\nM\u00171\n\t@\t\n@t\u0015\n: (C3)\nFor scalar \u0000, \u0000 \u000b;\f!\u0000\u000e\u000b;\f, the equation above becomes Eq. (B7).\nFrom Eqs. (C2) and (C3), one concludes that the stationary solution M\u0017of Eq. (C3) is independent of \u0000. In other\nwords, whether \u0000 depends on spin indices or not, the SW model (29) is left una\u000bected, also see the discussion in\nSection V of the main text.\nAppendix D: Description of magnetostriction\nFor a dipolar spinor BEC without quadratic Zeeman term, when there is no dissipation (\u0000 = 0), the mean-\feld\nequation in Eq. (3) can be written as\n\u0016\u000b(t) \u000b(r;t) =8\n<\n:\u0000~2\n2mr2+Vtr(r) +c0SX\n\f=\u0000Sj \f(r;t)j29\n=\n; \u000b(r;t)\u0000~fb\u0000bdd(r;t)g\u0001SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f \f(r;t)\n+SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSX\n\u000b1;\f1;\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b1;\f1\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f \u0003\n\u000b1(r;t) \f1(r;t) \f(r;t):\n(D1)\nwhere we have substituted i~@ \u000b(r;t)\n@t=\u0016\u000b(t) \u000b(r;t).\nSince we consider the homogeneous-local-spin-orientation limit, we may write  \u000b(r;t) = \t uni(r;t)\u0010\u000b(t). In this\nlimit, we have\nj (r;t)j2:= y(r;t) (r;t) =SX\n\u000b=\u0000S y\n\u000b(r;t) \u000b(r;t) =j\tuni(r;t)j2; (D2)\nsinceSX\n\u000b=\u0000Sj\u0010\u000b(t)j2= 1 from the de\fnition of \u0010\u000b(t) in Eq. (7). Thus j\tuni(r;t)j2is equal to the number density.\nThen Eq. (D1) can be written as\n\u0016\u000b(t)\u0010\u000b(t) \tuni(r;t) =\u001a\n\u0000~2\n2mr2+Vtr(r) +c0j\tuni(r;t)j2\u001b\n\u0010\u000b(t) \tuni(r;t)\n\u0000~fb\u0000bdd(r;t)g\u0001SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\u0010\f(t) \tuni(r;t)\n+SSX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSX\n\f=\u0000SM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t)\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f(t)j\tuni(r;t)j2\tuni(r;t):\n(D3)16\nNow, we decompose the chemical potential \u0016(t) as\u0016(t):=SX\n\u000b=\u0000S\u0016\u000b(t)j\u0010\u000b(t)j2. Then one obtains\n\u0016(t) \tuni(r;t) =\"\n\u0000~2\n2mr2+Vtr(r) +(\nc0+S2SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t))\nj\tuni(r;t)j2#\n\tuni(r;t)\n+ [\bdd(r;t)\u0000S~fb\u0001M(t)g] \tuni(r;t);\n(D4)\nwhere\n\bdd(r;t):=S2cdd2\n4Z\nd3r08\n<\n:X\n\u0017;\u00170=x;y;zM\u0017(t)Q\u0017;\u00170(r\u0000r0)M\u00170(t)9\n=\n;j\tuni(r0;t)j23\n5; (D5)\nis the dipole-dipole mean-\feld potential [41] following from the de\fnition of bddbelow Eq. (3) in the main text.\nDue toMx(t) = sin\u0012(t) cos\u001e(t),My(t) = sin\u0012(t) sin\u001e(t), andMz(t) = cos\u0012(t), from Eqs. (A4) and (A6), we\nhave\nX\n\u0017;\u00170=x;y;zM\u0017(t)Q\u0017;\u00170(\u0011)M\u00170(t) =\u00001\n\u00113r\n6\u0019\n5hn\nY2\n2(e\u0011)e\u00002i\u001e(t)+Y\u00002\n2(e\u0011)e2i\u001e(t)o\nsin2\u0012(t)\n\u0000n\nY1\n2(e\u0011)e\u0000i\u001e(t)\u0000Y\u00001\n2(e\u0011)ei\u001e(t)o\nsinf2\u0012(t)gi\n+1\n\u00113r\n6\u0019\n5Y0\n2(e\u0011)r\n2\n3\b\n3 sin2\u0012(t)\u00002\t\n; (D6)\nwhereYm\nl(e\u0011) are the usual spherical harmonics.\nBy using Eq. (A2), an alternative form of Eq. (D6) can be obtained:\nX\n\u0017;\u00170=x;y;zM\u0017(t)Q\u0017;\u00170(\u0011)M\u00170(t) =X\n\u0017;\u00170=x;y;z\u00112\u000e\u0017;\u00170\u00003\u0011\u0017\u0011\u00170\n\u00115M\u0017(t)M\u00170(t) =\u00112jM(t)j2\u00003f\u0011\u0001M(t)g2\n\u00115\n=\u00112\u00003f\u0011\u0001M(t)g2\n\u00115: (D7)\nThus, \bdd(r;t) can be written as\n\bdd(r;t) =S2cdd\"Z\nd3r0jr\u0000r0j2\u00003f(r\u0000r0)\u0001M(t)g2\njr\u0000r0j5j\tuni(r0;t)j2#\n=S2cdd\"Z\nd3\u0016r0j\u0016r\u0000\u0016r0j2\u00003f(\u0016r\u0000\u0016r0)\u0001M(t)g2\nj\u0016r\u0000\u0016r0j5j\tuni(\u0016r0;t)j2#\n=\u00003\n2S2cddsin2\u0012(t)Z\nd3\u0016\u0011j\tuni(\u0016\u0011+\u0016r;t)j21\n\u0016\u00115h\n\u0016\u00112\u0000\u0016\u00112\nz\u00002f\u0016\u0011xsin\u001e(t)\u0000\u0016\u0011ycos\u001e(t)g2i\n\u00003S2cddsinf2\u0012(t)gZ\nd3\u0016\u0011j\tuni(\u0016\u0011+\u0016r;t)j2\u0016\u0011z\n\u0016\u00115f\u0016\u0011xcos\u001e(t) + \u0016\u0011ysin\u001e(t)g\n+1\n2S2cdd\b\n1\u00003 cos2\u0012(t)\tZ\nd3\u0016\u0011j\tuni(\u0016\u0011+\u0016r;t)j21\n\u0016\u00115\u0000\n3\u0016\u00112\nz\u0000\u0016\u00112\u0001\n: (D8)\nwhere \u0016r:=r=LwithLbeing some length which scales r(so that \u0016ris a dimensionless vector). For example, in\nquasi-1D with trap potential being Eq. (4), L=l?. Note that, in the special case where M(t) =Mz(t)ez, the form\nof Eq. (D8) becomes identical to Eq.(6) in Ref. [40].\nSince we concentrate on quasi-1D gases, with trap potential given by Eq. (4) in the main text, we will explicitly\ncompute the form of \b dd(r;t) for the quasi-1D setup. By writing\nj\tuni(r;t)j2=e\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2; (D9)17\n-20 -10 10 20\n-0.4-0.20.20.40.60.81.0\n-60 -40 -20 20 40 60\n-0.4-0.20.20.40.60.81.0\nFIG. 6. Scaled dipole-dipole mean-\feld potential \u0016\bdd(z) as a function of zfor a quasi-1D box trap. (Left) Lz=l?= 10.\n(Right)Lz=l?= 30.\nand integrating out xandydirections, one can get the quasi-1D dipole-dipole-interaction mean-\feld potential \b dd(z;t)\nas follows (which is in Eq. (38)):\n\bdd(z;t) =cdd\n2l2\n?S2\b\n1\u00003M2\nz(t)\t\u001aZ1\n\u00001d\u0016zj\t (z+ \u0016zl?;t)j2G(j\u0016zj)\u00004\n3j\t (z;t)j2\u001b\n: (D10)\nNow, let us consider box trap in quasi-1D case, i.e. V(z) = 0 forjzj\u0014LzandV(z) =1forjzj>LzwhereV(z)\nis in Eq. (4). Then we may write\nj\t (z;t)j2=2\n64N\n2Lzforjzj\u0014Lz,\n0 forjzj>Lz,(D11)\nsinceV(z) = 0 for\u0000Lz\u0014z\u0014Lz. Thus, \b dd(z;t) can be written as\n\bdd(z;t) =2\n666664\u0016\bdd(t)(Z(Lz\u0000z)=l?\n\u0000(Lz+z)=l?d\u0016zG(j\u0016zj)\u00004\n3)\nforjzj\u0014Lz,\n\u0016\bdd(t)Z(Lz\u0000z)=l?\n\u0000(Lz+z)=l?d\u0016zG(j\u0016zj) forjzj>Lz,(D12)\nwhere \u0016\bdd(t):=NcddS2\b\n1\u00003M2\nz(t)\t\n=\u0000\n2Lzl2\n?\u0001\n. \bdd(z;t) is discontinuous at z=\u0006Lzbecause of the sudden change\nof the density at the boundary ( z=\u0006Lz) due to box trap potential.\nDe\fning the scaled density-density mean-\feld potential \u0016\bdd(z):= \bdd(z;t)=\u0016\bdd(t), we obtain Fig. 6, for two\ndi\u000berent axial extensions, Lz=l?= 10 and 30. As Fig. 6 clearly illustrates, in a box-trapped quasi-1D gas, \b dd(z;t)\nbecomes approximately constant for jzj<LcandLc!LzforLz=l?\u001d1. Depending on the value of M(t), \bdd(r;t)\nwill introduce either a repulsive or an attractive force. This force will however exist only near the boundary for a box\ntrap, where it can lead to a slight modi\fcation of the density of atomes. Its relative in\ruence decreases with increasing\nextension of the trapped gas along the zaxis, and can therefore be consistently neglected in the approximation of\nconstant particle-density.\nHowever, to assess whether signi\fcant magnetostriction occurs, one has to consider, in addition to \b dd, the trap\npotentialVtrand the `quasi' density-density interaction mean \feld potential \b 0de\fned as\n\b0(r;t):=(\nc0+S2SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t))\nj\tuni(r;t)j2: (D13)\nWe can coin \b 0(r;t) a `quasi' density-density interaction mean \feld potential because only c0is a density-density\ninteraction coe\u000ecient ( c2kare interaction coe\u000ecients parametrizing the spin-spin interactions for spin- Sgas wherek\nis an integer with 1 \u0014k\u0014S. For example, c2is the spin-spin interaction coe\u000ecient of a spin-1 gas). In our quasi-1D\ncase, this \b 0(r;t) potential is \b 0(z;t) where\n\b0(z;t):=(\nc0\n2\u0019l2\n?+S2SX\nk=1c2k\n2\u0019l2\n?X\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t))\nj\t (z;t)j2: (D14)18\nIn the main text, we assume that c0\u001dS2PS\nk=1c2kP\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t). For spin-123Na or87Rb,S= 1\nandc0'100jc2j[31, 34], so this is an appropriate assumption (note thatPS\nk=1P\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t) = 1).\nThe values of the c2kare not yet established for166Er. We therefore tacitly assume in the main text, when calculating\nconcrete numerical examples for166Er, that the above condition also still holds, despite the prefactor S2enhancing\nthe importance of spin-spin interactions in \b 0(z;t). 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Karis1\n1Department of Physics and Astronomy,\nUppsala University, Box 516, SE- 75120 , Uppsala, Sweden\n2School of Science and Technology,\nOrebro University, SE- 70182 Orebro, Sweden\n3Department of Materials Science and Engineering,\nUppsala University, Box 534, SE- 75121 , Uppsala, Sweden\nAbstract\nWe report on optically induced, ultrafast magnetization dynamics in the Heusler alloy Co 2FeAl,\nprobed by time-resolved magneto-optical Kerr e\u000bect. Experimental results are compared to results\nfrom electronic structure theory and atomistic spin-dynamics simulations. Experimentally, we \fnd\nthat the demagnetization time ( \u001cM) in \flms of Co 2FeAl is almost independent of varying structural\norder, and that it is similar to that in elemental 3d ferromagnets. In contrast, the slower process\nof magnetization recovery, speci\fed by \u001cR, is found to occur on picosecond time scales, and is\ndemonstrated to correlate strongly with the Gilbert damping parameter ( \u000b). Our results show\nthat Co 2FeAl is unique, in that it is the \frst material that clearly demonstrates the importance of\nthe damping parameter in the remagnetization process. Based on these results we argue that for\nCo2FeAl the remagnetization process is dominated by magnon dynamics, something which might\nhave general applicability.\n1arXiv:2002.12255v1  [cond-mat.mtrl-sci]  27 Feb 2020Studies of ultrafast demagnetization was pioneered by Beaurepaire et al. [1], who demon-\nstrated that the optical excitation of a ferromagnetic material - using a short pulsed laser\ncould quench the magnetic moment on sub-picosecond timescales. The exact underlying mi-\ncroscopic mechanisms responsible for the transfer of angular momentum have been strongly\ndebated for more than 20 years [2{4]. Ultrafast laser-induced demagnetization has now be-\ncome an intense \feld of research not only from fundamental point of view but also from\na technological aspect, due to an appealing possibility to further push the limits of oper-\nation of information storage and data processing devices [5]. Both experiment [4{15] and\ntheory [16{23] report that all of the 3d ferromagnets (Fe, Ni and Co) and their alloys, show\ncharacteristic demagnetization times in the sub-picosecond range, while 4f metals exhibit a\ncomplicated two-step demagnetization up to several picoseconds after the excitation pulse\n[4, 24].\nIn this work, we have made element speci\fc investigations of the ultrafast magnetiza-\ntion dynamics of a half-metallic Heusler alloy. This class of alloys has been investigated\nintensively, especially concerning the magnetic properties, ever since the discovery in 1903,\nwhen Heusler et al. reported that alloys like Cu 2MnAl exhibit ferromagnetic properties,\neven though none of its constituent elements was in itself ferromagnetic [25]. The ferro-\nmagnetic properties were found to be related to the chemical ordering [26]. One of the key\nfeatures of several Heusler alloys is their unique electronic structure, where the majority\nspin band-structure has a metallic character while the minority spin band is semiconducting\nwith a band gap. Such materials are also referred to as half-metallic ferromagnets (HMFs)\nand were initially predicted by de Groot et al. [27], based on electronic structure theory.\nHalf-metals ideally exhibit 100% spin-polarization at the Fermi level. This exclusive prop-\nerty makes them candidates to be incorporated in spintronic devices, e.g. spin \flters, tunnel\njunctions and giant magneto-resistance (GMR) devices [28{31]. One of the advantages of\nHeusler alloys with respect to other half-metallic system, like CrO 2and Fe 3O4, are their\nrelatively high Curie temperature ( Tc) and low coercivity ( Hc) [32, 33]. Heusler alloys are\nalso appealing for spintronic applications due to the low Gilbert damping, which allows for\na long magnon di\u000busion length [34{39]. It has been shown that the low value of the Gilbert\ndamping constant is related with the half-metallicity [37, 38]. The origin of the band gap\nand the mechanism of half-metallicity in these materials have been studied by using \frst\nprinciple electronic structure calculations [40{43]. The half-metallic property is furthermore\n2known to be very sensitive to structural disorder [41{45]. From a fundamental point of view,\nit is intriguing to ask, how the band gap in the minority spin channel e\u000bects the ultrafast\nmagnetization dynamics of Heusler alloys [46, 47]. It has already been reported that some\nof the half-metals like CrO 2and Fe 3O4exhibit very slow dynamics, involving time-scales of\nhundreds of picoseconds, [46, 47] while several Co-based Heusler alloys show a much faster\ndemagnetization, similar to the time-scales of the elemental 3d-ferromagnets [48{50]. The\nfaster dynamics of these Heuslers has been discussed in Ref.[46] to be due to the fact that\nthe band gap in the minority spin channel is typically around 0 :3\u00000:5 eV, which is smaller\nthan the photon energy (1 :5 eV) of the exciting laser. It is also smaller than the band gap\nof CrO 2and Fe 3O4. Importantly, the Heusler alloys o\u000ber the possibility to study magneti-\nzation dynamics, as a function of structural order, since they normally can be prepared to\nhave a fully ordered L21phase, a partially ordered B2 phase, and a completely disordered\nA2 phase. The structural relationships of these phases are described in the Supplemental\nMaterial (SM) [51].\nWe have here studied the optically induced, ultrafast magnetization dynamics of Co 2FeAl\n(CFA) \flms, using time-resolved magneto-optical Kerr e\u000bect (TR-MOKE) as described in\nRef. [52]. By control of the growth temperature, CFA alloy forms with varying degree of\nstructural order, in a continuous way between the A2 and B2 phases, as well as between\ntheB2 andL21phases [53, 54]. We present data from four CFA samples, grown at 300 K,\n573 K, 673 K, and 773 K respectively. We henceforth denote each sample by its growth\ntemperature as a subscript, e.g. CFA 300K. As evidenced by X-ray di\u000braction, the sample\ngrown at 300 K is found to exhibit the A2 phase, while the samples grown at 573 K and\n673 K predominantly exhibit the B2 phase. The sample grown at 773 K is found to exhibit a\npure B2 phase [54]. The value of the Gilbert damping \u000bis found to monotonously decrease\nwith annealing temperature and is thus lowest for the sample grown at 773 K [55].\nCalculations based on density functional theory (DFT) of the magnetic moment, Heisen-\nberg exchange interaction and the Gilbert damping parameter are described in detail in\n(SM) [51]. These parameters were used in a multiscale approach to perform atomistic mag-\nnetization dynamics simulations, described in Sec.S1 of (SM) [51]. Here we employed the\ntwo temperature model (2TM) for the temperature pro\fle of the spin-system. In the 2TM,\nthe spin temperature increases due to the coupling to the hot-electron bath, that is excited\nby the external laser pulse. In the simulations we used a peak temperature in the 2TM of\n30.70.750.80.850.90.951.01.051.1M/M(0)05 1 0 1 5A2CoFe05 1 0 1 5t( p s )B2\n05 1 0 1 5 2 0L21FIG. 1. (Color online) Simulations of ultrafast dynamics of Co 2FeAl in the di\u000berent structural\nphasesA2 (left panel), B2 (central panel), and L21(right panel). The demagnetization is shown\nelement resolved (blue line - Co, red line - Fe). The peak temperature is 1200 K. The dotted line\nindicates the equilibrium magnetization at T= 300 K.\n1200 K. A full description of the 2TM and the details of all spin-dynamics simulations are\ndescribed in Sec.S2 of SM [51].\nThe results of the simulations are shown in Fig. 1, for the A2,B2 andL21phases. It can\nbe seen that the di\u000berent phases react di\u000berently to the external stimulus. In general, this\nmodel provides a dynamics that is controlled by i)the temperature of the spin-subsystem,\nii)the strength of the magnetic exchange interaction and iii)the dissipation of angular\nmomentum and energy during the relaxation of the atomic magnetic moments (Gilbert\ndamping) [56]. Before continuing the discussion, we note that the average magnetization,\nM, of element Xis calculated as MX=P\nicX\niMX\ni=4P\nicX\ni, wherecX\niis the concentration of the\nparticular element Xin the particular phase and iruns over the four nonequivalent sites of\nthe unit cell. After the material demagnetizes, the spin temperature eventually drops and\nthe average magnetization returns to its initial value after 10 \u000020 ps (cf. Fig. 1).\nTo estimate the time constants of the demagnetization \u001cMand remagnetization ( \u001cR)\n4processes, in an element-speci\fc way, we \ft both the theoretical and experimental transient\nmagnetizations by a double exponential function [57]. We show results of \u001cMand\u001cRin\nFig. 2 for the A2 and B2 phase, as well as for alloys with intermediate degree of disorder\n(described in Sec.S2 of SM [51]).\n4.5\n4.0\n3.5\n3.0\n2.5\n2.0R (ps)\n0 20 50 80 100\nAmount of B2 order [%] Fe\n Co\n Remagnetization (b)2.0\n1.5\n1.0\n0.5\n0.0M (ps)\n Fe\n CoDemagnetization (a)\nFIG. 2. (Color online) Element resolved relaxation times of Co 2FeAl, from simulations of alloys\nwith varying amounts of A2!B2 phase. 0 corresponds to pure A2 phase while 100 corresponds to\npureB2 phase. Panel (a) shows the demagnetization time and panel (b) shows the remagnetization\ntime. Both time constants are obtained from \ftting the time trajectory of MX(t) by a double-\nexponential function (see text).\nThe theoretical demagnetization time is seen from Fig. 2 to typically be around 1 ps,\nwhereas the remagnetization time is 2 \u00005 ps. Going from the A2 to the B2 alloy, both\ntimes increase, albeit the simulations show a stronger increase of the remagnetization time\n5as function of alloy composition. We also note that the relevant time scale is somewhat\nlarger for Fe than for Co, and the ratio between them,\u001cFe=\u001cCo, grows when going from A2 to\nB2 phase.\nFigures 3 (a-d) shows the measured magnetization dynamics of CFA \flms that were\ngrown at di\u000berent temperatures (see SM, Sec.S3 for thin \flms synthesis, and Sec.S4 for\ndetails on the experimental measurements [51]). The inset shows the observed magnetization\ndynamics up to\u00181 ps. For all samples, the data for Fe (red) and Co (blue) show similar\ndemagnetization dynamics in the \frst few hundred femtoseconds, whereupon di\u000berences in\nthe magnetization dynamics become visible, especially on the picosecond timescale.\n1.000.950.900.850.800.75 Normalized Asymmetry (arb.units)1086420 Time Delay (ps) A2(a)\n1.000.950.900.850.800.75A/A00.80.40.0FeCo1086420 Time Delay (ps) 90% B2(b)\n1.000.950.900.850.800.75A/A00.80.40.0FeCo  1086420 Time Delay (ps) 90% B2(c)\n1.00.90.80.7A/A00.80.40.0 Fe Co1086420 Time Delay (ps) 100% B2(d)\n1.00.90.80.7A/A00.80.40.0FeCo\nFIG. 3. Measured element-speci\fc Fe (red) and Co (blue) magnetization dynamics of Co 2FeAl.\nSamples are denoted by the growth temperature in each case. The red and blue lines correspond\nto \ftted data (see text). (a) 300 K (100% A2phase), (b) 573 K (90% B2 phase), (c) 673 K (90%\nB2 phase), and (d) 773 K (100% B2 phase). The insets show the demagnetization dynamics up to\n\u00181 ps. All of the measurements were performed with similar pump-\ruence (for details, see Sec.S4\nof SM).\nFigure 4 (a-b) shows the measured values of the demagnetization and remagnetization\ntime constants, for the four di\u000berent growth temperatures, representing di\u000berent degree of\ndisorder in Co 2FeAl, along the alloy path A2!B2. It may be seen that the \u001cMfor Fe\nand Co is the same within the error bars for all four samples, regardless of the degree of\nstructural ordering (Fig. 4a). It may also be noted that the measured \u001cMfor CFA is similar\nto that of 3d transition metals [48, 49] and very much shorter than that of CrO 2or Fe 3O4.\nDemagnetization times that are independent on degree of structural ordering is interest-\n6ing, since it can be expected that the presence of structural disorder in Heusler alloys ought\nto result in a lower degree of spin polarization of the electronic states (i.e. an increased\ndensity of states (DOS) at the Fermi level in the minority band). This is expected to en-\nhance spin-\rip scattering, with an accompanying speed-up of the demagnetization dynamics\n[46, 47]. The electronic structure calculation of CFA also shows that the DOS at the Fermi\nlevel varies with di\u000berent structural phases (analyzed in the Sec.S1 of SM [51]). The A2\nphase has a large number of states at the Fermi level, while the L21phase, and to some\nextent the B2 phase, has a low amount [54]. Despite these di\u000berences in the electronic struc-\nture, the measured demagnetization dynamics shown in Fig. 4(a) is essentially independent\non degree of structural ordering.\nOn longer time-scales, there is a signi\fcant e\u000bect of structural ordering on the observed\nmagnetization dynamics, which becomes particularly relevant for the remagnetization pro-\ncess. As seen in Fig. 4b, there is a monotonous increase of remagnetization time, \u001cR, with\nincreasing growth temperature and hence the degree of ordering along the A2!B2 path.\nThe sample grown at 300 K with A2 phase, exhibits the fastest remagnetization dynam-\nics (\u001cR). With increasing growth temperature and corresponding increase in the structural\nordering along the A2!B2 path, a distinct trend of increasingly slower remagnetization\ndynamics is observed.\nThe most conspicuous behaviour of the measured magnetization dynamics, and its depen-\ndence on the degree of ordering, concerns the remagnetization time (Fig. 4b). The time-scale\nof the remagnetization process is su\u000eciently long to allow for an interpretation based on\natomistic spin-dynamics. Two materials speci\fc parameters should be the most relevant to\ncontrol this dynamics; the exchange interaction, as revealed by the local Weiss \feld, and\nthe damping parameter. In the Sec.S2 of SM [51], we report on the calculated Weiss \felds\nand damping parameters. It is clear from these results that the trend in the experimental\ndata shown in Fig. 4b, can not be understood from the Weiss \feld alone, whereas an expla-\nnation based on the damping is more likely. In order to illustrate this, we show in Fig. 5\nthe inverse of the measured remagnetizatiom time compared to the theoretically calculated\ndamping and experimental measured damping through ferromagnetic resonance (FMR) (de-\nscribed in Sec.S6 of SM)[51]. The \fgure shows that the damping is large in the completely\ndisordered A2 phase and for a large range of structural orderings, which comes out from\nboth theory and experiment. The \fgure also demonstrates that the inverse of the measured\n7300250200150100500 τM (fs)  Fe Co(a)\n3.02.52.01.5 τR (ps)CFA300 K CFA573 K CFA673 K CFA773 K (b)FeCo FIG. 4. Measured magnetization times for the investigated Co 2FeAl alloys. In (a) the demagneti-\nzation time, \u001cM, is shown and in (b) the remagnetization time, \u001cR, is plotted.\nremagnetization time scales very well with both the calculated damping and experimentally\nmeasured damping . According to the \fgure, a large damping parameter corresponds to\nfaster remagnetization dynamics in the measurements.\nCo2FeAl is, to the best of our knowledge, the \frst system where experimental observations\nand theory point to the importance of damping in the process of ultrafast magnetization\ndynamics. We note that this primarily is relevant for the remagnetization process; the\ninitial part of the magnetization dynamics (\frst few hundred fs) is distinctly di\u000berent. In\nthe demagnetization we observe a similar behaviour for Fe and Co in all samples, and\nan insensitivity of the demagnetization times in relation to structural ordering. Also, the\n80.550.500.450.400.350.300.25 (τr )-1 (1/ps) 43210Damping ( x 10-3)\n100806040200Amount of B2 ordering (%)Remagnetization timeTheoretical dampingExperimental dampingFIG. 5. The relationship of inverse of the measured remagnetization time (right y-axis) and the-\noretically calculated and experimentally measured Gilbert damping (left y-axis) in Co 2FeAl for\nvarying amount of B2 order along the A2!B2 path, i.e. 0 corresponds to pure A2 phase while\n100 corresponds to pure B2 phase.\nmeasured and theoretical demagnetization times evaluated from atomistic spin-dynamics\nsimulations, do not agree. Other mechanisms, of electronic origin, most likely play role in\nthis temporal regime.\nThe remagnetization process of Co 2FeAl alloys with varying degree of structural order,\nhighlights clearly the importance of the Gilbert damping and that magnon dynamics domi-\n9nates the magnetization at ps time-scales. The relevance of the Gilbert damping parameter\nfor ps dynamics is natural, since this controls angular momentum (and energy) transfer to\nthe surrounding. What is surprising with Co 2FeAl is the fact that other interactions (e.g.\nthe Weiss \feld) show such a weak dependence on the amount of structural diorder. This\nis fortuitous, since it allows to identify the importance of the Gilbert damping. A picture\nemerges from the results presented here, that the magnetization dynamics in general have\ntwo regimes; one which is primarily governed by electronic processes, and is mainly active\nin the \frst few hundered fs ( \u001cM), and a second regime where it is primarily magnons that\ngovern the remagnetiztion dynamics ( \u001cR).\nWe acknowledge support from the Swedish Research Council (VR, contracts 2019-\n03666,201703799, 2016-04524 and 2013-08316), the Swedish Foundation for Strategic Re-\nsearch, project SSF Magnetic materials for green energy technology under Grant No. EM16-\n0039, the Knut and Alice Wallenberg foundation, STandUP and eSSENCE, for \fnancial\nsupport. 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[54] the room temperature FMR data does not show the trend we are reporting here.\nThis has later been found to be related to the limited frequency range studied in that work. For\nthis work we have repeated the FMR measurements with a more extended frequency range,\nand now \fnd the same trends at room temperature as at low temperatures, i.e., Gilbert\ndamping decreases monotonously with increasing growth temperature.\n[56] O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik, Atomistic Spin Dynamics , Founda-\ntions and Applications (Oxford University Press, 2016).\n[57] U. Atxitia, O. Chubykalo-Fesenko, J. Walowski, A. Mann, and M. M unzenberg, Phys. Rev. B\n81, 174401 (2010).\n13"    },    {        "title": "2004.01245v2.Stable_solitons_in_a_nearly_PT_symmetric_ferromagnet_with_spin_transfer_torque.pdf",        "content": "arXiv:2004.01245v2  [nlin.PS]  6 Apr 2020Stable solitons in a nearly PT-symmetric ferromagnet with spin-transfer torque\nI V Barashenkov∗\nCentre for Theoretical and Mathematical Physics, Universi ty of Cape Town,\nRondebosch 7701, South Africa and Joint Institute for Nucle ar Research, Dubna, Russia\nAlexander Chernyavsky†\nCentre for Theoretical and Mathematical Physics, Universi ty of Cape Town,\nRondebosch 7701, South Africa and Department of Mathematic s and Statistics,\nUniversity of Victoria, Victoria, BC V8P 5C2, Canada\nWe consider the Landau-Lifshitz equation for the spin torqu e oscillator — a uniaxial ferromagnet\nin an external magnetic field with polarised spin current dri ven through it. In the absence of the\nGilbert damping, the equation turns out to be PT-symmetric. We interpret the PT-symmetry\nas a balance between gain and loss — and identify the gaining a nd losing modes. In the vicinity\nof the bifurcation point of a uniform static state of magneti sation, the PT-symmetric Landau-\nLifshitz equation with a small dissipative perturbation re duces to a nonlinear Schr¨ odinger equation\nwith a quadratic nonlinearity. The analysis of the Schr¨ odi nger dynamics demonstrates that the spin\ntorque oscillator supports stable magnetic solitons. The PTnear-symmetry is crucial for the soliton\nstability: the addition of a finite dissipative term to the La ndau-Lifshitz equation destabilises all\nsolitons that we have found.\nKeywords: spin torque oscillator; Landau-Lifshitz equati on; parity-time symmetry; nonlinear Schroedinger\nequation; solitons; stability\nI. INTRODUCTION\nConceived in the context of nonhermitian quantum me-\nchanics [1], the idea of parity-time ( PT) symmetry has\nproved to be useful in the whole range of applied disci-\nplines [2]. A PT-symmetric structure is an open system\nwhere dissipative losses are exactly compensated by sym-\nmetrically arranged energy gain. In optics and photon-\nics, systems with balanced gain and loss are expected to\npromote an efficient control of light, including all-optical\nlow-threshold switching [3, 4] and unidirectional invisi-\nbility [4–6]. There is a growing interest in the context\nof electronic circuitry [7], plasmonics [8], optomechanical\nsystems [9], acoustics [10] and metamaterials [11].\nThis study is concerned with yet another area where\nthe gain-loss balance gives rise to new structures and\nbehaviours, namely, the magnetism and spintronics. In\ncontrast to optics and nanophotonics, where the nonher-\nmitian effects constitute a well-established field of study,\nthe research into PT-symmetric magnetic systems is still\nin its early stages, with only a handful of models set up\nover the last several years.\nOne of the systems proposed in the literature com-\nprises two coupled ferromagnetic films, one with gain\nand the other one with loss [12]. (For an experimen-\ntal implementation of this structure, see [13].) A re-\nlated concept consists of a pair of parallel magnetic\nnanowires, with counter-propagating spin-polarized cur-\nrents [14]. In either case the corresponding mathematical\nmodel is formed by two coupled Landau-Lifshitz equa-\n∗Igor.Barashenkov@uct.ac.za\n†chernya@uvic.cations, with PTsymmetry being realised as a symme-\ntry between the corresponding magnetisation vectors. A\ntwo-spin Landau-Lifshitz system gauge-equivalent to the\nPT-symmetric nonlocal Schr¨ odinger equation is also a\nmember of this class of models [15].\nAn independent line of research concerned the dynam-\nics of a single spin under the action of the spin-transfer\ntorque. Projecting the magnetisation vector onto the\ncomplex plane stereographically and modelling the spin\ntorque by an imaginary magnetic field [16], Galda and\nVinokur have demonstrated the PT-symmetry of the re-\nsulting nonhermitian Hamiltonian [17]. (For the gener-\nalisation to spin chains, see [18]; the nonreciprocal spin\ntransfer is discussed in [19].) Unlike the two-component\nstructures of Refs [12–14], the PT-symmetry of the spin\ntorque oscillator of Galda and Vinokur is an intrinsic\nproperty of an individual spin. It results from the sys-\ntem’s invariance under the simultaneous time reversal\nand the imaginary magnetic field flip [17].\nThe structure we consider in this paper shares a num-\nber of similarities with the spin torque oscillator of Refs\n[17, 18]. (There is also a fair number of differences.) It\nconsists of two ferromagnetic layers separated by a con-\nducting film (Fig 1). The spin-polarised current flows\nfrom a layer with fixed magnetisation to a layer where\nthe magnetisation vector is free to rotate [16, 20].\nA one-dimensional uniaxial classical ferromagnet in\nthe external magnetic field is described by the Landau-\nLifshitz equation [16, 20] (also known as the Landau-\nLifshitz-Gilbert-S/suppress lonczewski equation in the current con-\ntext):\n˙M=−M×M′′−M×H−β(M·ˆz)M׈z\n−γM×M׈z+λM×˙M. (1)2\nFIG. 1. A schematic of the spin torque oscillator. An electri c\ncurrent flows through a nanowire with two ferromagnetic lay-\ners. In the thick layer (on the left) the magnetisation is fixe d\n(through large volume, large anisotropy or pinning by addi-\ntional underlayers). This causes a polarisation of the pass ing\nelectron spins. The polarised current exerts torque on the\nthin layer (on the right) where the magnetisation is governe d\nby the Landau-Lifshitz equation (1).\nHere the overdot stands for the time derivative and the\nprime indicates the derivative with respect to x. In equa-\ntion (1), the variables have been non-dimensionalised so\nthat the magnetisation vector M= (Mx,My,Mz) lies on\na unit sphere: M2= 1. The magnetic field is taken to\nbe constant and directed horizontally: H= (H0,0,0).\nThe anisotropy axis is z, with ˆz= (0,0,1). The positive\nand negative constant βcorresponds to the easy-axis and\neasy-plane anisotropy, respectively. (Note that the au-\nthors of [17, 18] considered the ferromagnet anisotropic\nalong thexaxis.) The fourth term in the right-hand side\nof (1) — the S/suppress lonczewski term — accounts for the spin\ntransfer by the current that passes through an external\nferromagnetic layer that has a fixed magnetisation in the\ndirection ˆz. The last term is the Gilbert damping term.\nThe damping coefficient λis positive; the field H0and the\ncurrent amplitude γcan also be chosen positive without\nloss of generality.\nIn this paper, we study the nonlinear dynamics of the\nlocalised solutions of the equation (1), both with small\nand finite-strength damping.\nA class of soliton solutions of the Landau-Lifshitz equa-\ntion (1) was obtained by Hoefer, Silva and Keller [21].\nThe solitons discovered by those authors are dissipative\nanalogs of the Ivanov-Kosevich magnon droplets [22].\n(For the experimental realisation, see [23].) Our setup\nhas a different geometry from the one of Hoefer et al.\nOne difference is that we consider the magnetic layer with\na parallel anisotropy while the magnon droplets require\na perpendicular one [21–23]. An additional distinction\nis that our vector His orthogonal to the direction of\nthe fixed magnetisation — while the magnetic field in\nRef [21] was not. Because of the different geometry, the\nLandau-Lifshitz equation of Ref [21] does not exhibit the\nPTinvariance. The dissipative magnon droplets are sus-\ntained through the competition of torque and damping,\nthe two actors represented by terms of different mathe-\nmatical form, rather than by a symmetric balance of twosimilar but oppositely-directed effects.\nAnother class of localised structures in the spin torque\noscillator is commonly referred to as the standing spin\nwave bullets . These have been theoretically predicted by\nSlavin and Tiberkevich [24] — outside the context of the\nLandau-Lifshitz equation. (For the experimental realisa-\ntion, see [25].) The spin wave bullets are found in the\nmagnetic layer with parallel anisotropy, when the mag-\nnetic field is directed parallel to the fixed layer’s mag-\nnetisation. The direction of the vector His what makes\nour geometry different from the setup considered in Ref\n[24]. Like the magnon droplets, the spin wave bullets are\nsustained by an asymmetric balance of the spin torque\nand finite-strength damping.\nThe paper is organised as follows. We start with the\ndemonstration of the gain-loss balance in the Landau-\nLifshitz equation with the vanishing Gilbert damping.\nThisPT-symmetric system and systems that are close\nto it will prove to have special properties in this paper,\nwhere we consider equations both with small and finite λ.\nIn section III we classify stability and bifurcation of four\nnonequivalent stationary states with uniform magnetisa-\ntion. Three of those states are found to be admissible\nas stable backgrounds for localised structures. In the\nvicinity of the bifurcation points, the dynamics of the lo-\ncalised structures are governed by quadratic Schr¨ odinger\nor Ginsburg-Landau equations, depending on whether\nthe Gilbert damping is weak or finite-strength (section\nIV). Despite the absence of the dissipative terms, our\nquadratic Schr¨ odinger equations are not conservative;\nhowever one of them obeys the PT-symmetry. Both\nGinsburg-Landau equations and their Schr¨ odinger coun-\nterparts — PT-symmetric or not — support two types\nof soliton solutions. We show that either of these types\nis only stable in the PT-symmetric situation (sections\nV-VI). Section VII summarises results of this study.\nII. GAIN-LOSS BALANCE IN THE ABSENCE\nOF GILBERT LOSSES\nThe equation (1) is nonconservative due to the pres-\nence of the spin torque and Gilbert’s dissipative term. In\nspin torque oscillators, solitons are expected to exist due\nto the energy supplied by torque being offset by finite-\nstrength dissipation [21]. However when λ= 0, the spin\nhamiltonian modelling our structure is PT-symmetric\n[17] and therefore some form of the gain-loss balance\nshould occur in this case as well, despite the absence of\nthe Gilbert damping. To uncover the gain-loss competi-\ntion intrinsic to the spin torque, we define two complex\nfields,u(x,t) andv(x,t), related to the magnetisation\nvectorMvia the Hopf map:\nMx=v∗u+u∗v, M y=i(u∗v−v∗u), Mz=|u|2−|v|2.\n(2)\nWhen the magnetisation is spatially uniform, ∂M/∂x=\n0, the equation (1) with λ= 0 can be reformulated as a3\nnonlinear Schr¨ odinger dimer:\niut+H0\n2v+β\n2(|u|2−|v|2)u=iγ|v|2u, (3)\nivt+H0\n2u−β\n2(|u|2−|v|2)v=−iγ|u|2v. (4)\nAccording to equations (3)-(4), the external energy is fed\ninto theu-mode and dissipated by its vcounterpart. The\nmagnetic field H0couplesutov, carrying out the energy\nexchange between the two modes.\nThe sustainability of the gain-loss balance in the sys-\ntem (3)-(4) is reflected by its invariance under the prod-\nuct of the PandTtransformations. Here the inversion\nPswaps the two modes around,\nP:u→v, v→u, (5)\nwhileTrepresents the reflection of time:\nT:t→ −t, u→u∗, v→v∗. (6)\nThese transformations admit a simple formulation in\nterms of the components of magnetisation (2):\nP:My→ −My, Mz→ −Mz (7)\nand\nT:t→ −t, M y→ −My. (8)\nThe involutions (7) and (8) remain relevant in the anal-\nysis of the equation (1) with the x-dependent magneti-\nsation. Here one can either leave the parity operation in\nthe form (7) or include the inversion of the xcoordinate\nin this transformation:\nP:x→ −x, M y→ −My, Mz→ −Mz. (9)\nWriting the vector equation (1) in the component form,\n\n\n˙Mx=MzM′′\ny−MyM′′\nz−βMyMz\n−γMxMz+λ(My˙Mz−Mz˙My),\n˙My=MxM′′\nz−MzM′′\nx−H0Mz+βMxMz\n−γMyMz+λ(Mz˙Mx−Mx˙Mz),\n˙Mz=MyM′′\nx−MxM′′\ny+H0My\n+γ(M2\nx+M2\ny) +λ(Mx˙My−My˙Mx),(10)\none readily checks that in the conservative limit ( γ=λ=\n0), the Landau-Lifshitz equation is invariant under the\nP- andT-involutions individually. The equation with\nthe spin torque term added ( γ/ne}ationslash= 0) is invariant under\nthe product ( PT) transformation only. Accordingly, the\nequation with the γ-term is a PT-symmetric extension of\nthe conservative Landau-Lifshitz equation. Finally, the\naddition of the Gilbert damping term ( λ/ne}ationslash= 0) breaks the\nPT-symmetry.III. UNIFORM STATIC STATES\nThe uniform static states are space- and time-\nindependent solutions of equation (1) satisfying M2= 1.\nThese are given by fixed points of the dynamical system\n\n\n˙Mx=−βMyMz−γMxMz+λ(My˙Mz−Mz˙My),\n˙My=(βMx−H0−γMy)Mz+λ(Mz˙Mx−Mx˙Mz),\n˙Mz=H0My+γ(M2\nx+M2\ny)+λ(Mx˙My−My˙Mx)\n(11)\non the surface of the unit sphere.\nOnce a fixed point M(0)has been determined, we let\nM=M(0)+δM, linearise the system (10) in δM, and\nconsider solutions of the form\nδM=meµt−ikx, (12)\nwherem= (mx,my,mz)Tis a real constant vector and k\na real wavenumber that may take values from −∞to∞.\nWe call the uniform static state unstable if at least one\nof the roots µof the associated characteristic equation\nhas a positive real part in some interval of k. Otherwise\nthe state is deemed stable.\nSince the equation (1) conserves the quantity M2, the\ndifference between/parenleftbig\nM(0)/parenrightbig2and the square of the vector\nM(0)+δMwill be time-independent:\n∂\n∂t/parenleftBig\n2δM·M(0)/parenrightBig\n= 0.\nSubstituting from (12) and assuming µ/ne}ationslash= 0, this gives\nm·M(0)= 0. (13)\nEquation (13) implies (a) that the time-dependent per-\nturbations of the uniform static states lie on the unit\nsphere; and (b) that the characteristic equation may not\nhave more than two nonzero roots, µ1andµ2. The third\nroot (µ3) has to be zero.\nApart from classifying stability of the uniform static\nstates, it is useful to know which of these solutions can\nserve as backgrounds to static magnetic solitons. To weed\nout a priori unsuitable cases, we set µ= 0 in the char-\nacteristic equation and consider k2as a new unknown\n(rather than a parameter that varies from 0 to ∞). If\nall roots (k2)nof the resulting equation are real positive,\nthere can be no localised solutions asymptotic to the uni-\nform static state M(0)asx→ ±∞ . On the other hand,\nif there is at least one negative or complex root, the solu-\ntionM(0)remains a candidate for solitons’ background.\nA. Equatorial fixed points on the unit sphere\nOne family of time-independent solutions of the system\n(11) describes a circle on the ( Mx,My)-plane:\nM2\nx+/parenleftbigg\nMy+H0\n2γ/parenrightbigg2\n=H2\n0\n4γ2, Mz= 0.4\nFIG. 2. The phase portrait of the dynamical system (11) with\nH0>/radicalbig\nβ2+γ2(a) andH0< γ(b). In (a), two dots on the\nequator of the unit sphere mark the fixed points of the vector\nfield: the western (blue) and eastern point (red). In (b), the\nblue dot indicates the northern and the red dot the southern\nfixed point. Apart from the fixed points, the figures show a\nfewrepresentativetrajectories; physically, thesecorre spondto\nspatially-uniform evolutions of magnetisation. (The port raits\nin (a) and (b) are for λ= 0.)\nImposing the constraint M2= 1 leaves us with just two\nmembers of the family:\nM(0)\nx=±/radicalBig\n1−γ2/H2\n0, M(0)\ny=−γ/H0, M(0)\nz= 0.\n(14)\nIn the system with γ/ne}ationslash= 0, these fixed points are born\nasH0is increased through the value H0=γ. Since the\npoints (14) lie on the equator of the unit sphere, we will\nbe referring to them simply as the equatorial fixed points,\nthe eastern ( M(0)\nx>0) and the western ( M(0)\nx<0) one.\nFig 2(a) depicts the equatorial fixed points in the phase\nportrait of the dynamical system (11).\nLinearising equation (1) about the uniform static state\ncorresponding to an equatorial fixed point, we obtain two\nnonzero stability eigenvalues\nµ1,2=−λ(2K−β)±/radicalbig\nλ2β2−4K(K−β)\n2(1 +λ2),(15a)\nK=k2+H0M(0)\nx.(15b)Making use of (15) it is not difficult to see that in the\neasy-plane or isotropic ferromagnet (i.e. in the situation\nwhereβ≤0), the eastern uniform static state ( M(0)\nx>0)\nis stable irrespective of the choice of γ,λandH0. On\nthe other hand, when the anisotropy is easy-axis ( β >0),\nthe eastern state is stable if\nH0≥/radicalbig\nβ2+γ2 (16)\nand unstable otherwise.\nTo check the suitability of the eastern uniform static\nstate as a background for solitons, we set µ= 0 in\nthe expression (15a); this transforms it into a quadratic\nequation for k2. Whenβ </radicalbig\nH2\n0−γ2, both roots\nof this equation are negative: k2=−/radicalbig\nH2\n0−γ2and\nk2=β−/radicalbig\nH2\n0−γ2. This implies that there is a pair of\nexponentials exp( −ik1,2x) decaying to zero as x→ −∞\nand another pair decaying as x→+∞. Therefore\nthe uniform static state (14) with M(0)\nx>0 can serve\nas a background to solitons for any set of parameters\nβ,γ,H 0,λin its stability domain.\nTurning to the west-point solution ( M(0)\nx<0 in equa-\ntion (14)), a simple analysis of the eigenvalues (15) indi-\ncates that there are wavenumbers ksuch that Re µ >0\nfor any quadruplets of β,γ,H 0andλ. Hence the west-\nern uniform static state is always unstable. We are not\nconsidering it any further.\nB. Latitudinal fixed points\nAnother one-parameter family of constant solutions\nof the equation (1) forms a vertical straight line in the\nMx,My,Mz-space:\nMx=H0β\nβ2+γ2, My=−H0γ\nβ2+γ2,−∞<Mz<∞.\nThe substitution of the above coordinates into M2= 1\nselects two fixed points on the unit sphere:\nM(0)\nx=H0β\nβ2+γ2, M(0)\ny=−H0γ\nβ2+γ2,\nM(0)\nz=±/radicalBigg\n1−H2\n0\nβ2+γ2. (17)\nSince these points lie above and below the equatorial\n(Mx,My)-plane, we will be calling them the latitudinal\nfixed points: the northern ( M(0)\nz>0) and the southern\n(M(0)\nz<0) point. See Fig.2(b).\nIn the anisotropic equation ( β/ne}ationslash= 0) the northern and\nsouthern points are born as H0is decreased through/radicalbig\nβ2+γ2. In this case, there is a parameter interval\nγ < H 0</radicalbig\nβ2+γ2where two pairs of fixed points,\nlatitudinal and equatorial, coexist.\nThe bifurcation diagram for the isotropic equation is\ndifferent. When β= 0, the latitudinal fixed points (17)5\nemerge as the eastern and western points (14) converge\nand split out of the equatorial plane. In this case, there\nis just one pair of uniform static states for any H0/ne}ationslash=γ:\nthe equatorial pair for H0> γ and the latitudinal pair\nforH0<γ.\nThe linearisation of equation (1) about the uniform\nstatic state corresponding to the latitudinal fixed-point\n(17) gives\nµ1,2=−λQ+γM(0)\nz±/radicalBig\n1\n4λ2β2h2−P(P−βh)\n1 +λ2,(18)\nwhere\nP=k2+β−γλM(0)\nz, Q =k2+β/parenleftbigg\n1−h\n2/parenrightbigg\nand\nh=H2\n0\nβ2+γ2. (19)\nA simple analysis demonstrates that when β≥0, the\nnorth-point solution ( M(0)\nz>0 in (17)) is stable regard-\nless of the values of λ≥0,H0andγ >0. As for the easy-\nplane anisotropy ( β < 0), the northern uniform static\nstate is stable only when the inequalities λ≤λcand\nH0≤Hcare satisfied simultaneously. Here\nλc=γ\n|β|√\n1−h\n1−h/2(20)\nand\nHc=/radicalBigg\n2γ(β2+γ2)\nβ2/parenleftBig/radicalbig\nβ2+γ2−γ/parenrightBig\n. (21)\nNote thatHcis smaller than/radicalbig\nγ2+β2; hence the re-\ngionH0≤Hclies entirely within the northern point’s\nexistence domain (defined by the inequality H0</radicalbig\nγ2+β2).\nFinally, we consider the eigenvalues pertaining to the\nsouthern uniform static state ( M(0)\nz<0 in (17)). Our\nconclusion here is that in the isotropic and easy-plane\nferromagnet ( β≤0), this solution is unstable regardless\nof the choice of other parameters. In the easy-axis situ-\nation (β >0), the southern state is stable if λ≥λcwith\nλcas in (20) — and unstable otherwise.\nTo determine the parameter region where the north-\nand south-point solutions can serve as backgrounds for\nsolitons, we set µ= 0 in (18). In each of the two cases,\nthe resulting quadratic equation for k2has two positive\nroots only if β <0 is satisfied along with the inequality\nH0> Hc, whereHcis as in (21). This is the only no-\ngo region for solitons. Outside this region, the quadratic\nequation has either two negative or two complex roots;\nthe corresponding uniform static states can serve as soli-\ntons’ asymptotes.\nThe bottom line is that either of the two latitudinal\nuniform static states is suitable as a background for soli-\ntons in its entire stability domain.C. Summary of uniform static states\nFor convenience of the reader, the stability properties\nof the constant solutions corresponding to the four fixed\npoints are summed up in Table I.\nBefore turning to the perturbations of these uniform\nstatic states, it is worth noting their symmetry proper-\nties. Each of the equatorial states is PT-symmetric in the\nsense that each of these two solutions is invariant under\nthe product of the transformations (9) and (8). In con-\ntrast, neither of the two latitudinal states is invariant; the\nPToperator maps the northern solution to southern and\nthe other way around. The different symmetry proper-\nties of the equatorial and longitudinal solutions will give\nrise to different invariances of equations for their small\nperturbations.\nFixed-point\nβ <0 β= 0 β >0\nsolution:\neastern stable stablestable if\nH0≥/radicalbig\nβ2+γ2\nwestern unstable unstable unstable\nnorthernstable if λ≤λcstable stable\nandH0≤Hc\nsouthern unstable unstablestable\nifλ≥λc\nTABLE I. Stability of four constant solutions of equation (1 ).\nIV. SLOW DYNAMICS NEAR BIFURCATION\nPOINTS\nA. Perturbation of equatorial fixed point\nConsider the eastern point of the pair of equatorial\nfixed points (14):\nM(0)=/parenleftBigg/radicalBigg\n1−γ2\nH2\n0,−γ\nH0,0/parenrightBigg\n. (22)\nWe assume that the parameters β,γ,λandH0lie in the\nstability domain of the uniform static state (22).\nThe plane orthogonal to the vector M(0)is spanned by\nthe vectors\nA= (0,0,1),B=/parenleftBigg\nγ\nH0,/radicalBigg\n1−γ2\nH2\n0,0/parenrightBigg\n.\nThe unit vector Mcan be expanded over the orthonormal\ntriplet{A,B,M(0)}:\nM=ηA+ξB+χM(0).6\nLettingM(x,t)→M(0)asx→ ±∞ , the coefficient fields\nη,ξandχhave the following asymptotic behaviour:\nη→0, ξ→0, χ→1 as|x| → ∞.\nThe complex field Ψ = ξ+iηsatisfies\ni˙Ψ =χΨ′′−Ψχ′′+λ(Ψ ˙χ−χ˙Ψ)−/radicalBig\nH2\n0−γ2Ψ\n+γ(χ−iηξ−1 +η2) +iβηχ, (23)\nwhereχ=/radicalbig\n1−|Ψ|2while the prime and overdot in-\ndicate the derivative with respect to xandt, respec-\ntively. Note that when λ= 0, the equation (23) is PT-\nsymmetric, that is, invariant under a composite transfor-\nmation consisting of three involutions: t→ −t,x→ −x,\nand Ψ→Ψ∗.\nAssume that H0is close to the bifurcation point of the\nuniform static state (22) — that is, H0is slightly greater\nthanγ. In this case, Ψ will depend on a hierarchy of\nslow times Tn=ǫntand stretched spatial coordinates\nXn=ǫn/2x, wheren= 1,3,5,...and the small parameter\nǫis defined by\nǫ2= 1−γ2\nH2\n0.\nIn the limit ǫ→0 the new coordinates become indepen-\ndent so we can write\n∂\n∂t=ǫD1+ǫ3D3+...;\n∂2\n∂x2=ǫ∂2\n1+ 2ǫ2∂1∂3+ǫ3(∂2\n3+ 2∂1∂5) +... ,\nwhereDn=∂/∂Tnand∂n=∂/∂Xn. Assume, in addi-\ntion, that the anisotropy constant βis of orderǫand let\nβ=ǫBwithB=O(1). Considering small ηandξ, we\nexpand\nΨ =ǫψ1+ǫ3ψ3+... .\nSubstituting the above expansions in (23), we equate\ncoefficients of like powers of ǫ. The order ǫ2gives a\nGinsburg-Landau type of equation with a quadratic non-\nlinearity:\n(i+λ)D1ψ−∂2\n1ψ+γ\n2ψ2=−γψ+B\n2(ψ−ψ∗).(24)\n(Hereψis just a short-hand notation for ψ1.)\nNote that in the derivation of (24) we took λto be\nO(1). If we, instead, let λ=O(ǫ), the dissipative term\nwould fall out of the equation (24) and we would end up\nwith a nonlinear Schr¨ odinger equation:\niD1ψ−∂2\n1ψ+γ\n2ψ2=−γψ+B\n2(ψ−ψ∗). (25)\nThe quadratic Schr¨ odinger equation (25) does not have\nthe U(1) phase invariance. However, the equation is PT-\nsymmetric, that is, invariant under the composite map\nt→ −t,x→ −x,ψ→ψ∗. As we will see in section V,\nthis discrete symmetry is enough to stabilise solitons.B. Perturbation of latitudinal fixed points\nChoosing the background in the form of one of the two\nlatitudinal fixed points\nM(0)=/parenleftbiggβ\nH0h,−γ\nH0h,±√\n1−h/parenrightbigg\n, (26)\nwe letM(x,t) approach the same point M(0)asx→ ±∞ .\nIn (26),his defined by the equation (19).\nAs in the previous subsection, we expand the magneti-\nsation vector over an orthonormal basis {A,B,M(0)}:\nM=ηA+ξB+χM(0), (27)\nwhere, this time,\nA=/parenleftbigg\n∓β\nH0/radicalbig\nh(1−h),±γ\nH0/radicalbig\nh(1−h),√\nh/parenrightbigg\nand\nB=/parenleftbiggγ\nH0√\nh,β\nH0√\nh,0/parenrightbigg\n.\nWe assume that H0is close to the bifurcation point\nwhere the northern and southern fixed points are born\n(that is,H0is slightly smaller than/radicalbig\nβ2+γ2) and define\na small parameter ǫ:\nh= 1−ǫ2.\nAs in the analysis of the equatorial fixed points, we let\nβ=ǫB, whereB=O(1). Assuming that the magnetisa-\ntionMis just a small perturbation of M(0), we expand\nthe small coefficients in (27) in powers of ǫ:\nη=ǫη1+ǫ3η3+..., ξ =ǫξ1+ǫ3ξ3+... .\nThe constraint η2+ξ2+χ2= 1 implies then\nχ= 1−ǫ2η2\n1+ξ2\n1\n2+... .\nSubstituting these expansions in the Landau-Lifshitz\nequation (10) and equating coefficients of like powers of\nǫ, the order ǫ2gives\nD1ξ1=∂2\n1η1−λD1η1−γ(η1ξ1±ξ1)\nand\nD1η1=λD1ξ1−∂2\n1ξ1+Bξ1∓γη1−γ\n2(η2\n1−ξ2\n1).\nThe above two equations can be combined into a single\nequation for the complex function ψ=ξ1+iη1:\n(i+λ)D1ψ−∂2\n1ψ+γ\n2ψ2=∓iγψ−B\n2(ψ+ψ∗).(28)\nThe Ginsburg-Landau equation (28) resembles the\nequation (24) governing the dynamics near the equato-\nrial uniform static state; however there is an important\ndifference. Namely, even if we let λ= 0 in (28) [that is,\neven if we assume that the damping is O(ǫ) or weaker\nin the Landau-Lifshitz-Gilbert equation (1)], the result-\ning nonlinear Schr¨ odinger equation will notbecomePT-\nsymmetric. This fact will have important repercussions\nfor the stability of solitons.7\n5\n4\n3\n2\n1\n0\n10\n5\n0-505\n2\n1.5\n1\n0.5\n0\n10\n5\n0-505\nFIG. 3. Instability of the fundamental soliton in the presen ce of damping. This evolution was obtained by the direct nume rical\nsimulation of the equation (30) with b= 0 and λ= 0.1. The initial condition was in the form of the soliton (31) pe rturbed by\na random perturbation within 5% of the soliton’s amplitude. The spatial interval of simulation was ( −58,58); in the plot it\nhas been cut down for visual clarity.\nV. SOLITON EXCITATIONS OF EQUATORIAL\nSTATE\nLetting\nu(x,t) =−1\n3ψ(X1,T1), x =√γ\n2X1, t =γ\n4T1,(29)\nthe Ginsburg-Landau equation (24) is cast in the form\n(i+λ)ut−uxx−6u2=−4u+b(u−u∗), (30)\nwhereb= 2B/γ. (We alert the reader that the scaled\nvariablesxandtdo not coincide with the original xandtof the Landau-Lifshitz equation (1). We are just re-\nemploying the old symbols in a new context here.)\nIn the present section we consider localised solutions of\nthe equation (30) approaching 0 as |x| → ∞ . Regardless\nofλ, the zero solution is stable if b≤2 and unstable\notherwise. This inequality agrees with the stability range\n(16) of the eastern uniform static state within the original\nLandau-Lifshitz equation. (Note that the term bu∗plays\nthe role of the parametric driver in (30) [26]; the above\nstability criterion states that the zero solution cannot\nsustain drivers with amplitudes greater than b= 2.)\nA. Fundamental soliton and its stability\nEquation (30) has a stationary soliton solution:\nus= sech2x. (31)\nTo distinguish it from localised modes with internal\nstructure, we refer to this solution as the fundamental\nsoliton — or simply sech mode . Letting\nu(x,t) =us(x) +ε[f(x) +ig(x)]eµt\nand linearising in small ε, we obtain an eigenvalue prob-\nlem\nµ(g−λf) =Hf, (32a)\n−µ(f−λg) = (H− 2b)g, (32b)\nwith the operator\nH=−d2/dx2+ 4−12 sech2x. (33)The vector eigenvalue problem (32) is reducible to a\nscalar eigenvalue problem of the form\n(H−b+µλ)2g+ (µ2−b2)g= 0.\nThe stability exponents µare roots of the quadratic equa-\ntion\n(E−b+µλ)2+µ2−b2= 0,\nwhereEis an eigenvalue of the operator H:Hy=Ey.\nThe two roots are\nµ(±)=λ(b−E)±/radicalbig\nλ2b2+E(2b−E)\n1 +λ2. (34)\nThe eigenvalues of the P¨ oschl-Teller operator (33) are\nE0=−5,E1= 0, and E2= 3, with the eigen-\nfunctionsy0= sech3x,y1= sech2xtanhxandy2=8\n1.5\n1\n0.5\n02 \u0000\n15\n105\n-505\n0\n0.1\n0.05\n0\n-0.05\n-0.1\u0001 \u0002\n15\n10\n5\n0-505\nFIG. 4. The evolution of the initial condition in the form of a gaussian, u(x,0) = exp( −x2), in the equation (30) with λ= 0\nandb= 0. Left panel: Re u; right panel: Im u. The emerging solution is a breather with a small imaginary p art and the real\npart close to the soliton (31). Note that the figure shows only a portion of the full simulation interval ( −58,58).\nsechx/parenleftbig\n1−5\n4sech2x/parenrightbig\n, respectively. The continuous spec-\ntrum occupies the semiaxis Econt≥4, with the edge\neigenfunction given by y3= tanhx/parenleftbig\n1−5\n3tanh2x/parenrightbig\n. For\neach eigenvalue En,n= 0,1,2, equation (34) yields two\nroots,µ(+)\nnandµ(−)\nn.\nIn the analysis of the roots (34) we need to distinguish\nbetween two situations: damped ( λ>0) and undamped\none (λ= 0). Assume, first, that λ >0 and let, in addi-\ntion,b≥0. It is not difficult to check that the root µ(+)\nn\nwill have a positive real part provided the corresponding\neigenvalueEnsatisfiesEn<2b. On the other hand, the\nset of three eigenvalues of the operator (33) does include\na negative eigenvalue ( E0) that satisfies E0<2bregard-\nless of the particular value of b≥0. Therefore the soliton\nhas an exponent µ(+)\n0with Reµ(+)\n0>0 for anyb≥0.\nIn the case where λ>0 butb <0, the root µ(+)\nnwill\nhave a positive real part provided EnsatisfiesEn<0.\nAs in the previous case, this inequality is satisfied by the\neigenvalueE0so that the soliton has an exponent with\nReµ(+)\n0>0 for anyb<0.\nWe conclude that the fundamental soliton of the equa-\ntion (30) is unstable in the presence of damping — re-\ngardless of the sign and magnitude of the anisotropy co-\nefficientb. Figure 3 illustrates the evolution of a weakly\nperturbed soliton in the Ginsburg-Landau equation with\nλ/ne}ationslash= 0.\nTurning to the situation with λ= 0 we assume, first,\nthatb>0. The equations (34) will give a pair of oppo-\nsite real roots µ(±)\nnif the corresponding eigenvalue sat-isfies 0< En<2band a pair of pure imaginary roots\notherwise. The only positive eigenvalue of the operator\n(33) isE2= 3; it satisfies the above inequality if b>3/2.\nIn the situation where λ= 0 butb <0, the pair of\nopposite exponents µ(±)\nnis real ifEnfalls in the interval\n2b < E n<0 and pure imaginary if Enlies outside this\ninterval. The only negative eigenvalue is E0=−5; it falls\nin the interval in question if b<−5/2.\nFinally, in the isotropic ferromagnet ( b= 0) the sta-\nbility exponents are all pure imaginary: µ(±)\nn=±iEn.\nCombining the intervals where all exponents are pure\nimaginary gives us the stability region of the undamped\nfundamental soliton in terms of the anisotropy to spin-\ncurrent ratio:\n−5\n2≤b≤3\n2. (35)\nB. Twisted modes in isotropic ferromagnet\nThe Ginsburg-Landau equation (30) with b= 0 admits\nan additional pair of localised solutions:\nuT= 2sech2(2x)±2isech(2x) tanh(2x). (36)\nThe modulus of uT(x) is bell-shaped while its phase\ngrows or decreases by πasxchanges from −∞ to +∞.\nThe solution looks like a pulse twisted by 180◦in the\n(Reu,Imu)-plane. In what follows, we refer to each of\nequations (36) as a twisted , or simply sech-tanh , mode.\nLinearising equation (30) about the twisted mode (36)\nand assuming that the small perturbation depends ontime aseµt, we arrive at an eigenvalue problem\nLf(X) =−µ\n4(λ+i)f(X) (37)9\nfor the Schr¨ odinger operator with the Scarff-II complex\npotential:\nL=−d2\ndX2+ 1−6sech2X∓6isechXtanhX. (38)\nIn (37)-(38), X= 2x.\nThePT-symmetric operator (38) has an all-real spec-\ntrum including three discrete eigenvalues [27]. Let yn\nbe the eigenfunction associated with an eigenvalue En:\nLyn=Enyn. The eigenvalue-eigenfunction pairs are\nthen given by\nE0=−5\n4, y0= (sech2X±isechXtanhX)3/2;\nE1= 0, y1= sechX(sechX±itanhX)2, (39)\nandE2= 3/4 with\ny2= (3±2isinhX)(sech2X±isechXtanhX)3/2.(40)\nEach of the eigenvalues Engives rise to a stability ex-\nponent\nµn= 4i−λ\n1 +λ2En\nin equation (37). When the dissipation coefficient λ>0,\nthe exponent pertaining to the negative eigenvalue E0\nhas a positive real part. Accordingly, the twisted modes\n(36) are unstable in the presence of damping. In con-\ntrast, when λ= 0, all exponents µn(n= 0,1,2) are pure\nimaginary so the twisted modes are stable.\nC. Oscillatory modes\nAn interesting question is whether there are any other\nstable localised structures — in particular, in the situa-\ntion where the equation (30) has zero damping. Figure\n4 illustrates the evolution of a gaussian initial condition\nu(x,0) = exp( −x2) that can be seen as a nonlinear per-\nturbation of the soliton (31). The gaussian evolves into\nan oscillatory localised structure (a kind of a breather)\nwhich remains close to the soliton (31) — but does not\napproach it as t→ ∞ . This observation suggests that\nequation (30) with λ= 0 has a family of stable time-\nperiodic spatially localised solutions, with the stationary\nsoliton (31) being just a particular member of the family.\nIt is fitting to note that the existence of breather fam-\nilies is common to nonlinear PT-symmetric equations\n[28]. Breathers prevail among the products of decay of\ngeneric localised initial conditions [28, 29].\n2\n1.5\n1\n0.5\n0\n5\n0\n-5-505\n2\n1.5\n1\n0.5\n0\n0\n1\n2\n3\n4\n5202224262830\nFIG. 5. Localised solutions of the quadratic Schr¨ odinger\nequation on the plane: the stationary soliton (a) and a\nbreather (b). Both figures were produced by direct numer-\nical simulations of equation (42) with b= 0. In panel (a), the\ninitial condition was taken in the form of the soliton (43) pe r-\nturbed by a random perturbation within 5% of the soliton’s\namplitude. After t= 100, the solution (shown in the panel)\nremains close to the soliton. In panel (b), the initial condi tion\nwas chosen as u= 1.6 exp(−r2). After an initial transient,\nthe solution settles to a localised oscillatory state shown in\nthe figure.\nD. Stable solitons in two dimensions\nWe close this section with a remark on the Landau-\nLifshitz-Gilbert-S/suppress lonczewski equation in two dimensions:\n∂M\n∂t=−M×∇2M−M×H−β(M·ˆz)M׈z\n−γM×M׈z+λM×∂M\n∂t.(41)\nHere∇2=∂2\n∂x2+∂2\n∂y2. Assuming that H0is only slightly\naboveγand that the anisotropy βand damping λare\nsmall, we consider a perturbation of the east-point uni-10\nform state (22). Following the asymptotic procedure out-\nlined in section IV A, the equation (41) is reducible in this\nlimit to a planar Schr¨ odinger equation:\niut=uxx+uyy+ 6u2−4u+b(u−u∗), (42)\nwhere\nb=2H0\nγβ/radicalbig\nH2\n0−γ2.\nLike its one-dimensional counterpart (25), equation\n(42) isPT-symmetric. The PT-operation can be cho-\nsen, for instance, in the form\nt→ −t, x→ −x, y→ −y, u→u∗.\nThe quadratic Schr¨ odinger equation (42) has a static\nradially-symmetric soliton solution,\nus(x,y) =R(r), (43)\nwhereR(r) is a nodeless (bell-shaped) solution of the\nboundary-value problem\nRrr+1\nrRr−4R+ 6R2= 0,\nRr(0) = 0,R(r)→0 asr→ ∞.\nPostponing the detailed stability analysis of the soliton\n(43) to future publications, we restrict ourselves to the\nsimplest case of isotropic ferromagnet, b= 0. A numeri-\ncal simulation of equation (42) with the initial condition\nin the form of the noise-perturbed soliton (43) indicates\nthat the soliton is stable against small perturbations.\n[See Fig 5(a).] On the other hand, generic localised initial\nconditions evolve into time-periodic breather-like states\n[Fig 5(b)]. This suggests that the quadratic Schr¨ odinger\nequation (42) [and hence the planar Landau-Lifshitz\nequation (41)] supports a broad class of stable stationary\nand oscillatory localised structures.\nVI. SOLITON EXCITATIONS OF\nLATITUDINAL STATE\nThe scaling transformation (29) takes the equations\n(28) to the nondimensional form\n(i+λ)ut−uxx−6u2=∓4iu−b(u+u∗). (44)\nAs in section V, b= 2B/γhere.\nIn what follows, we confine ourselves to the analysis of\nthe isotropic equations ( b= 0) as it is the only regime\nwhere we were able to obtain soliton solutions of (44). In\nthe isotropic case, the u= 0 solution of the top-sign equa-\ntion in (44) is stable and that of the bottom-sign equation\nunstable — regardless of whether λis zero or not. (This\nagrees with the stability properties of the north and south\nfixed-point solutions of the Landau-Lifshitz equation; see\nsection III B.) Hence we only keep the top-sign equation\nin what follows.A.sechmode\nLettingb= 0, the top-sign equation in (44) can be\nfurther transformed to\n(1−iλ)wt=wzz−4w+ 6w2, (45)\nwhere\nw(z,t) =−iu, z =eiπ/4x. (46)\nAn obvious static solution of the equation (45) is ws=\nsech2z; the corresponding solution of the original equa-\ntion (44) is\nus(x) =isech2/parenleftbig\neiπ\n4x/parenrightbig\n. (47)\nThe solution (47) decays to zero as x→ ±∞ and does\nnot have singularities on the real line. Similar to the\nsolution (31) over the equatorial background, we term\nthe solution (47) the sech soliton .\nTo classify the stability of the soliton (47), we linearise\nequation (45) about ws= sech2z. Assuming that the\nsmall perturbation depends on time as eµt, we obtain\nµ=−1 +iλ\n1 +λ2E, (48)\nwhereEis an eigenvalue of the P¨ oschl-Teller operator\nH=−d2\ndz2+ 4−12 sech2z.\nThe operator acts upon functions y(z) defined on the line\nz=eiπ/4ξ(−∞< ξ <∞) on the complex- zplane and\nsatisfying the boundary conditions y→0 asξ→ ±∞ .\nAs discussed in the previous section, the equation\nHy=EywithE=−5 has a solution y0= sech3z. The\nfunction sech3(eiπ/4ξ) is nonsingular for all −∞<ξ<∞\nand decays to zero as ξ→ ±∞ ; henceE0=−5 is a dis-\ncrete eigenvalue of the operator H. The corresponding\nexponentµin (48) has a positive real part regardless of\nλ. This implies that the sech soliton (47) is unstable\nirrespective of whether λis zero or not.\nB.sech-tanh modes\nApplying the transformation (46) to the solutions\nwT= 2sech2(2z)±2isech(2z) tanh(2z) (49)\nof the equation (45), we obtain a pair of localised solu-\ntions of the original equation (44):\nuT=∓2 sech(2eiπ/4x) tanh(2eiπ/4x) + 2isech2(2eiπ/4x).\n(50)\nBy analogy with solutions (36) over the equatorial back-\nground, we are referring to (50) as the sech-tanh modes .\nLinearising equation (45) about its stationary solutions\n(49) and assuming that the small perturbation depends11\non time as eµt, we obtain the following equation for the\nexponentµ:\nµ=−41 +iλ\n1 +λ2E.\nHereEis an eigenvalue of the Scarff-II operator:\nLy=Ey, (51a)\nL=−d2\ndZ2+ 1−6 sech2Z∓6isechZtanhZ, (51b)\nwithZ= 2z. The eigenvalue problem (51) is posed on\nthe line\nZ=eiπ/4ξ,−∞<ξ<∞ (52)\non the complex- Zplane, with the boundary conditions\ny→0 asξ→ ±∞ .\nThree solutions of the equation (51) are in (39)-(40).\nSinceyn(Z) (n= 0,1,2) are nonsingular and decay to\nzero asZtends to infinity in either direction along the\nline (52), these solutions are eigenfunctions of the opera-\ntorL— and the corresponding Enare eigenvalues. The\nexponentµ0pertaining to the eigenvalue E0=−5/4 has\na positive real part:\nµ0=−41 +iλ\n1 +λ2E0.\nConsequently, the sech-tanh modes (50) are unstable —\nno matter whether λis zero or not.\nC. Summary of one-dimensional solitons\nThe stability properties of six localised modes sup-\nported by the quadratic Ginsburg-Landau equations (30)\nand (44) are summarised in Table II. The Table includes\ntwosech solitons (the fundamental soliton (31) and its\nlatitudinal-background counterpart, equation (47)) and\nfoursech-tanh modes (the twisted modes (36) and their\nlatitudinal analogs (50)).\nNonlinear\nmodeover equatorial\nbackgroundover latitudinal\nbackground\n(withb= 0)\nsech stable if λ= 0\nunstable\nsoliton and−5\n2< b <3\n2\nsech-tanh exist ifb= 0;\nunstable\nmodes stable if λ= 0\nTABLE II. Stability of the stationary nonlinear modes in one\ndimension. Themiddlecolumnclassifies solutionsoftheequ a-\ntion(30)whiletheright-handcolumncorrespondstosoluti ons\nof (44).VII. CONCLUDING REMARKS\nWe have studied nonlinear structures associated with\nthe spin torque oscillator — an open system described\nby the Landau-Lifshitz-Gilbert-S/suppress lonczewski equation. In\nthe limit of zero damping ( λ= 0), this nonconservative\nsystem is found to be PT-symmetric. The nearly -PT\nsymmetric equation corresponds to small nonzero λ. In\nthis paper, we have considered both nearly-symmetric\nand nonsymmetric oscillators (small and moderate λ).\nThe spin torque oscillator has four stationary states of\nuniform magnetisation; they are described by four fixed\npoints on the unit M-sphere. Two of these states have\ntheir magnetisation vectors lying in the equatorial plane\nof the unit sphere while the other two correspond to\nfixed points in the northern and southern hemisphere, re-\nspectively. We have assumed that the external magnetic\nfieldH0has been tuned to values ǫ2-close to the bifurca-\ntion points of the “equatorial” and “latitudinal” uniform\nstatic states, and that the ferromagnet is only weakly\nanisotropic: β=O(ǫ). In that limit, small-amplitude lo-\ncalised perturbations of the uniform static states satisfy\nthe Ginsburg-Landau equations — equations (30) and\n(44), respectively.\nIf the damping coefficient λisO(ǫ) or smaller,\neach of the two Ginsburg-Landau reductions becomes a\nquadratic nonlinear Schr¨ odinger equation. Of the two\nSchr¨ odinger equations, the one corresponding to pertur-\nbations of the “equatorial” uniform static state turns out\nto bePT-symmetric. (Thus the asymptotic reduction of\nanearlyPT-symmetric Landau-Lifshitz system is exactly\nPT-symmetric.) This Schr¨ odinger equation proves to be\nquite remarkable. Indeed, despite both our Ginsburg-\nLandau reductions supporting soliton solutions, it is only\nin thePT-symmetric Schr¨ odinger limit that the solitons\nare found to be stable.\nThePT-symmetric Schr¨ odinger equation supports two\ntypes of stable solitons. The constant-phase solution (31)\nis stable in a band of β=O(ǫ) values, extending from the\neasy-axis to the easy-plane region. [The stability band is\ndemarcated by the inequality (35).] On the other hand, a\npair of stable solitons with the twisted phase, equations\n(36), are only supported by the nearly-isotropic ferro-\nmagnet:β=O(ǫ2) or smaller. In addition to stable\nstatic solitons, the PT-symmetric Schr¨ odinger equation\nexhibits stable breathers.\nIn the two-dimensional geometry, the Landau-Lifshitz\nequation for the spin torque oscillator admits an asymp-\ntotic reduction to a planar quadratic Schr¨ odinger equa-\ntion, equation (42). Like its one-dimensional counter-\npart, the PT-symmetric planar Schr¨ odinger equation has\nstable static and oscillatory soliton solutions.\nFinally, it is worth re-emphasising here that the\nPT-symmetric Schr¨ odinger equation is a reduction of\nthe whole family of nearly- PT symmetric Landau-\nLifshitz-Gilbert-S/suppress lonczewski equations with λ=O(ǫ) —\nand not just of its special case with λ= 0. Therefore\nour conclusion on the existence of stable solitons is12\napplicable to the physically relevant class of spin torque\noscillators with nonzero damping.\nACKNOWLEDGMENTS\nWe thank Boris Ivanov and Andrei Slavin for useful\ndiscussions. This project was supported by the NRF ofSouth Africa (grants No 105835, 120844 and 120467).\n[1] C.M. Bender and S. Boettcher, Phys. Rev. Lett. 805243\n(1998); C.M. Bender, S. Boettcher, and P.N. Meisinger,\nJ. Math. Phys. 402201 (1999); C.M. Bender, Contemp.\nPhys.46277 (2005); C.M. Bender, Rep. Prog. 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Balachandran4,7, Tim Mewes2,3, Satoru Emori1 \n1Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A.  \n2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, U.S.A.  \n3Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa,  \nAL 35487, U.S.A . \n4Department of Material Science and Engineering, University of Virginia,  \nCharlottesville, VA 22904, U.S.A. \n5Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061, \nU.S.A.  \n6Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, \nU.S.A.  \n7Department of Mechanical and Aerospace Engineering, University of Virginia,  \nCharlottesville, VA 22904, U.S.A.  \n \n  2 \n To develop low -moment, low -damping metallic ferromagnets for power -efficient spintronic \ndevices, it is crucial to understand how magnetic relaxation is impacted by the addition of \nnonmagnetic elements. Here, we compare  magnetic relaxation in epitaxial Fe films alloyed \nwith light nonmagnetic elements of V and Al. FeV alloys exhibit lower intrinsic damping  \ncompared to pure Fe,  reduced by nearly a factor of 2,  whereas damping in FeAl alloys \nincreases with Al content . Our experimental and computat ional results indicate that \nreducing the density of states at the Fermi level , rather than the average atomic number, \nhas a more significant impact in lowering damping in Fe alloyed with light elements . \nMoreover, FeV is confirmed to exhibit an intrinsic Gi lbert damping parameter of ≃0.001, \namong the lowest ever reported for ferromagnetic metals.  \n \nI. INTRODUCTION  \n The relaxation of magnetization dynamics (e.g., via Gilbert damping) plays important \nroles in many spintronic applications, including those based  on magnetic switching1,2, domain \nwall motion3,4, spin wave propagation5,6, and su perfluid -like spin transport7,8. For devices driven \nby spin -torque  precessional dynamics1,9,10, the critical current density for switching is predicted \nto scale with the produ ct of the Gilbert damping parameter and the saturation magnetization 2,11. \nThus, it is desirable to engineer magnetic materials  that possess both low damping and low \nmoment for energy -efficient operation . While some electrically insulating magnetic oxides have \nbeen considered for certain applications5,12,13, it is essential to engineer low -damping, low -\nmoment metallic  ferromagnets for robust electrical readout via giant magnetoresistance and \ntunnel magnetoresistance. Fe is the elemental ferromagnet with the lowest intrinsic Gilbert \ndamping  parameter ( ≃0.002)14,15, albeit with the highest saturation magnetization ( ≃2.0 T). 3 \n Recent experiments have reported that Gilbert damping can be further reduced by alloy ing Fe \nwith Co (also a ferromagnetic element), with Fe 75Co25 yielding an ultralow intrinsic Gilbert \ndamping parameter of ≃0.00116,17. However, Fe 75Co25 is close to the top of the Slater -Pauling \ncurve , such that its saturation magnetization is greater than that of Fe by approximately 20 %18. \nThere is thus an unmet need to engineer ferromagnetic alloys tha t simultaneously exhibit lower \ndamping and lower moment than Fe.  \n A promising approach towards low -damping, low -moment ferromagnetic metals is to \nintroduce nonmagnetic  elements into Fe . In addition to diluting the magnetic moment, \nnonmagnetic elements int roduced into Fe could influence the spin -orbit coupling strength ξ, \nwhich underlies  spin relaxation  via orbital and electronic degrees of freedom19–21. Simple atomic \nphysics suggests that ξ is related to the average atomic number <Z> of the alloy so that, \nconceivably, damping might be lowered by alloying Fe with lighter (lower -Z) elements. Indeed, \nmotivated by the premise of lowering damping through a reduced  <Z> and presumably ξ, prior \nexperiments have explored Fe thin films alloyed with V20,22,23, Si24, and Al25. However, the \nexperimentally reported damping parameters for these alloys are often a factor of >2 higher22,23 ,25 \nthan the theoretically predicted intrinsic Gilbert damping parameter of ≃0.002 in Fe26 and do not \nexhibit a significant dependence on the alloy composition20,23,24. A possible issue is that the \nreported damping parameters – obtained from the frequency dependence of ferromagnetic \nresonance (FMR) linewidth with the film magnetized in -plane – may include contributions from \nnon-Gilbert relaxation induced by inhomogeneity and defects (e.g., two -magnon scattering)27–36, \nwhich can be affected by the alloying. Therefore, how Gilbert damping in Fe is impacted by \nalloying with low -Z elements remains an open question.  4 \n Here, we investigate the compositiona l dependence of magnetic relaxation at room \ntemperature in  epitaxial thin films of ferromagnetic FeV and FeAl alloys. Both  alloys are \ncrystalline bcc solid solutions  and hence constitute excellent model systems. We employ two \nconfigurations of FMR measurem ents to gain complementary insights: (1) FMR with samples \nmagnetized in the film plane (similar to the prior experiments) to derive the “effective” Gilbert \ndamping parameter, 𝛼𝑒𝑓𝑓𝐼𝑃, which is found to include extrinsic magnetic relaxation due to two -\nmagnon scattering, and (2) FMR with samples magnetized perpendicular to the film plane to \nquantify the intrinsic Gilbert damping parameter, 𝛼𝑖𝑛𝑡, which is free of the two -magnon \nscattering contribution.  \nSince Al ( Z = 13) is a much lighter element  than V ( Z = 23), we might expect lower \nmagnetic relaxation in FeAl than FeV, if the smaller < Z> lowers intrinsic Gilbert damping via \nreduced ξ. Instead, we find a significant decrease in  magnetic relaxation by alloying Fe w ith V – \ni.e., yielding an intrinsic Gilbert damping parameter of ≃0.001, on par with the lowest values \nreported for ferromagnetic metals  – whereas damping in FeAl alloys  increases with Al content . \nThese experimental results , combined with density functi onal theory calculations, point to the \ndensity of states at the Fermi level D(EF) as a plausible dominant factor for the lower (higher) \nGilbert damping in FeV (FeAl). We thus find that incorporating a low -Z element does not \ngenerally lower damping and that, rather, reducing D(EF) is an effective route for lower damping \nin Fe alloyed wi th a nonmagnetic element. Our findings confirm that FeV is an intrinsically \nultralow -damping alloy, as theoretically predicted by Mankovsky et al.26, which also possesses a \nlower saturation magnetization than Fe and FeCo. The combination of low damping and low \nmoment makes FeV a highly promising material for practical metal -based spintronic \napplications.  5 \n II. FILM DEPOSITION AND STRUCTURAL PROPERTIES  \nEpitaxial Fe 100-xVx and Fe 100-xAlx thin films were grown using dc magnetron sputtering \non (001) -oriented MgO substrates. Prior to deposition, the substrates were annealed at 600 oC for \n2 hours37. The base pressure prior to deposition was <  5×10-8 Torr, and all film s were grown with \nan Ar pressure of 3 mTorr. Fe and V (Al) 2” targets were dc co -sputtered to deposit Fe 100-xVx \n(Fe 100-xAlx) films at a substrate temperature of 200 oC. By adjusting the deposition power, we \ntuned the deposition rate of each material (calibrated by X -ray reflectivity) to achieve the desired \natomic percentage x of V (Al). All FeV and FeAl films had a thickness of 25 nm, which is well \nabove the thickness regime where interfacial effects dominate31,38.  The FeV (FeAl) films were \ncapped with 3 -nm-thick V (Al) deposited at room temperature to protect against oxidation, \nyielding a film structure of MgO/Fe 100-xVx(25nm)/V(3nm) or MgO/Fe 100-xAlx(25nm)/Al(3nm).  \n We confirmed the epitaxial bcc structure of our thi n films using high resolution X -ray \ndiffraction. 2θ -ω scans show only the (002) peak of the film and the (002) and (004)  peaks of the \nsubstrate, as shown in Fig ure 1. Rocking curve scans of the film peaks show similar full -width -\nat-half-maximum values of  ≃ 1.3o irrespective of composition .  The epitaxial relation between \nbcc Fe and MgO is well known16,39:  the bcc film crystal is rotated 45o with respect to the \nsubstrate crystal , such that  the [100] axis of the film lies parallel to the [110] axis of the \nsubstrate. The absence of the (001) film peak indicates that our epitaxial FeV and FeAl films are \nsolid sol utions rather than B2 -ordered compounds40. \n  6 \n III. MAGNETIC RELAXATION  \n3.1. In -Plane Ferromagnetic Resonance   \nMany spintronic devices driven by precessional magnetization dynamics are based on in -\nplane magnetized thin films. The equilibrium magnetization also lies in -plane for soft \nferromagnetic thin films dominated by shape anisotropy (i.e., negligible perpendicular magnetic \nanisotropy), as is the case for our epitaxial FeV and FeAl films. We therefore first discuss FMR \nresults w ith films magnetized in -plane. The in -plane FMR results further provide a basis for \ncomparison with previous studies20,22,23,25.  \nSamples were placed with the film side facing a coplanar waveguid e (maximum \nfrequency 50 GHz) and magnetized by an e xternal field H (from a conventional electromagnet, \nmaximum field 1.1 T) along the in -plane [100] and [110] axes of the films.  Here, unless \notherwise stated, we show results for H || [110] of the film.  FMR spectra were acquired via field \nmodulation by sweeping H and fixing the microwave excitation frequency.  \nExemplary  spectra for Fe, Fe 80V20, and Fe 80Al20 are shown in Fig ure 2, where we \ncompare the peak -to-peak linewidths at a microwave excitation frequency of 20 GHz. We see \nthat the linewidth for Fe 80V20 shows a ≃ 25 % reduction compared to Fe. We further note that \nthe linewidth for the Fe 80V20 sample here is a factor of ≃ 2 narrower than that in previously \nreported FeV20; a possible origin of the narrow linewidth is discussed later . In contrast, Fe 80Al20 \nshows an enhancement in linewidth over Fe, which is contrar y to the expectation of lower \nmagnetic relaxation with a lower average atomic number.  \nThe FMR linewidth is generally governed not only by magnetic relaxation, but also by \nbroadening contributions from magnetic inhomogeneities28,41,42. To disentangle the magnetic 7 \n relaxation and inhomogeneous broadening contributions to the linewidth, the typical prescription \nis to fit the frequency f dependence of linewidth ∆𝐻𝑝𝑝𝐼𝑃 with the linear relation41 \n∆𝐻𝑝𝑝𝐼𝑃=∆𝐻0𝐼𝑃+ℎ\n𝑔𝜇𝐵𝜇02\n√3𝛼𝑚𝑒𝑎𝑠𝐼𝑃𝑓, (1) \nwhere h is the Planck constant, 𝜇𝐵 is the Bohr magneton, 𝜇0 is the permeability of free space, \nand 𝑔 is the g-factor obtained from the frequency dependence of the resonance field (see Section \nIV and Supplementa l Material). In Eq. (1), the slope is attributed to viscous magnetic damping, \ncaptured by the measured damping parameter 𝛼𝑚𝑒𝑎𝑠𝐼𝑃, while t he zero -frequency linewidth ∆𝐻0𝐼𝑃 is \nattributed to inhomogeneo us broadening. The fitting with Eq. (1) was carried out for f  10 GHz, \nwhere H was sufficiently large to saturate the films. As is evident from  the results in Fig ure 3, \nFe80V20 has lower linewidths across all frequencies and a slightly lower slope, i.e.,  𝛼𝑚𝑒𝑎𝑠𝐼𝑃. On the \nother hand, Fe 80Al20 shows higher linewidths and a higher slope.  \nThe measured viscous damping includes a small contribution from eddy currents, \nparameter ized by 𝛼𝑒𝑑𝑑𝑦 (Supplemental  Material) , and a contribution due to radiative damping43, \ngiven by 𝛼𝑟𝑎𝑑 (Supplemental Material).  Together these contributions make up ≃20 % of the total \n𝛼𝑚𝑒𝑎𝑠𝐼𝑃 for pure Fe and decrease in magnitude with increasing V or Al content . We subtract these \nto obtain  the effective in -plane Gilbert damping parameter,  \n 𝛼𝑒𝑓𝑓𝐼𝑃=𝛼𝑚𝑒𝑎𝑠𝐼𝑃−𝛼𝑒𝑑𝑑𝑦 − 𝛼𝑟𝑎𝑑.  (2) \nAs shown in Fig ure 4a, 𝛼𝑒𝑓𝑓𝐼𝑃 remains either invariant or slightly decreases in Fe 100-xVx up to x = \n25, whereas we observe a monotonic enhancement of 𝛼𝑒𝑓𝑓𝐼𝑃 with Al content in Figure 4b . These \nresults point to  lower (higher) damping in FeV (FeAl) and suggest a factor other than the average \natomic number governing magnetic relaxation in these alloys.  However, such a conclusion \nassumes that 𝛼𝑒𝑓𝑓𝐼𝑃 is a reliable measure of intrinsic Gilbert damping . In reality, 𝛼𝑒𝑓𝑓𝐼𝑃 may include 8 \n a contribution from defect -induced two -magnon scattering27–31,35,36, a well -known non -Gilbert \nrelaxation mechanism in in -plane magnetized epitaxial films27,32 –34,44. We show in the next \nsubsection that substantial two -magnon scattering  is indeed present in our FeV and FeAl alloy \nthin films.  \n Although Eq. (1) is not necessarily the correct framework for quantifying Gilbert \ndamping in in -plane magnetized thin films, we can gain insight into the quality (homogeneity) of \nthe films from ∆𝐻0𝐼𝑃. For our samples, μ0∆𝐻0𝐼𝑃 is below ≈ 1 mT (see Fig ure 4c,d), which implies  \nhigher film quality for our FeV samples than previously reported20. For example, Fe 73V27 in \nScheck et al.  exhibits μ0∆𝐻0𝐼𝑃 ≃ 2.8 mT20, whereas Fe 75V25 in our study exhibits μ0∆𝐻0𝐼𝑃 ≃ 0.8 \nmT. Although  𝛼𝑒𝑓𝑓𝐼𝑃 is comparable between Scheck et al. and our study, the small ∆𝐻0𝐼𝑃 leads  to \noverall much narrower linewidths  in our FeV films (e.g., as shown in Figs. 2 and 3) . We \nspeculate that the annealing of the MgO substrate prior to film deposition37 – a common practice \nfor molecular beam epitaxy – facilitates high -quality epitaxial film growth  and hence small ∆𝐻0𝐼𝑃 \neven by sputtering.  \n  \n3.2. Out -of-Plane Ferromagnetic Resonance  \nTo quantify intrinsic Gilbert damping, we performed broadband FMR with the film \nmagnetized out -of-plane, which is the configuration that suppresses two -magnon scattering28–31. \nSamples were placed in side a W-band shorted -waveguide spectrometer (frequency range 70 -110 \nGHz) in a superconducting electromagnet that enabled measurements at fields > 4 T. This high \nfield range is well above the shape anisotropy field of ≤2 T for our films and hence sufficient to \ncompletely saturate the film out -of-plane.   9 \n The absence of two -magnon scattering in broadband out -of-plane FMR allows us to \nreliably obtain the measured viscous damping parameter 𝛼𝑚𝑒𝑎𝑠𝑂𝑃 by fitting the linear frequency \ndependence of the linewidth ∆𝐻𝑝𝑝𝑂𝑃, as shown in Figure 5, with \n∆𝐻𝑝𝑝𝑂𝑃=∆𝐻0𝑂𝑃+ℎ\n𝑔𝜇𝐵𝜇02\n√3𝛼𝑚𝑒𝑎𝑠𝑂𝑃𝑓. (3)  \nWe note that the zero -frequency linewidth for the out -of-plane configuration ∆𝐻0𝑂𝑃 (Figure 6c,d) \nis systematically g reater than that for the in -plane configuration ∆𝐻0𝐼𝑃 (Figure 4c,d). Such a trend \nof ∆𝐻0𝑂𝑃>∆𝐻0𝐼𝑃, often seen in epitaxial films15,33,45, may be  explained by the stronger \ncontribution of inhomogeneity to the FMR field when the magnetic precessional orbit is circular, \nas is the case for out -of-plane FMR, compared to the case of the highly elliptical precession in \nin-plane FMR41; however, the detailed mechanisms contributing to the zero -frequency linewidth \nremain  the subject of future work . The larger ∆𝐻0𝑂𝑃 at high V and Al concentrations may be due \nto broader distributions o f anisotropy fields and saturation magnetization, or the presence of a \nsecondary crystal phase that is below the resolution of our X -ray diffraction results.   \nThe absence of two -magnon scattering in out -of-plane FMR allows us to quantify the \nintrinsic Gilbert damping parameter,  \n𝛼𝑖𝑛𝑡=𝛼𝑚𝑒𝑎𝑠𝑂𝑃−𝛼𝑒𝑑𝑑𝑦,  (4) \nby again subtracting the eddy current contribution 𝛼𝑒𝑑𝑑𝑦. Since we utilize a shorted  waveguide, \nthe contribution due to radiative damping does not apply.    \nFrom the compositional dependence of 𝛼𝑖𝑛𝑡 as summarized in Figure 6a1, a reduction in \nintrinsic Gilbert damping is evidenced with V alloying. Our observation is in contrast to the \nprevious experiments on FeV alloys20,22,23 where the reported damping parameters remain >0.002 \n \n1 We were unable to carry out out -of-plane FMR measurements for FeV with x = 20 (Fig. 2(c,d )) as the sample had \nbeen severely damaged during transit.  10 \n and depend weakly on the V concentration. In particular, the observed minimum of 𝛼𝑖𝑛𝑡≃0.001 \nat x ≃ 25-30 is approximately half of the lowest Gilbert damping parameter previously reported \nfor FeV20 and that of pure Fe15. The low 𝛼𝑖𝑛𝑡 here is also comparable to the lowest  damping \nparameters reported for ferromagnetic metals, such as  Fe75Co2516,17 and Heusler compounds46–48. \nMoreover, t he reduced intrinsic damping by alloying Fe w ith V is qualitatively consistent with \nthe computational prediction  by Mankov sky et al.26, as shown by the curve in Figure 6a. Our \nexperimental finding therefore confirms that FeV is indeed an intrinsically ultralow -damping \nferromagnet that possesses a smaller saturation magnetization than Fe.  \nIn contrast to the reduction of 𝛼𝑖𝑛𝑡 observed  in FeV alloys, FeAl shows an increase in \nintrinsic damping with increasing Al concentration, as seen in Figure 6b. Recalling that Al has an \natomic number of Z = 13 that is lower than Z = 23 for V, this trend clashes with the expectation \nthat lower < Z> red uces the intrinsic Gilbert damping through a reduction of the atomic spin -orbit \ncoupling. Thus, we are required to consider an alternative mechanism to explain the higher \n(lower) damping in FeAl (FeV), which we discuss further in Section V.  \n \n3.3. Magnetic Relaxation: Practical Consideration s \nFor both FeV and FeAl alloys, 𝛼𝑖𝑛𝑡 derived from out -of-plane FMR (Figure 6a,b) is \nconsistently lower than 𝛼𝑒𝑓𝑓𝐼𝑃 derived from in -plane FMR (Fig ure 4a,b). Th is discrepancy \nbetween 𝛼𝑖𝑛𝑡 and 𝛼𝑒𝑓𝑓𝐼𝑃 implies  a two-magnon scattering contribution  to magnetic relaxation in \nthe in-plane configuration (Figure 4a,b). For many applications including spin -torque oscillators \nand magnonic devices , it is crucial to minimize magnetic relaxation in in-plane magnetized thin \nfilms. While the in -plane magnetic relaxation ( 𝛼𝑒𝑓𝑓𝐼𝑃≃0.002) is already quite low for the FeV \nalloys shown here, the low intrinsic Gilbert damping ( 𝛼𝑖𝑛𝑡≃0.001) points to the possibility of 11 \n even lower relaxation and narrow er FMR linewidths by minimizing two -magnon scattering and \ninhomogeneous linewidth broadening. Such ultralow magnetic relaxation in FeV alloy thin films \nmay be achieved by optimizing structural properties through growth conditions16 or seed layer \nengineering49.  \nWhile ultralow intrinsic Gilbert damping values have been confirmed  in high -quality \nepitaxial FeV, it would be desirable for device integration to understand how magnetic relaxation  \nin FeV would be impacted by the presence of grain boundaries, i.e. in polycrystalline thin films. \nReports on polycrystalline FeCo49 suggest intrinsic damping values comparable to those  seen in \nepitaxial FeCo16,17. While beyond the scope of this study, our future work will explore the \npossibility of low damping in polycrystalline FeV thin films.  \n \nIV. SPECTROSCOPIC PARAMETERS  \nThe results presented so far reveal that magnetic relaxation is reduced by alloying Fe with \nV, whereas it is increased by alloying Fe with Al. On the other hand, FeV and FeAl alloys \nexhibit similar compositional dependence of the spectroscopic parameters: effective \nmagnetization Meff (here, equivalent to saturation magnetization  Ms), magnetocrystalline \nanisotropy field Hk, and the g-factor 𝑔 – all of which are quantified by fitting the frequency \ndependence of resonance field (Supplemental Material) . As shown in Fig ure 7a, there is a \nsystematic reduction in Meff with increasing concentration of V and Al. We also note in Fig ure 7b \na gradual reduction in magnitude of the in -plane cubic anisotropy. Both of these trends are \nexpected as magnetic Fe atoms are r eplaced with nonmagnetic atoms of V and Al. The reduction \nof Meff by ≃20% in the ultralow -damping Fe 100-xVx alloys with x = 25-30, compared to pure Fe, \nis of particular practical interest. The saturation magnetization of these FeV alloys is on par with 12 \n commonly used soft ferromagnetic alloys (e.g., Ni 80Fe2050, CoFeB51), but the damping parameter \nof FeV is several ti mes lower. Further, w hile FeV and FeCo in the optimal composition window \nshow similarly low intrinsic damping parameters, FeV provides the advantage of lower moment . \nWith the product 𝛼𝑖𝑛𝑡𝑀𝑒𝑓𝑓 approximately proportional to the  critical current densi ty to excite \nprecessional dynamics  by spin torque2,11, FeV is expected to be a superior material platform for \nlow-power spin tronic devices .  \nThe g-factor 𝑔=2(1+𝜇𝐿/𝜇𝑆) is related to the orbital moment 𝜇𝐿 and spin moment 𝜇𝑆; \nthe deviation from the spin -only value of 𝑔= 2.00 provides insight into the strength of spin -orbit \ncoupling ξ52. As seen in Figure 7c, 𝑔 increases  by 1-2% with both V and Al alloying, which \nsuggests that ξ increases slightly with the addition of these low -Z elements.  This finding verifies \nthat < Z> is not necessarily a good predictor of ξ in a solid. Moreover, the higher 𝑔 for FeV is \ninconsistent with the scenario for lower damping linked to a  reduced spin -orbit coupling. Thus, \nspin-orbit coupling alone cannot  explain the observed behavior of Gilbert damping in Fe alloyed \nwith low -Z elements.  \n \nV. DISCUSSION  \nIn contrast to what has been suggested by prior experimental studies20,22 –25, we have \nshown that the reduction of average atomic number by alloying with a light element (e.g., Al in \nthis case) does not generally lower the intrinsic Gilbert dampin g of Fe. A possible source for the \nqualitatively distinct dependencies of damping on V and Al contents is the density of states at the \nFermi level, D(EF): it has been predicted theoretically that the intrinsic Gilbert damping \nparameter is reduced with decr easing D(EF), since D(EF) governs the  availability of  states for \nspin-polarized electrons to scatter into21,26,53 –55. Such a correlation between lower damping and 13 \n smaller D(EF) has been reported by recent experiments on FeCo alloys17,50, FeRh alloys40, CoNi \nalloys56, and Heusler compounds46,48,57. The similarity in the predicted composition dependence \nof the Gilbert damping parameter for FeCo and FeV26 suggests that the low damping of FeV may \nbe correlated with reduced D(EF). However, no prior experiment has corroborated this \ncorrelation for FeV or other alloys of Fe and light elements.  \nWe therefore e xamine whether the lower (higher) damping in FeV (FeAl) compared to Fe \ncan be qualitatively explained  by D(EF).  Utilizing the Quantum ESPRESSO58 package to \nperform density functional theory calculations (details in Supplemental Material) , we calculated \nthe density of states  for Fe, Fe 81.25V18.75, and Fe 81.25Al18.75. It should be recalled that although \nFeV and FeAl films measured experimentally her e are single -crystalline, they are solid solutions \nin which V or Al atoms replace Fe atoms at arbitrary bcc lattice sites.  Therefore, f or each of the \nbinary alloys, we computed 6 distinct atomic configurations  in a 2×2×2 supercell , as shown in \nFigure 8 . The spin -split density of states for each unique atomic configuration is indicated by a \ncurve in Figure 9. Here, D(EF) is the sum of the states for the spin -up and spin -down bands, \naveraged over results from the 6 distinct atomic configurations.  \nAs summari zed in Fig ure 9 and Table 1, FeV has a smaller D(EF) than Fe, whereas FeAl \nhas a larger D(EF). These calculation results confirm a smaller  (larger ) availability of states for \nspin-polarized electrons to scatter into in FeV (FeAl), qualitatively consistent with the lower \n(higher) intrinsic Gilbert damping in FeV (FeAl).   \nWe remark that this correlation between damping and D(EF) is known to hold parti cularly \nwell in the limit of low electronic scattering rates 𝜏−1, where intra band scattering dominates21,54. \nGilmore et al.  have pointed out that at sufficiently high electronic scattering rates, i.e., when \nℏ𝜏−1 is large enough that inter band scattering is substantial, the simple correlation between the 14 \n strength of Gilbert damping and D(EF) breaks down. It is unclear whether our FeV and FeAl \nalloy films at room temperature are in the intraband - or interband -dominated regime. Schoen et \nal. have argued that polycrystalline FeCo alloy films – with higher degree of structural disorder \nand likely higher electronic scattering rates than our epitaxial films – at room temperature are \nstill well within the intraband -dominated regime17. On the other hand, a recent temperature -\ndependent study on epitaxial Fe suggests coexistence of the intraband and interband \ncontributions at room temperature15. A consistent explanation for the observed room -temperature \nintrinsic damping in our alloy films is that the interband contribution depends weakly on alloy \ncomposition; it appears re asonable to conclude that D(EF), primarily through the intraband \ncontribution, governs the difference in intrinsic Gilbert damping among Fe, FeV, and FeAl .  \n \nVI. SUMMARY  \nWe have experimentally in vestigated magnetic relaxation in epitaxial thin films of Fe \nalloyed with low -atomic -number nonmagnetic elements V and Al . We observe a reduction in the \nintrinsic Gilbert damping parameter to 𝛼𝑖𝑛𝑡≃0.001 in FeV films , comparable to the lowest -\ndamping ferromagnetic metals reported to date. In contrast, an increase in damping is observed \nwith the addition of Al, demonstrating that a smaller average atomic number does not necessarily \nlower intrinsic damping in an alloy . Furthermore, our  results on FeV and FeAl cannot be \nexplained by the change  in spin -orbit coupling through alloying . Instead, we conclude that the \ndensity of states at the Fermi level plays a larger role in determining the magnitude of damping \nin Fe alloyed w ith lighter elements. Our work also confirms FeV alloys as promising ultra low-\ndamping , low-moment metallic materials for  practical power -efficient spin -torque  devices.  \n  15 \n Acknowledgements:  \nThis research was funded in part by 4 -VA, a collaborative partnership for advancing the \nCommonwealth of Virginia, as well as by the ICTAS Junior Faculty Program. D.A.S. \nacknowledges support of the Virginia Tech Graduate School Doctoral Assistantship. A. Sapkota \nand C. M . would like to acknowledge support by NSF -CAREER Award No. 1452670,  A.R. and \nT.M. would like to acknowledge support by DARPA TEE Award No. D18AP00011,  and A. \nSrivastava would like to acknowledge support by NASA Award No. CAN80NSSC18M0023.  \n \nWe thank M.D. Stiles for helpful input regarding intrinsic damping mechanisms in alloys.  \n \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request.  \n \n Number of Spin -Up States  (eV-1) \nat EF Number of Spin -Down States  \n(eV-1) at EF \nFe 10.90 3.44 \nFe81.25V18.75 6.28 ± 1.80  4.61 ± 0.43  \nFe81.25Al18.75 6.81 ± 1.58  10.20 ± 3.03  \nTable 1:  Number of spin -up and spin -down states  at EF. For Fe81.25V18.75 and \nFe81.25Al18.75, the average and standard deviation of values for the 6 distinct atomic \nconfigurations (cf. Figure 8) are shown.    16 \n References:  \n1 A. Brataas, A.D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012).  \n2 J.Z. Sun, Phys. Rev. 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Phys.Condens. \nMatter 21, 395502 (2009).  \n  21 \n \n15 30 45 60 75 90\nMgO (004) MgO (004) MgO (004)MgO (002) MgO (002) \n MgO (002)BCC\nFe\n(002)\n Log(Intensity) (arb. units)BCC\nFe80V20\n(002)\n \n2q (deg)BCC\nFe80Al20\n(002) \nFigure 1:  (a) 2θ-ω X-ray diffraction scans showing (00 2) and (004)  substrate and (002) film \npeaks for bcc Fe, Fe 80V20, and Fe 80Al20.  \n  22 \n \n-15 -10 -5 0 5 10 15 \n \nFe 2.70 mT\n FMR Signal (arb. units)Fe80V20 2.04 mT\n \nm0(H - HFMR) (mT)Fe80Al203.20 mT \nFigure 2: FMR spectra at f = 20 GHz with the magnetic field H applied in the film plane, fitted \nusing a Lorentzian derivative (solid curve ) for Fe, Fe 80V20 and Fe 80Al20.   23 \n \n0 10 20 30 40 5002468 Fe\n Fe80V20\n Fe80Al20\n Scheck et al.m0DHIP\nPP (mT)\nFrequency (GHz) \nFigure 3:  FMR linewidths versus microwave frequency for the magnetic field applied within the \nplane of the film for three distinct alloys. The solid lines are linear fit s, described by Eq. (1), \nfrom which the effective damping parameter and zero frequency linewidth are determined.  The \ndashed line represents the result for Fe 73V27 from Scheck et al.20 \n  24 \n \n0 10 20 30 40246\n0 10 20 302468\n0 10 20 30 40024\n0 5 10 15 20 25 3001 Fe100-xVx\n Scheck et al.aIP\neff x 103\naIP\neff x 103 Fe100-xAlxm0DHIP\n0 (mT)\nAlloy Composition, x (%)(a) (b)\n(c) (d)m0DHIP\n0 (mT)\nAlloy Composition, x (%) \nFigure 4:  The effective damping parameter 𝛼𝑒𝑓𝑓𝐼𝑃 for (a) Fe 100-xVx and (b) Fe 100-xAlx and zero \nfrequency linewidth 𝜇0Δ𝐻0𝐼𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from in -plane FMR.  \nThe solid symbols in (a)  and (c) represent results reported by Scheck et al.20 \n  25 \n \n0 20 40 60 80 100 12001020304050\n Fe\n Fe70V30\n Fe70Al30m0DHOP\nPP (mT)\nFrequency (GHz) \nFigure 5:  FMR linewidths versus applied microwave frequency for the magnetic field applied \nperpendicular to the plane of the film for three distinct alloys. The line is a linear fit, described \nby Eq. (3), from which the intrinsic Gilbert damping parameter and zero frequency linewidth are \ndetermined.   \n  26 \n \n0 10 20 30 400123\n0 10 20 300246\n0 10 20 30 4001020\n0 10 20 3001020 Fe100-xVx\n Mankovsky et al.aint x 103 Fe100-xAlxaint x 103(a) (b)\n(c) (d)m0DHOP\n0 (mT)\nAlloy Composition, x (%)\nm0DHOP\n0 (mT)\nAlloy Composition, x (%) \nFigure 6:  The intrinsic Gilbert damping parameter 𝛼𝑖𝑛𝑡 for (a) Fe 100-xVx and (b) Fe 100-xAlx and \nzero frequency linewidth  𝜇0Δ𝐻0𝑂𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from out -of-\nplane FMR. In (a), the dashed curve  show s the predicted intrinsic damping parameter  computed \nby Mankovsky et al.26 \n  27 \n \n0.81.21.62.02.4\n204060\n0 10 20 30 402.082.102.122.14 Fe\n Fe100-xVx\n Fe100-xAlx \n m0Meff (T)(a)\n |m0Hk| (mT)(b)\n g-factor\nAlloy Composition, x (%)(c) \nFigure 7: (a) Effective magnetization, (b) in -plane cubic anisotropy field, and (c) g-factor versus  \nV and Al concentration.  The solid (open) markers represent data from in -plane (out -of-plane) \nmeasurements . \n  28 \n  \nFigure 8:  The six unique atomic configurations from the supercell program for mimicking the \nFe81.25V18.75  or Fe81.25Al18.75 solid solution.   \n29 \n \n-10010-10010\n-1.0 -0.5 0.0 0.5 1.0-10010 \n (a)\nFe81.25V18.75\n Density of States (eV-1)\n(b)Fe\n \nE - EF (eV)(c)Fe81.25Al18.75 \nFigure 9: Calculated spin-up (positive) and spin -down (negative) densit ies of states for (a) Fe, \n(b) Fe 81.25V18.75 and (c) Fe 81.25Al18.75. Results from the 6 distinct atomic configurations are shown \nin (b,c); the average densities of states at EF for Fe81.25V18.75 and Fe81.25Al18.75 are shown in \nTable  1.   \n "    },    {        "title": "2004.08082v1.Collective_coordinate_study_of_spin_wave_emission_from_dynamic_domain_wall.pdf",        "content": "arXiv:2004.08082v1  [cond-mat.mes-hall]  17 Apr 2020Collective coordinate study of spin wave emission from dyna mic\ndomain wall\nGen Tatara\nRIKEN Center for Emergent Matter Science (CEMS)\n2-1 Hirosawa, Wako, Saitama, 351-0198 Japan\nRubn M. Otxoa de Zuazola\nHitachi Cambridge Laboratory, J. J. Thomson Avenue,\nCB3 OHE, Cambridge, United Kingdom and\nDonostia International Physics Center, 20018 San Sebasti´ an, Spain\n(Dated: April 20, 2020)\nAbstract\nWe study theoretically the spin wave emission from a moving d omain wall in a ferromagnet.\nIntroducing a deformation mode describing a modulation of t he wall thickness in the collective\ncoordinate description, we show that thickness variation c ouples to the spin wave linearly and\ninducesspinwave emission. Thedominant emitted spinwave t urnsout tobepolarized in theout-of\nwall plane ( φ)-direction. The emission contributes to the Gilbert dampi ng parameter proportional\nto/planckover2pi1ωφ/K, the ratio of the angular frequency ωφofφand the easy-axis anisotropy energy K.\n1I. INTRODUCTION\nSpin wave (magnon) is an excitation playing essential roles in the tran sport phenomena\nin magnets, and its control, magnonics, is a hot recent issue. Beside s application interest\nfor devices, behaviours of spin waves have been drawing interests from fundamental science\nview points. Many theoretical studies have been carried out on gen eration of spin waves\nby dynamic magnetic objects such as a domain wall [1–6]. The subject is highly nontrivial\nbecause the wall is a soliton, which is stable in the absence of perturb ation, meaning that\nit couples to fluctuations, spin waves, only weakly in the ideal case, w hile in reality, various\nperturbations and dynamics leads to strong emission of spin waves. There are several pro-\ncesses that lead to the emission, and it is not obvious which is the domin ant process and\nhow large is the dissipation caused by the emission.\nThe low energy behavior of a domain wall in a ferromagnet is described in terms of\ncollective coordinates, its center of mass position Xand angle of the wall plane, φ0[7]. In\nthe absence of a pinning potential, a displacement of the wall costs n o energy owing to the\ntranslational invariance, and it is thus natural to regard Xas a dynamic variable X(t).\nThis is in fact justified mathematically; X(t) is a collection of spin waves that corresponds\nto the translational motion of the wall [8, 9]. It turns out that the c anonical momentum of\nthe ferromagnetic domain wall is the angle φ0. This is because the translational motion of\ncollective spins requires a perpendicular spin polarization, i.e., a tilting o f the wall plane.\nMathematically this is a direct consequence of the spin algebra, and is straightforwardly\nderived based on the equation of motion for spin (Landau-Lifshitz( -Gilbert) equation) [7] or\non the Lagrangian formalism [10]. In the absence of hard-axis anisot ropy energy, φ0is also\na zero mode. As zero modes, X(t) andφ0do not have linear coupling to the fluctuation,\nspin wave, and thus emission of spin wave does not occur to the lowes t order. In this case,\nthe second-order interactions to the spin wave give rise to the dom inant effect. In Ref. [1],\nthe coupled equations of motion for the wall and spin wave modes wer e solved classically\nand demonstrated that a damping indeed arises from the quadratic interaction. In the case\nof a strong hard-axis anisotropy, the plane of the wall is constrain ed near the easy-plane,\nφ0is frozen, resulting in a single variable system described solely by X(t) [9, 11]. The spin\nwave coupling and dissipation in this limit was discussed in Ref. [11].\nIn real materials, hard-axis anisotropy and pinning potential exist , andX(t) andφ0\n2are not rigorously zero modes. In other words, wall dynamics induc es a deformation and\nemission of spin wave is possible due to linear couplings. It was argued in Ref. [2] that there\nemerges a linear coupling when the wall driven by a spin-transfer tor que has a velocity ˙X\ndifferent from the steady velocity determined by the spin-transfe r torque, and the damping\ndue to spin wave emission was discussed. Numerical analysis of Ref. [3 ] revealed that spin\nwave emission occurs by the modulation of the wall thickness during t he dynamics. The\ncoupling to the wall velocity and second order in the spin wave was stu died analytically in\ndetail and dissipation was estimated in Ref. [6]. The energy dissipation proportional to the\nsecond-order in the wall velocity was found.\nIn this paper, we study the spin wave emission extending convention al collective coor-\ndinate representation of the wall [10]. As the domain wall is a soliton, t here is no linear\ncoupling of its center of mass motion to the spin wave field if deformat ion is ignored. We\nthus introduce a deformation mode of the wall, a change of the thick nessλ. This is a natural\nvariable in the presence of the hard-axis anisotropy energy, as th e thickness depends on the\nangleφ0as pointed out in Refs. [12, 13]. Following the prescription of spin wave expansion\n[9], we derive the Lagrangian for the three collective coordinates, t he center of mass position\nX(t), the angle of the wall plane φ0(t) and thickness λ(t), including the spin waves to the\nsecond order. It turns out that Xandφ0and their time-derivatives do not have linear\ncoupling to the spin wave, while ˙λdoes. This result is natural as Xandφ0are (quasi) zero\nmodes, and consistent with numerical observation [3]. It is shown th at the emitted spin\nwave is highly polarized; The dominant emission is the fluctuation of ang leφ, while that\nofθis smaller by the order of the Gilbert damping parameter α. The forward emission of\nwavelength λ∗∝v−1\nw, wherevwis the domain wall velocity, is dominant. The modulation of\nλis induced by the dynamics of φ0, and the contribution to the Gilbert damping parameter\ndue to the spin wave emission from this process is estimated from the energy dissipation\nrate. It was found to be of the order of αφ\nsw≃λ\na/planckover2pi1ωφ\nK, whereωφis the angular frequency of\nthe modulation of φ0,Kis the easy-axis anisotropy energy and ais the lattice constant.\nThis damping parameter contribution becomes very strong of the o rder of unity if /planckover2pi1ωφis\ncomparable to the spin wave gap, K, as deformation of the wall becomes significant in this\nregime.\n3II. COLLECTIVE COORDINATES FOR A DOMAIN WALL\nWe consider a one-dimensional ferromagnet along the x-axis with easy and hard axis\nanisotropy energy along the zandyaxis, respectively. The Lagrangian in terms of polar\ncoordinates ( θ,φ) of spin is\nL=LB−HS (1)\nwhere\nLB=/planckover2pi1S\na/integraldisplay\ndx˙φ(cosθ−1)\nHS=S2\n2a/integraldisplay\ndx/bracketleftbig\nJ[(∇θ)2+sin2θ(∇φ)2]+Ksin2θ(1+κsin2φ)/bracketrightbig\n(2)\nare the kinetic term of the spin (spin Berry phase term) and the Ham iltonian, respectively,\nJ >0,K >0andκK≥0being theexchange, easy-axis anisotropyandhardaxisanisotro py\nenergies, respectively, abeing the lattice constant. A static domain wall solution of this\nsystem is\ncosθ= tanhx−X\nλ0,φ= 0 (3)\nwhereλ0≡/radicalbig\nJ/Kis the wall thickness at rest. The dynamics of the wall is described\nby allowing the wall position Xandφas dynamic variables. This corresponds to treat\nthe energy zero mode of spin waves (zero mode) describing a trans lational motion and its\nconjugate variable φas collective coordinates [9]. This treatment is rigorous in the absenc e\nof pinning and hard-axis anisotropy but is an approximation otherwis e. Most previous\nstudies considered a rigid wall, where the wall thickness is a constant λ0. Here we treat\nthe wall thickness as a dynamic variable λ(t) to include a deformation and study the spin-\nwave emission. This treatment was applied in Ref. [13], but only static s olution of λwas\ndiscussed.\nAs demonstrated in Ref. [9], the spin wave around a domain wall in ferr omagnet is\nconveniently represented using\nξ=e−u(x,t)+iφ0(t)+η(x−X(t),t)(4)\nwhereφ0(t) is the angle of the wall,\nu(x,t) =x−X(t)\nλ(t)(5)\n4θ\nφ\nFIG. 1. Fluctuation corresponding to the real and imaginary part of the spin wave variable\n˜η= ˜ηR+i˜ηI. (a): The profile of ˜ ηantisymmetric with respect to the wall center, which turns\nout to be dominant excitation. (b): The real part ˜ ηRdescribes the deformation within the wall\nplane, i.e., modulation of θ, while the imaginary part ˜ ηIdescribes the out-of plane ( φ) fluctuation\nas shown in (c). Transparent arrows denotes the equilibrium spin configuration.\nandη(x−X,t) describes thespin-wave viewed inthemoving frame. Asitis obvious f romthe\ndefinition, the real and imaginary part of ηdescribe the fluctuation of θandφ, respectively.\nThe fluctuations antisymmetric with respect to the wall center, sh own in Fig. 1, turns out\nto be dominant. The ξ-representation of the polar angles are\ncosθ=1−|ξ|2\n1+|ξ|2, sinθsinφ=−iξ−ξ\n1+|ξ|2. (6)\nA. Domain wall dynamic variables\nWe first study what spin-wave mode the new variable λ(t) couples to, by investigating\nthe ’kinetic’ term of the spin Lagrangian, LB, which is written as\nLB=2i/planckover2pi1Sλ\na/integraldisplay\nduIm[ξ˙ξ]\n1+|ξ|2. (7)\nUsing Eq. (6) and\n∂tu=−1\nλ/parenleftBig\n˙X+u˙λ/parenrightBig\n, ∂ tξ=/parenleftbigg1\nλ/parenleftBig\n˙X+u˙λ/parenrightBig\n+i˙φ0+(∂t−˙X∇x)η/parenrightbigg\nξ,(8)\nwe have\n2iIm[ξ˙ξ] = 2i( ˙ηI+˙φ0−˙X∇xηI)|ξ|2(9)\n5sδ\nFIG. 2. Schematic figure showing the effect of asymmetric perpe ndicular spin polarization δs\ndue to the spin wave mode ϕ. The asymmetric torque (curved arrows) induced by asymmetr icδs\nrotates the spins within the wall plane, resulting in a compr ession of the wall, i.e., to ˙λ.\nwhereηi≡Im[η]. The kinetic term is expanded to the second order in the spin wave as\n(using integral by parts)\nLB=2/planckover2pi1S\na[φ0˙X+ϕ˙λ]+L(2)\nB (10)\nwhere\nϕ≡/integraldisplay\nduu\ncoshu˜ηI, (11)\nrepresents an asymmetric configuration of ˜ ηIand\nL(2)\nB≡2/planckover2pi1Sλ\na/integraldisplay\ndu/bracketleftbigg\n˜ηR↔\n∂t˜ηI−˙X˜ηR↔\n∇x˜ηI−2\nλtanhu/parenleftBig\n2˙X+u˙λ/parenrightBig\n˜ηR˜ηI/bracketrightbigg\n,(12)\nwhere ˜η≡η/(2coshu).\nWhen deriving Eq. (10), the orthogonality of fluctuation and the ze ro-mode,\n/integraldisplay\ndu˜η\ncoshu= 0, (13)\nwas used. Equation (10) indicates that ϕis the canonical momentum of λ. In fact, it\nrepresents the asymmetric deformation of angle φ, as the imaginary part of the spin wave,\n˜ηI, corresponds to fluctuation of φas seen in the definition, Eq. (4). Such an asymmetric\nconfiguration of φexerts a torque that induces a compression or expansion of the do main\nwall (Fig. 2), andthisiswhy ϕandλareconjugatetoeach other. Thecoupling ϕ˙λdescribes\nthe spin wave emission when thickness changes, as we shall argue lat er. The second term\nproportional to ˙Xin the bracket in Eq. (12) represents a magnon current induced in t he\nmoving frame (Doppler shift).\n6The Hamiltonian of the system is similarly written in terms of spin wave va riables to the\nsecond order as\nHS=KS2λ\na/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n+1+κsin2φ0/bracketrightBigg\n+2KS2λ\na/integraldisplay\ndutanhu\ncoshu˜ηR/bracketleftBigg\n−/parenleftbiggλ0\nλ/parenrightbigg2\n+1+κsin2φ0/bracketrightBigg\n+H(2)\nS,\n(14)\nwhere\nH(2)\nS≡2KS2λ\na/integraldisplay\ndu/bracketleftbigg\nλ2\n0[(∇˜ηR)2+(∇˜ηI)2]\n+ ˜ηR2/bracketleftbigg\n−λ2\n0\nλ2/parenleftbigg\n1−1\ncosh2u/parenrightbigg\n+/parenleftbigg\n2−3\ncosh2u/parenrightbigg\n(1+κsin2φ0)/bracketrightbigg\n+ ˜ηI2/bracketleftbiggλ2\n0\nλ2/parenleftbigg\n1−2\ncosh2u/parenrightbigg\n+κcos2φ0/bracketrightbigg\n+2κ˜ηR˜ηItanhusin2φ0/bracketrightbigg\n(15)\nIn the case of small κandλ≃λ0, the spin waves are described by a simple Hamiltonian as\nHsw≡2KS2λ\na/integraldisplay\ndu/bracketleftbigg\nλ2[(∇˜ηR)2+(∇˜ηI)2]+( ˜ηR2+ ˜ηI2)/parenleftbigg\n1−2\ncosh2u/parenrightbigg/bracketrightbigg\n+HD,(16)\nwhere\nHD≡2/planckover2pi1Sλ\na˙X/integraldisplay\ndu˜ηR↔\n∇x˜ηI, (17)\nis the Doppler shift term. For a constant wall velocity ˙X, it simply shifts the wave vector\nof the spin wave. Without the Doppler shift, the eigenfunction of th is Hamiltonian (16) is\nlabeled by a wave vector kas\nφk(u) =1√2π˜ωk(−ikλ+tanhu)eikλu, (18)\nwhere\n˜ωk≡1+(kλ)2(19)\nis the dimensionless energy of spin wave.\nDissipation function is\nW=α/planckover2pi1S\n2a/integraldisplay\ndx(˙θ2+sin2θ˙φ2)\n=/planckover2pi1Sλ\n2a\nα/parenleftBigg˙X\nλ/parenrightBigg2\n+α˙φ02+αλ/parenleftBigg˙λ\nλ/parenrightBigg2\n, (20)\n7whereαis the Gilbert damping parameter and αλ≡α/integraltext\nduu2\ncosh2u=π2\n12α.\nAsdrivingmechanisms ofadomainwall, weconsider amagneticfieldandc urrent-induced\ntorque (spin-transfer torque) [9, 14, 15]. A magnetic field applied a long the negative easy\naxis is represented by the Hamiltonian ( γ=e/mis the gyromagnetic ratio)\nHB=/planckover2pi1Sγ\naBz/integraldisplay\ndxcosθ. (21)\nUsing Eqs. (6)(13), we obtain\nHB=−2/planckover2pi1Sγ\naBz/parenleftbigg\nX+λ/integraldisplay\ndutanhu˜η2\nR/parenrightbigg\n. (22)\n(The first term is derived evaluating a diverging integral/integraltext\ndx1\n1+e2u(x)carefully introducing\nthe system size Las/integraltextL/2\n−L/2dx1\n1+e2u(x)and dropping a constant.) The magnetic field therefore\nexerts a force2/planckover2pi1Sγ\naBzon the domain wall.\nThe spin-transfer effect induced by injecting spin-polarized electr ic current is represented\nby a Hamiltonian having the same structure as the spin Berry’s phase termLB[9, 15]\nHSTT=−/planckover2pi1S\navst/integraldisplay\ndxcosθ(∇xφ), (23)\nwherevst≡aP\n2eSjis a steady velocity of magnetization structure under spin polarized current\nPj(Pis the spin polarization and jis the applied current density (one-dimensional)). The\nspin wave expression is\nHSTT=2/planckover2pi1S\navst/bracketleftbigg\nφ0+2/integraldisplay\ndx/parenleftbigg\n˜ηR∇x˜ηI+1\nλtanhu˜ηR˜ηI/parenrightbigg/bracketrightbigg\n. (24)\nAs has been known, a spin-transfer torque contributing to the wa ll velocity and does not\nwork as a force, as the applied current or vstcouples to φ0and not to X.\nThe equation of motion for the tree domain wall variables is therefor e obtained from Eqs.\n(10) (14) (20) and driving terms (22)(24) as\n˙X−αλ˙φ0=vcsin2φ0+2vcsin2φ0ζ+vst\n˙φ0+α˙X\nλ=˜Bz\nαλ˙λ\nλ=KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n−(1+κsin2φ0)/bracketrightBigg\n−˙ϕ−2KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n+(1+κsin2φ0)/bracketrightBigg\nζ,\n(25)\n8wherevc≡KSκ\n2/planckover2pi1λ,˜Bz≡γBzandϕ(Eq. (11)) and\nζ≡/integraldisplay\ndutanhu\ncoshu˜ηR, (26)\nare contributions linear in spin wave.\nIII. SPIN WAVE EMISSION\nIn this section we study the spin wave emission due to domain wall dyna mics. The\nemission is described by the linear coupling between the spin wave field a nd the domain wall\nin Eqs.(10) (14). Moreover, dynamic second-order couplings in Eqs . (12)(15) leads to spin\nwave excitation. In the first linear process, the momentum and ene rgy of the spin wave is\nsupplied by the dynamic domain wall, while the second process present s a scattering of spin\nwaves where the domain wall transfer momentum and energy to the incident spin wave.\nA. Linear emission\nWe here discuss the emission due to the linear interactions in Eqs.(10) (14) in the labo-\nratory (rest) frame. The laboratory frame is described by replac ingη(x−X(t),t) byη(x,t)\nin the derivation in Sec. IIA. It turns out that the Lagrangian Eq.(1 2) in the laboratory\nframe has no Doppler shift term and the term ˙X˜ηR˜ηIis half. The emitted wave has an\nangular frequency shifted by the Doppler shift from the moving wall. Using the equation of\nmotion, Eq. (25), the spin wave emission arises from the thickness c hange. The interaction\nHamiltonian reads in the complex notation ˜ η= ˜ηR+i˜ηI\nH(1)\nη(t) =˙λ(t)/integraldisplay\ndx(g˜η+g˜η), (27)\nwhere\ng(x)≡2/planckover2pi1S\na1\ncoshx−X(t)\nλ/parenleftbigg\n−αλtanhx−X(t)\nλ+ix−X(t)\nλ/parenrightbigg\n(28)\nLet us study here the emission treating λas a constant as its dynamics is taken account in\nthe first factor in the interaction Hamiltonian (27). The Fourier tra nsform of the interaction\n9is calculated using\n/integraldisplay∞\n−∞duei˜kuu\ncoshu=iπ2\n2sinhπ\n2˜k\ncosh2π\n2˜k\n/integraldisplay∞\n−∞duei˜kutanhu\ncoshu=π˜k\ncoshπ\n2˜k(29)\nas\nH(1)\nη(t) =−π2\n2λ˙λ(t)/summationdisplay\nk1\ncoshπ\n2kλeikX(t)/parenleftbigg\n˜ηIk(t)tanhπ\n2kλ+2\nπαλkλ˜ηRk(t)/parenrightbigg\n,(30)\nWe consider the case where the wall is approximated by a constant v elocityvw, i.e.,X(t) =\nvwt. The frequency representation of time-integral of Eq. (30) is\n/integraldisplay\ndtH(1)\nη(t) =−π2\n2/integraldisplaydΩ\n2π/integraldisplaydω\n2πλ˙λ(Ω)/summationdisplay\nk1\ncoshπ\n2kλ/parenleftbigg\n˜ηIk(t)tanhπ\n2kλ+2αλ\nπkλ˜ηRk(t)/parenrightbigg\nδ(ω−(kvw+Ω)),\n(31)\nIt is seen that the angular frequency of the emitted spin wave ( ω) iskvw+ Ω, i.e., that of\nthe thickness variation ˙λwith a Doppler shift due to the wall motion. The Doppler shift of\nangular frequency, δν≡kvw, is expected tobesignificant; For k= 1/λwithλ= 10−100nm\nandvw= 100 m/s, we have δν= 10−1 GHz. The function g(x) represents the distribution\nof the wave vector k, which has a broad peak at k= 0 with a width of the order of λ−1.\nTo have a finite expectation value ∝angb∇acketleft˜η∝angb∇acket∇ight, the angular frequency ωand wave vector kneeds\nto match the dispersion relation of spin wave, ω=ωk, i.e.,\nkvw+/planckover2pi1Ω =KS(1+(kλ)2). (32)\nThe angular frequency Ω is determined by the equation for λin Eq. (25), and is of the\norder of the angular frequency of φ0,ωφ. (See Sec. VA for more details.) Equation (32) has\nsolution for a velocity larger than the threshold velocity vth≡2KS\n/planckover2pi1λ/radicalBig\n1−/planckover2pi1ωφ\nKS. The emitted\nwave lengths k∗are (plotted in Fig. 4)\nk∗λ=/planckover2pi1vw\n2KSλ\n1±/radicalBigg\n1−/parenleftbiggvth\nvw/parenrightbigg2\n. (33)\nThe sign of k∗(direction of emission) is along the wall velocity, meaning that the emis sion is\ndominantly in the forward direction. The group velocity of the emitte d wave is of the same\n10vwλ∗\nFIG. 3. Schematic figure showing the spin wave emission from a domain wall with thickness\noscillation ( ˙λ) moving with velocity vw. The linear coupling leads to a forward emission of spin\nwave with wave length λ∗≡2π/k∗, wherek∗is defined by Eq. (33).\nvwk*\n•\n•\n•ω~=0.2\nω~=0.5\nω~=0.8\nFIG. 4. Plot of the wave length k∗of the emitted spin wave as function of wall velocity vwfor\n˜ω≡/planckover2pi1ωφ/KS= 0.2,0.5 and 0.8. Dotted line is k∗=/planckover2pi1\nKSλ2vw. Threshold velocity for the emission\nvthis denoted by circles.\norder as the wall velocity;\ndωk\ndk/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=k∗=2KS\n/planckover2pi1λ2k∗=vw\n1±/radicalBigg\n1−/parenleftbiggvth\nvw/parenrightbigg2\n. (34)\nThe dominant spin wave emission considered here is the antisymmetric excitation of the\nimaginary part ˜ ηIrepresenting the fluctuation of angle φ. The antisymmetric excitation of\nφis a natural excitation arising from the intrinsic property, the aniso tropy energy. The\neasy-axis anisotropy energy acts as a local potential VKfor each spin in the wall as in Fig.\n5. When the wall moves to the right, the spins ahead of the wall are d riven towards the\nhigh energy state, while the spins behind (left in Fig. 5) are towards lo w energy states. This\nasymmetry leads to an asymmetric local “velocity” of angle θ, and its canonical momentum\n11FIG. 5. The local potential VKfor spins in a domain wall arising from the easy axis anisotro py\nenergy,K. When the wall moves to the right, the spins right (left) of th e wall rotates towards\nhigh (low) energy states, resulting in an asymmetric local v elocity of rotation, exciting the angle\nφasymmetrically with respect to the wall center.\nφ. This role of Kto induce asymmetric φis seen in the equations of motion for polar angles\n[9]: Focusing on the contribution of the easy axis anisotropy, the ve locity of the in-plane\nspin rotation, sin θ˙φ=−KSsinθcosθis asymmetric with the wall center θ=π/2. Faster\nrotation in the left part of the wall (π\n2< θ < π) than the right part (0 < θ <π\n2) indicates\nthat the wall becomes thinner. In the equation of motion for λ(Eq. (25)), this effect is\nrepresented by the term −˙ϕon the right-hand side, meaning that asymmetric deformation\nmodeϕtends to compress the wall.\nB. Green’s function calculation\nWe present microscopic analysis of the spin wave emission using the Gr een’s function.\nWe consider here the slow domain wall dynamics limit compared to the sp in-wave energy\nscale and neglect the time-dependence of the variable uarising from variation of ˙X. The\ncalculation here thus corresponds to the spin wave effects in the mo ving frame with the\ndomain wall. The amplitude of the spin wave, ∝angb∇acketleft˜η∝angb∇acket∇ight, is calculated using the path-ordered\nGreen’s function method as a linear response to the source field ˙λ. The amplitude is\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−i/integraldisplay\nCdt′˙λ(t′)/integraldisplay\ndu′g(u′)/angbracketleftbig\nTC˜η(u,t)˜η(u′,t′)/angbracketrightbig\n(35)\nwhereCdenotes the contour for the path-ordered (non-equilibrium) Gre en’s function in the\ncomplex time and TCdetnoes the path-ordering. Evaluating the path-order, we obta in the\n12real-time expression of\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=/integraldisplay∞\n−∞dt′˙λ(t′)/integraldisplay\ndu′g(u′)Gr\nη(u,t,u′,t′) (36)\nwhere\nGr\nη(u,t,u′,t′)≡ −iθ(t−t′)/angbracketleftbig\n[˜η(u,t),˜η(u′,t′)]/angbracketrightbig\n(37)\nthe retarded Green’s function of ˜ η. The Green’s function is calculated expressing ˜ ηin terms\nof the orthogonal base for spin wave wave function [9] as\n˜η(u,t) =/summationdisplay\nkηk(t)φk(u), (38)\nwhereφkis the eigenfunction of Eq. (18) and ηkis the annihilation operator satisfying\n[ηk,ηk′] =δk,k′. The time-development of the operator is ηk(t) =e−iωktηk(0), where ωk≡\nKS˜ωkis the energy of spin wave. The retarded Green’s function thus is\nGr\nη(u,t,u′,t′) =−iθ(t−t′)/summationdisplay\nke−iωk(t−t′)φk(u)φk(u′)≡/integraldisplaydω\n2πe−iω(t−t′)Gr\nη(u,u′,ω) (39)\nwhere\nGr\nη(u,u′,ω) =/summationdisplay\nk1\nω−ωk+i0φk(u)φk(u′) (40)\nis the Fourier transform, + i0 denoting the small positive imaginary part. The Green’s\nfunction has a nonlocal nature in space, as seen from the overlap o f the spin wave function\n/summationdisplay\nkφk(u)φk(u′) =a\n2πλ/bracketleftbigg\nδ(u−u′)−1\n2/parenleftBig\ne−|u−u′|(1−tanhutanhu′)+sinh(u−u′)(tanhu−tanhu′)/parenrightBig/bracketrightbigg\n.\n(41)\nHere we use low-frequency approximation, namely, consider the eff ect of high-frequency\nmagnon compared to the wall dynamics and use Gr\nη(u,u′,ω)≃ −/summationtext\nk1\nωkφk(u)φk(u′). The\nretarded Green’s function then becomes local in time as Gr\nη(u,t,u′,t′) =δ(t−t′)Gr\nη(u,u′,ω).\nWe thus obtain\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−˙λ(t)/summationdisplay\nk1\nωkφk(u)/integraldisplay\ndu′g(u′)φk(u′) (42)\n13withuandu′havingX(t) of the equal time t. The integral/integraltext\ndu′g(u′)φk(u′) describing the\noverlap of spin-wave wave function and the domain wall is calculated u sing\n/integraldisplay\ndutanhu\ncoshuφk(u) =1√2π˜ωkπ\ncoshπ\n2kλ˜ωk\n2/integraldisplay\nduu\ncoshuφk(u) =1√2π˜ωkπ\ncoshπ\n2kλ(43)\nas\n/integraldisplay\ndug(u)φk(u) =2/planckover2pi1S\na1√2π˜ωkπ\ncoshπ\n2kλ/parenleftbigg\ni−αλ˜ωk\n2/parenrightbigg\n(44)\nThe spin wave amplitude emitted by the wall dynamics is therefore\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−˙λ(t)2/planckover2pi1\nKa/summationdisplay\nk1√2π˜ωkπ\ncoshπ\n2kλ1\n˜ωkφk(u)/parenleftbigg\ni−αλ˜ωk\n2/parenrightbigg\n(45)\nThe integral/summationtext\nk(˜ωk)−β1\ncoshπ\n2kλφk(u) (β=1\n2,3\n2) is real, and thus Re[ ∝angb∇acketleft˜η∝angb∇acket∇ight]/Im[∝angb∇acketleft˜η∝angb∇acket∇ight]≃α. As\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ightis odd in u, the emitted spin wave is an antisymmetric fluctuation of the angle\nφwith respect to the wall center (Fig. 1). (Because of low frequenc y approximation in\nderiving Eq. (42), the nonlocal nature (Eq. (41)) is smeared out in the result Eq. (45). )\nThe quantities representing the effects of spin wave emission on the wall dynamics in Eq.\n(25) are\nζ=/integraldisplay\ndutanhu\ncoshuRe[˜η] =˙λπ/planckover2pi1\n4Kaαλ/summationdisplay\nk1\ncosh2π\n2kλ≡αµζ˙λ\nλ\nϕ=/integraldisplay\nduu\ncoshuIm[˜η] =−˙λπ/planckover2pi1\nKa/summationdisplay\nk1\n˜ω2\nk1\ncosh2π\n2kλ≡µϕ˙λ\nλ(46)\nwhereµζ≡π3/planckover2pi1λ\n48Ka/summationtext\nk1\ncosh2π\n2kλandµϕ≡π/planckover2pi1λ\nKa/summationtext\nk1\n˜ω2\nk1\ncosh2π\n2kλ. The first integral is evaluated as\n/summationtext\nk1\ncosh2π\n2kλ=a/integraltextdk\n2π1\ncosh2π\n2kλ=2a\nπ2λand the second one is/summationtext\nk1\n˜ω2\nk1\ncosh2π\n2kλ≡2a\nπ2λγϕ, whereγϕ\nis a constant of the order of unity. The constants are therefore\nµζ=π/planckover2pi1\n24K\nµϕ=−2/planckover2pi1γϕ\nπK. (47)\nFrom Eq. (46), the averaged amplitude of the imaginary part of the emitted spin wave is\nof the order of/planckover2pi1˙λ\nKλ(the real part is a factor of αsmaller). As seen from Eq. (25), the time\nscale ofλdynamics is K//planckover2pi1, and thus the emitted spin wave amplitude can be of the order\nof unity if the modulation of λis strong, resulting in a significant damping. (See Eq. (61)\nbelow.)\n14C. Spin wave excitation due to second order interaction\nBesides emission due to the linear order interaction discussed above , spin waves are\nexcited also due to the second order interaction in Eqs.(10) (14) wh en the wall is dynamic.\nHere we focus on the effect of a dynamic potential in the Hamiltonian ( Eq. (16))\nV(x,t)≡4KS2\na1\ncosh2x−X(t)\nλ(48)\nand calculate the excited spin wave density in the laboratory frame b y use of linear response\ntheory. For a constant wall velocity, X(t) =vwt, the Fourier representation of the potential\nis\nVq(Ω) = 8π2KS2λ\naqλ\nsinhπ\n2qλδ(Ω−qvw), (49)\nThe potential thereforeinduces Dopplershift of qvwintheangular frequency ofthescattered\nspin wave. This dynamic potential induces an excited spin wave densit y asδn(x,t) =\niG<\nη(x,t,x,t), where G<\nηis the lesser Green’s function of spin wave. The linear response\ncontribution in the Fourier representation is\nδn(q,Ω) =i/summationdisplay\nk/integraldisplaydω\n2πVq(Ω)(n(ω+Ω)−n(ω))gr\nkωga\nk+q,ω+Ω (50)\nInthisprocess, theexcitedspinwave density hasthesamewavelen gth andangularfrequency\nof the driving potential Vq(Ω). This means that the excitation moves together with the\ndomain wall, and thus this is not an emission process. For slow limit, q≪kand Ω≪ω,\nusingn(ω+Ω)−n(ω) =n(ω+qvw)−n(ω)≃qvwn′(ω), we obtain a compact expression of\nδn(q,Ω) =i4πKS2\navw(qλ)2\nsinhπ\n2qλδ(Ω−qvw)/integraldisplaydω\n2π/summationdisplay\nkn′(ω)|gr\nkω|2(51)\nand the real space profile is\nδn(x,t) =δn0vw\nvatanhx−vwt\nλ\ncosh2x−vwt\nλ(52)\nwhereδn0=−4\nπ(KS)2/integraltextdω\n2π/summationtext\nkn′(ω)|gr\nkω|2andva≡Kλ//planckover2pi1is a velocity scale determined\nby magnetic anisotropy energy. The induced spin wave density has t hus an antisymmetric\nspatial profile with respect to the wall center and propagate with a domain wall velocity in\nthe present slowly varying limit. It is not therefore a spin wave emissio n, but represents the\ndeformation of the wall asymmetric with respect to the center.\n15IV. EQUATION OF MOTION OF THREE COLLECTIVE COORDINATES\nTheequationofmotion(25)including thespinwave emission effectsex plicitly istherefore\n˙X−αλ˙φ0=vcsin2φ0+2vcsin2φ0αµζ˙λ\nλ+vst (53)\n˙φ0+α˙X\nλ=˜Bz\nαλ˙λ\nλ=KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n−(1+κsin2φ0)/bracketrightBigg\n−µϕ¨λ\nλ−2KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n+(1+κsin2φ0)/bracketrightBigg\nαµζ˙λ\nλ.\n(54)\nThe spin-wave contribution of the first equation, the second term of the right-hand side, is of\nthe order αsmaller than the first term and is neglected. From the equations, we see that the\ndynamics of Xandφare not strongly coupled to the variation of the width. In particular ,\nwhenκis small, the dynamics of the wall center ( Xandφ) governed by the energy scale of\nK⊥=κKis much slower than that of a deformation mode λ, which is of the energy scale\nofK, and thus it is natural that the two dynamics are decoupled. Then κis not small, λ\naffects much the wall center dynamics.\nFor static case of λ, we have\nλ=λ0/radicalbig\n1+κsin2φ0, (55)\nas was argued in Refs. [12, 13]. Using this relation assuming slow dynam ics to estimate the\nspin-wave contribution in the equation for λ, we obtain\nµϕ¨λ+ ˜αλ˙λ=KS\n/planckover2pi1λ/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n−(1+κsin2φ0)/bracketrightBigg\n, (56)\nwhere ˜αλ≡αλ/parenleftbig\n1+2S\nπ/parenrightbig\n=π2\n12α/parenleftbig\n1+2S\nπ/parenrightbig\nis the effective damping for the width. The mass\nforλ,µϕ, was induced by the imaginary part of the spin-wave.\nV. DISSIPATION DUE TO SPIN WAVE EMISSION\nConsidering the action, which is a time-integral of the Lagrangian, a nd by use of integral\nby parts with respect to time, the linear interaction Hamiltonian, Eq. (27), is equivalent to\nH(1)\nη=−λFλ, where\nFλ≡2/integraldisplay\nduRe[g˙˜η], (57)\n16is a generalized force for variable λ. Using Eqs. (45)(43), it reads\nFλ=−¨λfλ, (58)\nwhere (neglecting the order of α2)\nfλ≡π/planckover2pi12S\nKa2/summationdisplay\nk1\n˜ω2\nk1\ncosh2π\n2kλ=2/planckover2pi12S\nπKλaγϕ. (59)\nThe energy dissipation rate due to the spin wave emission is therefor e\ndEsw\ndt≡ −˙λFλ=fλ\n2d\ndt˙λ2, (60)\nand thus Esw=fλ\n2˙λ2. As is seen from Eq. (56), the intrinsic energy scale governing the\ndynamics of λisK, and thus the intrinsic scale of ˙λ/λis of the order of K//planckover2pi1. The energy\ndissipation by an intrinsic spin-wave emission is estimated roughly as Ei\nsw≃Kλ\na, which is\nthe typical spin wave energy multiplied by the number of spin waves ex cited in the wall.\nThe quantitydEi\nsw\ndtcorresponds to a dissipation function Wi\nswinduced by the intrinsic spin\nwave emission. Considering the intrinsic frequency of λof the order of K//planckover2pi1, the Gilbert\ndamping parameter induced by the intrinsic emission is\nαi\nsw≃2aλ\n/planckover2pi1SfλK\n/planckover2pi1=4γϕ\nπ. (61)\nThis value is of the order of unity ( γϕis a constant), meaning that spin wave emission from\nthe thickness change is very efficient in dissipating energy from the w all. This result may\nnot be surprising if one notices that the intrinsic energy scale of thic kness change is that of\neasy-axis anisotropy energy K, which is the energy scale where significant deformation of\nthe wall is induced.\nA. Modulation of λdue toφ0dynamics\nIn most cases, the dynamics of λis driven by the time-dependence of φ0as seen in Eq.\n(56). Let us consider this case of a forced oscillation. We consider b y simplyfying φ0grows\nlinear with time, φ0=ωφt,ωφbeing a constant. Linearizing Eq. (56) using λ=λ+δλ,\nwhereλ≡λ0//radicalbig\n1+κ/2 is the average thickness, we have an equation of motion of a force d\noscillation,\nµϕ¨δλ+ ˜αλ˙δλ+µϕ(Ωλ)2δλ=KS\n2/planckover2pi1λκcos2ωφt, (62)\n17where Ω λ=K\n/planckover2pi1/radicalBig\nπS\nγϕ/parenleftbig\n1+κ\n2/parenrightbig\nis an intrinsic angular frequency of δλ. The solution having an\nexternal angular frequency of 2 ωφis\nδλ=δλcos(2ωφt−εφ), (63)\nwhere\nδλ≡κλπS\n4γϕ(K//planckover2pi1)2\n/radicalBig\n(Ω2\nλ−4ω2\nφ)2+4(˜αλωφ\nµϕ)2(64)\nis the amplitude of the forced oscillation and εφ≡tan−12˜αλωφ\nµϕ\nΩ2\nλ−4ω2\nφis a phase shift. A resonance\noccurs for ωφ= Ωλ/2. The energy dissipation rate for the emission due to forced oscillat ion\ninduced by dynamics of φis\ndEφ\nsw\ndt≃λ\na/parenleftbiggδλ\nλ/parenrightbigg2ω3\nφ\nK. (65)\nThe contribution to the Gilbert damping parameter is obtained from t he relationdEφ\nsw\ndt=\nαφ\nsw(˙λ/λ)2as\nαφ\nsw≃λ\na/planckover2pi1ωφ\nK. (66)\nLet us focus on the periodic oscillation of φ0, realized for large driving forces, namely, for\nBz> αKSκ\n2/planckover2pi1γ≡BW(γBz> αvc) for the field-driven case or j >eS2\n/planckover2pi1Pλ\naKκ≡ji(vst> vc) for\nthe current-driven case ( BWis the Walker’s breakdown field and jiis the intrinsic threshold\ncurrent [10]). The solution of the equation of motion (54) then read s\nφ0≃ωφt, (67)\nwhere (jis defined in one-dimension to have the unit of A=C/s)\nωφ≃˜Bz+αvst\nλ=γBz+aP\n2eSλαj. (68)\nTheGilbertdampingconstant duetospinwaveemission, Eq. (66), th usgrowslinearlyinthe\ndriving fields in this oscillation regime. Using current-induced torque f or a pinned domain\nwall would be straightforward for experimental observation of th is behaviour, although the\ncontribution to the Gilbert damping is proportional to αand not large, αφ\nsw≃α/planckover2pi1P\nej\nK(for\nS∼1,P∼1).\n18VI. SUMMARY\nWestudiedspinwaveemissionfromamovingdomainwallinaferromagne tbyintroducing\na deformation mode of thickness modulation as a collective coordinat e. It was shown that\nthe time-derivative of the thickness ˙λhas a coupling linear in the spin wave field, resulting\nin an emission, consistent with previous numerical result [3]. The domin ant emitted spin\nwave is in the forward direction to the moving domain wall and is strong ly polarized in the\nout-of plane direction, i.e., it is a fluctuation of φ. The dynamics of λis induced by the\nvariation of the angle of the wall plane, φ0, as has been noted [12, 13]. For a φ0with an\nangular frequency of ωφ, the Gilbert damping parameter as a result of spin wave emission\nisαφ\nsw≃λ\na/planckover2pi1ωφ\nK, whereKis the easy-axis anisotropy energy ( ais the lattice constant).\nThe present study is in the low energy and weak spin wave regime, and treating the\nhigher energy dynamics with strong spin wave emission is an important future subject.\nACKNOWLEDGMENTS\nGT thanks Y. Nakatani for discussions. This work was supported b y a Grant-in-Aid\nfor Scientific Research (B) (No. 17H02929) from the Japan Societ y for the Promotion of\nScience and a Grant-in-Aid for Scientific Research on Innovative Ar eas (No.26103006) from\nThe Ministry of Education, Culture, Sports, Science and Technolog y (MEXT), Japan.\n[1] D. Bouzidi and H. Suhl, Phys. Rev. Lett. 65, 2587 (1990).\n[2] Y. L. Maho, J.-V. Kim, and G. Tatara, Phys. Rev. B 79, 174404 (2009).\n[3] X. S. Wang, P. Yan, Y. H. Shen, G. E. W. Bauer, and X. R. Wang,\nPhys. Rev. 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Tatara, Physica E: Low-dimensional Systems and Nano structures 106, 208 (2019).\n20"    },    {        "title": "2005.05011v1.Manipulating_1_dimensinal_skyrmion_motion_by_external_magnetic_field_gradient.pdf",        "content": "Manipulating 1-dimensinal skyrmion  motion by external  magnetic field gradient  \nJaehun  Cho1, 2, Eiiti Tamura2, 3, 4, Chaozhe  Liu5, Soma Miki2, 3, Chun-Yeol You6, June-Seo Kim1, \nHikaru  Nomura2, 3, 5, Minori  Goto2, 3, Ryoichi Nakatani5, Yoshishige  Suzuki2, 3 \n \n1Division of Nanotechnology, Daegu Gyeongbuk Institute of Science and Technology (DGIST), \nDaegu, Republic of Korea  \n2Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan  \n3Center for Spintronics Research Network, Osaka University, Toyonaka, Osaka, Japan  \n4Department of Electronic Science and Engineering, Kyoto University, Kyoto, Kyoto, Japan  \n5Graduate School of Engineering, Osaka University, Suita, Osaka, Japan  \n6Department of Emerging Materials Science, Daegu Gyeongbuk Institute of Science and \nTechnology (DGIST ), Daegu, South Korea  \n \nAbstract  \nWe have investigated an analytic formula of the 1 -dimensional magnetic skyrmion dynamics \nunder external magnetic field gradient. We find excellent agreement between the analytical \nmodel and micromagnetic simulation results for various magnetic parameters su ch as the \nmagnetic field gradient, Gilbert damping constant. We also observe much faster velocity of the \nchiral domain wall  (DW)  motion . The chiral DW  is exist with smaller interfacial \nDzyaloshinskii -Moriya interaction energy density cases. These results p rovide to develop \nefficient control of skyrmion for spintronic devices.   \n \n \n \n \n \n \n \n \n  \nIntroduction  \nIn magnetic multilayer systems, the strong competition among Heisenberg exchange \ninteraction, Dzyaloshinskii -Moriya (DM) interaction, and magnetocrystalline  anisotropy can \nexhibit complex spin textures such as Skyrmions [ 1, 2], chiral magnetic domain walls (DWs) \n[3 - 5], Bloch lines [6, 7], and so on. A crucial contribution of interfacial DM interaction is \ndirectly related to a strong spin -orbit coupling at the interfaces between heavy metals and \nferromagnets combined with the broken inversion symmetry at the interfaces [8 - 12]. These \nexotic spin textures based on DM interaction are topologically stable, which are appropriate \nfor various applications as an inform ation carrier and manipulator [13, 14]. Indeed, numerous \ntheoretical and num erical studies have shown for the positive possibilities that magnetic \nskyrmions and chiral DWs could be essential ingredients for the next -generation spintronic \ndevices for storage devices and logic application [8, 14]. A single or bunch of these topological \nobjects are manipulated by laterally applied electrical currents due to  spin transfer torque  [15, \n16] or spin -orbit torque  [17, 18]. For the case below a critical current density, the magnetic \ntextures are fastened due to a  large pinning potential. When the electric current is larger than a \ncritical current density, the non -negligible displacements of topological objects occur. While \nthe electrical current injection technique is a promising method to drive multiple skrymions  \nand chiral DWs synchronously, a large critical current density caused by poor resistivities of \nmagnetic materials and the extremely narrow and long nanoscale wire architectures makes an \ninsurmountable obstacle which is so -called “Joule heating problem ” owing to Ohmic losses \n[19, 20]. Furthermore, the extra contributions such as Rashba effect and spin Hall effects lead \nto even more complex magnetization dynamics.  \nThe magnetic field driven chiral DWs and magnetic solitons  are received attentions because \nthe system is totally governed by the the Landau –Lifshitz (LL) equation , which is equivalent \nto the the Landau –Lifshitz –Gilbert  (LLG)  equation  when the magnetic damping constant of \nthe system is small enough. Moreover, vari ous manipulation idea such as DC or AC magnetic \nfield driven magnetic solitons, the transverse magnetic field pulse induced DW and skyrmion \nracetrack are demonstrated recently  [21, 22]. For realistic applications for information storage \ndevices or logic applications, the magnetic skyrmion or DW racetrack should be compatible \nfor the complementary metal -oxide -semiconductor (CMOS) architectures and the continuous \nminiaturization of CMOS architectures is essential for increasing the data capacity of the \ndevices. For the magnetic field driven skyrmion or DW motions in real spintronic d evices, the \nexternal magnetic field is applied from the ultrashort electrical current pulses passing through \nthe conduction lines adjacent skyrmion or DW racetracks to minimize the energy consumptions. \nNaturally, the applied magnetic field to the magnetic racetracks are not uniform due to Oersted \nlaw.  \nIn this work, the magnetic skyrmion and DW dynamics by applying gradient magnetic fields \nare systematically investigated by performing LLG simulations and Thiele approach . We described analytical and micromagnetic simulation studies of magnetic skyrmion dynamics in \na 1-dimensional nanowire, force by magnetic field gradient  along  the z-direction  while field \ngradient applied x-direction. According to the analytic model, the skyrm ion dynamics in the \nnanowire with magnetic field gradient is proportionality to the skyrmion width and radius, and \nits dynamics in good agreement with the analytic model and micromagnetic simulation results. \nIn micromagnetic simulations of DW dynamics, we observed much higher DW velocities than \nskyrmion one.  \n \nAnalytical model for magnetic field gradient driven skyrmions  \nWe briefly describe a simple theory for the skyrmion motion in our system. The motion of \nskyrmion in a two -dimensional film can be expressed by a Thiele ’s equations [ 23] for \nsufficiently slow varying and not too strong forces is  fellow:  \n𝑮×𝑹̇+𝛼𝒟𝑹̇=𝑭 (1) \nHere, R is the center coordinat e, G is the gyromagnetic coupling vector  with the winding \nnumber of the skyrmion q,  is the Gilbert damping constant, 𝒟 the dissipation dyadic and F \nthe external force e.g. by electric currents, magnetic field gradients, and thermal fluctuations.   \n The gyromagnetic coupling vector G is given by,  \n𝑮=∫𝑔̂𝑖𝑗𝑑𝑉𝑉. Here, 𝑔̂𝑖𝑗=𝑀𝑠\n|𝛾|𝒎⋅(𝜕𝒎\n𝜕𝑥𝑖×𝜕𝒎\n𝜕𝑥𝑗), Ms is the saturation magnetization and  is the \ngyromagnetic ratio . The components of the dissipative force, which is second term of Eq. (1), \n𝛼𝒟 describes the friction of the skyrmion, 𝒟=∫𝑑𝑖𝑗𝑑𝑉𝑉. Here, 𝑑𝑖𝑗=∫𝑀𝑠\n|𝛾|𝜕𝒎\n𝜕𝑥𝑖⋅𝜕𝒎\n𝜕𝑥𝑗𝑑𝑉𝑉.  \nWe study the effects of a non -uniform perpendicular magnetic field with a longitudinal \ndirection of nanowire . We neglect thermal fluctuation. As the skyrmion has a large magnetic \nmoment relative to the ferromagnetic nanowire, the field gradient leads to a force acting on the \nskyrmion. The force is given by  \n𝑭𝑔=−∫𝐻(𝑟)⋅𝜕𝑀\n𝜕𝑥𝑖𝑑2𝑟  \n=−𝑀𝑠∫𝐻(𝑟)⋅𝜕𝑛\n𝜕𝑥𝑖𝑑2𝑟. (2) \nBecause we apply the gradient magnetic field along  the z-direction  while field gradient \napplied x-direction,  force by magnetic field gradient can be expressed as  \n \n(𝑭𝑔(ℎ𝑔𝑧))\n𝑥=−𝑀𝑠ℎ𝑔𝑧∫𝑥⋅𝜕𝑛𝑧\n𝜕𝑥𝑑2𝑟 =𝑀𝑠ℎ𝑔𝑧∫𝑑𝑟𝑟2𝑠𝑖𝑛𝛩𝜕𝛩\n𝜕𝑟∞\n0∫𝑑𝜑𝑐𝑜𝑠2𝜑2𝜋\n0 \n=𝜋𝑀𝑠ℎ𝑔𝑧∫𝑑𝑟𝑟2𝑠𝑖𝑛𝛩𝜕𝛩\n𝜕𝑟∞\n0 \n=2𝜋𝑀𝑠ℎ𝑔𝑧𝑤2∫𝑑𝑡2𝑡2𝑠𝑖𝑛ℎ2(𝑥)𝑠𝑖𝑛ℎ(𝑡)𝑐𝑜𝑠ℎ(𝑡)\n[𝑠𝑖𝑛ℎ2(𝑥)+𝑠𝑖𝑛ℎ2(𝑡)]2∞\n0. (3) \nhere, ℎ𝑔𝑧  is perpendicular  magnetic field gradient , 𝑥=𝑅\n𝑤,  𝑡=𝑟\n𝑤 , where, R is the radius of \nskyrmion,  w is width of the skyrmion, and r is the position of magnetization.  \nThen, we consider the 1 -dimensional equation of skyrmion motion in Eq. (1). Using magnetic \nfield gradient force and dissipation above, we can calculate how t he velocity of skyrmion \ndepends  on the magnetic field gradient as below  \n(𝑅̇)𝑥=(𝑭𝑔)𝑥\n𝛼𝐷=𝑀𝑠ℎ𝑔𝑧∫𝑥𝜕𝑛𝑧\n𝜕𝑥𝑑𝑥𝑑𝑦\n𝛼𝑀𝑠\n|𝛾|∫𝜕𝑛⃗⃗ \n𝜕𝑥⋅𝜕𝑛⃗⃗ \n𝜕𝑥𝑑𝑥𝑑𝑦≅|𝛾|\n𝛼ℎ𝑔𝑧𝑤2𝑥2\n𝑥+𝑞2\n𝑥. (4) \nFor q = ± 1 and x ≫ 1, the velocity of skyrmion is simply calculated by  \n𝑣𝑥≈|𝛾|\n𝛼ℎ𝑔𝑧𝑤𝑅. (5) \nThe value of 𝑣𝑥 increase with increasing skyrmion radius . The magnetic field gradient force \nhas a large effect to large skyrmion because of large magnetic field differences from left and \nright side of skyrmion.  Surprisingly, the skyrmion velocity is proportional to not only radius of \nskyrmion but also width of skyrmion. The skyrmion width  and radius  are determined as \nskyrmion shape which is correlated with magnetic parameters, thereby, the skyrmion width is \nnot an independent variable in Eq. (5). Therefore, we find the relationship between the \nskyrmion velocities divided by skyrmion width ( 𝑣𝑥𝑤⁄) and the skyrmion radius as follows:  \n \n(𝑣𝑥\n𝑤)≈|𝛾|\n𝛼ℎ𝑔𝑧𝑅 (6) \n \nMicromagnetic simulations  \nThe micromagnetic simulations were performed by using the MuMax3 which is numerically \nsolve d the Landau -Lifshitz -Gilbert equation [ 24]. Here, we consider a nanowire  shaped with \n1000 -nm-length, 100 -nm-width, and 1.2 -nm-thick . The discretized cell for simulations is set to \nbe 2 × 2 × 1.2 nm3. In the simulation, a magnetic skyrmion is nucleated at the center of nanowire \nas shown in Fig. 1(a). As an example, here the mag netic parameters are  saturation \nmagnetization,  Ms = 560 kA/m , exchange stiffness constant, Aex = 12 pJ/m , perpendicular \nmagnetic anisotropy energy K = 1.1 MJ/m3, interfacial Dzyaloshinskii -Moriya interaction energy density D = 4.0 mJ/m2. To characterize the size and the shape of skyrmion, we take mz \nprofiles across the center of skyrmion. We approximate the line profile across a skyrmion  along \nthe longitudinal direction of nanowire  using a standard 360  o domain wall profile [ 25, 26] as \n𝑚𝑧=𝑐𝑜𝑠(2𝑎𝑟𝑐𝑡𝑎𝑛(𝑠𝑖𝑛ℎ((𝑟−𝑐\n𝑤))\n(𝑠𝑖𝑛ℎ(𝑅\n𝑤)))). (7) \nWhere r is the position of magnetization, c is the skyrmion center position, w is width of the \nskyrmion and R is the radius of skyrmion. The open black circles in Fig. 1(b) calculated values \nwith the fitting parameter using Eq. 7. The obtained R and w are 17.56  nm and 3.54 nm, \nrespectively . The material parameters in our simulations are chosen as table 1. The magnetic \nparameters selected for stable skyrmion conditions. In order to investigate skyrmion dynamics \nunder applied magnetic field gradient, ℎ𝑔𝑧=(𝐻final−𝐻initial)𝐿⁄ , is applied the whole \nnano wire along x - direction . We calculated the difference between the skyrmion center position \nat the initial time and the position at the final time as displacement . The final time is determined \nas the moment when the skyrmion stopped by nanowire edge. The determined skyrmion \nvelocity is defined as the skyrmion displacement with respect to time.  \n \nMagnetic field gradient driven skyrmions  in nanowire  \nThe skyrmion width a nd radius are determined by magnetic parameters such as Ms, Aex, K and \nD, the skyrmion width and radius cannot consider separate . Since Eq. (5) implies the skyrmion \nvelocity proportional to the skyrmion radius, however the skyrmion velocity are not matched \nwell with skyrmion radius as shown in inset of Fig. 2.  Because the values of skyrmion width \nare different with each skyrmions , the obtained skyrmion width are  from 3.57 to 5.78 nm  in the \nmicromagnetic simulation  results . The represented skyrmion radius dependence of the \nskyrmion velocities divided by skyrmion width ( 𝑣𝑥𝑤⁄) is plotted in Fig. 2. The black open \ncircles indicate the  simulation result for each skyrmion shown in table 1. The saturation \nmagnetization of a whole skyrmion is 560 kA/m. The red line is the theoretical calculation with \n = 176 GHz/T ,  = 0.3 , and ℎ𝑔𝑧 = 10.0 mT/ m. The value of 𝑣𝑥𝑤⁄ increase with increasing \nskyrmion radius . The micromagnetic results are  well matched as theoretical expectation results \nas shown in Eq. (6).   \nBased on the theoretical calculation, o ther magnetic parameters such as the damping constant \nand field gradient can also affect the skyrmion motion. In order to reveal the effects of skyrmion \ndynamics with various damping constant  () and field gradient  (ℎ𝑔𝑧 ), we perform \nmicromagnetic simulation s. Figure 3(a) and (b) indicate the 𝑣𝑥𝑤⁄  with various  and ℎ𝑔𝑧 , \nrespectively. In Fig. 3 (a) shows simulation results about the damping dependences of the \nskyrmion 𝑣𝑥𝑤⁄  for the skyrmion radius (open symbols) along with the 𝑣𝑥𝑤⁄  calculated \nwith Eq. (6) (solid lines).  The damping constant varied from 0.3 to 0.05  with  = 176 GHz/T, \nand ℎ𝑔𝑧  = 10.0 mT/ m. We find that the 𝑣𝑥𝑤⁄  value increase by decreasing the damping constant. The 𝑣𝑥𝑤⁄ for various ℎ𝑔𝑧 obtained by micromagnetic simulations (open symbols) \nand calculated  by Eq. (6) (solid lines)  as a function of skyrmion radius are displayed in Fig. 3  \n(b). The field gradient varied from 10. 0 to 2.5 mT/ m with  = 176 GHz/T,  = 0.05. As the \nfield gradient decrease, the value of 𝑣𝑥𝑤⁄ decreases. The agreements between the results of \nmicromagneti c simulation and Eq. (6) are excellent.  \nIn Fig. 4, we have plotted the 𝑣𝑥𝑤⁄  as a function of . Magnetic field gradient driven \nskyrmions  dependence  on the damping parameter is investigated at ℎ𝑔𝑧  = 10.0 mT/ m and \nskyrmion radius, R = 23.30 nm . Here the magnetic parameters are Ms = 560 kA/m, Aex = 12 \npJ/m, K = 1.1 MJ/m3, and D = 4.0 mJ/m2. The black open circles are micromagnetic simulation \nresults and red solid line is calculated values using Eq. (6). The results  show the inverse \nproportionality between 𝑣𝑥𝑤⁄ and  which is consistent with Eq. (6).  The decrease in alpha \nfrom 0.5 to 0.0125 resulted in an increase in 𝑣𝑥𝑤⁄ from 0.07 to 1.79 GHz.  \n \nSkyrmion dynamics for small interfacial DMI  \n \nWe set the DW center magnetization initially along the y direction and relax it with magnetic \nfield gradient , ℎ𝑔𝑧  as 10.0 mT/ 𝜇m. The iDMI energy density values coincides with the \nexperimentally determined values [refs.]. The snapshot of the magnetizations with Ms = 560 \nkA/m, Aex = 12 pJ/m, K = 1.1 MJ/m3,  = 0.1, and D = 0.1 mJ/m2, depicted in Fig . 5(a) and the \nchiral DW displacement s are shown in Fig. 5(b). As shown in colored lines in Fig. 5(b), the \nDW displacements are saturated after t = 35 ns for iDMI energy density is 0.1 mJ/m2 and t = \n33 ns for larger than 0.5 mJ/m2.  \n \nSummary  \nIn summary, we have investigated the skyrmion dynamics forced by a magnetic field gradient. \nThe velocity of skyrmion is predicted analytically  through the Thiele approach, which agrees \nwell with micromagnetic simulation results. The skyrmion dynamics is related with skyrmion \nshape, Gilbert damping, magnetic field gradient. Interestingly, skyrmion velocities divided by \nskyrmion width is proportional to th e skyrmion radius, magnetic field gradient and inverse \nGilbert damping constant. For the DW dynamics case which is small iDMI energy density, DW \nvelocity is much faster than the skyrmion velocities.  \n \n \n References  \n1. T. H. R. Skyrme, Nucl. Phys. 31, 556 (1962).,  \n2. U. K. Rössler, A. N. Bogdanov, and C. Pfleiderer, Nature 442 (7104), 797 (2006).  \n3. 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Vansteenkiste , Arne, Leliaert, Jonathan, Dvornik, Mykola, Helsen, Mathias, Garcia -\nSanchez, Felipe, and Waeyenberge, Bartel Van, The Design and verification of Mumax3, \nAIP Advances 4, 107133 (2014).  \n25. H.-B. Braun, Phys. Rev. B 50, 16485 (1994).,  \n26. A. Kubetzka, O. Pietzsch , M. Bode, and R. Wiesendanger,  Phys. Rev. B 67, 020401 (2003).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure captions  \nFig. 1 . (a) Spin structure of a hedgehog ( Néel-type) skyrmion texture in a nanowire film. (b)  \nLine profile across a skyrmion along the longitudinal direction of nanowire.  \n \nFig. 2 . The skyrmion radius dependence of skyrmion velocity divided by skyrmion width (𝑣𝑠∕𝑤) \nwith 𝛾 = 176 GHz/T, 𝛼 = 0.3, and magnetic field gradient ℎ𝑔𝑧  = 1.00 mT/ 𝜇m. inset: The \nskyrmion radius dependence of skyrmion velocity  \n \nFig. 3. The skyrmion radius dependent of skyrmion motion velocity divided by skyrmion width \n(𝑣𝑠∕𝑤) on the (a) damping constant, 𝛼 and (b)magnetic field gradient ℎ𝑔𝑧 dependence.  \n \nFig. 4 . The damping constant, 𝛼 dependent of skyrmion motion velocity divid ed by skyrmion \nwidth (𝑣𝑠 ∕𝑤) with 𝛾 = 176 GHz/T, 𝑅 = 23.39 nm, and magnetic field gradient ℎ𝑔𝑧 = 10.0 mT/ 𝜇m. \n \nFig. 5. (a) The snapshots of the magnetization with various time. (b) DW displacements as a \nfunction of time with various iDMI energy densities. All lines are the DW displacements \nanalyzed by Mz/Ms values.  \n \nTable caption  \nTable 1. magnetic parameters for chosen stable skyrmion. The saturation magnetization of a \nwhole skyrmions are 560 kA/m.  \n  \nFigure 1.  \n \nFigure 2.  \n \nFigure 3.  \n \nFigure 4.  \n \nFigure 5.  \n \n \n \n \n \nAex (pJ/m)  K (MJ/m3) D (mJ/m2) \n12 1.1 4 \n15 0.7 3 \n15 0.8 3.5 \n15 0.9 4 \n20 1.1 5 \n15 0.4 2 \nTable 1.  "    },    {        "title": "2005.07730v1.Slow_magnetosonic_wave_absorption_by_pressure_induced_ionization_recombination_dissipation.pdf",        "content": "Slow magnetosonic wave absorption by pressure induced\nionization-recombination dissipation\nTodor M. Mishonov1,a)and Albert M. Varonov1,b)\nInstitute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee Blvd., BG-1784 So\fa,\nBulgaria\n(Dated: 15 May 2020, 21:05)\nA new mechanisms for damping of slow magnetosonic waves (SMW) by pressure induced oscillations of the\nionization degree is proposed. An explicit formula for the damping rate is quantitatively derived. Physical\nconditions where the new mechanism will dominate are brie\ry discussed. The ionization-recombination\ndamping is frequency independent and has no hydrodynamic interpretation. Roughly speaking large area of\npartially ionized plasma are damper for basses of SMW while usual MHD mechanisms operate as a low pass\n\flter. The derived damping rate is proportional to the square of the sine between the constant magnetic \feld\nand the wave-vector. Angular distribution of the spectral density of SMW and Alfv\u0013 en waves (AW) created\nby turbulent regions and passing through large regions of partially ionized plasma is qualitatively considered.\nThe calculated damping rate is expressed by the electron impact cross section of the Hydrogen atom and in\nshort all details of the proposed damping mechanisms are well studied.\nI. SHORT INTRODUCTION\nBehind purely fundamental interest for plasma physics\npropagation of hydromagnetic (nowadays known as mag-\nnetohydrodynamic ) waves attracted signi\fcant attention\nand was strongly simulated by the development of the\nphysics of solar atmosphere and the eternal problems\nrelated to its heating.1{3It has already been con\frmed\nthat the magnetohydrodynamic (MHD) waves (both in-\ncompressible and compressible) are present in the solar\natmosphere and they have already been considered for\nheating of the solar chromosphere and corona.4{7Models\nadapted to study heating problems of solar atmosphere\ninclude two \ruids coupled through collisions and chemi-\ncal reactions, such as impact ionization and radiative re-\ncombination with imposed initial thermal and chemical\nequilibrium. Within this approach the plasma heating is\ndominantly wave-based and the main energy source for\nheating are the excited fast magnetosonic \ructuations,8\nwhile older studies of FMW heating can be found in\nRef. 9 for instance.\nIt is worthwhile to mention also works on overre\rection\nor swing ampli\fcation in shear \row of slow magnetosonic\nwaves (SMW); see for example Refs. 10 and 11.\nIn a review on partially ionized astrophysical plasmas12\nit is shown that viscosity plays no important role in the\ndamping of chromospheric Alfv\u0013 en waves and recently it is\nconcluded that the solar corona electrical resistivity has\nonly very small impact, while and thermal conduction\nand viscosity contribute equally.13Therefore, the ques-\ntion of chromospheric heating due to the ion-neutral in-\nteraction will require further studies in the future. A\ncomplete review on the problem requires many hundred\ncitations, but here we mention the importance of two\na)E-mail: mishonov@bgphysics.eu\nb)E-mail: varonov@issp.bas.bg\ruid approach for consideration of MHD waves in par-\ntially ionized plasmas.14\nA. Scenario\nWhen a slow magnetosonic wave (SMW) propagates\nthrough partially ionized plasma, the oscillations of the\npressure creates oscillations of the temperature and gen-\nerates small oscillations of the degree of ionization \u000b.\nThose pressure induced deviations of the chemical equi-\nlibrium gives an extra entropy production and energy\ndissipation of the SMW.\nThis additional mechanism does not work for the\nAlfv\u0013 en waves (AW) and in spite of common dispersion\nand damping of AW and SMW in (MHD) approach at\nsmall magnetics \feld their ionization-recombination ab-\nsorption can be completely di\u000berent.\nThe purpose of the present work is to present an ex-\nplicit formula for the chemical damping and to consider\nin short when the predicted new damping mechanism is\nimportant and dominates and how it can be observed.\nThe article is organized as follows. In order to create\nthe necessary system of notions and notations following\nLandau and Lifshitz3in the next Sec. II we will recall\nthe physics of SMW. Then we derive in Sec. III our new\nresult for ionization-recombination absorption. Finally\nwe will discus in Sec. IV\nII. RECALLING SMW\nIn this section we will repeat these details which are\ncommon for Alfv\u0013 en waves (AW) and SMW. The di\u000ber-\nences between AW and SMW which are our new result\nwe derive after that.arXiv:2005.07730v1  [physics.plasm-ph]  15 May 20202\nA. Dispersion of MHD Waves\nLow density hydrogen plasma we approximate as a\ncocktail of ideal gases of electrons, protons and neutral\natoms with pressure pand mass density \u001a\np=nT; n =ne+np+n0; (1)\n\u001a=n\u001aM; n \u001a=np+n0; (2)\nwhere temperature Tis written in energy units and M\nis the proton mass. The sound velocity is de\fned by the\nadiabatic compressibility\ncs=s\u0012@p\n@\u001a\u0013\ns; (3)\nfor which the standard expression from the averaged\natomic mass of the cocktail hMi\ncs=s\n\rpT\nhMi;hMi=npM+n0M+nem\nnp+n0+ne; (4)\ncv= 3=2; cp=cv+ 1 = 5=2; \r =cp=cv= 5=3 (5)\nwherecvandcpare the heat capacities per atom and m\nis the electron mass.\nSmall amplitude MHD waves we treat as small varia-\ntions of the magnetic \feld b, density\u001a0, pressurep0and\ntemperature T0\nB=B0+b; \u001a =\u001a0+\u001a0; (6)\np=p0+p0; T =T0+T0(7)\nfrom their constant values. The index 0 we omit where\nit is obvious. The variations of the pressure are related\nwith variations of the density\np0\u0019c2\ns\u001a0; (8)\naccording the de\fnition of the sound speed. Here we\nrecall also the equations of state pV\r= const and\nTV1=cv= const for S= const (constant entropy S) and\ngive the relations between variations of the temperature,\npressure and density15\nT0\nT=1\ncv\u001a0\n\u001a=1\ncpp0\np;\u001a0\n\u001a=1\n\rp0\np; \r\u0011cp\ncv=5\n3:(9)\nFor a weak amplitude plane wave the variations of all\nvariables are proportional to the imaginary exponent\n/ei(k\u0001r\u0000!t)and phase velocity u\u0011!=k the ratio be-\ntween the frequency !and the modulus k=jkjof the\nwave-vector. For a plane wave the time tand space r\nderivatives are reduced to multiplication\n@t=\u0000i!;r= ik (10)\nand the MHD equations, omitting index 0 and imaginary\nunit i, reads3\n\u0000!\u001av=\u0000c2\nsk\u001a0+B\u0002(k\u0002b)=\u00160; \u0016 0= 4\u0019;(11)\n\u0000!b=k\u0002(v\u0002B); !\u001a0=\u001ak\u0001v; \"0= 1=4\u0019;(12)where k\u0001b= 0. We use Gaussian units but in SI&C:\n\u00160\u0003= 10\u00007and\"0==c210\u00007, i.e. all formulae in the\npresent work are written in system invariant form. In\nHeaviside{Lorentz units \u00160= 1 and\"0= 1. Thex-axis\nis chosen along the wave-vector k=kex, andy-axis is in\nthe plane of the wave-vector and the constant magnetic\n\feld\nB=Bxex+Byey; Bx=Bcos\u0012; By=Bsin\u0012;\nsee Fig. 1. The unit vector along the external magnetic\nx,k B\nyθ\nvgr=VAcosθkB=|k·B|/|B|\nω=VAkB\nFIG. 1: Geometry of propagating of SMW in a constant\nexternal magnetic \feld B. The group velocity of the\npropagating wave packet vgris along the external\nmagnetic \feld. The wave vector kis along the normal\nof wave fronts (equiphase planes) shown here with lines\nparallel to the yaxis.\n\feld is\neB=B=B= cos\u0012ex+ sin\u0012ey (13)\nDividing by k, the nonzero components of the MHD equa-\ntions Eqs. 11 and 12 read\n\u001au\u0012\n1\u0000c2\ns\nu2\u0013\nvx=Byby=\u00160; (14)\n\u001auvy=\u0000Bxby=\u00160; (15)\nuby=Byvx\u0000Bxvy: (16)\nExpressing velocity components vxandvyfrom the \frst\ntwo equations and substituting it in the third one gives a\nquadratic equation for the phase velocity u=!=kwhich\nhave the solutions describing fast (f) and slow (s) mag-\nnetosonic waves3\nu2\nf;s=1\n2n\nV2\nA+c2\ns\u0006\u0002\n(V2\nA+c2\ns)2\u00004c2\n\u0012V2\nAc2\ns\u00031=2o\n:(17)\nFor small magnetic \felds for SMW wave we have\nu=us\u0019VAc\u0012\u001ccs; c\u0012\u0011jcos\u0012j; (18)3\nwhere\nVA\u0011Bp\u00160\u001a; \u001aV2\nA=B2=\u00160 (19)\nis the speed of AW and uA\u0011VAc\u0012is the modulus of\nits projection along the x-axis. In such a way for the\ndispersion of SMW we have\n!=jVA\u0001kj;vgr\u0011@!\n@k=VAsgn(B\u0001k): (20)\nThe frequency of SMW can be expressed by the projec-\ntion of Alfv\u0012 en speed along the wave vector uA\n!=uAk=VAkB; (21)\nu\u0019uA\u0011VAc\u0012\u001ccs; kB=kc\u0012 (22)\nor by projection of the wave-vector along the magnetic\n\feldkB. For SMW the last inequality substituted in\nEqs. (14) and (15) gives\n\u0012c2\ns\nu2\u00001\u0013\n\u0019c2\ns=u2\u001d1; (23)\nand we have approximate expressions for the components\nof the velocity\nvy=\u0000Bx\n\u00160\u001auby;k\u0001v=\u0000Bxk\n\u00160\u001auby (24)\nvx\u0019\u0000u\nc2sBy\n\u00160\u001aby; bx= 0: (25)\nThen from Eq. (12) we obtain the variation of the density\n\u001a0\n\u001a=k\u0001v\n!=Byby\n\u00160\u001ac2s; (26)\nand from Eq. (8)\np0=c2\ns\u001a0=\u0000Byby\n\u00160=\u0000B\n\u00160s\u0012by; s\u0012\u0011sin(\u0012);(27)\nwe express the variations of the pressure proportional to\nthe small wave component of the magnetic \feld\nby=b0cos(kx\u0000!t);b=b0cos(kx\u0000!t): (28)\nThe unit vector eb\u0011b0=b0=eyalong the oscillating\ncomponent of the magnetic \feld has angle \u0019=2\u0000\u0012with\nthe constant one\n(eb\u0001eB)2=s2\n\u0012: (29)\nNow we can express the averaged density of the wave\nenergy which according to the virial theorem is twice the\naveraged density of the magnetic energy\nE= 2\u001cb2\n2\u00160\u001d\n=b2\n0\n2\u00160;q=Evgr (30)and the averaged density of the pressure oscillations\nwhich is one important ingredient of the forthcoming\nanalysis\n\n(p0)2\u000b\n=B2\n\u00160s2\n\u0012b2\n0\n2\u00160=B2\n\u00160Es2\n\u0012: (31)\nWe mention that the pressure oscillations disappear for\nwave-vector parallel to the magnetic \feld ( kkBor\nsin\u0012= 0) and v\u0001k=b\u0001k= 0, the waves are purely\ntransverse.\nAfter the consideration of dissipationless wave prop-\nagation in the next subsection we recall the results for\nSMW damping.\nB. MHD Absorption\nIn the WKB approximation we suppose that wave am-\nplitudes have small exponential decay e\u0000\rttas a function\nof time or space extinction e\u0000\rxif we trace a traveling\nwave packet.\nFor the energy \rux and density we have quadratic de-\npendence e\u00002\rxxand the extinction\n\rx=QMHD\n2qx(32)\nis given by the ratio of the time averaged power of MHD\ndissipation QMHD and the energy \rux qx.3In dissipation-\nless approximation the substitution of\nby=b0cos(kx\u0000!t); bx=bz= 0 (33)\nand the derived velocity vyEq. (24) in the formula for\nthe Pointing vector in MHD\nq\u0019S\u0019B\u0002(v\u0002B)=\u00160 (34)\ngives\nS=VAE; (35)\nE=\nb2=2\u00160+\u001av2=2\u000b\n=b2\n0=2\u00160; (36)\nin agreement with Eq. (30). For the x-component we\nhave\nqx=uAE: (37)\nThe small dissipation is proportional to the dissipative\ncoe\u000ecients\nQMHD =\u0017k\u001a\u0012@v\n@x\u00132\n+\u0017m\u0012@b\n@x\u00132\n=\u00160 (38)\nparamererized by kinematic \u0017k=\u0011=\u001aand magnetic dif-\nfusivity\u0017m=\"0c2%, where\u0011is viscosity coe\u000ecient and %\nis the Ohmic resistivity. Expressing vyfrom Eq. (15) and\nassumingvx\u00190 from Eq. (14) meaning that div v\u00190\nafter some algebra we obtain\n\rx=\rt\nua; \rt=1\n2(\u0017k+\u0017m)k2: (39)4\nIf we trace a wave packet of SMW propagating along\nmagnetic force lines Bat distance l=x=c\u0012for the energy\ndamping/e\u00002\rllwe have the extinction\n\rl=QMHD\n2VAE=\rt\nVA=(\u0017k+\u0017m)k2\n2VA: (40)\nThis space damping rate does not depend on the angle\n\u0012. For AW we have velocity vzand magnetic \feld bz\noscillations only normal to the ( k-B) plane direction but\nfor small magnetic \feld VA\u001ccsthe dispersion and wave\ndamping are the same. The di\u000berence appears when we\nanalyze the chemical damping of SMW.\nAfter this recall of the well-known result we analyze in\nthe next section the chemical damping.\nIII. IONIZATION-RECOMBINATION ABSORPTION\nThe degree of the ionization\n\u000b=np\nn\u001a(41)\nis a result of the continuous balance of ionization and\nrecombination processes\ndnp\ndt=\fn0ne\u0000\rrecnpn2\ne (42)\nwith temperature dependent rates of electron impact ion-\nization\f(T) and two electron recombination \r(T):For\ndense enough plasma the radiative processes have neg-\nligible contribution, especially for optically thin plasma\nregions.16,17\nIn thermal equilibrium the degree of ionization is given\nby Saha equation15,18\n\u0016np\u0016ne\n\u0016n0=nS\u0011\u0012mT\n2\u0019~2\u00133=2\ne\u0000I=T; (43)\nwhereIis the ionization energy. The rates of ionization\nand recombination processes in equilibrium are equal\n\u0017=\f\u0016ne\u0016n0=\rrec\u0016n2\ne\u0016np: (44)\nThe variable \u0017(T) gives the number of reactions\n\f: H + e\u0000!p + e + e; (45)\n\rrec: p + e + e\u0000!H + e (46)\nper unit volume and unit time. From this rate one can\ncreate a temperature dependent variable\nQ\u0013\u0011T\u0017 (47)\nwith dimension of power density; energy per unit volume\nand unit time.\nWhen MHD waves propagate through the plasma os-\ncillations of the pressure p0, density\u001a0and the tempera-\ntureT0perturbate the chemical equilibrium and inducevariations of the chemical composition and ionization de-\ngree\u000b. This extra chemical chaos creates an additional\nmechanism of increasing of entropy and wave energy dis-\nsipation\nQion=Q\u0013\n\u001f2\u000b\n; (48)\nwhere brackets denotes wave period averaging.\nThe main detail of the chemical energy dissipation is\nthe deviation from the chemical equilibrium\n\u001f\u0011nenp\nn0nS\u00001 =nenp\nn0\u0016n0\n\u0016ne\u0016np\u00001 (49)\ndescribed in detail in a recent Ref. 19. We suppose that\nthe variations of the chemical chomposition are relatively\nsmall, and the frequency of SMW is high enough\n!\u001d\u000b(1\u0000\u000b)\fn\u001a; (50)\n\u0016ne= \u0016np=\u000b\u0016n\u001a;\u0016n0= (1\u0000\u000b)n\u001a: (51)\nIn equilibrium \u0016 \u001f= 0 and we have to calculate the small\nsmall change of the variable \u001fdescribing the deviation\nfrom the chemical equilibrium substituting in Eq. (49) all\nnecessary details\nne= \u0016ne+n0\ne; np= \u0016np+n0\np; n 0= \u0016n0+n0\n0;(52)\nnS(T+T0) =nS+n0\nS=nS(T) +dnS\ndTT0: (53)\nFor linearized waves and small j\u001fj\u001c1 we have\n\u001f\u0019n0\ne\nne+n0\np\nnp\u0000n0\n0\nn0\u0000n0\nS\nnS: (54)\nAll relative changes of the variables can be expressed by\nthe relative change of the pressure\nn0\ne\nne=n0\np\nnp=n0\n0\nn0=\u001a0\n\u001a=1\n\rp0\np(55)\nand the Saha density\nn0\nS\nnS=dnS\nnSdTT0=\u0012I\nT+cv\u0013T0\nT=\u0012I\nT+cv\u0013p0\ncpp:(56)\nDue to detailed text-book recalling of the SMW dynamics\nwe easily arrive at a simple result\n\u001f=\u0012I\ncpT+2\n\r\u0013p0\np(57)\nand its square can be easily averaged using Eq. (31)\n\n\u001f2\u000b\n=\u0012I\ncpT+2\n\r\u00132\n(p0)2\u000b\np2(58)\n=\u0012I\ncpT+2\n\r\u00132B2\n\u00160pE\nps2\n\u0012: (59)5\nMultiplying with the power density rate we \fnally derive\nthe main result of the present work: the mean energy\ndissipation of a SMW propagating in magnetized plasma\nQion=Q\u0013\u0012I\ncpT+2\n\r\u00132B2\n\u00160pE\nps2\n\u0012: (60)\nNow for the time damping we obtain\n~\rt=Qion\n2E=Q\u0013\np\u0012I\ncpT+2\n\r\u00132B2=2\u00160\nps2\n\u0012; (61)\n(62)\nand for the extinction at low temperatures T\u001cIwe\nhave an additional chemical term\n~\rl\u0019Qion\n2EVA=Q\u0013\npVA\u0012I\ncpT\u00132B2=2\u00160\nps2\n\u0012; (63)\nwhich disappears at small angles \u0012\u001c1:In the next \fnal\nsection we will discuss the di\u000berence between two damp-\ning mechanisms giving total SMW extinction\n\rtot=\rl+ ~\rl: (64)\nIV. DISCUSSION AND CONCLUSIONS\nThe angular dependence of the chemical damping ob-\ntained in Eq. (63) ~ \rl/sin2\u0012is the main di\u000berence be-\ntween the chemical damping and the MHD one. Here we\nwish to emphasize also that the derived new ionization-\nrecombination damping is frequency independent and has\nno hydrodynamic sense as second viscosity, for example.\nThe MHD damping according to Eq. (40) \rl/k2/!2is\nproportional to the square of the wave-vector and square\nfrequency.\nRoughly speaking MHD damping is a low pass \fl-\nter while ionization-recombination mechanism is a bass\ndamper.\nImagine that turbulence generates broad distribution\nof MHD waves and the angular distribution of the spec-\ntral density is almost constant at small angles between\nthe wave-vector and constant magnetic \feld\ncos\u0012=k\u0001B\nkB: (65)\nIf then SMW pass through a partially ionized region with\nlengthlthe chemical damping gives transmission coe\u000e-\ncient\n~TSMW = e\u00002~\rll= exp\u0012\n\u0000\u00122\n2\u00122\n0\u0013\n; (66)\n1\n2\u00122\n0=2\u0017Tl\npVA\u0012I\ncpT\u00132B2=2\u00160\nps2\n\u0012; (67)\n\u00120=cpT\n2Ir\npVA\fpl\n\u0017Tl\u001c1; \f pl\u0011p\npB; (68)\npB=B2=2\u00160; p = (\u0016ne+ \u0016np+ \u0016n0)T: (69)In other words, strong ionization-recombination absorp-\ntion gives a cumulative small angle distribution of SMW.\nWaves with signi\fcant angles \u0012are absorbed and domi-\nnantly AW will pass through large area of partially ion-\nized plasma.\nHow this can be checked by observations. Imagine that\nin an observation point we have a good record of the time\ndependence of the magnetic \feld B(t). Time averaging\ncan give mean value B0and orientation eBof the con-\nstant component of the magnetic \feld\nB0=hB(t)i;eB=B0=jB0j:\nThen we can make Fourier analysis and calculate the\nwave components of the magnetic \feld for all frequen-\ncies!\nb0\n!=h(B(t)\u0000B0) cos(!t)i;e0\n!=b0\n!=jb0\n!j;\nb00\n!=h(B(t)\u0000B0) sin(!t)i;e00\n!=b00\n!=jb00\n!j:\nFor the considered in Sec. II example we have\neB= cos\u0012ex+ sin\u0012ey;e0\n!=ey;eB\u0001e0\n!= sin\u0012:\nIn the general case we have di\u000berent angles for all Fourier\nfrequencies\nsin(\u00120\n!) =eB\u0001e0\n!;sin(\u001200\n!) =eB\u0001e00\n!;\nand it is worthwhile the study probability distribution\nfunction of angles \u0012. Our simple consideration predicts\nGaussian distribution\nP(\u0012)/exp(\u0000\u00122=2\u00122\n0) (70)\ncreated by long regions of partially ionized plasma.\nRoughly speaking SMW \fltered by large regions of par-\ntially ionized plasmas will be almost transverse.\nEvery similarity with phenomena in the magnetized at-\nmosphere even in the nearest star is random. We present\na purely academic study.\nLast but not least the ionization rate \f=hv\u001biis given\nby the Maxwell velocity vaveraging of the electron im-\npact ionization cross-section \u001band all details of the pro-\nposed new damping mechanisms are well studied.\nThe considered in the present work the chemical damp-\ning of the pressure oscillations in some sense belongs to\nthe notions of plasma multi-\ruid approach. Not only\nrelative velocity between di\u000berent \ruids12,14creates dis-\nsipation as a friction forces. Periodic oscillations around\nthe Saha equilibium for some MHD modes of partially\nionized plasmas can give even bigger dissipation and in-\ndispensably have to be taken into account in the arsenal\nof the plasma physics notions.\nACKNOWLEDGMENTS\nThe authors appreciate stimulating discussions cor-\nrespondence with Dantchi Koulova, Kamen Kozarev,\nHassan Chamati, Yavor Boradjiev, Nedko Ivanov, and\nStanislav Varbev.6\nDATA AVAILABILITY STATEMENT\nData sharing is not applicable to this article as no new\ndata were created or analyzed in this study, which is a\npurely theoretical one.\n1H. Alfv\u0013 en, \\Existence of Electromagnetic-Hydrodynamic\nWaves\", Nat. 150, 405 (1942).\n2H. Alfv\u0013 en, \\Granulation, magnetohydrodynamic waves, and the\nheating of the solar corona\", Mon. Not. Roy. Astr. Soc. 107, 211\n(1947).\n3L. D. Landau and E. M. Lifshitz, Electrodynamics in Continuous\nMedia in L. D. Landau and E. M. Lifshitz, Course of Theoretical\nPhysics, Vol. VIII (Pergamon Press, New York, 1960), Sec. 52\n\\Hydromagnetic waves\".\n4I. M. Rutkevich and M. Mond, \\Localization of slow magne-\ntosonic waves in the solar corona\", Phys. Plasmas 3(1), 392\n(1996).\n5L. M. B. C. Campos, \\On the viscous and resistive dissipation of\nmagnetohydrodynamic waves\", Phys. Plasmas 6(1), 57 (1999).\n6A. G. de Wijn, B. De Pontieu and R. J. Rutten, \\Chromospheric\nand Transition-Region Dynamics in Plage\" in The Physics of\nChromospheric Plasmas, 9-13 October, 2006, Coimbra, Portu-\ngal, ed. by P. Heinzel, I. Dorotovi\u0015 c, and R. J. Rutten, (Astronom-\nical Society of the Paci\fc Conference Series 368, San Francisco,\n2007);\n\\Fourier Analysis of Active-region Plage\", Astrophys. J 654,\n1128 (2007).\n7R. J. Morton, G. Verth, D. B. Jess, D. Kuridze, M. S. Ruderman,\nM. Mathioudakis and R. Erd\u0013 elyi, \\Observations of ubiquitous\ncompressive waves in the Sun's chromosphere\", Nat Commun 3,\n1315 (2012).\n8Y. G. Maneva, A. A. Laguna, A. Lani and S. Poedts, \\Multi-\n\ruid Modeling of Magnetosonic Wave Propagation in the Solar\nChromosphere: E\u000bects of Impact Ionization and Radiative Re-\ncombination\", Astrophys. J 836, 197 (2017).\n9W. Sahyouni Zh. Kiss'ovski and I. Zhelyazkov, \\Chromospheric\nand Coronal Heating Due to the Radiation and Collisional Damp-\ning of Fast Magnetosonic Surface Waves\", Z. Naturforsch. A\n42a(12), 1443 (1987).10Z. D. Dimitrov, Y. G. Maneva, T. S. Hristov and T. M. Mis-\nhonov, \\Over-re\rection of slow magnetosonic waves by homoge-\nneous shear \row: Analytical solution\", Phys. Plasmas 18, 082110\n(2011).\n11G. Gogoberidze, G. D. Chagelishvili, R. Z. Sagdeev, and\nD. G. Lominadze, \\Linear coupling and overre\rection phenomena\nof magnetohydrodynamic waves in smooth shear \rows\", Phys.\nPlasmas 11, 4672 (2004).\n12J. L. Ballester, I. Alexeev, M. Collados, T. Downes, R. F. Pfa\u000b,\nH. Gilbert, M. Khodachenko, E. Khomenko, I. F. Shaikhislamov,\nR. Soler, E. V\u0013 azquez-Semadeni, T. Zaqarashvili, \\Partially Ion-\nized Plasmas in Astrophysics\", Space Sci. Rev. 214, 58 (2018).\n13Anna Perelomova, \\On description of periodic magnetosonic per-\nturbations in a quasi-isentropic plasma with mechanical and ther-\nmal losses and electrical resistivity\" Phys. Plasmas 27, 032110\n(2020).\n14T. V. Zaqarashvili, M. L. Khodachenko, and H. O. Rucker, \\Mag-\nnetohydrodynamic waves in solar partially ionized plasmas: two-\n\ruid approach\", Astron. Astrophys. 529, A82 (2011).\n15L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Statistical\nPhysics Part 1 in L. D. Landau and E. M. Lifshitz, Landau-\nLifshitz Course on Theoretical Physics, Vol. V (3rd ed., Perga-\nmon Press, New York, 1980), Sec. 43, \\Ideal gas with constant\nheat capacity\", Sec. 45, \\Mono-atomic gas\", Sec. 46, \\Mono-\natomic gas. In\ruence of electronic momentum\", Eq. (46.1a),\nSec. 104, \\Ionization equilibrium\".\n16L. P. Pitaevskii, \\Electron Recombination in a Monatomic gas\",\nJETP 15(5), 919 (1962).\n17E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics in\nL. D. Landau and E. M. Lifshitz, Landau-Lifshitz Course on The-\noretical Physics, Vol. X (Pergamon, New York, 2002), Sec. 24,\n\\Recombination and ionization\".\n18M. N. Saha, \\On a physical theory of stellar spectra\", Proc. R.\nSoc. Lond. A, 99(697), 135-153 (1921).\n19T. M. Mishonov, A. M. Varonov, \\Sound Absorption in Par-\ntially Ionized Hydrogen Plasma and Heating Mechanism of Solar\nChromosphere\", arXiv:2005.05056 [physics.plasm-ph]."    },    {        "title": "2005.14153v2.Spintronics_meets_nonadiabatic_molecular_dynamics__Geometric_spin_torque_and_damping_on_noncollinear_classical_magnetism_due_to_electronic_open_quantum_system.pdf",        "content": "Spintronics meets nonadiabatic molecular dynamics: Geometric spin torque and\ndamping on noncollinear classical magnetism due to electronic open quantum system\nUtkarsh Bajpai and Branislav K. Nikoli´ c\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\nWe analyze a quantum-classical hybrid system of steadily precessing slow classical localized mag-\nnetic moments, forming a head-to-head domain wall, embedded into an open quantum system of\nfast nonequilibrium electrons. The electrons reside within a metallic wire connected to macroscopic\nreservoirs. The model captures the essence of dynamical noncollinear and noncoplanar magnetic\ntextures in spintronics, while making it possible to obtain the exact time-dependent nonequilib-\nrium density matrix of electronic system and split it into four contributions. The Fermi surface\ncontribution generates dissipative (or damping-like in spintronics terminology) spin torque on the\nmoments, and one of the two Fermi sea contributions generates geometric torque dominating in the\nadiabatic regime. When the coupling to the reservoirs is reduced, the geometric torque is the only\nnonzero contribution. Locally it has both nondissipative (or field-like in spintronics terminology)\nand damping-like components, but with the sum of latter being zero, which act as the counter-\nparts of geometric magnetism force and electronic friction in nonadiabatic molecular dynamics.\nSuch current-independent geometric torque is absent from widely used micromagnetics or atomistic\nspin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation,\nwhere previous analysis of Fermi surface-type torque has severely underestimated its magnitude.\nOne of the most fruitful applications of geometric (or\nBerry) phase [1] concepts is encountered in quantum-\nclassical hybrid systems where separation of time scales\nmakes it possible to consider fast quantum degrees of free-\ndom interacting with the slow classical ones [2, 3]. The\namply studied example of this kind are fast electrons in-\nteracting [4, 5] with slow nuclei in molecular dynamics\n(MD) [6–9] problems of physics, chemistry and biology.\nThe parameters driving adiabatic evolution of quantum\nsubsystem, with characteristic frequency smaller that its\nlevel spacing, are nuclear coordinates elevated to the\nstatus of dynamical variables. The electronic system\nthen develops geometric phase in states evolving out of\nan instantaneous energy eigenstate, while also acquiring\nshifts in the energy levels. Conversely, nuclei experience\nforces due to back-action from electrons. The simplest\nforce is the adiabatic Born-Oppenheimer (BO) force [4, 5]\nwhich depends only on the coordinates of the nuclei, and\nit is associated with electronic adiabatic potential sur-\nfaces [6, 7]. Even small violation of BO approximation\nleads to additional forces—the first nonadiabatic correc-\ntion generates forces linear in the velocity of the nuclei,\nand being Lorentz-like they are dubbed [2, 10] “geomet-\nric magnetism.” The “magnetism” is not a not a real\nmagnetic field, but an emergent geometrical property of\nthe Hilbert space [11], and akin to the true Lorentz force,\nthe emergent geometric force is nondissipative .\nAdditional forces appear upon making the quantum\nsystem open by coupling it to a thermal bath [10, 12]\n(usually modeled as an infinite set of harmonic oscilla-\ntors [13]) or to macroscopic reservoirs of particles [14].\nIn the latter case, one can also introduce chemical po-\ntential difference between the reservoirs to drive particle\nflux (i.e., current) through the quantum system which is,\nthereby, pushed out of equilibrium [14–16, 18, 19]. In\nFIG. 1. (a) Schematic view of a two-terminal system where\na single classical LMM, precessing steadily with frequency ω\nand cone angle θ, interacts with an open quantum system of\nconduction electron spins. The electrons hop along 1D infinite\ntight-binding chain which terminates into the left and right\nmacroscopic reservoirs kept at the same chemical potential µ.\nPanel (c) depicts 7 LMMs, M1–M7forming a head-to-head\nBloch domain wall, which precess with the same frequency\nbut are noncollinear andnoncoplanar . Both (a) and (c) can\nbe mapped in the rotating frame to a time-independent four-\nterminal system in (b) with an effective bias voltage ~ω/e\nbetween the left or right pair of leads.\nboth equilibrium and nonequilibrium cases, the energy\nspectrum of the quantum system is transformed into a\ncontinuous one, and frictional forces [8–10, 14–19] linear\nin the velocity of the nuclei become possible. Also, due\nto continuous spectrum, adiabaticity criterion has to be\nreplaced by a different one [14]. Stochastic forces also ap-\npear, both in equilibrium and in nonequilibrium, where\nin the former case [10, 12] they are due to fluctuations at\nfinite temperature while in the latter case they includearXiv:2005.14153v2  [cond-mat.mes-hall]  14 Jun 20202\nadditional contribution from nonequilibrium noise [14–\n16]. Finally, specific to nonequilibrium is the emergence\nof nonconservative forces [14–16, 18, 19]. The derivation\nof all of these forces is achieved by computing nonadia-\nbatic corrections to the density matrix (DM) [10, 12, 14–\n16, 18, 19]. This yields a non-Markovian stochastic\nLangevin equation, with nonlocal-in-time kernel describ-\ning memory effects [20], as the most general [16, 19] equa-\ntion for nuclei in nonadiabatic MD.\nThe analogous problem exists in spintronics, where the\nfast quantum system is comprised of conduction electron\nspins and slow classical system is comprised of localized-\non-atoms spins and associated localized magnetic mo-\nments (LMMs) described by unit vectors Mi(t). The\ndynamics of LMMs is accounted by the Landau-Lifshitz-\nGilbert (LLG) type of equation [21]\n∂Mi\ndt=−gM×Beff\ni+λMi×∂Mi\n∂t\n+g\nµM/parenleftBig\nTi/bracketleftBig\nISα\next/bracketrightBig\n+Ti[∂Mi/∂t]/parenrightBig\n. (1)\nThis includes phenomenological Gilbert damping, whose\nparameter λcan be measured or independently calcu-\nlated [22] by using electronic Hamiltonian with spin-orbit\ncoupling and impurities. It can also include Slonczewski\nspin-transfer torque (STT) term Ti/bracketleftBig\nISα\next/bracketrightBig\ndue to exter-\nnally supplied spin current ISα\next. The STT is a phe-\nnomenon [28] in which spin angular momentum of con-\nduction electrons is transferred to local magnetization\nnot aligned with electronic spin-polarization. Finally,\nsome analyses [23–25] also consider current-independent\ntorque Ti[∂Mi/∂t] as a back-action of electrons pushed\nout of equilibrium by time-dependent Mi(t). Neverthe-\nless, such effects have been deemed negligible [23, 26] or\neasily absorbed into Eq. (1) by renormalizing gandλ[23].\nHeregis the gyromagnetic ratio; Beff\ni=−1\nµM∂H/∂Mi\nis the effective magnetic field as the sum of external field,\nfield due to interaction with other LMMs and magnetic\nanisotropy field in the classical Hamiltonian Hof LMMs;\nandµMis the magnitude of LMM [21].\nThe STT vector, T=TFL+TDL, can be decomposed\n[Fig. 1(a)] into: ( i) even under time-reversal or field-like\n(FL) torque, which affects precession of LMM around\nBeff\ni; and ( ii) odd under time-reversal or damping-like\n(DL) torque, which either enhances the Gilbert damp-\ning by pushing LMM toward Beff\nior competes with\nGilbert term as “antidamping.” For example, negative\nvalues ofTDL=TDL·eDLin Figs. 2 and 3, where\neDL= (Mi×∂Mi/∂t)|Mi×∂Mi/∂t|−1, means that TDL\nvector points away from the axis of precession which is\nantidamping action. Similarly, TFL=TFL·eFL, where\neFL= (∂Mi/∂t)|∂Mi/∂t|−1, is plotted in Figs. 2 and 3.\nThe current-driven STT Ti/bracketleftBig\nISα\next/bracketrightBig\nacts as the coun-\nterpart of nonconservative force in nonadiabatic MD.\nThe Gilbert damping plus current-independent torque\nFIG. 2. The FL and DL components [Fig. (1)] of three spin\ntorques contributions in Eq. (4) exerted by nonequilibrium\nspin density of electrons onto a single localized precessing\nmagnetic moment in the setup of Fig. 1(a) as a function of\ncoupling to the leads. Black dotted line is the sum of the three\ntorques. In panels (a) and (c) Jsd= 0.1γ, while in panels\n(b) and (d) Jsd= 20γensures perfectly adiabatic regime [32],\nJsd/~ω/greatermuch1, for the chosen precession frequency ~ω= 0.001γ.\nTi[∂Mi/∂t] appear as the counterpart of electronic\nfriction [8, 9, 14–19], but Gilbert damping requires\nagents [22] other than electrons alone considered in nona-\ndiabatic MD. Thus, the geometric torque and damping,\nas counterparts of geometric magnetism force [2] and fric-\ntion [10], are absent from standard modeling of classi-\ncal magnetization dynamics. Geometric torque has been\nadded ad hoc into the LLG equation applied to spe-\ncific problems, such as spin waves within bulk magnetic\nmaterials [29–31]. A recent study [32] of a single clas-\nsical LMM embedded into a closed (i.e., finite length\none-dimensional wire) electronic quantum system finds\nthat nonequilibrium electronic spin density always gener-\nates geometric torque, even in perfectly adiabatic regime\nwhere electron-spin/LMM interaction is orders of mag-\nnitude larger than the characteristic frequency of LMM\ndynamics. It acts as a purely FL torque causing anoma-\nlous frequency of precession that is higher than the Lar-\nmor frequency. By retracing the same steps [14, 15] in\nthe derivation of the stochastic Langevin equation for\nelectron-nuclei system connected to macroscopic reser-\nvoirs, Ref. [33] derived the stochastic LLG equation [34–\n37] for a single LMM embedded into an open electronic\nsystem out of equilibrium. The novelty in this derivation\nis damping, present even in the absence of traditional\nspin-flip relaxation mechanisms [23, 25], while the same3\nFIG. 3. Spatial profile of FL and DL components of Tgeo\ni,Tsea\niandTsurf\nispin torques on precessing LMMs depicted in Fig. 1(c)\nfor closed or open electronic quantum system and for two different values of Jsd. Insets on the top of each row mark positions\nand static configuration of LMMs within the Bloch DW, with their x-component depicted by the colorbar next to panel (a).\nconclusion about geometric torque changing only the pre-\ncession frequency of LMM has been reached (in some\nregimes, geometric phase can also affect the stochastic\ntorque [38]). However, single LMM is a rather special\ncase, which is illustrated in Fig. 1(a) and revisited in\nFig. 2, and the most intriguing situations in spintronics\ninvolve dynamics of noncollinear textures of LMMs. This\nis exemplified by current- or magnetic-field driven dy-\nnamics of domain walls (DWs) and skyrmions [25, 37, 39–\n43] where a much richer panoply of back-action effects\nfrom fast electronic system can be expected.\nIn this Letter, we analyze an exactly solvable\nmodel of seven steadily precessing LMMs, M1(t)–M7(t)\n[Fig. 1(c)], which are noncollinear and noncoplanar and\nembedded into a one-dimensional (1D) infinite wire host-\ning conduction electrons. The model can be viewed as\na segment of dynamical noncollinear magnetic texture,\nand it directly describes magnetic field-driven [43] head-\nto-head Bloch DW [44] but without allowing it to prop-\nagate [41, 43]. Its simplicity makes it exactly solvable—\nwe fins the exact time-dependent DM via the nonequi-\nlibrium Green function (NEGF) formalism [45] and an-\nalyze its contributions in different regimes of the ratio\nJsd/~ωofsdexchange interaction Jsd[23] between elec-\ntron spin and LMM and frequency of precession ω. In\nboth Figs. 1(a) and 1(c), the electronic subsystem is an\nopen quantum system and, although no bias voltage is\napplied between the macroscopic reservoirs, it is pushed\ninto the nonequilibrium state by the dynamics of LMMs.\nFor example, electronic quantum Hamiltonian becomes\ntime-dependent due to M1(t) [Fig. 1(a)] or M1(t)–\nM7(t) [Fig. 1(c)], which leads to pumping [25, 27, 46]\n[Fig. 4(b),(c)] of spin current locally within the DW re-\ngion, as well as into the leads [Fig. 4(a)]. Pumping ofcharge current will also occur if the left-right symmetry\nof the device is broken statically [27] or dynamically [47].\nThe electrons are modeled on an infinite tight-binding\n(TB) clean chain with Hamiltonian in the lab frame\nˆHlab(t) =−γ/summationdisplay\n/angbracketleftij/angbracketrightˆc†\niˆcj−Jsd/summationdisplay\niˆc†\niˆ\u001bˆci·Mi(t). (2)\nHere ˆc†\ni= (ˆc†\ni↑,ˆc†\ni↓) and ˆc†\niσ(ˆciσ) creates (annihilates) an\nelectron of spin σ=↑,↓at sitei. The nearest-neighbor\nhoppingγ= 1 eV sets the unit of energy. The active re-\ngion in Figs. 1(a) or 1(c) consists of one or seven sites,\nrespectively, while the rest of infinite TB chain is taken\ninto account as the left (L) and the right (R) semi-infinite\nleads described by the same Hamiltonian in Eq. (2), but\nwithJsd= 0. The hopping between the leads and the\nactive region is denoted by γc. The leads terminate at\ninfinity into the macroscopic particle reservoirs with iden-\ntical chemical potentials µL=µR=EFdue to assumed\nabsence of bias voltage, and EF= 0 is chosen as the\nFermi energy. In contrast to traditional analysis in spin-\ntronics [23, 25], but akin to Refs. [32, 33], Hamiltonian in\nEq. (2) does not contain any spin-orbit or impurity terms\nas generators of spin-flip relaxation.\nThe spatial profile of Bloch DW is given by\nMx\ni=−sech[(hDW−zi)/W] tanh[(ZDW−zi)],My\ni=\nsech2[(ZDW−zi)/W] andMz\ni= tanh[(ZDW−zi)/W],\nwhereZDW= 4 andW= 0.9. Instead of solving LLG\nequations [Eq. (1)] for M1(t)–M7(t), we impose a so-\nlution where LMMs precess steadily around the z-axis:\nMx\ni(t) = sinθicos(ωt+φi);My\ni(t) = sinθisin(ωt+\nφi); andMz\ni(t) = cosθi. Using a unitary transfor-\nmation into the rotating frame (RF), the Hamiltonian\nin Eq. (2) becomes time-independent [25, 27], ˆHRF=4\nˆU†(t)ˆHlab(t)ˆU(t)−i~ˆU†∂ˆU/∂t =ˆHlab(t= 0)−~ωˆσα/2,\nwith LMMs frozen at t= 0 configuration from the lab.\nThe unitary operator is ˆU(t) = exp(−iωtˆσα/2) forα-axis\nof rotation. In the RF, the original two-terminal Lan-\ndauer setup for quantum transport in Figs. 1(a) and\n1(c) is mapped, due to ~ωˆσα/2 term, onto an effective\nfour-terminal setup [27] [illustrated for single LMM in\nFig. 1(b)]. Each of its four leads is an effective half-metal\nferromagnet which accepts only one spin species, ↑or↓\nalong theα-axis, and effective dc bias voltage ~ω/eacts\nbetween L or R pair of leads.\nIn the RF, the presence of the leads and macro-\nscopic reservoirs can be taken into account exactly us-\ning steady-state NEGFs [45] which depend on time\ndifferencet−t/primeand energy Eupon Fourier trans-\nform. Using the retarded, ˆG(E), and the lesser, ˆG<(E),\nGreen functions (GFs), we find the exact nonequilib-\nrium DM of electrons in the RF, ˆ ρRF=1\n2πi\u0001\ndEˆG<(E).\nHere the two GFs are related by the Keldysh equa-\ntion, ˆG<(E) = ˆG(E)ˆΣ<(E)ˆG†(E), where ˆΣ<(E) is\nthe lesser self-energy [45] due to semi-infinite leads and\nˆG(E) = [E−ˆHRF−ˆΣ(E,~ω)]−1with ˆΣ(E,~ω) =/summationtext\np=L,R,σ=↑,↓ˆΣσ\np(E−Qσ\nα~ω) being the sum of retarded\nself-energies for each of the four leads p,σin RF. We\nuse shorthand notation Q↑\np=−1/2 andQ↓\np= +1/2.\nSince typical frequency of magnetization dynamics is\n~ω/lessmuchEF, we can expand [48] ˆ ρRFin small ~ω/EF\nand then transform it back to the lab frame, ˆ ρlab(t) =\nˆU(t)ˆρRFˆU†(t) to obtain ˆ ρlab(t) = ˆρad\nt+ ˆρgeo(t)+ ˆρsea(t)+\nˆρsurf(t) where:\nˆρad\nt=−1\nπˆU+∞\u0002\n−∞dEImˆG0f(E)ˆU†, (3a)\nˆρgeo(t) =1\nπˆU+∞\u0002\n−∞dEIm/bracketleftbigg\nˆG0/parenleftbigg\ni~ˆU†∂ˆU\n∂t/parenrightbigg\nˆG0/bracketrightbigg\nf(E)ˆU†,(3b)\nˆρsea(t) =−~ω\n2πˆU/summationdisplay\np+∞\u0002\n−∞dEIm/bracketleftbigg\nˆG0/parenleftbigg∂ˆΣ↑\np\n∂E−∂ˆΣ↓\np\n∂E/parenrightbigg\nˆG0/bracketrightbigg\n×f(E)ˆU†, (3c)\nˆρsurf(t) =~ω\n4πˆU/summationdisplay\np+∞\u0002\n−∞dEˆG0(ˆΓ↑\np−ˆΓ↓\np)ˆG†\n0∂f\n∂EˆU†.(3d)\nWe confirm by numerically exact calculations [39] that\nthus obtained ˆ ρlab(t) is identical to ~G<(t,t)/icomputed\nin the lab frame. Here ˆG0(E) = [E−ˆHRF−ˆΣ(E,0)]−1\nisˆG(E) with ~ω= 0; ˆΓσ\np(E) =i[ˆΣσ\np(E)−ˆΣσ\np(E)†] is\nthe level broadening matrix due the leads; and fσ\np(E) =\nf(E−[EF+Qσ\nα~ω]) is the the Fermi function of macro-\nscopic reservoir p,σin the RF.\nThe total nonequilibrium spin density, /angbracketleftˆsi/angbracketright(t) =\nTr[ˆρlab(t)|i/angbracketright/angbracketlefti|⊗ˆ\u001b] =/angbracketleftˆsi/angbracketrightad\nt+/angbracketleftˆsi/angbracketrightgeo(t) +/angbracketleftˆsi/angbracketrightsea(t) +\n/angbracketleftˆsi/angbracketrightsurf(t), has the corresponding four contributions fromDM contributions in Eq. (3). Here /angbracketleftˆsi/angbracketrightad\ntis the equilib-\nrium expectation value at an instantaneous time twhich\ndefines ‘adiabatic spin density’ [23, 25, 30–32]. It is com-\nputed using ˆ ρad\ntas the grand canonical equilibrium DM\nexpressed via the frozen (adiabatic) retarded GF [14, 15,\n33], ˆGt(E) = [E−ˆHt−ˆΣ]−1, for instantaneous configu-\nration of Mi(t) while assuming ∂Mi/∂t= 0 [subscript\ntsignifies parametric dependence on time through slow\nvariation of Mi(t)]. The other three contributions—from\nˆρgeo(t) and ˆρsea(t) governed by the Fermi sea and ˆ ρsurf(t)\ngoverned by the Fermi surface electronic states—contain\nfirst nonadiabatic correction [14, 15, 33] proportional to\nvelocity∂Mi/∂t, as well as higher order terms due to\nˆρlab(t) being exact. These three contributions define STT\nout of equilibrium [23, 39, 48]\nTi=Jsd/angbracketleftˆsi/angbracketright(t)×Mi(t) =Tgeo\ni+Tsea\ni+Tsurf\ni.(4)\nEach term Tgeo\ni,Tsea\ni,Tsurf\nican be additionally sepa-\nrated into its own DL and FL components [Fig. 1(a)], as\nplotted in Figs. 2 and 3. Note that Tsea\niis insignificant\nin both Figs. 2 and 3, so we focus on Tgeo\niandTsurf\ni.\nTo gain transparent physical interpretation of Tgeo\niand\nTsurf\ni, we first consider the simplest case [32, 33]—a single\nM1(t) in setup of Fig. 1(a). The STT contributions as a\nfunction of the coupling γcto the leads (i.e., reservoirs)\nare shown in Fig. 2. We use two different values for Jsd,\nwhere large ratio of Jsd= 20 eV and ~ω= 0.001 eV is\nperfect adiabatic limit [30–32]. Nevertheless, even in this\nlimit and for γc→0 we find Tgeo\n1/negationslash= 0 in Fig. 2(b) as the\nonly nonzero and purely FL torque. This is also found\nin closed system of Ref. [32] where Tgeo\n1was expressed\nin terms of the spin Berry curvature. As the quantum\nsystem becomes open for γc>0,Tgeo\n1is slightly reduced\nwhile Tsurf\n1emerges with small FL [Fig. 2(b)] and large\nDL [Fig. 2(d)] components. The DL torque Tsurf,DL\n1\npoints toward the z-axis and, therefore, enhances the\nGilbert damping. In the wide-band approximation [49],\nthe self-energy ˆΣ(E) =−iΓˆI2is energy-independent for\nEwithin the bandwidth of the lead, which allows us to\nobtain analytical expression (at zero temperature)\nTgeo\n1(t) =~ω\n2π/bracketleftbigg\nπ−2 tan−1/parenleftbiggΓ\nJsd/parenrightbigg/bracketrightbigg\nsinθeφ(t).(5)\nHere eφ(t) =−sinωtex+ cosωtey. Thus, in per-\nfect adiabatic limit, Jsd/~ω→∞ , or in closed system,\nΓ→0,Tgeo\n1is independent of microscopic parameters\nas expected from its geometric nature [29]. The always\npresent Tgeo\ni/negationslash= 0 means that electron spin is never along\n‘adiabatic direction’ /angbracketleftˆsi/angbracketrightad\nt.\nSwitching to DW [Fig. 1(c)] embedded into a closed\nquantum system ( γc= 0) shows in Fig. 3(a)–(d) that\nonlyTgeo\ni/negationslash= 0, which also acquires DL component lo-\ncally with damping or antidamping action depending on\nthe position of LMM. Upon opening the quantum sys-\ntem (γc=γ), Fig. 3(e)–(h) shows emergence of ad-\nditional Tsurf\ni/negationslash= 0 which, however, becomes negligible5\nFIG. 4. (a) The z-component of total DL torques which act\non DW in Fig. 1(c) as a function of Jsdforγc=γ. Cir-\ncles show that sum of spin currents pumped into the leads\nmatches/parenleftbig/summationtext\niTsurf,DL\ni/parenrightbig\nz≡ISz\nL+ISz\nR. Panel (b) and (c),\nwhich correspond to Fig. 3(g), show spatial profile of lo-\ncal spin currents ISz\ni→jpumped between sites iandjfor\nJsd= 0.1γ, with their sum being identically zeroin panel (c).\nDashed black line in panels (a) and (b) is pumped local spin\ncurrent by SMF [24, 26], ISz\nSMF(x) =gµB~G0\n4e2[∂M(x,t)/∂t×\n∂M(x,t)/∂x]z, whereG0=G↑+G↓is the total conductivity.\n[Fig. 3(f),(h)] in the perfectly adiabatic limit Jsd/~ω/greatermuch1.\nAt first sight, Tgeo,DL\ni/negationslash= 0 violates Berry and Robbins\noriginal analysis [2] according to which an isolated quan-\ntum system, with discrete energy spectrum, cannot exert\nfriction onto the classical system. This apparent contra-\ndiction is resolved in Fig. 4(a) where we show that total/summationtext\niTgeo,DL\ni≡0 is always zero. Conversely, Fig. 4(a) con-\nfirms that total/parenleftBig/summationtextTsurf,DL\ni/parenrightBig\nz≡ISz\nL+ISz\nRis identical\nto net spin current pumped into the leads via which the\nconduction electrons carry away excess angular momen-\ntum of precessing LMMs [46]. Such identity underlies\nphysical picture where spin current generated by time-\ndependent magnetization becomes DL torque [24, 46].\nNote that pumped spin current ISz\ni→jdue to ˆρgeoor ˆρsea\nin Fig. 4(c) can be nonzero locally, but they sum to zero.\nThe nonuniform pumped spin current due to spatially\nand time varying magnetization has prompted propos-\nals [24, 26] to amend the LLG equation by adding the\ncorresponding DL torque M×D·∂M/∂twith 3×3 damp-\ning tensorDwhose spatial dependence is given by the so-\ncalled spin-motive force (SMF) formula. However, SMF\ncorrection was estimated to be small [26] in the absence\nof spin-orbit coupling in the band structure. We confirm\nits smallness in Fig. 4(a),(b) for our DW case, but this\nactually reveals that SMF formula produces incorrectly\nan order of magnitude smaller torque than obtained from\nour exact ˆρsurf(t). Due to possibly complex [40] time and\nspatial dependence of Tsurf\niandTgeo\ni, the accurate path\nto incorporate them is offered by self-consistent coupling\nof electronic DM and LLG calculations, as proposed in\nRefs. [39, 42, 50] and in full analogy to how electronic\nfriction is included in nonadiabatic MD [7–9, 14–19].This research was supported in part by the U.S. Na-\ntional Science Foundation (NSF) under Grant No. CHE\n1566074.\n[1] M. V. Berry, Quantal phase factors accompanying adia-\nbatic changes, Proc. R. Soc. Lond. A 392, 45 (1984).\n[2] M. V. Berry and J. M. Robbins, Chaotic classical and\nhalf-classical adiabatic reactions: geometric magnetism\nand deterministic friction, Proc. R. Soc. Lond. A 442,\n659 (1993).\n[3] Q. Zhang and B. 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Pirro 1 \n \n1 Fachbereich Physik and Landesforschungszentrum OPT IMAS, Technische Universität Kaiserslautern, 67663 \nKaiserslautern, Germany \n2 Faculty of Physics, University of Vienna, Boltzman ngasse 5, A-1090 Vienna, Austria \n \n3 Graduate School Materials Science in Mainz, Staudi ngerweg 9, 55128 Mainz, Germany \n \n4 Nano Structuring Center, Technische Universität Kai serslautern, 67663 Kaiserslautern, Germany \n \n5 INNOVENT e.V., Technologieentwicklung, Prüssingstr aße 27B, 07745 Jena, Germany \n \nRelaxation of linear magnetization dynamics is well  described by the viscous Gilbert damping pro- \ncesses. However, for strong excitations, nonlinear damping processes such as the decay via magnon-mag-\nnon interactions emerge and trigger additional rela xation channels. Here, we use space- and time-resol ved \nmicro-focused Brillouin light scattering spectrosco py and micromagnetic simulations to investigate the  \nnonlinear relaxation of strongly driven propagating  spin-waves in yttrium iron garnet nanoconduits. We  \nshow that the nonlinear magnon relaxation in this h ighly quantized system possesses intermodal feature s, \ni.e. magnons scatter to higher-order quantized mode s through a cascade of scattering events. We furthe r \nshow how to control such intermodal dissipation pro cesses by quantization of the magnon band in single -\nmode devices, where this phenomenon approaches its fundamental limit. Our study extends the knowledge \nabout nonlinear propagating spin-waves in nanostruc tures which is essential for the construction of ad - \nvanced spin-wave elements as well as the realizatio n of Bose-Einstein condensates in scaled systems.  \n \nRelaxation of magnons, the quanta of spin waves \n(SWs), due to magnetic damping is a complicated \nprocess and involves different (non)linear contribu - \ntions. Relaxation mechanisms which can be de- \nscribed by the phenomenological Gilbert damping  \ndrive the magnetization towards its equilibrium sta te \nby e.g. dissipating the energy to the lattice. It i s one \nof the key elements of performance in many practica l \ndevices and fundamental phenomena  [1–10]. \nDissipation of the energy can be more intricate \nfor strongly driven excitations, where nonlinear re - \nlaxation mechanisms via magnon-magnon interac- \ntions open up additional dissipation channels  [11–\n17]. Unlike the Gilbert damping, these types of in-\ntrinsic dissipation processes can redistribute the \nmagnon energy within the magnon spectrum [18–\n27]. \n       The classical works of Suhl predicted that l arge \namplitude uniform magnetization oscillations lead t o the onset of instability processes, allowing the no n- \nlinear relaxation of strongly driven magnons by a d e- \ncay into secondary magnon modes  [25]. In particu- \nlar, the common second-order Suhl instability pro- \ncess can be: ( i) a disadvantage since it comes along \nwith detrimental influence on the magnon transport \nand decay characteristics, potentially dominating t he \ncompeting linear damping  [17,22,28] , or, ( ii ) an ad- \nvantage by providing additional degrees of freedom \nof magnon transport for device architectures and \nquantum computing concepts  [23,29,30]. So far, \nmost of such investigations in scaled systems, whic h \nare of large interest for applications, have been c ar- \nried out for standing SW modes with vanishing mo- \nmentum ( k = 0), e.g. the Ferromagnetic resonance \n(FMR) mode. However, SWs carrying a momentum \nare not only essential for applications, but they p os- \nsess an enriched physics behind their nonlinear ins ta- \nbilities due to the increased amount of potential s cat- \ntering channels. Nevertheless, little investigation s \nhave been carried out in this direction yet. Recent development of ultra-low damping na- \nnoscale systems based on YIG, the most promising \nhosts for SWs, provides access to quasi-1D systems \nwith highly quantized magnon spectra  [31,32] . By \nimposing limitations on the available relaxation \nchannels due to the strong quantization of the mag-\nnon band, and a drastically modified SW character- \nistics including the SW dispersion relation, mode \nprofile and their ellipticity, nonlinear SW dynamic s \nin such devices can be different compared to contin - \nuous films and quasi-2D systems  [33-34]. Further- \nmore, recent experimental and theoretical studies o f \nSW dynamics and magnon condensates in nano- \nscopic systems  [32, 35,36] enforce us to better un - \nderstand nonlinear SW dynamics and magnon ther- \nmalization processes in nano-scaled 1D systems.  \n \nHere, we use space- and time-resolved mi- \ncro-focused Brillouin light scattering (µBLS) to un - \ncover the mechanism of nonlinear relaxation of \nstrongly driven propagating magnons via the second-\norder Suhl instability in YIG nanoconduits. We \ndemonstrate how magnons nonlinearly relax to other \nquantized modes via four-magnon scattering pro- \ncesses, and such nonlinear processes can be con- \ntrolled using quantization of the magnon band.  \n \nTo demonstrate the effect of quantization on \nthe nonlinear dynamics, we use two exemplary mag- \nnonic nanoconduits structured from a Liquid Phase \nEpitaxial (LPE) YIG film grown on top of a Gado- \nlinium Gallium Garnet (GGG) substrate  [37]. The \nmulti-mode nanoconduit with a lateral width of w = \n400 nm (Fig. 1a) and a thickness of d = 85 nm was \nfabricated using a hard mask and ion beam milling \nprocess  [31]. A comparative single-mode conduit \nwith a smaller width of w = 100 nm and d= 44 nm \nwas fabricated using a similar method (Fig. 1b). SW s \nin both devices are excited by a microwave antenna \nwhich is placed on top of the nanoconduits by elec-\ntron beam lithography and a lift-off process [31]. Ap- \nplying a microwave rf  current to the antenna gener- \nates a dynamic Oersted field which in return excite s \nSWs resonantly, see supplemental materials SM \n[40]. The detection of the generated SWs has been carried out using space- and time-resolved \nµBLS  [38]. An incident laser light with an effecti ve \nspot size of 300 nm (focused by a ×100 microscope \nobjective with a numerical aperture NA=0.85) is \nused to probe the SWs through the GGG substrate \nunder the antenna. The inelastically scattered ligh t \nwas analyzed using a tandem Fabry-Perot interfer- \nometer to obtain the frequency and intensity of the  \nmagnons.   \n \nFIG. 1. (a)-(b) SEM images of the w = 400 nm (multi-\nmode) and w =100 nm (single-mode) wide conduits \n(shaded in orange), respectively. (c)-(d) Magnon ba nd \nstructures of the multi-mode and single-mode condui ts, \nrespectively. Color plots are obtained by micromagn etic \nsimulations and dashed lines from analytical calcul ations. \nNote the different scales of the frequencies. (e)-( f) Meas- \nured spin-wave spectra of the multi-mode and single -\nmode conduits in the presence of different powers, respec- \ntively. The excited modes are represented by the ye llow \ndots in (c) and (d). \nA static external field (µ 0He = 60 mT) saturates \nthe nanoconduits along their length. Thus, the wave  \nvector of the propagating SWs is parallel to the ma g- \nnetization vector, /g2193 ‖  /g2169 , and waveguide (WG) \nmodes appears [32].  The width of the multi-mode \nwaveguide is large enough to ensure dipolar pinning  \nof the spins at the edges, while spins at the edges  of \nthe single-mode conduit are fully unpinned [32]. \nMoreover, due to the interplay between the contribu - \ntions of the dipolar and exchange energy to the SW \ndispersion, the different WG modes are well quan- \ntized on the frequency axis. The dispersion relatio n \nof the fundamental mode and the first two WG \nmodes are shown in Fig 1c-d, in which the dashed \nlines are analytical results based on method discus ses \nin Ref  [32], and the color plot is obtained by mic ro- \nmagnetic simulations using the MuMax 3.0 pack- \nage  [39, 40]. The fundamental mode and higher or- \nder WG modes are labeled as n = 0 and n = 1, 2 re- \nspectively. Please note that the spectrum is much \nmore dilute in the 100 nm wide conduit due to the \nhigher contribution of the exchange energy to the \nmagnon band structure, which leads to a strong quan - \ntization and the absence of degenerate states among  \nmodes (single-mode system for wave vectors below \napprox. 40 rad/µm).  \nWe first set the rf  frequency to f = 3.85 GHz \nwhere dipolar SWs having a wave vector of kx = 1.5 \nrad/µm are excited in the multi-mode device [40]. T o \ncharacterize the linear SW dynamics, we set the rf  \npower to P = 10 dBm and measure the intensity of \nthe generated magnons as displayed in Fig 1e (black  \ncircles). Up to  P = 18 dBm, only the frequency of the \nresonantly driven SW mode is observed (red and \ngreen triangles). A further increase in the rf  power up \nto P = 20 dBm (blue curve) leads to the appearance \nof two additional peaks in the SW frequency spec- \ntrum labeled as f – and f + in Fig. 1e. We refer to these \nmagnons as secondary magnons which are modes \npopulated by nonlinear scattering processes. They \nhave the lowest threshold for the observed instabil ity \nprocess and can fulfill the fundamental conservatio n \nlaws to permit the scattering process [22]. The en-\nergy and momentum conservation laws of these pro- \ncesses generally  read  [18,20,22,28], \n \n/g1858/g2869+ /g1858/g2870= /g1858/g2871+ /g1858/g2872,   /g2193/g2869+ /g2193/g2870= /g2193/g2871+ /g2193/g2872          (1) \n \nwhere two magnons with the frequencies f 1 & f 2 and \nmomenta k 1 & k 2 scatter to two magnons with the \nfrequencies f 3 & f 4 and momenta k 3 & k 4. Note that \nthe lateral component of the k vector is symmetric, \nand the out of plane component is zero in this fre-quency range due to the small thickness.   In our ex- \nperiments, two magnons with a frequency of f = 3.85 \nGHz scatter finally to two magnons with the frequen - \ncies of f 3 = f - = 3.25 GHz and  f 4 =  f + = 4.45 GHz. \nWe note that this process is not a special peculiar ity \nof the chosen spectral position, see SM [40]. \nFor comparison, we now investigate the \nsame nonlinear process in the comparative single-\nmode waveguide. We set the f = 3.71 GHz and meas- \nure the intensity of the driven mode as shown in Fi g. \n1f.  Clearly, even in the presence of high powers l ike \nP = 20 dBm, side peaks cannot be observed, evidenc- \ning the absence of a similar nonlinear dissipation \nprocesses. Here, only the µBLS intensity drops at \nhigh powers which is caused by the nonlinear fre- \nquency shift of the dispersion relation and possibl e \nimpacts of the higher temperature [31, 41]. In prin ci- \nple, the absence of side peaks demonstrates that su ch \nscattering processes can be efficiently suppressed in \nnarrower conduits where the magnon band structure \nis highly quantized and therefore, the fundamental \nconservation laws required for the scattering pro- \ncesses cannot be fulfilled. \nTo understand the fundamental differences \nbetween the two waveguide types, let us investigate  \nthe observed nonlinear dynamics in the multi-mode \nconduit in more detail. A nonlinear scattering inst a- \nbility is characterized by a clear threshold of the  ini- \ntial magnon intensity which is required for its on-\nset [18,22,24,42]. Neglecting SW radiation losses, \nthe threshold magnon amplitude is defined by the ef - \nfective relaxation frequency of the secondary mag- \nnons divided by the four-magnon coupling \nstrength  [16,22]. To investigate the threshold beh av- \nior in the multi-mode conduit in which the scatteri ng \nis observed, we sweep the rf  power for a fixed fre- \nquency f = 3.85 GHz as shown in Fig. 2. Once the \ninstability threshold is reached at P = 18 dBm (indi- \ncated by the black arrow), the growth rate of the d i- \nrectly excited magnon intensity as a function of mi - \ncrowave power drops. Increasing the power to P = \n19 dBm leads to an abrupt increase of the intensity  \nof the secondary magnons  labeled as f + and f – (indi- \ncated by the gray arrow). From this power ( P = 19 \ndBm) on, the intensity growth rate of the directly ex- \ncited mode with respect to the power is decreased, evidencing that the energy transfers to the seconda ry \nmagnon modes. \nFIG. 2. Spin-wave amplitude in the multi-mode  conduit as \na function of microwave excitation power when f = 3.85 \nGHz. The secondary magnons created by the second or der \nSuhl instability are denoted as f + and f -. The back and \ngray arrows indicate the onset of instability and t he rise of \nthe secondary magnons, respectively.  \nA closer look on Fig. 2 near the instability \nthreshold opens the question what happens when the \ninstability threshold is approached at P = 18 dBm \nand the µBLS intensity of the initially excited mod e \ndrops, while the amplitudes of the secondary mag- \nnons  at f+ and f- are still at the thermal level, implying \nthe absence of magnon scattering to these modes. \nWe perform micromagnetic simulations to \nuncover the wave vector of the scattered magnons \nand address the discussed question. Figure 3a shows  \nthe frequency spectrum of the simulated multi-mode \nconduit ( f = 3.85 GHz) in which different amplitude \nof the rf  currents are used to drive the system. For a \nsmall rf  current equal to irf  = 4 mA, only the reso- \nnantly excited SWs can be observed in the frequency  \nspectrum (black curve). The corresponding popula- \ntion of the magnon band is depicted in Fig. 3b show - \ning the wave vector of kx = 1.5 rad/µm of the directly \nexcited mode. Increasing the rf  current to a higher \nvalue of irf  = 8 mA increases the amplitude and the \nlinewidth of the resonant SWs (red curve in Fig \n3.a) [28]. As shown in Fig. 3c, this is related to the \nonset of a first-level four magnon scattering proce ss \nin which the frequency of the magnons is conserved.  \nSuch a process cannot be observed in the measured frequency spectrum of the conduit, but it can mani-\nfest itself in the observed drop of the directly ex cited \nmode intensity with power.  As evidenced by the sim- \nulations, two incoming magnons from the resonantly \ndriven mode with opposite momenta scatter to two \noutgoing magnons at the same frequency, but with \ndifferent momenta. The scattered magnons populate \nthe fundamental mode ( n = 0) at a higher wave num- \nber of /g1863/g3051= 30 /g1870/g1853/g1856/µ/g1865 , and two spectral position \nat the first WG mode ( n = 1). These frequency-con- \nserving  scattering processes which are similar to \nplane films  [25] are indicated by the pink arrows in \nFig. 3c, and can also be observed in the single-mod e \nconduit, see SM [40]. \nA further increase of the rf  current to irf  = 13 \nmA leads to the onset of the sideband peaks in the \nfrequency spectrum (blue curve in Fig 3.a), similar  \nto the experiments. As evidenced from the simulated  \nband structure (Fig. 3d), this is due to the second  \nlevel of the magnon scattering cascade. Once the \nmagnons scattered by the first level process to the  \nn=1 WG mode reach a critical amplitude, they un- \ndergo themselves another second order instability. In \nthis process, two magnons with the frequency of f = \n3.85 GHz  and identical momentum of  /g1863/g3051=\n10.7 /g1870/g1853/g1856/µ/g1865  at the first WG mode ( n = 1), scatter \nto two outgoing magnons with the frequencies of f - \n= 3.46 GHz and f + = 4.24 GHz at the fundamental \nmode ( n = 0) and the second WG mode ( n = 2), re- \nspectively. The simulated values are in very good \nagreement with the experimentally obtained frequen-\ncies. \nIn Fig. 3d, this type of frequency-noncon- \nserving  scattering is represented by the red arrows. \nThe scattered magnons feature /g1863/g3051/g2878= 14.3 /g1870/g1853/g1856/µ/g1865  \nand /g1863/g3051/g2879= 7.1 /g1870/g1853/g1856/µ/g1865 , assuring momentum conser- \nvation laws given by 2/g1863/g3051= /g1863 /g3051/g2878+ /g1863 /g3051/g2879. We note that \nthe second scattering step clearly shows that the f i- \nnite momentum of the ingoing magnon opens the op- \nportunity to scatter to two new, different frequenc ies \nand thus, to redistribute the magnon energy towards  \nthe bottom of the spectrum and to higher frequencie s \n(modes). Unlike the first level process, it involve s \nonly magnons of a single propagation direction (+ k \nor –k) and can only occur for propagating waves. \nThis is evidenced by the momentum and energy con- \nservation laws which require a finite sum of the mo - \nmenta of the two incoming magnons to allow for a \nfrequency non-degenerated splitting. This is a sign if- \nicant difference to the nonlinear instabilities of the \nFMR mode without momentum ( kx = 0) in which \nmagnon instabilities are always degenerated [43-44] . \nThus, if the FMR undergoes a second-order instabil-\nity, this process never leads to a redistribution o f the \nmagnon energy across the spectrum.  \n \n \nFIG. 3. Results of the micromagnetic simulations in  the \nmulti-mode conduit. (a) spin-wave frequency spectra  \nwhen the microwave current varies. (b-d) Magnon ban d \nstructures (linear scale) of the driven system corr espond \nto the black, red and blue curves in (a), respectiv ely. The \nscaling of b-d is independent from each other.  \nThe properties of the cascade-like magnon \nscattering events coupling different waveguide \nmodes in the multi-mode waveguide and the absence \nof this effect in the single mode waveguide also im - \nplies that thermalization of magnons is significant ly \nchanged in systems with strongly diluted spectra \ncompared to earlier investigations in systems which  \nquasi-continuous spectra. \nThe simulations also explain the observed \npeculiarity in the threshold curve of the experimen ts \nas were discussed in the context of Fig. 2. Indeed,  the \nmagnons scattered to higher wave numbers via the \nfirst level frequency-conserving scattering process  (Fig. 3c) cannot be detected experimentally due to \nthe maximum detectable momentum using µBLS \nspectroscopy, which is approximately kx ~ 21 rad/µm \nin our experiments [38]. This explains at least par - \ntially the decrease of the measured magnon intensit y \nat the driving frequency. Since the different level s of \nthe cascade process have different threshold powers , \nthe nonlinear scattering to the secondary magnon \nmodes at different frequencies is observed at a \nslightly higher power than the start of the drop of  the \nintensity at the directly excited frequency. In add i- \ntion, the limited wave vector sensitivity of the BL S \ncan pose inconsistency for the SW amplitude ob- \nserved in the simulations compared to the experi- \nments. \n \nFIG. 4. Time-resolved spin-wave amplitude measured by \nµBLS spectroscopy. (a) Beginning of the pulse. (b) End \nof the pulse. Black arrows indicate the onset and t he decay \nof the instability, respectively. Note that the dec ay rates \ncorrespond to the intensity of the magnons.  \nTo further characterize the impact of the \nnonlinear relaxation on the total relaxation of the  sys- \ntem  [16], we perform time-resolved µBLS measure- \nments in the multi-mode conduit. The measured in- \ntensity of the driven and secondary magnons at the \nbeginning and the end of a 1µs long microwave rf  \npulse ( f = 3.85 GHz and P = 24 dBm) at the meas- \nurement position are shown in Fig. 4a-b. Figure 4a \nillustrates that the resonantly driven SW mode (blu e \ncurve) undergoes the second-level four-magnon \nscattering after t ~ 4 ns, evidenced by the rise of the \nsecondary magnons (yellow and red curves). This is \nindicated by the black arrow in Fig. 4a. Note that the \ngrowth rate of the driven mode drops immediately \nwhen the rise of the secondary magnons sets in, evi - \ndencing the conservation of the energy in the nonli n- \near redistribution process.  \nThe decay of the magnons at the end of pulse \nis presented in Fig. 4b. In particular, the decay o f the \nsecondary magnons begins once the intensity of the \ndriven SW mode is decayed enough after t ~ 4 ns \n(indicated by the black arrow). More interestingly,  \nthe decay of the magnons at the resonantly driven \nfrequency to the thermal level includes two steps \nmanifesting the high nonlinearity of the dynamics. \nFirst, it decays with an exponential decay time of t1,d \n= 19 ns, which is accompanied by the decay of the \nsecondary magnons at f+ and f-. Afterwards, it de- \ncays with a longer exponential decay time of t2,d = 24 \nns suggesting a transition from a nonlinear relaxat ion \nto a linear relaxation with a lower decay rate. In other \nwords, the first decay includes an energy flow to t he \nsecondary magnons which acts as an additional dis- \nsipation channel for the driven magnons. After the \nsecondary magnons decayed to the thermal level, thi s \nadditional dissipation channel is switched off, whi ch \nleads to a slower decay time of the driven SWs. \n In summary, we explored the nonlinear re- \nlaxation of strongly driven propagating spin waves \nin nanodevices. The finite momentum of the mag- \nnons investigated in our study provides an addition al \nplayground for the nonlinear magnon instability pro - \ncesses. Furthermore, it was shown that such inter- \nmodal dissipation process is strongly suppressed in  \nsystems with a strongly quantized magnon band (sin-\ngle-mode systems), suggesting the fundamental lim- \nitation of this process in nanodevices.  This can o pen \na new avenue for coherent nonlinear nano-mag- \nnonics. The nonlinear dynamics studied in this lett er \nare general and thus, can be applied to devices bas ed \non other deposition techniques as well. Our study c an \nbe used for several device architectures, namely, f re- \nquency mixers  [45], squeezed states [46], signal a nd \ndata processing units [29, 47-50], and quantum com-\nputing concepts [23], and further open doors to eng i- \nneered dissipation of magnons in nanodevices.  Acknowledgments \n The authors thank Burkard Hillebrands for \nsupport and valuable discussions. This project is \nfunded by the Deutsche Forschungsgemeinschaft \n(DFG, German Research Foundation) - TRR 173 - \n268565370 (“Spin+X”, Project B01) and by the pro- \nject - 271741898, the European Research Council \nwithin the Starting Grant No. 678309 “MagnonCir- \ncuits” and by the Austrian Science Fund (FWF) \nthrough the project I 4696-N. 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Kinane,3and\nSubhankar Bedanta1,a)\n1)Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences,\nNational Institute of Science Education and Research (NISER), HBNI, P.O.- Jatni, 752050,\nIndia\n2)Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, TN 37830,\nUnited States\n3)ISIS, Harwell Science and Innovation Campus, Science and Technology Facilities Council,\nRutherford Appleton Laboratory, Didcot, Oxon OX11 0QX, United Kingdom\n(Dated: April 2019)\nWe present a comprehensive study on the magnetization reversal in Fe/NiFe bilayer system by alternating the\norder of the magnetic layers. All the samples show growth-induced uniaxial magnetic anisotropy due to oblique\nangle deposition technique. Strong interfacial exchange coupling between the Fe and NiFe layers leads to the\nsingle-phase hysteresis loops in the bilayer system. The strength of coupling being dependent on the interface\nchanges upon alternating the order of magnetic layers. The magnetic parameters such as coercivity HC, and\nanisotropy \feld HKbecome almost doubled when NiFe layer is grown over the Fe layers. This enhancement\nin the magnetic parameters is primarily dependent on the increase of the thickness and magnetic moment of\nFe-NiFe interfacial layer as revealed from the polarized neutron re\rectivity (PNR) data of the bilayer samples.\nThe di\u000berence in the thickness and magnetization of the Fe-NiFe interfacial layer indicates the modi\fcation\nof the microstructure by alternating the order of the magnetic layers of the bilayers. The interfacial magnetic\nmoment increased by almost 18 % when NiFe layer is grown over the Fe layer. In spite of the di\u000berent values\nof anisotropy \felds and modi\fed interfacial exchange coupling, the Gilbert damping constant values of the\nferromagnetic bilayers remain similar to single NiFe layer.\nIn an exchange-coupled soft/hard bilayers, one can \fnd\nhigh energy product ( BH)max value as the soft mag-\nnetic layer provides high saturation magnetization MS\nand the hard one provides intermediate coercivity HC1.\nThis soft-hard combination of the magnetic bilayers pro-\nvides an excellent research opportunity not just for their\npotential application in the \feld of permanent magnets1\nbut also for the sake of fundamental understanding of\nvarious magnetization reversal processes. Hard magnetic\nlayer gives large HCdue to its high magnetic anisotropy\nwhich is not desired in the application of the write head.\nFurther, the switching \feld of the hard layer can be re-\nduced by fabricating soft/hard magnetic bilayers which\nful\flls the requirement of write-head and simultaneously\nprovide excellent temperature stability2,3. The interface\nplays a very important role in tuning the HCof the bi-\nlayers by modifying the interfacial exchange coupling.\nIn literature, several techniques have been employed for\nmodi\fcations of the interfacial layer. Di\u000berent deposi-\ntion techniques and subsequent post-deposition anneal-\ning at di\u000berent temperatures have been widely employed\nfor modi\fcation of the interface2{4. Varying the thick-\nness of the soft layer and using the materials of di\u000ber-\nent crystallographic structures have also been considered\nto study the role of interfacial exchange coupling in a\nsoft/hard bilayer system1,2,5. The interfacial exchange\ncoupling between the hard and soft layers can be en-\na)Electronic mail: sbedanta@niser.ac.inhanced by interdi\u000busion6. Although there are several\nreports on various techniques for modi\fcation of the in-\nterface, there is still a continuous focus to understand the\ninterface properties7,8.\nIn order to account the role of interdi\u000busion on the\nmagnetic properties, several experiments and simulations\nhave been performed. Conversion Electron M ossbauer\nSpectroscopy has been used to \fnd the presence of in-\nterdi\u000busion in hard (FePt)/soft (Fe or Co/Fe) bilayers9.\nThe presence of a graded interface has been observed in\nSmCo/Fe system from Synchrotron x-ray scattering and\nelectron microscopy elemental mapping measurements10.\nTransmission electron microscopy and magnetic measure-\nments show an enhanced epitaxy in the postannealed\nCo/CoPt system11. Depth and element resolved x-ray\nresonant magnetic scattering (XRMS) measurements on\nSmCo/Fe bilayer show the presence of di\u000bused Co-atoms\nfrom Sm-Co layer in Fe magnetic layer12. Despite several\nexperimental techniques for studying the magnetic prop-\nerties of the individual layers, a quantitative knowledge\nabout the interface of a layered system always remains\nchallenging due to the very complex nature of the in-\nterface13. In this context, polarized neutron re\rectivity\n(PNR) is a very promising tool for a quantitative struc-\ntural and magnetic information about the interface.\nIn addition to the above study, materials with lower\nGilbert damping constant \u000bare being studied exten-\nsively for their application in spin-transfer torque-based\noscillators14, and also in spintronics devices15. Intrin-\nsic Gilbert damping in materials has its origin on thearXiv:2006.05756v1  [cond-mat.mtrl-sci]  10 Jun 20202\nspin-orbit coupling16. Extrinsic contributions can en-\nhance this damping. Several deposition methodologies\nsuch as di\u000berent oblique angle of deposition17, deposi-\ntion pressure18etc. have been employed for tuning of\nthe damping constant. It is desired to fabricate hard/soft\nmagnetic bilayers where anisotropy gets enhanced keep-\ning the damping value of the same order as that of the\nreference soft layer.\nIn the present paper, we report tuning of the interfacial\nexchange coupling by alternating the order of magnetic\nlayers in the hard/soft Fe/NiFe bilayers. We show that\nby alternating the order of layers the interface changes\nwhich results in tuning of the magnetic properties of the\nbilayers. In order to quantify the interface thickness and\nmoment, we have performed polarized neutron re\rectom-\netry on the bilayer samples. We also made a comparative\nstudy on the damping constants of the samples through\nFMR analysis.\nI. EXPERIMENTAL DETAILS:\nAll the samples are deposited by combination of dc\nmagnetron sputtering and e-beam deposition in a high\nvacuum chamber on naturally oxidized Si (100) sub-\nstrate. The base pressure was \u00186\u000210\u00008mbar. Prior\nto the deposition, the substrates were annealed for a pe-\nriod of 2 hr at 150\u000eC. The samples were prepared on to\nthe Si-substrates kept at 150\u000eC at an Ar pressure of \u0018\n5\u000210\u00003mbar. A capping layer of Au (3nm) was further\ndeposited by e-beam evaporation to protect the samples\nfrom oxidation. The rate of deposition of Fe, NiFe and\nAu were kept at 0.22, 0.17 and 0.1 \u0017A/sec, respectively.\nTable 1 shows the list of sample nomenclature, and struc-\nture.\nTABLE I. Details of sample name and structure for all the\nsamples.\nSample name Sample structure\nS1 Si (100)/NiFe (10 nm)/Au (3 nm)\nS2 Si (100)/Fe (5 nm)/Au (3 nm)\nS3 Si (100)/Fe (5 nm)/NiFe (10 nm)/Au (3 nm)\nS4 Si (100)/NiFe (10 nm)/Fe (5 nm)/Au (3 nm)\nWe performed x-ray re\rectivity (XRR) measurements\nto evaluate the thickness, density and roughness of\neach individual layers by using x-ray di\u000bractometer from\nRigaku with CuK \u000bradiation (\u0015= 0.154 nm). We have\nperformed PNR measurements at room temperature at\nPOLREF neutron re\rectometer, at Rutherford Appleton\nLaboratory, UK. In the PNR measurements, magnetic\n\feld was applied along the easy axis (EA) and exper-\niments were performed at saturation and near to coer-\ncive \feld of the bilayer samples. POLREF is a white\nbeam instrument and we have used a pulse of length 2-\n15\u0017A's with several varying angles. We plotted the ex-\nperimental data of re\rectivity vs perpendicular scatter-ing vectorQZ= 4\u0019sin\u0012/\u0015whereQZis the component\nof momentum transfer perpendicular to the sample sur-\nface, thus, giving samples layer-by-layer information19,20.\nWe always applied positive saturation \feld and then re-\nverse the \feld to the measurement \felds. The guiding\n\feld was -1 mT. Neutron re\rectivity can be spin \ripped\nor non-spin \ripped. We measured two non-spin \ripped\nscattering cross sections namely R++and R\u0000\u000019,20. In\nR++, the \frst + sign is for the incident neutron with\nup-spin polarization and the second + sign is for the\nre\rected neutron with up-spin polarization. Similarly,\nwe can explain R\u0000\u0000(down-down). The XRR and PNR\ndata were \ftted using GenX software21. We have per-\nformed longitudinal magneto-optic Kerr e\u000bect based mi-\ncroscopy to simultaneously measure hysteresis loops and\nimage the magnetic domains. Magnetic dynamic proper-\nties were studied using ferromagnetic resonance (FMR)\nsetup manufactured by Nano Osc.\nII. RESULTS AND DISCUSSION:\nA. PNR analysis\nWe have evaluated the quantitative structural informa-\ntion such as density, roughness and thickness of the sam-\nples from x-ray re\rectivity (XRR) measurement (data\nare not shown). The layer thickness and roughness ob-\ntained from XRR and PNR measurements are similar for\nthe bilayer samples.\nIn order to get quantitative information from the lay-\ners and interfaces in the sample stack, we have performed\npolarized neutron re\rectivity (PNR) measurement on the\nsamples. PNR has been proven to be an ideal tech-\nnique for providing layer-by-layer magnetization pro\fle\nin a multilayer stack. We have \ftted the PNR data by\nconsidering di\u000berent interface models to \fnd the best \fg-\nure of merit (FOM). Considering all other interface mod-\nels other than the three interface model, we found less\nvalue of FOM and the \ftting is not good. We found that\nbest FOM is achieved by considering a three interface\nmodel in samples S3 and S4. The interfaces are named\nas NiFe-Au, Fe-NiFe and Si O2-Fe for sample S3. Sim-\nilarly the interfaces are named as Fe-Au, NiFe-Fe and\nSiO2-NiFe for sample S4. The interfaces taken to \ft\nthe neutron re\rectivity data are shown in \fg. 1. FOMs\nof 4.60\u000210\u000002and 5.04\u000210\u000002are found in samples S3\nand S4. Here, we have used LOG type of FOM. Using\nthis type of FOM, we \ftted the data more easily and\nrobustly. LOG type of FOM takes into account the aver-\nage of the di\u000berence between the logarithms of the data\nand the simulation. Structural and magnetic parameters,\nobtained from PNR \ft, are shown in tables 2 and 3 for\nsamples S3 and S4, respectively. The magnetic moment\nof Fe and NiFe obtained from the PNR data of the sam-\nple S3 are 1.57 \u0016B/atom and 0.7 \u0016B/atom, respectively.\nThe observed deviation in the magnetic moment of Fe\nand NiFe from their bulk value is due to the transfer of3\nTABLE II. Structural and magnetic parameters obtained after \ftting the PNR experimental data using GenX software for\nsample S3.\nLayer description thickness (nm) roughness (nm) Magnetic moment ( \u0016B/atom) at -50 mT Magnetic moment ( \u0016B/atom) at -4 mT\nAu 3.79 1.99 { {\nNiFe-Au 2.99 0.94 -0.10 -0.003\nNiFe 8.39 1.09 -0.79 0.78\nFe-NiFe 2.31 1.20 -0.90 0.8\nFe 2.48 1.56 -1.57 1.57\nSiO2-Fe 1.99 0.99 -1.00 0.99\nTABLE III. Structural and magnetic parameters obtained after \ftting the PNR experimental data using GenX software for\nsample S4.\nLayer description thickness (nm) roughness (nm) Magnetic moment ( \u0016B/atom) at -50 mT Magnetic moment ( \u0016B/atom) at -1.2 mT\nAu 3.00 1.37 { {\nFe-Au 2.66 1.19 -0.52 0.20\nFe 3.24 0.85 -1.26 1.17\nNiFe-Fe 1.79 1.10 -0.76 0.76\nNiFe 8.99 1.29 -0.75 0.75\nSiO2-NiFe 1.66 0.9 -0.75 0.75\nmagnetic moment to the interface caused by interdi\u000bu-\nsion. Similarly, the Fe-NiFe interface of sample S3 has a\nmagnetic moment of 0.90 \u0016B/atom which is intermediate\nbetween Fe and NiFe layers, and has a thickness of 2.3\nnm. The Si O2-Fe interface has lesser magnetic moment\nof 1.00\u0016B/atom than Fe itself due to interdi\u000busion in\nsample S322. Interface roughness of the order of 1 nm\nmight also be a reason for lesser magnetic moment at\nFe-SiO2interface22. In contrast to the Si O2-Fe interface,\nthe NiFe-Au interface in the sample S3 has a relatively\nsmaller values of magnetic moment (0.10 \u0016B/atom) and\nthickness(2.99 nm), indicating high amount of interdif-\nfusion. Thus, a dead layer is formed at the NiFe-Au\ninterface of sample S3. Our XRR data also suggests dif-\nferent rougnness values of Fe and NiFe which are in direct\ncontact with Si O2and Au layers, respectively. Thus, we\ncan conclude that interface roughness might be a reason\nfor the di\u000berent values of magnetic moment at Si O2-Fe\nand NiFe-Au interfaces than the parent layers in sam-\nple S3. Interdi\u000busion and/or alloying at the interfaces\nare the result of high temperature (150\u000eC) deposition\nof the studied \flms. The formation of a magnetic dead\nlayer is also reported in the case of Fe/Ge system when\nFe is grown on Ge at 150\u000eC22. In contrast to the sam-\nple S3, the magnetic moment of the Fe layer, NiFe layer\nand the NiFe-Fe interfacial layer in the sample S4, are\n1.26\u0016B/atom, 0.75 \u0016B/atom, and 0.76 \u0016B/atom, respec-\ntively. This indicates that the sample S3 has relatively\nhigher magnetic moment values of its constituent lay-\ners Fe, NiFe and Fe-NiFe interface as compared to that\nof the S4. This is further con\frmed from the SQUID\ndata of the samples where the S3 has a higher satura-\ntion magnetization value (762 emu. cc\u00001) in comparisonto that of S4 (636 emu. cc\u00001). We observed that all\nFIG. 1. Schematic of all the interfaces and thin \flm layers in\nsamples (a) S3 and (b) S4.\nmagnetic layers including Si O2-Fe interface are reversed\ncompletely at -4 mT of magnetic \feld in sample S3. 88\n% of Fe-NiFe interface magnetic moments are reversed\nfrom positive saturation state at -4 mT \feld in sample\nS3. Further, 92 % of magnetic moments of Fe have re-\nversed their direction near to coercive \feld (-1.2 mT)\nfrom positive saturation state in sample S4. Again, 38\n% of the magnetic moment at the Fe-Au interface has\nreversed direction in sample S4 whereas all other layers\nhas reversed completely. Further, the thicknesses of all\nthe interdi\u000bused interface layers of sample S3 are higher\nthan that of sample S4. Thus, larger interdi\u000busion might\nbe a reason for the di\u000berence in magnetic properties of\nsamples S3 and S4. We found from tables 2 and 3 that\nthe roughness of Fe in sample S3 is higher than sam-4\nFIG. 2. Polarized neutron re\rectivity (PNR) data for sample S3 at room temperature with saturation magnetic \feld of -50 mT\n(a) and -4 mT of magnetic \feld which is near to coercivity (b) are applied along EA. The open circles are the experimental\ndata points and the solid lines are \ftted data for the non-spin \rip (NSF) re\rectivities R++(red colour), R\u0000\u0000(blue colour),\nrespectively.\nFIG. 3. Polarized neutron re\rectivity (PNR) data for sample S4 at room temperature measured at saturation magnetic \feld of\n-50 mT (a) and -1.2 mT of magnetic \feld near to coercivity (b), along EA. The open circles are the experimental data points\nand the solid lines are \ftted data for the non-spin \rip (NSF) re\rectivities R++(red colour), R\u0000\u0000(blue colour), respectively.\nple S4 whereas the magnetic moment of Fe is higher in\nsample S3. Similarly, Fe-NiFe interface of sample S3 has\nhigher roughness and magnetic moment than sample S4.\nThe thickness of NiFe magnetic layer is higher in sample\nS4, and hence, higher roughness in comparison to sample\nS3. We found high values of interdi\u000busion layer thickness\nand magnetic moment at Fe-NiFe interface in sample S3\nas compared to sample S4. Also, a dead layer is created\nat the NiFe-Au interface in sample S3 whereas no dead\nlayer is formed in the Fe-Au interface in sample S4. The\npresence of high exchange coupling may be a possible\nreason for the higher value of coercivity and anisotropy\n\feldHKin sample S3 than S4 (see table 4).\nThe nuclear scattering length density (NSLD) is found\nto be of 0.2 fm/ \u0017A3near to SiO2-Fe interface whereas zero\nNSLD is found above NiFe-Au interface in sample S3.\nHowever, we found zero magnetic scattering length den-sity (MSLD) near to Si O2-Fe and NiFe-Au interfaces of\nsample S3. Similar trends of NSLD and MSLD is found\nnear to the interfaces Si O2-NiFe and Fe-Au. Also, we\nfound the similar trend of the NSLD and MSLD pro\fles\nnear to saturation and HC\feld values for the samples\nS3 and S4. Comparing the SLD pro\fles of samples S3\nand S4, we found a sharp drop in NSLD for Fe magnetic\nlayer of sample S4 whereas SLD is almost constant for Fe\nand NiFe layers of sample S3. Also, we found the change\nin sign of the MSLD's for the samples S3 and S4 near\nto the coercive \feld HCand this is due to magnetic \feld\nhistory. We can not say that depolarisation is responsible\nfor the sign chnage of MSLD because the PNR measure-\nment \felds (shown in the \fgures 6 (c) and (d) with green\ncoloured square symbols) and guide \feld are along the\nsame direction.\nWe can calculate the MSLD from the MSobtained5\nFIG. 4. Nuclear and magnetic scattering length densities (NSLD and MSLD) vs layer thickness (z) of the sample S3 at\nsaturation \feld of -50 mT (a) and near to HCat -4 mT (b).\nFIG. 5. Nuclear and magnetic scattering length densities (NSLD and MSLD) vs layer thickness (z) of the sample S4 at\nsaturation \feld of -50 mT (a) and near to HCat -1.2 mT (b).\nfrom SQUID using the relation MSLD=C. MS, where\nC=2.853\u000210\u00009(\u0017A\u00002).(cm3/emu). We found MSLD\nof 0.22 and 0.18 fm/ \u0017A3for the samples S3 and S4. These\nMSLD values obtained from SQUID match well with the\nvalues found from PNR (see \fgures 5 and 6).\nB. Kerr microscopy and magnetometry analysis\nTABLE IV. HCalong EA and HA and HKfor all the samples.\nSample name HC(EA) (mT) HC(HA) (mT) HK(mT)\nS1 0.80 0.38 4.28\nS2 0.74 0.32 2.44\nS3 5.10 1.47 7.10\nS4 1.45 0.79 4.00\nHysteresis loops were measured using longitudinalmagneto optic Kerr e\u000bect (LMOKE) based magnetome-\ntry at room temperature along \u001e= 0\u000e(EA), 30\u000e, 60\u000e, 90\u000e\nw.r.t. EA for all the samples, which are shown in \fgure 6.\nWe observed square-shaped loops along EA and s-shaped\nloops along HA for all the samples. This indicates, mag-\nnetization reversal is occuring via domain wall motion\nalong EA and coherent rotation along HA. From the hys-\nteresis loops, it is also concluded that the samples exhibit\nuniaxial magnetic anisotropy due to oblique angle of de-\nposition. It is reported in the literature that anisotropic\nsamples give high energy product value ( BH)maxthan\nthe isotropic samples23. This is because sample with\nmagnetocrystalline anisotropy gives high coercivity with\nsquare shaped loop, and thus, high ( BH)maxvalue. Al-\nthough the thickness of sample S1 is twice of S2, we found\nsimilar coercivity and di\u000berent anisotropy \feld values in\nsamples S1 and S2. We found the enhancement of co-\nercivity in magnetic bilayers than the reference single\nmagnetic layers. Interfacial exchange coupling might be6\nFIG. 6. (a)-(d): Hysteresis loops measured by LMOKE at room temperature along \u001e= 0\u000e, 30\u000e, 60\u000e, and 90\u000efor samples\nS1-S4.\nFIG. 7. Magnetic domain images of sample S1-S4 along \u001e= 0\u000e(EA), 30\u000e, 60\u000eand 90\u000e(HA) recorded in LMOKE based\nmicroscopy at room temperature.\na reason for this enhancement of coercivity. Also, we\nfound the tuning of coercivity by alternating the order ofmagnetic layers. This indicates the presence of di\u000berent\ninterfacial exchange coupling strength in the samples S37\nand S4. The magnetization reversal behaviour of samples\nS3 and S4 is like a rigid magnetic system because the soft\nand hard phases reverse with a single coercive \feld.\nMagnetic domain images of samples S1 ((a)-(d)), S2\n((e)-(h)), S3 ((i)-(l)) and S4 ((m)-(p)) along \u001e= 0\u000e(EA),\n30\u000e, 60\u000eand 90\u000e(HA) are shown in \fgure 7. We found\nbig branch domains along EA for all the samples. Fur-\nther, magnetization reversal is occuring via 180\u000edomain\nwall motion. We found the nucleation and propagation of\ndomain walls in all the samples. Magnetization reversal\nof samples S1, S2 and S4 away from EA occurs via big\ndomains indicating the presence of anisotropy inhomo-\ngeneity24. However, magnetization reversal of sample\nS3 occurs via small domains indicating strong uniaxial\nanisotropy in this sample. The absence of magnetic do-\nmains is found along HA in all the samples, thus, the\nmagnetization reversal occurs via coherent rotation.\nC. FMR analysis\nIn order to understand the anisotropy symmetry, we\nhave performed in-plane angle ( \u001e) dependent FMR mea-\nsurements at an interval of 10\u000e.\nWe can write the magnetic free energy density as the\nequation given below24,25;\nE=HMS[sin\u0012Hsin\u0012cos(\u001e\u0000\u001eH) + cos\u0012Hcos\u0012]\n\u00002\u0019(MS)2(sin\u0012)2+KP(sin\u0012)2\n+Kin(sin\u0012)2(sin(\u001e\u0000\u001e0)2(1)\nwhere, perpendicular uniaxial anisotropy and in-plane\ntwo-fold uniaxial anisotropy constants are de\fned as KP\nandKin, respectively. The angles of applied magnetic\n\feldHand saturation magnetization MSwrt z-axis are\ndenoted as \u0012Hand\u0012, respectively. \u001eHis the angle of\nprojection of MSin x-y plane wrt x-axis. \u001eis the angle\nof the projection of Hin the x-y plane wrt x-axis. \u001e0is\nthe two-fold EA direction wrt the x-axis. The directions\nofMS,Hand the two fold EA \u001e0can be found in our\nprevious work by Mallick et al.24.\nIt should be noted that the magnetic \feld was applied\nin the \flm plane. Therefore we have used the follow-\ning dispersion relation to \ft the angle dependent Hresin\norder to \fnd the values of HKandhu24.\n(!\n\r)2= [Hrescos(\u001e\u0000\u001eH)\u0000hU\n+HK(sin(\u001e\u0000\u001e0))2][Hrescos(\u001e\u0000\u001eH)\n+HK+ 2HK(sin(\u001e\u0000\u001e0))2] (2)\nwhere,hu=2KP\nMS- 4\u0019MSandHK=2Kin\nMS. In-plane\nangle dependent FMR measurements are performed at a\n\fxed frequency of 9 GHz. FMR measurement con\frms\nthe presence of uniaxial magnetic anisotropy in all the\nsamples. Figure 8 shows the plot of in-plane angle de-\npendentHres. The solid scattered data points are the\nexperimental data whereas the solid continuous line isthe \ftted data using eq. 2. We could not \fnd the fer-\nromagnetic resonance signal of sample S2, therefore the\nplot ofHresvs\u001eof this sample has not been shown. HK\nvalues of 0.0036 T, 0.0082 T and 0.0041 T are evaluated\nfor samples S1, S3 and S4, respectively, by \ftting the ex-\nperimental data (\fg. 8) using eq. 2. Figure 9 shows the\nFMR frequency ( fFMR ) vsHresand line width (\u0001 H), re-\nspectively. The e\u000bective demagnetization \feld (4 \u0019Meff),\ne\u000bective anisotropy \feld ( HKeff ) and the gyromagnetic\nratio\r=g\u0016B\n~values have been extracted by \ftting ex-\nperimental data (\fg. 9 (a)) using the following Kittel\nequation26,27:\nfFMR =\r\n2\u0019p\n(4\u0019Me\u000b+Hres+HKe\u000b)(Hres+HKe\u000b) (3)\nSimilarly, the Gilbert damping constant value \u000bis ob-\ntained by \ftting the line width (\u0001 H) vsfFMR (\fg. 9\n(b)) using the following equation27,28;\n\u0001H= \u0001H0+4\u0019\u000bf FMR\n\r(4)\nwhere, \u0001H0is the inhomogeneous linewidth broaden-\ning.\nDue to large linewidth broadening, we could not mea-\nsure the FMR spectra of sample S2, and hence, the Hres\nand \u0001Hvalues. It is theoretically reported that line\nwidth value depends on anisotropy \feld HKand the in-\nterlayer exchange coupling of two ferromagnetic layers\nseparated by a non-magnetic layer29. We found di\u000berent\nvalues ofHKin the bilayer samples S3 and S4 from Kerr\nmicroscopy measurements. This result indicates di\u000berent\ninterfacial exchange coupling strength from the ferromag-\nnetic bilayers. The large increase in the linewidth value of\nsample S3 in comparison to S4 may be due to the change\nin interfacial exchange coupling of the bilayers. Also, lit-\ntle deviation in the Hresvalue from all other samples is\nobserved in sample S4 and this may be due to modi\fed\nexchange coupling at the interface of the ferromagnetic\nlayers. We reported earlier that direct exchange coupling\nbetween magnetic layer leads to the enhancemet of the\nGilbert damping constant \u000bvalue30. We are getting si-\nmultaneously high coercivity and less \u000bvalues in sample\nS3 which is good for FMR applications. Omelchenko et\nal., reported the tuning of damping by alternating the\norder of Py/Fe bilayers deposited on Si substrate with\nTa as seed layer31. However, in this study, damping re-\nmains similar by alternating the order of magnetic layers\nwhich is useful for potential applications. Table 5 shows\nthe list of values of g, 4 \u0019Meff,HKeff ,\u000b, \u0001H0, andKS\nof all the samples.\nIn a crystalline material, due to symmetry in crystal\nlattice, average value of orbital angular momentum is\nzero. But, the orbital contribution of magnetic moment\n\u0016Lis non-zero leading to the g-factor greater than 2 fol-\nlowing the relation g'2 (1 + (\u0016L/\u0016S))32. As the\nsurfaces and interfaces break inversion symmetry, that\nleads to crystal \feld no longer symmetric. Therefore,\ng-factor is less than 2 and follows the relation g'28\nFIG. 8. (a)-(c): The plot of resonance magnetic \feld ( Hres) vs in-plane angle \u001efor samples S1, S3 and S4, respectively. Solid\nsymbols are the experimental data while solid lines are the best \ft using eq. 2.\nFIG. 9. (a) Hres, (b) \u0001HversusfFMR plot and their \fts using eqs. (3) and (4) for the samples S1, S3 and S4.\nTABLE V. List of values of the magnetic parameters g, 4 \u0019Meff,HKeff,\u000b, \u0001H0, andKSfor all the samples.\nSample name g \u001604\u0019Meff(mT)\u00160HKeff (mT)\u000b \u0016 0\u0001H0(mT)KS(erg:cm\u00002)\nS1 1.956 \u00060.007 636.25 \u00068.69 8.44 \u00060.34 0.0160 \u00060.0005 0.31 \u00060.41 -0.042 \u00060.002\nS3 2.032 \u00060.016 630.47 \u000615.83 2.35 \u00060.47 0.0150 \u00060.0006 21.59 \u00060.48 -0.148 \u00060.007\nS4 2.060 \u00060.002 731.63 \u00062.73 2.43 \u00060.05 0.0180 \u00060.0003 2.92 \u00060.28 -0.025 \u00060.001\n(1 - (\u0016L/\u0016S))32. We observed large value of the inho-\nmogeneous linewidth broadening \u0001 H0in sample S3 and\nthis value is higher than sample S4. We evaluated the\nvolume anisotropy \feld HKfor all the samples using\nKerr microscopy measurements. However, using FMR\nspectroscopy, we can evaluate surface induced anisotropy\nknown as perpendicular surface anisotropy constant KS.\nThe e\u000bective demagnetization \feld (4 \u0019Meff) and satura-\ntion magnetization MSvalues follow the below relation;\n4\u0019Meff= 4\u0019MS+2KS\nMStFM(5)\nWe foundMSvalue of 762 emu:cc\u00001in sample S3\nwhich is higher than sample S1 (639 emu:cc\u00001). Thus,\ndirect exchange coupling between the magnetic layers re-\nsults higher value of MS. Also, we found MSvalue of 860\nemu:cc\u00001in sample S2. Again, samples S3 and S4 ( 636\nemu:cc\u00001) have dissimilar MSvalues indicating the tai-\nloring of the interfacial exchange coupling by alternating\nthe order of magnetic layers.Therefore it is observed that in sample S3 i.e. when\nNiFe layer is grown on top of Fe, the sample exhibits\nhigh coercive \feld and anisotropy. Further this sample\nalso exhibits damping value comparable to the reference\nsingle NiFe \flm.\nIII. CONCLUSIONS:\nWe have studied the role of interface modi\fcation\non the magnetization reversal of the Fe/NiFe bilayer\nsystem fabricated by magnetron sputtering. Kerr Mi-\ncroscopy data showed single-phase hysteresis loops, indi-\ncating strong interfacial exchange coupling. Quantitative\nanalysis of the Kerr loops revealed the enhancement of\nthe magnetic parameters such as coercive \feld HCand\nanisotropy \feld HKwhich get almost doubled when NiFe\nlayer grows over the Fe layer. Further, this bilayer sample\nshowed smaller domains away from the EA, con\frming\nthe presence of high uniaxial magnetic anisotropy in it.\nThe presence of uniaxial magnetic anisotropy was also9\nrevealed from the in-plane angle-dependent FMR study.\nBy comparing the PNR results of the bilayer samples,\nwe observed the modi\fcation of the Fe-NiFe interfacial\nlayer upon changing the order of the magnetic layers.\nThe strength of the interfacial exchange coupling was\nhigher when the NiFe layer is grown over the Fe layer.\nDespite di\u000berent values of the anisotropy \feld and mod-\ni\fed interfacial exchange coupling, the Gilbert damping\nconstant of the bilayer systems remains similar to sin-\ngle NiFe layer. In summary, interchanging the order of\nmagnetic layers plays a key role in tuning the interfacial\nexchange coupling through modi\fcation of interdi\u000busion\nlayer thickness and magnetic moment. In this respect\nPNR has been proven to be an ideal technique to reveal\nthe interface magnetic properties. Tuning of fundamen-\ntal magnetic properties is possible by this methodology\nwhereas the Gilbert damping constant remains similar\nwhich is good for applications.\nACKNOWLEDGMENTS:\nWe acknowledge the \fnancial support from Depart-\nment of Atomic Energy (DAE), Department of Sci-\nence and Technology- Science and Engineering Research\nBoard (SB/S2/CMP-107/2013) and Department of Sci-\nence and Technology (DST) Nanomission (SR/NM/NS-\n1018/2016(G)), Government of India for providing the\n\fnancial support. SB and SSD acknowledge the \fnan-\ncial support from Newton Fund to carry out the PNR\nexperiments at Rutherford Appleton Laboratory. BBS\nacknowledges DST for INSPIRE faculty fellowship. ISIS\nDOI for the data set along with the RB number of the\nexperiment is 10.5286/ISIS.E.RB1520249.\n1Wei, L.; Xiong-Hua, L.; Wei-Bin, C.; Wen-Jie, G.; and Zhi-Dong,\nZ.Chin. Phys. 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The\nmajor challenge is to experimentally generate the negative Gilbert damping, which was thought to be highly\nunlikely but is demanded by the PT symmetry. In this work, we study the magneto-optical interaction of\ncircularly-polarized lasers with a submicron magnet placed in an optical cavity. We show that the o \u000b-resonant\ncoupling between the driving laser and cavity photon in the far-blue detuning can induce the magnetic gain (or\nnegative damping) exactly of the Gilbert type. A hyperbolic-tangent function ansatz is found to well describe\nthe time-resolved spin switching as the intrinsic magnetization dissipation is overcome. When the optically\npumped magnet interacts with a purely lossy one, we observe a phase transition from the imbalanced to passive\nPTsymmetries by varying the detuning coe \u000ecient. Our findings provide a feasible way to manipulate the sign\nof the magnetic damping parameter and to realize the EP in cavity optomagnonics.\nIntroduction. —One of the most fundamental principles in\nquantum mechanics is that a physical observable should be\ndescribed by a Hermitian operator to guarantee real eigenval-\nues [1]. However, Bender and Boettcher [2] reported a class\nof non-Hermitian Hamiltonians that allow entirely real spec-\ntrum as long as the combined parity ( P) and time (T)-reversal\nsymmetries are respected. By tuning system parameters, both\nthe eigenvalues and eigenstates of the PT-symmetric Hamil-\ntonian simultaneously coalesce [3, 4], giving rise to a non-\nHermitian degeneracy called exceptional point (EP). The na-\nture around the EP that is accompanied by a phase transi-\ntion can trigger many intriguing phenomena, such as unidi-\nrectional invisibility [5, 6], loss-induced laser suppression and\nrevival [7] and optical transparency [8], laser mode selection\n[9], and EP enhanced sensing [10–13]. Over the past decades,\nthe experimental observation of EPs has been realized in a\nbroad field of photonics [14–17], acoustics [18, 19], and elec-\ntronics [20–22]. Very recently, the concept of PTsymmetry\nis attracting significant attention in spintronics and magnonics\n[23–31]. The simplest way to obtain a PT-symmetric system\nconsists in coupling two identical subsystems, one with gain\nand the other with equal amount of loss. The composite sys-\ntem isPT symmetric because space reflection interchanges\nthe subsystems, and time reversal interchanges gain and loss.\nIndeed, aPT-symmetric magnetic structure composed of two\nidentical ferromagnets with balanced gain and loss was first\nproposed by Lee et al. [23] and subsequently investigated by\nYang et al. [27]. One recent breakthrough was made by Liu\net al. [32] who reported EP in passive PT-symmetric devices\nin the form of a trilayer structure with two magnetic layers\nof di\u000berent (positive) Gilbert damping. However, the exper-\nimental observation of genuine PT symmetry for magnons\n(the quanta of spin waves)—as elementary excitations in or-\ndered magnets—is still elusive. The di \u000eculty lies in that the\nGilbert damping can hardly be tuned to be negative [33, 34].\nThe past ten years have witnessed the development and\napplication of spin cavitronics, allowing cavity photons res-\nonantly coupled to magnons with the same microwave fre-quency [35–46]. One recent trend beyond microwaves is the\nrealization of the parametric coupling between optical lasers\nand magnons, that would generate interesting new opportuni-\nties. Tantalizing physics indeed has been demonstrated, such\nas nonreciprocal Brillouin light scattering [47], microwave-\nto-optical converting [48, 49], optical cooling of magnons\n[50], etc. In these studies, considerable interests have been\ndrawn to the scalar properties of magnons, e.g., magnon num-\nber (population), temperature, and chemical potential, which\nis successful to describe the small-angle spin precession. In\ncontrast, their vectorial behavior, i.e., the full time-evolution\nof the magnetic moment driven by optical lasers, remains\nlargely unexplored, with few exceptions [51]. It has been\nshown that a ferromagnetic-to-antiferromangetic phase tran-\nsition may emerge in the vicinity of the magnonic EP [27]. In\nsuch case, the magnetic moment would significantly deviate\nfrom its equilibrium direction, and a vectorial field descrip-\ntion becomes more relevant than a scalar one.\nS\nz\nx y\nCircularly polarized \nlaser beamOptical cavity(a) (b)\n(c)ωlas > ωcav\nωlas‘ωm\nRed-detuningBlue-detuning\nωlas < ωcav\nωlas‘\nωm\nFIG. 1: (a) Schematic illustration of a macrospin Sinteracting\nwith three orthogonally propagating circularly-polarized lasers (red\nbeams) in an optical cavity. O \u000b-resonant coupling between the driv-\ning laser (!las) and the cavity photon ( !cav) mediated by magnons\n(!m\u001c!cav) in the blue (b) and red (c) detuning regimes.\nIn this Letter, we propose to realize the negative GilbertarXiv:2006.16510v1  [cond-mat.mtrl-sci]  30 Jun 20202\ndamping by considering the optomagnonic interaction be-\ntween three orthogonally propagating circularly-polarized\nlasers and a submicron magnet placed in an optical cavity [see\nFig. 1(a)]. By solving the coupled equations of motion and\nintegrating the photon’s degree of freedom, we derive the an-\nalytical formula of the optical torque acting on the macrospin.\nIn the far-blue detuning, we find that the optical torque exactly\ntakes the Gilbert form \u0000\u000bopt\nS˙S\u0002Swith\u000bopt>0 (see below).\nThe total Gilbert damping becomes negative when the intrin-\nsic dissipation is overcome. In such case, a hyperbolic-tangent\nfunction ansatz is found to well describe the time-resolved\nspin switching. We further study the optically pumped spin\ninteracting with a purely lossy one, and observe a phase transi-\ntion from the imbalanced to passive PTsymmetries by vary-\ning the detuning parameter.\nModel. —The proposed setup is schematically plotted in\nFig. 1(a). Three circularly-polarized laser beams propagat-\ning respectively along x;y;zdirections drive the parametric\ncoupling with a macrospin S=(ˆSx;ˆSy;ˆSz) inside the optical\ncavity. The Hamiltonian reads\nH=\u0000~!0ˆSz\u0000~X\nj=x;y;z\u0010\n\u0001j\u0000gjˆSj\u0011\nˆcy\njˆcj+Hdr; (1)\nwhere!0=\rB0is the Larmor frequency around the exter-\nnal magnetic field B0pointing to the negative z-direction with\n\rbeing the gyromagnetic ratio, \u0001j=!las;j\u0000!cavis the de-\ntuning between the laser frequency !las;jand the cavity reso-\nnant frequency !cav, and ˆ cy\nj(ˆcj) is the creation (annihilation)\noperator of the optical cavity photons, with j=x;y;z. The\ncoupling strength gjbetween the spin and optical photon orig-\ninates from the Faraday-induced modification of the electro-\nmagnetic energy in ferromagnets [52]. The last term describes\nthe interaction between the driving laser and the cavity pho-\ntonHdr=i~P\nj(Ajˆcy\nj\u0000h:c:), where Aj=(2\u0014jPj=~!las;j)1=2is\nthe field amplitude, with \u0014jthe laser loss rate and Pjbeing the\ndriving power.\nThe Heisenberg-Langevin equations of motion for coupled\nphotons and spins are expressed as ( o\u0011hˆoi),\n˙cj=(i\u0001j\u0000\u0014j)cj\u0000igjSjcj+Aj; (2a)\n˙Sx=!0Sy+gynySz\u0000gznzSy; (2b)\n˙Sy=\u0000!0Sx\u0000gxnxSz+gznzSx; (2c)\n˙Sz=\u0000gynySx+gxnxSy; (2d)\nwhere nj=hˆcy\njˆcjiis the average photon number in the cav-\nity. Because the spin dynamics usually is much slower than\noptical photons, one can expand the cavity photon operator as\ncj(t)\u0019cj0(t)+cj1(t)+\u0001\u0001\u0001, in orders of ˙Sj. Equation (2a) then\ncan be recast in series\n0=(i\u0001j\u0000\u0014j)cj0\u0000igjSjcj0+Aj; (3a)\n˙cj0=(i\u0001j\u0000\u0014j)(cj0+cj1)\u0000igjSj(cj0+cj1)+Aj;(3b)\nby keeping up to the first-order terms. We can therefore derivethe formula of photon number in the cavity\nnj(t)\u0019 jcj0j2+2Re[ c\u0003\nj0cj1]\n=A2\nj\n(\u0001j\u0000gjSj)2+\u00142\nj\u00004\u0014jA2\njgj(\u0001j\u0000gjSj)\nh\n(\u0001j\u0000gjSj)2+\u00142\nji3˙Sj:(4)\nSubstituting (4) into Eqs. (2b)-(2d), we obtain\n˙S=\u0000\rS\u0002Be\u000b+\u000b\nS(˙S\u0002S)\u0000\fopt\u0002S; (5)\nwhere the e \u000bective magnetic field Be\u000b=\u0000B0ez+Boptin-\ncludes both the external magnetic field and the optically in-\nduced magnetic field\nBopt=X\nj\r\u00001gjA2\nj\n(\u0001j\u0000gjSj)2+\u00142\njej; (6)\nwhich is the zeroth-order of ˙Sj. The second term in the right\nhand side of (5) is the intrinsic Gilbert damping torque, with\nS=jSjthe total spin number and \u000b > 0 being the intrinsic\nGilbert damping constant. The last term in (5) represents the\noptical torque with the anisotropic e \u000bective field\n\fopt=X\nj4\u0014jA2\njg2\nj(\u0001j\u0000gjSj)\nh\n(\u0001j\u0000gjSj)2+\u00142\nji3˙Sjej; (7)\nwhich is linear with the first-order time-derivative of Sj. Be-\nlow, we show that the anisotropic nature of (7) can be smeared\nout under proper conditions.\nNegative Gilbert damping. —To obtain the optical torque of\nexactly the Gilbert form, we make two assumptions: (i) the\nthree laser beams are identical, i.e., Aj=A;gj=g;\u0014j=\u0014;\nand\u0001j= \u0001; (ii) the optomagnonic coupling works in the far\ndetuning regime, i.e., j\u0011j\u001d1 with\u0011= \u0001=(gS), which allows\nus to drop the gjSjterms in Eq. (7). The optically induced\ne\u000bective fields then take the simple form\nBopt=\r\u00001gA2\n\u00012+\u00142X\njej; (8)\nand\n\fopt=\u000bopt\nS˙S;with\u000bopt=4\u0014A2g2S\u0001\n(\u00012+\u00142)3(9)\nbeing the laser-induced magnetic gain or loss that depends the\nsign of the detuning \u0001. Based on the above results, we finally\nobtain the optically modulated spin dynamics\n˙S=\u0000\rS\u0002Be\u000b+\u000be\u000b\nS(˙S\u0002S); (10)\nwith\u000be\u000b=\u0000\u000bopt+\u000b. One can observe that a negative ef-\nfective Gilbert constant ( \u000be\u000b<0) emerges in the far-blue de-\ntuning regime, i.e., 1 < \u0011 < \u0011 c. In case of the red detun-\ning (\u0011 < 0), we have \u000bopt<0, which indicates the enhance-\nment of the magnetic attenuation. In the deep-blue detuning3\nηηηPTηC\nηC=7.11η P (W)P (μWĎ\nr (m)αopt /α Bopt (μT)αeff =0\nBopt =333 μT(a)\n(b)(c)\n(d)\nηPTηCBopt =440 μTαeff =-α\n×\nFIG. 2: Optically induced magnetic gain (a) and magnetic field (b)\nvs. the optical detuning parameter \u0011. (c)\u0011PT(orange) and \u0011C(green)\nas a function of the driving laser power. (d) Radius dependence of\nthe laser power at the compensation point \u0011C=7:11.\nregime (\u0011>\u0011 c), driving lasers can still generate the magnetic\ngain (\u000bopt>0) but cannot compensate the intrinsic dissipa-\ntion, i.e., 0 < \u000b opt< \u000b. Here\u0011cis the critical detuning pa-\nrameter at which the e \u000bective Gilbert damping vanishes. The\nphysics can be understood from the diagram plotted in Figs.\n1(b) and 1(c): In the blue detuning regime ( !las> ! cav), mi-\ncrowave magnons are emitted in the non-resonant interaction\nbetween the driving laser and the cavity photon, representing\na magnetic gain. On the contrary, they are absorbed in the red\ndetuning (!las< ! cav), manifesting a magnon absorption or\ncooling. Below we discuss practical materials and parameters\nto realize this proposal.\nMaterials realizations. —For a ferromagnetic insulator like\nyttrium ion garnet (YIG), the intrinsic Gilbert constant \u000btyp-\nically ranges 10\u00003\u001810\u00005[53–55]. We take \u000b=10\u00004\nin the following calculations. The magneto-optical coupling\nstrength is determined by the Faraday rotation coe \u000ecient\u0012F\nof the materials gS'c\u0012F=p\u000fr, with cthe speed of light and\n\u000frthe relative permittivity (for YIG, we choose \u000fr=15 [56]\nand\u0012F=188\u000e=cm [57]). We thus have gS=2\u0019\u00191 GHz. The\noptical cavity is set at the resonant frequency !cav=2\u0019=100\nTHz with the loss rate \u0014=2\u0019=1 GHz. For a YIG sphere of\nradius r=10 nm and spin density \u001as\u00191028m\u00003, we esti-\nmate the total spin number S=\u001asr3\u0019104and the coupling\nstrength g=2\u0019\u00190:1 MHz. Materials parameters are summa-\nrized in Table I. Because g\u001c\u0014, all interesting physics occurs\nin the weak coupling regime. A negative \u000be\u000bis demanded for\nrealizing thePTsymmetry in magnetic system. Considering\nthe driving laser with a fixed power P=1\u0016W, the e \u000bective\nTABLE I: Parameters for optical cavity and YIG.\n!cav=2\u0019 \u0014= 2\u0019 ! 0=2\u0019 gS=2\u0019 r\u000b\n100 THz 1 GHz 10 GHz 1 GHz 10 nm 10\u00004Gilbert-type magnetic gain is \u000be\u000b=\u0000\u000bat\u0011PT'6:16, and\nthe critical gain-loss point \u000be\u000b=0 occurs at\u0011C'7:11, indeed\nsatisfying the large-detuning condition j\u0011j\u001d1 in deriving (9).\nFigure 2(a) shows the monotonically decreasing dependence\nof the optically induced magnetic gain \u000bopton the detuning pa-\nrameter\u0011. The\u0011-dependence of the optical field is plotted in\nFig. 2(b), showing that it monotonically decreases with the in-\ncreasing of the detuning, too. Enhancing the laser power will\npush the two critical points \u0011Cand\u0011PTinto the deep detuning\nregion, as demonstrated in Fig. 2(c). For a magnetic sphere\nof larger volume (1 \u0016m)3\u0018(1 mm)3that contains a total spin\nnumber S=1010\u00181019with the reduced magneto-optical\ncoupling strength g=2\u0019=10\u00001\u001810\u000010Hz, the required laser\npower then should be 6 \u001815 orders of magnitude higher than\nthe nm-scale sphere case, as shown in Fig. 2(d).\nTime-resolved spin flipping. —To justify the approximation\nadopted in deriving the Gilbert-type magnetic gain, we di-\nrectly simulate the time evolution of the unit spin components\n(sj\u0011Sj=S) based on both Eq. (5) and Eq. (10). Numer-\nical results are, respectively, plotted in Figs. 3(a) and 3(b)\nfor the same detuning parameter \u0011=1:8 (corresponding to\nan e\u000bective magnetic gain \u000be\u000b=\u00000:0453) and!0=2\u0019=10\nGHz. Both figures show that the very presence of the negative\nGilbert damping can flip the spin in a precessional manner,\nwith similar switching curves. The fast Fourier transforma-\ntion (FFT) analysis of the spatiotemporal oscillation of sxalso\nconfirms this point (see the insets). Although the analytical\nform of sz(t) by solving (5) generally is unknown [58, 59], we\nfind an ansatz that can well describe the time-resolved spin\nswitching\nsj sj\nszη=1.8 (αeff =-0.0453 )\n(a) (c)\n(d) (b)szsysx\nττ τ\nηEq. (5)\nEq. (10)τ0=98.9 (a) fitting\n(b) fitting\nTheoryτ0=107.8’’\nτ0=100.4’τp=22.1τp=22.4’’τp=14.9’ tanh(-      )τ-τ0τp\nτ0’\nτ0’’\nTheory\nτp’\nτp’’\nTheory46810121401020 9.779.71\nFrequency (GHz)Frequency (GHz)FFT of s x FFT of s x46810121401020\nFIG. 3: Time evolution of unit spin components ( sx;sy;sz) at de-\ntuning\u0011=1:8 based on Eq. (5) (a) and Eq. (10) (b). Insets\nshow the FFT spectrum of sx. (c) Theoretical fittings of szusing\nthe hyperbolic-tangent ansatz (11) (dashed curves). The solid green\ncurve is the analytical formula without any fitting. (d) Numerical re-\nsults of the \u0011-dependence of the two characteristic times \u001c0and\u001cp,\ncomparing with formula (12) (solid curves).4\nsz(\u001c)'tanh \n\u0000\u001c\u0000\u001c0\n\u001cp!\n; (11)\nwhich is reminiscent of the Walker solution for modeling the\nprofile of 180\u000emagnetic domain wall [60] by replacing the\ntime coordinate \u001cwith the space coordinate x. Here\u001c0is\nthe switching time, \u001cprepresents the life-time of uniform\nmagnons, and \u001c=!0t. From perturbation theory, we derive\nthe analytical form of these two parameters\n\u001cp=\u00001+\u000b2\ne\u000b\n\u000be\u000b;and\u001c0=\u001cptanh\u00001vt\n1\u00004B2\nopt\nB2\ne\u000b:(12)\nFigure 3(c) shows the time evolution of sz. Symbols repre-\nsent the numerical results, dashed curves label the theoretical\nfittings of ansatz (11), and the solid curve is the analytical for-\nmula without fitting. The fitted switching time \u001c0\n0=100:4\n(\u001c00\n0=107:8) and magnon life-time \u001c0\np=14:9 (\u001c00\np=22:4)\nfrom from Eq. (5) [Eq. (10)] compare well with the analyt-\nical formula (12) which gives \u001c0=98:9 and\u001cp=22:1. We\nfurther show that the analytical ansatz agrees excellently with\nnumerical results in a broad range of detuning parameters, as\nplotted in Fig. 3(d).\nPhase transition in spin dimers. —We have shown that un-\nder proper conditions, one can realize the Gilbert-type mag-\nnetic gain which is essential for observing PT-symmetry in\npurely magnetic structures. Next, we consider the optically\npumped spin Sinteracting with a lossy one S0, as shown in\nFig. 4(a). The coupled spin dynamics is described by the\nLandau-Lifshitz-Gilbert equation\n˙s=\u0000\rs\u0002Be\u000b+!exs\u0002s0+\u000be\u000b˙s\u0002s; (13a)\n˙s0=\u0000\rs0\u0002B0\ne\u000b+!exs0\u0002s+\u000b˙s0\u0002s0; (13b)\nwhere s(0)\u0011S(0)=Sis the unit spin vector. Since the optically\ninduced magnetic field is the same order of magnitude with\nthe geomagnetic field (much smaller than B0), it can be safely\nignored. Spin s0is exchange coupled to the optically pumped\nspins, and su \u000bers an intrinsic Gilbert damping. If \u000be\u000b=\u0000\u000b,\nthe two-spin system satisfies the PT-symmetry: Eqs. (13) are\ninvariant in the combined operation of the parity P(s$s0\nandBe\u000b$B0\ne\u000b) and the time reversal T(t!\u0000 t,s!\u0000s,\ns0!\u0000s0,Be\u000b!\u0000Be\u000b, and B0\ne\u000b!\u0000B0\ne\u000b).\nAssuming a harmonic time-dependence for the small-angle\nspin precession sx;y(t)=sx;yei!twithjsx;yj \u001c 1, one can\nsolve the eigenspectrum of Eqs. (13). By tuning the spin-\nspin coupling strength !ex, we observe a transition from exact\nPT phase to the broken PT phase, separated by the EP at\n!c\nex=2\u0019=1 MHz for\u0011=\u0011PT=6:16, as shown in Figs. 4(b)\nand 4(c). Interestingly, the unequal gain and loss, i.e., \u000be\u000b<0\nand\u000be\u000b,\u0000\u000b, leads to an imbalanced parity-time ( IPT )-\nsymmetry. In this region ( \u0011>\u0011 IPT=5:66), the eigenfrequen-\ncies have di \u000berent real parts but share the identical imaginary\none, as plotted in Fig. 4(d). A passive parity-time ( PPT )-\nsymmetry is further identified when \u000be\u000b>0. In such case\nRe[(ω-ω0)/2π] (MHz)(b)(a)\n(d)\n(c)\nωex /2π (MHz)Im[(ω-ω0)/2π] (MHz)\nηη=ηPT\nηPTηPPT ηIPTωex \n2πc\n=1 MHz(e)\nωexs s’\nωex \n2π=1.5 MHzFIG. 4: (a) Spin dimmer consisting of an optically pumped spin sand\na purely lossy one s0. Evolution of eigenfrequencies vs. the exchange\ncoupling (b,c) at the detuning \u0011PT=6:16, and vs. the detuning pa-\nrameter (d,e) at the exchange coupling !ex=2\u0019=1:5 MHz.\n(\u0011 > \u0011 PPT=7:11), the imaginary part of both branches is\nsmaller than their intrinsic damping [see Fig. 4(e)].\nDiscussion. —In the above derivation, we focus on the case\nthat the intrinsic Gilbert damping is isotropic. Our approach\ncan also be generalized to treat the case when the intrinsic\ndamping is anisotrpic [61, 62]. The three propagating lasers\nthen should be accordingly adjusted to match the tensor form\nof the intrinsic magnetic damping, by modulating the driving\npower or the frequency of each beam, for instance. The red-\ndetuning region is appealing to cool magnons to the subtle\nquantum domain. Inspired by PT-symmetric optics [19], we\nenvision a giant enhancement of the magnonic gain and an\nultralow-threshold magnon lasing in a two-cavity system with\nbalanced optical gain and loss, which is an open question for\nfuture study. While the magnonic passive PTsymmetry has\nbeen observed by Liu et al. [32], the exact and imbalanced\nPTphases are still waiting for the experimental discovery.\nConclusion. —To summarize, we have proposed an opto-\nmagnonic method to generate the negative Gilbert damp-\ning in ferromagnets, by studying the parametric dynamics\nof a macrospin coupled with three orthogonally propagating\ncircularly-polarized lasers in an optical cavity. We analyti-\ncally derived the formula of the optical torque on the spin\nand identified the condition for the magnetic gain exactly in\nthe Gilbert form. 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Phys. 14, 490\n(2018)."    },    {        "title": "2007.04372v2.Finite_frequency_spin_susceptibility_and_spin_pumping_in_superconductors_with_spin_orbit_relaxation.pdf",        "content": "Finite-frequency spin susceptibility and spin pumping in superconductors with\nspin-orbit relaxation\nM.A. Silaev1, 2, 3\n1Department of Physics and Nanoscience Center, University of Jyv askyl a,\nP.O. Box 35 (YFL), FI-40014 University of Jyv askyl a, Finland\n2Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia\n3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia\n(Dated: July 24, 2020)\nStatic spin susceptibility of superconductors with spin-orbit relaxation has been calculated in the\nseminal work of A.A. Abrikosov and L.P. Gor'kov [Sov. Phys. JETP, 15, 752 (1962)]. Surprisingly\nthe generalization of this result to \fnite frequencies has not been done despite being quite important\nfor the modern topic of superconducting spintronics. The present paper \flls this gap by deriving\nthe analytical expression for spin susceptibility. The time-dependent spin response is shown to\nbe captured by the quasiclassical Eilenberger equation with collision integrals corresponding to\nthe ordinary and spin-orbit scattering. Using the developed formalism we study the linear spin\npumping e\u000bect between the ferromagnet and the adjacent superconducting \flm. The consequences\nfor understanding recent experiments demonstrating the modi\fcation of Gilbert damping by the\nsuperconducting correlations are discussed.\nI. INTRODUCTION\nSpin transport and spin dynamics in superconductors\nhave attracted signi\fcant attention recently1{7. Quite in-\nteresting experimental results have been obtained for the\nspin pumping e\u000bects which in general play the central\nrole in spintronics8{10. Ferromagnet/ superconductor\nmultilayers were found recently to demonstrate changes\nof the ferromagnetic resonance (FMR) frequency and\nlinewidth11{19due to the superconducting correlations.\nDespite signi\fcant e\u000borts theoretical understanding of\nthese e\u000bects is not complete yet. For example, puz-\nzling experimental result has been obtained for the fer-\nromagnetic insulator/superconductor multilayers where\nthe pronounced peaks in the temperature dependence of\nGilbert damping have been observed16.\nThe enhancement of Gilbert damping due to the\nmetal spin sink can be calculated using the linear re-\nsponse approximation20which involves the momentum\nand frequency-dependent spin susceptibility \u001fh(k;\n) of\nthe metal spin sink. Hence, to understand the mod-\ni\fcation of Gilbert damping due to the spin pumping\nin superconducting \flms it is necessary to the calculate\nthe corresponding function \u001fh(k;\n) in the presence of\nspin relaxation mechanisms like the spin-orbit scatter-\ning. Quite surprisingly, this calculation has not been\never performed correctly. Recent papers which have ad-\ndressed this topic in connection with spin pumping21,22\nreport \fnite zero-temperature dissipation at low frequen-\ncies: Im\u001fh(q;\n)=\n6= 0 at \n!0. This result contradicts\nphysical intuition because there can be no dissipation\nat \n<2\u0001 and in the absence of thermal quasiparti-\ncles which are frozen out in superconductors at T\u001c\u0001,\nwhere \u0001 is the superconducting energy gap. As we show\nbelow this inconsistency comes from neglecting the im-\nportant contributions while performing analytical contin-\nuation procedure.The \frst purpose of the present paper is to report the\nanalytical expression for the \fnite-frequency spin sus-\nceptibility of superconductors with spin-orbit relaxation\nmechanism. This result is a generalization of the classi-\ncal work of Abrikosov and Gor'kov23who have considered\nthe static spin susceptibility to explain the \fnite Knight\nshift in superconductors at T\u001c\u0001. We analyse di\u000berent\ncharacteristic regimes including large and strong spin re-\nlaxation as well as the behaviour for various values of the\nDynes parameter24.\nThe second purpose is to study the spin pumping in\nsuperconductor/ferromagnet systems in the framework\nof the interfacial exchange model20. The expressions for\nGilbert damping are derived for the \fnite thickness of\nthe spin sink layer. Also we consider the system with an\nadditional perfect spin absorber which can be realized\nexperimentally by adding the layer of material with very\nstrong spin relaxation. The derived general expressions\ncan be parametrized in terms of the dimensionless param-\neter characterizing the strength of the interfacial coupling\nbetween the ferromagnet and adjacent superconductor.\nSystems with elevated values of this parameter are pre-\ndicted to feature pronounced shift of the ferromagnetic\nresonance line induced by superconducting correlations.\nII. GENERAL FORMALISM\nA. Diagrammatic formalism\nWe describe the interaction of electrons with Zeeman\n\feldh=h(r;t) using the following Hamiltonian\n^VP=^\u001bh (1)\nwhere ^\u001b= (^\u001bx;^\u001by;^\u001bz) is the vector of spin Pauli matri-\nces. Besides that we assume the presence of disorder\ndescribed by the Gaussian impurity potential. It hasarXiv:2007.04372v2  [cond-mat.supr-con]  23 Jul 20202\n(a)\n(b)\nFIG. 1. (Color online) (a) Bubble diagram for the linear\nresponse of spin polarization generated by the time-dependent\nZeeman \feldh\nei\nt+iqrshown by the wavy line. Circles show\nspin vertices ^\u001b. The shaded region shows impurity ladder.\n(b) Diagrammatic equation for the impurity ladder. The blue\nand red dashed lines correspond to the ordinary and spin-\norbit scattering potentials averaged over the random impurity\ncon\fguration.\nboth the usual Vimpand the spin-orbit Vsoscattering\namplitudes\n^V(p;p0) = (2)\nu0X\nroeiro(p\u0000p0)+uso\np2\nF^\u001b\u0001(p\u0002p0)X\nrsoeirso(p\u0000p0);\nwhereroandrsodenote the random impurity coordi-\nnates corresponding to the ordinary and spin-orbit scat-\ntering respectively. We assume this coordinates to be\nindependent and thus neglect the magnetoelectric ef-\nfects arising from the combined ordinary and spin-orbit\nscattering25.\nThe spin polarization as a function of the imaginary\ntimet2[0;\f] where\f= 1=Tis given by\nS(t;r) =1\n4Tr[^\u001b^G](r;r;t1;2=t) (3)\nwhere ^G(r1;r2;t1;2) is the imaginary time Green's func-\ntion (GF). The stationary propagators depend only on\nthe relative time and coordinate. In the frequency and\nmomentum representation they are given by23,26\n^G0(!;p) =~\u0001^\u001c2\u0000i~!^\u001c0+\u0018p^\u001c3\n~\u00012+ ~!2+\u00182p(4)\n~!=!~s(!)\ns(!);~\u0001 = \u0001~s(!)\ns(!); (5)\nwhere\u0018p=p2=2m\u0000\u0016is the deviation of the kinetic en-\nergy from the chemical potential \u0016and ^\u001c1;2;3are the Pauli\nmatrices in Nambu space. We denote s=p\n!2+ \u00012and\n~s=s+1=2\u001cimpwhere the scattering time is given by the\nsuperposition \u001c\u00001\nimp=\u001c\u00001\no+\u001c\u00001\nso. We denote the usual\u001c\u00001\no= 2\u0019n\u0017u 0and spin-orbit \u001c\u00001\nso= 2\u0019n\u0017uso=3 scat-\ntering rates. The propagator (4) is averaged over the\nrandomly disordered point scatterers con\fgurations.\nWe are interested in the spin polarization induced by\nthe external Zeeman \feld h(t;r) =h\nei\nt+iqr. The in-\nduced spin polarization as given by the diagram shown\nin Fig.1a can be written as follows\nS\n=\u001fh(\n;q)h\n (6)\nThe linear spin susceptibility is de\fned by substituting\ninto the Eq.3 ^Ghwhich is the \frst-order correction to\nthe GF induced by the Zeeman \feld. The diagrammatic\nequation for this correction which includes the summa-\ntion of impurity ladder corrections is shown Fig.1b. The\nshaded region denotes impurity ladder corresponding to\nthe ordinary and spin-orbit impurity scattering averaged\nover the random disorder con\fguration. The red and\nblue dashed lines correspond to the spin-orbit and or-\ndinary impurity scattering potentials averaged over the\nrandomly distributed impurities. Analytical expression\nfor the diagrammatic equation in Fig.1b reads as follows\n^Gh(12) =\u0000^G0(1)^\u001bh\n^G0(2)+ (7)\n^G0(1)^\u001bh^ghi^\u001b^\u001c3\n6i\u001cso^G0(2) + ^G0(1)h^ghi^\u001c3\n2i\u001co^G0(2)\nwhere we have introduced the notation\n^gh=i\n\u0019 \nd\u0018p^\u001c3^Gh: (8)\nWe use the condensed notation ^G0(2) = ^G0(!;p) and\n^G0(1) = ^G0(!\u0000\n;p+q). The correction depends on the\ntwo frequencies and momenta ^Gh(12) = ^Gh(!1;p;!2;p+\nq). The angular brackets denote average over the mo-\nmentum directions on the Fermi sphere so that in total\nh^ghi= (i=\u0019\u0017)\u0001\nd3p^Gh, where\u0017is the density of states at\nthe Fermi level. Diagrammatically the equation for impu-\nrity ladder (7) is shown in Fig.1b. The second and third\nterms in Eq.7 corresponding to the spin-orbit and ordi-\nnary scattering are shown by blue and red dashed lines,\nrespectively. As we see below the momentum-integrated\ncorrection ^Ghcoincides with the solution of quasiclassi-\ncal Eilenberger equation27with collision integrals corre-\nsponding to the ordinary and spin -orbit scattering28.\nB. Quasiclassical formalism\nUnder quite general conditions the non-equilibrium\nstate of a metal involves perturbations of spectrum and\ndistribution function in the vicinity of the Fermi level.\nFor that the external \felds should have frequencies much\nsmaller than the Fermi energy and spatial scales much\nlarger than the Fermi wave length. Both these require-\nments are satis\fed for the spin pumping systems. Hence3\nwe can use the theory formulated in terms of the quasi-\nclassical propagator27\n^g(r;np;t;t0) =i\n\u0019 \nd\u0018p^\u001c3^G: (9)\nThe calculation can be performed either using the\nimaginary time formalism of the real-time formalism. In\nthe imaginary time domain the quasiclassical propagator\nis determined by the Eilenberger equation with collision\nintegrals describing the impurity scattering27\n(vFr)^g\u0000if^\u001c3@t;^ggt=i[^\u001c3^H;^g]t+ [(^\u0006o+^\u0006so)\u000e;^g]t\n(10)\n^\u0006so= (^\u001bh^gi^\u001b)=6\u001cso (11)\n^\u0006o=h^gi=2\u001co: (12)\nHere ^\u0006oand ^\u0006soare the self-energies corresponding to\nthe ordinary and spin-orbit scattering, respectively29and\n^H= \u0001^\u001c2+^\u001bh. We denote the commutators [ X;g]t=\nX(t1)g(t1;t2)\u0000g(t1;t2)X(t2) and the convolution h^gi\u000e\n^g=\u0001\f\n0dth^gi(t1;t)^g(t;t2). The angle-averaging over the\nFermi surface is given by hgi. The spin polarization is\ngiven by\nS(t;r) =\u0000i\u0019\u0017\n4Tr[^\u001c3^\u001bh^g(t;t;r)i] (13)\nThe quasiclassical equations are supplemented by the\nnormalization condition ^ g\u000e^g= 1.\nC. Analytical continuation\nIn order to \fnd the real-frequency response we need to\nimplement the analytic continuation of Eq. (13). The\n\frst-order correction to the quasiclassical GF can be\nwritten as ^gh(t1;t2) =TP\n!e\u0000i!1t1+i!2t2g(!1;!2) where\n!2=!and!1=!\u0000\n are the fermionic Matsubara fre-\nquencies shifted by the Bosonic frequency \n of the exter-\nnal Zeeman \feld. The analytic continuation of the sum\nis determined according to the general rule30\nTX\n!gh(!1;!2)! (14)\n\u0002d\"\n4\u0019in0(\"1)\u0002\ngh(\u0000i\"R\n1;\u0000i\"A\n2)\u0000gh(\u0000i\"A\n1;\u0000i\"A\n2)\u0003\n+\n\u0002d\"\n4\u0019in0(\"2)\u0002\ngh(\u0000i\"R\n1;\u0000i\"R\n2)\u0000gh(\u0000i\"R\n1;\u0000i\"A\n2)\u0003\nwheren0(\") = tanh(\"=2T) is the equilibrium distribution\nfunction. In the r.h.s. of (14) we substitute \"1=\"\u0000\n,\n\"2=\"and\"R=\"+i\u0000,\"A=\"\u0000i\u0000. Here the term with\n\u0000>0 is added to shift the integration contour into the\ncorresponding half-plane. At the same time, \u0000 can be\nused as the Dynes parameter31to describe the e\u000bect of\ndi\u000berent depairing mechanisms on spectral functions in\nthe superconductor. We implement the analytical contin-\nuation in such a way that s(\u0000i\"R;A) =\u0000ip\n(\"R;A)2\u0000\u00012assuming that the branch cuts run from (\u0001 ;1) and\n(\u00001;\u0000\u0001).\nEquilibrium GF in the imaginary frequency domain\nis given by ^ g0(!) = (^\u001c3!+ ^\u001c1\u0001)=s(!). The real-\nfrequency continuation reads ^ gR;A\n0(\") = (^\u001c3\"R;A+\ni^\u001c1\u0001)=p\n(\"R;A)2\u0000\u00012.\nThus the linear response spin polarization is given by\n\u001fh+ 1 = (15)\u0002d\"\n4\u0019i\u001f(\u0000i\"R\n1;\u0000i\"A\n2) [n0(\"1)\u0000n0(\"2)] +\n\u0002d\"\n4\u0019i\u0002\nn0(\"2)\u001f(\u0000i\"R\n1;\u0000i\"R\n2)\u0000n0(\"1)\u001f(\u0000i\"A\n1;\u0000i\"A\n2)\u0003\nwhere we denote \u001f(!1;!2) = (\u000e=\u000eh)Tr[\u001b^gh(!1;!2)]. In\nthe l.h.s. of Eq. 15 we subtract the o\u000b-shell contribution\nto the spin polarization due to the band edge shift by the\nZeeman \feld.\nIt is interesting to note that in the superconducting\nstate both the \frst and the second terms in the r.h.s. of\n(15) contribute to the dissipative part of spin suscepti-\nbility With that we obtain physically correct behaviour\nin the low-temperature limit Im \u001fh(\n)=\n!0 atT!0\nand small frequency \n \u001c\u0001. This is in contrast to pre-\nvious calculations21,22which take into account only the\n\frst term in (15) and obtained physically incorrect \fnite\ndissipation in the absence of quasiparticles at T= 0.\nIII. SPIN SUSCEPTIBILITY\nA. Diagram summation\nFirst, we demonstrate connection between response\nfunctions determined by the diagram Fig.1a and by the\nsolution of time-dependent Eilenberger equation (10). In-\nstead of using the usual approach of calculating the ver-\ntex function23we use the alternative route and solve di-\nrectly the equation for the \frst-order correction 7.\nWe use the general approach suggested recently32for\nderiving equation for the momentum-integrated propaga-\ntors ^ghstarting from the general equation for the exact\nGF (7). The key idea of this derivation is based on the\nfollowing trick. Let us multiply the function ^Gh(12) by\n^G\u00001\n0(1) from the left and by ^G\u00001\n0(2) from the right, sub-\ntract the results and integrate by \u0018p. We use that Eq.(4)\nyields the relations ^G\u00001\n0(j) =~\u0001j^\u001c2+i~!j^\u001c0+\u0018p(pj)^\u001c3and\n~\u0001j^\u001c2+i~!j^\u001c0=i(sj+ 1=2\u001cimp)^g0(!j)^\u001c3. Then we elim-\ninate o\u000b-shell contributions in the momentum integrals\nto express the result through quasiclassical propagators\n\u0002d\u0018p\n\u0019h\n^G\u00001\n0(1)^Gh\u0000^\u001c3^Gh^G\u00001\n0(2)^\u001c3i\n= (16)\n~s1^g0(1)^gh\u0000~s2^gh^g0(2)\u0000i(vFq)^gh\nNext let us derive the l.h.s. of the equation for ^ gh.\nUsing the diagram Fig.1b or the Eq.7 we get that4\n^G\u00001\n0(1)^Gh\u0000^\u001c3^Gh^G\u00001\n0(2)^\u001c3= (17)\n^\u001c3^G0(1)(^h\n+ih^ghi^\u001c3=2\u001co+i\u001bh^ghi\u001b^\u001c3=6\u001cso)^\u001c3\n\u0000(^h\n+ih^ghi^\u001c3=2\u001co+i\u001bh^ghi\u001b^\u001c3=6\u001cso)^G0(2)+\nwhere we denote ^h\n=^\u001bh\n. Then combining with Eq.16\nwe obtain the following equation with collision integrals\n^Ioand^Iso\ns1^g0(1)^gh\u0000s2^gh^g0(2)\u0000i(vFq)^gh (18)\n=\u0000i[^g0(1)^h\n^\u001c3\u0000^h\n^\u001c3^g0(2)] + ^Iso+^Io\n^Io= [^g0(1)h^ghi+h^ghi^g0(2)\u0000 (19)\nh^ghi^g0(2)\u0000^g0(1)^gh]=2\u001co\n^Iso= [^g0(1)\u001bh^ghi\u001b+ 3h^ghi^g0(2)\u0000 (20)\n\u001bh^ghi\u001b^g0(2)\u00003^g0(1)h^ghi]=6\u001cso\nThis Eq.(18) coincides with the Eilenberger Eq. (10) ex-\npanded for the \frst-order correction ^ gh. This proves that\nthe time-dependent spin response in metals is captured\nby the Eilenberger equation with corresponding collision\nintegrals.\nB. Susceptibility of the spatially homogeneous\nsystem\nFirst, we consider the spatially homogeneous system\nwhen the Zeeman \feld depends only on time and not\non the spatial coordinate so that q= 0. The spatial\ndispersion of susceptibility is discussed in in the di\u000busive\nlimit in Sec.III C. In the homogeneous case the ordinary\nscattering drops out from Eq.18 since ^Io= 0. Then Eq.18\ncan be solved analytically yielding the frequency-resolved\nsusceptibility \u001f(12) = (\u000e=\u000eh)Tr[\u001b^gh(12)]\n\u001f(12) =\u00012+s1s2\u0000!1!2\ns1s2(s1+s2+ 4=3\u001cso); (21)\nwhere!are fermionic Matsubara frequencies, !1=!\u0000\n,\n!2=!,s1;2=q\n!2\n1;2+ \u00012. Substituting this expression\nto the analytical continuation rule (15) we obtain the\nfrequency dependent spin susceptibility \u001fh=\u001fh(\n). It\nis interesting to note that this response function (21) is\nidentical to that which determines the \fnite-frequency\nconductivity of a superconductor.\nWe can obtain analytical results in several important\nlimiting cases. For the (i) normal metal \u0001 = 0\nEqs.(21,15) yield (see detailed calculation in Appendix\nSec.B)\n\u001fh(\n) =2(2=3\u001cso+ \u0000)\n2(2=3\u001cso+ \u0000)\u0000i\n(22)In this case the only contribution is provided by the \frst\nterm in Eq.15. As one can in the absence of spin re-\nlaxation \n\u001cso! 1 and \u0000!0 the susceptibility is\nvanishes. Physically this result is quite transparent be-\ncause without relaxation the spin projection on the os-\ncillating Zeeman \feld remains a good quantum number.\nLet us check that this result remains valid in the su-\nperconducting state. For that we consider the limit of\n(ii) superconductor without spin relaxation . In\nthis case using following relations s2\n1\u0000s2\n2=!2\n1\u0000!2\n2and\n2(!1!2\u0000\u00012\u0000s1s2) = (!1+!2)2\u0000(s1+s2)2Eq.21 can\nbe simpli\fed as follows, see details in Appendix A\n\u001f(12) =2\n\n\u0012!1\ns1\u0000!2\ns2\u0013\n(23)\nThus making the analytical continuation and neglecting\nterms of the order \u0000 =\n we obtain\n\u001fh(\n) =\u00001\u0000\u00021\n\u00001d\"\n2\n[N(\"1)n0(\"1)\u0000N(\"2)n0(\"2)]\n(24)\nwhereN(\") is the normalized DOS, \"1=\"\u0000\n and\"2=\n\". One can see that this expression yields \u001fh(\n) = 0\nirrespective of the particular energy dependence of DOS.\nThis result can be qualitatively explained by the fact that\nin the absence of spin relaxation spin projection on the\noscillating Zeeman \feld axis is a conserved quantity.\nForm this limiting case one can clearly see that to ob-\ntain the correct result it is necessary to take into ac-\ncount all parts in the Eq.15. Indeed, the contribution of\nthe \frst term in Eq.15 is proportional to\u0001\nd\"[^gR\n0(1)\u0000\n^gA\n0(2)]@\"n0\u00192\n=\u0001 at low temperatures. This contri-\nbution is cancelled by the second term in Eq.14 to yield\n\u001fh(\n) = 0 for \u001c\u00001\nso= 0.\nAs we have obtained in the normal metal limit, the\ncontribution of the \frst term in spin susceptibility (15)\nis of the order \n \u001csfor weak spin relaxation \n \u001cs\u001d1.\nThus when \u001cso\u0001\u001d1 the contribution of second term can\nbe neglected. For stronger spin-orbit relaxation such an\napproximation which has been used in as it has been done\nin previous works21,22is inaccurate. Below we con\frm\nthis conclusion by evaluation Eq.15 numerically.\nLet us now considered the opposite limit of (iii) su-\nperconductor with strong spin relaxation \u001cso\u0001\u001c1\nand small frequencies \n \u001c\u0001. In this case from the gen-\neral Eq.21 we obtain\n\u001f(12) =3\u001cso\n4\u0012\u00012\u0000!1!2\ns1s2+ 1\u0013\n; (25)\nSubstituting this expression into the analytical continu-\nation rule (15) after some algebra we get\n4\n3\u001csoIm\u001fh\n\n=\u00021\n\u00001d\"\u0000\n\u00012=\"2+ 1\u0001\nN2@\"n0 (26)\nFrom this expression one can see analytically that the\ndissipative part of the susceptibility vanishes in the zero-\ntemperature limit.5\n(a)\u001csoTc= 100\n (b)\u001csoTc= 10\n (c)\u001csoTc= 1\n (d)\u001csoTc= 0:1\nFIG. 2. Comparison of the contributions to the dissipative spin response Im \u001fhgiven by the both terms in Eq.15 (solid blue\nlines) and only the \frst term in Eq.15 (red dashed lines). The parameters are \u0000 = 0 :001Tc, \n = 0:01Tcand spin-orbit scattering\ntime\u001csoTcis (a)100, (b) 10, (c) 1, (d) 0 :1.\n(a)\u001csoTc= 100\n\u0000=Tc\n(b)\u001csoTc= 10\n (c)\u001csoTc= 1\n (d)\u001csoTc= 0:1\nFIG. 3. Temperature dependencies of the dissipative part of spin susceptibility Im \u001fhat small frequency \n = 0 :01Tc. In each\npanel curves from top to bottom correspond to the Dynes parameter values \u0000 =Tc= 0:0001; 0:01; 0:1. The spin-orbit scattering\ntime\u001csoTcis (a)100, (b) 10, (c) 1, (d) 0 :1.\nGeneral case. Now let us consider the behaviour\nof spin susceptibility in the wide range of parameters by\nevaluating numerically the integral in Eq.15. First, we\ncompare the results given by the full Eq.15 with the con-\ntribution of only the \frst term. The sequence of plots\nin Fig.2 show temperature dependence of the dissipative\npart Im\u001fhat \n = 0:01Tc, Dynes parameter \u0000 = 0 :0001Tc\nand several values of the spin-orbit scattering rate. The\ndependencies given by the full Eq.15 are shown by the\nblue solid curves while the dependencies given only by\nthe \frst term in Eq.15 are shown by the red dashed\ncurves. One can see that for weak spin-orbit scatter-\ning\u001csoTc\u001d1 these curves coincide, according to the\nconclusion we have made based on the analysis of lim-\niting cases above. However, there is a large discrepancy\nfor stronger spin-orbit relaxation \u001csoTc<1. Note that\nthe behaviour of dashed curves is similar to that which\nhas been obtained for the dissipation signal in previous\nworks21. That is, at \u001csoTc<1 they signi\fcantly deviate\nfrom zero at T!0. As we have noted, the \fnite value of\nIm\u001fhat in the low-temperature limit is physically incor-rect. On the other hand, the solid curves always demon-\nstrate the correct behaviour going to zero in the limit\nT!0. Thus, the numerical analysis also con\frms that\nboth terms in the Eq.15 contribute to the dissipative part\nof the spin response in the superconducting state.\nNext, let us consider how the temperature dependen-\ncies of Im\u001fhat \n = 0:01Tcchange with the Dynes pa-\nrameter. The sequence of plots for the three values of\n\u0000=Tc= 0:001; 0:01; 1 is shown in Fig.3 for di\u000berent val-\nues of the spin-orbit relaxation rate. One feature demon-\nstrated by these curves is that the peak in the temper-\nature dependencies becomes less pronounced and disap-\npears for weak spin relaxation. At the same time there\nrelative hight of the peak almost does not change between\nstrong\u001csoTc= 1 (Fig.3c) and very strong \u001csoTc= 0:1\n(Fig.3d) spin relaxation. Besides that, one can see that\nthe height of the peak is strongly suppressed by increas-\ning Dynes parameter. For the realistic value in the super-\nconductor NbN \u0000 = 0 :1Tcthe relative hight of the peak\nis about 0:2\u00000:5 of the normal metal value at T > Tc.\nThis increase is by the order of magnitude weaker than6\n(a) \u0000 = 0:1Tc\n\u001csoTc\n(b) \u0000 = 0:01Tc\nFIG. 4. Temperature dependencies of the dissipative part\nof spin susceptibility at \n = 0 :01Tcand di\u000berent values of\nthe Dynes parameter (a) \u0000 = 0 :1Tc; (b) \u0000 = 0 :01Tc. Curves\nfrom top to bottom in each panel correspond to the spin-orbit\nscattering times \u001csoTc= 0:1; 1; 5; 10.\nthe relative peak heights of 2 \u00003 observed in spin pump-\ning experiment in GdN/NbN bilayers16. Therefore one\ncan assume that there should be a di\u000berent explanation\nof the this experiment rather than the peaked behaviour\nof spin susceptibility21.\nNow let us consider the behaviour of spin susceptibility\nat larger frequencies comparable with superconducting\nenergy scales \n\u0018Tc. In this case it is interesting to con-\nsider both the dissipative and the non-dissipative parts\nof spin susceptibility. As we show below they are respon-\nsible for the damping and \feld-like spin torque contri-\nbutions to the spin dynamics. In Fig.5 we plot the rele-\nvant quantities Im \u001fh(\n)=\n which contributes to the ex-\ncess Gilbert damping and Re \u001fh(\n)\u0000Re\u001fh(0) which con-\ntributes to the shift of the ferromagnetic resonance cen-\ntral frequency. First, we notice that the non-monotonic\ntemperature dependence of the dissipative part (left pan-\nels in Fig.) disappear at the frequencies much larger\nthan the Dynes parameter \n \u001d\u0000. For such frequencies\nIm\u001fhmonotonically decreases with temperature and \f-\nnally disappears at T!0 provided that \n <2\u0001. For\n\n>2\u0001 there a non-zero signal even at T= 0 due to the\nexcitation of quasiparticles across the gap.\nC. Spatial dispersion of the susceptibility\nIn general, due to the presence of anisotropic term in\nEq.18 the analytical solution is not possible for q6= 0.\nHowever, we can still get the analytical solution in the\nexperimentally relevant di\u000busive limit when the ordinary\nscattering rate is very large ( Tc\u001co)\u00001\u001d1. In this case\nEq. 10 can be simpli\fed by averaging over momentum di-\nrections. The isotropic part of the GF satis\fes Keldysh-\nUsadel equation\n\u0000if^\u001c3@t;\u0014ggt+Dr(\u0014g\u000er\u0014g) =i[^\u001c3^H;\u0014g]t+[\u0014\u0006so\u000e;\u0014g]t(27)\nwhereD=\u001cov2\nF=3 is the di\u000busion coe\u000ecient.\n(a) \u0000 = 0:1Tc\n(b) \u0000 = 0:001tc\n(b) \u0000 = 0:001Tc\nFIG. 5. Imaginary (left row) and real (right row) parts of the\nspin susceptibility as functions of Tand \n, normalized to the\nzero-temperature gap \u0001( T= 0). The Dynes parameters are\n(a) \u0000 = 0:1Tcand (b) \u0000 = 0 :001Tc. The spin-orbit scattering\ntime is\u001csoTc= 1.\nThe spin response to the spatially-inhomogeneous Zee-\nman \feldh\nei\nt+iqzcan be calculated analytically in\nthe di\u000busive limit using Usadel Eq.27. Using the imag-\ninary time representation and searching the solution in\nthe form ^gh(12)eiqzei(!1t1\u0000!2t2)we obtain the linearized\nUsadel equation\n(s1+Dq2)^g0(1)^gh\u0000s2^gh^g0(2) = (28)\ni(h\n^\u001b)[^g0(1)^\u001c3\u0000^\u001c3^g0(2)]\nThe solution of this equation yields susceptibility in the\nform (21) with the substitution of e\u000bective spin relax-\nation time 4 =3\u001cso!4=3\u001cso+Dq2\n\u001f(12) =\u00012+s1s2\u0000!1!2\ns1s2(s1+s2+Dq2+ 4=3\u001cso)(29)\nThis expression together with Eq.15 can be used to\nstudy various phenomena related to the spin dynamics\nin superconductors with spin-orbit relaxation. For exam-\nple, it is possible to study the e\u000bect of spin relaxation on\nthe nuclear magnetic resonance33,34and electron param-\nagnetic resonance35in superconductors. It is interesting\nthat the peak in spin relaxation observed in these exper-\niments is robust against even the very strong spin-orbit\nscattering as it follows from Fig.3d and Fig.4. In the limit\nof weak spin relaxation there is no peak, i.e. the temper-\nature dependence is monotonous as shown in Fig.2a and\n3a.7\nD. Keldysh formalism and kinetic equations\nIn the general case the procedure of analytical continu-\nation is not possible and one has to consider the real time\nequations from the very start. This brings extra compli-\ncation related to the matrix structure of the contour-\nordered propagator \u0014 g=\u0012\n^gR^gK\n0 ^gA\u0013\nhaving the spectral\nretarded (advanced) ^ gR(A)and the Keldysh component\n^gK. The matrix GF satis\fes Keldysh-Usadel equation\nwhich is formally identical to the Eq.10 or 27 with the\nsubstitution @t!\u0000i@t. Using the normalization con-\ndition ^gR\u000e^gK+ ^gK\u000e^gA= 0 one can introduce the\nparametrization of the Keldysh component in terms of\nthe distribution function ^ gK= ^gR\u000e^f\u0000^f\u000e^gA. Local\nspin density given by\nS(t) =\u0000\u0019\u0017\n4Tr[^\u001b^\u001c3^gK(t;t)] (30)\nThe driven state of superconductor is described by the\ndeviation of the Keldysh function from equilibrium which\nconsists of the parts with perturbations of spectral func-\ntions\u000e^gR;Aand the non-equilibrium part of distribution\nfunction\u000e^f. In the linear response regime one can write\n\u000e^gK(12) = (31)\n[^gR\n0(1)\u0000^gA\n0(2)]\u000e^f+\u000egR(12)n0(2)\u0000\u000egA(12)n0(1)\nComparing expressions (31,37) with 15 one can see that\nthe \frst term here yields the \frst term in the r.h.s. of\nEq. 15 and ^f/n(\"1)\u0000n(\"2).\nIn the low-frequency limit one can calculate the correc-\ntions to distribution function using the kinetic equation2\nwith the driving term obtained from the gradient expan-\nsion of the mixed product in the analytical continuation\nof Eq.27\n[^H;^f]t=i^\u001b@th@\"n0 (32)\nParametrizing the spin-dependent distribution function\nas^f=^\u001bfwe get the kinetic equation which for the\nspatially homogeneous system is given by\n@tf+ (2\u0000 +\u001c\u00001\ns)f=@\"n0@th (33)\n\u001c\u00001\ns= (1=3\u001cso)N\u00001Tr(1\u0000^gR^gA) (34)\nwhereN= ReTr[^\u001c3^gR]=2 is the normalized density of\nstates. At the subgap energies j\"j<\u0001 the spin relaxation\nrate (34) is not de\fned if the density of states is strictly\nzeroN= 0. However, for the \fnite Dynes parameter\nN/\u0000 so that\u001c\u00001\ns/\u0000. The solution of the Eq.33 yields\nthe contribution to the spin density\n\u001fkin+ 1 =\n2\u00021\n\u00001d\"N@\"n0\n\n\u0000i(2\u0000 +\u001c\u00001s); (35)\nwhich coincides with the \frst term in Eq.(15) in the low-\nfrequency limit.IV. SPIN PUMPING IN SUPERCONDUCTING\nFILMS\n(a)\n(b)\nFIG. 6. (Color online) Schematic setup with the interface\nbetween metallic spin sink (M) and ferromagnetic \flm (F) of\nthe widths dManddF, respectively. The constant external\nmagnetic \feld is H0x. The magnetization precession m\nei\nt\nis driven by the external magnetic \feld H\nei\nty. It generates\nspin currenti\npumped from F to M. (a) M has interface with\nvacuum; (b) M has interface with the perfect spin absorber.\nWith the general expression for spin susceptibility in\nhand we can study e\u000bects of spin pumping from the fer-\nromagnet into the adjacent metallic \flm. The schematic\nsetups are shown in Fig.6. The metallic spin sink M has\nan interface with (a) vacuum and (b) perfect spin ab-\nsorber. The correposnding boundary conditions are (a)\nvanishing spin current and (b) vanishing non-equilibrium\nspin polarization at z=dM. To quantify the spin pump-\ning e\u000bect we consider the interfacial exchange interaction\nbetween the localized spins in F and conduction elections\nin M. Within this model the local spin polarization close\nto the interface S(t) acts as e\u000bective \feld for the local-\nized magnetic moments. This process can be taken into\naccount by introducing the additional term i(t) into the\nLandau-Lifshitz-Gilber equation\n(1 +\u000bm\u0002)@tm+\rm\u0002Heff=i=SF0dF (36)\ni(t) =JsdS(t)\u0002m(t) (37)\nHereSF0is the equilibrium spin density in F, dFis the\nFI \flm thickness, Heffis the e\u000bective \feld and \u000bis the\nintrinsic Gilbert damping coe\u000ecient. The term i(t) can\nbe interpreted as the spin current between F and M.\nTo calculateS(t) we use the spin susceptibility (6) with\nthe Zeeman \feld determined by the interfacial exchange8\nh=Jsdm\u000e(z). In the linear regime the local spin polar-\nization near F interface can be written as follows\nS\n=\u0017heff\u001fmm\n (38)\n\u001fm(\n) =1X\nn=0\u001fh(qn;\n) (39)\nwhere we introduce the e\u000bective exchange \feld heff=\nJsd=dMand the local spin susceptibility \u001fMwhich deter-\nmines the response to the delta-functional Zeeman \feld.\nThe summation in Eq.39 runs over the discrete set of\nmomenta given by qn=n\u0019=dMfor the vacuum interface\nFig.(6a) which is determined by the zero boundary con-\ndition for the spin current at the interface with vacuum\nz=dM. For the strong spin sink interface Fig.6b we\nhaveqn= (n+ 1=2)\u0019=dMwhich is determined by the\nzero boundary of the non-equilibrium spin polarization\nwhich is suppressed by the strong spin sink at z=dM.\nDerivation of this result is given in Appendix D.\nTaking into account the Eq.29 one can see that the\nonly di\u000berence introduced by the spin absorber Fig.6b is\nthe modi\fcation of spin relaxation rate to \u001c\u00001\nso!\u001c\u00001\nso+\nD(pi=2dM)2. Therefore hereafter we will not distinguish\nthese two cases implying that the e\u000bective spin relaxation\nis used.\nThe Fourier components of the spin current (37) is\ngiven by\ni(\n) =\u0017h2\neffdM[\u001fm(\n)\u0000\u001fm(0)]m\u0002m\n (40)\nFor the con\fguration in Fig.6 the e\u000bective \feld is given\nbyHeff=H\nei\nty+B0xwhereB0=H0+ 4\u0019M. In\nthis case the eigen frequencies of LLG Eq.36 satisfy the\nequation\n\n =p\n(\rB0+\u000e!)(\rH0+\u000e!) (41)\n\u000e!=i\n\u000b+ [\u001fm(\n)\u0000\u001fm(0)]TcC (42)\nC=heff\nTc\u0017heff\nSF0dM\ndF(43)\nThe extra dissipation, that is the imaginary part of\n\n in (41) can be considered resulting from the e\u000bective\nGilbert damping constant increase\n\u000e\u000b=CTcIm\u001fm=\n (44)\nIn case if the \flm thickness is small dM< min (lso;\u0018)\nwherelso=pD\u001csois spin relaxation length and \u0018= p\nD=Tcis the zero-temperature coherence length , only\nthe contribution with n= 0 in the sum (39) is impor-\ntant. In this case the spin pumping e\u000bect is totally deter-\nmined by the homogeneous spin-orbit relaxation so that\n\u001fm(\n)\u0019\u001fh(\n;q= 0). For larger \flm thickness we need\nto take into account several terms in Eq.(39). Only for\nthe very large thickness dM\u001dmin(lso;\u0018) the expression\nused in previous works20,21is recovered in the form\n\u000e\u000b=\u0017J2\nsd\ndFSF0\u00021\n\u00001dq\n\u0019Im\u001fh(q;\n)\n\n: (45)Temperature dependencies of the normalized excess\nGilbert damping are shown in Fig.5. One can see that\nthese dependencies are qualitatively similar to that ob-\ntained in the absence of spin relaxation for in\fnite super-\nconducting \flms36. They are also qualitatively similar\nto the temperature dependencies of the NMR33,34and\nEPR37,38linewidths in superconductors. Note that for\nrelatively large Dynes parameter \u0000 = 0 :1Tcthe peak in\nthe temperature dependencies of Gilbert damping is al-\nmost absent (red curves in Fig.7) and superconductivity\nleads to the monotonous suppression of the spin pumping\ndissipative signal. This result reproduces theoretically\nthe behaviour observed in FMR experiments with Py/Nb\nbilayers11. Using large Dynes parameter \u0000 \u0018Tcone can\ndescribe qualitatively the e\u000bect of superconducting gap\nsuppression near the surface of metallic ferromagnet such\nas Fe or Ni. At the same time the Dynes parameter\n\u0000 = 0:1Tccorresponds to the superconductors with large\nelectron-phonon relaxation rate such as NbN. Therefore,\nprovided the mechanism of spin pumping between the FI\nand NbN superconductor is correctly described by the\nEq.45 or Eq. 44 the Gilbert damping behaviour should\ncorrespond to the red curves in Fig.7 with rather weak\npeaks. The amplitude of these peaks is much smaller\nthan has been observed in the experiment16. This dis-\ncrepancy shows the presence of some other yet unknown\nmechanism of spin pumping which can yield more pro-\nnounced peaks. The identi\fcation of such a mechanism\nis however beyond the scope of the present paper.\n(a)dM= 3\u0018\n (b)dM=\u0018=2\n\u0000=Tc\nFIG. 7. Temperature dependencies of the additional Gilbert\ndamping coe\u000ecient \u000e\u000bEq.44 at small frequency \n = 0 :01Tc.\nIn each panel curves corresponding to the Dynes parameter\nvalues \u0000=Tc= 0:001; 0:1 are shown. The spin-orbit scatter-\ning time\u001csoTc= 4 corresopnding to the normal state spin\nrelaxation length lso=\u0018=2. The metallic \flm thickness is (a)\ndM= 3\u0018, (b)dM= 0:5\u0018.\nQuite interestingly, spin relaxation and superconduct-\ning correlations lead to the pronounced frequency depen-\ndence of the real part of the susceptibility Re \u001fhas shown\nin Fig.5, right panels. This leads to the additional contri-\nbution to the spin pumping having the form of the \feld-\nlike spin torque, that is additional frequency-dependent\ne\u000bective \feld acting on the magnetization of the ferro-9\n(a)C= 0:01\n (b)C= 0:1\nFIG. 8. Normalized amplitude of the FMR response signal\nas a function of the constant external magnetic \feld H0and\ntemperature T. The magnetic \feld is measured in the units\nHp= \u0001(T= 0)=\r. The spin relaxation time is \u001csoTc= 1 and\nthe frequency is \n = Tc. We consider (a) weak C= 0:0:1 and\n(b) relatively large C= 0:1 values of the interfacial coupling\nparameter (43).\nmagnetm. This leads to the shift of the FMR central\nfrequency which can be obtained from Eq.41 as follows\n\u000e\n =CTcRe[\u001fm(\n)\u0000\u001fm(0)]\n2\n\r(B0+H0) (46)\nThis shift is negligible at small frequencies \n Tc\u001c1\nand \n\u001cso\u001c1 and small interfacial coupling between F\nand M \flms measured by the dimensionless parameter\n(43). However, it becomes signi\fcant for higher frequen-\ncies and larger C.\nTo quantify the superconductivity-induced FMR fre-\nquency shift we consider the system with not very strong\nspin relaxation \u001csoTc= 1. The normalized FMR response\nfunction which according to Eqs.(36,41) is proportional\nto [\n2\u0000(\rB0+\u000e!)(\rH0+\u000e!)]\u00001. In Fig.8 we nor-\nmalize this response function of its largest value at each\nfrequency, so that it is possible to see the transformation\nof the FMR line as a function of temperature.\nOne can see two pronounced e\u000bects which appear with\nincreasing the coupling parameter. First, comparing\nFig.8a and 8b at T >Tcone can see a signi\fcant growth\nof the normal state resonance linewidth. Given the fact\nthe in the experiment16with FMR in FI/S multilayers\nthe resonance is well-de\fned at \n \u00190:01Tcone can con-\nclude that the coupling parameter is C\u00180:01 corre-\nsponding to the Fig.8a. In this case there is no noticeable\nshift of the FMR resonance line as a function of temper-\nature.\nAs follows from its de\fnition (43) the coupling param-\neterC/(dFdM)\u00001can be increased by decreasing either\nthe thickness of the metal \flm dMor the ferromagnetic\n\flmdF. By doing so and reaching the value of C= 0:1\none would be able to see that the superconducting cor-\nrelations produce signi\fcant shifht oincrease of the tem-\nperature dependence of the resonant \feld H0.V. CONCLUSION\nWe have derived and analysed the general expression\nfor the time-dependent linear spin response in the super-\nconductor with spin-orbit relaxation. The homogeneous\nspin susceptibility is found for any amount of the ordi-\nnary disorder. In the spatially-inhomogeneous case the\ndi\u000busive limit is considered. We show that the e\u000bective\nspin relaxation rate is given by the sum of the spin-orbit\nscattering rate and the di\u000busive term. At low frequencies\n\n\u001cTcincreasing the e\u000bective spin relaxation leads to\nthe formation of the peak in the temperature dependence\nof the dissipative part of spin susceptibility. This peak is\nstrongly suppressed by increasing the Dynes parameter\nwhich models the smearing of the gap edge singularities\nin the superconductors due to the inhomogeneities or the\ninelastic phonon scattering.\nUsing this result and the model of interfacial exchange\ninteraction we examined the spin pumping from the fer-\nromagnet with magnetization precession into the adja-\ncent superconducting \flm. In the low-frequency regime,\ncorresponding to the recent experiments11{19we have\nanalysed the temperature dependence of the additional\nGilbert damping parameter induced by the spin pump-\ning. For realistic values of the Dynes parameter in such\nmaterials as NbN this temperature dependence is al-\nmost monotonic. This result indicates that there should\nexist some other mechanism for producing large peaks\nobserved recently in S/FI structures16. The regime of\nlarge Dynes parameters can be also considered to model\nthe spectral smearing which occurs due to the spatial\ninhomogenuity of the order parameter in systems with\nmetallic ferromagnets. The monotonic suppression of\nthe Gilbert damping parameter in this case corresponds\nto experimentally observed behaviour of FMR in Py/Nb\nsystems11. Similar behaviour is also reproduced by the\nscattering theory formalism39.\nFor larger frequencies, comparable with the supercon-\nducting gap and enhanced interfacial couplings we get\nsigni\fcant shifts of the FMR line. These shifts act to-\nwards increasing the resonant \feld H0at a given fre-\nquency. This behaviour is opposite to the one found in\nrecent experiments at low frequencies14,17{19.\nVI. ACKNOWLEDGEMENTS\nThis work was supported by the Academy of Finland\n(Project No. 297439) and Russian Science Foundation,\nGrant No. 19-19-00594.10\nAppendix A: Absence of spin response without spin\nrelaxation\nIn the absence of spin-orbit scattering \u001c\u00001\nso= 0 and\nq= 0 the susceptibility can be written as follows\n\u001fh(\n;q= 0) =\u0019TX\n!\u00012+s1s2\u0000!1!2\ns1s2(s1+s2)\nWe can use following relations s2\n1\u0000s2\n2=!2\n1\u0000!2\n2and\n2(!1!2\u0000\u00012\u0000s1s2) = (!1+!2)2\u0000(s1+s2)2so that\nX\n!(!1+!2)2\u0000(s1+s2)2\ns1s2(s1+s2)=\nX\n!\u0014(!1+!2)2\ns1s2(s1+s2)\u0000s1+s2\ns1s2\u0015\n=\nX\n!\u0014(!1+!2)\n(!1\u0000!2)\u0000\ns\u00001\n2\u0000s\u00001\n1\u0001\n\u0000s\u00001\n1\u0000s\u00001\n2\u0015\n=\n\u00001\n\nX\n!\u0002\n(!2\u0000!1)(s\u00001\n1+s\u00001\n2)\u0000(!1+!2)\u0000\ns\u00001\n2\u0000s\u00001\n1\u0001\u0003\n=\n2\n\nX\n!\u0002\n!1s\u00001\n1\u0000!2s\u00001\n2\u0003\n=2\n\nX\n![sgn(!1)\u0000sgn(!2)]\n(A1)\nThus after analytical continuation we can write\n\u001fh(\n) + 1 =\u00021\n\u00001d\"\n2\n[n0(\"+ \n)\u0000n0(\")] = 1\nso that\u001fh(\n) = 0 at \n6= 0.\nAppendix B: Normal metal limit\nIn the normal metal limit \u0001 = 0 and \u00181;2=j!1;2j.\nThen\n\u001fh+ 1 =\u0019TX\n!1\u0000sign(!1)sign(!2)\n(j!1j+j!2j+ 4=3\u001cso)(B1)\nAnalytical continuation is implemented as follows\n\u001fh+ 1 =\n\u00021\n\u00001d\"\n4i[n0(\"\u00001)\u0000n0(\")][1\u0000sign(!1)Rsign(!2)A]\n(j!1jR+j!2jA+ 4=3\u001cso)\nwhere we have used that j!1jR!s(\u0000i\"R\n1) =i(\"\u0000\n)+\u0000\nandj!2jA!s(\u0000i\"A\n2) =\u0000i\"+\u0000, so thatj!1jR+j!2jA!\n\u0000i(\"\u0000\n) +i\"+ 2\u0000 =i\n + 2\u0000\nThen we obtain\n\u001fh+ 1 =\u00021\n\u00001d\"\n2i[n0(\")\u0000n0(\"+ \n)]\n(i\n + 2\u0000 + 4=3\u001cso)=\n\n\n\u00002i(2=3\u001cso+ \u0000)(B2)\nFrom this we obtain Eq.22.Appendix C: Derivation of the strong spin\nrelaxation limit Eq.26\nSubstituting Eq.(21) obtained assuming the strong\nspin relaxation to the general analytical continuation rule\n(14) we obtain\n8\n3i\u001cso\u001fh= \u00012\u0002\nd\"\u0014F1(\"\u0000\n)\n\u0018A(\")+F1(\"+ \n)\n\u0018R(\")\u0015\n+ (C1)\n\u0002\nd\"\u0014F2(\"\u0000\n)\"\n\u0018A(\")+F2(\"+ \n)\"\n\u0018R(\")\u0015\n+\n\u0002\nd\"\u0014F2(\")(\"+ \n)\n\u0018A(\"+ \n)\u0000F2(\"\u0000\n)\"\n\u0018A(\")\u0015\nwhere\u0018R;A(\") =p\n(\"R;A)2\u0000\u00012,F1=n0(\")N(\")=\",\nF2=n0(\")N(\"), andN= Re(\"=\u0018R) is the DOS. The\ncontribution of last term can be calculated to be equal\n\u0000i\n using asymptotic F2(\"\u00061) =\u00061 and\"=\u0018A(\")!\u00001\nat large energies. The \frst two terms can be calculated\nusing expansions F(\"\u0006\n) =F(\")\u0006\n@\"Fwhich yields\n2\n3\u001csoIm\u001fh\n\n=\u00021\n\u00001d\"N\n\"(\u00012@\"F1+\"@\"F2)\u00001 (C2)\nIntegrating by parts this equation can be rewritten as\nEq.26 in the main text.\nAppendix D: Calculation of local spin susceptibility\nin the \flm of \fnite thickness\nTo take into account \fnite metallic \flm thickness we\nincorporate the interfacial exchange \feld as the boundary\nconditions to the non-stationary Usadel equations\nD\u0014g\u000e@z\u0014g(z= 0) =iJsd[\u001bm;^g]t (D1)\nMathematically it is more convenient to consider the\nequivalent problem incorporating the interfacial ex-\nchange \feld as the point source to the Usadel equation\n\u0000if^\u001c3@t;\u0014ggt+D@z(\u0014g\u000e@z\u0014g) = (D2)\ni[^\u001c3^\u001c2\u0001;\u0014g] + [\u0014\u0006so\u000e;\u0014g]t+iJsd\u000e(z)[ ^m;\u0014g]t\nThis equation is considered in the interval jzj< dM. In\ncase if atz=\u0006dMare the interfaces with vacuum the\ncurrent vanishes\n\u0014g\u000e@z\u0014g(z=\u0006dM) = 0 (D3)\nIn case if at z=dMare the interfaces with very strong\nspin sink the correction to GF vanishes\n\u0014gh(z=\u0006dM) = 0 (D4)\nWe assume that magnetization depends on time as\nm(t) =m\ne\u0000i\ntand search for the corrections to the\nGF in the form\n^g(t;t0) =TX\n![^g0(1)e\u0000i!1(t1\u0000t2)+ ^gh(12)e\u0000i(!1t1\u0000!2t2)]\n(D5)11\nwhere!2=!1\u0000\n and ^ghrepresents the correction to\nthe \frst order of the oscillating \feld m\n. To satisfy\nboundary conditions we search the solution in the form\n^gh(12) =1X\nn=0gqn(12) cos(qnz) (D6)\nwithqn=n\u0019=dMin case of the vacuum interface\n(D3) andqn= (n+ 1=2)\u0019=dMin case of the strong\nspin sink interface (D4) . Using the expansion \u000e(z) =\n(2dM)\u00001P\nncos(qnz) We have the equation for the cor-\nrection\n(~s1+Dq2)^g0(1)^gq(12)\u0000~s2^gq(12)^g0(2) = (D7)\ni(h\n^\u001b)[^g0(1)^\u001c3\u0000^\u001c3^g0(2)]\nwhereh\n= (G\"#\ni=2\u0017dM)m\n. Using the commutation\nrelation ^g0(1)^gk(12)+^gk(12)^g0(2) = 0 we get the solutionis given by\n^gq(12) =i(h\n^\u001b)^\u001c3\u0000^g0(1)^\u001c3^g0(2)\ns1+s2+ 4=3\u001cso+Dq2(D8)\nThe spin polarization at the M/F interface which can\nbe written in terms if the susceptibility\nS(z= 0) =\u0017heff\u001fm(\n)m\n (D9)\nSubstituting the solution (D8) to the expression for the\nspin polarization\nS(t;z) =\u0000i\u0019\u0017\n4Tr[^\u001b^\u001c3^g]jt1;2=t: (D10)\nwe get the imaginary frequency local susceptibility of the\n\fnite-thickness \flm (39) .\n1J. Linder and J. W. A. Robinson, Nat Phys 11, 307 (2015).\n2F. S. Bergeret, M. Silaev, P. Virtanen, and T. T. Heikkil a,\nRev. Mod. Phys. 90, 041001 (2018).\n3W. Han, S. Maekawa, and X.-C. 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Caianiello”, Università di Salerno, Via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy\nbCentro de Física de Materiales, Centro Mixto CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain\ncDonostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain\nARTICLE INFO\nKeywords :\nFerromagnetic Josephson junction\nAnomalous Josephson effect\nMagnetization reversal phenomenon\nLandau–Lifshitz–Gilbert (LLG) equa-\ntion\nResistivelyshuntedjunction(RSJ)model\nThermal noise effectsABSTRACT\nWediscusstheeffectsofthermalnoiseonthemagneticresponseofalateralferromagneticJosephson\njunctionwithspin-orbitcouplingandout-of-planemagnetization. Thedirectionofthemagneticmo-\nmentintheferromagneticlayercanbeinvertedbyusingcontrolledcurrentpulses. Thisphenomenon\nis due to the magnetoelectric effect that couples the flowing charge current and the magnetization of\nthe ferromagnet. We investigate the magnetization reversal effect versus intrinsic parameters of the\nferromagnet, such as the Gilbert damping and strength of the spin-orbit coupling. We estimate the\nmagnetization reversing time and find the optimal values of the parameters for fast switching. With\nthe aim of increasing the operation temperature we study the effects induced by thermal fluctuations\non the averaged stationary magnetization, and find the conditions that make the system more robust\nagainst noise.\n1. Introduction\nIn the past few years many efforts have been devoted to\nthe theoretical study of the magnetic response of ferromag-\nnetic anomalous Josephson junctions (JJs) [30, 52, 40, 48,\n50, 3, 51, 49, 39, 2, 41, 19, 44, 37], thus offering a path\nof concrete applications based on the electrical control of\nthe magnetization in a so-called '0–junction. A realization\nof such junction consists essentially of a superconductor-\nferromagnet-superconductor(SFS)Josephsonjunctionwith\nan intrinsic spin-orbit coupling (SOC). Its ground state cor-\nresponds to a finite phase shift, 0<'0<\u0019, in the current-\nphase-relation. Recently, such anomalous phase has been\nobserved experimentally in hybrid Josephson devices fabri-\ncated with a topological insulator Bi2Se3and Al/InAs het-\nerostructuresandnanowires[56,1,36,55]. Asdemonstrated\ntheoretically[9,28,52],themagnetizationoftheFlayercan\nbe electrically controlled. In fact, in a '0–junction, due to\nthe magnetoelectric effect, a charge current induces an in-\nplanemagneticmoment[12,13,35,5,29,7],whichinturn\nacts as a torque on the out-of plane magnetization of the F\nlayer,inducingeventuallyitsswitching. Alternatively,inthe\nplace of an electric current, the magnetization reversal can\nbe driven by a magnetic field in a SQUID setup [50], i.e.,\nadevicewidelyusedfordetectinganomalousJosephsonef-\nfects [56, 1, 36, 55, 38, 20].\nThe magnetization reversal phenomenon might eventu-\nallyfindanapplicationindifferentfieldsofsuperconducting\nspintronics[34,14,18]. Nevertheless,anyconcreteapplica-\ntionsbasedonthemagnetizationofa '0junctionhastodeal\nwiththeeffectsonthemagneticresponsestemmingfromthe\nunavoidable thermal fluctuations. In fact, the temperature\n<Corresponding author\ncguarcello@unisa.it (C. Guarcello); sebastian_bergeret@ehu.eus\n(F. Bergeret)\nORCID(s): 0000-0002-3683-2509 (C. Guarcello); 0000-0001-6007-4878\n(F. Bergeret)at which the system resides can significantly influence the\nevolutionofthemagneticmoment,eveninducingunwanted\ntransitions or hindering the inversion of the magnetization.\nTheimpactofstochasticthermalfluctuationsonthemag-\nnetization reversal phenomenon was addressed in Ref. [19].\nThatworkdescribesalsothefeasibility,initiallysuggestedin\nRef. [52], to employ a current-biased SFS Josephson junc-\ntion as a memory element, with the information encoded in\nthe magnetization direction of the F layer.\nInthepresentwork,wetakeacuefromRef.[19]andfo-\ncus in more detail on the noise effects on the magnetization\ndynamics, studying how the value of some system parame-\nterscaninfluencetherobustnessagainstthermalfluctuations\nofthecurrent-inducedmagnetizationreversalphenomenon.\nIn particular, we demonstrate that the values of the Gilbert\ndamping parameter and strength of the spin-orbit coupling\ncanbeconvenientlychosentomakethesystemmorestable,\nto increase the range of suitable working temperatures.\nThe work is organized as follows: In Sec. 2, we present\nthe theoretical model used to describe the time evolution of\nboth the magnetic moment and the Josephson phase of a\ncurrent-driven SFS junction. In Sec. 3 we discuss the mag-\nnetic configuration of the junction after applying a current\npulse. We look the stationary magnetization as a function\noftheGilbertdampingparameterandtheSOCstrength;ad-\nditionally, we show a simple way to predict the overall re-\nsponseofthestationarymagnetization. Moreover,weinves-\ntigate the full temporal evolution of the magnetization and\nwe demonstrate the possibility to minimize the magnetiza-\ntionswitchingtimebyadjustingspecificsystemparameters.\nInSec.3.1,weshowtheeffectsofstochasticthermalfluctu-\nations on the magnetization dynamics. Finally, in Sec. 4 we\npresent our conclusions.\n:Preprint submitted to Elsevier Page 1 of 10Figure 1: S/F/S Josephson junction driven by a rectangular\nbias current pulse, Ibias. The cartoon depicts the inversion of\nthez-component of the magnetization of the F layer after the\npassage of the current pulse Ibiasthrough the junction.\n2. The model\nThe setup that we consider consists of an SFS junction,\nsee Fig. 1, with a thin ferromagnetic film with an out-of-\nplane magnetic anisotropy and a Rashba-like SOC [19].\nDue to the interplay between the exchange field and the\nSOC, the current-phase relation of the SFS Josephson junc-\ntion readsI'=Icsin.'*'0/, whereIcis the critical cur-\nrentofthejunction, 'istheJosephsonphasedifference,and\n'0is the anomalous phase shift. The latter depends on sev-\neralparametersofthesystem,suchustheRashbacoefficient\n\u000b[45,10],thetransparencyofS/Finterfaces,thespinrelax-\nation, and the disorder degree of the system. For our pur-\nposes,theexactdependenceof '0ontheseparametersisnot\nas important as the geometry of the device. If we assume a\ntwo-dimensional SOC with momenta in the plane of the F\nfilm, and the charge current flows in x-direction, the phase\nshift'0is proportional to the y-component of the magnetic\nmoment according to [9, 28, 29, 5]\n'0=rMy\nM; (1)\nwhereM=t\nM2\nx+M2\ny+M2\nzisthemodulusofthemag-\nnetization vector, and the parameter rquantifies the SOC\nstrength and encloses the \u000b-dependence. Equation (1) es-\ntablishes a direct coupling between the magnetic moment\nand the supercurrent, which eventually makes the current-\ninduced inversion of the magnetization possible.\nThetimeevolutionofthemagnetizationcanbedescribed\nintermsoftheLandau–Lifshitz–Gilbert(LLG)equation[32,\n16]\ndM\nd\u001c=\r\nM\u0018\nMdM\nd\u001c\u0019\n*grMHeff; (2)\nwheregrdenotes the gyromagnetic ratio. The first term on\ntheright-handsideaccountsthedissipationthroughthephe-\nnomenologicaldimensionlessGilbertdampingparameter \r,\nwhilethesecondtermrepresentstheprecessionaround Heff,\nwhich components can be calculated as [33]\nHeff;i=*1\nV)F\n)Mi;withi=x;y;z: (3)HereVisthevolumeoftheFlayerand Fisthefreeenergy\nof the system, which reads as follow\nF=*EJ'Ibias+Es.';'0/+EM: (4)\nHere,EJ= \b0Ic_.2\u0019/(with\b0being the flux quantum),\nIbiasistheexternalcurrentinunitsof Ic,Es.';'0/=EJ[1*\ncos.'*'0/], andEM= *KV\n2\u0018Mz\nM\u00192\nis the magnetic en-\nergy that depends on the anisotropy constant K. In the fol-\nlowing, we indicate the ratio between the energy scales of\nthe system with the parameter \"=EJ_.KV/.\nFrom Eq. (3), we obtain the effective magnetic field\nHeff=K\nM\u0004\"rsin\u0000'*rmy\u0001 y+mz z\u0005; (5)\nwheremx;y;z=Mx;y;z_Marethenormalizedcomponentsof\nthemagnetizationthathavetosatisfythecondition m2\nx+m2\ny+\nm2\nz=1. The LLG equations can be conveniently expressed\ninsphericalcoordinates[47],sothat mx;y;zcanbewrittenin\nterms of the polar and azimuthal angles \u0012and\u001eas\nmx.\u001c/ = sin\u0012.\u001c/cos\u001e.\u001c/\nmy.\u001c/ = sin\u0012.\u001c/sin\u001e.\u001c/ (6)\nmz.\u001c/ = cos\u0012.\u001c/:\nWe can also define the \u0012and\u001ecomponents of the normal-\nized effective field as\nHeff;\u0012=\"rsin.'*rmy/cos\u0012sin\u001e*mzsin\u0012(7)\nHeff;\u001e=\"rsin.'*rmy/cos\u001e: (8)\nThus,bynormalizingthetimetotheinverseoftheferromag-\nnetic resonance frequency !F=grK_M, that ist=!F\u001c,\ntheLLGequationsinsphericalcoordinatesreducetothefol-\nlowing two coupled equations [47]\nd\u0012\ndt=1\n1+\r2\u0018\nHeff;\u001e+\rHeff;\u0012\u0019\n(9)\nsin\u0012d\u001e\ndt=1\n1+\r2\u0018\n\rHeff;\u001e*Heff;\u0012\u0019\n:(10)\nThedynamicsofanoverdampedSFSJosephsonjunction\ncan be described in terms of the resistively shunted junc-\ntion (RSJ) model [4, 26, 54, 27] generalized to include the\nanomalous phase shift '0=rmy[19]. Including also ther-\nmal fluctuations accounted by the Gaussianly distributed,\ndelta correlated stochastic term, Ith.t/, the dynamics of the\nJosephsonphase '.t/canbedescribedbytheequation[19]:\nd'\ndt=!\u0004Ibias.t/*sin\u0000'*rmy\u0001+Ith.t/\u0005+rdmy\ndt:(11)\nHere, the time is still normalized to the inverse of the ferro-\nmagnetic resonance frequency and !=!J_!F, with!J=\n2\u0019IcR_\b0beingthecharacteristicfrequency[4]ofthejunc-\ntion with a normal-state resistance R.\n:Preprint submitted to Elsevier Page 2 of 10r=0.1\n0.0 0.1 0.2 0.3 0.4 0.50.0450.0500.0550.060\nγΔUγ=0.10.0 0.1 0.2 0.3 0.4 0.5rFigure 2: Average washboard potential barrier, \u0001U.\r;r/, cal-\nculated during the current pulse with intensity Imax=0:9and\nduration\u0001t= 10, versus\rat a fixedr= 0:1(blue-dashed\ncurve) and versus rat a fixed\r= 0:1(red-solid curve). The\nhorizontal gray dashed line indicates the washboard potential\nbarrier\u0001U.ib= 0:9/. The other parameters are: \"= 10and\n!=1.\nThe noise term Ith.t/is a sort of “thermal current” with\nthe usual white-noise statistical properties that, in normal-\nized units, can be expressed as [4, 23, 21]\nêIth.t/ë= 0 (12)\u0010Ith.t/Ith.t¨/\u0011= 2DI\u000e\u0000t*t¨\u0001; (13)\nwhere we introduced the dimensionless amplitude of ther-\nmal current fluctuations defined as\nDI=kBT\nR!F\nI2\nc=1\n!kBT\nEJ: (14)\nIn this work we assume only thermal fluctuations directly\naffectingthephasedynamics. Instead,thecaseofastochas-\ntic “thermal field” Hth, with an intensity DH, included also\nin Eq. (2) [8, 11, 47, 42] was discussed in Ref. [19]. Since\nthe normalized intensities of the two noise sources are pro-\nportional, i.e.,DH= .\r\"!/DI[19], one could, in princi-\nple, optimize the system parameters in such a way to make\ntheimpactofthethermalfieldnegligiblewithrespecttothe\nthermal current.\nInourwork,weassumeanSFSjunctionbiasedbyarect-\nangular current pulse, Ibias, centered at tc:\nIbias.t/=<\nImax; tc*\u0001t_2ftftc+\u0001t_2\n0; elsewhere:(15)\nHere,\u0001tis the width and Imaxis the intensity, in units of\nIc,ofthepulse,sothatthecondition Imax<1meansabias\ncurrent lower than the critical value, Ic.\nOften,itisusefultodepicttheresponseofaJJintermsof\ntheevolutionofaparticle,representingthesuperconducting\nphase difference 'across the JJ, in a cosine “washboard”\npotentialU.ib/ =EJ\u00041*cos.'/*ib'\u0005, with the current\nibflowing through the junction being the slope of this po-\ntential. The resulting activation energy barrier \u0001U.ib/ =\nFigure 3: (a) Stationary value, mst\nz, [computed by numeri-\ncal solution of Eqs. (9)-(11)] and (b) expected behavior, mth\nz,\n[calculated through Eq. (17)] of thez-component of the mag-\nnetization as a function of rand\r, in the absence of noise\nfluctuations, DI=0. The other parameters are: \"=10,!=1,\nImax=0:9,\u0001t=10, andmz.t=0/=+1 . The colored circles\nin panel (a) indicate the regions around the .\r;r/combinations\nused to obtain the curves in Fig. 4. The legend in panel (b)\nrefers to both panels.\n24t\n1*i2\nb*ibarcsin.ib/5\nconfinesthephase 'inapoten-\ntial minimum.\nForcompleteness,weobservethatthemagnetizationterm\nincluded in the modified RSJ model affects also the height\nof the potential barrier. In fact, the total current contribu-\ntion in Eq. (11) can be assumed to be the sum of the cur-\nrent pulseIbias.t/and the additional term stemming from\nthe time derivative of the ycomponent of the magnetiza-\ntion,Imy.t/ =r\n!dmy\ndt. This means that during the current\npulse, the phase particle “sees” a time-dependent potential\nbarrier that depends on \randr. In Fig. 2 we show the av-\neragepotentialbarrier, \u0001U.\r;r/,calculatedduringacurrent\npulse with intensity Imax= 0:9and duration \u0001t= 10, as a\nfunction of\randr. It is evident that Imy.t/tends to reduce\ntheaveragebarrierheightwithrespecttothevalue \u0001U.Imax/\ncalculated neglecting the additional magnetic contribution,\nsee the gray dashed line. This means that, by increasing \r\nandr, the phase particle would experience an effective po-\ntentialbarrierslightlyreducedfortheenhancingofthetotal\nbias current due to the magnetic contribution. This mecha-\nnismshouldbetakenintoaccountwhencomparingthenoise\nintensity at which the system becomes unstable with the ef-\nfectivepotentialbarrier. Infact,thismodulationofthewash-\nboardpotentialaccountstheresponsetonoisyfluctuationsat\nlow values of \randr, see Sec. 3.1.\nIn the following, we investigate how thermal noise af-\nfects the magnetization reversal, setting specific combina-\ntions of the system parameters. Specifically, we explore the\nresponse of the magnetization, with and without taking into\naccount noise effects, by varying \randrin suitable ranges.\nThe energy and timescales ratios are fixed to the values \"=\n10and!=1, respectively.\n:Preprint submitted to Elsevier Page 3 of 10Figure 4: Time evolution of Josephson phase and current, top panels, and magnetization\ncomponents, bottom panels, at different .\r;r/values, in the absence of noise, DI=0. The\nyellow shaded regions indicate the time window in which the current pulse, with Imax=0:9\nand\u0001t=10, is switched on. Legends in panel (a) refer to all panels.\n3. Results\nInthissection,wefirststudythemagneticresponseinthe\nabsenceofnoise, i.e.,weimposeDI=0. Weassumethatthe\nmagnetizationinitiallypointstowardsthe z-direction,thatis\nM=.0;0;1/att=0. With this initial condition, we solve\nEqs. (9)-(11) self-consistently, choosing different values of\nthe system parameters. From the solution we determine the\nmagnetization direction after the current pulse, and in par-\nticular we focus on the stationary magnetization mst\nz, that is\nthe value of mzlong after the current pulse is turned off (at\nthe timetmax=100).\nInthefollowingweanalyzealsothenormalizedmagne-\ntizationswitchingtime, \u001cSW,whichisdefinedasthetime(in\nunits of!*1\nF) thatmztakes to switch from the initial value\n+1to*1, after the current is turned off. This characteristic\ntimecloselydependsonthevaluesof rand\r;inparticularit\nispossibletofindspecificcombinationsoftheseparameters\nmaking\u001cSWshorter. Thismeansthepossibilitytominimize\nthe magnetization switching time by adjusting specific sys-\ntemparameters,suchas rand\r,bychoosingmaterialswith\nthe proper characteristics, for instance with a larger SOC to\nmakerlarger. Thisaspectbecomescrucialdesigningappli-\ncations based on the magnetization reversal phenomenon in\na'0–junction,inwhichahighresponsespeedcanplayakey\nrole,e.g., superconducting memory applications [22, 19].\nIn Fig. 3(a) we show the overall behavior of mst\nzas a\nfunction of both the Gilbert damping parameter, \r, and the\nSOC strength, r, in response to a current pulse with inten-\nsityImax= 0:9and width\u0001t= 10. This contour plot is\nformedbyadark-bandspattern,namely,weobserveregions\nofthe.\r;r/parametricspaceinwhichthemagnetizationre-\nversal systematically occurs, i.e., in whichmst\nz= *1, andother regions in which this effect is systematically lacking,\ni.e.,mst\nz=+1. In other words, when increasing rat a fixed\n\r, we observe a regular sequence of mst\nz=+1andmst\nz=*1\nvalues. Similar mst\nzpatterns were observed and discussed\nalso in Refs. [2, 19]. Here we go a step forward, and clarify\nhow the magnetic response depends on the choice of .\r;r/\ncombinations by analyzing different regions of the density\nplot in Fig. 3(a). Essentially, we proceed as follows: i) first,\nwechoose.\r;r/layingindifferentregionsofthedensityplot\n(i.e., we fix\rand changer) andii) then, we focus on .\r;r/\npoints situated on the same region ( i.e., we fixrand change\n\r). For more clarity, in Fig. 3(a) we also highlight with col-\nored circles the regions around the specific .\r;r/combina-\ntions chosen to fully characterize the temporal evolution of\nthe magnetization and to explore noise effects.\nThefulltimeevolutionoftheobservablesofinterest, i.e.,\nthe Josephson phase, the supercurrent, and the magnetiza-\ntion components, in response to a current pulse with ampli-\ntudeImax=0:9and width\u0001t=10is shown in Fig. 4. We\nconvenientlyexaminefirstthebehaviorofthephaseandthe\nJosephson current. In all panels of Fig. 4, we observe that\nduring the current pulse, i.e., within the yellow shaded re-\ngion,theJosephsonphasefirstincreases,andthenitgoesto\nzeroasthepulseisturnedoff. Infact,inthewashboard-like\npicture [4], the tilting imposed by the current pulse, even\nconsidering the Imy.t/contribution, is not enough to allow\nthe “particle” to overcome the nearest potential barrier and\nto switch the system to the finite voltage “running” state.\nThis means that the phase-particle remains confined within\na potential minimum, so that when the current is turned off,\nthe tilting of the washboard potential goes back to zero and\nthe phase restores its initial position, i.e.,'0. Also\nthe Josephson current follows a similar evolution, which is\n:Preprint submitted to Elsevier Page 4 of 10characterizedbyarapidexponentialincrease(decrease),ap-\nproaching the value Imax(0), when the pulse is switched on\n(off).\nTheanalysisofthetemporalevolutionofallthecompo-\nnents of magnetization can give us a deeper understanding\nofhow,andwhy,thephenomenonofreversalofmagnetiza-\ntionoccurs. Thus,inthetoppanels(a)-(d)ofFig.4,weshow\nhowthemagnetizationswitchingdevelopsbychoosing .\r;r/\ncombinations lying on different dark bands of the contour\nplotinFig.3(a). ThismeanstokeepfixedtheGilbertdamp-\ningparameter(inparticular,weset \r=0:1)andtoadjustthe\nrvalue,e.g., we impose r=^0:15;0:23;0:3;0:38`. During\nthe current pulse, i.e., within the yellow shaded regions, mx\nandmzundergoexponentiallydampedoscillationsarounda\nzero value. Instead, myfollows an exponential growth rip-\npled by small, damped oscillations. In other words, the cur-\nrent pulse temporarily causes the precession of the magne-\ntization around the y-direction. Subsequently, as the pulse\nisswitchedoff,thedynamicsofthemagnetizationmoments\nchange,indeedboth mxandmyundergodampedoscillations\naround a zero value, while the z-component, after a tran-\nsient regime, flips definitively to the value mz= *1. We\nalso observe that the oscillating behavior of myis reflected\nintheevolutionoftheJosephsonphase ',whichshowsmin-\nima/maxima in the same position as my.\nFrom the top panels of Fig. 4, it is evident that, as rin-\ncreases, also the oscillation frequency of both mxandmz\nincreases. Furthermore, at a larger r, the curve of myap-\nproaches earlier the value 1and its oscillations are more\ndampened. Thus, we can speculate that the dark bands in\nFig. 3(a) “differ” essentially in the number of oscillations\nmade bymzbefore reversing completely.\nNow, in order to understand how the Gilbert damping\nparameter affects the magnetization response, we consider\n.\r;r/combinations lying on the same band of the contour\nplot in Fig. 3(a), see panels of (e)-(h) of Fig. 4. To this aim,\nwe imposerí0:1and\r=^0:1;0:2;0:3;0:4`. An increase\nof\rtends to dampen more the oscillations of both mxand\nmy,especiallywhenthecurrentisturnedoff. Moreover,the\nvalue of\raffects also the time the magnetization needs to\nfully reverse. In fact, from Fig. 4 one can also estimate the\nnormalized magnetization switching time, \u001cSW.\nInFig.5(a)and(b)weillustratethebehaviorof \u001cSWasa\nfunctionofratafixed\r=0:1andasafunctionof \ratafixed\nr=0:1, respectively. The gray shaded fringes in this figure\nhighlighttherangesof rand\rvalueswithinwhichthemag-\nnetizationreversaltakesplace. Inotherwords,eachgrayre-\ngioncorrespondstoadarkbandinFig.3(a). InFig.5(a)we\ncannotethat,withineachfringe, \u001cSWbehavesnon–monotonically,\ninasmuch it grows significantly at the edges of the gray re-\ngionandreachesalocalminimumroughlyinitsmiddlepoint.\nMoreover, the value of this \u001cSWminimum tends to increase\npassing from a fringe to the next one at higher rvalues. A\nnon-monotonic trend emerges also looking the behavior of\n\u001cSWas a function of \r, see Fig. 5(b). In particular, the \u001cSW\nversus\rcurveisasymmetricandreachestheminimumvalue\n\u001cmin\nSWô11at\rô0:3. Thus,byassumingatypicalferromag-\n0.1 0.2 0.3 0.4 0.5203040506070\nrτsw\nγ=0.1\n0.1 0.2 0.3 0.4 0.510152025303540\nγτsw\nr=0.1(a)\n(b)Figure 5: Magnetization switching time \u001cSW, as a function of\nrat a fixed\r= 0:1, (a), and as a function of \rat a fixed\nr=0:1, (b). In both panels, the gray shaded bands represent\nthe range of rand\rvalues within which the magnetization\nreversal phenomenon takes place.\nnetresonancefrequencyequalto !Fô10GHz[51],weob-\ntainaminimumswitchingtimeoftheorderofnanoseconds.\nFinally, Fig. 4 suggests also that the stationary value of\nthe magnetization depends on the state of mzwhen the cur-\nrentpulseisswitchedoff. Infact,themagnetizationreversal\nseems to occur when mzis in the first half of its oscillation\nperiod just when current is turned off. This simple obser-\nvation allows us to roughly predict the overall response of\nthe stationary magnetization without solving Eqs. (9)-(11)\nnumerically. In fact, according to the dynamics illustrated\nin Fig. 4, one could in principle assume for the x– andz–\ncomponentsofthemagnetizationanoscillating,dampedtime\nevolution,suchas mx.t/ôe*t\n\u001cdsin.\nt/andmz.t/ôe*t\n\u001cdcos.\nt/\n(where\nistheoscillationfrequencyand \u001cdistheexponential-\ndecay constant). Inserting these tentative solutions in the\nequation for  mythat can be derived from Eq. (2), we obtain\nthe following equation for the frequency, \n.\r;r/\n\r\n1+\r2sin.2\nt/\n2+\n=Imax\"r\n1+\r2: (16)\nToobtainthisequationweassumethat \u001cd¸1andsin\u0000'*rmy\u0001í\nImax. Thesetwoassumptionscorrespond,essentially,tosup-\nposeasmall randapulsedurationlongerthantheexponen-\ntialriseoftheJosephsoncurrent,seeFig.4. Finally,bycom-\nparingtheoscillationperiod, T\n.\r;r/=2\u0019_\n.\r;r/,andthe\npulse duration \u0001t, we can give an estimate of the stationary\n:Preprint submitted to Elsevier Page 5 of 1010-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0mxstr=0.15-γ=0.1\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0myst\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\nDImzst10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0r=0.23-γ=0.1\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\nDI10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0r=0.3-γ=0.1\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\nDI10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0r=0.38-γ=0.1\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\nDI(a) (b) (c) (d)Figure 6: Averagestationarymagnetizations, mst\nx,mst\ny, andmst\nz, asafunctionofthethermal\nnoise intensity, DI, at the.\r;r/-values used to obtain the top panels of Fig. 4, calculated\nby averaging over Nexp=103independent numerical repetitions. The vertical dashed lines\nindicate the average washboard potential barrier \u0001U.\r;r/.\nmagnetization as following\nmth\nz.\r;r/=h\nn\nl\nnj*1if Mod\u001c\n\u0001t;T\n.\r;r/\u001d\nfT\n.\r;r/_2\n+1if Mod\u001c\n\u0001t;T\n.\r;r/\u001d\n>T\n.\r;r/_2;\n(17)\nwhere Mod[a;b]gives the remainder on division of abyb.\nThe behavior of mth\nzas a function of \randris shown in\nFig. 3(b), at Imax= 0:9,\u0001t= 10, and\"= 10. This con-\ntour plot recalls closely that obtained solving Eqs. (9)-(11)\nnumerically and shown in Fig. 3(a), especially at high .\r;r/\nvalues. Thismeansthatthesimpleassumptionsmadetoob-\ntain Fig. 3(b) allow to grasp, in our case, the essential fea-\ntures behind the magnetization reversal phenomenon. As a\nclosingremark,weobservethatananalyticalsolutionforthe\nmagnetization dynamics induced by a current pulse was re-\ncently proposed in Ref. [37]. In this work, Mazanik et al.\nformulate criteria for magnetization reversal in a '0junc-\ntions,obtainingagoodagreementbetweennumericalresults\nandanalyticalprediction,inspecificrangesofthesystempa-\nrameters.\n3.1. Noise effects\nInthissectionwediscusstheeffectsproducedbyanon-\nnegligible thermal noise source on the magnetization dy-\nnamics. The temperature can influence significantly the re-\nsponseofthesystem,sincethermalfluctuationsmayeventu-\nallyinduceanunwantedmagnetizationswitchorpreventthe\nmagnetizationreverse. Weinvestigatetheeffectsofstochas-\nticthermalfluctuationsinthephasedynamics,byincluding\na Gaussian noise source in the RSJ model, see Eq. (11).\nIn the following, we focus on the average components\nof the stationary magnetization, mst\nx,mst\ny, andmst\nz, which arecomputed by averaging over Nexp= 103independent nu-\nmerical repetitions in the presence of a non-negligible ther-\nmal noise intensity, DI0.\nFigure 6 shows the behavior of the average magnetiza-\ntion as a function of the noise intensity DI, obtained by fix-\ning\r=0:1and varying r; in particular we choose the .\r;r/\ncombinations used to obtain the curves in the top panels of\nFig. 4. In this way we explore the noise effects by focusing\nonthedifferentgraybandsinFigs.3and5atagiven \r. We\nindicate with D0\nIthe noise amplitude at which mst\nzstarts to\ndeviatesignificantlyfromthevalue *1andwithDIthenoise\namplitudeatwhichtheswitchingprocessisfullysuppressed,\nthat is when mst\nzapproaches the zero value. In all panels of\nFig.6wealsomarkwithadashedverticallinethe DIvalue\ncoinciding with the average barrier height, \u0001U.\r;r/.\nr=0.133\nr=0.135\nr=0.14\nr=0.15\nr=0.16\nr=0.166r=0.169\n10-40.001 0.010 0.100-1.0-0.8-0.6-0.4-0.20.0\nDImzstγ=0.1\nFigure 7: Average stationary magnetization, mst\nz, as a function\nof the thermal noise intensity, DI, at\r=0:1andrË.0:133*\n0:17/, calculated by averaging over Nexp= 103independent\nnumerical repetitions. Lines in the figure are guides for the\neye.\n:Preprint submitted to Elsevier Page 6 of 1010-1.0-0.50.00.51.0mxstr=0.09-γ=0.1\n10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0myst\n10-4 0.001 0.01   0.1  1 10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\nDImzst10-1.0-0.50.00.51.0r=0.1-γ=0.2\n10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\n10-4 0.001 0.01   0.1  1 10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\nDI10-1.0-0.50.00.51.0r=0.11-γ=0.3\n10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\n10-4 0.001 0.01   0.1  1 10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\nDI10-1.0-0.50.00.51.0\n10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\n10-4 0.001 0.01   0.1  1 10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\nDI(a) (b) (c) (d)r=0.12-γ=0.4Figure 8: Average stationary magnetizations, mst\nx,mst\ny, andmst\nz, as a function of the\nthermal noise intensity, DI, at the.\r;r/-values used to obtain the bottom panels of Fig. 4,\ncalculated by averaging over Nexp=103independent numerical repetitions. The vertical\ndashed lines indicate the average washboard potential barrier \u0001U.\r;r/.\nFrom Fig. 6(a), which is obtained for r=0:15and\r=\n0:1, one sees that mst\nzô *1only forDI¿ D0\nI= 0:01.\nFor higher noise intensities both mst\nzand the error bars in-\ncrease, approaching a zero value of mst\nzexactly forDIô\n\u0001U.\r;r/=0:051. Byincreasingfurtherthenoiseintensity,\ni.e., forDIÀDI,mst\nzstill remains close to zero, showing\nhowever quite large error bars that indicate a highly fluctu-\natingresponsetotallydrivenbynoise. Thevaluesof mst\nxand\nmst\nyhover around zero at each noise intensity, despite their\nerror bars tend to enlarge when DIÀDI.\nIncreasingr, both the value of D0\nIandDIreduces sig-\nnificantly, see Fig. 6(b-d), so that the greater r, the moreDI\ndeviatesfrom \u0001U. Thismeansthat,increasing r,thesystem\nis more sensitive to noise and the maximum temperature at\nwhichitcanreside,withoutsignificanteffectsonthestation-\nary magnetization, reduces. In other words, the robustness\nagainstthermalfluctuationsofthemagnetizationreversalef-\nfect is damaged by a high rvalue.\nToobtainthecurvesinFig.6,wefixed \rwhilerchanges\ninsuchawaytoexplorethesystemresponseinthedifferent\ngraybandsofFig.5(a),morespecifically,wechosethe rval-\nueslyingexactlyinthemidpointofeachband. Ontheother\nhand, we can demonstrate that, if we restrict now to just a\nsingle gray band, the impact of thermal noise can change\nsignificantly even for a small variation of r. For instance,\nin Fig. 7 we show how mst\nzversusDImodifies by setting\nrË.0:13;0:17/,with\r=0:1,thatiswefocusonthedarkest\nfringeinFig.5(a). Weobservethatimposing r=0:15noise\neffects are kept at a minimum, while thermal fluctuations\naffect more the magnetization switching if ris slightly in-\ncreased,ordecreased. Inparticular,thevalueof DIdependslittleonr,unlikethevalueof D0\nIwhichchangessignificantly\nby changing r. Specifically, for r=^0:15,0:14, and0:135`\nwe obtain the values D0\nIô^10,3, and0:5`10*3, respec-\ntively. If we suppose a temperature–dependent critical cur-\nrent, withIc= 10\u0016A at low temperatures1, these noise\nintensitiesD0\nIcorrespond to the normalized temperatures\nT_Tcô^0:85,0:58,and0:12`,respectively. Insummary,at\nafixed\r,theoptimalvalueof rcorrespondstothemidpoint\nofagrayband,whereasjustasmallchangeof risenoughto\nundermine the stability of the system.\nFinally, we discuss how an increase of \rcan influence\nthe magnetization reversal. In Fig. 8 we present the average\nstationary magnetizations mst\nx,mst\ny, andmst\nzversusDI, im-\nposing the.\r;r/combinations used to obtain the curves in\nthe bottom panels of Fig. 4. For \r= 0:1,mst\nzapproaches a\nzero value only for DIô\u0001U.\r;r/ = 0:054, see Fig. 8(a) .\nForhighernoiseintensities, mst\nzremainsclosetozero,show-\ning quite large error bars. The same happens for the xand\ny*components of the magnetization. The \rcoefficient acts\nasafrictiononthemagnetizationdynamics,sothatalarger\n\rmeans a system more “stiff” and, therefore, less sensitive\nto noisy disturbances. This is why the noise intensity DI,\nat whichmst\nzapproaches a zero value, increases with \r, see\nFigs.6(b)-(d). Inparticular,athigh \rvalues,weobservethat\nDIis always well above \u0001U.\r;r/. This means that, in view\nof a possible application based on a '0–junction, a larger \r\ncould allow a higher working temperature, still preserving\nthe magnetization reversal phenomenon. In principle, one\n1In the case of weak proximity effect and large exchange field in F,\nthe critical current temperature-dependence is proportional to Ic.T/ ×\n\u0001.T/tanh\u0004\u0001.T/_.kBT/\u0005[6], where\u0001.T/is the superconducting gap.\nThus, to find the temperature corresponding to a given noise intensity, we\nuse this relation, with a zero-temperature value Ic.0/=10\u0016A.\n:Preprint submitted to Elsevier Page 7 of 10couldthinktooptimizethesystemparametersinsuchaway\ntomake,forinstance,theswitchingtimeminimal,seeFig.5,\nstill keeping the device at a suitable working temperature.\n4. Conclusions\nInconclusion,inthispaperwediscussthebistablemag-\nnetic response of a current-biased '0–junction, that is a su-\nperconductor–ferromagnet–superconductorJosephsonjunc-\ntionwithaRashba-likespin-orbitcoupling. Thedirectionof\nthemagnetizationoftheferromagneticlayercanbeinverted\nvia controlled current pulses. We study the temporal evolu-\ntion of all the components of the magnetization in different\nconditions. We determine the values of intrinsic system pa-\nrameters, such as the Gilbert damping and strength of the\nspin-orbitcoupling,correspondingtoaminimumswitching\ntime of the magnetization. We also suggest a way to grasp\nreadilytheessentialfeaturesbehindthemagnetizationrever-\nsalphenomenonthroughsimpleassumptions,withoutfacing\nthenumericalsolutionsofthedifferentialequationsforboth\nthe magnetization and the Josephson dynamics.\nWe also explore how the magnetization switching time\nstrongly depends on the .\r;r/values. We observe a switch-\ningtimesofordernanoseconds,whichcanbeacceptablefor\nqubits[46],butcouldbetooslowforconventionallowtem-\nperature electronics. Also the specific shape of the driving\npulse and the orientation of the switching field can signifi-\ncantly affect the magnetization switching time [43]. How-\never, in this work, we aim to demonstrate that the switch-\ning time can strongly depend on the value of the parame-\nter.\r;r/. From the other side, we assumed fixed energy\nand timescales ratios. Usually, the energy ratio \"ranges\nfrom\"í 100[28], if the magnetic anisotropy is weak, to\n\"í 1[51], in the case of a stronger anisotropy. In our\ncalculation we choose an intermediate value, \"= 10. The\ntypical ferromagnet resonance frequency is !Fí 10GHz,\nwhilethecharacteristicJosephsonfrequencyisusuallyinthe\nrange!JË [10*100] GHz. In our work, we conserva-\ntively choose !=!J_!F= 1, but we could reasonably\nimposealsoahighervalueforthefrequencyratio !. Inthis\ncase, one can expect a strong suppression of the magneti-\nzation switching time (e.g., see Ref. [37]. Please note that\nin this paper the frequency ratio is defined as w=!F_!J,\nso thatw=!*1). Since!J= 2\u0019_\b0IcR, the magne-\ntization switching time could be reduced by adjusting the\nvalues of the junction parameters IcandR. Moreover, in\nthisworkweestimateamagnetizationswitchingtimeofthe\norderofnanosecondsbyassumingaferromagnetresonance\nfrequency equal to !Fí10GHz; since!F×K_M(with\nKandMbeing the anisotropy constant and the modulus of\nthe magnetization vector, respectively) one could envisage\ntoreducefurtherthisratiotoobtainashortermagnetization\nswitching time.\nFinally,weexploretherobustnessofthecurrent-induced\nmagnetizationreversalagainstthermalfluctuations,inorder\nto find the regime of system parameters in which the mag-\nnetization switching induced by a current pulse is more sta-ble. In particular, we demonstrate that the choice of a low r\nand/orahigh \rvaluecanbeconvenienttokeepthermalfluc-\ntuationsatbay,inordertoincreasethetemperatureatwhich\nthe system can reside still preserving the magnetization re-\nversal effect.\nTheinvestigationofthermaleffectsonhybridsupercon-\nductive/ferromagneticstructuresisimportantinapplications\nofnovelelectronicdevices,suchasspintronics[34,17],qubits\nandsuperconductinglogicelements[15],anddetectorswith\nelectron cooling [31]. Moreover, the role of noise can be-\ncomecrucialinhigh-speedswitchingelectronics,wheresta-\nbilizationeffectsduetonoisecanleadtoenhancementofthe\nswitchingtime,theso-callednoise-delayedswitchingeffect.\nIndeed, noise-enhanced stabilization effects can play a rele-\nvant role by reducing the size of the magnetic element [53]\nand are also demonstrated to be important in the switching\ndynamics of Josephson devices [57, 24, 25].\n5. 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URL: https:\n//doi.org/10.1038/nphys3742 , doi: 10.1038/nphys3742 .\n[57] Valenti, D., Guarcello, C., Spagnolo, B., 2014. Switching times in\nlong-overlap Josephson junctions subject to thermal fluctuations and\nnon-Gaussiannoisesources. Phys.Rev.B89,214510. URL: https://\nlink.aps.org/doi/10.1103/PhysRevB.89.214510 ,doi: 10.1103/PhysRevB.89.214510 .\n:Preprint submitted to Elsevier Page 10 of 10"    },    {        "title": "2007.08580v1.Linearized_wave_damping_structure_of_Vlasov_Poisson_in___mathbb_R_3_.pdf",        "content": "arXiv:2007.08580v1  [math.AP]  16 Jul 2020Linearized wave-damping structure of Vlasov-Poisson in R3\nJacob Bedrossian†Nader Masmoudi‡and Cl´ ement Mouhot§\nJuly 20, 2020\nAbstract\nIn this paper we study the linearized Vlasov-Poisson equation for loc alized disturbances of\nan infinite, homogeneous Maxwellian background distribution in R3\nx×R3\nv. In contrast with the\nconfined case Td\nx×Rd\nv, or the unconfined case Rd\nx×Rd\nvwith screening, the dynamics of the\ndisturbance are not scattering towards free transport as t→ ±∞: we show that the electric\nfield decomposes into a very weakly-damped Klein-Gordon-type evo lution for long waves and a\nLandau-damped evolution. The Klein-Gordon-type waves solve, to leading order, the compress-\nible Euler-Poisson equations linearized about a constant density sta te, despite the fact that our\nmodel is collisionless, i.e. there is no trend to local or global thermaliza tion of the distribution\nfunction in strong topologies. We prove dispersive estimates on the Klein-Gordon part of the\ndynamics. The Landau damping part of the electric field decays fast er than free transport at\nlow frequencies and damps as in the confined case at high frequencie s; in fact, it decays at the\nsame rate as in the screened case. As such, neither contribution t o the electric field behaves as\nin the vacuum case.\nContents\n1 Introduction 2\n1.1 The problem at hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\n1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2 Decomposition of the electric field 8\n2.1 Volterra equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2.2 Asymptotic expansions and lower bounds on the dispersion funct ion . . . . . . . . . . . . . . 8\n2.3 Construction of the branches of poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n2.4 Spectral surgery and extraction of Klein-Gordon waves . . . . . . . . . . . . . . . . . . . . . . 14\n2.5 Another look at the long wave hydrodynamic behavior . . . . . . . . . . . . . . . . . . . . . . 18\n3 Electric field estimates 20\n3.1 Landau damping estimates on the electric field . . . . . . . . . . . . . . . . . . . . . . . . . . 20\n3.2 Dispersive estimates of the electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n†Department of Mathematics, University of Maryland, Colleg e Park, MD 20742, USA jacob@math.umd.edu . J.B.\nwas supported by NSF CAREER grant DMS-1552826 and NSF RNMS #1 107444 (Ki-Net)\n‡NYUADResearch Institute, NewYorkUniversityAbuDhabi, PO Box 129188, AbuDhabi, UnitedArabEmirates.\nCourant Institute of Mathematical Sciences, New York Unive rsity, 251 Mercer Street, New York, NY 10012, USA.\nmasmoudi@cims.nyu.edu . The work of N. M is supported by NSF grant DMS-1716466 and by T amkeen under the\nNYU Abu Dhabi Research Institute grant of the center SITE.\n§Centre for Mathematical Sciences, University of Cambridge .c.mouhot@dpmms.cam.ac.uk Partially funded by\nERC grant MAFRAN.\n14 Decomposition and scattering for the distribution functi on 23\nReferences 25\n1 Introduction\n1.1 The problem at hand\nOne of the fundamental equations in the kinetic theory of pla smas is the Vlasov-Poisson equations\nfor an infinitely extended plasma (see e.g. [12, 7]),\n\n\n∂tf+v·∇xf+E(t,x)·∇vf= 0,\nE(t,x) =−∇xW∗xρ(t,x),\nρ(t,x) =/integraldisplay\nR3f(t,x,v)dv−n0,\nf(t= 0,x,v) =fin(x,v),(1.1)\nfor the time-dependent probability density function f(t,x,v)≥0 of the electrons in the phase space\n(x,v)∈R3×R3,n0is the number density of the constant ion background, and whe reWis the\nkernel of the Coulomb interaction1.\nW(x) =q2\n4πǫ0me|x|\nwithqthe electron charge, methe electron mass, and ǫ0the vacuum permittivity. We will consider\n(1.1) linearized about the homogeneous Maxwellian backgro und with fixed temperature T\nf0(v) :=n0\n(2πT)3/2e−me|v|2\n2T. (1.2)\nIn 1930s and 1940s, Vlasov [39, 38] suggested to neglect coll isions and derive the so-called Vlasov-\nPoisson equation for long-range interactions, independen tly from Jeans’ derivation [23] in stellar\ndynamics. Motivated by the mathematical understanding of t he plasma oscillations previously\ntheorized in particular by Langmuir, he studied the lineari zed approximation of (1.1) (see (1.5)\nbelow) formally searching for eigenmodes in the form of plan ar waves e−iωt+ikxF(v), given some\nvelocity distribution F, and computed various dispersion relations (see for instan ce equation (50)\nin [39]). He asserted, not quite correctly but almost correctly as we shall see and clarify in this\npaper, that in the long-wave limit |k| ≪1, where kis the Fourier variable in space, one has the\ndispersion relation\nω2=ω2\np+3T\nme|k|2+O/parenleftBig\n|k|4/parenrightBig\nask→0,where ωp:=q2n0\nǫ0me(1.3)\nis thecold plasma frequency . The relation ω2=ω2\np+3T\nme|k|2is called the Bohm-Gross dispersion\nrelation2, and arises fromthefollowing compressibleEuler-Poisson for thespatial density n(t,x)≥0\n1Note that we are slightly abusing notation as Ein (1.1) actually denotes the acceleration of the electrons due to\nthe electric field, not the electric field itself. This conven tion is taken for the remainder of the paper.\n2Technically, the relation seems to have first appeared in Vla sov’s earlier work [39] before appearing in [6], however,\nit is likely that access to Vlasov’s work was difficult at that t ime. See also e.g. pg 260 of [12].\n2and macroscopic velocity u(t,x)∈R3\n\n\n∂tn+∇x·(nu) = 0\nmen[∂tu+(u·∇x)u] =qnE(t,x)−3T∇xn\nǫ0∇x·E(t,x) =qn\nwhen linearized around a constant density state n0>0,u0= 0:\n\n\n∂tn+n0∇x·u= 0\nmen0∂tu=qn0E(t,x)−3T∇xn\nǫ0∇x·E(t,x) =qn.(1.4)\nThis model is sometimes referred to as a warm plasma model in the physics literature see e.g.\n[Chapter 16; [12]] for more detail. Vlasov’s prediction was shown to be incomplete by Landau in\n1946[25], whoshowedthatinfactthelinearizedelectricfie lddecaystozeroas t→ ∞, aphenomenon\nnow known as Landau damping , and in particular that non-trivial planar waves never sati sfy the\ndynamics. The damping arises due to mixing/filamentation in phase space, not unlike a scalar\nquantity being stirred in a fluid (as first pointed out in [37]) . Landau damping was later observed\nin experiments [26, 27] and is considered one of the most impo rtant properties of collisionless\nplasmas [32, 12, 35].\nOnT3×R3, the linearized electric field decays rapidly provided the i nitial data is regular,\nspecifically, its decay in time is comparable to the decay in F ourier variable associated with vof the\ninitial data [28, 4]. Analogous estimates are also true for t he density on R3×R3if one uses a model\nwith Debye shielding (this arises when studying the ion dist ribution function), i.e. where Wis the\nfundamental solution to the elliptic problem −∆W+αW=δfor some constant α >0 (see [5, 19]\nand the references therein). However, such rapid decay esti mates on the linearized electric field are\nfalse inR3×R3for Vlasov-Poisson. Landau himself predicted an extremely slow Landau damping\nof long waves approximately solving the dispersion relatio n (1.3) (see also e.g. [12, 35] for modern\nexposition in the physics literature), something that we wi ll make much more precise below. This\nlack of significant Landau damping was proved rigorously by G lassey and Schaefer [10, 11], though\na precise characterization of the dynamics was still lackin g.\nIn this work, we precise the linearized dynamics and prove th at the linearized electric field\ncan be split into two contributions: one contribution at lon g spatial waves k∼0 that is a Klein-\nGordon-like propagation matching (1.3) to leading order wi th very weak Landau damping with rate\n“O(|k|∞)” and another contribution, which is properly Landau dampi ng and decays at a rate faster\nthan kinetic free transport for long-waves k∼0, i.e. faster than Vlasov-Poisson equation linearized\naroundf0= 0,n0= 0. In particular, our work shows that the hydrodynamic desc ription (linearized\nEuler-Poisson) in fact is the leading order description of t he electric field at long waves, despite the\nlack of collisional effects. As such, we remark that Vlasov was essentially correct to leading order\nin his prediction of the long wave dynamics of (1.5). For long waves, we show that the distribution\nfunction decomposes (to leading order) into two pieces: a te rm that factorizes as ˜E(t,x)·∇vf0(v)\n(wheref0is the Maxwellian background) where ˜Esolves a Klein-Gordon-type equation (including\nan additional tiny damping) and a separate contribution tha t scatters to free transport in Lp\nx,vfor\np >6. Hence, the hydrodynamic behavior arises from a large-sca le collective motion of the plasma\nthat is insensitive to the filamentation in phase space norma lly associated with Landau damping.\nUnderstanding the relationship between Landau damping (or other kinetic effects) and the ob-\nserved large-scale hydrodynamic behavior in collisionles s plasmas has been an area of research in\nthe physics literature for some time (see e.g. [17, 16, 33, 20 ] and the references therein). Our work\n3provides a precise description of the hydrodynamic behavio r and its leading order corrections due\nto Landau damping and other kinetic effects, for the simple lin earized problem; studies of more\nphysical settings (e.g. external and/or self-consistent m agnetic fields, inclusion of ions, inhomoge-\nneous backgrounds etc) and/or the inclusion of nonlinear effe cts is an interesting direction of further\nresearch.\nOur work is only linear, but we remark that much progress has b een made at the nonlinear level\nin recent years in other settings. After the earlier work of [ 8] (see also [21]), the major breakthrough\ncame in [28], when Landau damping for the nonlinear problem w as shown on T3×R3for all\nsufficiently small and smooth initial perturbation of Landau -stable stationary solutions; the actual\nsmoothness required being Gevrey or analytic. We also refer to [4, 13] for simplified proofs and [3]\nfor a study of the problem with collisions. The work [2] shows that the results therein do not hold in\nfinite regularity (see also [14]). Our previous work [5] stud ied the nonlinear problem with shielding\nonR3×R3; see also [19, 29] for an alternative approach and some refine ments. Other works, old\nand new, have studied the nonlinear Cauchy problem near the v acuum state, that is without the\npresence of a non-zero spatially homogeneous background eq uilibrium f0, see e.g. [1, 22]. As can\nbe seen from our results below, the dynamics are significantl y different in this case.\n1.2 Main results\nWe linearize the Vlasov-Poisson equation (1.1) on R3\nx×R3\nvaround an infinitely-extended, homoge-\nneous background f0(v)≥0. This models a spatially localized disturbance of an infini te plasma in\nwhich collisions can beneglected, a fundamental problem in the kinetic theory of plasmas [12, 7, 35].\nFor a localized disturbance h:=f0−f, thelinearized Vlasov-Poisson equations for the electron\ndistribution function are given by\n\n\n∂th+v·∇xh+E(t,x)·∇vf0= 0,\nE(t,x) =−∇xW∗xρ(t,x),\nρ(t,x) =/integraldisplay\nR3h(t,x,v)dv,\nh(t= 0,x,v) =hin(x,v),(1.5)\nwhere we take the charge neutrality assumption/integraltext\nR3×R3hin(x,v)dxdv= 0. Define the standard\nplasma constants (number density, plasma frequency and tem perature):\nn0:=/hatwiderf0(0), ω2\np:=q2n0\nǫ0me, T:=me\n3n0/integraldisplay\nR3|v|2f0(v)dv,\nand as stated above, we assume the Maxwellian distribution b ackground (1.2).\nOurmain resultsare asymptotic decomposition of theelectr ic fieldand thedistributionfunction;\nwe give slightly simplified statements for readability and r efer to the main body of the paper for\nmore detailed expansions. See Subsection 1.3 for the notati on/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht,/a\\}b∇acketle{tx,y/a\\}b∇acket∇i}ht,/a\\}b∇acketle{t∇/a\\}b∇acket∇i}ht, andWk+0,p\nw.\nWe remark that another preprint proving similar results wit h somewhat different methods [18]\nhas recently been completed as well. These works have been co mpleted totally independently.\nTheorem 1. Suppose/integraltext\nR3×R3hindxdv= 0and lethandEsolve(1.5). There is δ0>0and a\ndecomposition of the electric field E=EKG+ELDbetween a ‘Klein-Gordon’ and ‘Landau damped’\nparts, with EKGsupported in spatial frequencies |k|< ν0and:\n•ELDsatisfies the following Landau-damping-type decay estimates for anyσ∈N:\n/ba∇dbl/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσELD(t)/ba∇dblL2x/lessorsimilar1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht5/2/ba∇dblhin/ba∇dbl\nWσ+3\n2+0,1\n0\n4/ba∇dbl/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσELD(t)/ba∇dblL∞x/lessorsimilar1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4/ba∇dblhin/ba∇dblWσ+3+0,1\n0. (1.6)\n•EKGfurther decomposes as EKG=E(1)\nKG+E(2)\nKGwith the pointwise-in-time estimates :\n/vextenddouble/vextenddouble/vextenddoubleE(1)\nKG(t)/vextenddouble/vextenddouble/vextenddouble\nL2x/lessorsimilar/ba∇dblEin/ba∇dblL2+/ba∇dblvhin/ba∇dblL2x,v+/ba∇dblhin/ba∇dblW0,1\n4/vextenddouble/vextenddouble/vextenddoubleE(2)\nKG(t)/vextenddouble/vextenddouble/vextenddouble\nL2x/lessorsimilar/ba∇dblhin/ba∇dblW0,1\n5/vextenddouble/vextenddouble/vextenddoubleE(2)\nKG(t)/vextenddouble/vextenddouble/vextenddouble\nL∞x/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3/2/ba∇dblhin/ba∇dblW3/2,1\n5\n/vextenddouble/vextenddouble/vextenddoubleE(1)\nKG(t)/vextenddouble/vextenddouble/vextenddouble\nLp\nx/lessorsimilarp/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3/parenleftBig\n1\n2−1\np/parenrightBig\n/ba∇dbl/a\\}b∇acketle{tx/a\\}b∇acket∇i}hthin/ba∇dblW0,1\n4(∀2≤p <∞)\n/vextenddouble/vextenddouble/vextenddouble∇xE(1)\nKG(t)/vextenddouble/vextenddouble/vextenddouble\nL∞x/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3/2/ba∇dblhin/ba∇dblW0,1\n4.\n•E(1)\nKGsolves a weakly damped Klein-Gordon type equation in the fol lowing sense: there are\nbounded, smooth functions λ,Ω,k∈B(0,δ0)such that (for errors independent of t),\n/hatwideE(1)\nKG(t,k) =/hatwideEin(k)e−λ(k)tcos(Ω(k)t)+e−λ(k)tik\n|k|2/parenleftBig\nk·∇η/hatwiderhin(k,0)/parenrightBigsin(Ω(k)t)\nΩ(k)\n+O(|k|2)e−λ(k)t+iΩ(k)t+O(|k|2)e−λ(k)t−iΩ(k)t,\nwhere λ(k)>0, λ(k) =O(|k|∞),Ω2(k) =ω2\np+3T\nme|k|2+O/parenleftBig\n|k|4/parenrightBig\nask→0.\nFurthermore, here holds for all 2< p≤ ∞,\n/vextenddouble/vextenddouble/vextenddouble(−∆x)−1E(2)\nKG(t)/vextenddouble/vextenddouble/vextenddouble\nLp/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3/parenleftBig\n1\n2−1\np/parenrightBig\n/ba∇dblhin/ba∇dblW3/2,1\n5,\nwhich further emphasizes that for small k,E(1)\nKGis much larger than E(2)\nKG.\nRemark 1. TheE(2)\nKGelectric field is essentially a Klein-Gordon-type evolutio n subjected to a\nLandau damping external forcing. See Section 2 for more deta ils.\nRemark 2. For frequencies bounded away from zero, i.e. for any δ >0,P≥δE, Landau damps at a\npolynomial rate /a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−σprovided the initial data is Wσ,1, hence the extremely slow Landau damping\nofEKGmanifests only at k= 0. As expected, for frequencies bounded from zero, the elec tric field\n(bothELDandEKG) damps exponentially fast if the initial data is analytic (a nd analogously e−/angb∇acketleftkt/angb∇acket∇ights\nfor Gevrey initial data).\nRemark 3. As the electric field is not shielded, it is too restrictive to assume that Ein∈L1,\nregardless of how well localized the initial hinis. Ifxhin∈L1and/integraltext\nR3\nX×R3vhindxdv= 0, then\nEin(x)≈ |x|−3asx→ ∞(hence,Ein∈Lpforp >1 but not p= 1) but one cannot obtain faster\ndecay at infinity unless one has higher zero-moment conditio ns, such as/integraltext\nR3x×R3vxαhindxdv= 0 and\nthe latter are not propagated by the semi-group.\nRemark 4. In the case of the kinetic free transport, the electric field d ecays only as /ba∇dblEFT(t)/ba∇dblL∞x/lessorsimilar\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−2, which is a full power of tslower than (1.6). For frequencies boundedaway from zero, b oth the\nfree transport electric field EFTand our ELDare Landau-damped exponentially fast for analytic\ndata (and polynomial for Sobolev data); the difference in deca y rates comes from the contribution\nof long waves.\n5Remark 5. There is a clear distinction between Landau damping and disp ersive decay above: for\nthe Landau damping contributions, each derivative buys one power of time (a structure that can\nbe guessed from free transport), whereas for the Klein-Gord on-like contributions, there is no such\nbehavior.\nRemark 6. We have not endeavored to get the sharpest dependence on the i nitial data that could\nbe possible. This could be important for nonlinear extensio ns.\nWe deduce a similar decomposition at the level of the distrib ution function:\nTheorem 2. The solution to (1.5)decomposes as h=hKG+hLDwithhKG=˜EKG(t,x)·∇vf0(v)\nfor some effective electric field ˜EKG, with the following estimates:\n•for allm≥0,r∈[1,∞]and allp∈[2,∞), (˜EKGis essentially a phase-shifted version of\nE(1)\nKGand satisfies the same estimates)\n/ba∇dbl/a\\}b∇acketle{tv/a\\}b∇acket∇i}htmhKG(t)/ba∇dblL2xLrv/lessorsimilarm/ba∇dblhin/ba∇dblW0,1\n4(1.7)\n/ba∇dbl/a\\}b∇acketle{tv/a\\}b∇acket∇i}htmhKG(t)/ba∇dblLp\nxLrv/lessorsimilarp,m/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3/parenleftBig\n1\n2−1\np/parenrightBig\n/ba∇dbl/a\\}b∇acketle{tx/a\\}b∇acket∇i}hthin/ba∇dblW0,1\n4\n/ba∇dbl/a\\}b∇acketle{tv/a\\}b∇acket∇i}htm∇xhKG(t)/ba∇dblL∞xLrv/lessorsimilarm/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3/2/ba∇dblhin/ba∇dblW0,1\n4;\n•hLDscatters to free transport in all Lp\nx,v,p >6(provided one has enough initial regularity).\nIn particular, if /a\\}b∇acketle{t∇/a\\}b∇acket∇i}htσ/a\\}b∇acketle{tv/a\\}b∇acket∇i}htmhinis integrable for σ,m∈Nlarge enough, then for all p >6, there\nexists an h∞∈Lp\nx,vsuch that\n/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{tv/a\\}b∇acket∇i}htm/bracketleftBig\nhLD(t,x−vt,v)−h∞(x,v)/bracketrightBig/vextenddouble/vextenddouble/vextenddouble\nLp\nx,vt→+∞− −−− →0.\nRemark 7. Using the Strichartz estimates [9] for the transport equati on one obtains\n/ba∇dblhLD/ba∇dblLq\ntLp\nxLrv/lessorsimilar/ba∇dblhin/ba∇dblWσ,1\nm\nfor all (p,q,r,a) satisfying\n1\nr−1\nn<1\np≤1\nr≤1,1≤1\np+1\nr,2\nq=n/parenleftbigg1\nr−1\np/parenrightbigg\n,1\na=1\n2/parenleftbigg1\nr+1\np/parenrightbigg\n, a >6.\nTheorem 1 shows that the leading-order term in the asymptoti cs of the distribution hfactorises\nbetween a function of ( t,x) and a fixed function of v, which is reminiscent of a hydrodynamical\nlimiteven though the equations are collisionless . One can actually push this intuition further: a\nlong-wave re-scaling of the electric field converges weakly (in negative Sobolev spaces) to the electric\nfield solving the linearized Euler-Poisson system (1.4). Se e Subsection 2.5 for a proof.\nTheorem 3. Consider an initial data hin(x,v) =ǫ3H0(ǫx,v)such that H0has zero average and\nH0∈W5+0,1\n5, and denote\nE0=q2n0\nǫ0me∇x(−∆x)−1/integraldisplay\nR3H0(·,v)dv.\n6LetEbe the unique solution in the natural energy space E ∈L∞\ntH1\nxand∂tE ∈L∞\ntL2\nxto the\nfollowing Klein-Gordon equation,\n\n\n∂2\ntE+/parenleftbigg\nω2\np−3T\nme∆/parenrightbigg\nE= 0\nE(0,x) =E0(x)\n∂tE(0,x) =−n0∇x·/parenleftbigg/integraldisplay\nvhindv/parenrightbigg\n.(1.8)\nThen, for all s∈(5/2,7/2),0< ǫ≪1, for all0< t < ǫ−N, there holds\n/vextenddouble/vextenddouble/vextenddoubleE/parenleftBig\nt,·\nǫ/parenrightBig\n−E/parenleftBig\nt,·\nǫ/parenrightBig/vextenddouble/vextenddouble/vextenddouble\nH−s/lessorsimilarǫ2/parenleftBig\nǫ2+ǫs−3\n2−0/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/parenrightBig\n/ba∇dblH0/ba∇dblW2,1\n5.\nMoreover, given such initial data with appropriate scaling , both the electric fields in the linearized\nVlasov-Poisson and the linearized Euler-Poisson system ar e asymptotic to the Bohm-Gross disper-\nsion relation at large scale: E(t,x) =E++E−andE(t,x) =E++E−with the following weak L2\nx\nlimit on t∈[−T,T]for anyTfixed finite\n1\nǫ2e∓iωpt\nǫ2E±/parenleftbiggt\nǫ2,x\nǫ/parenrightbigg\nweak− −− →\nL2x1\n2e±3T\nmei∆tE0\n1\nǫ2e∓iωpt\nǫ2E±/parenleftbiggt\nǫ2,x\nǫ/parenrightbigg\nweak− −− →\nL2x1\n2e±3T\nmei∆tE0.\n1.3 Notation\nDenote/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht= (1+|x|2)1/2and/a\\}b∇acketle{t∇/a\\}b∇acket∇i}htthe Fourier multiplier defined by\n/hatwide/a\\}b∇acketle{t∇/a\\}b∇acket∇i}htf(ξ) =/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht/hatwidef(ξ);\nwe define other Fourier multipliers in the analogous way. We d efineP≤Nto be the Littlewood-Paley\nlow-pass filter in x, in particular, define χ∈C∞\nc(B(0,2)) with χ(x) = 1 for |x| ≤1, and define\n/hatwiderP≤Nf(k) =χ/parenleftbiggk\nN/parenrightbigg\n/hatwidef(k).\nDenote the weighted Sobolev spaces (note that the weight ind ex refers to the variable v)\n/ba∇dblf/ba∇dblLp\nx,v:=/parenleftbigg/integraldisplay\nR3×R3|f(x,v)|pdxdv/parenrightbigg1/p\n,/ba∇dblρ/ba∇dblLp\nx:=/parenleftbigg/integraldisplay\nR3|ρ(x)|pdx/parenrightbigg1/p\n/ba∇dblf/ba∇dblWσ,p\nm:=/ba∇dbl/a\\}b∇acketle{tv/a\\}b∇acket∇i}htm/a\\}b∇acketle{t∇x,v/a\\}b∇acket∇i}htσf/ba∇dblLp.\nFor inequalities involving norms Wσ+0,p\nmwe use the notation +0 if we mean that the estimate holds\nforσ+δwithδ∈(0,1) with a constant that depends on δ. Forp∈[1,∞] we denote p′=p\np−1the\nH¨ older conjugate. Let f: [0,∞)→Csatisfye−µtf(t)∈L1for some µ∈R. Then for all complex\nnumbers ℜz≥µ, we can define the Fourier-Laplace transform via the (absolu tely convergent)\nintegral\nˆf(z) :=1\n2π/integraldisplay∞\n0e−ztf(t)dt. (1.9)\n7This transform is inverted by integrating the ˜falong a so-called ‘Bromwich contour’ via the inverse\nFourier-Laplace transform: let γ > µand define:\nˇf(t) :=/integraldisplayγ+i∞\nγ−i∞eztf(z)dz.\n2 Decomposition of the electric field\n2.1 Volterra equation\nThe most important property of the linearized Vlasov equati ons is that one can reduce the problem\nto a Volterra equation for the density separately for each sp atial frequency. We now recall how this\nis done. Writing a Duhamel representation along free transp ort gives\nh(t,x,v) =hin(x−vt,v)+/integraldisplayt\n0(∇xW∗xρ)(t,x−v(t−s))·∇vf0(v)ds.\nTaking the Fourier transform in xand integrating in vgets (with w0:=ω2\npn−1\n0)\nˆρ(t,k) =/hatwiderhin(k,kt)−w0/integraldisplayt\n0(t−τ)/hatwiderf0(k(t−τ))ˆρ(τ,k)dτ. (2.1)\nTaking the Fourier-Laplace transform (1.9) in time for ℜzsufficiently large gives\n˜ρ(z,k) =H(z,k)+L(z,k)˜ρ(z,k), (2.2)\nwhereH(z,k) is the Fourier-Laplace transform of t/ma√sto→/hatwiderhin(k,kt) and the dispersion function is\nL(z,k) :=−w0/integraldisplay+∞\n0t/hatwiderf0(kt)e−ztdt=−w0\n|k|2/integraldisplay+∞\n0e−z\n|k|ss/hatwiderf0/parenleftBig\nˆks/parenrightBig\nds, (2.3)\nwith thestandard notation ˆk=k/|k|. This change of variable allows for the function s/ma√sto→s/hatwiderf0(ˆks) to\nhaveregularity boundsindependentlyof thesizeof k. Ourassumption(1.2)implies that z/ma√sto→ L(z,k)\nis an entire function. Moreover, we see that away from z= 0,L(z,k) is also a smooth function of\nk(even at k= 0 as we shall see in the expansions below).\n2.2 Asymptotic expansions and lower bounds on the dispersio n function\nSolving (2.2) for ρworks, formally at least, except where L(z,k) gets close to one. In the case of\nnot-so-small spatial frequencies k, the treatment is similar to that of the periodic domain x∈T3,\nand so we merely sketch the argument here; see [28, 4] for more details.\nLemma 2.1 (Resolvent estimates for non-small frequencies) .There exists a λ >0such that for\nanyν0>0,∃κ >0(depending on ν0) such that\n∀|k|> ν0,inf\nℜz>−λ|k||1−L(z,k)|> κ. (2.4)\nFurthermore, the following estimate holds uniformly on the critical vertical line of this region\n∀|k|> ν0, ω∈R,|L(λ|k|+iω,k)|/lessorsimilarλ1\n1+|k|2+ω2. (2.5)\n8Proof.The estimate (2.4) is proved in e.g. [28, 4]. To see (2.5), we u se integration by parts and the\nanalyticity of f0to get\n|L(λ|k|+iω,k)|=w0\n|k|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\n0eλs−iω\n|k|ss/hatwiderf0/parenleftBig\nˆks/parenrightBig\nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=w0\n|k|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\n01/parenleftBig\nλ−iω\n|k|/parenrightBig2∂2\ns/parenleftBig\neλs−iω\n|k|s/parenrightBig\ns/hatwiderf0/parenleftBig\nˆks/parenrightBig\nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤w0\n(λ2|k|2+ω2)/bracketleftbigg\n|n0|+/integraldisplay∞\n0/parenleftBig\n2/vextendsingle/vextendsingle/vextendsingle∂s/hatwiderf0/parenleftBig\nˆks/parenrightBig/vextendsingle/vextendsingle/vextendsingle+s/vextendsingle/vextendsingle/vextendsingle∂2\ns/hatwiderf0/parenleftBig\nˆks/parenrightBig/vextendsingle/vextendsingle/vextendsingle/parenrightBig\nds/bracketrightbigg\n/lessorsimilarλ1\n1+|k|2+ω2,\nwhich completes the proof.\nNext we turn to the low frequency estimates, which are more ch allenging, and contain the\nlong-wave dispersive structure. For δ,δ′>0, define the following region in the complex plane:\nΛδ,δ′:=/braceleftBig\nz=λ+iω∈C:λ >−min/bracketleftBig\n(1−δ)|ω|,δ′|k|/bracketrightBig/bracerightBig\n.\nΛδ,δ′ℜz=−δ′|k|\nℜz=−(1−δ)|ℑz|ℜz=−(1−δ)|ℑz|\nFigure 1: The region Λ δ,δ′\nWe will next show that L(z,k) stays uniformly away from one in the region Λ δ,δ′\\{|z±iωp|< ǫ}\nwhereωpis the cold plasma frequency. The proof relies on two represe ntations: (1) an expansion\nobtained by the stationary phase method (i.e. successive in tegrations by parts in time) meaningful\nfor large values ofz\n|k|, (2) an approximation argument using the explicit formula o btained from the\nPlemelj formula at the imaginary line ℜz= 0, that provides estimates near this line. The first\nrepresentation is given by the following lemma.\nLemma 2.2 (Asymptotic expansion of L).Givenδ′>0sufficiently small depending only on f0,\n∀z∈Λδ,δ′,L(z,k) =−ω2\np\nz2/bracketleftBigg\n1+3T|k|2\nmez2+O/parenleftBigg\n|k|4\n|z|4/parenrightBigg/bracketrightBigg\n,as|z|\n|k|→ ∞. (2.6)\n9Note that this expansion only contains information for freq uencies|k| ≪ |z|.\nProof.Sinces/ma√sto→sˆf0(ˆks) is odd, observe that ∂j\ns(s/hatwiderf0(ˆks))|s=0= 0 for all even j. Therefore,\nintegrating by parts repeatedly in sin (2.3) gives\nL(z,k) =−w0\nz2/hatwiderf0(0)−w0\nz2/integraldisplay∞\n0e−z\n|k|s∂2\ns/parenleftBig\ns/hatwiderf0/parenleftBig\nˆks/parenrightBig/parenrightBig\nds\n=−w0\nz2/hatwiderf0(0)−3w0|k|2\nz4/bracketleftBig\nˆk⊗ˆk:∇2/hatwiderf0(0)/bracketrightBig\n−w0|k|2\nz4/integraldisplay∞\n0e−z\n|k|s∂4\ns/parenleftBig\ns/hatwiderf0/parenleftBig\nˆks/parenrightBig/parenrightBig\nds\n=:−w0\nz2/hatwiderf0(0)−3w0|k|2\nz4/bracketleftBig\nˆk⊗ˆk:∇2/hatwiderf0(0)/bracketrightBig\n−w0|k|2\nz4ζ(z,k).\nSincew0/hatwiderf0(0) =w0n0=ω2\npand 3n0T:=meˆk⊗ˆk:∇2/hatwiderf0(0) gives the leading order terms in (2.6).\nNote that we have\nζ(z,k) :=/integraldisplay∞\n0e−z\n|k|s∂4\ns/parenleftBig\ns/hatwiderf0/parenleftBig\nˆks/parenrightBig/parenrightBig\nds.\nIt remains to show that for z∈Λδ,δ′, there holds\n|ζ(z,k)|/lessorsimilar|k|2\n|z|2.\nFirst consider the region ℜz≥ −δ′|k|: integrating by parts two more times, we obtain\n∀ℜz≥ −δ′|k|,|ζ(z,k)|/lessorsimilar|k|2\n|z|2/parenleftbigg\n1+/integraldisplay∞\n0eδ′s/vextendsingle/vextendsingle/vextendsingle∂6\ns/parenleftBig\ns/hatwiderf0/parenleftBig\nˆks/parenrightBig/parenrightBig/vextendsingle/vextendsingle/vextendsingleds/parenrightbigg\n/lessorsimilarδ′|k|2\n|z|2,\nwhere the last line followed by the analyticity of f0.\nTurnnexttotheregion ℜz <−δ′|k|withℜz >−(1−δ)|ℑz|. Observethenarg z2∈[π\n2+β,3π\n2−β]\nfor a small β >0 depending on δ. Write\nζ(z,k) =/integraldisplay∞\n−∞e−z\n|k|s∂4\ns/parenleftBig\ns/hatwiderf0(ˆks)/parenrightBig\nds−/integraldisplay0\n−∞e−z\n|k|s∂4\ns/parenleftBig\ns/hatwiderf0(ˆks)/parenrightBig\nds=:ζ1(z,k)+ζ2(z,k).\nOn the one hand, ζ2is bounded as in the region ℜz≥ −δ′|k|due to the now advantageous sign of\nthe exponent. On the other hand, ζ1is a true Fourier-Laplace transform, and due to (1.2),\nζ1(z,k) =/integraldisplay∞\n−∞e−z\n|k|s∂4\ns/parenleftBig\ns/hatwiderf0(ˆks)/parenrightBig\nds=z4\n|k|4/integraldisplay∞\n−∞e−z\n|k|s/hatwiderf0(ˆks)ds\n=n0\nm3/2\nez4\n|k|4/integraldisplay∞\n−∞e−z\n|k|s−s2T\n2meds=n0\nm3/2\nez4\n|k|4emez2\n2T|k|2.\nDue to arg z2∈[π\n2+β,3π\n2−β], it holds ℜz2/lessorsimilarδ−|ℑz|2/lessorsimilarδ−|z|2and this term vanishes (to infinite\norder) in terms of|z|\n|k|→ ∞. This completes the proof.\nLemma 2.2 suffices to estimate the resolvent in much of the area s of interest. The next lemma\nestimates the resolvent for low frequencies kin the half-plane ℜz≥ −δ′|k|(assuming δ′to be small\nenough) and away from the cold plasma frequencies ±iωp. Givenǫ >0, define the following region:\nHǫ,δ′:=/braceleftbig\nz=λ+iω∈C:λ >−δ′|k|and|z±iωp| ≥ǫ)/bracerightbig\n.\n10Hǫ,δ′ℜz=−δ′|k||z−iωp|< ǫ\n|z+iωp|< ǫ\nFigure 2: The region Hǫ,δ′\nLemma 2.3 (Low frequency resolvent estimates) .Givenε,δ′>0, there are ν0,κ >0such that\n∀|k|< ν0,∀z∈Hǫ,δ′,|1−L(z,k)| ≥κ.\nProof.LetR >0 be fixed large depending on f0but independent of k.\nCase 1: |z|> R|k|.In this region the estimate follows from (2.6) taking R >0 sufficiently large.\nCase 2: |z| ≤R|k|.In this region the asymptotic expansion (2.6) is no longer us eful and we use\nthePlemelj formula instead. Writing z=λ+iω, it is classical that for λ= 0, one has (see e.g.\n[31, 28] for explanations),\nL(iω,k) =w0\n|k|2/integraldisplay\nR(f0\nk)′(r)\nr−ω\n|k|dr+iw0π\n|k|2(f0\nk)′/parenleftbiggω\n|k|/parenrightbigg\n,\nwhere, for any k/\\e}atio\\slash= 0, the partial hyperplane average is defined as\n∀r∈R, f0\nk(r) :=/integraldisplay\nk\n|k|r+k⊥f0(v∗)dv∗.\nMoreover, observe that for any z∈Csuch that |z| ≤R|k|andℜz∈(−δ′|k|,0],\n|∂zL(z,k)|/lessorsimilarw0\n|k|3/integraldisplay∞\n0seδ′s/vextendsingle/vextendsingle/vextendsingle/hatwiderf0/parenleftBig\nˆks/parenrightBig/vextendsingle/vextendsingle/vextendsingleds/lessorsimilar1\n|k|3(2.7)\nwhere we have used the analyticity of f0and taken δ′small enough.\nSubcase 2.1: |z| ≤R|k|andc|k| ≤ |ω| ≤R|k|.Given any c >0, we deduce from decay, smoothness,\nradial symmetry, and monotonicity of f0that\ninf\nc<|ω|\n|k|<Rw0π\n|k|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle(f0\nk)′/parenleftbiggω\n|k|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/greaterorsimilarc,R1\n|k|2,\n11and therefore, using (2.7), |ℑL(z,k)|/greaterorsimilar1 holds uniformly for all ℜz∈(−δ′|k|,0).\nSubcase 2.2: |z| ≤R|k|and|ω|< c|k|.Observe that(f0\nk)′(r)\nris integrable since ( f0\nk)′(0) = 0 and f0\nsmooth, and if we denote\n/integraldisplay\nR(f0\nk)′(r)\nrdr=c0/\\e}atio\\slash= 0,\nwe deduce that if c≤|c0|\n2is small enough then\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR(f0\nk)′(r)\nr−ω\n|k|dr−c0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleω\n|k|/integraldisplay\nR(f0\nk)′(r)\nr/parenleftBig\nr−ω\n|k|/parenrightBigdr/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar|ω|\n|k|≤|c0|\n2\nand thus\n|1−L(iω,k)|/greaterorsimilar1\n|k|2.\nUsing again (2.7), for δ′small enough we deduce that for ℜz∈(−δ′|k|,0],\n|1−L(z,k)|/greaterorsimilar1.\nThese different cases above together prove that 1 /|1− L|is bounded from above on the strip\nℜz∈(−δ′|k|,0] and outside B(0,R|k|)∩{ℜz >0}, and since thereare no poles within theremaining\nregionB(0,R|k|)∩ {ℜz >0}, the function is holomorphic in this region and the upper bou nd is\nalso valid there by the maximum principle, which completes t he proof.\n2.3 Construction of the branches of poles\nFrom Lemma 2.2, we have a pole at |k|= 0 at the cold plasma frequency: L(±iωp,0) = 1. It follows\nfrom Rouch´ e’s theorem that if |k|if small enough, exactly two poles persist in respective nei ghbor-\nhoods of ±iωp: Givenǫ >0, the two functions F(z) := 1−L(z,0) andG(z) :=L(z,k)−L(z,0) are\nholomorphicontheset |z∓iωp|lǫ, andLemma2.2 implies |1−L(z,0)|/greaterorsimilarǫand|L(z,0)−L(z,k)|/lessorsimilar\n|k|2on|z∓iωp|=ǫ. Therefore, F(z) = 1−L(z,0) andF(z)−G(z) = 1−L(z,k) have the same\nnumber of poles in |z∓iωp|< ǫprovided that |k|is sufficiently small relatively to ǫ.\niωp\n−iωpp+(k) =i/parenleftBig\nωp+3T\nmeωp|k|4+O(|k|2)/parenrightBig\n+Error(k)\np−(k) =−i/parenleftBig\nωp+3T\nmeωp|k|2+O(|k|4)/parenrightBig\n+Error(k)Error(k) =O(|k|∞)<0\nFigure 3: The branches of poles k/ma√sto→p±(k)\n12However knowing just the approximate location of the poles i s not enough to deduce dispersive\nestimates. We next use the implicit function theorem to cons truct the branches of solutions p±(k).\nLemma 2.4. There are ǫ,ν0>0such that for all |k|< ν0, there are unique p±(k)∈Csolution to\nL(p±(k),k) = 1in{|z∓iωp|< ǫ}andk/ma√sto→p±(k) =:−λ(k)±iΩ(k)are smooth (but not analytic)\nand satisfy λ(k)>0andp±(k)∼k→0±iωpwith the following expansions as k→0:\nΩ(k)2=ω2\np+3T\nme|k|2+O/parenleftBig\n|k|4/parenrightBig\n(2.8)\n∇Ω(k) =i3T\nmeωpk+O/parenleftBig\n|k|3/parenrightBig\n(2.9)\n∇2Ω(k) =i3T\nmeωpId+O/parenleftBig\n|k|2/parenrightBig\n(2.10)\n/vextendsingle/vextendsingle∇jλ(k)/vextendsingle/vextendsingle/lessorsimilarj,N|k|Nfor anyj,N∈N. (2.11)\nRemark 8. This expansion of ℑp±(k) =±Ω(k) provides the rigorous justification for the Bohm-\nGross dispersion relation in kinetic theory. Regarding the real part ℜp±(k) =−λ(k), physicists\nassert that (see e.g. [12, pp.419]),\nλ(k)≈√πm3/2\neω4\np\n|k|3T3/2exp/parenleftBigg\n−meω2\np\n4|k|2T/parenrightBigg\n,\nhowever at this time we lack a mathematically rigorous expla nation for this exact prediction.\nProof.Since/hatwiderf0is real,p+=p−and it is enough to build the branch near + iωp. By the implicit\nfunction theorem applied to the function Lof (z,k)∈C×Rd, the result follows by verifying\n∂zL(iωp,0)/\\e}atio\\slash= 0, since Lis smooth in ( z,k) and analytic in zin this neighborhood. Roughly\nspeaking we want to take derivatives of the expansion (2.6). From (2.3) and integrating by parts as\nin the proof of Lemma 2.2,\n∂zL(z,k) =w0\n|k|3/integraldisplay∞\n0e−z\n|k|ss2/hatwiderf0/parenleftBig\nˆks/parenrightBig\nds=w0\nz2|k|/integraldisplay∞\n0e−z\n|k|s∂2\ns/parenleftBig\ns2/hatwiderf0/parenleftBig\nˆks/parenrightBig/parenrightBig\nds\n=2w0n0\nz3+w0\nz3/integraldisplay∞\n0e−z\n|k|s∂3\ns/parenleftBig\ns2/hatwiderf0/parenleftBig\nˆks/parenrightBig/parenrightBig\nds\n=2w0n0\nz3+2w0|k|\nz4/bracketleftBig\nˆk·∇/hatwiderf0/parenleftBig\nˆks/parenrightBig/bracketrightBig\n+w0|k|\nz4/integraldisplay∞\n0e−z\n|k|s∂4\ns/parenleftBig\ns2/hatwiderf0/parenleftBig\nˆks/parenrightBig/parenrightBig\nds\n=2w0n0\nz3+w0|k|\nz4ζ′(z,k)\nwithζ′(z,k) that is uniformly bounded for |z−iωp|< ǫand|k|< ν0(using the analyticity of f0).\nHence∂zL(iωp,0) =2ω2\np\n(iωp)3=2i\nωp/\\e}atio\\slash= 0. The implicit function theorem then implies the existenc e of a\nunique smooth solution k/ma√sto→p+(k) to the equation L(p+(k),k) = 1 in a neighborhood of iωp.\nTo get more precise information on the behavior of the poles n eark= 0, we need to compute\nderivatives in kas well of L. Expansion (2.8) immediately follows from Lemmas 2.2:\np2\n+=p2\n+L(p+,k) =ω2\np−3Tω2\np\nmep2\n+|k|2+O/parenleftbig\n|k|4/parenrightbig\n=ω2\np+3T\nme|k|2+O/parenleftbig\n|k|4/parenrightbig\n.\n13Next, observe that\n∇p+=−(∇kL)(p+,k)\n(∂zL)(p+,k), (2.12)\n∇2p+=−(∂zL)(p+,k))−1/bracketleftbig\n(∇2\nkL)(p+,k)+(∇k∂zL)(p+,k)·∇p++(∂2\nzL)(p+,k)|∇p+|2/bracketrightbig\n.(2.13)\nBy the same integration by parts method used in Lemma 2.2, we o btain\n∇kL(z,k) =−w0\n|k|3/integraldisplay∞\n0e−z\n|k|ss2∇/hatwiderf0(ˆks)ds=2w0k\nz4/bracketleftBig\nˆk⊗ˆk:∇2/hatwiderf0(0)/bracketrightBig\n+O/parenleftBig\n|k|3/parenrightBig\n.\nUsing∇2/hatwiderf0(0) = 3n0T\nmeId andw0=ω2\npn−1\n0, this yields\n∇kL(p+(k),k) =6T\nmeω2pk+O/parenleftBig\n|k|3/parenrightBig\n,\nwhich implies (2.9). The proof of (2.10) is similar, using no w (2.13) instead of (2.12); the lengthier\ncalculations are omitted for the sake of brevity. In these ca lculations a clear pattern emerges: in all\nderivatives ∇m\nkL(z,k) (respectively ∇m\nk∂zL(z,k)), form∈N, the leading order as k→0 is an even\n(resp. odd) power of z−1, and thus at z=±iωpall derivatives ∇m\nkL(iωp,0) (resp. ∇m\nk∂zL(iωp,0))\nare purely real (resp. purely imaginary). This proves all de rivatives ∇mp+(0) are purely imaginary\nand implies thus (2.11), i.e. ℜp+=−λvanishes to infinite order at k= 0. Observe that since\nnevertheless ℜp+<0 ask∼0 (by Lemma 2.1), the function k/ma√sto→p+(k) differs from its Taylor\nseries at k= 0 and is therefore not analytic.\n2.4 Spectral surgery and extraction of Klein-Gordon waves\nThrough (2.2), the solution to the Volterra equation (2.1) i s classically [15] given formally as:\nˆρ(t,k) =/hatwiderhin(k,kt)+/integraldisplayt\n0R(t−τ,k)/hatwiderhin(k,kτ)dτ, (2.14)\nwhere the resolvent kernel Ris given by the inverse Laplace transform\nR(t,k) =1\n2iπ/integraldisplayγ+i∞\nγ−i∞eztL(z,k)\n1−L(z,k)dz,\nfor a suitable Bromwich contour such that z/ma√sto→L(z,k)\n1−L(z,k)is holomorphic for ℜz > γ−0.\nThe calculations in the two previous Subsections 2.2-2.3, s how that for |k|< ν0sufficiently\nsmall,L(·,k)\n1−L(·,k)is holomorphic in the region Hǫ,δ′represented in Figure 2 (the half-plane ℜz≤ −δ′|k|\nminusǫ-discs around for the two poles), with one isolated pole p±(k) in each disc, depending on k\nas studied in the last Subsection. Therefore, by Cauchy’s Re sidue theorem,\nR(t,k) =/parenleftBigg\nep+(k)t\n−∂zL(p+(k),k)+ep−(k)t\n−∂zL(p−(k),k)/parenrightBigg\n+1\n2iπ/integraldisplayγ′+i∞\nγ′−i∞eztL(z,k)\n1−L(z,k)dz\n=:R+\nKG(t,k)+R−\nKG(t,k)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nRKG(t,k)+RRFT(t,k),\n14for some γ′∈(−δ′,−ǫ) sothat the vertical line is to the leftof the poles p±(k) butstill in Hǫ,δ′where\nL/(1−L) is meromorphic. This decomposes the resolvent R=RKG+RLDinto aKlein-Gordon\npart and a remainder free transport part, which then yields a corresponding decomposition of th e\ndensity ˆρ(t,k) through (2.14):\nˆρ(t,k) =/hatwiderhin(k,kt)+/integraldisplayt\n0RKG(t−τ,k)/hatwiderhin(k,kτ)dτ+/integraldisplayt\n0RRFT(t−τ,k)/hatwiderhin(k,kτ)dτ\n=: ˆρFT(t,k)+ ˆρ+\nKG(t,k)+ ˆρ−\nKG(t,k)+ ˆρRFT(t,k).\nWe first prove a general expansion of ˆ ρ±\nKG(t,k) by successive integrations in time (producing addi-\ntional powers of k). Terms in this expansion are either comparable to solution s to free transport\n(with additional Fourier multipliers) or to solutions to a K lein-Gordon-like evolution equation.\nLemma 2.5 (Expansion of the Klein-Gordon density) .For all|k|< ν0and allℓ∈N, we have\nˆρ±\nKG(t,k) =ℓ/summationdisplay\nj=0ep±(k)tA±\nj(k)/bracketleftBig\nk⊗j:∇j\nη/hatwiderhin(k,0)/bracketrightBig\n−ℓ/summationdisplay\nj=0A±\nj(k)/bracketleftBig\nk⊗j:∇j\nη/hatwiderhin(k,kt)/bracketrightBig\n+/integraldisplayt\n0RKG(t−τ,k)A±\nℓ+1(k)/bracketleftBig\nk⊗(ℓ+1):∇ℓ+1\nη/hatwiderhin(k,kτ)/bracketrightBig\ndτ (2.15)\nwhere∇η/hatwiderhin(k,η)is the differential in the second Fourier variable, and with t he notation\nA±\nj(k) :=−J±(k)\np±(k)j+1andJ±(k) :=−1\n∂zL(p±(k),k)).\nRemark 9. Note that the Fourier multipliers A±\njare smooth and bounded for |k|< ν0.\nProof.Integrating by parts in time gives\nˆρ±\nKG(t,k) =/integraldisplayt\n0R±\nKG(t−τ,k)/hatwiderhin(k,kτ)dτ\n=/integraldisplayt\n0J±(k)ep±(k)(t−τ)/hatwiderhin(k,kτ)dτ=−/integraldisplayt\n0J±(k)\np±(k)∂τ/parenleftBig\ne−λ(k)(t−τ)±iΩ(k)(t−τ)/parenrightBig\n/hatwiderhin(k,kτ)dτ\n=J±(k)\np±(k)H(k,kt)−J±(k)\np±(k)ep±(k)t/hatwiderhin(k,0)+/integraldisplayt\n0J±(k)\np±(k)ep±(k)(t−τ)/bracketleftBig\nk·∇η/hatwiderhin(k,kτ)/bracketrightBig\ndτ,\nand iterating finitely many times yields the result.\nNote that by symmetry J+(k) =J−(k) andA−\nj(k) =A+\nj(k), and the calculations of Lemma 2.4\ngive the expansion J±(k) =∓ωp\n2i+O(|k|2) which allows to expand the coefficients A±\nj(k) in (2.15).\nDenoting p±(k) =−λ(k)±iΩ(k) withλ(k)>0 and Ω( k) =ωp+O(|k|2), it immediately implies:\nLemma 2.6 (Klein-Gordon coefficients at low frequencies) .One has as k→0,\nA+\n0(k)+A−\n0(k) = 1+O(|k|2) (2.16)\nA+\n1(k)+A−\n1(k) =O(|k|2)\nep+(k)tA+\n0(k)+ep−(k)tA−\n0(k) =e−λ(k)t/bracketleftBig\ncos[Ω(k)t]+O(|k|2)eiΩ(k)t+O(|k|2)e−iΩ(k)t/bracketrightBig\nep+(k)tA+\n1(k)+ep−(k)tA−\n1(k) =e−λ(k)t/bracketleftbiggsin[Ω(k)t]\nΩ(k)+O(|k|2)eiΩ(k)t+O(|k|2)e−iΩ(k)t/bracketrightbigg\n,\nwhere the O(|k|2)in the above represent infinitely differentiable, bounded fu nctions of kwhich are\nindependent of time.\n15Remark 10. Note crucially that (2.16) cancelsthe leading order of the free transport evolution\nfor long-waves as k→0, which leads to an improved decay of the Landau damping cont ribution of\nthe electric field for such long-waves.\nWe are now able to precise our decomposition of the electric fi eld.\nDefinition 2.1. We define the variable precision decomposition,\n/hatwideE(1;ℓ)\nLD(t,k) =w0ik\n|k|2/bracketleftbig\n1−A+\n0(k)−A−\n0(k)/bracketrightbig/hatwiderhin(k,kt)−w0ik\n|k|2ℓ/summationdisplay\nj=1A±\nj(k)/parenleftBig\nk⊗j:∇j\nη/hatwiderhin(k,kt)/parenrightBig\n/hatwideE(2)\nLD(t,k) =w0ik\n|k|2/integraldisplayt\n0RRFT(t−τ,k)/hatwiderhin(k,kτ)dτ,\n/hatwideE(1;ℓ)\nKG(t,k) =w0ik\n|k|2ℓ/summationdisplay\nj=0ep±(k)tA±\nj(k)/parenleftBig\nk⊗j:∇j\nη/hatwiderhin(k,0)/parenrightBig\n/hatwideE(2;ℓ)\nKG(t,k) =w0ik\n|k|2/integraldisplayt\n0RKG(t−τ,k)A±\nℓ+1(k)/bracketleftBig\nk⊗(ℓ+1):∇ℓ+1\nη/hatwiderhin(k,kτ)/bracketrightBig\ndτ,\nand accordingly define the particular decomposition we shal l use in the sequel (setting ℓ= 4)\nE(1)\nLD:=E(1;4)\nLD, E(2)\nLD:=as above\nE(1)\nKG:=E(1;4)\nKG, E(2)\nKG:=E(2;4)\nKG,\nELD:=E(1)\nLD+E(2)\nLD, EKG:=E(1)\nKG+E(2)\nKG.\nNext, we estimate the ‘remainder free transport part’ of the resolvent. The gain in powers of k\npresent in 2.7 is critical to the high quality decay rate of th e Landau damping electric field.\nLemma 2.7 (Remainder free transport resolvent at low frequencies) .There exists λ0>0such that\nfor all|k|< ν0there holds,\n∀ |k|< ν0,|RRFT(t,k)|/lessorsimilar|k|3e−λ0|k|t.\nProof.We add and subtract by the expected leading order behavior as k→0 (using that the\nintegration path is away from z= 0), hence define for α:= 3Tω2\npm−1\ne,\nQ(z) =ω2\np\nz2+ω2p+α|k|2\n(z2+ω2p)2.\nThe function z/ma√sto→eztQ(z,k) is holomorphic and decaying in the lefthalf-plane ℜz <−γ′|k|, and\nhence we deduce, deforming the contour suitably,\nRRFT(t,k) =1\n2iπ/parenleftbigg/integraldisplay\nΓ0+/integraldisplay\nΓ++/integraldisplay\nΓ−/parenrightbigg\nezt/parenleftbiggL(z,k)\n1−L(z,k)+Q(z,k)/parenrightbigg\ndz\n=:R0\nRFT+R+\nRFT+R−\nRFTwith\nΓ0=/braceleftBig\nz=λ+iω:λ=−δ|k|,ω∈(−R|k|,R|k|)/bracerightBig\nΓ+=/braceleftBig\nz=λ−i(1+δ)λ+i/bracketleftbig\nR−(1+δ)δ/bracketrightbig\n|k|:λ∈(−∞,−δ|k|]/bracerightBig\nΓ−=/braceleftBig\nz=λ+i(1+δ)λ−i/bracketleftbig\nR−(1+δ)δ/bracketrightbig\n|k|:λ∈(−∞,−δ|k|]/bracerightBig\n.\n16Γ0Γ+\nΓ−R|k|\n−R|k|−δ|k|∂Λδ,δ′\nFigure 4: The contour of integration (note that it is slightl y modified as compared to ∂Λδ,δ′)\nWe separate cases as in Lemma 2.3. Consider first z∈Γ0with|ℑz|< δ′|k|. As in the proof of\nLemma 2.3, in this region, there holds the following expansi on, valid for δ′sufficiently small,\n∀z∈Γ0with|ℑz|< δ′|k|,L(ℑz,k) =−ω2\np\n|k|2+O/parenleftbigg|ℑz|\n|k|3/parenrightbigg\n,\nand for all |z|/lessorsimilar|k|,\n|L(z,k)|/lessorsimilar|k|−2,|∂zL(z,k)|/lessorsimilar|k|−3.\nTherefore, for δandδ′sufficiently small, we have L(z,k)≈ |k|−2forz∈Γ0with|ℑz|< δ′|k|. Next\nconsider the case z∈Γ0withδ′|k| ≤ |ℑz| ≤R|k|. By similar arguments as above and in Lemma\n2.3, we have |1−L(z,k)|/greaterorsimilar|k|−2and|L(z,k)|/lessorsimilar|k|−2. Therefore, on Γ 0, the integrand O(|k|2),\nresulting in the estimate\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΓ0ezt/parenleftBigg\nz2L(z,k)+ω2\np\n(1−L(z,k))(z2+ω2p)+α|k|2\n(z2+ω2p)2/parenrightBigg\ndz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar|k|3e−δ|k|t.\nThis completes the estimates on Γ 0.\nWe next turn to Γ +,Γ−. We need only consider Γ +, Γ−is analogous. We use now |z| ≫ |k|and\nthe decomposition from Lemma 2.2:\nL(z,k) =−ω2\np\nz2−3ω2\npT|k|2\nmez4+|k|4\nz6ζ(z,k)\nwithζ(z,k) uniformly bounded on Γ +and decaying at infinity. Using this expansion we have\nL(z,k)\n1−L(z,k)+Q(z,k) =z2L(z,k)+ω2\np\n(1−L)(z2+ω2p+α|k|2\n(z2−ω2p)(z2+ω2p)\n17=αk2(z2L−ω2\np\n(z2−z2L)(z2+ω2p)2+ζ(z,k)k4\nz2(z2−z2L)(z2+ω2p)\n=αk2(αk2+ζ(z,k)k4z−2\n(z2−z2L)(z2+ω2p)2+ζ(z,k)k4\nz2(z2−z2L)(z2+ω2p).\nUsing the uniform boundedness of ζ(z,k) we integration then gives\n/vextendsingle/vextendsingleR+\nRFT/vextendsingle/vextendsingle/lessorsimilare−δ|k|t/integraldisplay∞\nR|k||k|4\nx2dx=|k|3e−δ|k|t.\nThis completes the proof of Lemma 2.7.\nWe finally estimate the whole resolvent at frequencies bound edaway from zero, which is simpler.\nLemma 2.8 (Non-small frequencies resolvent estimate) .Given any ν0>0there isλ1>0such\nthat\n∀|k| ≥ν0\n2,R(t,k)/lessorsimilar1\n|k|e−λ1|k|t.\nProof of Lemma 2.8. Chooseλ >0 as in Lemma 2.1 and deform the contour to get (there are no\npoles inℜz≥ −λ|k|when|k| ≥ν0/2)\nR(t,k) =1\n2iπ/integraldisplay−λ|k|+i∞\n−λ|k|−i∞eztL(z,k)\n1−L(z,k)dz\nand using the estimate of Lemma 2.1 gives\n|R(t,k)|/lessorsimilare−λ|k|t/integraldisplay∞\n−∞1\n|k|2+|ω|2dω/lessorsimilar1\n|k|e−λ|k|t.\n2.5 Another look at the long wave hydrodynamic behavior\nIn this subsection we prove Theorem 3 on the basis of the decom position of Definition 2.1 and the\nprevious estimates of this section.\nProof of Theorem 3 .We prove the first part of the theorem, as the second part is see n directly\nfrom the decomposition 2.1, together with some basic estima tes that are similar or easier than what\nis required to prove the first part. For any field F, define\nFǫ(t,x) :=1\nǫ3F/parenleftBig\nt,x\nǫ/parenrightBig\n,with Fourier transform /hatwiderFǫ(t,k) =/hatwideF(t,ǫk).\nNote that we have defined hinsuch that for all ǫ >0,hin,ǫ(x,v) =H0(x,v). Define the initial\nmacroscopic flux (recall that n0is the total mass of f0(v))\njin(x) :=1\nn0/integraldisplay\nR3vhindvwith Fourier transform /hatwiderjin(k) =1\nn0i∇η/hatwiderhin(k,0).\nDenote,\n/hatwideJ(k) =1\nn0i∇η/hatwiderH0(k).\n18The rescaled electric field Eǫin the linearized Euler-Poisson system (1.8) satisfies\nǫ/hatwideEǫ(t,k) =ik\n|k|2/hatwiderH0(k)cosΩ KG(ǫk)t+ǫω2\npk\n|k|2/parenleftBig\nk·/hatwideJ(k)/parenrightBigsinΩKG(ǫk)t\nΩKG(ǫk),\nwhere we define the exact Euler-Poisson imaginary phase Ω KG(k)\nΩKG(k) :=/radicalbigg\nω2p+3T\nm|k|2.\nConsider E(1)\nKGfirst defined in the decomposition of Definition 2.1, since it i s the contribution which\nis asymptotic to Eǫ. Lemma 2.6 shows that\nǫ/hatwideE(1)\nKG,ǫ(t,k) =e−λ(ǫk)tik\n|k|2/hatwiderH0(k)cos[Ω(ǫk)t]+e−λ(ǫk)tǫω2\npik\n|k|2/parenleftBig\nk·/hatwideJ(k)/parenrightBigsin[Ω(ǫk)t]\nΩ(ǫk)\n+O(ǫ|ǫk|2)e−λ(ǫk)t/ba∇dblH0/ba∇dblW0,1\n4.\nNote that the errors depend on time, but in a uniformly bounde d way, and that they depend on\nv-moments of hinup to order 5. By the expansion of Ω in Lemma 2.4, we have\n|Ω(ǫk)−ΩKG(ǫk)|/lessorsimilar/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalbigg\nω2p+3T\nm|ǫk|2+O(|ǫk|4)−/radicalbigg\nω2p+3T\nm|ǫk|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar|ǫk|4\nand thus/vextendsingle/vextendsingle/vextendsinglecos[Ω(ǫk)t]−cos[ΩKG(ǫk)t]/vextendsingle/vextendsingle/vextendsingle=O/parenleftBig\n|ǫk|4t/parenrightBig\n,/vextendsingle/vextendsingle/vextendsinglesin[Ω(ǫk)t]−sin[ΩKG(ǫk)t]/vextendsingle/vextendsingle/vextendsingle=O/parenleftBig\n|ǫk|4t/parenrightBig\n,\nand any N >0,/vextendsingle/vextendsingle1−e−λ(ǫk)t/vextendsingle/vextendsingle=O/parenleftBig\n|ǫk|Nt/parenrightBig\n, and therefore we deduce,\nǫ/vextendsingle/vextendsingle/vextendsingle/hatwideE(1)\nKG,ǫ−/hatwideEǫ/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftBig\n|ǫk|4t+ǫ|ǫk|2/parenrightBig\n/ba∇dblH0/ba∇dblW0,1\n4.\nand thus\nǫ/vextenddouble/vextenddouble/vextenddoubleE(1)\nKG,ǫ(t)−Eǫ(t)/vextenddouble/vextenddouble/vextenddouble\nH−s/lessorsimilar/ba∇dblH0/ba∇dblW0,1\n4/parenleftBigg/integraldisplay\n|ǫk|/lessorsimilar1ǫ2|ǫk|4+|ǫk|8t2\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2sdk/parenrightBigg1\n2\n/lessorsimilar/ba∇dblH0/ba∇dblW0,1\n4ǫs−3\n2−0/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht\nfors∈(3\n2,7\n2). Turn next to /hatwideE(2)\nKG; from its definition\nǫ/vextenddouble/vextenddouble/vextenddoubleE(2)\nKG,ǫ(t)/vextenddouble/vextenddouble/vextenddouble\nH−s/lessorsimilarǫ/integraldisplayt\n0/parenleftBigg/integraldisplay\n|ǫk|/lessorsimilar11\n|ǫk|2/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2s|ǫk|10/vextendsingle/vextendsingle/vextendsingle∇5\nη/hatwiderH0(k,kτ)/vextendsingle/vextendsingle/vextendsingle2\ndk/parenrightBigg1/2\ndτ/lessorsimilar/ba∇dblH0/ba∇dblW0,1\n4ǫs−1\n2−0,\nfors∈(3\n2,19\n2). This completes the treatment of the ‘Klein-Gordon parts’ of the electric field.\nWe turn next to the Landau damping contributions, which in fa ct dominate the error. It is\nconvenient to subdivide the Landau damping field as in Definit ion 2.1. The contribution of E(1)\nLDis\nstraightforward, indeed,\nǫ/vextenddouble/vextenddouble/vextenddoubleE(1)\nLD,ǫ(t)/vextenddouble/vextenddouble/vextenddouble\nH−s/lessorsimilarǫ\n4/summationdisplay\nj=0/integraldisplay\nR3/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht−2s|ǫk|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ǫk|j/hatwide∇j\nηH0(k,kt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndk\n1\n2\n/lessorsimilar/ba∇dblH0/ba∇dblW2,1\n4ǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht,\nfors >5\n2. Turn finally to E(2)\nLD, which produces the dominant error. Then,\nǫ/vextenddouble/vextenddouble/vextenddoubleE(2)\nLD,ǫ(t)/vextenddouble/vextenddouble/vextenddouble\nH−s/lessorsimilarǫ/integraldisplayt\n0/parenleftBigg/integraldisplay\nR3min(|ǫk|2,|ǫk|−1)\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2se−λ0|ǫk|(t−τ)/vextendsingle/vextendsingle/vextendsingle/hatwideH0(k,kτ)/vextendsingle/vextendsingle/vextendsingle2\ndk/parenrightBigg1/2\ndτ/lessorsimilarǫ2/ba∇dblH0/ba∇dblW2,1\n0,\nfors >5\n2. This completes the proof of Theorem 3.\n193 Electric field estimates\n3.1 Landau damping estimates on the electric field\nIn this section we provide estimates for ELD. We start with the optimal decay estimates for the\ndensity for the kinetic free transport (optimal in terms of t ime decay, not in the dependence on the\ninitial data). Denote the spatial density of the solution to the free transport equation\nH(t,x) :=/integraldisplay\nR3hin(x−tv,v)dvwith Fourier transform /hatwideH(t,k) =/hatwiderhin(k,kt).\nLemma 3.1. For allσ≥0,\n/ba∇dbl/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσH(t,·)/ba∇dblL1x(3.1)\n/lessorsimilar/ba∇dblhin/ba∇dblWσ,1\n0\n/ba∇dbl/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσH(t,·)/ba∇dblL2x\n/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3/2/ba∇dblhin/ba∇dblWσ+3/2+0,1\n0\n/ba∇dbl/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσH(t,·)/ba∇dblL∞x/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3/ba∇dblhin/ba∇dblWσ+3+0,1\n0. (3.2)\nMore generally, for all 1≤p≤ ∞,j≥0,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσ/parenleftbigg\n∇⊗j\nx:/integraldisplay\nR3v⊗jhin(·−tv,v)dv/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLp\nx/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−j−3/parenleftBig\n1−1\np/parenrightBig\n/ba∇dblhin/ba∇dblWσ+3+j+0,1\nj.(3.3)\nProof.The inequality (3.1) is clear. To see (3.2) note that\n/vextenddouble/vextenddouble∇j\nxH(t,·)/vextenddouble/vextenddouble\nL∞x/lessorsimilar/integraldisplay\nR3|k|j/vextendsingle/vextendsingle/vextendsingle/hatwideH(t,k)/vextendsingle/vextendsingle/vextendsingledk/lessorsimilar/parenleftBigg/integraldisplay\nR3|k|j\n/a\\}b∇acketle{tk,kt/a\\}b∇acket∇i}ht3+j+dk/parenrightBigg\n/ba∇dblhin/ba∇dblWj+3+,1\n0/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3−σ/ba∇dblhin/ba∇dblWj+3+,1\n0.\nThe proof of (3.3) follows similarly (using also interpolat ion).\nNext, we turn to estimates on the damped part of the electric fi eld. By Lemmas 2.6 and 3.1,\nE(1)\nLDsatisfies the estimates claimed in Theorem 1.\nTurn next to obtaining estimates on E(2)\nLD.\nLemma 3.2. There holds the following estimates\n/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσE(2)\nLD(t)/vextenddouble/vextenddouble/vextenddouble\nL2x/lessorsimilar1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht5/2/ba∇dblhin/ba∇dblWσ+2+0,1\n0(3.4)\n/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσE(2)\nLD(t)/vextenddouble/vextenddouble/vextenddouble\nL∞x/lessorsimilar1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4/ba∇dblhin/ba∇dblWσ+3+0,1\n0. (3.5)\nProof.Consider first the low spatial frequencies |k|/lessorsimilarν0. Compute using Lemma 2.7, for any a >0,\n/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσE(2)\nLD(t)/vextenddouble/vextenddouble/vextenddouble\nL∞x≤/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∇x,t∇x/a\\}b∇acket∇i}htσ|∇x|−1RRFT(t−τ)∗xH(τ,·)/vextenddouble/vextenddouble/vextenddouble\nL∞xdτ\n/lessorsimilar/integraldisplayt\n0/integraldisplay\nR3/a\\}b∇acketle{tk,(t−τ)k/a\\}b∇acket∇i}htσ|k|−1|RRFT(t−τ,k)|/a\\}b∇acketle{tk,τk/a\\}b∇acket∇i}htσ/vextendsingle/vextendsingle/vextendsingle/hatwiderhin(k,τk)/vextendsingle/vextendsingle/vextendsingledkdτ\n/lessorsimilar/bracketleftbigg/integraldisplayt\n0|k|2/parenleftbigg/integraldisplay\nR3/a\\}b∇acketle{tτk/a\\}b∇acket∇i}ht−3−a/a\\}b∇acketle{t(t−τ)k/a\\}b∇acket∇i}ht−3−adk/parenrightbigg\ndτ/bracketrightbigg\n/ba∇dblhin/ba∇dblWσ+3+a,1\n0.\n20We split the integral\n/integraldisplayt\n0/parenleftbigg/integraldisplay\nR3|k|2/a\\}b∇acketle{tτk/a\\}b∇acket∇i}ht−3−a/a\\}b∇acketle{t(t−τ)k/a\\}b∇acket∇i}ht−3−adk/parenrightbigg\ndτ=/parenleftBigg/integraldisplayt\n2\n0+/integraldisplayt\nt\n2/parenrightBigg/parenleftbigg/integraldisplay\nR3|k|2/a\\}b∇acketle{tτk/a\\}b∇acket∇i}ht−3−a/a\\}b∇acketle{t(t−τ)k/a\\}b∇acket∇i}ht−3−adk/parenrightbigg\ndτ,\nand change variables k′=τkork′= (t−τ)kin each one to obtain\n/integraldisplayt\n0/parenleftbigg/integraldisplay\nR3|k|2/a\\}b∇acketle{tτk/a\\}b∇acket∇i}ht−3−a/a\\}b∇acketle{t(t−τ)k/a\\}b∇acket∇i}ht−3−adk/parenrightbigg\ndτ/lessorsimilart−4.\nNote that the |k|2in the numerator is crucial for obtaining this sharp rate. wh ich concludes the\nproof of (3.5) at low frequencies. At frequencies bounded aw ay from zero, the estimate is similar\nthough more straightforward and with less loss of regularit y. TheL2case (3.4) follows similarly\nand is omitted for brevity.\n3.2 Dispersive estimates of the electric field\nWe now consider the ‘Klein-Gordon part’ of the electric field in Definition 2.1.\nConsider first the following useful dispersive estimates:\nLemma 3.3 (Dispersive estimates for weakly damped poles) .We have 1≤p≤2there holds for\nt≥0andf=f(x)∈Lp(R3\nx),\n/vextenddouble/vextenddouble/vextenddoubleep±(∇)tP≤ν0f/vextenddouble/vextenddouble/vextenddouble\nLp′/lessorsimilart−3/parenleftBig\n1\np−1\n2/parenrightBig\n/ba∇dblf/ba∇dblLp. (3.6)\nRemark 11. The linear propagator ep±(∇)tis not a unitary operator, and the standard TT∗argu-\nment as in e.g. [24, 36] do not apply. Hence, it is not as trivia l to obtain the homogeneous Strichartz\nestimates or the full range of expected inhomogeneous Stric hartz estimates (although some inho-\nmogeneous Strichartz estimates follow immediately from (3 .6) and the O’Neil Young convolution\ninequality [30]).\nProof.The case p= 2 is immediate. For the case p= 1, write\nep±(∇)tP≤ν0f=K(t)∗xf,\nwith the integral kernel\nK(t,x) :=1\n(2π)d/2/integraldisplay\nR3eix·k+itΩ(k)−λ(k)taν0(k)dk,\nwhereaν0is a Schwartz class function compactly supported in a ball of radius≤2ν0corresponding\nto the Littlewood-Paley projection. Hence the p= 1 case follows from the pointwise kernel estimate\n/ba∇dblK(t)/ba∇dblL∞/lessorsimilart−d/2(∀t≥0). (3.7)\nThe intermediate exponents p∈(1,2) are then obtained by the Riesz-Thorin interpolation theo rem.\nLet us prove (3.7). Despite the complex phase, Kis essentially a standard oscillatory integral,\nand we may easily adapt the standard arguments as in e.g. [Pro position 6, pg 344 [34]]. Let us first\nexplain the argument assuming that Ω( k) =|k|2(the case Ω( k) =ω2\np+3T\nme|k|2is the same). In this\ncase, we make the change of variables y=k+x\n2tand we write (for some scale ǫchosen below),\nK(t,x) =ei|x|2/4t2\n(2π)d/2/integraldisplay\nR3eit|y|2−λtχ/parenleftBigy\nǫ/parenrightBig\naν0/parenleftBig\ny−x\n2t/parenrightBig\ndy\n21+ei|x|2/4t2\n(2π)d/2/integraldisplay\nR3eit|y|2−λt/bracketleftBig\n1−χ/parenleftBigy\nǫ/parenrightBig/bracketrightBig\naν0/parenleftBig\ny−x\n2t/parenrightBig\ndy\n=:KS(t,x)+KNS(t,x),\nwhereχ∈C∞\nc(B(0,1)) with χ(z) = 1 for |z| ≤1/2, and where we have split Kinto a ’stationary’\npart and a ’non-stationary’ part. For the stationary part KSwe simply bound the integrand and\nestimate the volume of integration:\n|KS(t,x)|/lessorsimilarν0ǫd. (3.8)\nFor the non-stationary part KNSwe integrate by parts using the non-vanishing of the phase. I n\nparticular observe that (note that |·|2=x∗xforx∈Cn),\nt|2iy−∇λ|2eit|y|2−λt= (−2iy−∇λ)·∇yeit|y|2−λt.\nTherefore, if we define the differential operator\nDf:=∇y·/parenleftbigg−2iy−∇λ\n|2iy−∇λ|2f/parenrightbigg\n,\nthen by repeated integration by parts we have,\n|KNS|/lessorsimilar1\ntN/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3eit|y|2−λtDN/parenleftBig\n(1−χ(y\nǫ))aν0(y−x\n2t)/parenrightBig\ndy/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nWe next show that the integrand is bounded by |y|−2N; for the proof in three dimensions, we only\nneedN= 1,2 though it holds for all N. The case N= 1 is easily checked. Indeed,\n∂j/parenleftbigg−2iy−∇λ\n|2iy−∇λ|2/parenrightbigg\n=−−2iej−∂j∇λ\n|2iy−∇λ|2=O(|y|−2),\nwhich is sufficient as the terms in which the derivative lands e lsewhere are only better (estimates\non∇nλare provided by Lemma 2.4). For N= 2 we analogously have\n∂ℓ/parenleftbigg/parenleftbigg−2iy−∇λ\n|2iy−∇λ|2/parenrightbigg−2iej−∂j∇λ\n|2iy−∇λ|2/parenrightbigg\n=−/parenleftbigg−2ieℓ−∂ℓ∇λ\n|2iy−∇λ|2/parenrightbigg/parenleftbigg−2iej−∂j∇λ\n|2iy−∇λ|2/parenrightbigg\n+/parenleftbigg−2iy−∇λ\n|2iy−∇λ|2/parenrightbigg/parenleftbigg∂ℓj∇λ\n|2iy−∇λ|2−(−2ieℓ−∂ℓ∇λ)(2iy−∇λ)·(−2iej−∂j∇λ)\n|2iy−∇λ|4/parenrightbigg\n=O(|y|−4),\nwhich is similarly sufficient (note the pattern that selects a particular dominant term whereas the\nmore complicated error terms are smaller, hence the desired estimates hold for all N). Therefore,\nprovided we choose N≥2,\n|KNS|/lessorsimilar1\ntN/integraldisplay\n|y|≥ǫ1\n|y|2Ndy/lessorsimilar1\ntNǫd−2N.\nHence making the choice ǫ∼t−1/2gives the result when combined with (3.8). The case using\nthe true Ω( k) follows by Morse’s lemma due to (2.10) and the other estimat es in Lemma 2.4 (see\n[Proposition 6, pg 344 [34]] for more details). This complet es the main dispersive estimates (3.6).\n22The estimates on E(1)\nKGin Theorem 1 follow immediately from (3.6) and the decomposi tion 2.1\nupon observing that for/integraltext\nR3ρindx= 0 we have for all 1 < p,\n/vextenddouble/vextenddouble∇x(−∆x)−1P≤ν0ρin/vextenddouble/vextenddouble\nLp/lessorsimilarp,ν0/ba∇dbl/a\\}b∇acketle{tx/a\\}b∇acket∇i}htρin/ba∇dblW0,1\n0.\nNext, we will prove the pointwise-in-time decay estimates o nE(2)\nKG.\nLemma 3.4. There holds for all 2≤p≤ ∞,\n/vextenddouble/vextenddouble/vextenddoubleE(2)\nKG/vextenddouble/vextenddouble/vextenddouble\nLp\nx/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−3/parenleftBig\n1\n2−1\np/parenrightBig\n/ba∇dblhin/ba∇dblW3/2,1\n5.\nProof.DefineTas above:\nT(t,x) =/integraldisplay\nR3v⊗5hin(x−tv,v)dv,/hatwideT(t,k) =i5∇⊗5\nη/hatwiderhin(k,0).\nBy Minkowski’s inequality (and boundednessof the prefacto r multipliers on Lpas they are Schwartz\nclass functions),\n/vextenddouble/vextenddouble/vextenddoubleE(2)\nKG/vextenddouble/vextenddouble/vextenddouble\nLp\nx/lessorsimilar/summationdisplay\n+,−/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddoubleep±(∇x)(t−τ)P≤ν0|∇x|−1∇5\nx:T/vextenddouble/vextenddouble/vextenddouble\nLp\nxdτ.\nOn the one hand, by Bernstein’s inequality, for any p≥r′>2, it follows from (3.6) that we have\n/vextenddouble/vextenddouble/vextenddoubleep±(∇x)(t−τ)|∇x|−1/parenleftbig\n∇⊗5\nx:T/parenrightbig/vextenddouble/vextenddouble/vextenddouble\nLp\nx/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleep±(∇)(t−τ)P≤ν0|∇x|−1/parenleftbig\n∇⊗5\nx:T/parenrightbig/vextenddouble/vextenddouble/vextenddouble\nLr′\nx\n/lessorsimilar1\n|t−τ|3(1\nr−1\n2)/vextenddouble/vextenddouble/vextenddoubleP≤ν0|∇x|−1/parenleftbig\n∇⊗5\nxT/parenrightbig/vextenddouble/vextenddouble/vextenddouble\nLrx.\nOn the the other hand, we similarly have\n/vextenddouble/vextenddouble/vextenddoubleep±(∇x)(t−τ)|∇x|−1/parenleftbig\n∇⊗5\nx:T/parenrightbig/vextenddouble/vextenddouble/vextenddouble\nLp\nx/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleP≤ν0|∇x|−1/parenleftbig\n∇5\nx:T/parenrightbig/vextenddouble/vextenddouble/vextenddouble\nL2x.\nTherefore, by Lemma 3.1, we have for r >1,\n/vextenddouble/vextenddouble/vextenddoubleE(2)\nKG/vextenddouble/vextenddouble/vextenddouble\nLp\nx/lessorsimilar/ba∇dblhin/ba∇dblW3/2,1\n5/summationdisplay\n+,−/integraldisplayt\n0min/parenleftBig\n/a\\}b∇acketle{tτ/a\\}b∇acket∇i}ht−3/2−3/2,|t−τ|−3(1\nr−1\n2)/a\\}b∇acketle{tτ/a\\}b∇acket∇i}ht−3(1−1\nr)−3\n2/parenrightBig\ndτ,\nwhich integrates to imply the stated result\n4 Decomposition and scattering for the distribution functi on\nIn this section we prove Theorem 2. Denote the solution in the free transport moving frame\ng(t,x,v) :=h(t,x+vt,v),\nwhich satisfies ∂tg=−E(t,x+tv)·∇vf0(v). Therefore we have on the Fourier side (note /hatwideh(t,k,η) =\n/hatwideg(t,k,η+kt)),\n/hatwideg(t,k,η) =/hatwiderhin(k,η)−/integraldisplayt\n0/hatwideE(τ,k)·/hatwide∇vf0(η−kτ)dτ.\n23Consider the contribution E(1)\nKG:\n/integraldisplayt\n0/hatwideE(1)\nKG(τ,k)·/hatwide∇vf0(η−kτ)dτ\n=w0ik\n|k|2ℓ/summationdisplay\nj=0/summationdisplay\n+,−/integraldisplayt\n0ep±(k)τA±\nj(k)/parenleftBig\nk⊗j:∇j\nη/hatwiderhin(k,0)/parenrightBig\n·/hatwide∇vf0(η−kτ)dτ\n=w0ik\n|k|2ℓ/summationdisplay\nj=0/summationdisplay\n+,−ep±(k)tA±\nj(k)\np±(k)/parenleftBig\nk⊗j:∇j\nη/hatwiderhin(k,0)/parenrightBig\n·/hatwide∇vf0(η−kt)\n−w0ik\n|k|2ℓ/summationdisplay\nj=0/summationdisplay\n+,−A±\nj(k)\np±(k)/parenleftBig\nk⊗j:∇j\nη/hatwiderhin(k,0)/parenrightBig\n·/hatwide∇vf0(η)\n−w0ik\n|k|2ℓ/summationdisplay\nj=0/integraldisplayt\n0ep±(k)τA±\nj(k)\np±(k)/parenleftBig\nk⊗j:∇j\nη/hatwiderhin(k,0)/parenrightBig\n·∇η/hatwide∇vf0(η−kτ)k dτ\n=:/hatwidestgKG+/hatwideg2+/hatwideg3,\nwithhKG,h2,andh3defined analogously. Note that since /hatwidesthKG(t,k,η) =/hatwidestgKG(t,k,η+kt),\n/hatwidesthKG(t,k,η) =˜EKG(t,k)·/hatwide∇vf0(η),\nwhere we define\n˜EKG(t,k) :=w0ik\n|k|24/summationdisplay\nj=0/summationdisplay\n+,−ep±(k)tA±\nj(k)\np±(k)/parenleftBig\nk⊗j:∇j\nη/hatwiderhin(k,0)/parenrightBig\n.\nArguing as for the E(1)\nKGestimates in Theorem 1, hKGsatisfies (1.7). Similarly, we define\n/hatwidesthLD(k,η) =/hatwiderhin(k,η)+/hatwideg2(k,η)+/hatwideg3(t,k,η)−/integraldisplayt\n0/parenleftbigg\n/hatwideEKG(2)+/hatwidestELD/parenrightbigg\n(τ,k)·/hatwide∇vf0(η−kτ)dτ.(4.1)\nThe term g2is constant in time and in Lpfor allp≥2 by the assumptions on the initial data. The\ntermg3on the physical side is written in the form\nh3(t,x,v) =/integraldisplayt\n0∇x˜E3(τ,x+τv) :/parenleftbig\nv⊗∇vf0(v)/parenrightbig\ndτ,\nfor a suitable ˜E3. By straightforward variations of the arguments used to est imateE(1)\nKGabove, we\nhave for any 6 < p,\n/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddouble∇x˜E3(τ,x+τv) :/parenleftbig\nv⊗∇vf0(v)/parenrightbig/vextenddouble/vextenddouble/vextenddouble\nLp\nx,vdτ/lessorsimilar/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddouble∇x˜E3(τ,·)/vextenddouble/vextenddouble/vextenddouble\nLp\nxdτ\n/lessorsimilar/ba∇dblhin/ba∇dblW3+0,1\n5/integraldisplayt\n0/a\\}b∇acketle{tτ/a\\}b∇acket∇i}ht−3/parenleftBig\n1\n2−1\np/parenrightBig\ndτ,\nand hence h3converges in Lp\nx,vfor allp >6 ast→ ∞. Due to the decay estimates in Lemma 3.3,\nthe term in (4.1) involving E(2)\nKGsimilarly converges in Lp\nx,vfor allp >6.\n24Next we prove that hLDconverges in Lp\nx,vforp >6. The easiest contribution is from ELD.\nRecall the decomposition ELD=E(1)\nLD+E(2)\nLDfrom Subsection 3.1. From a straightforward variant\nof Lemma 3.1, we have for all 2 ≤p≤ ∞\n/ba∇dblELD(t)/ba∇dblLp/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−4+3\np/ba∇dblhin/ba∇dblW4+0,1\n0,\nand hence, for p≥2,\n/integraldisplayt\n0/vextenddouble/vextenddoubleELD(τ,x+τv)·∇vf0(v)/vextenddouble/vextenddouble\nLp\nx,vdτ≤/integraldisplayt\n0/ba∇dblELD(τ)/ba∇dblLp\nx/vextenddouble/vextenddouble∇vf0/vextenddouble/vextenddouble\nLp\nvdτ/lessorsimilar/ba∇dblhin/ba∇dblW4+0,1\n0,\nand so the corresponding contribution converges as t→ ∞.\nReferences\n[1] C. Bardos and P. Degond. Global existence for the Vlasov- Poisson equation in 3 space variables\nwith small initial data. In Annales de l’IHP Anal. non-lin. , volume 2, pages 101–118, 1985.\n[2] J. Bedrossian. Nonlinear echoes and Landau damping with insufficient regularity. To appear:\nTunis. J. of Math. , 2016.\n[3] J. Bedrossian. Suppression of plasma echoes and landau d amping in Sobolev spaces by weak\ncollisions in a Vlasov-Fokker-Planck equation. 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Duke Mathematical Journal , 30(1):129–142,\n1963.\n[31] O. Penrose. Electrostatic instability of a uniform non -Maxwellian plasma. Phys. Fluids , 3:258–\n265, 1960.\n26[32] D. Ryutov. Landau damping: half a century with the great discovery. Plasma physics and\ncontrolled fusion , 41(3A):A1, 1999.\n[33] P. Snyder, G. Hammett, and W. Dorland. Landau fluid model s of collisionless magnetohydro-\ndynamics. Physics of Plasmas , 4(11):3974–3985, 1997.\n[34] E. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality , and Oscillatory Integrals .\nPrinceton University Press, 1993.\n[35] D. G. Swanson. Plasma Waves . Elsevier, 2012.\n[36] T. Tao. Nonlinear dispersive equations. CBMS Regional Conference Series in Mathematics,\n106, 2006.\n[37] N. van Kampen. On the theory of stationary waves in plasm as.Physica, 21:949–963, 1955.\n[38] A. Vlasov. On the kinetic theory of an assembly of partic les with collective interaction. Russ.\nPhys. J., 9:25–40, 1945.\n[39] A. A. Vlasov. The vibrational properties of an electron gas.Zh. Eksp. Teor. Fiz. , 291(8), 1938.\nIn Russian, translation in English in Soviet Physics Uspekhi , vol. 93 Nos. 3 and 4, 1968. Avail-\nable athttps://books.google.co.uk/books?id=ZRX0F8nynRAC&pr intsec=frontcover .\n27"    },    {        "title": "2008.02728v1.Quantum_sensing_of_open_systems__Estimation_of_damping_constants_and_temperature.pdf",        "content": "Quantum sensing of open systems: Estimation of damping constants and temperature\nJ. Wang,1L. Davidovich,1, 2, 3and G. S. Agarwal1, 4\n1Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA\n2Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-972, Brazil\n3Hagler Institute for Advanced Study and Institute for Quantum Science and Engineering,\nTexas A&M University, College Station, Texas 77843, USA\n4Institute for Quantum Science and Engineering and Department of Biological and Agricultural Engineering,\nTexas A&M University, College Station, Texas 77843, USA\nWe determine quantum precision limits for estimation of damping constants and temperature of\nlossybosonicchannels. Adirectapplicationwouldbetheuseoflightforestimationoftheabsorption\nandthetemperatureofatransparentslab. Analyticlowerboundsareobtainedfortheuncertaintyin\nthe estimation, through a purification procedure that replaces the master equation description by a\nunitary evolution involving the system and ad hoc environments. For zero temperature, Fock states\nare shown to lead to the minimal uncertainty in the estimation of damping, with boson-counting\nbeing the best measurement procedure. In both damping and temperature estimates, sequential\npre-thermalization measurements, through a stream of single bosons, may lead to huge gain in\nprecision.\nIntroduction. The quest for better precision in the\nestimation of parameters is common to many areas of\nscience, ranging from probing weak electric and mag-\nnetic fields, temperature, pressure, and small rotations\nand displacements, to high-resolution spectroscopy and\nmagnetic resonance, with applications to atomic clocks,\ngeophysics, medicine, and biology. Fundamental limits\nof precision have been established, within the realm of\nclassical physics, by Cramér, Rao, and Fisher [1, 2]. The\nusual procedure involves measuring a probe, prepared in\na convenient initial state, after it has interacted with the\nsystem under investigation, and then obtaining from the\nmeasurement results an estimation of the parameter of\ninterest, through some convenient estimator. Through a\ngeneralization of the classical framework to quantum me-\nchanics [3–6], it has been realized that quantum probes,\nprepared in states with features like squeezing and en-\ntanglement, help to increase the precision of the esti-\nmation, for the same amount of resources (which could\nbe the number of atoms or photons used in the estima-\ntion). This has been relevant, for instance, for extending\nthe coverage of gravitational-wave interferometers, with\nthe use of squeezed light [5, 7] or of entangled states [8],\nfor increasing the magnetic sensitivity with spin squeez-\ning [9], for optimal thermometry [10], for detecting weak\nelectric fields with superpositions of Rydberg states [11],\nfor achieving quantum-enhanced contrast and resolution\nin biological microscopy [12, 13], and for superresolution\nof spatial separation and frequency [14]. Quantum sens-\ning [15, 16] involves the exploration of subtle quantum\neffects to increase the precision of parameter estimation.\nQuantum sensors have become one of the most promising\napplications of quantum technologies [17–19], involving\nsingle- or multi-parameter estimation [20, 21].\nThe unavoidable interaction between these systems\nand their environments may reduce the advantage of us-\ning quantum states, due to the fragility of these resourcesinthepresenceofnoisyprocesses, likedampinganddiffu-\nsion. However, sometimes these processes may yield im-\nportant information on the system. The damping rate of\na particle moving in a medium may allow the estimation\nof the quantum memory time and radiation properties\n[22]. Absorption spectroscopy has a wide range of appli-\ncations, in remote sensing [23], in chemistry and atomic\nphysics [24], in astronomy [25], and in the characteriza-\ntion of materials, not only at the macroscopic level, but\nalsoformicroscopicsystems, likecellsandorganelles[26].\nMoreover, tasks like the precise estimation of phases in\nan interferometer must necessarily include a precise esti-\nmation of photon damping and phase diffusion.\nHere we derive the uncertainties in the estimation of\nboth damping and temperature of a lossy bosonic chan-\nnel, with boson-counting as the measurement procedure.\nThis is of great interest for several areas of science, the\nmost prominent example being the use of light to inves-\ntigate absorption and temperature of samples [27, 28].\nThe precision in the estimation is limited both by the\nuncertainty in the number of bosons in the probe and\nby the noise introduced in the boson distribution by the\nprobed system. This suggests that one should mini-\nmize the variance of the boson-number distribution of\nthe probe, so incoming Fock states should render better\nresults, as opposed to what happens in noiseless phase\nestimation, when the variance should be maximized, for\na given amount of resources (in this case, input photons).\nWe discuss the advantages of using single-boson states\nand boson-counting measurements for damping and tem-\nperature estimation and compare our results within lit-\nerature [29–31]. Sequential pre-thermalization measure-\nments with single-boson streams are shown to lead to a\nhuge increase in the precision. We also obtain analytic\nlowerboundsfortheuncertaintyintheestimationofboth\ndamping and temperature, through a purification proce-\ndure that replaces the master equation description by aarXiv:2008.02728v1  [quant-ph]  6 Aug 20202\nunitary evolution involving the system and ad hoc envi-\nronments. These bounds are shown to be tight in two\nlimiting cases, both involving boson-counting measure-\nments: zero temperature for damping estimation, and\nvacuum input for temperature estimation. For other sit-\nuations, and for the range of parameters here considered,\nthey are very close to the exact numerical solutions.\nThe usual procedure in parameter estimation con-\nsists in obtaining the uncertainty in the parameter,\nfor a given initial state, from the Fisher information\n[1, 2]. For a complete set of measurement results fjg,\non a probe that carries information about the param-\neterXto be estimated, and for unbiased estimators,\nso thathXiequals the true value of the parameter,\nthe standard deviation in the estimation of Xis given\nby the Cramér-Rao expression \u000eX\u00151=p\nNF(hXi),\nwhereF(X)is the Fisher information, given by F(X) =P\nj[1=Pj(X)][dPj(X)=dX]2,Nis the number of repeti-\ntions of the experiment, and Pj(X)is the probability of\ngetting the experimental result jif the value of the pa-\nrameter isX. As shown by Fisher, the lower bound can\nbe reached asymptotically for N!1. The ultimate\nprecision in the estimation of a parameter, for a given\ninitial state, is obtained by maximizing F(X)over all\npossible measurements: this defines the quantum Fisher\ninformation (QFI) FQ(X). In the absence of noise, an-\nalytic expressions can be obtained for the QFI [3, 4].\nFor a parameter-dependent unitary evolution U(X)of\nthe probe,FQ(X)is equal to four times the variance\n(\u0001G)2, calculated in the initial state of the probe, with\nG\u0011i(dUy(X)=dX)U(X)being the generator of U(X).\nHowever, thisisnotsoforopensystems, whichrequire, in\ngeneral, the diagonalization of the parameter-dependent\ndensity matrix of the probe, usually a cumbersome task\nfor high-dimensional systems.\nA general method for obtaining an upper bound for\nthe quantum Fisher information of an open system was\nintroduced in [32]. It consists in purifying the open sys-\ntem, by considering the joint unitary evolution of sys-\ntem+environment. There is an infinite number of purifi-\ncations, which must satisfy the criterion that the reduced\ndescription of the system – obtained by tracing out the\nenvironment – should coincide with the one given by the\nmaster equation. The quantum Fisher information of\nthe purified system should be larger or at least equal to\nthe QFI of the system, since allowing measurements on\nsystem+environment yields no less information on the\nparameter than measuring the system alone. If the envi-\nronment is chosen in such a way that measurements on\nsystem+environment do not give more information than\nmeasurements on the system, the corresponding upper\nbound is tight. In [32], it was shown that this can always\nbeaccomplished. Findingthebestpurificationcouldpro-\nvide therefore an alternative to the involved procedures\nthatdealdirectlywiththeopensystem. Thismethodhas\nled to exact solutions for the estimation of forces acting\nFigure 1. Purification of the master equation for finite tem-\nperature by introducing two environments bandc, initially\nin the vacuum state. The outgoing operators are obtained by\napplying a two-mode squeezing operation and a beam-splitter\ntransformation to the incoming operators. Tracing out modes\nbandcrecovers the master equation (1).\non damped harmonic oscillators [33] and very good ap-\nproximations for the estimation of transition frequencies\nin atomic spectroscopy in the presence of dephasing, and\nphases in optical interferometers, subject to damping [32]\nand diffusion [34]. In the following, this method is ap-\nplied to the estimation of damping and temperature with\nbosonic probes.\nEstimation of damping . Boson damping can be described\nby the master equation\nd\u001a\ndt=\r(nT+ 1)(2a\u001aay\u0000aya\u001a\u0000\u001aaya)\n+\rnT(2ay\u001aa\u0000aay\u001a\u0000\u001aaay);(1)\nwhere\u001ais the density matrix of the bosonic probe, \r\nis the damping constant, nTis the number of thermal\nbosons, and aandayare boson annihilation and creation\noperators, with [a;ay] = 1.\nA possible purification of the corresponding evolution\nwas derived in [33]. This is done by adding two indepen-\ndent environments, which can be represented by a beam-\nsplitter and a two-mode squeezing operation, as shown\nin Fig. 1.\nWehavethen, with BandScorrespondingrespectively\nto the beam-splitter and squeezing transformations:\nj\t(T)i=SBj\t0ij0ibj0ic; (2)\nwhere\nB= exp[\u00121(aby\u0000ayb)];S= exp[\u00122(aycy\u0000ac)];(3)\nand\n\u00121(t) = arccos\u0014r\u0011\nnT(1\u0000\u0011) + 1\u0015\n; (4)\n\u00122(t) = arccoshhp\nnT(1\u0000\u0011) + 1i\n; \u0011=e\u00002\rt:(5)\nThe corresponding operators are transformed as Oout=3\nBySyOinSB, as shown in Fig. 1. We have then [35]\naout= (aincos\u00121\u0000binsin\u00121) cosh\u00122+cy\ninsinh\u00122:(6)\nEquation (2) leads to an upper bound to the QFI of\nthe system. One should note that other purifications\nare possible. Indeed, addition of further unitary opera-\ntions, depending only on the operators bandc, still lead\nto the same master equation. Variational parameters in\nthese additional unitary transformations can be used to\nminimize the corresponding upper bound, so that it gets\ncloser to the QFI of the system [32–34]. Here, however,\nwe adopt the simpler procedure of using the purification\n(2), comparing the corresponding bound with the QFI of\nthe open system.\nForT= 0,\u00122= 0andS= 1, so modecgets decou-\npled from modes aandb, implying that the correspond-\ning master equation is purified with just a beam-splitter\n[29, 31, 35], with transmission coefficient \u0011= exp(\u00002\rt)\nandB= exp\u0002\n\u001e(ayb\u0000aby)\u0003\n;cos2\u001e=\u0011. From the corre-\nsponding generator G(\r) =i[dBy(\r)=d\r]B(\r), one gets\n\u000e\r=\r\u0015\u000e\rmin=\r=\u0000\ne2\rt\u00001\u00011=2\n2\rt\u0016N1=2\nin; (7)\nwhere\u000e\rmin, obtainedfrom G(\r), isalowerboundforthe\nuncertaintyintheestimationof \r, andtistheinteraction\ntime between the bosonic probe and the sample.\nA simple way to estimate the standard deviation \u0001\ris\nto use the error-propagation sensitivity expression \u0001\r=\n\u0001Nout=j@\u0016Nout=@\rj, where (\u0001Nout)2=h(ay\noutaout)2i\u0000\n\u0016N2\noutis the variance in the boson distribution after the\ndamping, and \u0016N=hay\noutaoutiis the average number of\nbosons at the output. From (6), with \u00122= 0, one gets\n(the subscript Sstands for sensitivity):\n\u0001\rS\n\r=[(\u0001N)2\nin+ (e2\rt\u00001)\u0016Nin]1=2\n2\rt\u0016Nin\u0000\u0000\u0000\u0000!\n\u0001N!0\u000e\rmin=\r:\n(8)\nThis expression shows that the uncertainty in \rhas two\ncontributions, theterm (\u0001N)2\ninstemmingfromtheinitial\nvariance in the bosonic number of the incoming probe,\nand the remaining terms corresponding to the random\ntransmission of the incoming bosons. It is clear that,\nin order to minimize (8), one must have (\u0001N)2\nin= 0,\nimplying that the incoming bosons should be in a Fock\nstate. In this case, \u0001\rS=\rbecomes identical to the lower\nbound in (7)! The presence of \u0016N1=2\nin– where \u0016Ninis now\njust the number of bosons in the Fock state – in the\ndenominator of the right-hand side of (7) implies that\nthe same result would be obtained with a stream of N\nindependent single bosons. We note that \u000e\r!1when\nt!0ort!1, corresponding respectively to no action\nof the damping and to complete absorption, leading to\nno information on \r(quantum Fisher information equalto zero). The minimum value of (7) is obtained for\n\rtopt\u00190:8)\u000e\ropt\nmin=\r= 1:24=p\u0016Nin:(9)\nThis defines the optimal interaction time. Better preci-\nsion can be obtained, however, by adopting a “divide and\nconquer” strategy. Instead of estimating the damping\nthrough a single measurement for an interaction time t,\none applies sequential measurements, for instance with a\nsingle-photon stream, such that tis divided into Ninter-\nvals of length \u001c, which could be taken as the interaction\ntime between each single photon and the probed sam-\nple. We replace then, in the right-hand side of (7), tby\n\u001cand \u0016NinbyN=t=\u001c. The corresponding expression\nis minimized for \u001c!0. However, any other \u001csmaller\nthantwould lead to a result better than measuring just\nat timet. For\r\u001c\u001c1,\u000e\r=\r\u00191=p2\rt, which is much\nsmaller than (9) if \rt\u001d1. We note that this strategy\nnot only leads to better precision, but could be manda-\ntory for thin or fragile samples, for which the interaction\ntime with the probe should necessarily be smaller than\nthe thermalization time.\nConfirmation of this result is obtained by explicitly\ncalculating the quantum Fisher information for incom-\ning Fock states. The general expression for the quan-\ntum Fisher information for estimation of a parameter\nXis expressed in terms of the density operator of the\nprobe asFQ(X) = Tr\u0002\n\u001a(X)L2(X)\u0003\n, where the sym-\nmetric logarithmic derivative is defined by the equation\nd\u001a(X)=dX = [\u001a(X)L(X) +L(X)\u001a(X)]=2. FindingLre-\nquires, in general, the diagonalization of the density op-\nerator, for a given initial state [5, 29]. However, for in-\ncoming Fock states the density matrix is diagonal, and\ntherefore the singular logarithmic derivative is given by\nLnn= (1=pn)(dpn=dX), wherepn\u0011\u001annis the boson-\nnumber probability distribution. It follows then that\nFQ(\r) = Tr\u0000\n\u001aL2\u0001\n=X\nn(1=pn)(dpn=d\r)2;(10)\ncoinciding with the Fisher information associated to\nthe measurement of the bosonic population distribution,\nwhich is thus shown to be the best measurement in this\ncase. On the other hand, the boson-number distribution\nfor the outgoing bosons is identical to the beam-splitter\nbinomialdistribution, pn(\r) =\u0000N\nn\u0001\n(1\u0000\u0011)N\u0000n\u0011n. Replac-\ning this expression in (10) leads precisely to (7). Further-\nmore, asN\u001d1(which could apply to a Fock state or\na stream of single photons), the combinatorial distribu-\ntion goes to a Gaussian distribution, with width given\nby the lower bound in (7), so this bound is actually satu-\nrated by these states. This completes our demonstration\nthat Fock states lead to the minimal uncertainty in the\nestimation of \r[36].\nForT6= 0, one gets a lower bound \u0001\rG\nmin(T)from the\nunitary transformation in (2) (details in supplementary4\nFigure 2.\u000e\r=\ras a function of the bath thermal photon num-\nber, for two different values of \u0011\u0011exp(\u00002\rt). The solid and\ndotted curves correspond to single-photon and thermal state\ninputs, this last one with an average photon number equal\nto one. The dashed curve corresponds to the bound (11),\nobtained from the purification procedure. For \u0011= 0:9, and\nsingle-photon input, precision increases with temperature, for\nthe range here considered.\nmaterial sec I):\n\u000e\rG\nmin(T)=\u000e\rmin\n=nT(1\u0000\u0011) + 1p\nnT(1 +\u00112) + 1 + (nT=\u0016Nin)\u0011[nT(1\u0000\u0011) + 1];(11)\nwhere\u000e\rminis defined in (7). Calculations also show\nthat (11), for any T, is lower than the bound calculated\nusing error propagation sensitivity (see supplementary\nmaterial Sec II).\nThe QFI of the system, for incoming Fock states, can\nbe calculated numerically, from the number probability\ndistribution given in [37, 38] – see Eq. (S20) in the Sup-\nplementary Material. It coincides with (11) when there is\nno input i.e. \u0016Nin= 0. In this case, only thermal photons\ncontribute to the estimation of \r(supplementary mate-\nrialsecIII).Fig.2showsthebehaviorof \u000e\r=\rfor\u0016Nin= 1\nand two values of \u0011= exp (\u00002\rt), namely\u0011= 0:9and\n\u0011= 0:7. As expected, say from (8), the incoming ther-\nmal state is a poor choice for estimation of \r. In case\nof initial thermal state with \u0016Nin=nT, there is no time\nevolution of the incoming state, and hence the quantum\nFisher information vanishes, which leads to the divergent\nbehavior of the dotted curve in Fig. 2.\nEstimation of temperature . The simplest situation corre-\nsponds to no incoming photons. In this case, the beam\nsplitter in (2) does not play a role, and the purification\nis given byj\t(t)i=Sj0iSj0iR1j0iR2. From the gener-\natorG(nT) =i[dSy(nT)=dnT]S(nT), one gets then an\nupper bound for the quantum Fisher information, from\nwhich it follows a lower bound for the uncertainty in the\nestimation of nT:\n\u000enT=s\nnT\u0012\nnT+1\n1\u0000\u0011\u0013\n\u0000\u0000\u0000!\nt!1p\nnT(nT+ 1):(12)\nFornoincomingphotonthesensitivityexpressionandthe\nFigure 3. Uncertainty \u000enTin the measurement of tempera-\nture, normalized by the steady state value (\u000enT)st, for differ-\nent values of \u0011= exp(\u00002\rt). Each curve is labelled by the\nphoton number Nin the incoming Fock state of the probe. In\nthelimitt!1, sothat\u0011!0, onehas\u000e\u0011T=(\u000e\u0011T)st= 1. The\ngraph suggests that the best measurement occurs for large t\n(or small\u0011). For\u0011= 0:9(\rt\u00180:1), and fornT\u00144:5, single-\nboson Fock state leads to better precision than the vacuum\nstatej0i. Sequentialmeasurementsmayleadhowevertomuch\nbetter precision, as shown in the text of the article.\nQFI yield for \u000enTthe same result. Therefore, in this case\nthe lower bound for the uncertainty coincides with the\nexact result. As the interaction time between probe and\nsample increases, \u000enTis reduced, attaining the steady-\nstate limit (\u000enT)st=p\nnT(nT+ 1)whent!1, which\ncoincides with the quantum-mechanical uncertainty for a\nthermal field. The numerical results from the solution of\nthe master equation for an incoming single photon state\nare shown in Fig. 3. The Fock state j1ileads to better\nprecision for small times and low temperatures, as com-\npared the vacuum state j0i.\nAs in the estimation of damping, an increase in\nprecision can be obtained by applying sequential pre-\nthermalization measurements, through a stream of single\nbosons. The measurement time tis divided into \u0017inter-\nvalsoflength \u001c, correspondingtotheinteractionbetween\nasinglebosonandtheprobedsystem. Thecorresponding\nQFIFQ(nT;\u001c)canbeobtainedfrom(1)inthesmall-time\nlimit\r\u001c(nT+1)\u001c1, and the corresponding uncertainty\nis\u000enT= 1=p\n(t=\u001c)FQ(nT;\u001c). It turns out that the best\nresult is obtained when \u001c!0, but any other \u001csmaller\nthantwould lead to a better result then measuring at\nt. In the limit \r\u001c(nT+ 1)\u001c1, we get (supplementary\nmaterial Sec IV)\n\u000enT!s\nnT(nT+ 1)\n(3nT+ 2)2\rt\u0000\u0000\u0000\u0000!\nnT\u001c1p\nnT=4\rt:(13)\nWhen\rt\u001d1, this expression is much smaller than5\n(\u000enT)st, implying a huge gain in precision, as compared\nto measurement at time t. The effect on the protocol\nby timing errors can be easily accounted for, since the\nabove expression depends only on the total time t. For\n\u0001t=t\u001c1, then the extra uncertainty in the temperature\nestimation, \u0001(\u000enT), will be much smaller than \u000enT.\nConclusion . We have established the quantum preci-\nsion limits for the estimation of damping constants and\ntemperature, when bosons are used as probes. Bosonic\nprobes occupy a prominent place in science, especially\nin view of the large number of processes involving light\nor microwave fields to obtain information on absorption\ncoefficients or the temperature of transparent samples.\nLower analytic bounds for the uncertainty in the estima-\ntion of these parameters have been obtained, through a\npurification procedure that involves replacing the mas-\nter equation by a unitary transformation composed by\na beam splitter and a squeezing operator, acting on the\nbosonic mode and two auxiliary environments. These\nbounds were shown to be tight, for some specific condi-\ntions, and, more generally, close to the numerical solu-\ntions. We have shown that sequential pre-thermalization\nmeasurements with single-photon streams can lead to\nhuge gain in precision, both for damping and tempera-\ntureestimation. Thisresultisespeciallyrelevantformea-\nsurements on thin or fragile samples. We believe these\nfindings should stimulate experimental work on physical\nand biological systems.\nLD acknowledges the support of the Brazilian agen-\ncies CNPq, CAPES, FAPERJ, of the National Institute\nof Science and Technology for Quantum Information,\nand of the Hagler Institute for Advanced Study of the\nTexasA&MUniversity. G.S.A.thanksthesupportofAir\nForce Office of Scientific Research (Award NoFA-9550-\n18-1-0141) and the Robert A Welch Foundation (A-1943-\n20180324).\nSUPPLEMENTARY MATERIAL\nI. Lower bounds on uncertainties in the estimation\nof damping and temperature\nHere we provide the derivation of lower bounds on the\nuncertainties in the estimation of damping and tempera-\nture by using the purification procedure described in the\ntext, corresponding to Fig. 1. If the probe is in the initial\nstatej\t0i, and interacts with the probed system during\na timet, then the purified output state is\nj\t(t)i=SBj\t0ij0ibj0ic; (S1)\nwhere the two environments bandcare assumed to be\ninitially in the vacuum state. The operators SandBare\ndefined by Eqs. (3)-(5) in the main text. The operator\nG(X)\u0011i[dUy(X)=dX]U(X), for an arbitrary parameterX, whereU(X) =S(X)B(X), is given by\nG(X) =\u0000if(aby\u0000ayb)d\u00121\ndX+ [cy(aycos\u00121\u0000bysin\u00121)\n\u0000c(acos\u00121\u0000bsin\u00121)]d\u00122\ndXg; (S2)\nwhere\u00121and\u00122are defined by Eqs. (4) and (5) in\nthe main text, and a,b, andcare annihilation operators\ncorresponding respectively to the original bosonic mode\nand the additional environments bandc. On applying\nthis operator to the initial state j\t0ij0ibj0ic, one gets:\nGj\t0ij0ibj0ic=\u0000i[aj\t0ij1ibj0icd\u00121\ndX\n+ (ayj\t0ij0ibj1iccos\u00121\u0000j\t0ij1ibj1icsin\u00121)d\u00122\ndX]:\n(S3)\nThe expectation value of the operator GyGin the state\nj\t0ij0ibj0icis therefore\n\nGy(X)G(X)\u000b\n=Nin\u0014\n(d\u00121\ndX)2+ cos2\u00121(d\u00122\ndX)2\u0015\n+(d\u00122\ndX)2;\n(S4)\nwhereNin=\naya\u000b\nis the average number of photons\nin the input state j\t0i. After simplification we get the\nfinal expressions for X=\randX=nT, where\ris\nthe damping coefficient and nTis the thermal photon\nnumber, as functions of \u0011,nTandNin:\n\nGy(\r)G(\r)\u000b\n=(2t\u0011)2\n4fNin1 +nT(1 +\u00112)\n\u0011(1\u0000\u0011)[1 +nT(1\u0000\u0011)]2\n+nT\n(1\u0000\u0011)[1 +nT(1\u0000\u0011)]g; (S5)\n\nGy(nT)G(nT)\u000b\n=1\n4f(1\u0000\u0011)\n[1 +nT(1\u0000\u0011)]nT\n+Nin\u00112(1\u0000\u0011)fnT(1\u0000\u0011)[1 +nT] + (1 +nT)g\n[1 +nT(1\u0000\u0011)]3(1 +nT)nTg:\n(S6)\nSince the Quantum Fisher Information FQ= 4(\u0001G)2=\nGy(nT)G(nT)\u000b\n\u0000hGi2, the lower bound for the uncer-\ntainty in the estimation of damping and temperature can\nbe calculated from \u000eXG\nmin=F\u00001=2\nQ:\n\u000e\rG\nmin=1 +nT(1\u0000\u0011)\n2t\u0011[\u0011(1\u0000\u0011)]1=2\n\u0002fNin[1 +nT(1 +\u00112)] +\u0011[1 +nT(1\u0000\u0011)]nTg\u00001=2;\n(S7)6\n\u000enTG\nmin=f[1 +nT(1\u0000\u0011)]3(1 +nT)nTg1=2\n\u0002fNin\u00112[nT(1\u0000\u0011)(2 +nT\u0000\u0011nT)) + (1\u0000\u0011)]\n+ [1 +nT(1\u0000\u0011)]2(1 +nT)(1\u0000\u0011)g\u00001=2:(S8)\nComparing (S7) with the one at zero temperature,\n\u000e\rmin=p\n1=\u0011\u00001\n2tp\nNin, we get Eq. (11) in the main text,\n\u000e\rG\nmin\n\u000e\rmin= [1 +nT(1\u0000\u0011)]f[1 +nT(1 +\u00112)]\n+\u0011[1 +nT(1\u0000\u0011)]nT=Ning\u00001=2:(S9)\nFor vacum input, the expression for \u000enTG\nminbecomes\nEq.(12) in the main text,\n\u000enT=s\nnT[1 +nT(1\u0000\u0011)]\n1\u0000\u0011: (S10)\nII. Sensitivity calculations using the master equation\nThe error-propagation expression for the uncertainty\nin the estimation of a parameter Xis given by \u0001X=\n\u0001Nout=@Nout\n@X, where (\u0001Nout)2=h(ay\noutaout)2i\u0000N2\nout.\nFromEq.(1)inthemaintext, wecanstudytheevolution\nof an operator Aby@\n@tA=Tr[@\u001as\n@tA]. For any operator\nA,\n@\n@tA=\u0000\r(1 +nT)<Aaya\u00002aAay+ayaA>\n\u0000\rnT<aayA\u00002ayAa+Aaay> : (S11)\nSinceTr[AB] =Tr[BA];we have\n@\n@tA=\u0000\r(1 +nT)<[A;ay]a+ay[a;A]>\n\u0000\rnT<[A;a]ay+a[ay;A]> : (S12)\nTakingA= (aya)i;(i= 1;2), one gets\n@\n@tN=\u00002\rN+ 2\rnT; (S13)\n@\n@tN2=\u00004\rN2+ 2\r(4nT+ 1)N+ 2\rnT:(S14)\nIntegrating these equations on both sides one gets\nN(t) =e\u00002\rt(N(0)\u0000nT) +nT;(S15)N2(t) =e\u00004\rtN2(0) +e\u00002\rt(4nT+ 1)(1\u0000e\u00002\rt)N(0)\n+ 2n2\nT(1\u0000e\u00002\rt)2+nT(1\u0000e\u00002\rt):(S16)\nNote thatNout=N(t)andNin=N(0). Using (S15)\nand (S16) we find\n4\r= [\u00112(4Nin)2+\u0011(2nT+ 1)(1\u0000\u0011)Nin\n+ (nT+ 1\u0000\u0011nT)nT(1\u0000\u0011)]1=2[2t\u0011(Nin\u0000nT)]\u00001:\n(S17)\nFor givenNin,\u0011, andnT, the minimal uncertainty 4\rmin\nis achieved for (4Nin)2= 0, indicating that Fock states\nlead to the best estimation of \r.\nWe compare now (S17), for Fock states, so that Nin=\nNmin, with the bound \u000e\rG\nmin, obtained in Sec I using pu-\nrification. The ratio of 4\rminand\u000e\rG\nmin.\n4\rmin\n\u000e\rG\nmin= (nT(1\u0000\u0011) + 1)Nin\nNin\u0000nT\n\u0002s\nNin\u0011(2nT+ 1) +nT[nT(1\u0000\u0011) + 1]\nNin\u0011[nT(1 +\u00112) + 1] +\u0011nT[nT(1\u0000\u0011) + 1]:\n(S18)\nSincenT(1\u0000\u0011) + 1\u00151,Nin\u0011(2nT+ 1)\u0015Nin\u0011[nT(1 +\n\u00112) + 1], andnT[nT(1\u0000\u0011) + 1]\u0015\u0011nT[nT(1\u0000\u0011) + 1], one\ngets\n4\rmin\u0015\u000e\rG\nmin: (S19)\nTheequalitysignin(S19)holdsonlywhen nT= 0, which\ncoincides with the discussion after Eq. (8) in the main\ntext: at zero temperature, the error-propagation formula\nfor the estimation uncertainty coincides with the lower\nbound. It may be noted that expressions like (S17) are\nnot meaningful when Ninapproaches nT. In this limit\nthe output photon number becomes independent of the\nparameter\r, which we had set out to determine. In such\ncasespostprocessingofsignalisneeded. Itmaybeadded\nthat the full master equation solution for the input Fock\nstate has no such divergence as the bound is calculated\nusing full photon number distribution. For thermal input\nwith input photon number equal to nT, master equation\nsolution gives divergence [Fig.2] because as noted there\nthe Fisher information becomes zero and not meaningful\nas the system does not evolve then.\nIII. Master equation result for the QFI with no\nincoming photons\nThe solution of the master equation given by Eq. (1)\nwas studied numerically in the paper for both Fock states\nand thermal states. However it is possible to get the\nanalytical result for vacuum input. From [37] we get the7\nprobability of seeing nphotons at the output state with\ninput Fock statejmi:\npn=(1\u0000e\u00002\rt)n+m(e\f!\u00001)em\f!\n(e\f!\u0000e\u00002\rt)n+m+1\n\u0002F[\u0000n;\u0000m;1 :e\f!+e\u0000\f!\u00002\ne2\rt+e\u00002\rt\u00002];(S20)\nwheree\f!= 1 +n\u00001\nTandFis the hypergeometric func-\ntion. For m= 0,F[\u0000n;0;1 :z] = 1, thus we have\npn=(1\u0000\u0011)nn\u00001\nT\n(1+n\u00001\nT\u0000\u0011)n+1. Letn(t) =nT(1\u0000\u0011), thenpncan\nbe written as the Bose-Einstein distribution\npn=n(t)n\n(1 +n(t))n+1: (S21)\nFrom (S21) we obtain the Quantum Fisher Information\nFQ=n2\nT(2t\u0011)2\nn(t)(1 +n(t)): (S22)\nWith\u000e\r\u0015q\nFQ\u00001, we get the lower bound for \ras\n\u000e\rG(T) =1\n2t\u0011s\n(1\u0000\u0011)[1 +nT(1\u0000\u0011)]\nnT:(S23)\nThis coincides with \u000e\rG\nminin (S7) with Nin= 0obtained\nin section I with purification.\nIV. Estimation of bounds for the uncertainty in\ntemperature estimation with a stream of single\nphotons\nWe consider now the bound for the uncertainty in \u000enT\nwithastreamofsinglephotons, eachoneinteractingwith\nthe probed system during a time interval of \u001c, so that the\ntotal interaction time is divided into \u0017intervals, with\nt=\u0017\u001c. From the master equation, we get the dynamics\nofpn, the probability of detecting nbosons, after they\nhave interacted with the sample:\ndpn\ndt= 2\r(nT+ 1)[(n+ 1)pn+1\u0000npn]\n+ 2\rnT[npn\u00001\u0000(n+ 1)pn]: (S24)\nHere if we have a single-boson input at each time interval\n\u001c, sopn(0) =\u000en;1. We integrate equation (S23) assuming\n\r\u001c(nT+ 1)\u001c1, so that\npn(\u001c)'\u000en;1+ 2\r\u001c(nT+ 1)[(n+ 1)pn+1\u0000npn]\n+ 2\r\u001cnT[npn\u00001\u0000(n+ 1)pn]; (S25)\nwherepnon the right side gives the distribution at \u001c= 0.\nFigure 4. A huge increase in the estimation precision can be\nobtained with a stream of Nsingle bosons, each one interact-\ning with the probed material for a time \u001cmuch smaller than\nthe thermalization time t.\nFrom (S25) we then obtain\np0(\u001c) = 2\r\u001c(nT+ 1);\np1(\u001c) = 1\u00002\r\u001c(nT+ 1)\u00004\r\u001cnT;\np2(\u001c) = 4\r\u001cnT:(S26)\nFor a total interaction time t=\u0017\u001c, corresponding to\n\u0017=t=\u001csingle-boson interactions, the Quantum Fisher\nInformation is then\nFQ(t)=\u0017FQ(\u001c)=\u0017X\nn1\npn\u0012dpn\ndnT\u00132\n=t\n\u001c[(2\r\u001c)2\n2\r\u001c(nT+ 1)+(6\r\u001c)2\n1\u00002\r\u001c(nT+ 1)\u00004\r\u001cnT+(4\r\u001c)2\n4\r\u001cnT]\n\u001c\u00010\u0000!2\rt3nT+ 2\nnT(nT+ 1): (S27)\nThe corresponding lower bound for the uncertainty in\nthe estimation of the thermal photon number is obtained\nfrom\u000enT= 1=q\nt\n\u001cFQ(\u001c):\n\u000enT\u001c\u00010\u0000!s\nnT(nT+ 1)\n(3nT+ 2)2\rt: (S28)\nIn the low temperature limit nT\u001c1, we have\n\u000enTjnT\u001c1\u001c\u00010\u0000!rnT\n4\rt: (S29)\nThe limits (S28), (S29) are discussed in the main text.\n[1] H.Cramér, “MathematicalMethodsofStatistics,” p.500,\nPrinceton University, Princeton, NJ, USA (1946).\n[2] R. 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Wolf, North Holland, Amsterdam (1973).\n[38] S. Chaturvedi and V. Srinivasan, Journal of Modern Op-\ntics38, 777 (1991)."    },    {        "title": "2008.06253v1.Large_enhancement_of_spin_pumping_due_to_the_surface_bound_states_in_normal_metal_superconductor_structures.pdf",        "content": "Large enhancement of spin pumping due to the surface bound states in normal\nmetal/superconductor structures\nM.A. Silaev1, 2, 3\n1Department of Physics and Nanoscience Center, University of Jyv askyl a,\nP.O. Box 35 (YFL), FI-40014 University of Jyv askyl a, Finland\n2Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia\n3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia\nWe show that the spin pumping from ferromagnetic insulator into the adjacent metallic spin\nsink can be strongly stimulated by the superconducting correlations. The key physical mechanism\nresponsible for this e\u000bect is the presence of quasiparticle surface states at the ferromagnetic insu-\nlator/superconductor interface. We consider the minimal model when these states appear because\nof the suppressed pairing constant within the interfacial normal layer. For thin normal layers we\nobtain a strongly peaked temperature dependence of the Gilbert damping coe\u000ecient which has been\nrecently observed in such systems. For thicker normal layers the Gilbert damping monotonically\nincreases down to the temperatures much smaller than the critical one. The suggested model paves\nthe way to controlling the temperature dependence of the spin pumping by fabricating hybrid normal\nmetal/superconductor spin sinks.\nPACS numbers:\nIntroduction Spin transport and spin dynamics\nin superconductors have attracted signi\fcant attention\nrecently1{7. Quite interesting experimental results have\nbeen obtained for the spin pumping e\u000bects8{18which\nin general play the central role in spintronics19{21. It\nwas found that superconducting correlations can lead ei-\nther to the signi\fcant suppression8or to the signi\fcant\nenhancement9{13,17of Gilbert damping (GD) coe\u000ecient\nFIG. 1: Schematic setup of the ferromagnetic insulator (FI)\n\flm with the adjacent metallic spin sink consisting of of nor-\nmal (N) and superconducting (S) layers. The constant ex-\nternal magnetic \feld is H0x. The magnetization precession\nm(t) is driven by the external magnetic \feld H\nei\nty. It\ngenerates spin current i\npumped from F to the spin sink.\nUpper panel shows the coordinate dependencies of the order\nparameter \u0001( x) and local density of states N(x) at the energy\n\"= 0:5\u00010fordN= 0:2\u00180,dS= 3\u00180,T= 0:7Tc.in systems consisting of superconducting and ferromag-\nnetic layers, such as in the generic example shown in\nFig.1. The basic mechanism for changing GD in such\nsystems is the spin pumping e\u000bect. This mechanism\nis based on the spin angular momentum transfer from\nthe ferromagnet into the the adjacent metallic \flm via\nthe pumped spin current i(t) generated by the time-\ndependent magnetization m(t). The spin relaxation in\nthe metallic spin sink leads to the damping-like spin\ntorque and modi\fes the e\u000bective GD coe\u000ecient of the\nsystem.\nIn this way the suppression of GD with decreasing\ntemperature T < Tcin systems with superconducting\nspin sink8can be qualitatively understood as result-\ning from the the freezing out of quasiparticles in the\nsuperconductor22. However, the strong increase of GD\nwith lowering temperature9{13,17seems to be counter-\nintuitive and its understanding requires further theoret-\nical e\u000borts.\nIn ferromagnetic insulator (FI) /superconductor (S) bi-\nlayers GdN/NbN the peaked behaviour of GD as a func-\ntion of temperature has been observed13. The maximal\nGD reached at about T\u00190:7Tcis several times larger\nthan in the normal state \u000e\u000b=\u000e\u000bN\u00182\u00003, where\u000e\u000bis\nthe spin-pumping related change of GD. Because of the\nseveral reasons such behaviour cannot be explained23by\nthe coherence peak of spin susceptibility in homogeneous\nsuperconductors24. First, this peak occurs at T\u00190:9Tc\nand for the realistic values of the Dynes parameter25\n\u0000\u00190:1Tcin NbN its magnitude is23\u000e\u000b=\u000e\u000bN\u00180:2\u00000:3.\nSuch behaviour is typical for the line widths of nu-\nclear magnetic resonance26,27and electronic paramag-\nnetic resonance28in superconductors. It is clearly di\u000ber-\nent from the observed behaviour of GD in FI/S systems13\nwhich has an order of magnitude larger peak \u000e\u000b=\u000e\u000bN\u0018\n2\u00003 at signi\fcantly lower temperatures T\u00190:7Tc.\nIn this Letter we suggest a minimal theoretical modelarXiv:2008.06253v1  [cond-mat.supr-con]  14 Aug 20202\nwhich explains the large enhancement of GD in FI/S\nstructures. The key physical mechanism responsible for\nthis e\u000bect is the existence of quasiparticle states localized\nat the FI/S interface. Such states appear due to the sup-\npressed pairing within the interfacial normal layer29{32\n(N) as illustrated in Fig.1. Shown on top of the Fig.1\nare the spatial pro\fles of the order parameter \u0001( x) and\nthe local density of states (DOS) N(x) at the subgap\nenergy\"= 0:5\u00010, where \u0001 0is the bulk energy gap at\nT= 0. The overall N/S \flm thickness is dS= 3\u00180, where\n\u00180=p\nDS=Tc0is the coherence length, DSis the di\u000bu-\nsion constant in S, Tc0is the bulk critical temperature.\nNear the interface at x= 0 the DOS is enhanced due to\nthe subgap quasiparticle states which are formed in the\nN/S structure33{36and occupy the certain energy interval\nbetween the bulk gap and Thouless energy DN=d2\nNwhere\nDNis the di\u000busion coe\u000ecient and dNis the thickness of\nN. The existence of surface bound states in N/S struc-\ntures is demonstrated37in Fig.2a,c where the N(x;\") pro-\n\fles are shown to have a maximum at x= 0 and energies\nwhich depend on dN. The order parameter and DOS\nin Figs.1,2 are calculated within the Usadel theory38as\nexplained below. In Fig.1 we choose identical di\u000busion\ncoe\u000ecient in N and S layers DN=DS=Dwhile in\nFig.2DN= 0:05DS.\nAt low frequencies \n \u001c\u00010the DOS enhancement\nleads to the increased probability of the magnon ab-\nsorption by conductivity electrons in the N/S layer.\nQualitatively, at a given energy level this probability\nis determined by number of available states for transi-\ntionN(\")N(\"+ \n)\u0019N2(\") and the di\u000berence of oc-\ncupation numbers n0(\"+ \n)\u0000n0(\")\u0019\n@\"n0where\nn0(\") = tanh(\"=2T) is the equilibrium distribution func-\ntion. The product of these factors leads to the energy-\nresolved magnon absorption probability Pm= \nN2@\"n0.\nIn Fig.2b,d one can that of Pm(\") atT= 0:7Tc0is en-\nhanced at the boundary of N layer x= 0 (red curves)\nas compared to x=dS(blue curves). Besides that, the\nlocalization of surface states is qualitatively equivalent\nto the decrease of the spin sink volume which and the\ncorresponding increase of the non-equilibrium spin po-\nlarization. As we show by an exact calculation below\nthese mechanisms lead to the large enhancement of spin\npumping in the N/S \flms.\nInterestingly, besides explaining the large peak of the\nspin pumping for dN\u001c\u00180the model described above\nyields also the qualitatively di\u000berent regime with almost\nmonotonic increase of GD down to the temperatures\nT\u001cTc. This behaviour is obtained for dN\u0018\u00180when\nthe bound states are pushed down to lower energies as\nshown in Fig.2c and the absorption probability us en-\nhanced for quasiparticles with \"\u001c\u00010which are not\nfrozen out down to the signi\fcantly low temperatures\ndetermined by the Thouless energy Tth\u0019DN=d2\nN. Sim-\nilar behaviour of GD has been observed experimentally\nin Py/Nb/Pt superconducting heterostructures12,17, al-\nthough its physical origin can be di\u000berent.\nModel of spin pumping To quantify the spin\n(a)\n (b)\n(c)\n (d)\nFIG. 2: (a,c) Density of states pro\fle N(\";x) in the N/S\nstructure. The position of N/S boundary shown by the dashed\nline is at (a) dN= 0:2\u00180and (c)dN= 0:8\u00180.T= 0:7Tc0,\n\u0000 = 0:1Tc0,dS= 5\u00180,DN= 0:05DS. Plots for other dSare\nshown in Appendix37. (b,d) Magnon absorption probability\nPm(\") = \n@\"n0N2for the frequency \n = 0 :02Tc0, Red and\nblue curves are taken at x= 0 andx=dS, respectively.\nParameters are the same as in (a,b).\npumping e\u000bect we consider the microscopic model of\nthe spin-dependent scattering of electrons at the FI\ninterface37,39,40. As we show below, it formally yields\nthe spin current identical to the one given by the inter-\nfacial exchange interaction between the localized spins\nin FI and conduction elections in the adjacent metal41.\nWithin this model the local spin polarization close to the\ninterfaceS(t) acts as e\u000bective \feld for the localized mag-\nnetic moments. This process can be taken into account\nby introducing the additional term i(t) into the Landau-\nLifshitz-Gilber equation\n(1 +\u000bm\u0002)@tm+\rm\u0002Heff=i=SF0dF (1)\ni(t) =JsdS(t)\u0002m(t) (2)\nHereSF0is the equilibrium spin density in F, dFis the\nF \flm thickness, Heffis the e\u000bective \feld and \u000bis the\nintrinsic Gilbert damping coe\u000ecient. The term i(t) can\nbe interpreted as the spin current between FI and metal.\nTo calculateS(t) we need to \fnd the spin response of\nthe superconductor to the interfacial exchange \feld. In\nthe linear regime it is given by\nS\n=\u0017heff\u001fmm\n (3)\nwhere we introduce the e\u000bective exchange \feld heff=\nJsd=dS, normal metal DOS at the Fermi level \u0017and the\nlocal spin susceptibility \u001fm.3\nThe spin-pumping related change of the GD is deter-\nmined by the dissipative part of the susceptibility\n\u000e\u000b=CTc0Im\u001fm=\n (4)\nwhere the dimensionless coe\u000ecient determining the cou-\npling strength between the FI and metallic \flms is23\nC=heff\nTc0\u0017heff\nSF0dS\ndF(5)\nFrom there one can see that since heff/1=d2\nSthe cou-\npling coe\u000ecient is C/1=dS. Localization of surface\nstates provides the e\u000bective decrease of dSwhich leads\nto the increase of Cand the spin response.\nCalculation of the time-dependent spin re-\nsponse. What is left is to calculate the local spin\nsusceptibility \u001fmin the Eq.4 for the FI/N/S structure\nin Fig.1. We do so by developing the microscopic ki-\nnetic theory of spin pumping generalizing the quasiclas-\nsical approach2,40,42,43to the time-dependent situation.\nThe magnetization of conduction electrons is deter-\nmined by spin accumulation and can be written in terms\nof the Keldysh quasiclassical Green's function (GF) as\nS(t) =\u0000\u0017Tr [^\u001c3^\u001bgK(t;t)]=8 (6)\ngKis the (2\u00022 matrix) Keldysh component of the qua-\nsiclassical GF matrix \u0014 g=\u0012\n^gR^gK\n0 ^gA\u0013\nwhich depends\non two times and a single spatial coordinate variable\n\u0014g= \u0014g(t1;t2;r). GF \u0014gobeys the Usadel equation\nf^\u001c3@t;\u0014ggt+r(D\u0014g\u000er\u0014g) = \u0001[^\u001c1;\u0014g]+[\u0014\u0000;\u0014g]\u0000[\u0014\u0006so;\u0014g]t:(7)\nwhere ^\u001bk;^\u001ck,k= 0;1;2;3 are Pauli matrices, Dis the\ndi\u000busion coe\u000ecient. The commutator operator is de\fned\nas [X;g]t=X(t1)g(t1;t2)\u0000g(t1;t2)X(t2), similarly for\nanticommutatorf;gt. The symbolic product operator is\ngiven by (A\u000eB)(t1;t2) =R\ndtA(t1;t)B(t;t2).\nSpin relaxation is determined by the spin-orbital scat-\ntering self energy\n^\u0006so=\u001b\u0001^g\u001b=(6\u001cso) (8)\nThe self-consistency equation for the gap function is\n\u0001 =\u0015Tr[^\u001c1^gK]=4 (9)\nwhere\u0015is the pairing coe\u000ecient. In our model we assume\nthe pairing constant to be suppressed in the N region\n\u0015(x<dN) = 0:05\u0015(x>dN) as compared to its value in\nS. We scan over the values of the di\u000busion coe\u000ecient in\nthe N layer DNwhile keeping it \fxed in S layer DS. The\ninelastic scattering is described by the Dynes44param-\neter which enters to the Eq.7 as the matrix in Nambu-\nKeldysh space with ^\u0000R;A=\u0006\u0000^\u001c3which described both\nthe DOS singularity broadening and the relaxation of\nnon-equilibrium distribution functions as described be-\nlow. Note that this terms conserves the total spin in ac-\ncordance with the general property of spin-independent\nelectron-phonon scattering.Eq.7 is supplemented by the dynamical boundary con-\nditions atx= 0 describing the spin splitting and pump-\ning induced by the electron scattering at the FI inter-\nface with time-dependent magnetization. These bound-\nary conditions are derived37from the spin-dependent\nscattering matrix ^Sconnecting the incident ^ iand re-\n\rected ^ relectronic waves ^ r=^S(t)^ i. For frequen-\ncies small compared to the exchange \feld in FI we use\nthe adiabatic approximation which yields the expression\n^S=ei(m^\u001b)^\u001c3\u0002=2, where \u0002 is the time-independent spin-\nmixing angle. Then, assuming that j\u0002j\u001c1 and\nD\u0014g\u000e@x\u0014g(x= 0) =iJsd[\u001bm^\u001c3;^g]t (10)\nwherem=m(t) is the time-dependent magnetiza-\ntion. Within the minimal band model of the FI39,40\nthe interfacial exchange constant is expressed through\nthe spin-mixing angle as Jsd=\u0017vF\n4R1\n\u00001d^pxj^pxj\u0002(^px),\nwhere ^pxis the electron momentum projection on the\ninterface normal. Eq.10 generalizes the static bound-\nary condition at the spin-active interface39,40,43,45to the\ncase of time-pendent magnetization. The induced spin\ncurrent is obtained using the general expression i(t) =\n\u0019\u0017DTr[^\u001b\u0014g\u000e@x\u0014g](t;t). With the help of Eqs.(10,6) it\nyields the phenomenological Eq.(2).\nIntroducing the usual parametrization of quasiclassical\nKeldysh function in terms of the distribution function\n^gK= ^gR\u000e^f\u0000^f\u000e^gAwe can identify the terms which are\nessential to calculate linear response in the low-frequency\nlimit. Expanding the energy representation of ^ gKto the\n\frst order in \n we obtain the non-equilibrium correction\n\u000e^gK= (^\u001bm \n)\u0014\n(^gR\n0\u0000^gA\n0)fh+\n@\"n0\n2(gR\nh+gA\nh)\u0015\n(11)\nwhere we parametrise the spin-dependent corrections\nas follows ^f= (^\u001bm \n)fhand\u000egR;A= (^\u001bm \n)\u000egR;A\nh.\nIn contrast to stationary non-equilibrium situations42\nwhen only the \frst term in (11) is important the time-\ndependent case requires taking into account also the sec-\nond term with the corrections of spectral functions23. In\nthe low-frequency limit the calculation is simpli\fes by ne-\nglecting the frequency dependence of the perturbed spec-\ntral GF in (11). Using (11) we write the time-dependent\nspin polarization in the metallic \flm as follows\nS\n=i\nm\nZ1\n\u00001d\"[2Nfh+ (gR\n3h+gA\n3h)@\"n0] (12)\nwhereN= Tr(^\u001c3^gR)=2 is the local DOS and gR;A\n3h=\nTr(^\u001c3^gR;A\nh)=2 . Equations for zero-order spectral func-\ntion ^gR;A\n0(\";x), corrections ^ gR;A\nh(\";x) and the distribu-\ntion function fh(\";\n;x) are obtained straightforwardly37\nfrom Eqs.(7, 10). The zero-order GF ^ gR;A\n0(\";x) are calcu-\nlated in the N/S structure self-consistently together with\nthe order parameter 9. This gives in particular the \u0001( x)\nandN(\";x) pro\fles shown in Fig.1,2. The corrections fh\nand ^gR;A\nhare determined by the linear equation37.4\nDN=DS, \u0000 = 0:1Tc0\ndN\n\u00180=\n(a)\nDN= 0:05DS, \u0000 = 0:1Tc0\n(b)\nDN=DS, \u0000 = 0:01Tc0\n(c)\nDN= 0:05DS, \u0000 = 0:01Tc0\n(d)\n(e)\n (f)\n (g)\n (h)\nFIG. 3: Upper row: temperature dependencies of the GD \u000e\u000b(T) in FI/N/S systems. The three curves in each plot correspond\ntodN=\u00180= 0:8; 0:2; 0. Lower row: color plots of the functions \u000e\u000b(dN;T)=\u000e\u000bN. Horizontal lines in each panel are positioned\nas guide for eyes at dN=\u00180= 0:8; 0:2; 0 corresponding to the curves in the upper plot. The four columns correspond to various\nDynes parameters \u0000 =Tc0= 0:1; 0:01 and ratios of di\u000busion coe\u000ecients in N and S layers DN=DS= 1; 0:05 speci\fed on top of\nthe panels. Common parameters are dS= 3\u00180,\u001csnTc0= 1, \n = 0:02Tc0.\nResults and discussion Using the described formal-\nism we calculate the non-equilibrium spin polarization\n(12) in the N/S structure shown in Fig.1. This gives us\nthe local susceptibility (3) and the excess GD (4 ). The\nresulting temperature dependencies of \u000e\u000b(T) are shown\nin Fig. 3 for various parameters. The \frst column in\nFig.3 corresponds to \u0000 = 0 :1Tc0and identical di\u000busion\ncoe\u000ecients in N ans S layers. In the absence of N layer\ndN= 0 there is a usual coherence peak at T\u00190:9Tc\nwith the small amplitude \u000e\u000b=\u000e\u000bN\u00191:4. Adding the\nthin N layer with dN>0:1\u00180leads to the increase of the\npeak amplitude to \u000e\u000b=\u000e\u000bN\u00191:9 and shifting to lower\ntemperatures.\nThe peak is enhanced by decreasing the di\u000busion coef-\n\fcientDNin the normal layer. Qualitatively, this leads\nto better localization of surface bound states and hence\nto the increase of surface DOS. As shown in the second\ncolumn of Fig.3 for DN= 0:05DSand \u0000 = 0:1Tc0the\npeak is enhanced to \u000e\u000b=\u000e\u000bN\u00192:5 reached at T\u00190:7Tc\nwithdN= 0:2\u00180. This behaviour is quite similar to the\nexperimental observation13. For larger dN>0:5\u00180the\ntemperature dependence is qualitatively changed to the\nmonotonic increase down to the low temperatures. As\nshown by the yellow curve with dN= 0:8\u00180the increase\ncontinues to T\u00190:1Tc.\nEven larger increase is obtained for smaller Dynes pa-\nrameters \u0000 = 0 :01Tc0as shown in the third and fourth\ncolumns of the Fig. 3. For DN=DSwe obtain the max-imal value\u000e\u000b=\u000e\u000bN= 3. ForDN= 0:05DSwe obtain the\nmaximal value \u000e\u000b=\u000e\u000bN= 4:8. For all values of \u0000 we note\nthat forDN\u001cDSthe monotonically increasing \u000e\u000b(T) is\nobtained down to the threshold temperature of the order\nof Thouless ennergy Tth\u0019DN=d2\nN. As one can see in\nthe color plots Fig.3f,h for increasing dNit can be rather\nsmallTth\u001cTc.\nThe introduced model can explain the observed spin-\npumping enhancement in GdN/NbN system13assuming\nthat there is a naturally formed thin normal layer at\nthe FI/S interface. The pairing suppression at the inter-\nface can result from various reasons, including magnetic\ndisorder46,47, strong usual disorder48or the band struc-\nture modi\fcation49. It is straightforward to check our\nprediction of the enhanced GD by fabricating arti\fcial\nFI/N/S structures with various parameters.\nThe behaviour of \u000e\u000b(T) obtained in Figs.3b,d with\ndN= 0:8\u00180is qualitatively similar to the one observed ex-\nperimentally in Py/Nb/Pt heterostructures12,17. In the\nequilibrium state of our model the spin-triplet supercon-\nductivity is absent. Therefore the monotonic increase\nof GD due to the supercondducting correlations is not\nin principle an exclusive feature of the system with spin\nsuper-currents. However, the spin-triplet correlations are\ngenerated in the non-equilibrium case (11) providing23\nsigni\fcant contribution to the spin response (12).\nThe developed quasiclassical theory of spin pumping\ncan be generalized to the case of metallic ferromagnets5\nby introducing the \fnite spin-dependent tunnelling prob-\nability through the F/S interface43,50,51to the boundary\ncondition (10). This provides the way to study charge\nand heat transport induced by the magnetic precession\nas well as spin torques induced by voltage and tempera-\nture biases52{56.\nConclusions We have developed the general\nformalism to calculate spin-pumping in spatially-\ninhomogeneous metallic \flms with spin-active interfaces.\nAs an example we have considered the FI/N/S structure\nand found that the the presence of quasiparticle bound\nstates localized near the spin-active interface provides\nstrong enhancement of spin pumping which shows up in\nthe strong increase of the GD coe\u000ecient with decreasing\ntemperature below Tc. The model explains large peak\nof GD in Gd/NbN structures and shows the way to\ncontrolling spin pumping properties in superconducting\nsystems.\nAcknowledgements This work was supported by the\nAcademy of Finland (Project No. 297439) and Russian\nScience Foundation, Grant No. 19-19-00594. I thank\nYakov Fominov for comments.\nAppendix A: Stationary spin-mixing scattering\nmatrix\nNear the \rat FI/M surface we write wave functions in\nthe form kkeikkrwherekk=kzz+kyyis the conserved\nmomentum parallel to the interface. Along zcoordinate\nwe have 1D Shrodinger equations\ni@t = (^H\u0000\"F?) (A1)\n^H=\u0000@2\nx=2m+ [\"F+V+ (m^\u001b)Vs]\u0012(\u0000x) (A2)\nwherem=m(t).\nLet us \frst \fnd the frozen scattering matrix which de-\npends adiabatically on time. In this case the energy of\nincoming and scattered electrons coincide so that writing\n /ei\"twe get stationary 1D Shrodinger equation\n^H = (\"+\"F?) (A3)\n^H=\u0000@2\nx=2m+ [\"F+V0+ (m^\u001b)Vs]\u0012(\u0000x) (A4)\nwhere\"F?=\"F\u0000k2\nk=2m. For the energy we have \"=\nk2=2m\u0000\"Fwherek2=k2\nx+k2\nk. First, we \fnd the\nscattering matrix writing solutions\n kk=A+eikxx+A\u0000e\u0000ikxx(A5)\n kk=Bex=\u0015\u001b(A6)\nwhere\u0015\u00002\n\u001b= 2mV\u001b\u0000k2\nxandV\"(#)=V0+ (\u0000)Vsare\nthe spin-up (down) band energies in FI. The re\rection\ncoe\u000ecientS\u001b=A+=A\u0000is then\nS\u001b=ei'ei\u001b\u0002=2=1 +ikx\u0015\u001b\n1\u0000ikx\u0015\u001b(A7)Since we are interested in spin-dependent re\rection phase\nwe get the spin-mixing angle\nei\u0002=1 +k2\nx\u0015+\u0015\u0000+ikx(\u0015+\u0000\u0015\u0000)\n1 +k2x\u0015+\u0015\u0000\u0000ikx(\u0015+\u0000\u0015\u0000)(A8)\nwhich yields\n\u0002=2 = arcsin \nkx(\u0015+\u0000\u0015\u0000)p\n(1 +k2x\u0015+\u0015\u0000)2+k2x(\u0015+\u0000\u0015\u0000)2!\n(A9)\nFinally, the spin-dependent part of the scattering ma-\ntrix connecting the incident ^ iand re\rected ^ relectronic\nwaves written in the basis-independent form\n^ r=^S^ i (A10)\n^S=ei(m^\u001b)^\u001c3\u0002=2(A11)\nAppendix B: Time-dependent boundary conditions\nat the FI/metal interface\nHere we derive boundary conditions () starting from\nthe scattering theory of the interface between FI and\nmetal, either normal or superconducting one. The main\ndi\u000berence from the previous works deriving boundary\nconditions at FI/M interface is that the magnetization\nof FI depends on time m=m(t).\nWe consider matrix GF de\fned in a Keldysh-Nambu-\nspin space\n\u0014G(r1;r2;t1;t2) =\u0012^GR^GK\n0^GA\u0013\n(B1)\nwhere retarded, advanced and Keldysh parts are de\fned\nin a standard way as follows\n^GR(r1;r2;t1;t2) =\u0012(t1\u0000t2)\u0002 (B2)h\nh^\t(r1;t1)^\t+(r2;t2)i+h^\t(r1;t1)^\t+(r2;t2)ii\n^GA(r1;r2;t1;t2) =\u0012(t2\u0000t1)\u0002 (B3)h\nh^\t(r1;t1)^\t+(r2;t2)i+h^\t(r1;t1)^\t+(r2;t2)ii\n^GK(r1;r2;t1;t2) = (B4)\nh^\t(r1;t1)^\t+(r2;t2)i+h^\t(r1;t1)^\t+(r2;t2)i\nwhere the \feld operators ^\t = ( ^ \";^ #;\u0000^ +\n#;^ +\n\") satisfy\nthe equations of motion\ni@t^\t = ^H(t)^\t (B5)\nand the Hamiltonian has time-dependent order parame-\nter \u0001 = \u0001( t), boundary potential V=V(t)\n^H(t) = ^\u001c3(k2=2m\u0000\"F) + ^\u001c2\u0001(t) +^V(t) (B6)6\nThe GF satis\fes Gor'kov equations\n[i@t1\u0000^H(t1;r1)]^G=\u000e(t12)\u000e(r12) (B7)\n^G[\u0000i@t2\u0000^H(t2;r2)] =\u000e(t12)\u000e(r12) (B8)\nwherer12=r1\u0000r2andt12=t1\u0000t2. Assuming the \rat\nFI/M interface we consider transverse momentum com-\nponentskz;yas conserved quantities. The perpendicular\ncomponent kxchanges to the opposite one upon elec-\ntron re\rection. We are interested in the components of\nGF which are slowly varying as function of the center of\nmass coordinate r= (r1+r2)=2 and thus can be written\nas follows\n\u0014Gkk(x1;x2;t1;t2) =Z\ndr12e\u0000ikkr12\u0014G(r1;r2;t1;t2)\nThe GF satis\fes Gor'kov equations\n[i@t1\u0000^H(t1;z1)]\u0014Gkk=\u000e(t12)\u000e(x12) (B9)\n\u0014Gkk[\u0000i@t2\u0000^H(t2;x2)] =\u000e(t12)\u000e(x12) (B10)\n^H(t;x) =\u0000(@2\nx=2m+\"?)^\u001c3+ ^\u001c2\u0001(t) +^V(t) (B11)\nwhere\"?=\"F\u0000k2\nk=2m.\nLet's consider the Fourier expansion\n\u0014Gkk(x1;x2) =X\nk1;2ei(k1x1\u0000k2x2)\u0014Gkk(k1;k2) (B12)\nNear the M/FI interface z= 0 we can establish the con-\nnection between amplitudes\n\u0014Gkk(\u0000k1;k2) =^S(t1)\u0014Gkk(k1<0;k2) (B13)\n\u0014Gkk(k1;\u0000k2) =\u0014Gkk(k1;k2<0)^S+(t2) (B14)\nFrom these two relations we get\n\u0014Gkk(\u0000k1;\u0000k2) =^S(t1)\u0014Gkk(k1<0;k2<0)^S+(t2)\n(B15)\nRelations (B13,B14) can be obtained as follows. First,\nconsider the vicinity of interface jx1;2j\u001c\u0018where\u0018=\nvx=\u0001. In this case we can use the simpli\fed equation for\nGF neglecting the time derivative and order parameter\n[(@2\nx1=2m+\"?)^\u001c3\u0000^V(t1;x1)]\u0014G(x1<x2) = 0 (B16)\n(@2\nx2=2m+\"?)\u0014G(x2<x1)^\u001c3\u0000\u0014G(x2<x1)^V(t2;x2)] = 0\n(B17)\nThese are two independent equations identical to the\nShrodinger equation (A3) at \"= 0. Thus we can write\nthe solution\n\u0014Gkk(x1<x2) =X\nk1>0[e\u0000ik1x1+eik1x1^S(t1)]\u0014F2(x2)\n(B18)\n\u0014Gkk(x2<x1) =X\nk2>0\u0014F1(x1)[eik2x2+e\u0000ik2x2^S+(t2)]\n(B19)where ^F1;2(x) in principle can be arbitrary functions.\nComparing these relations with the general Fourier ex-\npansion (B12) we get Eqs. (B13,B14).\nThe quasiclassical GF in general is introduced accord-\ning to the following general procedure\n\u0014gp(r) =1\n\u0019Z1\n\u00001d\u0018pZ\ndr12e\u0000ipr12^\u001c3\u0014G(r1;r2)\nNear the \rat surface we have only the z-dependence\n^gp(x) =1\n\u0019Z1\n\u00001dqe\u0000iqxZ1\n\u00001d\u0018p^\u001c3^Gkk(kx+q;kx\u0000q)\nwhere we denote r12=r1\u0000r2,\u0018p= (k2\nz+k2\nk)=2m\u0000\"F.\nThen atx= 0 we have\n\u0014gp(x= 0) =1\n\u0019ZZ1\n\u00001dqd\u0018p^\u001c3\u0014Gkk(\u0000kx\u0000q;\u0000kx+q)\n\u0014gp(x= 0) =1\n\u0019ZZ1\n\u00001dqd\u0018p^\u001c3\u0014Gkk(kx+q;kx\u0000q)\n(B20)\nThen using relations B15\n\u0014gp(x= 0) =1\n\u0019ZZ1\n\u00001dqd\u0018p^\u001c3\u0014Gkk(kx+q;kx\u0000q) =\n1\n\u0019ZZ1\n\u00001dqd\u0018p^S(t1)^\u001c3\u0014Gkk(\u0000kx\u0000q;\u0000kx+q)^S+(t2)\u0019\n^S(t1)^gp(z= 0) ^S+(t2)\nwhere in the last relation we assume that ^Sdoes not de-\npend onq. Finally we get the time-dependent boundary\ncondition for quasiclassical functions\n\u0014gp(x= 0) = ^S(t1)\u0014gp(x= 0) ^S+(t2) (B21)\nExpanding ^S(t)\u00191 +i\u0002^\u001bm(t)=2 in Eq.B21 we get\nthe matrix current at the M/FI boundary\n^I(t1;t2) =\u0000Zd\np\n4\u0019(n\u0001vF)\u0014gp(t1;t2) = (B22)\n\u0000vFZ1\n0d^px^pz[\u0014gp(t1;t2)\u0000\u0014gp(t1;t2)]\u0019\n\u0000ivF\n2Z1\n0d^px^px\u0002(^px)[^\u001bm^\u001c3;\u0014g]t\nwhere we denote [ ^X;\u0014g]t=^X(t1)\u0014g(t1;t2)\u0000\u0014g(t1;t2)^X(t2),\nn=zis the normal to FI interface and denote the inci-\ndent ^p\u0001n<0 and re\rected ^p\u0001n>0 momenta.\nThis expression can be simpli\fed even more if we as-\nsume that due to the impurity scattering the anisotropic\nparts of GF are small. Then we can use two lowest order\nterms in the spherical harmonics expansion\n\u0014gp= \u0014g+p\u0001\u0014ga=p (B23)7\nKeeping only the s-wave term we get for the matrix cur-\nrent (B22)\n\u0014I(t1;t2) =i\u0017\u00001Jsd[^\u001c3^\u001bm;\u0014g]t (B24)\nwhere the conductance is given by\nJsd=\u0017vF\n4Z1\n\u00001d^pxj^pxj\u0002(^px) (B25)\nWe can \fnd the spin current using the general expres-\nsion\ni(t) =\u0019\u0017Tr4[^\u001b^IK(t;t)]: (B26)\nTaking into account the de\fnition of the spin density\nS(t) =\u0000\u0017Tr [^\u001c3^\u001bgK(t;t)]=8 (B27)\nthe spin current B26 \rowing from FI to the spin sink can\nbe written as\ni(t) =JsdS(t)\u0002m(t) (B28)\nAppendix C: Equation for the spectral and\ndistribution functions\nKinetic equation From the Keldysh-Usadel equation\nin the main text we obtain the \fnite-frequency kinetic\nequation\nr(Drfh) = [\u001c\u00001\nso+ 2(2\u0000 +i\n)N]fh (C1)\nD@xfh(x= 0) =\u00002iheffN@\"n0 (C2)\n@xfh(x=dS) = 0 (C3)\nwhereD=DTr(1\u0000^gR^gA)=2 and\u001c\u00001\nso= 4D=3D\u001csn. The\nsystem (C1, C2, C3) is linear with the coe\u000ecients de-\ntermined by the zero-order spectral function. Solving it\nwe \fnd the spin-dependent non-equilibrium distribution\nfunction generated by the dynamical spin-active inter-\nface.Spectral functions\nIn the adiabatic approximation we \fnd the spectral\nfunctions from the stationary Usadel equation\ni[(\"+i\u0000)^\u001c3;^g] +@x(D^g@x^g) = \u0001[^\u001c1;^g]\u0000[^\u0006so;^g] (C4)\nwith the boundary conditions\nD^g@x^g=iJsd[^\u001c3^\u001bm(t);\u0014g] (C5)\nUsing the normalization condition (^ gR)2= 1 we use the\nfollowing parametrization for equilibrium GF and correc-\ntions in the low-frequency adiabatic approximation\n^gR\n0= cos\u00120^\u001c3+ sin\u00120^\u001c1 (C6)\n^gR\nh= (\u0000sin\u00120^\u001c3+ cos\u00120^\u001c1)\u0012h (C7)\nThen we get the following equations for the parameters\n\u00120,\u0012h\ni(\"+i\u0000) sin\u00120+ \u0001 cos\u00120+@x\u0012D\n2@x\u00120\u0013\n= 0 (C8)\n@x\u00120(x= 0;dS) = 0 (C9)\n\u0012h\u0014\ni(\"+i\u0000) cos\u00120\u00002\n3\u001cso\u0000\u0001 sin\u00120\u0015\n+@x\u0012D\n2@x\u0012h\u0013\n= 0\n(C10)\nDN@x\u0012h(x= 0) = 2iheffsin\u00120; (C11)\n@x\u0012h(x=dS) = 0 (C12)\nSolving the nonlinear Eq.(C8,C9) together with the self-\nconsistency equation for \u0001 we obtain the zero-order spec-\ntral functions in the N/S structure. 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Woltersdorf\nInstitut für Physik, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle, Germany\nG. Schmidt\u0003\nInstitut für Physik, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle, Germany and\nInterdisziplinäres Zentrum für Materialwissenschaften, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle, Germany\n(Dated: November 28, 2021)\nWe present a novel process that allows the transfer of monocrystalline yttrium-iron-garnet microstructures\nonto virtually any kind of substrate. The process is based on a recently developed method that allows the fabri-\ncation of freestanding monocrystalline YIG bridges on gadolinium-gallium-garnet. Here the bridges’ spans are\ndetached from the substrate by a dry etching process and immersed in a watery solution. Using drop casting the\nimmersed YIG platelets can be transferred onto the substrate of choice, where the structures finally can be reat-\ntached and thus be integrated into complex devices or experimental geometries. Using time resolved scanning\nKerr microscopy and inductively measured ferromagnetic resonance we can demonstrate that the structures\nretain their excellent magnetic quality. At room temperature we find a ferromagnetic resonance linewidth of\nm0DHHWHM \u0019195\u0016T and we were even able to inductively measure magnon spectra on a single micron-sized\nyttrium-iron-garnet platelet at a temperature of 5 K. The process is flexible in terms of substrate material and\nshape of the structure. In the future this approach will allow for new types of spin dynamics experiments up to\nnow unthinkable.\nI. INTRODUCTION\nThe growth of high quality thin film yttrium-iron-garnet\n(YIG) is very challenging. Even today very low Gilbert\ndamping ( a\u00145\u000210\u00004) is only achieved for deposition on\ngadolinium-gallium-garnet (GGG) which is almost perfectly\nlattice matched to YIG (see overview in Schmidt et. al.1).\nNevertheless, for many experiments a GGG substrate is not\nsuitable. GGG exhibits a strong paramagnetism that even2in-\ncreases below 70K. This results in an enlarged Gilbert damp-\ning in a thin YIG film due to the coupling of the YIG with the\nsubstrate. As a consequence, many experiments which aim\nfor example for the investigation of the strong coupling of\nmagnons and microwave photons3,4are limited to bulk YIG\nfabricated by liquid phase epitaxy (LPE) or to macroscopic\nYIG spheres. Up to now, this problem prevents experiments in\nhybrid quantum magnonics on YIG microstructures. Further-\nmore, experiments using YIG microstructures and integrated\nmicrowave antennae on GGG are difficult because of its large\ndielectric constant ( e\u001930). Unfortunately, there also hasn’t\nbeen any successful attempt to grow high quality YIG with\nreasonably low Gilbert damping on other substrates. Thus,\na method to fabricate thin high quality YIG microstructures\non GGG along a subsequent transfer on a different substrate\nwould lead the way towards many new promising experiments\nand applications. We have developed a process that allows us\nto transfer YIG microstructures from GGG onto other sub-\nstrates. Although the process is not suitable for mass fabrica-\ntion it nonetheless enables a new class of experiments which\nuntil today seemed unthinkable.\nFIG. 1. Patterning process flow: (a) Array of monocrystalline YIG\nbridges5. (b) The AlOx mask is deposited by e-beam lithography,\nevaporation, and lift-off. (c) The bridges are detached from the sub-\nstrate by argon ion milling. (d) The AlOx is dissolved in ammonia\nwater releasing the remaining YIG platelets into the liquid.\nII. PROCESSING\nOur method is based on a fabrication process5using room-\ntemperatue (RT) pulsed laser deposition (PLD), lift-off and\nannealing, which yields freely suspended YIG structures,\nwhereby we apply the process in order to fabricate bridges or\ndoubly clamped beams. The suspended parts of these struc-\ntures exhibit extraordinary magnetic properties. For these\nstructures a ferromagnetic resonance (FMR) linewidth at 9.6\nGHz of m0DHHWHM=140\u0016T and a Gilbert damping of a\u0019\n2\u000210\u00004were demonstrated. Using this process we fabri-\ncate an array of 500,000 bridges of 1 :5\u00025\u0016m2span-size on a\nGGG substrate. We then mask the spans of the bridges by alu-\nminum oxide using electron beam lithography, e-beam evap-arXiv:2008.09390v1  [cond-mat.mes-hall]  21 Aug 20202\noration, and lift-off. [Fig. 1 (b)]. Using argon ion milling al-\nlows to remove the part of the bridge that connects the span\nto the substrate leaving the masked YIG as a micro slab like\nplatelet embedded in the aluminum oxide (AlOx). [Fig. 1 (c)].\nDissolving the mask in ammonia water lifts the 500,000 YIG\nmicro platelets from the substrate and immerses them in the\nsolution. The wet etchant is then stepwise replaced by water\nyielding a watery suspension of uniform monocrystalline YIG\nplatelets [Fig. 1 (d)]. By drop-casting the YIG platelets can\nnow be transferred to any substrate. After drying, the platelets\nstick to the substrate and even stay in place during subsequent\nspin-coating of further resist layers. With the help of addi-\ntioanl lithography the platelets can be integrated in complex\ndevices or applications.\nHere we show one example how a YIG platelet can be inte-\ngrated into a coplanar waveguide geometry to achieve in-plane\nexcitation and high sensitivity in FMR. As a substrate we use\nsapphire onto which 150nm of Au with a Ti adhesion layer\nwere deposited by electron beam evaporation. Sapphire is\nchosen because of its excellent properties for high frequency\nmeasurements. Before the drop-casting, a layer of PMMA is\nspun onto the sample. The suspension is exposed for a few\nseconds to ultrasonic agitation to ensure a homogeneous sus-\npension of the YIG platelets and by using a pipette a single\ndrop of the suspension is then put onto the sample. After the\ndrop-casting the YIG platelets are typically flat on the sample\nsurface but randomly oriented. Once a suitable YIG platelet is\nidentified we heat the sample up to 250\u000eC which is well above\nthe glass transition temperature of the PMMA6causing the\nYIG platelet to slightly sink into the PMMA film [Fig. 2 (a)].\nBy electron beam lithography we then crosslink the PMMA\nat the end of the bridge, defacto welding the bridge to the Au\nsurface [Fig. 2 (b)]. Using the PMMA layer under the YIG\nhas several advantages compared to direct deposition on the\nAu surface. No spin coating is required before the bridge\nis fixed and after removing the non-crosslinked PMMA the\nsample surface is now also clean from possible residue of the\ndrop-casting process. It should be noted that there is most\nlikely a gap of 10 \u000040nm between YIG and Au so the system\ncorresponds rather to a bridge with a YIG platelet as a span\nand two pedestals of PMMA as posts. To realize the final test\nstructure we now use electron beam lithography, AlOx evap-\noration and lift-off to mask the intended area of the CPW and\nthe YIG platelet itself. By Argon ion milling we remove the\nunmasked Au and Ti. After removing the AlOx mask we end\nup with a CPW perfectly aligned with the YIG platelet and\nideally suited in terms of size and shape for the FMR charac-\nterization of the YIG platelet [Fig. 2 (c)]. The final structure is\nshown in Fig. 3 as an false-color SEM image.\nIII. MAGNETIC PROPERTIES\nIn order to assess the sensitivity of our experiment we now\nperform FMR measurements. The samples are bonded onto a\nsample holder that fits into a4He bath cryostate. The cryostate\nis placed inside an electromagnet that can be rotated in the\nsample plane. The external magnetic field can be modulated\nFIG. 2. (a) The YIG drop-cast on the PMMA sinks into the poly-\nmer during heating. (b) The PMMA at the ends of the platelet is\ncrosslinked to fix the YIG to the Au. (c) Electron beam lithography\nand dry etching are used to pattern the CPW.\nFIG. 3. False-color SEM image of a transferred YIG platelet (ma-\ngenta) fixed with crosslinked PMMA (green) on top of a Ti/Au CPW\n(yellow). The bridge has a span length of 4 :5\u0016m, a width of 1 :5\u0016m\nand a nominell YIG layer thickness of approximately 160nm.\nusing an air coil of a few turns of Cu wire wound around the\nsample holder inside the cryostate. For our measurements the\nexternal magnetic field is oriented along the long side of the\nplatelet. RF excitation is done by applying an RF signal with a\npower of \u000021dBm. Measurements are performed by sweep-\ning the magnetic field at constant RF frequency. The transmit-\nted RF signal is rectified and the modulation of the external\nfield allows for lock-in detection to increase sensitivity. With\nthe YIG platelet centered on the waveguide the exciting RF\nfield is oriented in the sample plane and homogeneous over\nthe YIG platelet. As a consequence we can only excite stand-\ning spin wave modes with an uneven number of antinodes that\nhave non-zero magnetization.\nFig. 4 shows two resonance curves obtained at 4GHz at\nroom temperature and at 5K respectively. In both cases we\nobserve an extended spin-wave spectrum with a large number\nof backward-volume modes (BVMs). These discrete modes\nare caused by the finite size of the YIG platelet and corre-\nspond to standing spin wave modes as observed in a previ-\nous experiment5. Because of the complexity of the spectrum\nand the overlap of multiple modes it is difficult to obtain a\nlinewidth or even extract a Gilbert damping from measure-3\nFIG. 4. FMR spectra for a frequency of 4GHz at (a) 5K and (b)\n295K showing the occurance of several spin wave modes in the YIG\nbridge. The extended spin-wave spectra even for low temperatures\nsuggests a very low Gilbert damping.\nFIG. 5. Spatial resolved measurements acquired at a frequency of\n4GHz at different respective magnetic fields. The TRMOKE im-\nages show standing BVMs in the span of the bridge for m0Hextof\n(a) 74mT, (b) 80mT, (c) 84 :5mT and (d) 88 :5mT. The dotted lines\nserve as a guide to the eye to indicate the approximate sample posi-\ntion. m0Hextis applied along the x-direction.\nments at different respective frequencies. A closer look at the\nshape of the main resonance line indicates that it is not a sin-\ngle line but composed from at least two separate lines if not\nmore [Fig 6]. At 5K the spectrum is more noisy than at room\ntemperature but still the details of the spectrum are similar to\nthose at room temperature. The major difference to the room\ntemperature measurement is the change in resonance field that\ncan be attributed to the change in saturation magnetization7.\nWe perform TRMOKE experiments on the YIG in order\nto obtain more detailed information about the local struc-\nture of the excited modes. Further details of this technique\nare described in the work of Tamaru et. al.8and Neudecker\net. al.9. Again the measurements are performed with the exter-\nnal magnetic field oriented along the long side of the platelet.\nTRMOKE allows to locally image magnon modes in terms of\nboth intensity and phase5. To perform the spatially resolved\nimaging the frequency was set to 4 GHz at an RF amplitude of\n-25 dBm. The real and imaginary part of the dynamic suscep-\ntibility were detected in pointwise fashion while the magnetic\nfield was kept constant for each picture [Fig 5].\nThe spatially resolved measurements show several stand-\ning BVM with the fundamental mode with only one antin-\node [Fig. 5 (a)] and three standing BVMs with antinodes dis-\ntributed along the bridge in Fig. 5 (b)-(d)5. As expected all ob-\nserved modes exhibit an uneven number of antinodes. Again,\nFIG. 6. Main FMR line as composition of two separate lines for\na single transferred YIG platelet of 1 :5\u00024:5\u0016m2. The linewidth is\nm0DHHWHM=195\u0016T.\nit is not possible to extract a precise value for the line width for\nthis sample. Another platelet from the same batch was trans-\nferred into the gap of a coplanar waveguide. In this geometry\nthe out-of-plane RF field allows for TRMOKE measurements\nwith the external field applied perpendicular to the long side\nof the platelet. This results in a larger spacing between the\nresonance lines and yields the spectrum shown in Fig. 6. The\nresonance field is slightly shifted compared to the measure-\nments shown in Fig. 5. At 4 GHz we observe two superim-\nposed lines which can be fitted by two lorentzian line shapes.\nWe obtain a linewidth of m0DHHWHM\u0019195\u0016T. To the best of\nour knowledge even for large area thin films there are only two\npublications from other groups that show a smaller linewidth\nat this frequency10,11. For untransferred bridges (on GGG)\nwe have already measured a smaller linewidth, however, it is\nunclear whether the original sample produced for the drop-\ncasting was of similar quality. In any case the magnetic qual-\nity is only weakly affected by the transfer, if at all.\nIV . OUTLOOK\nThe presented process opens up a large number of options.\nAs we have shown in5the 3D patterning process is not limited\nto linear bridges. Besides we can also make frames, rings, cir-\ncular drums, tables, or other arbitrariliy shaped flat structures\nwhich would allow us to use the transfer technique presented\nhere. The main restriction is merely the size. With increasing\nstructure size the yield of the initial 3D patterning process is\nreduced and also the writing time increases linearly with the\narea. On the other hand we need a large number of structures\nto have enough statistical hits in the drop-casting process. A\nlow concentration of YIG structures in the suspension would\nmake the drop-casting a hopeless procedure. Beyond that, are\neven more options. Before the masking with AlOx we can\nperform additional processing on the bridges. We can for ex-4\nample deposit a thin metal film on top. After detachment and\ndrop-casting we have a 50:50 chance that the metal film ends\nup at the bottom of our platelet. A second evaporation step\ncould then be used to create a double side metallized YIG\nfilm as has been used in12for the demonstration of magnon\ndrag. In our case, however, we have no limitations as to the\nmetals that we want to use and their respective thicknesses.\nFurthermore we may even be able to nanopattern the metal\nbefore detaching the bridges and finally achieve a piece of\nYIG thin film with lithographically nanopatterned metal on\nboth sides. Our structures may even be suitable for hybrid\nquantum magnonics at mK temperatures. As van Loo et. al.13\nand Mihalceanu et. al.14have shown, the damping of thin film\nYIG increases at low temperatures, mainly because of inter-\naction with the GGG substrate. In our case the YIG platelet\nis no longer on the substrate. Even more it has never been in\ndirect contact with GGG so also contamination effects can be\nexcluded, making high performance at mK temperatures even\nmore likely. And finally these isolated structures may also\nbe suitable for the formation of magnon-based Bose-Einstein\ncondensates15.V . CONCLUSION\nWe have demonstrated that it is possible to transfer high\nquality thin film YIG microstructures onto other substrates\nand to integrate them in complex experiments. The magnetic\nquality is only slightly affected by the process, if at all.\nNotably, we are able to measure FMR spectra at 5K with\nmany details. This process opens up new routes towards a\nmultitude of experiments which formerly seemed completely\nout of reach.\nVI. DATA A VAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\nACKNOWLEDGMENTS\nWe wish to acknowledge the support of TRR227 project\nB02 WP3 and project B01.\n\u0003georg.schmidt@physik.uni-halle.de\n1G. Schmidt, C. Hauser, P. Trempler, M. Paleschke, and E. T.\nPapaioannou, physica status solidi (b) 257, 1900644 (2020).\n2V . Danilov, Y . V . Lyubon’ko, A. Y . Nechiporuk, S. Ryabchenko,\net al. , Soviet Physics Journal 32, 276 (1989).\n3H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein,\nA. Marx, R. Gross, and S. T. Goennenwein, Physical Review\nLetters 111, 127003 (2013).\n4Y . Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and\nY . Nakamura, Phys. Rev. Lett. 113, 083603 (2014).\n5F. Heyroth, C. Hauser, P. Trempler, P. Geyer, F. Syrowatka,\nR. Dreyer, S. Ebbinghaus, G. Woltersdorf, and G. Schmidt, Phys.\nRev. Applied 12, 054031 (2019).\n6M. Mohammadi, H. fazli, M. karevan, and J. Davoodi, European\nPolymer Journal 91, 121 (2017).\n7P. Hansen, P. Röschmann, and W. Tolksdorf, Journal of Applied\nPhysics 45, 2728 (1974), https://doi.org/10.1063/1.1663657.8S. Tamaru, J. Bain, R. Van de Veerdonk, T. Crawford, M. Coving-\nton, and M. Kryder, Journal of Applied Physics 91, 8034 (2002).\n9I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gub-\nbiotti, and C. Back, Journal of Magnetism and Magnetic Materi-\nals307, 148 (2006).\n10O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef,\nC. Hahn, A. H. Molpeceres, C. Carrétéro, E. Jacquet, C. Deranlot,\nP. Bortolotti, et al. , Applied Physics Letters 103, 082408 (2013).\n11J. Ding, T. Liu, H. Chang, and M. Wu, IEEE Magnetics Letters\n11, 1 (2020).\n12J. Li, Y . Xu, M. Aldosary, C. Tang, Z. Lin, S. Zhang, R. Lake,\nand J. Shi, Nature communications 7, 1 (2016).\n13A. F. van Loo, R. G. E. Morris, and A. D. Karenowska, Phys.\nRev. Applied 10, 044070 (2018).\n14L. Mihalceanu, V . I. Vasyuchka, D. A. Bozhko, T. Langner, A. Y .\nNechiporuk, V . F. Romanyuk, B. Hillebrands, and A. A. Serga,\nPhys. Rev. B 97, 214405 (2018).\n15S. O. Demokritov, V . E. Demidov, O. Dzyapko, G. A. Melkov,\nA. A. Serga, B. Hillebrands, and A. N. Slavin, Nature 443, 430\n(2006)."    },    {        "title": "2008.12221v3.Nutation_Resonance_in_Ferromagnets.pdf",        "content": "1 \n Nutation  Resonance in Ferromagnets  \nMikhail Cherkasskii1,*, Michael Farle2,3, and Anna Semisalova2 \n1 Department of General Physics 1 , St. Petersburg State University , St. Petersburg , 199034, Russia  \n2 Faculty of Physics and Center of Nanointegration (CENIDE), University of Duisburg -Essen, Duisburg, 47057, Germany  \n3 Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Russia  \n \n* Corresponding author: macherkasskii@hotmail.co m \n \n \nThe inertial  dynamic s of magnetization in a ferromagnet is investigated  theoreticall y. The analytically derived dynamic \nresponse upon microwave excitation shows two pea ks: ferromagnetic and nutation resonances. The exact analytical expressions \nof frequency and linewidth of  the magnetic  nutation resonance are deduced from the frequency dependent susceptibility \ndetermined by the i nertial Landau -Lifshitz -Gilbert equation.  The study  shows that the dependence of nutation linewidth on the \nGilbert precess ion damping  has a minimum , which becomes more expressive with increas e of the applied magnetic field.  \n \nPACS numbers: 76.50.+g, 78.47.jp, 75.50. -y \n \nI. INTRODUCTION  \nRecently, the effects of inertia in the spin dynamics of \nferromag nets were reported to cause nutation resonance  [1-\n12] at frequencies  higher  than the conventional ferromagnetic \nresonance . It was shown  that inertia  is responsible for the \nnutation , and that this type of motion should be considered  \ntogether with magnetization precession in the applied \nmagnetic field . Nutation in ferromagnets was confirmed \nexperimentally only recently  [2], since  nutation and \nprecession operate at substantially different time scales , and \nconventi onal microwave ferromagnetic resonance (FMR) \nspectroscopy techniques do not easily reach the high -\nfrequency (sub-Terahertz) regime re quired to observe the \ninertia effect which in addition yields a much weaker signal . \nSimilar to any other oscillatory system, t he magnetiza tion \nin a ferromagnet has resonant frequencies  usually studied by  \nferromagnetic resonance  [13,14]. The resonant \neigenfrequency is determined by  the magnetic parameters  of \nthe material  and applied magnetic field . Including inertia of \nthe magnetization in th e model description shows that  nutation \nand precession are complementary to each other  and several \nresonances can be generated . In this Letter , we concentrate on \nthe investigation of the resonance characteristics of nutation.  \nThe investigation of nutation is connected to the progress \nmade  in studies of the  spin dynamics at ultrashort  time \nscales  [15,16] . These successes led to the rapi d development \nof a new scientific field , the so -called ultrafast magnetism  [17-\n25]. The experimental as well as theoretical investigation of \nthe inertial  spin dynamics is at the very beginning , although it \nmight be of significance for future high speed spintronics \napplications  including ultrafast magnetic switching .  Besides nutation driven by magnetization inertia, several  \nother origins  of nutation have been reported . Transient \nnutations (Rabi oscillations) have been widely investigate d in \nnuclear magnetic resonance  [26] and electron spin \nresonance  [27-29], they were  recently addressed in \nferromagnets  [30]. A complex dynamics and Josephson \nnutation of a local spin \n1/ 2s  as well as large spin cluster \nembedded in the tunnel junction between ferromagnetic leads \nwas shown to occur  due to  a coupling to Josephson \ncurrent  [31-33]. Low-frequency nutation was observ ed in \nnanomagnets exhibiting a non -linear FMR  with the large -\nangle precession of magnetization where the onset of spin \nwave instabilities can be delay ed due to geometric \nconfinement  [34]. Nutation dynamics due to inertia of \nmagnetization in ferromagnetic thin films was observed for \nthe first time by Neeraj et al.  [2]. \nThe microscopic derivation of the magnetization  inertia \nwas performed in  ref. [3-7]. A relation between the Gilbert \ndamping constant  and the inertia l regime  characteristic time  \nwas elaborated in ref. [3]. The exchange interaction, damping, \nand moment of inertia can be calculated from first principles \nas shown in  [7]. The study of inertia spin dynamics  with a \nquantum approach  in metallic ferromagnets  was performed \nin [8]. In addition, nutation  was theoretically analyzed as a \npart of magnetization dynamics in ferromagnetic \nnanostructure  [9,10] and nanoparticles  [11]. Despite these \nadvances, exact analytical expressions for the high-frequency \nsusceptibility including inertia had not been derived yet.  \nIn [35], the inertial regime  was introduced in the \nframework of the mesoscopic nonequilibrium \nthermodynamics theory , and it was shown to be responsible \nfor the nutation superimposed on the precession of \nmagnetization . Wegrowe and Ciornei  [1] discussed  the  \n2 \n equivalence between the inertia in the dynamics of uniform \nprecession and a spinning top within the framework of the  \nLandau –Lifshitz –Gilbert equation generalized to the inertial \nregime. This equation was  studied  analytical ly and \nnumerical ly [12,36]. Although the se reports provide \nnumerical tools for obtaining resonance characteristics, the \ncomplexity of the numerical solution of differential equations \ndid not allow to estimate the nutation frequency and linewidth \naccurately . Also in a recent  remarkable paper  [37] a novel \ncollective excitation  – the nutation wave  – was reported,  and \nthe dispersion characteristics  were derived wit hout discussion \nof the nutation  resonance lineshapes  and intensities.  \nThus, at present, there is a necessity to study  the resonance \nproperties of nutation in ferromagnet s, and this paper is \ndevoted to this study. We performed the investigation based \non the Landau -Lifshitz -Gilbert equation with  the addition al \ninertia term  and provide an  analytical solution.   \nIt is well known that the Landau -Lifshitz -Gilbert equation \nallow s finding the susceptibility as the ratio between  the time-\nvarying magnetization and the time-varying driving magnetic \nfield (see for exampl e [38,39] and references therein). This \nsusceptibility describes  well the magnetic response of a \nferromagnet in the linear regime, that is  a small cone angle of \nthe precession . In this description , the ferromagnet usually is \nplaced in a  magnetic field  big enough to align all  atomic \nmagnetic moments along the field , i.e., the ferromagnet is in \nthe saturated state and the  magnetization  precess es. The \napplied  driving magnetic field  allows one to obser ve FMR  as \nsoon as the driving field frequency coincides with  \neigenfrequency of precession. Using the expression for \nsusceptibility, one can elaborate  such resonance \ncharacteristics as eigenfrequency and linewidth.  We will \npresent similar expressions for the dynamic susceptibility, \ntaking nutation into account.  \nII. SUSCEPTIBILITY  \nThe ferromagnet  is subjected to a uniform bias magnetic \nfield \n0H  acting along the z -axis and being strong enough to \ninitiate  the magnetic saturation  state. The small time -varying \nmagnetic field \nh  is superimposed on the bias field. The \ncoupling between impact and response, taking into account \nprecession, damping, and nutation, is given by the Inertial  \nLandau -Lifshitz -Gilbert ( ILLG)  equation  \n \n2\n2\n0,effd d d\ndt M dt dt       M M MMH   (1) \nwhere \n  is the gyromagnetic ratio, \nM  the magnetization  \nvector , \n0M  the magnetization at saturation, \neffH  the vector \nsum of all magnetic fields, external and internal, acting upon \nthe magnetization , \n the Gilbert damping , and \n  the inertial \nrelaxatio n time. For simplicity, we assume that the \nferromagnet is infinite, i.e. there is no demagnetization  correction , with negligible magnetocrystalline anisotropy , and \nonly the  externally applied field s contribute  to the total  field. \nThus, the bias magnetic field \n0H  and signal field \nh  are \nincluded in \neffH . We assume that the signal is small \n0,hH\n hence the magnetization is directed along \n0.H  \nOur interest is to study the correlated  dynamics of nutation \nand precession  simultaneously;  therefore we write the \nmagnetization  and magnetic field  in the general ized form  \nusing  the Fourier transformation  \n \n 01ˆ  ,\n2itt M z d e\n\n\n Mm   (2) \n \n 01ˆ  ,\n2it\nefft H z d e\n\n\n Hh   (3) \nwhere \nˆz  is the unit vector along the z -axis. If we  substitute  \nthese expressions  in the ILLG equation  and neglect the small \nterms, it leads to  \n \n\n \n 00\n211 \n22\nˆˆ\nˆˆ .i t i td i e d e\nM z H z\ni z z   \n\n   \n   \n\n     \n    \n        m\nhm\nmm   (4) \nBy performing the Fourier transform and changing the order \nof integration , equation  (4) becomes   \n \n\n\n \n00\n21 2\n1 2\nˆˆ\nˆˆ ,it\nitd dt i e\nd dt e\nM z H z\ni z z\n  \n\n   \n    \n\n \n\n\n   \n \n     \n       \nm\nhm\nmm   (5) \nwhere the integral representation of the Dirac delta  function  \ncan be found. With the delta function,  the equation (5)\nsimplifies to  \n \n  \n 00\n2ˆˆ\nˆˆ .i M z H z\ni z z     \n      \n   m h m\nmm   (6) \nBy projecting to Cartesian coordinates and  introducing the \ncircular variables  for positive and negative circular \npolarization  \n,xy m m im  \n,xy h h ih one obtains  \n \n  \n 2\n20,\n0,HM\nHMm m i m m h\nm m i m m h   \n\n        \n          \n        (7) \nwhere \n0 H H  is the precession frequency and \n0.M M\n The small -signal susceptibility follows  from \nthese equations :  \n3 \n  \n2\n2,\n,\n.M\nH\nM\nHim\nih\n\n   \n\n  \n   \n\n  \n  \n\n   (8) \nIt is seen that the susceptibility (8) is identical with  the \nsusceptibility for LLG equation , if one drops  the inertial term , \nthat is  \n0.  \nLet us separate dispersive and d issipative  parts  of the \nsusceptibility \n,i       \n \n \n 2\n2,\n,\n,\n,MH\nM\nMH\nMD\nD\nD\nD\n\n   \n\n   \n\n\n\n\n\n\n  \n\n\n   (9) \n \n 2 2 4 3\n2 2 22\n22 , 1   \n     \n    H H HD   (10) \n \n 2 2 4 3\n2 2 22\n22 . 1   \n     \n    H H HD   (11) \n \nThe frequency dependence of the dissipative parts of \nsusceptibilit ies \n and \n  is shown in the Fig. 1. The plus \nand minus subscripts correspond to right -hand and left -hand \ndirection of rotation. Since the denominators \nD  and \nD  are \nquartic polynomials, four critical points  for either  \n or \n \ncan be expected . Two of them  that are extrema with  a clear \nphysical meaning are plotted.  In Fig. 1(a) the extremum , \ncorresponding  to FMR  at \n0 H H  is shown . Due to the \ncontribution  of nutation , the frequency and linewidth of this \nresonance are slightly different from the ones of usual FMR . \nThe resonance occurs  for right -hand precession, i.e. positive \npolarization.   \nIn Fig. 1(b) the nutation resonance  possessing negative \npolarization is presented.  Note  that the polarizations of \nferromagnetic and nutation resonances are reverse d. \nIII. APPROXI MAT ION FOR  NUTATION \nFREQUENCY  \nLet us  turn to the description  of an approximation  of the \nnutation resonance frequency. If we equate the denominator \nD\n to zero, solve the resulting equation, we obtain the \napproximation from the real part of  the roots. This is reasonable , since the numerator of \n  is the linear function of \n\n , and we  are interested in \n1.  Indeed , the equation  \n \n 2 2 4 3 2 2\n202 2 1\n2H\nHH      \n     \n    (12) \nhas four roots that are complex conjugate in pairs  \n \n1,221 1 4 2,2H\nFMRiiw   \n       (13) \n \n1,221 1 4 2.2H\nNiiw   \n       (14) \n \nFIG. 1. (Color online) (a) The FMR peak with nutation. (b) \nThe nutation resonance. The calculation was performed for \n1/ 2 28  GHz T ,\n \n00 1 T, M  \n00 100 mT, H  \n0.0065\n and \n1110  s.  \n \nOne should choose  the same sign from the \n  symbol  in each \nformula , simultaneously . The real part of expression  (13) \ngives the approximate frequency for FMR , but in negative \nnumbers, so the sign should be inversed . The approximate \nfrequency of FMR in positive numbers can be derived from \nequation \n0. D  The approximate nutation frequency  is \nobtained  by the real part of  the expression (14). One takes half \nthe sum of two conjugate roots  \n1,2,Nw neglect s the high \n  \nterms , and obtains  the nutation resonance frequency  \n \n1 1 2\n.2NH\nw \n\n   (15) \nNote  that the expression of \nNw  is close  to the approximation  \ngiven in  [36] at \n1/ , H    namely  \n \nweak\nnu1\n.H \n\n   (16) \nThe similarity of both approximations b ecomes clear , if we \nperform a Taylor series expansion  and return to the notation \n,H\n \n\n2\n2 2 3 3 3\n2\nweak\nnu\n2 2 3 3 31 1 2 1\n2 2 4\n1,4\n1 1\n2\n1.18\n6H HH\nN\nH\nH HH\nHw\nO\nO  \n \n    \n   \n       \n\n   \n\n \n \n4 \n IV. PRECISE  EXPRESSIONS FOR  FREQUENCY \nAND LINEWIDTH OF NUT ATION  \nThe analytical approach proposed in this Letter yields  \nprecise values of the frequency  of nutation resonance  and the \nfull width at half maximum (FWHM) of the peak . The \nfrequency is found by extremum, when the derivative of the \ndissipative part of susceptibilities (9) is zero \n \n0.\n   (17) \nIt is enough to determine  zeros of the n umerator of th e \nderivative , that are given by  \n \n 2 2 4 3 2 2 23 4 2 1 0.HH                (18) \nLet us use Ferrari's solution  for this quartic equation  and \nintroduce the notation:  \n \n22\n2\n2\n2\n2\n3\n23\n24\n343\n4,\n2 1,  \n3,8\n,28,\n3.16 25,\n6rH\nrH\nrr\nr r\nr r r\nrr\nr r r r\nr\nr rrr\nr\nr\nrC\nE\nC\nC\nCEcA\nB\nBaA A\nBBbAA\nBB\nA AA\n\n \n\n\n\n\n\n\n  \n\n   (19) \nIn Ferrari's method , one should determine a root of the nested \ndepressed cubic equation . In the investigated case , the root is \nwritten  \n \n5,6r\nr r ray U V     (20) \nwhere  \n \n32\n3\n2\n32,27 4 2\n,3\n12\n1,\n.3 108 8r r r\nr\nr\nr\nr\nr\nrr\nrr\nr r rP Q QU\nPVU\nPc\nQaa\nabc   \n\n \n     (21) \nThus, the precise value of the nutation frequency is given by  \n \n2\n42\n2 13 2 .2 2rr r\nN\nr\nr\nrr\nrry\nA\nbaa\nay\nyB  \n   \n   (22) \nThe performed analysis shows that approximate value of \nnutation resonance frequency is close to precise value.  The linewidth of the nutation resonance is found at  a half \npeak height. If one denotes the maximum by \n,N X \n the equation  which determines \nfrequencies at half magnitude  is \n \n 2 2 4 3 2 2\n212 2 12\n2 0.H\nH H MX      \n      \n\n   (23) \nWe repeat the procedure for finding solu tions with Ferrari's \nmethod  introducing the  new notations  \n \n 22\n2\n2\n2\n2\n3\n23\n24\n3 4 21\n2\n,\n12 1 ,2\n1\n2\n3,8\n,28\n3.16 2,\n56,\n4lw\nlw\nlw\nlw\nlw\nlw lw\nlw\nlw lw\nlw lw lw\nlw\nlw lw\nlw lw lw lw lw lw\nlw\nlw lwH\nHM\nH\nlw\nlw\nlw lwA\nB\nBaA A\nBBbAA\nB B B D\nA AX\nX\nCX\nDX\nEX\nC\nCD\nA\nCEcA A\n\n \n \n\n\n\n\n\n\n  \n\n\n\n  \n  \n\n\n   (24) \nA root of the nested depressed cubic equation \nlwy  must be \nfound in the same way as provided in (20) with the \ncorresponding replacement of variables, i.e. subscript r is \nreplaced by lw. The difference between two adjacent roots \ngives the nutation linewidth  \n \n23 2 .\n2lw\nN lw lw\nlw lwbay\na y   \n   (25) \nThe explicit expression for the linewidth can be written using \nthe equations  (19)-(25). \n \n \nFIG. 2. (Color online) The dependence of the nutation \nlinewidth on the inertial relaxat ion time for \n00 100 mT, H\n \n00 1 T, M  and \n0.0065.  \n \n5 \n  \nThe effect of the inertial relaxation time on the nutation \nlinewidth is shown in Fig.  2. One can see that increasing \ninertial relaxation time leads to narrowing of the linewidth. \nThis behavior is expected and is consistent with the traditional \nview that decreasing of losses results in narrowing of \nlinewidth.  \n \n \nFIG. 3. (Color online) The dependence of nutation \nresonance linewidth on precession Gilbert damping \nparameter at different magnetic fields \n0H  for \n00 1 T M\nand \n1110  s.  \n \nSince the investig ated oscillatory system implemen ts \nsimultaneous two types of  motions , it is of interest to study the \ninfluence of the Gilbert precession  damping parameter \n  on \nthe nutation  resonance linewidth. The result is presented in \nFig. 3 and is valid for ferromagnets with vanishing anisotropy.  \nOne sees that the dependence of \nN  on \n  shows a \nminimum that becomes more expressive with increasing  bias \nmagnetic field.  In other words, t he linewidth is parametrized \nby the magnitude of field. This non-trivial  behavior of \nlinewidth relates with the nature of th is oscillatory system, \nwhich performs  two coupled  motions.  \nTo elucidate the non-trivial behavior , one can consider  the \nsusceptibility  (9) in the same way as it is usually performed \nfor the forced harmonic oscillator with damping  [40]. For this \noscillator , the linewidth can be direct ly calculated from the \ndenominator of the response expression once the driving \nfrequency is equal to eigenfreq uency. In the investigated case \nof magnetization with inertia , the response expression is (9) \nwith denominator s (10) and (11) written as  \n \n 2 2 4 3\n2 2 22\n21 . 2H H HD   \n      \n   \n   (26) \nSince the applied magnetic field is included in this expression  \nas \n0,H H the linewi dth depends on the field.  \nThe obtained result can be generalized to a fin ite sample \nwith magnetocrystalline anisotropy with method of effective \ndemagnetizing factors  [41,42] . In this case the bias magnetic field \n0H  denotes an external field and in the final expressions \nthis field should be replaced by \n 0 0 0ˆˆ ,i a d NN  H H M  \nwhere \nˆ\naN  is the anisotropy demagnetizing tensor and \nˆ\ndN  is \nthe shape demagnetizing tensor.  \nV. CONCLUSION  \nIn summary, we derived a general analytical expression for \nthe linewidth and f requency of nutation resonance in \nferromagnets, depending on magnetization, the Gilbert \ndamping, the inertial relaxation time and applied magnetic \nfield.  We show the nutation linewidth can be tuned by the \napplied magnetic field , and this tunability breaks the direct \nrelation between losses and the linewidth. This for example  \nleads to the appearance of a minimum  in the nutation \nresonance linewidth for the damping parameter  around  \n0.15.\n The obtained results are valid for ferromagnets with \nvanishing anisotropy . \nACKNOWLEDGEMENTS  \nWe thank  Benjamin Zingsem for helpful discussions . In \npart funded by Research Grant No. 075 -15-2019 -1886 from \nthe Government of the Russian Federation, the Deutsche \nForschungsgemeinschaft (DFG, German Research \nFoundation) – projects  405553726 (CRC/ TRR 270 ), and \n392402498 (SE 2853/1 -1).  \n \n[1] J.-E. Wegrowe and M. -C. Ciornei, Am. J. Phys. 80, 607 \n(2012).  \n[2] K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Zhou \nHagström, S. S. P. K. Arekapudi, A. Semisalova, K. Lenz, \nB. Green, J. C. Deinert, I. Ilyakov, M. Chen, M. Bawatna, \nV. Scalera, M. d’Aquino, C. Serpico, O. Hellwig, J. E. \nWegrowe, M. Gensch, and S. Bonetti, Nat. Phys. (2020).  \n[3] M. Fähnle, D. Steiauf, and C. Illg, P hys. Rev. B 84, \n172403 (2011).  \n[4] D. Thonig, O. Eriksson, and M. Pereiro, Sci. Rep. 7, 931 \n(2017).  \n[5] R. Mondal, M. Berritta, and P. M. Oppeneer, J. Phys. \nCondens. Matter 30, 265801 (2018).  \n[6] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppeneer, \nPhys. Rev. B 96, 024425 (2017).  \n[7] S. Bhattacharjee, L. 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A 64, 968 (1951).  \n "    },    {        "title": "2008.13061v3.Exploring_a_quantum_information_relevant_magnonic_material__Ultralow_damping_at_low_temperature_in_the_organic_ferrimagnet_V_TCNE_x.pdf",        "content": " \n 1  \nExploring a quantum-information-relevant magnonic material: ultralow \ndamping at low temperature in the organic ferrimagnet V[TCNE] x \n \nH. Yusuf*1, M. Chilcote*1,2, D. R. Candido3, S. Kurfman1, D. S. Cormode1, Y. Lu1, M. E.  \nFlatté3, E. Johnston-Halperin1 \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210 \n \n2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853 \n \n3Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, 52242 \n \n* These authors contributed equally to this work. \n \nAbstract:  Quantum information science and engineering requires novel low-loss magnetic \nmaterials for magnon-based quantum-coherent operations. The search  for low-loss \nmagnetic materials, traditionally driven by applications in microwave electronics near \nroom-temperature, has gained additional constraints from the need to operate at cryogenic \ntemperatures for many applications in quantum information science and technology. \nWhereas yttrium iron garnet (YIG) has been the material of choice for decades, the \nemergence of molecule-based materials with robust magnetism and ultra-low damping has \nopened new avenues for exploration. Specifically, thin-films of vanadium \ntetracyanoethylene (V[TCNE] x) can be patterned into the multiple, connected structures \nneeded for hybrid quantum elements and have shown room-temperature Gilbert damping \n(α = 4 × 10-5) that rivals the intrinsic (bulk) damping otherwise seen only in highly-polished \nYIG spheres (far more challenging to integrate into arrays). Here, we present a \ncomprehensive and systematic study of the low-temperature magnetization dynamics for \nV[TCNE] x thin films, with implications for their application in quantum systems. These \nstudies reveal a temperature-driven, strain-dependent magnetic anisotropy that \ncompensates the thin-film shape anisotropy, and the recovery of a magnetic resonance \nlinewidth at 5 K that is comparable to room-temperature values (roughly 2 G at 9.4 GHz). \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 2 We can account for the se variations of the V[TCNE] x linewidth within the context of \nscattering from very dilute paramagnetic impurities, and anticipate additional linewidth \nnarrowing as the temperature is further reduced.  \n \nThe search for low-loss magnetic materials dates to the early days of radio and \nmicrowave electronics [1–3], and the study of elementary excitations, or magnons, in these \nmagnetically-ordered materials has proven to be a rich area of research for both \nfundamental physics and their potential technological applications. More recently, interest \nin these low-loss systems has expanded to include applications in the field of quantum \ninformation technology such as quantum sensing and quantum transduction [4 –7], wherein \nlow-temperature operation allows for the freeze-out of thermal excitations and access to \nthe single-quantum regime . In this regime the field of quantum magnonics utilizes hybrid \narchitectures for coupling magnons to other quantum degrees of freedom, such as \nmicrowave photons, with the aim of extending their functionality in the quantum limit \n[8,9]. It has been demonstrated that magnons can be resonantly excited over a wide range \nof microwave frequencies, allowing for precise control of qubit states mediated by coherent \nexchange via cavity-mode photon excitations [4 ,7]. Magnons also exhibit the potential to \ncoherently couple localized spin-qubits with high cooperativity [10] . However, while \nmagnons exist in a wide range of materials, the same delocalized electrons that are most \noften responsible for stabilizing ferromagnetic order also contribute to electron-magnon \nscattering [11], leading to substantial losses in most metallic ferromagnets. As a result, the \nstudy of low-dissipati on magnon dynamics for quantum applications has focused on \ninsulating ferromagnets and ferrimagnets, with yttrium iron garnet (YIG) and its close \nrelatives holding pride of place as the benchmark low-loss materials for more than 50 \nyears  [ 4,12–14]. As a result, despite these longstanding and emerging needs, applications \nare still constrained by the materials limitations of YIG; namely the need for growth or \nannealing at high temperatures (typically 800° C)  [15–17] and the resulting difficulty in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 3 integrating and patterning YIG thin-films with other microwave electronic structures and \ndevices.   \nIn this context, the emergence of the molecule-based ferrimagnet vanadium \ntetracyanoethylene (V[TCNE] x) has dramatically expanded the playing field for low-loss \nmagnets. Despite what one might expect from its molecular building blocks, V[TCNE] x \nhas a magnetic ordering temperature of over 600 K and shows sharp hysteresis at room-\ntemperature  [ 18–20]. Moreover, its dynamic properties are exceptional, showing ultra-\nnarrow ferromagnetic resonance (FMR) linewidth (typically ~ 1 – 1.5 G at 9 .4 GHz) with \na Gilbert damping parameter,  of 4 × 10-5 for thin-films [ 18,21]. As a comparison, the \nbest YIG thin-films typically show  = 6.5 × 10-5  [22] and a value of 4 × 10-5 is competitive \nwith the intrinsic  damping of bulk YIG  = 3 × 10-5  [15,23]. From an applications \nperspective, V[TCNE] x has been shown to deposit on a wide variety of substrates without \ncompromising material quality  [24–26], facile encapsulation allows for direct integration \nwith pre-patterned microwave structures for operation under ambient conditions [ 27], and \nrecent work has demonstrated patterning at length scales down to 10 m without increased \ndamping  [21]. However, while these properties clearly establish the potential of \nV[TCNE] x for new applications in traditional microwave electronics, very little is known \nabout its low-temperature magnetization dynamics and therefore its potential for \napplications in quantum information science and engineering (QISE ). \nHere we present a detailed study of the low-temperature magnetic resonance of \nV[TCNE] x films. We identify two regimes. In the high-temperature regime, extending from \n300 K down to 9 K, we observe a monotonic shift in the resonance frequency consistent \nwith a temperature-dependent strain. This strain results in a crystal-field anisotropy that \nincreases with decreasing temperature with a magnitude of at least 140 Oe and the same \nsymmetry, but opposite sign, to the shape anisotropy of the thin-film.  In addition, we \nobserve an increase in linewidth consistent with magnon scattering from paramagnetic \nimpurities similar to what has been observed in YIG  [23,28,29], but with an amplitude 3 \ntimes smaller ( i.e. an increase in linewidth by 9 times in V[TCNE] x as compared to 28 \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 4 times in YIG  [23, 30]). In the low-temperature regime, starting at 9 K and extending to 5 \nK, we observe a discontinuous change in both anisotropy and linewidth: the anisotropy \nabruptly reverts to the room-temperature symmetry (in-plane easy-axis) and the linewidth \napproaches room-temperature values (2.58 G) at 5 K. This linewidth variation can be \nexplained using a model for scattering between magnons and paramagnetic impurities that \ntakes into account the finite spin-lifetime of the impurity spins  [23,31]. At high \ntemperatures (above 100 K) the spin-lifetime is sufficiently short that changes in \ntemperature do not lead to significant changes in scattering rate, and at low-temperatures \n(below 9 K) the spin-lifetime becomes long with respect to the spin-magnon scattering \ntime, resulting in a saturation of the excited state. At intermediate temperatures (from 9 K \nto 100 K) this spin-magnon scattering dominates relaxation due to the increase of the \nground state impurity population, which results in a local maximum in the linewidth that \nis 9 times larger than the room-temperature value. These results are extremely promising \nfor low-temperature applications of V[TCNE] x magnonics, promising low-temperature \nmagnon resonators with unprecedented low-loss that can be integrated on-chip into \nmicrowave electronic circuits and devices [20,21]. \nFor this study, thin-films of V[TCNE] x are deposited on sapphire (Al 2O3 (0001)) \nsubstrates using chemical vapor deposition (CVD) growth process consistent with prior \nreports [18,19].  Briefly, argon gas transfers the two precursors tetracyanoethylene (TCNE) \nand vanadium hexacarbonyl (V(CO) 6) into the reaction zone of a custom-built CVD reactor \n(Fig. 1(a)) where V[TCNE] x is deposited onto polished sapphire substrates. The system is \ntemperature controlled to maintain the TCNE, V(CO) 6 and the reaction zones at 65° C, 10° \nC and 50° C respectively . After growth the sample is mounted on a custom, microwave-\ncompatible sample holder and sealed using a septa cap in an electron paramagnetic \nresonance (EPR) grade quartz tube in an argon environment. When the sample is not being \nmeasured, it is stored in a - 35° C freezer housed in an argon glovebox and is stable for over \none month [ 27].  \nFerromagnetic resonance (FMR) measurements are performed using a Bruker EMX \nPlus X-band EPR spectrometer at temperatures ranging from 300 K down to 5 K. The \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 5 microwave frequency of the spectrometer is tuned between 9 and 10 GHz for optimal \nmicrowave cavity performance before the measurement, and then the frequency is fixed \nwhile the DC field is swept during data collection.  Figure 1(b) shows a representative \nroom-temperature FMR measurement of a typical V[TCNE] x thin-film with the external \nmagnetic field applied in the plane of the sample. The resonance feature is consistent with \npreviously reported high-quality V[TCNE] x thin-film growth, showing a peak- to-peak \nlinewidth of 1.5 G at 9.4 GHz  [18,19].  \nComparing this data to FMR measurements at temperatures of 80 K and 40 K \n(Figure 1(c)) shows an increase in the resonance field of over 40 G (roughly half of the \nsaturation magnetization, 4𝜋𝑀 𝑠) as the temperature decreases. Since the applied \nmicrowave frequency is held constant at 9.4 GHz , this shift must arise from fields internal \nto the V[TCNE] x film, i.e. magnetic anisotropy fields. Note that since the value of the DC \napplied field varies between 3350 G and 3450 G, well above 4𝜋𝑀 𝑠, changes in the \nmagnetization of the film are not expected to contribute to this field shift.  In a similar \nfashion, changes in the shape-dependent anisotropy fields can be ruled out, leaving only \nchanges to the crystal-field anisotropy as a potential source of this phenomenon. Crystal-\nfield anisotropy originates from the interaction of a material’s mean exchange field and the \nangular momenta of neighboring atoms (ions) in the material , indicating that there is a \ntemperature dependence to the local atomic environment within the V[TCNE] x films , e.g. \ndue to a temperature-dependent strain within the film.  \nIn order to more comprehensively map out this phenomenon angle dependent FMR \nmeasurements are performed to quantitatively track changes in the magnetic anisotropy at \ntemperatures of 300 K, 80 K, and 40 K (Fig. 2). Variation of the magnetic resonance field \nas a function of the angle between the applied field and the princip al axes of the film can \nbe modeled by considering the free energy of the magnetic system with anisotropic \ncontributions. If we consider the case of a uniaxial anisotropy with the hard-axis \nperpendicular to the easy-axis, and where the magnetization is parallel to the external field \n(i.e. external field is much larger than the saturation magnetization, as is the case here) the \ntotal magnetostatic energy is as follows  [ 32]: \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 6   \n                                              𝐸 = −𝑴 · 𝑯 + 2𝜋 (𝑴 · 𝒏 )𝟐− 𝐾 (𝑴 · 𝒖 𝑀 ⁄)2                          (1) \n \nwhere M is the magnetization, H is the applied magnetic field, n is the unit vector parallel \nto the normal of the magnetic sample , u is the unit vector parallel to the easy-axis and K is \nan anisotropy constant. For the case of in-plane uniaxial anisotropy, this simplifies to \n \n           𝐸 = − 𝑀𝐻 (sin𝜙sin2𝜃 + cos2𝜃)+ 2𝜋𝑀2cos2𝜃 − 𝐾 sin2𝜃sin𝜙2               (2) \n    \nwhere 𝜃 is the angle between M and the sample normal and 𝜙  \nis the azimuthal angle. Minimizing the magnetostatic energy with respect to 𝜃 , one will \nfind that the easy-axis orientation occurs when 𝜃 = 2 𝑛𝜋±  𝜋\n2, where n is an integer . Using \nthis simple symmetry analysis, we can see that the data in Fig. 2 indicates that the easy-\naxis lies in-plane at a temperature of 300 K (i.e. the resonance field is smallest when the \napplied magnetic field lies in-plane) and out- of-plane at a temperature of 40 K (i.e. the \nresonance field is smallest when the applied magnetic field is out- of-plane). In this context, \nthe lack of variation in resonance field at 80 K indicates a nearly isotropic magnetic \nresponse. This switch in magnetic easy-axis from in-plane to out- of-plane further supports \nthe proposition that there is an additional temperature-dependent crystal-field contribution \nto the magnetic anisotropy.  \nIn previous studies, templated growth of V[TCNE] x resulting in nanowire \nmorphologies induced an additional in-plane magnetic anisotropy with easy-axis \nperpendicular to the long-axis of the nanowires, strongly suggesting the presence of a \nstrain -dependent contribution to the crystal-field anisotropy  [ 33]. In the thin-films studied \nhere, such a strain-dependent crystal-field effect would be expected to generate anisotropy \nparallel to the surface normal, i.e. in the out- of-plane direction. The anisotropy field would \nthen be parallel to the expected shape anisotropy from a thin-film, though not necessarily \nwith the same sign. As a result, if there is a difference in the coefficient of thermal \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 7 expansion between the V[TCNE] x film and the sapphire substrate then the temperature \ndependence of magnetic anisotropy can potentially be understood as a proxy for a \ntemperature dependence of strain in the thin- film; such variations in strain leads to changes \nin the local atomic structure, leading to the observed changes in magnetic anisotropy. We \nnote that while the coefficient of thermal expansion for V[TCNE] x has not yet been \nmeasured, the value for sapphire is 5.4 ppm/K and typical values for molecular-based solids \ncan range somewhere between 28 –500 ppm/K [ 34]. Assuming no strain at room-\ntemperature, this would then imply a compressive strain between 0.67% to 15% at the \nsapphire –V[TCNE] x interface at 5 K , leading to an out- of-plane distortion whose symmetry \nis consistent with the observed anisotropy.    \nA schematic describing how these two anisotropy fields would be expected to \ninteract as a function of temperature can be found in Fig. 3(a). At a temperature of 300 K \n(Fig. 3(a), upper panels), the orientation of the easy-axis is determined by the shape \nanisotropy, resulting in an in-plane easy-axis for thin-films. But at a temperature of 40 K \n(Fig. 3(a) lower panels), there is an additional crystal-field anisotropy, 𝐻⊥, proposed that \ndominates the shape anisotropy, reorienting the easy-axis to be out-of-plane. This \nsymmetry analysis also explains the lack of orientation dependence at a temperature of 80 \nK, which is apparently the temperature at which the strain-driven crystal-field anisotropy \nperfectly cancels out the shape anisotropy. We note that similar phenomenology is also \nobserved in vanadium methyl tricyanoethylenecarboxylate (V[MeTCEC] x) thin-films (see \nsupplementary materials), indicating that this temperature- and strain-dependent \nanisotropy is a general property of this class of metal-ligand ferrimagnets. \nThe fact that the shape and proposed crystal-field anisotropies have the same \nsymmetry make it challenging to distinguish between the two; therefore, an effective field \nis defined as 𝐻𝑒𝑓𝑓= 4𝜋𝑀 𝑒𝑓𝑓= 4𝜋𝑀 𝑠− 𝐻 ⊥, where 𝑀𝑠 is the saturation magnetization and \n𝐻⊥is the crystal-field anisotropy. Figure 2 shows the effects of this net anisotropy field in \nthe form of resonance field shifts and a change in the easy-axis orientation . Quantitatively \nextracting the magnitude and direction of this anisotropy field provides detailed insight \ninto the role of crystal-field anisotropy in tuning the magnetic response of V[TCNE] x thin-\nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 8 films. To this end, each scan is fit to the sum of the derivatives of absorption and dispersion \nfrom a Lorentzian function to extract the resonance frequency and linewidth (experimental \ndata are obtained using a modulated-field technique that yields the derivative of the \nexpected Lorentzian resonance lineshape). For scans showing an out- of-plane easy-axis a \nsingle derivative sum provides good agreement with the data, while for scans showing in-\nplane easy-axis more complex structure is observed requiring the addition of up to three \nderivative sums. In the results discussed below we focus on the behavior of the primary , \ni.e. largest amplitude, peak (a full description of the fitting and resulting phenomenology \ncan be found in the supplemental material).  \nFigure 3(b) shows the extracted resonance field plotted against sample rotation \nangle for the high-and low-temperature data shown in Fig. 2, 300 K and 40 K, respectively . \nTaking into account a uniaxial out- of-plane anisotropy defined by 𝐻𝑒𝑓𝑓, as described \nabove, the angular dependence for in-plane to out- of-plane rotation of a thin-film sample \nis given by  [19,35,36]: \n \n𝜔\n𝛾 =  √(𝐻 − 𝐻 𝑒𝑓𝑓cos2𝜃)(𝐻 − 𝐻 𝑒𝑓𝑓cos 2𝜃)\n=  √(𝐻 − (4𝜋𝑀 𝑠− 𝐻 ⊥) cos2𝜃)(𝐻 − (4𝜋𝑀 𝑠− 𝐻 ⊥)cos 2𝜃)  \n                                        \nwhere  is the resonance frequency and  is the gyromagnetic ratio. As a result, the \nphenomenology of the data presented in Fig. 2 can be understood as an 𝐻𝑒𝑓𝑓 that is positive \nat 300 K and negative at 40 K, as 𝐻⊥ increases with decreasing temperature, consistent \nwith the mechanism for anisotropy switching described in Fig. 3(a). This qualitative \nunderstanding can be made quantitative by fitting the data in Fig. 2 using Eq. (3) to extract \n𝐻𝑒𝑓𝑓= 4𝜋𝑀 𝑒𝑓𝑓 of 91.2 G  1.6 G and -22.8 G  0.4 G, respectively. \nFigure 3 (c) shows this 𝐻𝑒𝑓𝑓 plotted against temperature over the temperature range \nfrom 300 K to 5 K, extracted from angular dependencies such as the measurements \npresented in Fig. 2 . It should be noted that each anisotropy point in Figure 3(c) represents (3) \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 9 a fit to a complete angular dependence such as the data shown in Figure 3(b).The effective \nfield mak es a smooth transition through zero from positive (in-plane) to negative (out- of-\nplane) at a temperature of roughly 80 K . This behavior is qualitatively consistent with the \nphenomenological model presented above and reveals a magnitude of the variation in 𝐻𝑒𝑓𝑓, \nfrom +91 .2 G ± 1.6 G at 300 K to - 45.2 G ± 1.1 G at 10 K, that is roughly 150% of the \nroom-temperature value.  \nNotably, this more comprehensive study also reveals new phenomenology at the \nlowest temperature of 5 K, where the anisotropy abruptly shifts back to in-plane with a \nvalue of +26.2 G ± 0.6 G (roughly 25% of the room-temperature value). This behavior \nreproduces across all samples measured and is quantitatively reproduced upon temperature \ncycling of individual films. The abruptness of this change is distinct from the broad and \nmonotonic behavior observed for temperatures greater than 9 K . The origin of this abrupt \nchange is unclear, but there are two potential explanations consistent with this \nphenomenology. First, it is possible that the increase in strain results in an abrupt relaxation \nthrough the creation of structural defects. This explanation would require some level of \nself-healing upon warming in order to explain the reproducibility of the transition. Given \nthe lack of long-range structural order in V[TCNE] x films as-grown [ 37] it is possible that \nany residual structural defects do not contribute to additional magnetic loss (damping). \nSecond, it is possible that there exist paramagnetic spins in the system that magnetically \norder at temperatures below 9 K. If such spins were preferentially located in an interface \nlayer, for example, their ordering could create an exchange bias that would then pull the \neasy-axis back to an in-plane orientation. \nThe temperature dependence of the linewidth of the magnetic resonance provides \nan additional avenue for evaluating these potential explanations. Figure 4 shows the \nlinewidth for the in-plane magnetic resonance from 300 K to 5 K, with additional data to \nmore clearly resolve the sharp change between 5 K and 9 K. The linewidth data presented \nin Figure 4 is extracted from a single (in-plane applied magnetic field orientation) scan. As \na result, the initial dataset underlying Figure 3 was supplemented by a second temperature \ndependent scan at fixed angle in Figure 4. This data reveals a monotonic increase in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 10 linewidth with decreasing temperature from 300 K down to 9 K followed by a dramatic \ndecrease in linewidth between 9 K and 5 K, coincident with the abrupt change in magnetic \nanisotropy. We note that in studies of YIG thin-films broadly similar phenomenology is \nobserved, though with a maximum in linewidth that is both higher amplitude (roughly 28 \ntimes the room-temperature value) and at higher temperature (typically 25 K) than is \nobserved here [23,30]. Prior work [23,28 ] has explained this behavior using a model of \nmagnon scattering from paramagnetic defect spins (also referred to as two-level \nfluctuators, TLF) wherein the scattering cross-section at high temperature increases with \ndecreasing temperature as the thermal polarization of the spins increases. This \nphenomenology competes with magnon-pumping of the paramagnetic spins into their \nexcited state, a process that saturates as the spin-lifetime of the defects becomes long \nrelative to the spin-magnon scattering time. The competition between these two processes \nyields a local maximum in the damping (linewidth) that depends on the temperature \ndependent spin lifetime, ts, the energy separation between majority and minority spin states, \nℏωeg, and the difference between that energy splitting and the uniform magnon energy,  \n(ℏω - ℏωeg ). \nIn this model, the linewidth expression is proportional to the square of the exchange \ninteraction energy between V[TCNE] x atoms and the impurity level ( ℏωint)2 ~ (ℏωeg)2, a \nline-shape factor accounting for the finite spin lifetime, 1/𝑡 𝑠/(ℏ2/𝑡𝑠2 + (ℏω - ℏωeg )2ts2), and \nthe ratio between the ground and excited impurity states for fast impurity relaxation , \ntanh(ℏω/2k BT)  [23, 28], \n \nΔ𝐻 =  𝑆\n𝛾 𝑁𝑖𝑚𝑝\n𝑁 (ℏ𝜔 𝑖𝑛𝑡)2 1/𝑡 𝑠 \nℏ2/𝑡𝑠2+ (ℏ𝜔 − ℏ𝜔 𝑒𝑔)2 𝑡𝑠2tanh (1\n2 ℏ𝜔\n𝑘𝐵𝑇) + 𝐻 𝑂 \n \nwhere Nimp/N is the ratio between number of impurit ies and number of V[TCNE] x atoms, \nand S is the averaged V[TCNE] x spin per site , 𝛾 is the gyromagnetic ratio and 𝐻𝑂 is a \nconstant offset due to other relaxation mechanisms. In addition, we assume spin lifetime ts (4) \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 11 = t∞ 𝑒𝐸𝑏𝑘𝐵𝑇⁄ [31, 38, 39] where t∞ is the spin lifetime limit at very high temperatures, and \nEb is a phenomenological activation energy. Figure 4 includes a fit of Eq. (4) to the \nexperimental linewidth (orange line) that yields for S ~ 1 and ωint ~ ωeg the parameters: \nωegt∞ = 0.98, Eb = 1meV, ω egNimp/N = 36.5GHz and 𝐻𝑜= 1 G . Interestingly, if we assume \na reasonable value for ℏωeg of 1.3 meV, a value of Nimp/N = 0.1 follows, thus indicating \nthat V[TCNE] x is an exceptional low-loss magnetic material even if we assume an impurity \nconcentration as high as 10%. This observation is consistent with the hypothesis of \ninsensitivity to structural defects discussed above. \nHowever, it is important to note that the peak in linewidth coincides with the abrupt \nreversion in anisotropy from an out- of-plane easy-axis to an in-plane easy-axis. This \nchange in magnetic anisotropy has the potential to have a substantial impact on spin-\nmagnon scattering efficiency. For example, this change will result in a shift of the energy \nof the magnon bands (see Eq. 1), and if this change involves a commensurate change in the \nstrain there will also be a modification to the spin-orbit coupling and exchange parameters \nat the paramagnetic defects. It should be noted that although this reentrant anisotropy is an \nintriguing feature, the fits to our model for TLFs in Figure 4 are able to reproduce our \nlinewidth data without reference to this effect. As a result, we interpret this fit as an upper \nbound on Eb. This is represented by the additional fits shown in Figure S7 within the \nSupplemental Material wherein we assume a lower temperature for the nominal peak in \nlinewidth occurring due to spin-magnon scattering that is experimentally preempted by the \nchange in magnetic anisotropy. These alternate fits agree with experimental observations \nat temperatures above 9 K, and therefore must be considered as possible mechanisms. \nMoreover, if the residual paramagnetic spins are ordered at temperatures below 9 K, one \nwould require a large amount of energy (>> ℏω) to populate their excited states, which is \nunlikely to happen. Hence, magnetic ordering of the paramagnetic spins would also \nenhance the suppression of spin-magnon scattering, resulting in the sharp linewidth \nsuppression for T < 9 K.  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 12 When considering the expected behavior as the temperature is further reduced below \n5 K, as would be the case for many applications in QISE, it is useful to consider recent \nmilliKelvin-range measurements of YIG films [40]. That work confirms the expected \ncontinued narrowing down to 500 mK followed by a modest increase from 500 mK down \nto 20 mK, for an overall line narrowing of roughly a factor of 2. The model of scattering \nfrom TLFs described above is consistent with this result in YIG if one supposes a second \npopulation of TLFs that are dipole coupled to the magnons rather than exchange coupled , \nfor example dilute magnetic impurities in the substrate or environment. We note that \nextending this model into V[TCNE] x requires taking into account: i) the substantial \ndifference in structure and chemistry between V[TCNE] x and YIG, and ii) the fact that Ms. \nin V[TCNE] x is roughly 20 times smaller than in YIG. The former consideration indicates \nthat the presence of these dipole coupled TLFs need not correlate between the two systems, \nwhile the latter predicts that any relaxation associated with their presence should be \nreduced by a factor of 20 from Ref. [40]. As a result, the overall factor of 2 decrease in \nlinewidth observed in YIG between temperatures of 5 K and 20 mK should be taken as an \nextremely conservative lower bound on the performance of V[TCNE] x. Given that the \nlinewidth in V[TCNE] x at 5 K is already on par with its room temperature value, these \nresults firmly establish the suitability for this material for applications in quantum \nmagnonics and related aspects of QISE. \nIn conclusion, this work presents the first systematic study of the magnetization \ndynamics of V[TCNE] x at low temperatures. A strong variation in resonance frequency and \nanisotropy with temperature is observed , and attributed to a temperature-dependent strain \narising from the mismatch in thermal expansion coefficients between V[TCNE] x films and \ntheir sapphire substrates. The resonance linewidth of these films is found to increase with \ndecreasing temperature up to a maximum value of 15 G (roughly 9 times the room-\ntemperature value) and is well fit by a model based on magnon scattering from \nparamagnetic defect spins. At 5 K the magnetic anisotropy reverts to in-plane, coinciding \nwith a nearly complete recovery of the resonance linewidth to room-temperature values; \nquantitative modeling suggests the linewidth behavior arises from scattering from \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 13 paramagnetic defect spins that is suppressed at very low-temperature. This suppression of \nspin-magnon scattering is expected to strengthen as temperature is further decreased into \nthe milli-Kelvin range due to freeze-out of thermal magnons and phonons, providing a \ncompelling case for the utility of V[TCNE] x for low-temperature microwave applications, \nsuch as those emerging in the field of quantum information science and technology. \n \nAcknowledgements:  The authors would like to thank A. Franson for providing a \nsoftware suite for fitting FMR spectra as well as general fitting assistance, and G. Fuchs \nfor fruitful discussions . The work presented in the main text, both experiment and theory, \nwas primarily supported by the U.S. Department of Energy, Office of Basic Energy \nSciences, under Award Number DE-SC0019250. S. Kurfman was supported by NSF \nEFMA-1741666 and grew V[TCNE] x calibration samples used for preliminary \nmeasurements not explicitly included in this paper . Work on V[MeTCEC] x presented in \nthe supplementary material was performed by M. Chilcote and Y. Lu with the support of \nNSF Grant No. DMR- 1741666. \n \nData availability statement:  See supplementary material at URL will be inserted by AIP \nPublishing for datasets pertaining to temperature-dependent anisotropy of V[MeTCEC] x, \nmethod for extracting linewidth of V[TCNE] x from FMR scans and additional fits to \nexperimental data highlighting temperature dependence of V[TCNE] x linewidth. \n \nReferences:  \n \n[1] A. Raveendran, M. T. Sebastian, and S. Raman, “Applications of Microwave \nMaterials: A Review” J. Electron. Mater. 48, 2601 (2019). \n[2] Ü. Özgür, Y. Alivov, and H. Morkoç, “Microwave ferrites, part 1: Fundamental \nproperties” J. Mater. Sci. Mater. Electron. 20, 789 (2009). \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 14 [3] J. M. Silveyra, E. Ferrara, D. L. Huber, and T. C. 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M iller, “A \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 17 Room-Temperature Molecular/Organic- Based Magnet” Science (80 -. ). 252, 1415 \nLP (1991). \n[26] K. I. Pokhodnya, A. J. Epstein, and J. S. Miller, “Thin -Film V[TCNE]x Magnets” \nAdv. Mater. 12, 410 (2000). \n[27] I. H. Froning, M. Harberts, Y. Lu, H. Yu, A. J. Epstein, and E. Johnston-Halperin, \n“Thin -film encapsulation of the air-sensitive organic-based ferrimagnet vanadium \ntetracyanoethylene” Appl. Phys. Lett. 106, (2015). \n[28] P. E. Seiden, “Ferrimagnetic resonance relaxation in rare -earth iron garnets” Phys. \nRev. 133, A728 (1964). \n[29] A. M. Clogston, “Relaxation Phenomena in Ferrites” Bell Syst. Tech. J. 34, 739 \n(1955). \n[30] C. L. Jermain, S. V. Aradhya, N. D. Reynolds, R. A. Buhrman, J. T. Brangham, M. \nR. Page, P. C. Hammel, F. Y. Yang, and D. C. Ralph, “Increased low -temperature \ndamping in yttrium iron garnet thin films” Phys. Rev. B 95, 174411 (2017). \n[31] W. A. Yager, J. K . Galt, and F. R. Merritt, “Ferromagnetic resonance in two nickel -\niron ferrites” Phys. Rev. 99, 1203 (1955). \n[32] H. Puszkarski and M. Kasperski, On the Interpretation of the Angular Dependence \nof the Main FMR/SWR Line in Ferromagnetic Thin Films  (2012). \n[33] M. Chilcote, M. Harberts, B. Fuhrmann, K. Lehmann, Y. Lu, A. Franson, H. Yu, \nN. Zhu, H. Tang, G. Schmidt, and E. Johnston- Halperin, “Spin -wave confinement \nand coupling in organic- based magnetic nanostructures” APL Mater. 7, (2019).    \n[34]   Y. Mei, P. J. Diemer, M. R. Niazi, R. K. Hallani. K. Jarolimek, C. S. Day, C. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 18 Risko, J. E. Anthony, A. Amassian and O. D. Jurchescu, “Crossover from band -\nlike to thermally activated charge transport in organic transistors due to strain-\ninduced traps” P NAS 114, 33 (2017). \n[35] H. Suhl, “Ferromagnetic Resonance in Nickel Ferrite Between One and Two \nKilomegacycles” Phys. Rev. 97, 555 (1955). \n[36] J. Smit and H. G. Beljers., “Ferromagnetic resonance absorption in BaFe 12O19” \nPhilips Res. Rep. 10, 113 (1955). \n[37] M. Chilcote, Y. Lu, and E. Johnston-Halperin, Organic-Based Magnetically \nOrdered Films  (World Scientific, 2018). \n[38]   J. K. Galt and E. G. Spencer, “Loss Mechanism in Spinel Ferrites ” Phys. Rev. 127, \n1572, 1962. \n[39]   H. Maier-Flaig , S. Klingler, C. Dubs, O. Surzhenko, R. Gross, M. Weiler, H. \n          Huebl, and S. T. B. Goennenwein , “Temperature dependent damping of yttrium \n          iron garnet spheres ” Phys. Rev. B 95, 214423 (2017). \n[40]   S. Kosen, A. F. van Loo, D. A Bozhko, L. Mihalceanu, R. Gross, and A. D. \n          Karenowska , “Microwave magnon damping in YIG films at millikelvin \n          temperatures ” APL Mater. 7, 101120  (2019 ). \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 19 Figure Legends:  \n \nFigure 1  \n \n(a) Schematic (planar view) of the CVD growth system; (b) FMR scan of V[TCNE] x \nthin film at 300 K with the applied magnetic field applied in the plane (IP) of the \nsample with 𝜃 = 90𝑜 and resonance frequency of 9.4 GHz. ΔH pp denotes the peak-\nto-peak linewidth measured as the difference between the positive and negative peak \npositions; (c) FMR line scans for in-plane field orientation at 300 K, 80 K and 40 K \nwith 𝜃 = 90𝑜 and resonance frequency of 9.4 GHz. \n \nFigure 2  \n \nAngle-dependent FMR spectra at temperatures of 300 K, 80 K and 40 K at different \nfield orientations with respect to the sample normal. Nominally the sample is rotated \nfrom  𝜃 = − 10𝑜 to  𝜃 =  100𝑜 in increments of 10𝑜, where 𝜃 =  90𝑜 and 𝜃 =  0𝑜 \nare in-plane and out- of-plane field orientations respectively. Angle corrections have \nbeen taken into account (through fitting with Eq. ( 3)) to reflect the actual rotation \nangles, denoted by the black arrows to the right of each of the temperature-labeled \npanels. \n \nFigure 3  \n \n(a) Schematic of the changes in anisotropy at 300 K and 40 K. 𝑯𝒂𝒑𝒑 denotes the \nexternal magnetic field, 𝑯𝒅𝒆𝒎𝒂𝒈  represents the demagnetizing field of the \nV[TCNE] x film and 𝑯𝒄𝒓𝒚𝒔𝒕𝒂𝒍  is the crystal-field anisotropy. It should be noted that \na finite thin-film has a (negligibly) small demagnetization field when the external \nfiled is applied in the plane since this is not a truly infinite film; (b) Resonance field \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 20 at different field orientations plotted against sample rotation angles for 300 K (open \ncircles)  and 40 K (filled circles) and fits to Eq. (3) (dashed and solid line, \nrespectively) to extract the effective field 𝑯𝒆𝒇𝒇; (c) 𝑯𝒆𝒇𝒇 plotted against temperature \nranging from 300K – 5K. The inset shows the FMR lineshapes at 300 K and 5 K; \nfitting the data to extract the linewidth at FWHM gives 1.63 G and 2. 58 G \nrespectively , this shows that the two linewidths are indeed comparable with the \nlinewidth at 5 K only about 1.66 times larger than the room-temperature value . For \nboth (b) and (c), experimental errors are smaller than the point size.   \n \nFigure 4  \n \nV[TCNE] x linewidth as a function of temperature (black points) and \ncorresponding curve fit (orange line) using Eq. (4). \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 21  \n \n \n \n \n \n \n \n \n \n  \n  \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n     Figure 1 H. Yusuf  et al. \n  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 22  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2 H. Yusuf  et al. \n \n \n \n \n \n \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 23  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3 H. Yusuf  et al. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 24  \n \n \n \n \n \n \n \nFigure 4 H. Yusuf  et al. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 \n 25  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1116/5.0044193 1 Supplementary Materials for “ Exploring  a quantum -information -relevant  \nmagnonic  material:  ultralow  damping  at low temperature  in the organic  \nferrimagnet  V[TCNE] x” \nH. Yusuf *1, M. Chilcote *1,2, D. R. Candido3, S. Kurfman1, D. S. Cormode1, Y. Lu1, M. E.  \nFlatté3, E. Johnston -Halperin1 \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210  \n \n2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853  \n \n3Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, 52242  \n \n* These authors contributed equally to this work.  \n \n1. Temperature -dependent anisotropy of V [MeTCEC ]x \nHere, we investigate the magnetic properties of  vanadium methyl tricyanoethylene  \ncarboxylate  V[MeTCEC ]x thin-films using temperature -dependent cavity ferromagnetic \nresonance (FMR). The MeTCEC ligand is similar to the TCNE described in the main text, \nand these results demonstrate that strain -dependent anisotropy is a general feature of this \nclass of metal -ligand materials. Figure  S1a shows the molecular structures of both the \nTCNE molecule and the MeTCEC molecule discussed below.  Figure  S1b shows \ntemperature -dependent magnetization  data for zero field -cooled (ZFC; open black squares) \nand field -cooled (FC; open red circles) measurements , and electron transport data (filled \nblack squares) collected for V[MeTCEC] x thin-films on the same temperature axis. Notice \nthat the maximum in the ZFC magnetization curve – sometimes referred to as the blocking \ntemperature[13,14] – corresponds to the rapid rise observed in the resistance data. This \nchange in electronic and magnetization properties has been associated with carrier freeze  \nout and a magnetic phase transition in related materials such as magnatites, but in light of  2 the results presented in the main text we note \nthat a structural transition associated with \nincreased strain in the films may also play a role \nin these measurements.  \nV[MeTCEC] x samples are deposited on \nAl2O3(0001) substrates using a previously \nreported synthesis and chemical vapor \ndeposition (CVD) growth process.[4,16] During \nthe deposition, argon gas carries the two \nprecursors, MeTCEC and V(CO) 6, into the \nreaction zone where V [MeTCEC ]x is deposited \nonto one or more substrates. The system \nemploys three independently temperature -\ncontrolled regions for the MeTCEC , V(CO) 6, \nand reaction zone with typical setpoints of \n55 °C, 10 °C, and 50  °C, respectively and with \ntypical flow rates for each precursor of 50 \nsccm. Sample growth, manipulation, and \nhandling is p erformed in an argon glovebox \n(O2 < 1.0 ppm; H 2O < 1.0 ppm).   \nAfter growth, samples are mounted onto custom microwave compatible sample \nholders in the appropriate orientation, protected from undesired rotation, and flame -sealed \nin evacuated electron paramagnetic resonance (EPR) grade quartz tubes without exposure \nto air. When not being measured, the sealed samples are stored in a -55 °C freezer and are \nfound to be stable for weeks.  \nFigure  S2 shows four ferromagnetic resonance (FMR) spectra of V[MeTCEC] x \noriented both in  plane (90 °; see inset to Fig.  3c) and out of plane (0 °) at 140  K and 80  K. \nThe FMR response of magnetic materials is sensitive to the local field environment of the \nFigure S1 (a) The molecular structures of \ntetracyanoethylene (TCNE) and \ntricyanoethylenecarboxylate (MeTCEC). (b) \nMagnetization vs. temperature curves for zero field -\ncooled (ZFC; open black squares) and zero field -\ncooled (FC; open red circles) measurements. On the \nsame  temperature axis, resistance vs. temperature \ndata is shown for a V[MeTCEC]x thin -film (filled \nblack squares). The corresponding dependent -axis is \nshown on the right axis. Note the maximum in the \nmagnetization data corresponds to the rapid rise \nobserved in  the resistance data.   3 sample and therefore allows for sensitive characterization of the anisotropy fields in \nV[MeTCEC] x. FMR measurements are performed using a Bruker electron paramagnetic \nresonance spectrometer setup for X -band measurements with 200 µW of applied \nmicrowave power and fitted with an Oxford Instruments ESR900 cryostat insert. The \ncryostat is cooled by flowing liquid nitrogen and operates at tempe ratures ranging from \n80 K to 300  K with better than 50  mK stability during FMR measurements. In standard \noperation, the microwave frequency of the spectrometer is tuned between 9 and 10  GHz \nfor optimal microwave cavity performance before the measurement, a nd then the frequency \nis fixed while the DC field is swe pt during the measurement.   \nFigure  S2a shows FMR spectra collected at 140  K for the magnetic field applied in  \nplane (𝜃 = 90°) and out  of plane ( 𝜃 = 0°). Consistent with prior FMR measurements of \norganic -based magnetic  materials,[6,16,17] the center field associated with th e resonant \nfeature in the in -plane spectrum is at a lower field than that of the out -of-plane spectrum, \nand therefore the easy magnetization axis is oriented in the plane of the film. This easy -\naxis orientation is the expected outcome resulting from the sh ape anisotropy present in \nthin-film samples. Figure  S2b also shows FMR spectra collected with the magnetic field \napplied in  plane ( 𝜃 = 90° ) and out of plane (𝜃 = 0°).  However, this data is collected at 80  K, \nfurther below the maximum in the V [MeTCEC ]x ZFC magnetization curve than the data \nshown in Fig . S2a. Surprisingly, the center field of the dominant resonance feature in the \nin-plane spectrum is at a higher field than that of the out -of-plane spectrum. This behavior \nseems to indicate that th e sample has an easy  axis oriented out of the plane of the sample; \nthe spectra show signs of a switch in the magnetic easy  axis from in  plane to out  of plane \nas it is cooled from 140  K to 80  K.  4 To investigate this behavior in greater \ndetail, angular -dependent data is collected in 10 ° \nincrements as the ap plied field is rotated from in  \nplane ( 𝜃 = 90° ) to out of plane (𝜃 = 0°)  of the \nsample. The data is shown in Figs . 3a and b for \n140 K and 80  K respectively. A gray dashed line \nis overlaid on the data to serve as a guide to the \neye. The field shifts shown in Figs . S3a and S3b \nare consistent with those shown in Fig . 2 above. \nFigure  S3c shows the center fields extracted from \nthe two -angle series, emphasizing the magnitude \nof the change in the anisotropy.  \nThis switch in the magnetic easy  axis from \nin plane to out  of plane present in the data \nsuggests the presence of an additional \ncontribution to the anisotropy beyond simply \nshape anisotropy. Previously, given the isotropic \nin-plane response of thin  films at room  \ntemperature, additional contributions to  the anisotropy had been excluded. However, the \nresults here warrant the inclusion of an additional term 𝐻#, which is responsible for \ninducing perpendicular anisotropy in thin  films. This phenomenology is consistent with the \nmeasurements of V[TCNE] x thin films presented in the main text. Following that \ndevelopment,  the angular  dependence of the FMR response for in  plane to out of plane \nrotation of a thin -film sample can therefore be described by,[17–19] \n𝜔\n𝛾='(𝐻−4𝜋𝑀-..cos2𝜃)\t(𝐻−4𝜋𝑀-..\tcos2𝜃)\t\n='(𝐻−(4𝜋𝑀6−𝐻7)cos2𝜃)\t(𝐻−(4𝜋𝑀6−𝐻7)\tcos2𝜃), (1) \nFigure S2 (a) Single FMR line scans at 140  K for \na sample oriented in -plane (90°) and out of plane \n(0°) with respect to the externally applied magnetic \nfield. (b) Single FMR line scans at 80  K for a \nsample oriented in -plane (90°) and out of plane \n(0°) with respect to the externally applied magnetic \nfield.   5 where ω is the resonance fequency, γ is the gyromagnetic ratio, 𝐻 is the applied field, and \n𝜃 is the polar angle of the magnetization.  The FMR resonance fields are more than an order \nof magnitude larger than the typical saturation field for V[TCNE] x, and therefore we have \nassumed that the magnetization is effectively p arallel to the applied magnetic field (i.e. \n𝜙≈𝜙> and 𝜃≈𝜃? where 𝜃, 𝜙 and 𝜃?, 𝜙? are the polar and azimuthal angles of the \nmagnetization 𝑀 and the applied bias field 𝐻, respectively). Also, note that the in -plane \nFMR response remains isotropi c, with the 𝜙 dependence dropping out:  \n𝜔\n𝛾='𝐻\t(𝐻+4𝜋𝑀-..)\t\n='𝐻\t(𝐻+(4𝜋𝑀6−𝐻7)) \nThe data in Fig.  S3c are then fit according to the dispersion relation in Eq.  1. The fits \nare shown as solid and dashed lines in Fig.  3c. The 140  K data yields an 𝐻eff=4π𝑀eff= \n15.4 Oe ± 0.1 Oe while fi tting to the 80  K data result in an 𝐻eff value of  -28.3 Oe ± 1.0 Oe. \nThe negative value of 𝐻eff for the 80  K data means that 𝐻#>4π𝑀S and that the film has \nperpendicular magnetic anisotropy. Note that the magnetic energy landscape, and therefore \nthe angular dependence contained in Eq.  1, does not allow for the easy  magnetization axis \nto take on an intermediate vector between in  plane or  out of plane for this set of anisotropy \nfields. This result also implies that prior measurements of the anisotropy of thin  films are \nin fact measuring 4π𝑀eff rather than the bare 4π𝑀S as previously assumed.[17,20] However,  (2) \n  (𝜃=90°).   6 as with previous studies of uniform thin  films, \nit is challenging to disentangle this form of \nanisotropy from 4π𝑀S, leading us to use the \nmore general 𝐻eff=4π𝑀eff. Temperature -\ndependent FMR studies combined with careful \nDC magnetization measurements provide a \npromising avenue to decoupling the two \nanisotropy fields.  \nIn comparing the data shown in Fig.  S3a \nand b, also note that at lower temperatures, the \nresonance response becomes markedly multi -\nmodal and appears to broaden. To investigate \nthis behavior in greater detail, FMR data is \ncollected over a range of temperatures with the \napplied field oriented in the plane of the \nsample. The dat a is shown in Fig.  S4a. Note \nthe clear shift of the resonant features towards \nhigher field at lower temperatures as the in -\nplane orientation, which is the geometry being \nmeasured in this data set, changes from the \neasy magnetization axis to the hard \nmagnet ization axis.  \nThe effective magnetization, 4π𝑀eff, \nextracted from the data shown in Fig.  S3 \ncontains contributions from a perpendicular \nmagnetic  anisotropy energy. This 𝐻# does not \narise from shape anisotropy in thin  films and  \nFigure S3 (a) Shows FMR spectra as the sample is \nrotated from in -plane ( 𝜃 = 90°) to out of plane \n(𝜃 = 0°) with respect to the externally applied \nmagnetic field at 140  K. (b)  Shows the FMR \nspectra as the sample is rotated from in -plane \n(𝜃 = 90°) to out of plane ( 𝜃 = 0°) with respect to \nthe externally applied magnetic field at 80  K. (c) \nShows the extracted center fields from the angular \nseries shown in (a) and (b) with fits shown as solid \nand dashed lines. The inset shows the coordinate \nsystem with respect to the sam ple geometry.  0.1 Oe \n1.1 Oe  7   \n  \n  \n  \nmust instead come  from a crystal -field anisotropy wherein the local exchange vector \nacquires some anisotropy due to some combination of lattice symmetry and strain. Given \nthe large differences in the coefficients of thermal expansion for organic and in organic \nmaterials (often varying by an order of magnitude or more) , stain due to differential thermal \nexpansion at the interface between the substrate and organic -based materials is likely \ncreating an anisotropic strain field in the magnetic material. As the sample temperature is \nlowered, this strain field increases until 𝐻# becomes larger in magnitude than 4π𝑀S and \n4π𝑀eff takes on a negative value. The result is a magnet with an easy -axis out of plane as \nshown in Fig.  S3b.  \nWe note that qualitatively similar results were obtained for vanadium ethyl \ntricyanoethylene carb oxylate ( V[ETCEC ]x). V[ETCEC ]x is a third member of this class of \nmetal ligand ferrimagnets[23,24], supporting the thesis that strain -dependent anisotropy is a \ncommon feature of this class of materials.  \n2. Method for extracting linewidth from FMR scans  \nThe FMR scans are obtained through phase -sensitive detection, where in addition to the \nstatic DC magnet ic field the sample sees a sinusoidally modulated field component that is  Figure S4 (a) Shows FMR spectra of V[MeTCEC ]x sample mounted  in-plane \n(𝜃 = 90°) with respect to the externally applied magnetic field as a function of \ntemperature . (b)  Shows the extracted peak -to-peak linewidths from the \ntemperature -dependent spectra shown in (a)  \n 8 varied  at the same frequency as the amplitude modulation  of the microwaves reflected from \nthe cavity. If there is an EPR signal , that signal is converted into a sine wave whose \namplitude is proportional to the  derivative of  the signal  (change in microwave power \nrelative to field modulation)  and appears as the first derivative of a Lorentzian function . In \naddition, it should be noted that some FMR scans show multi  peaks (for examp le, the 300 \nK scans shown in Figure 2 of the main  text) and a possible reason for that could be \ninhomogeneous strain. As discussed in our main text, strain in our films is induced by \ndifference in thermal expansion coefficients between V[TCNE] x and the substrate. Given \nthat we have taken no special precautions to prevent it, we believe it is likely that this strain \nwill be inhomogeneous, resulting in regions of our sample with differing magnetic \nanisotropy, and therefore the potential for additi onal peaks in FMR spectra. It has been \nreported that strain -induced distortions can alter the local electronic and crystal -field \nenvironment by changing the orbital occupancy, tilt angle between neighboring spins[25] or \nmagnetocrystalline anisotropy[26,27], for instance, leading to local changes in magnetic \nanisotropy which result in the appearance of additional resonance peaks . \n Since the asymmetry of the FMR lineshape and the multi -peaks need to be \naccounted for , scans are  not simply fit by the derivative of a symmetric Lorentzian . In \nphase -sensitive measurements the microwave electric field generates oscillati ng electric \ncurrents in the sample ; the oscillating magnetization due to the microwave magnetic field \nresults in oscilla ting angles between the current flow and magnetization, leading to local \nlattice distortions  which may c ause the observed asymmetry in signal lineshape  due to \ninhomogeneous broadening[28,29]. Another possible source of this asymmetry could be the \nresult of high cavity loading[30] and the resulting phase error introduced by the automatic \nfrequency controller of the EPR spectrometer when the sample is resonantly excited. This \nwarrants the inclusion of a dispersion or antisymmetric term that takes int o account this \nasymmetry, therefore the FMR scans are fitted to the sum of the derivative of an absorption \n(symmetric  term) and dispersion (antisymmetric term ) from a Lorentzian . The derivative s \nhave the following form : (3) (4)  9 absorption derivative =\t−32\t√3\t𝐴\t𝐹𝑊𝐻 𝑀K(𝐵−𝐵M)\n9\t[FWHM \t2+4(𝐵−𝐵M)\t2]\t2 \n \ndispersion derivative =\t−4\t𝐷\t𝐹𝑊𝐻𝑀 \t(𝐵−𝐵M)\nFWHM \t2+4(𝐵−𝐵M)\t2 \nwhere FWHM  is the full -width at half -max, A is the height of the absorption derivative, D \nis the height of the dispersion derivative, 𝐵M is the location of the resonance (center) field  \nand B is the amplitude of the magnetic field that is being swept at each data point. \nTherefore, the resulting li ne shape depends on the relative contributions of these two terms .  \nFor scans with an out -of-plane easy  \naxis fitting with a single  derivative sum \nprovides good agreement with the data  \n(Figure S5). But for scans with in -plane \neasy axis, due to the appearance of a \nmodest satellite peak , obtaining a good fit \nto the data requires addition of up to three \nderivative sum s. For FMR scans in the \nrange 9 K  –  80 K (out -of-plane easy  axis \nbetween 9 K – 100 K and negligible \nanisotropy at 80 K) the date is well fit with a single derivate sum . On the other hand,  fits \nfor scans in the high temperatures between 120 K – 300 K (in -plane easy  axis) give good \nagreement with data when  two derivative sums  are used , a few requiring up to three \nderivative sums (Figure  S6b).  However, FMR scans at 5 K (in-plane easy  axis) and 6 K \n(out-of-plane easy  axis) mimic the high temperature fits by requiring two derivative sums.  \nFor the purposes of this study, which explores the fundamental FMR mode, in scans \nshowing multiple peaks we focus on the contribution fr om the peak that persists to low Figure S5 Shows a single derivative fit to the \nFMR data collected at 2 2 K \n 10 temperatures.  If we plot the individual Lorentzian \ncomponents of the FMR fit, we find that the first \ncomponent YL1 (component with the highest \noverall peak -to-peak magnitude) is present in all \nthe temperatures being consid ered in the range 300 \n– 5 K. Therefore, the linewidth date plotted against \ntemperatures in Figure 4 of the main text is the \nlinewidth at f ull width  half max (FWHM) of YL1.  \nIn Figure S6a it can be seen that fitting the \nFMR scan at 300 K with a single derivative sum \ndoes not provide a great fit to the data. However , \nfrom  Figure S6b it becomes clear that fitting the \nsame data with the superposition of three \nderivative sums or components (each with their \ndistinct A, D and FWHM ) gives a decent  fit. In \nFigure S6c the amplitude of each individual \ncomponent is plotted against magnetic field sweep \nrange to provide a visual understanding of how \neach component contributes to the overall FMR \nline shape.  \n3. Temperature dependent linewidth  \nThe V [TCNE ]x linewidth dependence on \ntemperature can be well explained from the \ninteraction between magnons and defects or Figure S6 (a) Shows a single derivative fit to the \nFMR data collected at 300 K. (b) Shows FMR \nscan at 300 K fitted to superposition of three \nLorentzian derivative  sums. (c) Amplitude of \neach sum or component plotted against magnetic \nfield sweep range.  YL1, YL2 and Y L3 are the \nfirst, second and third components respectively.  \n 11 impurities in V [TCNE ]x.  The \ndefects or impurities are \nconsidered to be a two -level spin \nsystem s. These experience spin -\nflip transitions excited by the \nannihilation of a uniform -magnon \nmode  [31,32]. This process \nintroduces a finite magnon \nlifetime, which in turn leads to the \nlinewidth expression Eq. (4)  in the \nmain  text. In Fig. S7, we use four \ndiffere nt parameter sets to fit the \nhigh temperature experimental \ndata using Eq. (4). All the different \nsets yield a good fitting for T> 9 K, \nalthough the smaller the E b, the \nsmaller the nominal peak in \nlinewidth. As discussed in the main text, this imposes an upp er bound on E b ~ 1meV.   \nHowever, it is important to note that the peak in linewidth coincides with the abrupt \nreversion in anisotropy from an out -of-plane easy  axis to an in -plane easy  axis. This change \nin magnetic anisotropy has the potential to have a substantial impact on spin -magnon \nscattering efficiency. For example, this change will result in a shift of the energy of the \nmagnon bands (see Eq. 1  in main text ), and if this change involves a commensurate change \nin the strain there will also be a modifi cation to the spin -orbit coupling and exchange \nparameters at the paramagnetic defects. This is represented by the fits shown in Fig. S 7 \nwherein we assume a lower temperature for the nominal peak in linewidth occurring due \nto spin -magnon scattering that is experimentally preempted  by the change in magnetic \nanisotropy.  As a result, we interpret th e fit in Fig. 4 of the main text as an upper bound on \nFigure S7 (a) (b), (c) and (d) show V [TCNE ]x linewidth as a \nfunction of temperature and the corresponding fit curves using \nfitting parameters of Eq. (4)  \n 12 Eb. The alternate fits presented in Fig. S7 agree with experimental observations at \ntemperatures above 9 K, and therefore must be considered as possible mechanisms. \nMoreover, if the residual paramagnetic spins are ordered at temperatures below 9 K, one \nwould require a large amount of energy (>>  ℏω) to populate their excited states, which is \nunlikely to happen. Hence, magnetic ordering of the paramagnetic spins  would also \nenhance the suppression of spin -magnon scattering, resulting in the sharp linewidth \nsuppression for T < 9 K. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  13 References  \n[1] C. J. Brabec, Sol. Energy Mater. Sol. Cells  2004 , 83, 273.  \n[2] H. Shirakawa, E. J. Louis, A. G. MacDiarmid, C. K. Chiang, A. J. Heeger, J. Chem. \nSoc. Chem. Commun.  1977 , 578.  \n[3] C. W. Tang, S. A. Vanslyke, Appl. Phys. Lett.  1987 , 51, 913.  \n[4] Y. Lu, M. Harberts, C. -Y. Y. Kao, H. Yu, E. Johnston -Halperin, A. J. Epstein, Adv. \nMater.  2014 , 26, 7632.  \n[5] Y. Lu, H. Yu, M. Harberts, A. J. Epstein, E. Johnston -Halperin, J. Mater. Chem. C  \n2015 , 3, 7363.  \n[6] Y. Lu, H. Yu, M. Harberts, A. J. Epstein, E. Johnston -Halperin, RSC Adv.  2015 , 5, \n82271.  \n[7] J. L. Arthur, S. H. Lapidus, C. E. Moore, A. L. Rheingold, P. W. Stephens, J. S. \nMiller, Adv. Funct. Mater.  2012 , 22, 1802.  \n[8] J. P. Fitzgeral d, B. B. 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Celinski and B. Heinrich , Journal of Applied Physics  1991 , 70, 5935 .  \n[29]    Y. Li, F. Zeng, S. -L. Zhang, H. Shin, H. Saglam, V. Karakas, O. Ozatay, J. E.  \n           Pearson, O. G. Heinonen, Y. Wu, A. Hoffman and W. Zhang , Phys. Rev. Lett.  \n           2019 , 122, 117203 .  \n [30]    I. B. Goldberg and H. R. Crowe, Anal. Chem. 1977 , 49, 9, 1353 -1357.  \n [31]    M. Sparks, Ferromagnetic -Relaxation Theory (McGraw Hill, New York, 1964 ). \n [32]    P. E. Seiden, Phys. Rev.  1964 , 133, A728.  \n \n \n "    },    {        "title": "2009.00600v2.Quantum_Brownian_Motion_for_Magnets.pdf",        "content": "Quantum Brownian Motion for Magnets\nJ. Anders,1, 2,\u0003C.R.J. Sait,1and S.A.R. Horsley1\n1Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, UK.\n2Institut f ur Physik und Astronomie, University of Potsdam, 14476 Potsdam, Germany.\nSpin precession in magnetic materials is commonly modelled with the classical phenomenological\nLandau-Lifshitz-Gilbert (LLG) equation. Based on a quantized spin+environment Hamiltonian, we here\nderive a general spin operator equation of motion that describes three-dimensional precession and damp-\ning and consistently accounts for e\u000bects arising from memory, coloured noise and quantum statistics.\nThe LLG equation is recovered as its classical, Ohmic approximation. We further introduce resonant\nLorentzian system{reservoir couplings that allow a systematic comparison of dynamics between Ohmic\nand non{Ohmic regimes. Finally, we simulate the full non-Markovian dynamics of a spin in the semi{\nclassical limit. At low temperatures, our numerical results demonstrate a characteristic reduction and\n\rattening of the steady state spin alignment with an external \feld, caused by the quantum statistics\nof the environment. The results provide a powerful framework to explore general three-dimensional\ndissipation in quantum thermodynamics.\nThe continued miniaturisation of critical components\nin consumer electronics and neighbouring technologies\nwill require a deeper understanding of thermal noise\nand general thermodynamic principles beyond the clas-\nsical macroscopic world. Quantum thermodynamics [1{\n3] has emerged as a \feld addressing the conceptual\nchallenges related to the exchange of energy and in-\nformation at the nanoscale. Recent advances include\nstudies of heat transport in quantum systems [4{11],\nthe characterisation of memory e\u000bects in their dynam-\nics [12{19], and clari\fcation of the impact of quan-\ntum coherence and correlation on thermodynamic pro-\ncesses [20{24]. The establishment of a generalised\nthermodynamic framework, valid for nanoscale systems\nthat strongly couple to environmental modes, is well\nunder way [25{33], and for magnetic molecules an\nenvironment-induced renormalisation of the anisotropy\nhas been predicted [34]. Two open quantum sys-\ntems models have served as the workhorse for many\nof these conceptual studies; the Caldeira-Leggett model\nfor quantum Brownian motion [8, 35, 36] and the spin-\nboson model of a spin (or many spins) coupled to a\none-dimensional harmonic bath [5, 6, 33, 37{39]. These\ndescribe a very wide range of physical situations extend-\ning to studies of quantum e\u000bects in bio-chemical reac-\ntions [40], where they are used to model exciton-phonon\ninteractions [41].\nUntil now few nanoscale technologies have required\nthe use of advanced open quantum systems techniques.\nBut advances in engineering magnetic materials for\nmagnetic hard drives at unprecedented length and time-\nscales [42] are likely to require a more detailed picture\nof spin dynamics including memory and quantum sig-\nnatures. Here we introduce a three-dimensional open\nquantum system model to characterise the quantum\nBrownian motion of spins in magnetic materials.\nMagnetic behaviour has been studied extensively\nbased on the classical phenomenological Landau{\nLifshitz{Gilbert (LLG) equation [43{47]\n@M\n@t=\rM\u0002\u0014\nBe\u000b\u0000\u0011G@M\n@t\u0015\n; (1)\n\u0003janet@qipc.orgwhich is routinely solved with micromagnetic and atom-\nistic simulations. Here Mis the magnetic moment, \r\nis the gyromagnetic ratio and Be\u000bis an e\u000bective mag-\nnetic \feld which includes the external \feld [106], ex-\nchange and anisotropy e\u000bects, as well as stochastic mag-\nnetic noiseb/p\nTstemming from an environment at\ntemperature Tthat was added by Brown [107] in 1963\n[48]. The \fnal term on the right of (1) is the so{called\n\\Gilbert damping\" term and the positive constant \u0011Gis\nthe damping parameter [108], which is often rewritten\nas\u0011G=\u0011=jMjj\rjwith a unit-free \u0011.\nGilbert damping is not derived from microscopic prin-\nciples, but chosen as the simplest term that could\nserve to align the magnetic moment with the applied\n\feld [43]. As we will see, it contains no memory which\nis increasingly seen as a limitation [49, 50]. Advances\nin engineering magnetic materials at the nanoscale\nand manipulating them on ultrafast timescales indicate\nthat a theory beyond the classical LLG equation is re-\nquired [47]. Early attempts have pursued a path integral\nderivation of a quantum spin dynamics equation [51], as\nwell as other conceptually related classical and quantum\nderivations [52, 53]. These derivations were not directly\napplied to the calculation of magnetization dynamics\nor steady states, nor have they been connected to re-\ncent generalizations of Gilbert damping that include in-\nertial terms [49, 50, 54{59] or provided an assessment\nof quantum e\u000bects.\nHere we go further and develop a comprehensive and\nquantum-thermodynamically consistent theory suitable\nto describe the quantum dynamics of spins in magnetic\nmaterials including non{Markovian damping, coloured\nnoise and quantum zero-point \ructuations. Unlike\nthe conceptually pioneering Caldeira-Leggett model that\nhas few experimental realisations, the developed three-\ndimensional quantum spin model is directly applicable\nfor atomistic spin dynamics simulations [47, 60], ultra-\nfast magnetism experiments [61], and systems exhibiting\nanisotropic damping [62].\nThe paper is organised as follows: In section I the\ngeneral quantum spin dynamics equation for spin oper-\nator precession in three dimensions is derived. For the\nsimplest, Ohmic, coupling this equation is found to re-\nduce to the memory-free LLG equation. In section II we\nintroduce Lorentzian couplings as a systematic method\nfor exploring non-Markovian dynamical regimes in gen-arXiv:2009.00600v2  [quant-ph]  7 Jul 20212\neral open quantum systems, including spins. Finally,\nin section III we detail a numerical method to simulate\nnon-Markovian dynamics, and present results for a sin-\ngle classical spin that illustrate the di\u000berences between\nspin dynamics and steady states arising with non-trivial\nmemory, coloured noise, and quantum bath statistics in\ncomparison to those obtained with the memory-free LLG\nequation. Conclusions and open questions are discussed\nin section IV.\nI. Quantum spin dynamics equation\nA. System+environment Hamiltonian\nWe begin by introducing the quantized Hamiltonian\ndescribing the di\u000berent contributions to the total energy\nof the system, consisting of spins as well as environmen-\ntal degrees of freedom (e.g. electrons and phonons),\ngiven by\n^H=^HS+^HR+^Vint; (2)\nwhere ^HSis the bare spin Hamiltonian operator which\ncaptures the spin energy in external \felds and interac-\ntions between spins, ^HRis the environmental or reser-\nvoir Hamiltonian, and ^Vintis the interaction between\nthe spins and the reservoir.\nWe choose ^HSas the sum of the interaction with\na homogeneous external \feld Bext[109] and the ex-\nchange interaction between three-dimensional spin vec-\ntor operators ^S(n)= (^S(n)\n1;^S(n)\n2;^S(n)\n3) at sitesnof a\nlattice [110],\n^HS=\u0000\rX\nn^S(n)\u0001Bext\u00001\n2X\nn;m6=n^S(n)\u0001J(nm)^S(m):(3)\nHereJ(nm)is the exchange tensor for spin pairs (n;m)\n[111], which can include the Dzyaloshinskii-Moriya in-\nteraction [47]. It is straightforward to include additional\nenergetic terms in the bare spin Hamiltonian, such as en-\nergies associated with magnetic anisotropy. Instead of\nthe magnetic moment Mused in Eq. (1), we will here\nwork with the spin angular momentum Sproportional\ntoM,M=\rS, where\ris the gyromagnetic ratio.\nIn the following we will assume the gyromagnetic ratio\n\r=\u0000ge\u0016B=~=\u00001:76\u00011011s\u00001T\u00001for an electron.\nThe reservoir Hamiltonian is commonly modelled as\na set of harmonic oscillators [35, 64], and we here fol-\nlow the continuous reservoir approach by Huttner and\nBarnett [64], taking the reservoir Hamiltonian as\n^HR=1\n2X\nnZ1\n0d!\u0014\u0010\n^\u0005(n)\n!\u00112\n+!2\u0010\n^X(n)\n!\u00112\u0015\n:(4)\nIt describes a continuous frequency reservoir at each lat-\ntice siten, where ^\u0005(n)\n!and^X(n)\n!are (three-dimensional)\nmomentum and position operators of the reservoir os-\ncillator with frequency !. The position operators ^X(n)\n!\nphysically represent variations in the environment to\nwhich the spin at site nresponds, see illustration Fig. 1,\nas for example, in magnon{phonon mediated loss [65].\nUnlike most system+environment Hamiltonians which\nassume one-dimensional coupling, we here take the spin-\nreservoir interaction to be of the three-dimensional form\n^Vint=\u0000\rX\nn^S(n)\u0001Z1\n0d!C(n)\n!^X(n)\n!: (5)This coupling allows angular momentum transfer as well\nas energy transfer between the spins and the environ-\nment. HereC(n)\n!is a three-dimensional coupling tensor\nand a function of frequency !. At each!, the coupling\ntensor determines the strength of the coupling of each\nspin to its reservoir oscillators at frequency !, thus act-\ning as a frequency \flter. As we shall see, the choice of\nthe couplingC(n)\n!will determine the damping of the spin\ndynamics as well as the stochastic noise experienced by\nthe spins.\nFor readers concerned about time-reversal symmetry\nof^Vintin (5), we note that ^X(n)\n!should be interpreted\nas an e\u000bective magnetic \feld seen by the spins due to\ntheir interaction with the environment, which has the\nsame time symmetry as ^S(n)[112]. Indeed, the the-\nory of magnetic materials developed here on the basis\nof the system+environment Hamiltonian ^His analo-\ngous to macroscopic QED, an e\u000bective medium theory\nwhich successfully describes quantum electromagnetism\nin dielectric materials [64, 66, 67]. Instead of trying to\ngive a fully microscopic description that accounts for\nevery light-matter interaction in the material, macro-\nscopic QED characterizes electromagnetic materials in\nterms of two measured frequency dependent suscepti-\nbilities. The quantum Hamiltonian is then written in\nterms of these susceptibilities, and can be used in ap-\nplications from predicting the Lamb shift to the Casimir\ne\u000bect [66, 68].\nB. Equations of motion\nHaving set up the full Hamiltonian (2) of the spins\nand environment degrees of freedom allows the study of\nthe spins' reduced state dynamics ^\u001aS(t) = trR[^\u001aSR(t)]\nof the total state\n^\u001aSR(t) =ei^Ht=~(^\u001aS(0)\n^\u001aR)e\u0000i^Ht=~; (6)\nwhere the reservoir state ^\u001aR=e\u0000\f^HR=tr[e\u0000\f^HR]is\nthermal at some inverse temperature \f= 1=kBT. In\nwhat follows it will be more convenient to work instead\nin the Heisenberg picture where the state is station-\nary,^\u001aSR(0), while the time dependence of an opera-\ntor^O(t)is governed by the commutator d^O(t)=dt=\n(i=~)[^H;^O(t)]. Expectation values at time tcan then\nbe obtained as\ntr[^\u001aSR(t)^O(0)] = tr[^\u001aSR(0)^O(t)]: (7)\nFIG. 1. Illustration of Hamiltonian model. In addition to\nprecessing in an external \feld Bext, and coupling to its spin\nneighboursmwith strengthJ(nm), each spin ^S(n)couples\nto its environmental mode (phonons, electrons) at frequency\n!with a coupling function C(n)\n!. All environmental modes\nare assumed to be thermal at the same temperature T.3\nUsing the standard commutation relations for the spin\noperators (and orbital angular momentum operators\nin general), [^S(n)\nj;^S(m)\nk] = i ~\u000emnP\nk\u000fjkl^S(n)\nl, and\nfor the position/momentum operators, [^X(n)\n!;j;^\u0005(m)\n!0;k] =\ni~\u000enm\u000ejk\u000e(!\u0000!0), we obtain the following equations\nof motion for the spin operators ^S(n)(t)\nd^S(n)\ndt=^S(n)\u00022\n4\r\u0010\nBext+^B(n)\nenv\u0011\n+X\nm6=n\u0016J(nm)^S(m)3\n5;\n(8)\nwhere \u0016J(nm)= (1=2)[J(nm)+ (J(mn))T]is the sym-\nmetrized exchange tensor and ^B(n)\nenv=R1\n0d!C(n)\n!^X(n)\n!\nis a magnetic \feld operator generated by the reservoir\noscillator positions ^X(n)\n!at siten.\nIn turn, the equations of motion for these operators\nare\nd2^X(n)\n!\ndt2+!2^X(n)\n!=\rC(n) T\n!^S(n); (9)\ni.e. the reservoir oscillators are driven by the motion\nof the spins, with the (transposed) coupling tensors\nC(n) T\n! governing the degree of driving for each of the\ncontinuum of oscillators. We assume retarded bound-\nary conditions so that the reservoir responds only to\nthe past behaviour of the spins. The retarded Green\nfunction,G!(t\u0000t0) = \u0002(t\u0000t0) sin(!(t\u0000t0))=!obeys\n(@2\nt+!2)G!=\u000e(t\u0000t0), and Eq. (9) can then be solved\nexactly by\n^X(n)\n!(t) =r\n~\n2!\u0010\n^a(n)\n!e\u0000i!t+^a(n)y\n!e+i!t\u0011\n+\rZ1\nt0dt0G!(t\u0000t0)C(n) T\n!^S(n)(t0):(10)\nHere, ^a(n)\n!and^a(n)y\n! are (vectors of) bosonic ladder op-\nerators with their components obeying [^a(n)\n!;j;^a(m)y\n!0;k] =\n\u000enm\u000ejk\u000e(!\u0000!0). Classically these correspond to the\ntwo integration constants for the di\u000berential equation\n(9) which set the initial amplitude and velocity of the\noscillator.\nSubstituting the reservoir solutions (10) into the\nequations of motion for the spins (8), we obtain the \frst\nresult: The Heisenberg{Langevin equation that governs\nthree-dimensional quantum spin dynamics under the in-\n\ruence of memory and coloured quantum noise is\nd^S(n)(t)\ndt=^S(n)(t)\u0002\u0014\n\rBext+X\nm6=n\u0016J(nm)^S(m)(t)\n+\r^b(n)(t) +\r2Zt\nt0dt0K(n)(t\u0000t0)^S(n)(t0)\u0015\n:(11)\nThe term ^b(n)(t)is a Hermitian magnetic noise operator\nfor siten,\n^b(n)(t) =Z1\n0d!r\n~\n2!C(n)\n!\u0012\n^a(n)\n!e\u0000i!t+h:c:\u0013\n;(12)which plays the role of the stochastic noise \frst de-\nscribed by Brown [48]. Here it arises from the spin's\ninteraction with its reservoir. As we will see below, the\nbath noise can be coloured and contain quantum zero-\npoint \ructuations. In addition to the coloured noise\n^b(n), a kernel tensorK(n)(t\u0000t0)appears in Eq. (11),\nwhich captures the damping of the spins. It arises from\nthe coupling tensor C!and is given by\nK(n)(\u001c) = \u0002(\u001c)Z1\n0d!C(n)\n!C(n) T\n!\n!sin (!\u001c);(13)\nwhere\u001c=t\u0000t0. Here \u0002is the Heaviside function\nwhich makes the spin's dynamics at time t, see (11), a\nfunction of the spin's state at previous times t0\u0014t0<t\n[113].\nThe three-dimensional spin dynamics equation (11)\ndescribes the evolution of spin operators and ex-\nplicitly includes memory of the past dynamics (non-\nMarkovianity). This contrasts with previous derivations\nof quantum spin dynamics in the form of a master equa-\ntion [69] which can be solved numerically. But to obtain\nthe master equation a range of simplifying assumptions\nwhere made, including as weak spin-environment cou-\npling, as well as the Markov and secular approximations.\nThese approximations are quite strong and may not al-\nways be justi\fed for a given spin system. We further\nremark that a di\u000berent method of including (classical)\ncoloured noise is based on the Miyazaki-Seki approach\n[70]. Similar to (8), this model assumes that the equa-\ntion of motion of the spins is coupled to a stochastic\nequation of motion for the lattice enabling transfer of\nenergy and angular momentum between the lattice and\nthe spin systems. Di\u000berent coupling potentials, such as\nharmonic and Morse potentials, have been considered\nand the spin-lattice coupling impact on the magnetisa-\ntion has been characterised [71].\nThe above spin+environment Hamiltonian and its\nresulting equations of motion share many similarities\nwith the well-known Caldeira-Leggett model for har-\nmonic quantum Brownian motion [26, 35] and the spin-\nboson model [33, 37, 39]. A key di\u000berence is the three-\ndimensional nature of the reservoir interaction in (5)\nwhich leads to the cross product in (11). The spin-boson\nmodel is recovered as a special case for spin 1=2oper-\nators and rank-1 coupling tensors, see Appendix A1.\nC. Fluctuation-Dissipation Relation and ^S2\nThe presence of the kernel in (11) gives rise to spin\ndamping that can signi\fcantly di\u000ber from Gilbert damp-\ning, as explored further in section II. Previous generali-\nsations to Gilbert damping have considered the spins' in-\nteractions with the lattice and electron motion [54, 71{\n73], and included inertial terms [49] and further memory\nterms within a damping kernel [55{58]. First measure-\nments have recently con\frmed the presence of inertial\ncorrections [50, 59]. But these studies have not con-\nsidered how the noise may have to change from the\nstandard white magnetic noise that is included in the\ne\u000bective magnetic \feld in the LLG equation (1).\nA separate strand of reasoning has argued that to ex-\ntend the LLG equation to the quantum setting requires\nthat the stochastic noise included in the e\u000bective \feld4\nBe\u000bshould have a quantum distribution rather than\nthe classical Boltzmann one [60, 74{77]. These studies\npartially reproduce experimentally measured magnetiza-\ntion over temperature curves. But the noise considered\nvanishes at low temperatures, whereas the full Bose-\nEinstein distribution famously has non-zero quantum\n\ructuations even at T = 0 K. Other pioneering inves-\ntigations have discussed the e\u000bect of coloured noise and\nnon-Markovian e\u000bects on the ultrafast demagnetization\nrate [70], while not requiring the \ructuation-dissipation\ntheorem (FDT) to hold and not including quantum ef-\nfects.\nThe quantum spin dynamics equation (11) brings\nthese pieces, the damping kernel, quantum statistics and\ncoloured noise, together in a quantum thermodynami-\ncally consistent framework. To see this we now prove\nthat the reservoir's two signatures - the stochastic \feld\n^b(n)and memory kernel K(n)- always ful\fl the (quan-\ntum) \ructuation-dissipation theorem, for any choice of\ncoupling tensorC(n)\n!.\nThe reservoir power spectrum ~P(!)is de\fned as the\n(quantum symmetrised [114]) expectation value of the\nautocorrelation function of the magnetic noise ^b(n)(t)\nin the thermal reservoir state ^\u001aR=e\u0000\f^HR=tr[e\u0000\f^HR],\nand then taking the Fourier transform from the time to\nthe frequency domain [115], i.e.\n~P(nm)\njk(!) =Z1\n\u00001d\u001cei!\u001cDn\n^b(n)\nj(t);^b(m)y\nk(t\u0000\u001c)oE\n\f\n2;(14)\nwheref:;:gis the anti-commutator. Inserting (12),\none \fnds that the only non-trivial contributions to\nthis expression come from the bosonic ladder operator\nexpectation values hf^a(n)\n!;l;^a(m)y\n!0;l0gi\f=\u000enm\u000ell0\u000e(!\u0000\n!0) coth(\f~!=2). Meanwhile by Eq. (13) the square of\nthe coupling tensor, C(n)\n!C(n) T\n! , is related to the imagi-\nnary part of the Fourier transform of the damping kernel\nK(n)(\u001c), i.e.\nC(n)\n!C(n) T\n! =2!\n\u0019Im[~K(n)(!)]: (15)\nHence one obtains for allchoices of the interaction ten-\nsorC(n)\n!in Eq. (5) the quantum \ructuation-dissipation\ntheorem (FDT)\n~P(nn)\nqu(!) =~Im[~K(n)(!)] coth~!\n2kBT; (16a)\n~P(nn)\ncl(!) =2kBT\n!Im[~K(n)(!)]forkBT\u001d~!\n2;(16b)\nwhere the last line is the well-known classical high-\ntemperature (or low frequency) approximation, and the\n\frst line is the low-temperature limit where the Bose-\nEinstein distribution of the reservoir oscillator modes\nbecomes important [116].\nFinally, the spin dynamics equation (11) leaves the\nsquare of the spin operators a constant of motion, since\nd\f\f^S(n)\f\f2\ndt=i\r~\n2X\nl[^S(n)\nl;B(n)\ne\u000b;l] = 0; (17)see Appendix A3. Here ^B(n)\ne\u000b=Bext+^B(n)\nenv+\n1\n\rP\nm6=n\u0016J(nm)^S(m)is an e\u000bective magnetic \feld at\nsitensee Eq. (8), which commutes with all compo-\nnents of ^S(n). This con\frms that the spin length j^S\f\fis\na constant in time, e.g. for a spin-1/2 ^S=~\n2^\u001bwith ^\u001bj\nthe Pauli matrices, one has a constant ^S2=~2\n4^\u001b2=\n3~2\n412.\nII. Coupling functions\nThe general quantum spin dynamics equation (11) is\nspeci\fed completely by the coupling tensor C!in (5)\nthat sets the interaction between spins and reservoirs.\nHere we will show that the standard LLG equation (1)\narises as a special case of (11), for a particular choice\nof couplingC!. We will further introduce a class of\nLorentzian coupling functions which allow a systematic\nexploration of spin dynamics behaviours beyond the LLG\nequation.\nFor simplicity from here on we will drop the lattice\nsite superscripts and consider isotropic coupling tensors\nC!=C! 13withC!a scalar function, and similarly for\nthe corresponding kernels K(\u001c) =K!(\u001c) 13and power\nspectra ~P(!) =~P(!) 13. These choices are appropriate\nfor 3D materials in which all spins couple to the environ-\nment in the same manner in all spatial directions. For\nother materials, such as 2D layers, non-isotropic cou-\npling tensors can be considered.\nA. Ohmic coupling and recovery of LLG equation\nIn the open quantum systems literature, coupling\nfunctions that are linear in frequency are referred to as\n\\Ohmic\", while those proportional to higher and lower\npowers of!are called super- and sub-Ohmic, respec-\ntively. For the magnetic system considered here, Ohmic\ncoupling means\nCOhm\n!=r\n2\u0011Gcos(!\u000f+)\n\u0019!; (18)\nwhere\u0011Gis a positive constant with units of kg =A2m2s,\nas in Eq. (1), and \u000f+is an in\fnitesimal positive constant\nwhich we take to zero at the end of the calculation. The\ncorresponding kernel (13) is close to instantaneous,\nKOhm(\u001c) =\u0011G\n\u0019\u0002(\u001c)Z1\n0d!![sin(!(\u001c+\u000f+))\n+ sin(!(\u001c\u0000\u000f+))]\n=\u0000\u0011G\u0002(\u001c)d\nd\u001c\u000e(\u001c\u0000\u000f+); (19)\nfrom which we can see that a positive value of \u000f+en-\nsures the response kernel KOhmis causal. For Ohmic\ncoupling (18), the damping term in (11) reduces to\n\r2^S(t)\u0002Zt\nt0dt0KOhm(t\u0000t0)^S(t0)\n=\u0000\r2\u0011G^S(t)\u0002d^S(t\u0000\u000f+)\ndt;(20)\ni.e. Ohmic coupling results in a damping term propor-\ntional to a \frst order time derivative of the system vari-\nable. In the magnetic case this is ^S, while for quantum5\nBrownian motion it would be the oscillator position [35].\nInserting (20) into the general quantum spin dynam-\nics equation (11), we see that it recovers (the quan-\ntum version of) the LLG equation (1) with \u0011Gbeing\nidenti\fed as the damping parameter. This becomes the\nwell{known classical equation in the limit of large spins.\nConsequently we will also refer to the Ohmic coupling\nfunction (18) as \\LLG coupling\" [117].\nGiven a coupling function it is straightforward to \fnd\nthe corresponding power spectrum (16a) governing the\ntime correlations of the quantum noise. For COhm\n! the\nquantum power spectrum is\n~POhm\nqu(!) =\u0011G~!coth\u0012~!\n2kBT\u0013\n; (21)\nwhere we have taken the limit \u000f+!0. The Ohmic\nquantum power spectrum (21) is proportional to the\ndamping parameter \u0011G, and tends to a linear (diverg-\ning) function of frequency in the low temperature limit\n~!\u001dkBT, see Fig. 2g+h. However, power spectra\ndiverging with frequency are unphysical. Coupling any\nsystem to environmental modes must go to zero at large\nenough frequencies, such as the interaction of spins with\nlattice phonons or the e\u000bect of conduction electrons\nscattering o\u000b the spins. A common way of integrat-\ning this physical information into Ohmic coupling is to\nintroduce a cut-o\u000b, i.e. an upper frequency to which the\ncoupling grows linearly, and after which it is set to 0 or\ndecays to 0 in either algebraic or exponential form [38].\nA di\u000berent approach was taken in [60, 74] for the mod-\nelling of spin dynamics, where a semi{quantum Ohmic\npower spectrum similar to (21) was chosen, but with the\nzero{point noise subtracted ensuring that at T= 0K it\nvanishes for all frequencies.\nIn the high temperature limit kBT\u001d~!, Eq. (21)\nreduces to the classical power spectrum\n~POhm\ncl(!) = 2\u0011GkBT; (22)\nwhich is frequency independent (white) noise. Thus\nOhmic coupling plus the high temperature approxima-\ntion to the power spectrum recovers the LLG equation\n(1) commonly used to simulate magnetic materials over\na wide range of temperatures.\nAs a \frst step to unravel how the dynamics predicted\nby the quantum spin equation (11) can deviate from\nthat predicted by the LLG equation (1), one can expand\nthe spin vector (operator) at time t0in Eq. (11) around\nthe end point of integration to arbitrary order,\n\r2^S(t)\u0002Zt\nt0dt0K(t\u0000t0)^S(t0)\n=\r21X\nm=1\u0014m^S(t)\u0002@m\nt^S(t):(23)\nThe0-th order coe\u000ecient, \u00140does not appear in the ex-\npansion of the damping operator (23), as ^S(t)\u0002^S(t) =\n0. For the quantum operators ^S(t)this is ensured by\nthe angular momentum commutation relations, which\nmake this cross product anti{Hermitian. As outlined in\nAppendix A2, when using any truncated form of ex-\npansion (23) one must employ an explicitly Hermitianform of Eq. (11), and hence any anti-Hermitian con-\ntributions drop out. The higher expansion coe\u000ecients\n\u0014m>0are proportional to the mthone{sided moment of\nthe memory kernel\n\u0014m=(\u00001)m\nm!Z1\n0d\u001c\u001cmK(\u001c); (24)\nwhere we have assumed that the dynamics has been\nrunning for some time longer than the kernel decay time,\nfor which one can replace the initial time t0by\u00001.\nFor the Ohmic kernel only the \frst moment is non-\nzero and corresponds to the (negative) damping param-\neter,\n\u0014Ohm\n1=\u0011GZ1\n0d\u001c\u001cd\nd\u001c\u000e(\u001c\u0000\u000f+) =\u0000\u0011G;(25)\nwhile\u0014Ohm\nm>1= 0 . This complete lack of higher mo-\nments, which would maintain a certain degree of mem-\nory in the dynamics, shows that Ohmic coupling dynam-\nics can only be an approximation to any real dynamics.\nFor example within magnetism, the memory-free form\nof damping (20) is known as Gilbert damping [43], and\nis almost universally used to describe magnetization dy-\nnamics through the LLG equation (1). However, \\iner-\ntial\" corrections to such dynamics have been proposed\n[49] and their presence was recently con\frmed experi-\nmentally [50].\nB. Lorentzian coupling\nHere we provide a tool to systematically study dy-\nnamics beyond the Ohmic case, allowing one to include\nmemory and coloured noise e\u000bects in a manner consis-\ntent with the quantum \ructuation dissipation theorem\n(16a). We consider the class of Lorentzian coupling\nfunctions\nCLor\n!=s\n2A\u0000\n\u0019!2\n(!2\n0\u0000!2)2+!2\u00002; (26)\nwhereAis a coupling amplitude, with the following\nproperties: i) for small !,CLor\n!grows linearly with !\nand can be approximated by an Ohmic coupling func-\ntion, ii) at large !,CLor\n!smoothly decays to zero, and\niii) at some intermediate frequency !0,CLor\n!has a reso-\nnant peak with some width \u0000. This peak characterises\nthe con\fned range and relative strength of system-\nenvironment interaction with two parameters. Alter-\nnative \\peaks\" such as Gaussians or top hat functions\ncould be considered, but here we chose the Lorentzian\nshape due to the fact that many expressions can be\nsolved analytically and, as we will demonstrate in section\nIII, Lorentzian couplings allow us to e\u000eciently simulate\nnon-Markovian dynamics.\nWe call the above functions \\Lorentzian coupling\"\nsince the corresponding damping kernel in the frequency\ndomain is the widely studied Lorentzian response\nKLor(!) =A\n!2\n0\u0000!2\u0000i!\u0000; (27)\nwhere the imaginary part is obtained from (15) and the\nreal part is determined using the usual Kramers{Kronig\nrelations. In the time-domain the Lorentzian memory\nkernel is\nKLor(\u001c) = \u0002(\u001c)Ae\u0000\u0000\u001c\n2sin(!1\u001c)\n!1; (28)6\nwhere!1=q\n!2\n0\u0000\u00002\n4and\u0000=2can now be interpreted\nas the kernel decay rate. For the coupling function\n(26), the collective response of the environment is thus\nequivalent to a single harmonic oscillator of resonant\nfrequency!0and damping rate \u0000=2[78]. From the\nquantum FDT (16a) it follows that the corresponding\npower spectrum is\n~PLor\nqu(!) =A\u0000~!\n(!2\n0\u0000!2)2+!2\u00002coth\u0012~!\n2kBT\u0013\n;(29)\nwhich takes its largest values at frequencies close to\n!0and tends to zero as !\u00003at large!. This power\nspectrum di\u000bers from classical Ohmic noise (22) in two\nimportant respects. Firstly, the quantum mechanical\ntreatment means that the low temperature noise is not\nproportional to temperature, and does not vanish at\nzero temperature. Secondly, even in the high temper-\nature limit, the noise spectrum is frequency dependent\n(coloured), unlike the white noise of Eq. (22). The pre-\nsented theory thus captures both of these aspects in a\nconsistent quantum thermodynamic framework.\nUnlike Ohmic coupling (18) which depends on a single\nparameter\u0011G, the Lorentzian coupling function (26),\nand hence its kernel and spectrum, depends on three\nparameters. These allow a systematic study of di\u000berent\nregimes of the environment response. Speci\fcally, the\nmemory time of the environment can be continuously\nvaried by changing !0and\u0000, which can lead to very\ndi\u000berent magnetic behaviour. Beyond the spin dynam-\nics explored here, the proposed Lorentzian coupling may\nalso be a useful tool for the characterisation of quan-\ntum Brownian motion of a variety of systems, including\noscillators and free particles.\nC. Two coupling regimes\nTo better understand the relation between the Ohmic\nand Lorentzian coupling functions, one may consider\ntheir kernel moments \u0014min expansion (23). In con-\ntrast to the Ohmic case, for the Lorentzian kernel (28)\nall\u0014mare non-zero and given by\n\u0014Lor\nm=(\u00001)mA\n!1!2(m+1)\n0Im\"\u0012\u0000\n2+ i!1\u0013m+1#\n:(30)\nThe \frst relevant two moments are\n\u0014Lor\n1=\u0000A\u0000\n!4\n0and\u0014Lor\n2=A\n!6\n0(\u00002\u0000!2\n0);(31)\nand when comparing to the Ohmic case, one \fnds that\nthe \frst Lorentzian moment can be identi\fed with minus\nthe damping parameter \u0011G, i.e.\u0014Lor\n1=\u0000A\u0000\n!4\n0=\u0000\u0011G.\nFor a material with a given damping parameter \u0011Gthis\n\fxes one of the Lorentzian parameters, i.e.\nCLor\n!=s\n2\u0011G!2\n\u0019!4\n0\n(!2\n0\u0000!2)2+!2\u00002; (32)\nwhich now only depends on the two parameters !0and\n\u0000. For a speci\fc material these may be approximately\ndetermined through information contained in the den-\nsity of states of the environment to which the spins\ncouple [79].Inserting expansion (23) with Lorentzian moments\n(30) in the quantum spin dynamics equation (11) one\ncan distinguish two di\u000berent dynamical situations.\nOhmic regime: When the resonant frequency !0and\nthe damping rate \u0000of the reservoir coupling is much\nlarger than the spin operators' typical frequency of mo-\ntion,!0\u001d!, each successive term in expansion (23) is\nsmaller by an extra factor of (!=! 0). In the limit of in\f-\nnite!0but \fnite damping parameter \u0011G, the Lorentzian\ndamping term in (11) thus tends to the Ohmic one (20),\n\u0000\r2\u0011G^S(t)\u0002@t^S(t).\nNon-Ohmic regime: When the environmental fre-\nquency!0is comparable to typical spin motion frequen-\ncies,!0\u0019!the Ohmic approximation to the Lorentzian\nkernel begins to fail. The \frst deviation in (11) is a new\nterm proportional to \u0014Lor\n2. I.e. in addition to the Gilbert\ndamping term, one adds the term \r2\u0014Lor\n2^S(t)\u0002@2\nt^S(t)\ncontaining a second time derivative of the spin operator\n^S. By analogy with the classical equation of motion for\na massive body, this is known as an \\inertial\" modi\fca-\ntion to the spin dynamics [49]. As the ratio \u00142=\u00141has\nthe dimensions of time, one may introduce an \\inertial\ntimescale\"\u001cin[49], which for the Lorentzian is\n\u001cin=\u0014Lor\n2\n\u0014Lor\n1=!2\n0\u0000\u00002\n!2\n0\u0000: (33)\nA large inertial time \u001cinindicates the presence of non-\nMarkovian dynamics, i.e. dynamics that has a certain\ndegree of memory. For a high quality factor resonance\n!0\u001d\u0000the inertial time is (half) the kernel's decay\ntime,\u001cd= 2=\u0000. For increasing resonance width \u0000, the\ninertial time \u001cindecreases and memory e\u000bects become\nless important. In magnetic systems this timescale de-\ntermines the time over which nutation oscillations are\nobserved in the precession of the spin. Such inertial cor-\nrections to standard magnetism have very recently been\nobserved for the \frst time in ultrafast experiments on\nthin \flms [50]. Curiously, the inertial timescale becomes\nnegative when \u0000>! 0, a fact that may be the subject\nof future investigation.\nSimilarly to the kernel expansion (23), one can also\nexpand the Lorentzian power spectrum in frequency, see\nAppendix A4. Generally, only the odd moments \u00142m+1\nappear in the power spectrum. This implies that if a ker-\nnel only has non-trivial \frst ( \u00141) and second ( \u00142) mo-\nments, while higher moments vanish ( \u0014m>2= 0) then\nthe power spectrum will still be given by the (quantum)\nOhmic one (21), with \u0011G=\u0000\u00141. When third or higher\nmoments are non-zero, then the power spectrum of the\nnoise will deviate from the Ohmic case at all tempera-\ntures.\nTo summarise, here we have demonstrated that\nLorentzian coupling functions, kernels and power spec-\ntra provide a systematic framework to explore sys-\ntem dynamics that arises from inertial terms and other\nmemory e\u000bects, while recovering the standard Ohmic\nlimit whenever the Lorentzian resonance frequency !0\nis much larger than the typical system frequencies.\nD. Unit-free variables and Lorentzian parameters\nIn section III we perform semi{classical simulations\nof the dynamics of Eq. (11), for the Lorentzian kernel\nof Sec. II B. For this purpose we re{write expressions in7\n01 4 7 ω/ωL00.511.5cωa)\nLorentzian with\nω0=7ωL\nt/prime= t - 15ω−1\nLt00.30.6k(t−t/prime)c)\n01 4 7 ω/ωL03.06.0˜p(ω)\ne)\n0.6LLG+cl.\nLLG+qu.\nLor+qu. \n01 4 7 ω/ωL00.150.3˜p(ω)\ng)\n0.003\n01 4 7 ω/ωL00.511.5cωb)\nLorentzian with\nω0=1.4ωL\nt/prime= t - 15 ω−1\nLt00.30.6k(t−t/prime)d)\n01 4 7 ω/ωL03.06.0˜p(ω)\nf)T=200 K\n0.6LLG+cl.\nLLG+qu.\nLor+qu.\n01 4 7 ω/ωL00.150.3˜p(ω)\nh)T=1 K\n0.003\nFIG. 2. Comparison of coupling functions, memory kernels and power spectra: Top panels show (a) coupling function c!,\n(c) time dependent damping kernel k(t\u0000t0), and magnetic noise power spectrum ~p(!)at (e)T= 200 K, and (g)T= 1 K, for\nLorentzian coupling with parameter Set 1 (34a) (solid blue). Typical spin dynamics frequencies !2[0;2:5!L]are highlighted\n(yellow shading) in the frequency domain. Bottom panels shows the same quantities for Lorentzian coupling with parameter\nSet 2 (34b) (solid red). In (a{b) the Lorentzian coupling functions are compared to the LLG (Ohmic) approximation (magenta\ndash-dotted). In (e{h) the Lorentzian power spectrum is compared to the quantum LLG approximation (cyan squares), and\nits high temperature white noise limit (magenta dash-dotted).\nthe operator equation (11) in terms of a unit-free set of\nquantities. The time coordinate tis replaced with the\nunit-free coordinate !Lt, where!L=j\rjjBextjis the\nLarmor frequency. In addition the spin operator ^Swith\nlargest eigenvalue S0is re{written in terms of a unit-free\noperator ^swith largest eigenvalue 1,^S= sign(\r)S0^s.\nThe sign of the gyromagnetic ratio is included in the\nde\fnition of ^sso that ^saligns with the magnetic \feld,\nwhatever the sign of \r.\nFrom Eq. (11) we can see that the damping ker-\nnel has dimensions of magnetic \feld squared divided\nby angular momentum, which leads us to identify a\nunit-free damping kernel k(t\u0000t0)throughK(t\u0000t0) =\njBextj2S\u00001\n0k(t\u0000t0). Similarly, the unit-free coupling\nfunctionc!is de\fned through C!=jBextjS\u00001=2\n0c!\nand the unit-free spectral functions ~pthrough ~P=\n~jBextj2S\u00001\n0!\u00001\nL~p.\nLooking at the Lorentzian kernel K(t\u0000t0)in (27),\nthe pulling of dimensions can be achieved by setting\nthe kernel amplitude to A=jBextj2S\u00001\n0\u000bwhere\u000b\nnow is a frequency. For the simulations we choose the\nLorentzian parameters !0;\u0000and\u000bto be independent\nof spin length S0. Through (5) this implies an inter-\naction energy scaling of ^Vint/S0p\nA/pS0which\nsets the scaling of the interaction versus self-energy to\n^Vint=^HS/1=pS0. Apart from it being implied by\ndimensional analysis, such scaling is heuristically plau-\nsible in many physical contexts. E.g. it is similar to\nthe increasing ratio of the surface (where reservoir in-\nteraction occurs) to volume (self-energy) for decreasing\nsystem sizes. For microscopic systems for which ^Vintis\nno longer small in comparison to ^HSa thermodynamic\ntreatment beyond the weak coupling limit [26, 27, 81], a\nlimit tacitly assumed in standard thermodynamics, may\nbe required.\nSimilarly for Ohmic coupling leading to the LLG equa-tion (1), the above scaling choice amounts to choos-\ning\u0011G/A/1=S0implying that \u0011=\u0011G\r2S0=\n\u000b\u0000!2\nL=!4\n0is assumed to be independent of S0. In phys-\nical situations where this assumption is not justi\fed,\none may instead choose \u0011and\u000bto depend on the spin\nlengthS0.\nFor a spin in an external \feld Bextthe typical fre-\nquency of the dynamics is set by the Larmor frequency,\n!L. In the simulations discussed in section III we will\nuse the following two sets of Lorentzian parameters, all\nexpressed in terms of !L,\nSet 1):!0= 7:0!L\u000b= 10:0!L\u0000 = 5:0!L\n\u0011= 0:02\u001cin= 0:1!\u00001\nL\u001cd= 0:4!\u00001\nL(34a)\nSet 2):!0= 1:4!L\u000b= 0:16!L\u0000 = 0:5!L\n\u0011= 0:02\u001cin= 1:7!\u00001\nL\u001cd= 4!\u00001\nL:(34b)\nIn the second rows we have also listed the equivalent\nunit-free Gilbert damping \u0011, the inertial timescale \u001cin,\nand the memory kernel decay time \u001cdfor the Lorentzian\nkernel. Figs. 2-5, show plots obtained with Lorentzian\ncoupling functions (26) with parameter Sets 1 and 2,\nwhich are shown in blue and red, respectively. The\nOhmic LLG approximation is shown in magenta when\nthe classical reservoir (22) is considered, and in cyan\nwhen the quantum reservoir (21) is considered. The\nexternal \feld is set to Bext= 10 Tezwithezthe unit-\nvector inz-direction throughout.\nParameter Set 1 has been chosen to have a reso-\nnant frequency !0much larger than the characteristic\nspin precession frequency !L(Ohmic regime). Conse-\nquently we can truncate the series given in Eq. (23) to\nleading order and recover the Ohmic form (20) typically\nconsidered in magnetism theory. The validity of this ap-\nproximation is demonstrated in the top row of Fig. 2.\nFig. 2a) shows that the Lorentzian coupling function cLor\n!\nis well approximated by LLG (Ohmic) coupling for the8\nrelevant frequency range, while Fig. 2c) shows that the\nkernel is approximately instantaneous on the timescale\n!\u00001\nL, in line with Ohmic damping for which \u0014Ohm\nm>1= 0.\nFig. 2e) shows that at high temperature ( T= 200 K)\nand for relevant frequencies !\u0019!L, the power spec-\ntrum ~pLoris well approximated by the quantum Ohmic\n(LLG) power spectrum (21), and its classical limit (22).\nFig. 2g) shows that at lower temperatures ( T= 1 K)\nquantum noise becomes important. Here the Ohmic ap-\nproximation (21) remains valid, while its classical limit\n(22) is invalid.\nParameter Set 2 is chosen such that it has a resonant\nfrequency!0comparable to the precession frequency !L\n(non-Ohmic regime). Here it is inaccurate to truncate\nthe series (23) and the damping will be fundamentally\nnon{Ohmic. To directly compare with Ohmic dynam-\nics generated by Lorentzian coupling with Set 1, both\nparameter sets have been chosen to correspond to the\nsame unit-free Gilbert damping parameter, \u0011= 0:02.\nThe failure of the Ohmic approximation is demonstrated\nin the bottom row of Fig. 2. Fig. 2b) shows that the\nlinear approximation to the coupling function fails in\nthe relevant frequency range. For this set of parame-\nters the damping kernel now exhibits signi\fcant mem-\nory and Fig. 2d) shows that the response persists over\na timescale of several !\u00001\nL. This memory kernel im-\nplies, through the FDT (16), a coloured quantum noise\npower spectrum ~pLor\nqu. As shown in Fig. 2h), this coloured\nLorentzian ~pLor\nqu(red) di\u000bers from the LLG counterpart,\n~pOhm\nqu (cyan). Furthermore, Fig. 2f) shows that in the\nhigh temperatures ~pLor\nqu(red) also di\u000bers very signi\f-\ncantly from the LLG power spectrum ~pOhm\ncl (magenta).\nThe presence of both memory and coloured quantum\nnoise for the Lorentzian with parameter Set 2 are both\nsignatures of a thermostat that substantially deviates\nfrom the classical Ohmic assumptions and, as we will\nsee in the next section, leads to markedly di\u000berent short\ntime dynamics and steady state of sz.\nIII. Semi-classical spin dynamics simula-\ntions\nThe general spin dynamics equation (11) is an oper-\nator equation for quantum spins in a lattice, each inter-\nacting with neighbouring spins and with a bosonic reser-\nvoir. Solving the quantum dynamics using, for example,\nLorentzian coupling, kernel and spectrum, is rather di\u000e-\ncult without approximations, even numerically, and such\nexploration is left for future work.\nTo make progress here, we will numerically solve\nthe full non-Markovian for a semi-classical version of\nEq. (11), while including coloured quantum noise and\nmemory e\u000bects arising from the coupling to the envi-\nronment. It replaces the quantum spin operator ^Swith\na classical spin vector S, and the quantum stochastic\nnoise \feld vector ^bwith a stochastic classical \feld vec-\ntorbwith statistics that obey the quantum \ructuation{\ndissipation theorem (16a). This semi{classical approach\nis currently used in the theory of molecular and ionic\ndynamics [82, 83], and was perhaps \frst applied by\nKoch [84] to include the e\u000bects of quantum \ructuations\nin Josephson junctions. It has been justi\fed through an\nexpansion of a forward{backward path integral [85, 86](note the remark of Caldeira and Leggett on pg. 589\nof [35]), and is valid when the potential energy can be\nexpanded in the path integral to \frst order in the devi-\nations from the average path. The validity of applying\nthis approach to the decay of metastable states was in-\nvestigated in detail in [87].\nHere we simulate a single spin allowing us to illustrate\nthe impact of memory e\u000bects and the reservoir's quan-\ntum statistics on the spin dynamics and steady state.\nThe simulation details presented below can readily be\nextended to multiple interacting spins and could be in-\ntegrated in sophisticated atomistic spin dynamics simu-\nlations such as those used in [47, 60].\nA. How to simulate coloured noise and memory\nkernel\nHere we detail how to e\u000eciently simulate non-\nMarkovian dynamics that arises as a result of Lorentzian\ncoupling (26) for the example of spins vectors. Numer-\nical implementation of (11) requires both - the integra-\ntion of the kernel with the spin state of previous time\nsteps and the inclusion of coloured noise as follows.\nDropping the spin index and for simplicity assuming\nany isotropic kernel K(\u001c) =13K(\u001c), the three vector\ncomponents bj(t)forj= 1;2;3of the magnetic noise\n(12) are implemented as [88]\nbj(t) =Z1\n\u00001dt0F(t\u0000t0)\u0018j(t0); (35)\nwhere\u0018j(t0)is standard white Gaussian noise for the j-\nth component, which is delta correlated h\u0018j(t)\u0018k(t0)i=\n\u000ejk\u000e(t\u0000t0). The \\coloured noise\" comes from choosing\nF(t\u0000t0)as the Fourier transform of the square root of\nthe power spectrum associated with the kernel through\n(16), i.e.\nF(t\u0000t0) =Z1\n\u00001d!\n2\u0019e\u0000i!(t\u0000t0)q\n~P(!); (36)\nwhich can be implemented using a fast Fourier trans-\nform. To simulate the e\u000bect of a Lorentzian damping\nkernel (27) we numerically integrate [89] the following\nset of \frst order coupled di\u000berential equations for the\nspin vectorSand two dummy vectors VandW:\ndS(t)\ndt=\rS(t)\u0002(Bext+b(t) +V(t));\ndV(t)\ndt=W(t); (37)\ndW(t)\ndt=\u0000!2\n0V(t)\u0000\u0000W(t) +A\rS(t):\nThe integrated values of the dummy vectors and the\nspin are separated by the time step dt. Solving these\nequations is equivalent to solving the integro{di\u000berential\nequation (11) for a Lorentzian kernel, see Appendix A5,\nbut is numerically more straightforward to implement.\nB. Single trajectories for di\u000berent couplings and\nnoises.\nWe wish to illustrate on a single trajectory level, the\ndi\u000berences between the dynamics predicted by (11) with\neither an approximately Ohmic (Set 1) or non-Ohmic\n(Set 2) Lorentzian coupling function, as well as the dy-\nnamics predicted by the standard LLG equation. At9\n0 6π 12π 18π-1-0.500.51sz,sx,|s|a)T=1 K S=1/2,\n0 6π 12π 18π-1-0.500.51sz,sx,|s|c)T=200 K S=200/2,\nsz for LLG+cl. +0.1\nsz for LLG+qu. +0.05\n|s| for Lor(Set1)+qu.\nsx for Lor(Set1)+qu.\nsz for Lor(Set1)+qu.\n0 6π 12π 18π\nωLt-1-0.500.51sz,sx,|s|b)T=1 K S=1/2,\n0 6π 12π 18π\nωLt-1-0.500.51sz,sx,|s|d)T=200 K S=200/2,\nsz for LLG+cl. +0.1\nsz for LLG+qu. +0.05\n|s| for Lor(Set2)+qu.\nsx for Lor(Set2)+qu.\nsz for Lor(Set2)+qu.\nFIG. 3. Sample of stochastic short-time spin dynamics for di\u000berent couplings and noises. Stochastic short-time dynamics of\nsz(blue in top row & red in bottom row), sx(green) andjsj(black dashed) according to Eq. (11) with Lorentzian coupling\nfunctionCLor\n!, for a classical spin initially in state s= (\u00001;0;0). All traces in the four panels are generated from the same\nsample of Gaussian noise, enabling a direct comparison. Shown are the dynamics for Set 1 (top row) and Set 2 (bottom row),\nand two spin+temperature pairs: S0= 1~=2andT= 1 K (left column), and S0= 200 ~=2andT= 200 K (right column).\nAlso shown in all four panels are the sz-dynamics according to the LLG equation Eq. (1) with damping parameter \u0011\u00190:02\nfor two types of noise: the classical \rat white noise power spectrum Eq. (22) (magenta) and the quantum noise power\nspectrum Eq. (21) (cyan). All cyan and magenta plots are o\u000b-set by +0.05 and +0.1, respectively, to avoid overlapping.\nThe external magnetic \feld is set to Bext= (0;0;10T)setting the timescale to !\u00001\nL\u00190:57\u000110\u000012s, and the simulation\ntime interval is dt= 0:15!\u00001\nL.\n\frst, because the dynamics is intrinsically stochastic,\ntrajectories will naturally di\u000ber and cannot readily be\ncompared. However, looking at the noise generation in\nEqs. (35) and (36), one can see that the same white\nnoise\u0018j(t)forj= 1;2;3may be used as a seed to cre-\nate comparable \\stochastic\" noise for di\u000berent power\nspectra ~P(!).\nFig. 3 shows the stochastic short time dynamics of\na single classical spin for two pairs of spin length and\ntemperature, S0= 1~=2atT= 1 K (left panel) for a\nsingle electron, and S0= 200 ~=2atT= 200 K (right\npanel) for a mesoscopic cluster of spins with a combined\nlarger e\u000bective spin.\nThe dynamics is obtained according to Eq. (11)\nfor Lorentzian coupling (26) with S0-scalingA=\njBextj2S\u00001\n0\u000b, for parameter sets Set 1 (top panel, blue)\nand Set 2 (bottom panel, red), and with the quan-\ntum coloured noise given by (29). For comparison we\nalso show the short time dynamics according to the\nLLG equation (1) with S0-scaling\u0011G=A\u0000=!4\n0=\njBextj2S\u00001\n0!\u00002\nL\u0011with the Gilbert damping parameter\n\u0011= 0:02common to both Lorentzian parameter sets,\nsee (34). That implies that the top and bottom LLG\nplots are identical. For the standard LLG equation two\ntypes of noise are considered - high-temperature classi-\ncal noise (magenta) see Eq. (22), and quantum noise\n(cyan) see Eq. (21). Since the same white noise time\nseries is used as a seed for producing the stochastic noise\nfor all traces, we can compare them directly. We willhere focus on sz, the component of s=sign(\r)S=S0\naligned with the external \feld Bext.\nThree features stand out in Fig. 3: i) as expected\nfrom section II, the dynamics generated with Eq. (11)\nwith Lorentzian Set 1 (top, blue) closely matches the dy-\nnamics obtained with the LLG equation (1) with quan-\ntum noise (cyan) for both spin-temperature pairs, ii)\nthe quantum statistics of the reservoir (cyan) at low\ntemperatures (left) introduces di\u000berences to the LLG\ndynamics compared to the LLG dynamics obtained with\nclassical noise (magenta), and iii) memory e\u000bects that\nare present for Lorentzian Set 2 (bottom, red) result in\nsigni\fcantly di\u000berent dynamics from that arising with\nthe memory-free Lorentzian Set 1 (top, blue).\nWe remark that due to the spin/temperature ratio\nbeing the same for the two spin-temperature pairs, the\nLLG equation with classical noise (magenta) integrates\nto exactly the same dynamics in left and right panel, see\nAppendix A8. This scaling relation ceases to be true for\nthe LLG equation with quantum noise (cyan). Another\ndi\u000berence to note is that in Fig. 3a{d) the dynamics for\nLorentzian parameter Set 1 (blue) varies more rapidly\nin time than for Lorentzian parameter Set 2 (red). This\nis due to the high frequency content of Set 1's power\nspectrum, see Fig. 2e+g).\nFinally, the spin component sx(green) and the spin-\nvector lengthjsj(black) are shown for the Lorentzian\ncoupling with Set 1 and Set 2 in the top and bottom\npanels of Fig. 3, respectively. The plots of jsjshow that10\n012π 36π 60π 84π\nωLt00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\n a)\n0.84\n0.28\n0.27\n0.29S0=1/2,\nT=1 K\n012π 36π 60π 84π\nωLt00.250.50.751/angbracketleftbig\nsz/angbracketrightbigb)\n0.84\n0.84\n0.85\n0.85S0=200/2,\nT=200 K\n/angbracketleftbig\nsz/angbracketrightbig\n for LLG+cl./angbracketleftbig\nsz/angbracketrightbig\n for LLG+qu./angbracketleftbig\nsz/angbracketrightbig\n for Lor(Set1)+qu./angbracketleftbig\n|s|/angbracketrightbig\n for Lor(Set1)+qu./angbracketleftbig\nsz/angbracketrightbig\n for Lor(Set2)+qu.\nFIG. 4. Ensemble averaged spin relaxation dynamics. Averaged dynamics hszi, averaged over 500 stochastic traces up to\ntimetmax= 2\u0019\u000248!\u00001\nL. Shown are the dynamics according to Eq. (11) with Lorentzian coupling functions for parameter\nSet 1 (blue) and Set 2 (red). Also shown is the ensemble averaged dynamics according to the LLG equation with damping\nparameter\u0011\u00190:02for two types of noise: classical white noise (magenta) and full quantum noise (cyan). The two panels\nshow two spin+temperature pairs: S0= 1~=2andT= 1 K (left), and S0= 200 ~=2andT= 200 K (right). Note that\nblue, cyan and magenta curves lie on top of each other in b). As discussed in section II D, the plots assume that both Aand\n\u0011Gscale as/S\u00001\n0with spin size S0, making\u000band\u0011the same for the two spin sizes. The external magnetic \feld is set to\nBext= (0;0;10T)and the simulation time interval is dt= 0:15!\u00001\nL.\nthe numerical integration of Eq. (11) indeed leads to a\nconstant spin-vector length jsj= 1, i.e. no renormali-\nsation is required.\nC. Ensemble-averaged hszitrajectories.\nFig. 4 shows the ensemble averaged hsziover time,\naveraged over 500 stochastic trajectories. We now high-\nlight two important features in Fig 4. Firstly, at low\ntemperatures (left) the quantum statistics of the reser-\nvoir (blue, red, cyan) results in a much depleted value of\nhszi, roughly at around 0:28, in comparison to that ob-\ntained with the LLG equation with classical noise (ma-\ngenta), ca 0:85. This indicates that for this particu-\nlar choice of spin length and temperature the quantum\ncharacter of the reservoir strongly a\u000bects the value of\nhszi, as further discussed below, and the classical high-\ntemperature limit taken in (16b) would not be appropri-\nate. For the high temperature T= 200 K + larger spin\npairS0= 200 ~=2(right), the di\u000berence between clas-\nsical and quantum statistics of the reservoir can be ne-\nglected andhszisettles at 0:85independent of whether\nthe spin dynamics integration was done for Eq. (11) with\neither Set 1 (blue) or Set 2 (red), or for Eq. (1) with\neither classical (magenta) or quantum noise (cyan).\nSecondly, for the large spin-temperature pair (right),\nthere is clear evidence of a much quicker relaxation\nto steady state (by a factor of a third) for Lorentzian\nSet 2 (red) compared to the other plots (blue, cyan,\nmagenta). This is a non-Markovian e\u000bect that arises\nbecause the memory kernel for Set 2 has an appreciable\nmemory over time, see Fig. 2d), while the other memory\nkernels are (close to) instantaneous. This quicker equi-\nlibration occurs because the non-Markovian kernel leads\nto a smoother dynamics which in turn is more quickly\nsampled by the dynamical system.\nD. Steady state hszias a function of tempera-\nture.\nFig. 5 shows the average steady state spin value hszi\nas a function of temperature T, found by time-averaging\na single trajectory szover late times, from 0:75tmax to\ntmax= 2\u0019\u00027200!\u00001\nL. There are two key observationsto make in Fig. 5. Firstly, for both the small spin (left)\nand the large spin (right) the steady state hsziobtained\nwith the LLG equation and classical noise (22) matches\nthe standard statistical physics prediction hszistat phys =\ncoth\u0010\nS0!L\nkBT\u0011\n\u0000kBT\nS0!L, see Appendix A6.\nSecondly, for simulations that include the full quan-\ntum noise (cyan, blue, red) in the dynamics of the small\nspin at low temperatures (left), we observe reduced hszi\nvalues in the range 0.2-0.4 at T= 0 K, i.e., well be-\nlow the classical value of 1. This arises because the\npower spectrum ~PLor\nqu, given through the quantum FDT\n(16a), includes quantum \ructuations which remain even\nforT!0K. The steady state curves with quantum\nnoise also show a characteristic \\\rattening\" compared\nto the steep decay of the steady state with tempera-\nture for classical noise (magenta). Qualitatively speak-\ning, this quantum zero point noise, when compared to\nclassical noise, is as if thermal noise is \\on\" even at\n0K. I.e. taking the Larmor frequency as the relevant\nfrequency, and setting ~POhm\nqu(0K) =~POhm\ncl(Tcl)for the\nOhmic coupling, for example, de\fnes a classical temper-\nature ofTcl= 6:7K \\equivalent\" to the quantum zero-\ntemperature case. For S0= 1~=2the classical statisti-\ncal physics steady state value at Tclishszistat phys\u00190:3.\nThis indeed is of comparable size to the hszivalues ob-\ntained with quantum noise at T= 0K. The correspond-\ning steady state value for the large spin ( S0= 200 ~=2)\nishszistat phys\u00190:995\u00191, see Fig. 5c).\nGenerally, for the classical temperature Tcl=~!L\n2kBwhich is \\equivalent\" to the quantum zero-point noise,\none obtainshszistat phys = coth\u00002S0\n~\u0001\n\u0000~\n2S0, which only\ndepends on the spin length S0while being independent\nof the \feld strength jBextj. With increasing S0this\nfunction rises very sharply from \u00190:3to1. For ex-\nample for spin length S0= 5~=2, the quantum zero\ntemperature value is hszistat phys\u00190:8and its decay\nwith increasing temperate is shown in Fig. 6b) in Ap-\npendix A7.\nThe middle panel, Fig. 5b), gives an alternative illus-\ntration of the steady state value for the small spin as a\nfunction of environment temperature T. It shows the11\n01 4 8 12 16 20\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\na)\nS0=1/2\nstat. phys.\nLLG+cl.\nLLG+qu.\nLor(Set1)+qu.\nLor(Set2)+qu.\n01 4 8 12 16 20\nT / K00.250.50.751m(T)\nb)\nS0=1/2\n0 800 1600 2400 3200 4000\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\nc)\nS0=200/2\nstat phys\nLLG+cl.\nLLG+qu.\nLor(Set1)+qu.\nLor(Set2)+qu.\nFIG. 5. Steady statehsziof a spin interacting with a (quantum) reservoir at temperature T.For a spin in an external\nmagnetic \feld Bext= (0;0;10T)and interacting with a thermal reservoir, the time averaged value of hsziis obtained\nby integrating Eq. (11) for Lorentzian parameter Set 1 (blue crosses) and Set 2 (red dots), as well as by integrating the\nLLG equation with damping parameter \u0011= 0:02for two types of noise: classical white noise (magenta dash-dotted) and\nfull quantum noise (cyan dashed). The spin lengths are S0= 1~=2(panel a and b), and S0= 200 ~=2(panel c). For\nthe small spin in panel a), the three curves with quantum noise (cyan, blue, red) start at hszi-values in the range 0.2-0.4\natT= 0 K. For both S0values, the steady state for the LLG equation with classical noise (magneta) coincide well with\nhszistat phys (black), derived from classical statistical mechanics for a thermal distribution, see Appendix A6. Note that all\ncurves lie on top of each other in panel c). Panel b) shows the same plot as panel a) - but with the y-axis \\rescaled\"\nasm(T) =hsz(T)i=hsz(0)iso that all plots start at 1 at T= 0 K. While the magenta curve remains the same as in\na), the rescaled blue, red and cyan curves now show a \rattened decay behaviour that somewhat resembles the corrected\nmagnetization curves for real materials analysed in [90]. Error bars for the four simulations (blue, red, cyan, magenta) are\nindicated at a) T= 1 K and c)T= 200 K. The simulation time interval is dt= 0:15!\u00001\nL.\nsame plot as panel a), but with the y-axis rescaled as\nm(T) =hsz(T)i=hsz(0)i. All plots now start at 1 at\nT= 0 K, independent of whether the dynamics was in-\ntegrated with quantum or classical noise. Interestingly,\nthe overarching behaviour of the resulting curves bears\nsome resemblance with heuristically rescaled magneti-\nzation curves that match experimental data [90, 91].\nRunning high-end atomistic simulations of Eq. (11),\ninstead of (1), for multiple interacting classical spins\nwould answer if the quantum power spectrum's impact\non their low temperature magnetisation behaviour as\nwell as their Curie temperature is the reason for the\napparent rescaling.\nWe note that larger numerical uncertainties arise for\nthe quantum noise because an additional scale is present\nin comparison to classical noise, see Appendix A8. Er-\nror bars obtained from an ensemble of simulations are\nindicated at one temperature value in a) and c) for all\nfourhszicurves in Fig. 5.\nIV. Conclusions and open questions\nWe have derived a general quantum spin dynam-\nics equation, Eq. (11), capable of describing three-\ndimensional precession and damping. The terms aris-\ning from the reservoir interaction are treated in a quan-\ntum thermodynamically consistent manner, by tracing\nthe origin of both the memory kernel, K(\u001c), and the\nstochastic noise, ^b(t), to a single coupling function,\nC!. Secondly, Lorentzian coupling functions were pro-\nposed and shown to provide a systematic means to in-\nvestigate di\u000berent dynamical regimes - from Ohmic to\nnon-Ohmic dynamics which is subject to memory and\ncoloured noise. We showed that only in the Ohmic\nregime, the standard LLG equation with Gilbert damp-\ning, widely used in magnetism, is recovered. Finally, we\nprovided details of how to include Lorentzian memory\nand coloured noise in numerical simulations of open sys-tem dynamics. For the example of a single spin vector,\nwe illustrated that a non-Ohmic Lorentzian kernel leads\nto a faster equilibration time of hsziin comparison to\nthe Ohmic (LLG) regime. We also discussed the steady\nstate di\u000berences that arise when the full quantum ther-\nmostat with quantum zero-point noise is employed, in\ncontrast to classical white noise.\nThe above three ingredients provide a complete\nframework for the simulation of damped three-\ndimensional precession including memory and coloured\nnoise. It can readily be adapted in atomistic spin dy-\nnamics simulations [47, 60] that solve the dynamics of\nmillions of interacting spins.\nThe theory presented here will be a useful tool for\ninvestigating non{Markovian behaviour, opening up a\nnumber of avenues for future research at the intersec-\ntion of quantum thermodynamics, magnetic materials\nand beyond. For example, it is an open question to\nclarify the connection between the three-dimensional\nprecession described by the spin equation (11) and ro-\ntational Brownian motion. The orientation of a non-\nsymmetric rotating body behaves analogously to the\nthree-dimensional spin vector, and the motion and vis-\ncosity of a gas surrounding a rotating body simulta-\nneously act on its motion while obeying the FDT as\ndiscussed in recent work [92, 93].\nWithin magnetism, for particular materials of inter-\nest, detailed models of the coupling functions C!can\nbe developed that are based on an understanding of\nthe interactions between spins, phonons, and electrons\nin the material [79]. Coupling to optical modes may\nfurther be included to describe, for example, whispering\ngallery photon-magnon coupling which leads to an e\u000bec-\ntive Gilbert damping term that can take either sign [94].\nA direct experimental characterisation of a material's\ndamping kernelKthat determines memory and noise\nin (11) may be attempted, for example with high \feld\nexperiments such as those recently reported in [50]. Of12\nparticular interest are dynamical features beyond the in-\nertial kernel approximation, which will also modify the\nnoise spectrum at larger temperatures.\nWhile we here discussed scalar couplings to the en-\nvironment in depth, Eq. (11) does hold for any real\n3x3 matrixC!describing the spin-environment interac-\ntion in three dimensions. Anisotropic coupling tensors\ncan be implemented, suitable for describing magnetiza-\ntion dynamics within thin \flms [62], where one direction\nis coupled di\u000berently to environmental modes than the\nother two. One simpli\fcation of our three-dimensional\nmodel is to choose a coupling tensor such that spins in-\nteract with only one-dimensional environmental modes.\nThis reduces the theory to the spin-boson model, see\nAppendix A1, whose quantum thermodynamic proper-\nties have been discussed very extensively, recently for\nexample in [33].\nMicroscopic heat transport in spin systems can also be\nanalysed by allowing non{equilibrium situations where\nindividual reservoir modes at frequencies !and for spins\nnare thermal - but at di\u000berent temperatures. This will\nresult in spin dynamics that shu\u000fes energy from one\nreservoir mode to another, and could result in two- and\nmore-temperature models. For example, the possibility\nof di\u000berent phonon modes, each with their own tem-\nperature, to couple with di\u000berent strengths to electrons\nhas recently been analysed in [95] for a magnetic system\nexcited by an ultra-short laser pulse. Furthermore, in de-\nriving the FDT we have assumed bosonic environmental\nmodes but it would be insightful to identify changes to\nthe properties of equation (11) that arise when the spins\ncouple directly to electrons, or fermionic modes in gen-\neral [14, 15, 96].\nBeyond the quantum character of the reservoir, it\nwill be important to numerically solve the full quan-\ntum dynamics according to Eq. (11), including spin\noperators interacting with neighbouring spin operators.\nAdvanced quantum numerical methods such as Hier-\narchical Equations Of Motion (HEOM) [97], and the\nrecently proposed time-evolving matrix product oper-\nators (TEMPO) method [98] will be required to e\u000e-\nciently describe the time evolution of even just a sin-gle quantum spin coupled to a non-Markovian environ-\nment. For multiple interacting spins at low tempera-\ntures one can expect entanglement between the spins\nbeing present during the short-time dynamics, and even\nin steady state [99{101]. Unfortunately, evaluating such\nproperties will very quickly become a numerically hard\nproblem, requiring advanced numerical techniques such\nas density-matrix renormalisation group (DMRG) [102]\nto \fnd realistic approximate solutions. Vice versa, solv-\ning (11) within the classical spin vector approximation\nwhile including a full quantum power spectrum for the\nenvironmental modes, may prove insightful and numer-\nically tractable in the context of \fnding suitable mod-\nels for noise in quantum computing hardware, such\nas superconducting qubits that are held in the mK\nrange [103]. The results may also inform implementa-\ntions of Young's double slit experiment with a levitated\nsingle magnetic domain nanoparticle using the Einstein-\nde Haas e\u000bect [104, 105].\nAcknowledgments\nWe thank Karen Livesey, Richard Evans, Marco Berritta,\nStefano Scali, Federico Cerisola, Luis Correa, James\nCresser, Claudia Clarke, Ian Ford and Rob Hicken\nfor inspiring discussions, Carsten Henkel and Richard\nEvans for comments on a draft of this manuscript,\nand Somayyeh Nemati for iron's Lorentzian parame-\nters mentioned in [79]. SARH thanks Paul Kinsler\nfor pointing out the stupidity of numerically solving\nan integro{di\u000berential equation when an ODE will do.\nSARH also acknowledges funding from the Royal So-\nciety and TATA (RPG-2016-186). CRJS and JA ac-\nknowledge support and funding from the EPSRC Centre\nfor Doctoral Training in Electromagnetic Metamaterials\nEP/L015331/1. 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(1) can be eliminated and the\nequation becomes @M=@t=\r0M\u0002Be\u000b\u0000\u0015M\u0002\n(M\u0002Be\u000b)with\r0and\u0015functions of \r;\u0011GandjMj.\n[109] We use SI rather than Gaussian units, so we have B\n(units mass=(charge\u0002time) ) rather than an H-\feld\n(units charge=(length\u0002time) ).\n[110] Here the spins are discrete and positioned on a lattice,\nbut one could also use a continuum description, as in\nmicromagnetics [44].\n[111] Note that tensors and vectors are set in calligraphic\nand bold font, respectively, and that scalar products\nbetween vectors are indicated with \u0001, while a tensor\nfollowed by a tensor or a vector is to be understood as\nmatrix multiplication.\n[112] Note that in contrast to what is typically done in\nCaldeira{Leggett type models[35] no counter term has\nbeen included here. In any case, coupling to a spin\nwould result in a term proportional to S2/ 1, which\nwill incur an o\u000bset in the overall Hamiltonian that does\nnot a\u000bect the dynamics.[113] The Fourier transform ~K(n)(!)of the kernelK(n)(\u001c)\nautomatically satis\fes the Kramers{Kronig rela-\ntions [63], connecting the dissipative and reactive parts\nof the response kernel, as is required for any causal re-\nsponse.\n[114] The autocorrelation function of two reservoir operators\nAandByin the thermal reservoir state ^\u001aRis de\fned\nas the expectation value of the Hermitian operator,\nh\b\nA(t);By(t\u0000\u001c)\t\ni\f=2. In the classical case Aand\nBycommute at all times, removing the need for this\ndistinction.\n[115] The Fourier transform is here de\fned as\n~f(!) =R1\n\u00001d\u001ce+i!\u001cf(\u001c), with the inverse\nf(\u001c) =R1\n\u00001d!\n2\u0019e\u0000i!\u001c~f(!).\n[116] The power spectrum given in (16) is the correct gen-\neral version for any kernel K(n)(t), ful\flling the full\nquantum FDT [63]. For a Gilbert damping kernel a\npower spectrum proportional to ~!=(exp ( ~!=kBT)\u0000\n1)was given in [60, 74{76]. This is missing the quan-\ntum ground state contribution of ~!=2, which acts\nas stochastic noise on the spin system even at zero\ntemperature.\n[117] Note that although (20) is not an explicitly Hermitian\noperator, Appendix A2 shows that Eq. (11) is never-\ntheless equivalent to a Hermitian equation of motion.16\nAppendix\nA1. Recovery of the spin-boson model\nThe spin-boson model is recovered as a special case\nof the three-dimensional Hamiltonian (2), and hence its\ndynamics is also given by Eq. (11). To see this one may\ndrop the site index n, and choose the external \feld as\nBext=B0(cos(\u0012)ez\u0000sin(\u0012)ex); (38)\nfor some angle \u0012. Taking spin 1=2operators ^S=\n(~=2)\u001band the coupling tensor as (C!)jk=C!\u000ejk\u000ej1\nwith a scalar coupling function C!, one recovers from\n(2) the one-dimensional spin-boson Hamiltonian\n^H=\u0000\rB0(cos(\u0012)^Sz\u0000sin(\u0012)^Sx)\n\u0000\r^SxZ1\n0d! C!^x! (39)\n+1\n2Z1\n0d!h\n(^px;!)2+!2(^x!)2i\n+^H2D\nR;\nwhere ^H2D\nRis a decoupled two-dimensional reservoir\nthat can be dropped from the dynamics.\nA2. Hermiticity of the quantum spin dynamics\nequation\nThe quantum spin dynamics equation (11) is not writ-\nten in an explicitly Hermitian form. The integral term\ncontaining the damping kernel K(n)includes an operator\nproduct that does not equal its conjugate transpose\n^S(n)(t)\u0002^S(n)(t0)6=\u0000^S(n)(t0)\u0002^S(n)(t): (40)\nNevertheless equation (11) is Hermitian, as it is the time\nintegral that commutes with ^S(n)(t)\n\u0014Zt\nt0K(n)(t\u0000t0)^S(n)(t0);^S(n)(t)\u0015\n= 0 (41)\nThis can be veri\fed from an observation that Eq. (11)\nis simply a re{written form of the explicitly Hermitian\nequation (8).\nAny confusion can be avoided through re{writing\nEq. (11) in an equivalent but explicitly Hermitian form\nd^S(n)(t)\ndt=\r\n2[^S(n)(t)\u0002^B(n)\ne\u000b(t)\n\u0000^B(n)\ne\u000b(t)\u0002^S(n)(t)](42)\nwhere the e\u000bective magnetic \feld operator at time tand\nsitenis given by\n^B(n)\ne\u000b(t) =Bext+1\n\rX\nm6=n\u0016J(nm)^S(m)(t) +^b(n)(t)\n+\rZt\nt0dt0K(n)(t\u0000t0)^S(n)(t0):(43)\nIn the Hermitian form (42) it is clear, for example\nthat any term proportional to ^S(n)(t)appearing in ^B(n)\ne\u000b\ndoes not a\u000bect the evolution of the spin operator, even\nthough the operator cross product\n^S(n)(t)\u0002^S(n)(t) = i~^S(n)(t) (44)is non{zero. A consequence of this result is that the\nzeroth order term in the expansion of the damping op-\nerator (23) does not contribute to the evolution of the\nspin operator.\nWe note that Eq. (41) implies that only the sum of\nallthe terms in the damping kernel expansion (23) com-\nmutes with the spin operator. When using a truncated\nform of the expansion (23) we must therefore use the\nexplicitly Hermitian equation of motion (42).\nA3. ^S2is a constant of motion of Eq. (11)\nTo evaluate the derivative of (^S(n)(t))2we \frst ex-\npress Eq. (11) in explicitely Hermitian form (42). Drop-\nping site index and time for simplicity, we \fnd\ndj^S(t)j2\ndt=X\nj \n^Sjd^Sj\ndt+d^Sj\ndt^Sj!\n=\r\n2X\njkl\u000fjkl\u0010\n^Sj^Sk^Bl+^Bl^Sk^Sj\u0011\n=i~\r\n2X\nl[^Sl;^Bl]\n= 0; (45)\nwhere we have applied the angular momentum com-\nmutation relations, interchanged indices, and used the\nanti-symmetric property of \u000fjkm. The \fnal line follows\nfrom the fact that the spin and the e\u000bective magnetic\n\feld commute.\nA4. Lorentzian power spectrum expansion\nSimilar to the damping kernel term expansion (23),\nin moments (30) and time-derivatives, the Lorentzian\npower spectrum (29) can be expanded in powers of fre-\nquency!, as\n~PLor\nqu(!) =1X\nm=0(\u00001)m+1!2m+1\u0014Lor\n2m+1coth\u0012~!\n2kBT\u0013\n;\n(46)\nwhere we have kept the quantum coth unexpanded. The\n\u0014Lor\nmare the same coe\u000ecients as those given in (30).\nFor small frequencies !the \frst term in the series (46)\ndominates and the power spectrum takes the (quantum)\nOhmic form\n~PLor\nqu(!)\u0019\u0000!\u0014Lor\n1coth\u0012~!\n2kBT\u0013\n; (47)\nwhere comparison with (21) again shows that \u0000\u0014Lor\n1is\nthe e\u000bective Gilbert damping constant.\nBeyond the Ohmic regime, one can see in (46) that\nonly the odd moments \u0014Lor\n2m+1contribute. Therefore\nthe inertial term \u0014Lor\n2, which is the \frst deviation of the\ndamping kernel from Ohmic behaviour, does not change\nthe quantum \ructuations in (16). Only when the third\norder time derivative of the spin operator contributes\nsigni\fcantly to equation (11), will memory e\u000bects begin\nto colour the spectrum away from the (quantum) Ohmic\nform (21).\nA5. Set of equations for kernel simulation\nHere we show that the simulation of the kernel in\nEq. (11) can be achieved by numerically integrating a set17\n01 4 8 12 16 20\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\na)\nS0=1/2\n0520 40 60 80 100\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\nb)\nS0=5/2\n0 800 1600 2400 3200 4000\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\nc)\nS0=200/2\nstat phys\nLLG+cl.\nLLG+qu.\nLor(Set1)+qu.\nLor(Set2)+qu.\nFIG. 6. Same notation and Lorentzian parameters as in Fig. 5 but here also showing spin length S0= 5~=2in panel b).\nPanels a) and c) show hsziforS0= 1~=2andS0= 200 ~=2, respectively, as in main text. Each curve is obtained by, for each\ntemperature T,time-averaging over the late times of a single stochastic trajectory, from 0:75tmax totmax= 2\u0019\u00028000!\u00001\nL.\nof \frst order coupled di\u000berential equations. We assume\na single spin and rewrite Eq. (11) as\ndS(t)\ndt=\rS(t)\u0002\u0014\nBext+b(t) +V(t)\u0015\n; (48)\nwhere we have de\fned V(t) =\rRt\nt0dt0K(t\u0000t0)S(t0).\nFurthermore de\fning W(t) =dV(t)\ndt, now leads to a\ndi\u000berential equation for W(t):\ndW(t)\ndt=\rZt\nt0dt0d2K(t\u0000t0)\ndt2S(t0); (49)\nwhere we have assumed K(0) = 0 and _K(0) = 0 . Ex-\npressingK(t\u0000t0)through its Fourier transform ~K(!),\nchoosing a Lorentzian kernel (27) and considering the\nexpressionZ(t) :=dW(t)\ndt+\u0000W(t)+!2\n0V(t), we obtain\nZ(t)=A\rZt\nt0dt0\u000e(t\u0000t0)S(t0): (50)\nRearranging gives\ndW(t)\ndt=\u0000\u0000W(t)\u0000!2\n0V(t) +A\rS(t); (51)\nas stated in the main text. (Note that the assumption\nK(0) = 0 and _K(0) = 0 is ful\flled for the Lorentzian\nkernel, (28), since the Heaviside function \u0002(\u001c) = 1 for\n\u001c >0, and zero elsewhere.)\nA6. Statistical physics prediction for hszias func-\ntion of temperature\nFor a classical spin Sof lengthS0=n~=2in an\nexternal \feld Bext=Bezthe thermal average hszi\nis determined by the Boltzmann distribution for the\nHamiltonian H=\u0000\rS\u0001Bextat inverse temperature\n\f= 1=kBT,\nhSzistat phys =Z+S0\n\u0000S0dSzSze\u0000\f(\u0000\rSzB)\nZa=@alnZa;(52)\nwithZa=R+S0\n\u0000S0dSzeaSzwherea=\f\rB . This gives\nZa=2 sinh(aS0)\na; (53)\nand hence\nhSzistat phys\nS0= coth (\f\rBS 0)\u00001\n\f\rBS 0;(54)\nhszistat phys = coth\u0012n~!L\n2kBT\u0013\n\u00002kBT\nn~!L; (55)where!L=j\rBjandsz=sign(\r)Sz\nS0. In the mag-\nnetism literature, sometimes a reduced temperature ex-\nperienced by a spin with n6= 1 is de\fned as Tred=T=n ,\ni.e. the temperature is e\u000bectively reduced in comparison\nto the temperature experienced by a spin with n= 1.\nA7. Steady state hsziplot for spin S0= 5~=2\nAs discussed in the main text, the impact of the quan-\ntum zero-point noise on the steady state hszivalue at\nT= 0K is very highly spin length dependent. For some\nmaterials a fundamental spin value of S0= 1~=2will\nnot be appropriate. For example Iron (III) has 5 elec-\ntrons in the outer dshell, and then from Hund's rules\nthe spin is maximized to S= 5=2, and the orbital an-\ngular momentum is zero, L= 0. Therefore J=S,\nLand\u0013 e g{factor equals 2, and the gyromagnetic ratio re-\nmains the electron gyromagnetic ratio. Fig. 6b) shows\nthe steady statehsziplot as a function of temperature\nfor spinS0= 5~=2, next to those for spin S0= 1~=2\n(a) andS0= 200 ~=2(c). Thehszivalue is below 1, at\n\u00190:8, but the reduction is far less severe than for the\nspin-1/2.\nA8. Scales in classical and for quantum ther-\nmostats\nHere we establish the set of scales determining the\ndynamics described by Eqs. (1) and (11) with either\nclassical orquantum power spectra.\nFor a single spin, i.e. ignoring exchange terms etc.,\none may rescale the LLG equation (1) using M=\rS=\nj\rjS0swith spin lengthjSj=S0. One obtains\nds\ndt=\rs\u0002\u0014\n^Bcl\ne\u000b(t)\u0000j\rjS0\u0011Gds\ndt\u0015\n; (56)\nwhere the spin length S0and\u0011Gappear together, set-\nting the \frst scale. Furthermore the e\u000bective \feld in-\ncluding classical stochastic noise with power spectrum\n(22), is given through (35) and (36) by\n^Bcl\ne\u000b(t) =Bext+r\n2S0\u0011GkBT\nS0\u0018(t): (57)\nHere we have introduced an S0so that\u0011Gappears to-\ngether with it, and we \fnd the second scale to be given\nbyT=S 0. The third scale is clearly set by the strength\nof the external \feld, Bext. We note that if one chooses\nthe sameS0\u0011Gvalue for di\u000berent spin lengths, i.e. as-\nsumes\u0011Gscales as 1=S0, then only two scales are left,\nBextandT=S 0.18\nHowever, for the quantum Ohmic power spectrum the\ncomponents of the stochastic noise can be written as\nbj=Z1\n\u00001dt0Z!c\n\u00001d!\n2\u0019e\u0000i!(t\u0000t0)\n\u0002s\nS0\u0011G~!\nS0coth\u0012~!\n2kBT\u0013\n\u0018j(t0); (58)\nClearly, in the quantum case the temperature Tnow\nappears separately from spin length S0, thus introduc-\ning an additional scale in comparison to the classical\ncase. Moreover the fact that the frequency integration\nfor the stochastic \feld does not simplify as in (57) means\nthat relaxation to the steady state at low temperatures(where the cothxcannot be approximated as 1=x) will\nbe much more noisy than in the high temperature case.\nThus in our simulations, this additional scale leads to\nlarger uncertainties in the steady state results, as seen\nin Fig. 5a).\nFinally, we note that for the integration of the quan-\ntum Ohmic power spectrum in (58) we have introduced\na frequency cut-o\u000b !cby hand, which is necessary at\nlow temperatures to avoid the integral diverging. At low\ntemperatures, this cut-o\u000b will set an additional, some-\nwhat arti\fcial, scale of the problem. Importantly, such\ncut-o\u000b is not required for the Lorentzian coupling since\nthe power spectrum (29) decays at high frequencies,\neven at low T."    },    {        "title": "2009.03196v3.Spin_pumping_in_d_wave_superconductor_ferromagnet_hybrids.pdf",        "content": " Spin pumping in d -wave superconductor/ferromagnet hybrids  \nS. J. Carreira1,*, D. Sanchez -Manzano1,2, M.-W. Yoo1, K. Seurre1, V. Rouco1, A. Sander1, J. \nSantamaría2, A. Anane1, and J. E. Villegas1 \n \n1Unité Mixte de Physique, CNRS, Thales, Université Paris -Saclay, 91767, Palaiseau, France.  \n2Grupo de Física de Materiales Complejos (GFMC). Dpto. de Física de Materiales. Facultad de \nCiencias Físicas, UCM, Plaza Ciencias, 1 28040, Madrid, Spain.  \nSpin-pumping across  ferromagnet/superconductor (F /S) interfaces has attracted  much \nattention  lately.  Yet the focus has been mainly on  s-wave superconductors -based  systems  \nwhereas (high -temperature) d -wave superconductors such as YBa 2Cu3O7-d (YBCO) have \nreceived scarce attention  despite their fundamental and technological interest .  Here we use \nwideband ferromagnetic resonance to study spin -pumping effects in bilayers that combine a \nsoft metallic Ni80Fe20 (Py) ferromagnet  and YBCO. We evaluate  the spin conductance in YBCO  \nby analyzing the magnetization dynamics in Py. We find that the Gilbert damping exhibits a \ndrastic drop  as the heterostructures are cooled across  the normal -superconducting transition and \nthen, depending on the S/F interface morphology , either stays co nstant or shows a strong upturn . \nThis unique  behavior is explained considering quasiparticle density of states  at the  YBCO  \nsurface , and is a direct consequence of zero-gap nodes for particular directions in the \nmomentum space. Besides showing  the fingerprint of d -wave superconductivity in spin -\npumping, our result s demonstrate the potential of high -temperature superconductors for fine \ntuning of the magnetization dynamics in ferromagnets  using k -space degrees of freedom of d-\nwave/F interface s. \n \n*Corresponding author: santiago.carreira@cnrs -thales.fr  \n  Introduction  \nSpin injection into superconductors constitutes a very active research topic within the \nnascent field of superconducting spintronics , aiming at expanding spintronic functionalities by \nexploiting the dissipationless electron transport and quantum coherence characteristic of \nsuperconductivity   [1–5]. \nTheory and experiments have shown that spin currents can flow into s -wave \nsuperconductors carried by equal -spin triplet Cooper pairs   [1,2,6 –9] or by superconducting \nquasiparticles   [10,11] , whose lifetime can exceed those of spin -polarized electrons in the \nnormal state   [12–16]. Spin-polarized quasiparticles can be  efficiently injected into the \nsuperconductor ( S) using an adjacent ferromagnet (F) by applying across the S/F interface a \nbias voltage that exceeds the superconducting gap   [10,17] . This mechanism has been \nextensively explored in transport experiments with spin valves   [13,18 –21]. Another \nmechanism for inducing a non -equilibrium spin accumulation in superconductors is spin -\npumping   [22] using  the resonant excitation  of the ferromagnet’s magnetization   [23,24]  as \nsource of pure spin current . In these ferromagnetic resonance (FMR) experiments, the \nsupercon ductor‘s efficiency as a spin -sink is evaluated via spin hall effect   [25] or microwave \nabsorption measurements   [8,25 –29], by monitoring the evolution of the resonant peak ’s \nlinewidth across the superconducting transition. The assumption is that the changes of the \nmagnetic damping (which lead to a narrow ing/broadening of the resonance linewidth   [23,24] ) \nreflect variations in the spin relaxation rate  when the superconducting gap opens, because this \nalters both the spin trans mission  across the superconductor/ferromagnet interfac e and the \nrelaxation mechanisms  within the superconductor . Pioneering experiments performed on \nNi80Fe20/Nb (Py/Nb) bilayers have found that the opening of the superconducting gap induces \nan abrupt  FMR linewidth narrowing when temperature is swept across the superconducting \ntransition   [26]. This was explained by considering that the opening of the superconducting gap leads to a vanishing density of states at the Fermi level, thereby hindering the  transmission  of \nspin polarized electrons  across the interface . More recent work on  GdN (F) / NbN ( S) \nmultilayers has found a different behavior, in which the Gilbert damping initially peaks across \nthe superconducting transition, and diminishes below the normal -state value upon further \ntemperature decrease   [30]. That behavior was associated to the presence of spin -orbit scattering \nat the interface   [31]. In contrast to the two examples mentioned above, studies carried out on \nPy/Nb multilayers with an adjacent stron g spin -orbit coupling metal (Pt) found a steady \nbroadening of the linewidth below T C, which was interpreted in terms of enhanced spin \ntransport across the superconductor due to the generation of equal -spin triplet \nsuperconductivity   [7,8] . Adding a new piece to the puzzle, a recent theory  shows that , if the  \nsuperconducting gap is suppressed near  the S/F interface, the presence of quasiparticle surface \nstates can also produce a n enhancement of spin transport in to the superconductor below \nTC  [32]. The strikingly wide variety of observed behaviors illustrate s the complexity of the \nunderlying physics, the importance of  the interfacial properties , and the fact that the conditions \nfor predominance  and interplay of  the different proposed scenarios  (quasiparticles and triplet \nsuperconductivity) is far from being fu lly understood . Beyond raising the se fundamental \nquestions, it is interesting  that the experimental investigations have evidenced that \nsuperconductivity may be exploited for tuning magnetization dynamics.  \nThe experiments discussed so far are based on conventional (low -Tc) s-wave \nsuperconductors, which present an isotropic superconducting gap.  In contrast, in \nunconventional  (high -Tc) d-wave ones the gap is suppressed along particular directions in the \nmomentum space, and there exists a  superconducting -phase shift between d -wave lobes   [33–\n35]. While spin diffusion effects in d -wave superconductors have been discussed in the c ontext \nof electrical measurements   [36–41], to our knowledge spin -pumping and the effects of the \nonset of superconducting pairing on the spin -sink behavior of d-wave cuprates remain unexplored. Notice that, at variance to s -wave superconductors, the presence of zero -gap nod es \nmay provide channels for injection of spin -polarized electrons , even in the superconducting \nstate. Consequently, the effects of superconductivity on spin -pumping and magnetization \ndynamics are expectedly  different  in the case  of s-wave superconductors. Here we \nexperimentally investigate  this issue using c-axis YBCO/Py heterostructures with different \ninterface structure.  In all cases, we observe an abrupt linewidth narrowing across the \nsuperconducting transition, similar to that observed in Py/Nb s -wave system  [26], which \nsuggests that , right below the critical temperature,  the opening of the d -wave gap  significantly \nsuppress spin -pumping. However, upon further temperature decrease , the behavior of the \nlinewidth depends  on the YBCO surface morphology. For the smoother YBCO  films , we \nobserve no further evolution of the linewidth. However,  in the presence of a faceted YBCO \nsurface s, the linewidth  monotonically widens as the temperature is decreased  below Tc. This \nbehavior  can be explained considering the  interfacial density o f quasiparticle state s, which \ndepend s on the YBCO surface morphology due to the anisotropic character of d -wave \nsuperconductivity . These results thereby provide a fingerprint of d-wave superconductivity in \nin the physics of spin-pumping . At the same time, they underline the need of a theor etical  \nframework  that specifically addresses the role of the mechanisms at pl ay (quasiparticle  density \nof states   [32,42] , changes in the spin -imbalance relaxation  [43] and dynamic generation of \ntriplet pairs   [44,45] ) in the context of d -wave superconductivity.  Finally,  this work  \ndemonstrate s the potential of high -temperature superconductors for manipulating the \nmagnetization dy namics of metallic ferromagnets, in a way that could be engineered by \nchoosing the orientation of the d -wave/F interface . \nExperimental  \nWe have studied different multilayers, namely c -axis YBa 2Cu3O7 (30 nm)/Ni 80Fe20 (15 nm)/Al \n(3 nm) grown on (001) SrTiO 3 (one sample) and on (001) NdGaO 3 (two samples)  − respectively  referred to as STO// S/F, NGO/ /S/F #1 and NGO//S/F #2  − and YBa 2Cu3O7 (30 nm)/Au (5 \nnm)/Ni 80Fe20 (15 nm)/Al (3 nm) on STO  −referred to as STO// S/Au/F. The YBCO films were \ngrown by pulsed laser deposition (PLD ) using an excimer laser ( = 305 nm) at a temperature \nof 700 °C and oxygen pressure of 0.36 mbar. Optimum oxygenation was ensured by raising the \nO2 pressure to 760 mbar during cooldown.  Where applicable, the Au interlayer (aimed at \npreventing and assessing the impact of eventual redox reactions between YBCO and Py) was \nsubsequently grown in -situ by PLD, at room temperature and in pure Ar atmosphere. Under \nthese growth condition s, the onset of the superconducting transition  determined  by resistivity \nmeasurements is typically around T c ~ 85 K, regardless of the substrate and presence of an Au \ninterlayer . \n \n   \nFIG. 1. AFM images measured on a 5x5 m2 area of a YBCO thin film grown on (a) STO (001) and (b) NGO \n(001). The height profile shown in (c) was measured along the oblique line 3 m long indicated in (a) and (b) \nrespectively . \n \nThe structural properties of the as-grown YBCO films w ere studied by high-angle  X-\nray diffraction , whic h confirmed c-axis (001) epitaxial growth on both substrates STO and \nNGO , as well as the absence of parasitic phases  (see Fig. S1 in Supplemental  Material).   \nHowever , we found that the YBCO ’s surface morphology is different  depending of the \nsubstrate.  Atomic Force Microscopy (AFM) images displayed in Fig. 1 show that YBCO on \nSTO Fig. 1 (a) present s a relatively  smooth surface  (rms roughness  ~ 2 nm) ,  while YBCO on \n(a) (b) \n(c) NGO  [Fig. 1 (b)] presents a high density of ~ 50 nm tall crystallites [see profile in Fig. 1 (c)] \non top of an otherwise similar background topography . The Py layer  and Al capping (aimed at \npreventing Py surface oxidatio n) were subsequently deposited on the YBCO ex-situ, using rf -\nsputtering in pure Ar atmosphere  at room temperature, without breaking vacuum between each \nlayer deposition.  Control s amples consisting of single Py films grown on both SrTiO 3 and \nNdGaO 3 (labeled  as STO//F, STO/ /Au/F , NGO//F  #1 and NGO//F #2 ) were studied. The \nsamples’ size is in all cases 5 5 mm2.  \n \nFIG. 2. Sketch of the multilayer structure and experimental  \ngeometry for the FMR experiments.  \n \nThe experimental geometry considered for the FMR experiments is sketched in Fig. 2. A DC \nmagnetic field H is applied parallel to the sample plane in order to saturate the magnetization \nof the Py , whose  precession  is excited  by applying and  a radiofrequency  (RF)  magnetic field  \nhRF perpendicular to the DC field, using a coplanar waveguide. A magnetic field modulation at \nlow frequency ( < 2 kHz) is used to measure the derivative of the absorbed power dP/dH with \nrespect to the DC magnetic field  H, as this is swept  around the resonance field H res where  the \ndynamical susceptibility peak s. A typical measurement is shown in Fig. 3 (a). This type of \nmeasurements were done for a number of fixed frequencies in the range 4 GHz  f  40 GHz.  \nFor each fr equency, the peak -to-peak linewidth ΔH pp and the resonance field H res were  \n \n \n \n \nFIG. 3. Typical (a) FMR absorption spectrum and fit, (b) f vs 0Hres and (c) 0Hpp vs f obtained for the sample \nSTO//S/Au/F at 30 K. The fits in (b) and (c) follows the FMR equations (1) and (2).  \n \ndetermined by fitting the dP/dH vs. the applied field H to the derivative of a Lorentzian function, \nas is shown in the example of Fig. 3 (a) . This allows extracting the values of the resonance field \nHres and linewidth ΔH pp versus the frequency, which are shown in Fig s. 3 (b) and (c)  for the \nexample in (a) .  The relationship  between the resonant microwave frequency f and field Hres is \ngiven by the Kittel formula  [46] which, neglecting the small magnetic anisotropy of Py,  is \n𝒇=𝜸𝝁𝟎√𝑯𝒓𝒆𝒔(𝑯𝒓𝒆𝒔+𝑴𝒆𝒇𝒇)    (Eq. 1)  \nwhere  is the gyromagnetic factor and 𝑀𝑒𝑓𝑓 is the effective  magnetization.  The linewidth is \nwell described by the linear expression  [24], \n𝝁𝟎∆𝑯𝒑𝒑=𝟐𝜶𝒇\n√𝟑𝜸+𝝁𝟎∆𝑯𝟎     (Eq. 2)  \nwhere 0H0 is the frequency -independent contribution o r inhomogeneous broadening and  \nis the Gilbert damping factor. Similarly as in the example shown in Fig. 3, the data for all the \nstudied samples is well described by Eq. 1 and Eq. 2. This allowed us to obtain  the te mperature \ndependent   and 0H0 for the series of samples, with error bars calculated from the linear \nregression  of the fits . Notice that, b ased on the linear behavior observed in 0Hpp vs. f for a \n(a) (b) (c) broadband frequency range  in all of the studied samples , we consider that the 2 -magnon \nscattering can be ruled out as a dominant relaxation mechanism [47] in all of them .  \n \n \nResults  \nFig. 4 (a) shows , as an example, a typical  series of the temperature -dependent FMR \nlinewidth  0ΔH pp measured f or different  frequencies , which corresponds to a NGO//F/S sample . \nThe background trend −a steady linewidth broadening  with decreasing temperature , with a drop \nbelow  ~ 20 K  for the measuremen ts at highest frequencies − is similar to that of the NGO//F \nreference samples (see Fig. S2(a) in Supp lemental  Materials ) and to the behavior  observed in  \nearlier  FMR experiments on  single  Py thin films  [47–50]. On top of that background , we \nobserve another feature, a “kink” around T ~ 85 K, which is not present in the reference samples \nand, as discussed below, is related to superconductivity. However, the fact that 0ΔH pp  results  \nfrom the addition of the (frequency independent) inhomogeneous  broadening  and the \n(frequency dependent)  magnetic damping , makes such feature evident  only for f  > 18 GHz . \nFIG. 4. ( a) Temperature dependence of the FMR linewidth, 0Hpp, measured at all frequencies from 4 \nGHz to 40  GHz in steps of 2 GHz for the sample NGO//S/F  #2. (b) 0Hpp - 0H0 vs f for the sample NGO//S/F  \n#2 obtained at temperatures just above (88 K) and below (83 K) the superconducting critical temperature of the \nYBCO. The straight lines correspond to linear fits of the data points.  (a) (b) This feature  indeed corresponds to a drop of the damping factor   across the superconducting \ntransition, as evidenced in Fig. 4 (b) where  the linewidth (after subtraction of the frequency -\nindependent broadening 0H0) is plotted as a function of the frequency. One can see that the \ndamping (slope of the straight  lines) is different above ( 88 K) and below ( 83 K) the \nsuperconducting transition of YBCO.  \nThe above example makes it evident that broadband measurement s are crucial to finely \nquantify the linewidth changes across the superconducting transition, and to univocally  ascribe \nthem to a variation of the damping factor. Thus, in what follows, we will compare samples \nbased o n the temperature dependence of the damping coefficien t (T), which can be obtained together with the temperature dependent inhomogeneous broadening 0H0(T) by applying the \nanalysis described above to a series of 0Hpp vs f measured at different temperatures.   \n \nFIG. 5. Temperature dependence of the [(a) and (b) ] magnetic damping  and [(c) and (d) ] inhomogeneous \nbroadening 0H0 for the samples STO// S/Au/F and STO//Au/F [(a) and (c)]  and NGO// S/F and NGO//F [(b) and \n(d)]. In (b) and (d) we plot the results obtained for two samples with the same nominal composition , #1 (filled \nsymbols) and #2 (open symbols). Data in circles corresponds to the samples with YBCO as a bottom layer and the \ncontrol samples without YBCO are denoted with triangles. The inset in (a) shows  vs T for the sample STO// S/F. \nThe d ash lines are g uides to the eye.  \n \nIn Fig. 5 (a) we show (T) for superconducting multilayers  STO//S /Au/F (red circles, \nmain panel) and STO//S /F (inset) , together with the data (black triangles ) for a single Py film \n(sample STO/ /Au/F) used as reference. One can see that, when Py is combined with the \nsuperconductor, and regardless of the presence of an Au interlayer , (T) drops by ~ 10-15% \nbetween 90 K and 70 K. Upon further temperature decrease (T) stays nearly constant. That \nis, 𝛼 drops across the superconducting  transition , and remains constant thereafter. This \n(a) (b) \n(c) (d) contrast s with the behavior of the STO/ /Au/F  sample used as reference (black dots), which \nshows no clear change of  around that temperature range . Notice also  that the damping  level \n ~ 4.5 10-3 in the temperature range in which  the YBCO is in the normal state (T > 90 K)  is \ncomparable  for the superconducting (STO//S/Au/F) and reference (STO//Au/F ) samples . Fig. 5 \n(c) shows that 0H0(T) behaves very similar ly in the superconducting and reference sample s. \nThis implies  that the presence of the YBCO does not create additional  magnetic  \ninhomogeneities  in Py , and unambiguously demonstrates a decrease of the Gilbert damping \nacross the  superconducting  transition . This effect  can also be observed in the NGO //S/F #1 and \n#2 bilayer s [see Fig. 5 (b)] for which (T) shows a ~ 10% drop across the superconducting \ntransition (red circles ) not observed in the reference NGO//F sample (black triangles ). As was \npointed out for  the STO substrate, the inhomogeneous broadening is not significantly affected \nby the presence of the YBCO layer, see Fig. 5 (d). However , there are two main differences \nwhen comparing samples grown on STO and on NGO.  First,  for NGO//S/F the  damping  level \n ~ 6.5 10-3 in the normal -state (T > 90 K) is significantly higher than for the reference sample \nNGO//F  [Fig. 5 (d)]. Second,  for NGO//S/F the magnetic dumping (T) does  not remain \nconstant  below the superconducting transition,  but show s instead  an upturn with decreasing \ntemperature .  \nDiscussion  \nThe central observation  is that the magnetic damping (T) of  Py in  YBCO/Py \nheterostructures drops across the YBCO superconducting transition  and that, upon further \ntemperature decrease , (T) either  stays constant or shows an upturn depending  on the substrate  \n(STO or NGO)  on which the heterostructures  are grown.  The initial drop across the transition  \nis reminiscent  of that observed  in earlier experiments with s -wave superconductors  [26], which \nwas explained based on the idea that , as the superconducting gap in the electronic density of \nstates opens  [48], the decrease of  electrons states at the Fermi level  impedes  spin injection.  Such blocking effect strengthens as temperature is lowered  further from T C, because this makes  the \nsuperconducting gap widen  and the quasiparticle population diminish  [48]. While such effect  \nis consistent with the behavior of (T) for  heterostructures grown on STO, it can not fully \naccount for the behavior of the samples grown on NGO:  these  show an upturn of the damping \nfactor , which at low temperature reaches values higher than those observed above TC [Fig. 5 \n(b)]. A similar  enhancement of spin -pumping in the superconducting phase was observed in \nS/F interfaces  [7,8]  in the presence of a heavy metal (Pt) , and was explained by the generation \nof equal -spin triplet  pairs. However,  in the present experiments we have no arguments no r \nevidence to  support such scenario. Instead, we have considered a different  situation recently \nstudied theoretically  [32], in which an enhancement of spin -pumping in the superconducting \nphase is explained  the presence of a quasiparticle states (Andreev bound states ) at the interface \nwith the F . In Ref. [32] s-wave superconductors were considered, for which the emergence of \nAndreev bound states stems from  the interfacial suppression of the superconducting gap due to \ninverse proximity effect . However, i n the case of  d-wave superconductors  quasiparticle  \n(Andreev ) surface bound states appear intrinsically , due to  the existence of zero-gap nodes \nalong particular k-space direction s [49]. As we detail below, the quasiparticle density depends \non the interface orientation.  This provides a possible  scenario to  explain  the distinct behaviors  \nof samples grown in STO and NGO  based on their different surface topography.  \nFollwing  [32], the s pin-pumping into the S depends  on the surface  density of \nquasiparticle sta tes: the larger the density of state s, the larger the spin injection  efficiency . \nExtending the full calculations  existing for  s-wave superconductros  [32] to the case of d -wave \nis out of the  present work’s scope . However , a qualitative explanation for experime ntal results  \nis at reach b y considering  the density of quasiparticle states  at d-wave/normal metal interface s \nwith finite tra nsparency . Following  [50], the normalized density of quasiparticle states is:  \n𝜌𝑆0𝜌𝑁⁄ (𝐸)= 1−(𝜎𝑁−1)2|Γ+Γ−|2\n|1+(𝜎𝑁−1)Γ+Γ−exp  (𝑖𝜙−−𝑖𝜙+)|2     (Eq. 3) where N is the normal -state electron density of states,  𝜎𝑁= 1\n1+𝑍2  with Z  the barrier strength \nat the interface,  Γ±=𝐸− √𝐸2−|Δ±|2\n|Δ±|  with E t he quasiparticle energy with respect to the Fermi \nlevel, and 𝜙+(respectively 𝜙−) is the effective phase of the anisotropic pair potential Δ+(Δ−). \nTemperature effects in the quasiparticle population can be taken into account by considering \nthe gap amplitude  Δ(𝑇)= Δ0tanh  (𝑏√𝑇𝐶\n𝑇−1) and by  convoluting  𝜌𝑆0𝜌𝑁⁄ (𝐸) with the \nderivative of the Fermi -Dirac distribution 𝑓𝐹𝐷(𝐸,𝑇)  [51], \n𝜌𝑆/𝜌𝑁 (𝐸,𝑇)= ∫𝜌𝑆0𝜌𝑁⁄ (𝐸′)𝜕𝑓𝐹𝐷\n𝜕𝐸(𝐸−𝐸′,𝑇) 𝑑𝐸′                  (Eq. 4) \nCalculations  of the normalized density of states  𝜌𝑆/𝜌𝑁 (𝐸,𝑇) for interfaces facing a d-\nwave gap  lobe (g = 0) and facing a gap node (g = /4) are shown in Fig. 6 (a) and ( c) \nrespectively , considering a moderate interface transparency Z = 2.5  (Fig. S 3 of the \nSupplemental  Materials demonstrates that, except for very transparent interfaces Z<1, the \neffects discussed thereafter are qualitatively similar for any Z) . The different  behavior s in Fig. \n6(a) and 6( c) result from  the anisotropic nature of the density of states at the YBCO  surface . \nFor a g = 0 surface, we observe  at low energies (𝐸<Δ ) that the opening of the \nsuperconducting gap leads to a fast reduction  of the density of states  upon decreasing \ntemperature, similarly as in s -wave superconductors . On the contrary , for the g = /4 case [ Fig \n6 (c)] we observe the emergence  of Andreev bound states around 𝐸=0, whose population \ngradually increases  upon  decreasing temperature , leading to a peak in the density of state s. In \nour experiments , the microwave energy  ℏ𝑓≪∆, and thus the relevant quantity is  the density \nof states near the Fermi level ( 𝐸~0) [32]. This is shown in Fig. 6 (b) for the two cases g =  \nand g = /4 .   \nFIG. 6. Calculated density of states for an interface (a) facing a d -wave gap lobe g = 0 and (b) facing at d -wave \ngap node g = /4 for different temperatures. The sketches in (a) illustrate the possible directions for the spin \ninjection according to the surface morphology . In (b) we show the temperature depende nce of the density of states \nfor quasiparticles injected along the g = 0 and g = /4 directions and in (d) we plot  the resulting density of states \nwhen 10% / 90% contributions  of the g = 0 and g = /4 are considered for the spin injection.  \n \nBased on th e above , and considering the  different topography of the STO and NGO \nsamples, a  possible interpretation for the different α(T) emerges. As sketched  in the inset of \nFig. 6 (a), in the case of STO the effects along  the out of plane direction  dominate , because of \nthe smoother S/F interface . In this situation, the density of quasiparticle surface states is  as in \nFig. 6 (a) [52] and, as was observed for s-wave supercoductors  [26], we expect that (T) decays  \nacross the superconducting transition , in agreement with our experimental findings  [Fig.  5 (a)]. \nHowever, for samples grown on NGO the presence of crystallites at the surface  allow s spin \npumping into the YBCO  basal (ab) plane [ sketch in the inset of Fig. 6 (a)], which provide s \naccess to a large r density of zero-energy quasiparticle states. If we consider that this results in  \n(a) (b) \n(c) (d) an effective density of state s in which the contribution of directions presenting a large density \nof Andreev bound states weigth s 10%, the calculated ρS/ρN (E,T)  [Fig 6 (d)] qualitatively \nreproduce  the behavior of (T) in the experiments [red in Fig. 5 (b) ]:  an abrup t drop across the \ntransition, followed by an upturn upon further tem perature decrease.  A 10% weight of \ndirections with large zero -energy quasiparticle density is reaso nable for the samples  grown  on \nNGO  considering  the lateral area of the crystallites , which  can be estimated from the AFM \nimages.  As discussed in the Sup plemental Material , the ratio between the lateral  surface area \n(normal to the  ab plane ) and the horizontal one ( normal to the c -axis) is between 1 % and 1.7 \n% depending on the criterion used for the estimate . Their contribution nee ds to be corrected d ue \nthe large electronic anisotropy of YBCO, because the conduc tivity  in the basal ( ab) plane  is up \nto 10 times larger than along the c -axis [53–55]. Thus, the  90%/10% contribution that allows \nreproduc ing the experimetna l results  seem  reasonable.  We stress nevertheles  that the discussed \nmodel aims a provinding a qualitative explanation of the observed behavior, and that th e \nnumerical estimates are made just to verify that the size of the effects are of the right order of \nmagnitude.    \nConsistenly  with the  scenario  discussed above , we observe that for the NGO //S/F  \nsamples  the normal -state damping  is signi ficantly higher than for the reference sample  (see F ig. \n5 (b) for T   90 K ), as Py contacts  the YBCO  not only on the c-axis surface but also on the \nmore conducting basal (ab) plane. This result s in a higher interfacial conductance  than for the \nfilm grown on STO, which enhances the spin absorption and therefore the overall damping.  \nA final word concerning the impact of the Au interlayer. When the Au layer is deposited \non YBCO, w e observe  no major effect on (T), which indicates that its pres ence does not \nsignificantly change the interface transparency and is consistent with the fact the spin the \ndiffusion length of Au (≈ 50 nm at 10 K)   [56] is larger than the Au layer  thickness.  In the \ncontrol (non -superconducting) samples, the presence of an Au interlayer between Py and the insulati ng substrate enhances the magnetic d amping, which reflects that Au is a more efficient \nspin-sink than the substrate.  \nIn summary, we have found that in d-wave superconductor /ferromagnet  YBCO/Py \nmultilayers , the opening of the superconducting gap  reduces  the spin-sinking  efficiency  and \nresults in a significant  drop of the magnetic damping across  the superconducting trans ition. \nHowever, upon further temperature decrease different behavior s are observed (either a plateau \nor an upturn ), which can be associated  with the YBCO’s surface morphology . In particular, the  \nlow-temperature upturn can be explained by  the large density  of quasiparticle bound s tates \ncharacteristic of d -wave superconductivity . Our hypothesis  is that those  states  are accessible \nvia YBCO crystallites at the surface , that directly expose the YBCO ab plane to the interface  \nwith t he ferromagnet .  This suggest s that spin-pumping into quasipartic le bound states could be \nfurther enhanced b y engineering the YBCO surface, for example by growing YBCO in different \ncrystallographic directions, or by creating vicinal surfaces . This, together with further \ntheoretical developments -for instance an extension of Ref .  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Moodera1,8 \n1 Plasma Science and Fusion  Center, and Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology , Cambridge,  Massachusetts 02139, USA  \n2 Física de Materiais , Escola Politécnica de Pernambuco , Universidade de Pernambuco , Recife, Pernambuco 50720 -001, Bra sil \n3 Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670 -901, Brasil  \n4 International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.  \n5 Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.  \n6 Depart amento de Física, Universidade Federal de Viçosa , Viçosa, Minas Gerais 36570 -900, Brasil  \n7Facultad de Física, Pontificia Universidad Católica de Chile, Casilla  306, Santiago, Chile  \n8Department of Physics, Massachusetts Institute of Technology , Cambridge, Massachusetts 02139, USA  \n \nElectronic mail: gilvania.vilela@upe.br  \nAbstract    \nThe control of pure spin currents carried by magnons in magnetic insulator (MI) garnet films with a robust perpendicular \nmagnetic anisotropy (PMA) is of great interest to spintronic technology as they can be used to carry, transport and process \ninformation . Garnet films with PMA p resent labyrinth domain magnetic structures that enrich the magnetization dynamics, and \ncould be employed  in more efficient wave -based logic and memory computing devices.  In MI/NM bilayers, where NM being \na normal metal providing a strong spin -orbit coupli ng, the PMA benefits the spin -orbit torque (SOT) driven magnetization's \nswitching by lowering the needed current and rendering the process faster, crucial for developing magnetic random -access \nmemories (SOT -MRAM). In this work, w e investigated the magnetic  anisotropies in thulium iron garnet  (TIG)  films with PMA  \nvia ferromagnetic resonance measurements , followed by the excitation  and detection  of magnon -mediated pure spin currents  \nin TIG/Pt  driven by microwave s and heat currents.  TIG films presented a Gilbert  damping  constant  𝛼 ≈0.01, with resonance \nfields above 3.5 kOe and half linewidth s broader than 60 Oe , at 300 K and 9.5 GHz . The spin-to-charge current conversion  \nthrough  TIG/Pt  was observed  as a micro -voltage generated at the edges of the Pt film. The obtained spin Seebeck coefficient  \nwas 0.54 𝜇𝑉/K, confirm ing also the high interfacial spin  transparenc y.  \n \nSpin-dependent phenomena in systems composed by \nlayers of magnetic insulators (MI) and non-magnetic  heavy \nmetals ( NM) with strong spin -orbit coupling have been \nextensively explored  in the insulator -based  spintronics [ 1-6]. \nAmong th e MI materials, YIG  (Y3Fe5O12) is widely  employed  \nin devices for  generation  and transmission of pure spin \ncurrents.  The main  reason is its  very low magnetic damping  \nwith Gilbert parameter on the order  of 10-5, and its large spin \ndecay length which permits spin waves to travel distances of \norders of centimeters  inside it before they vanish [7-9]. When \ncombined with  heavy metals  such as Pt, Pd, Ta, or W, many \nintriguing spin -current related phenomena emerge , such as  the \nspin pumping  effect (SPE)  [10-14], spin Seebeck  effect (SSE)  \n[7, 15 -18], spin Hall effect  (SHE) [19-21], and  spin-orbit \ntorque  (SOT)  [22-25]. The origin of these effects  relies  mainly \non the  spin diffusion length , and the quantum -mechanical \nexchange  and spin -orbit interaction s at the interface and inside \nthe heavy metal [26]. All of these effects  turn out  the MI/NM  \nbilayer  into a fascinating  playground  for exploring  spin-orbit \ndriven  phenomena at interfaces  [27-30].  \nWell investigated for many years, intrinsic YIG(111)  \nfilms on GGG(111) , (GGG = Gd 3Ga5O12) exhibit in -plane \nanisotropy. To obtain  YIG single -crystal films with \nperpendicular magnetic anisotropy  (PMA)  it is necessary to \ngrow them on top of a different substra te or partially substitute \nthe yttrium ions  by rare-earth ions , to cause strain -induced \nanisotropy  [31-33]. Even so, it is well -known that magnetic \nfilms with PMA  play an important role in spintronic \ntechnology. The PMA enhances the spin-switching efficiency , which reduces the current density for observing the spin -orbit \ntorque ( SOT) effect , and it is  useful for developing SOT based \nmagnetoresistive random access memory ( SOT-MRAM ) [34-\n36]. Besides that, PMA  increase s the information density in \nhard disk drives  and magnetoresistive random access \nmemories  [37-39], and it is crucial  for breaking the time -\nreversal symmetry in topological insulators (TIs) aiming  \ntowards quantized anomalous Hall state in MI/TI [ 40-42]. \nRecent ly, thin films of another rare-earth iron garnet, \nTIG (Tm 3Fe5O12), have caught the attention of researchers due \nto its large negative magnetostriction constant , which favors \nan out -of-plane easy axis [ 4, 43, 44 ]. TIG is a ferrimagnetic \ninsulator with  a critical temperature of 549 K, a crystal \nstructure  similar to YIG , and a Gilbert  damping parameter on \nthe order  of 𝛼~ 10−2 [4, 45 ]. Investigations of spin transport \neffects have been reported in TIG/Pt  [45, 46 ] and TIG/TI  [42, \n47], where the TIG was fabricated by pulsed laser deposition \n(PLD) technique. The results showed a strong spin mixing \nconductance at the interface of these materials  that made it \npossible to observe spin Hall magnetoresistance, spin \nSeebeck , and spin -orbit tor que effects.  \nIn this paper,  we first present a  study  of the \nmagnetocrystalline and uni axial anisotropies , as well as  the \nmagnetic damping  of sputtered epitaxial TIG thin films using \nthe ferromagnetic resonance  (FMR)  technique . For obtaining \nthe cubic and uniaxial anisotropy fields, w e analyze d the  \n 2  \n dependence of the  FMR spectr a on the film thickness  and the \norientation of the dc applied magnetic field  at room \ntemperature  and 9.5 GHz . Then, we swept the microwave \nfrequency for getting the ir magnetic damping  at different \ntemperatures . Subsequently , we focused this investigation on \nthe excitation of magnon -mediated pure spin currents in \nTIG/Pt via the spin pumping and spin Seebeck mechanisms \nfor different orientations of the dc applied magnetic field  at \nroom temperature . Pure spin currents transport spin angular  \nmomentum without carrying charge currents . They are free of \nJoule heating  and could lead to spin -wave based  devices  that \nare energetically more efficient . Employing  the inverse spin \nHall effect (ISHE)  [12], we observed th e spin-to-charge \nconversion  of these currents  inside the Pt film which was \ndetected as a developed  micro -voltage .  \nTIG films with thickness ranging from 15 to 60 nm \nwere deposited  by rf sputtering  from  a commercial target with  \nthe same nominal composition o f Tm 3Fe5O12, and a purity  of \n99.9 %.  The deposition process was performed at room \ntemperature, in pure argon working pressure of 2.8  mTorr , at \na deposition rate of 1.4 nm/min. To improve the crystallinity \nand the magnetic ordering, the films were post -growth \nannealed for 8 hrs at 800 ℃ in a quartz tube in flowing  \noxygen. After the thermal treatment, the films yield ed a \nmagnetization saturation of  100 emu/cm3, and an RMS \nroughness below 0.1 nm confirmed using a superconducting \nquantum interference device ( SQUID) and high -resolution X -\nray diffraction measurements,  as detailed in our recent article  \n[44]. Moreover, the  out-of-plane  hysteresis loops  showed \ncurved shapes which might be related with labyrinth domain structures very common in garnet films  with PMA  [48]. The \nnext step of sample preparation  consist ed of an ex -situ \ndeposition of a 4 nm -thick Pt film over the post -annealed TIG \nfilms using the dc sputtering technique . Platinum  films were  \ngrown under a n Ar gas pressure of 3. 0 mTorr , at room \ntemperature , and a deposition rate of 10 nm/min . The Pt films \nwere  not patterned.   \nFerromagnetic resonance (F MR) is a well -\nestablished technique for study  of basic magnetic  properties \nsuch as saturation magnetization, anisotropy energies and \nmagnetic relaxation mechanisms . Furthermore , FMR has been \ncentral to the investigat ion of microwave -driven spin-\npumping phenomena in FM/NM bilayers  [11, 12, 49 ]. First, \nwe used a homemade FMR spectrometer running  at a fixed \nfrequency of 9.5 GHz,  at room temperature , where t he sample s \nwere  placed in the middle of  the back wall of  a rectangular \nmicrowave cavity operating in the TE102 mode  with a Q factor \nof 2500. Field scan spectra of the derivative  of the absorption \npower  (𝑑𝑃𝑑𝐻⁄) were acquired  by modulating the  dc applied  \nfield 𝐻⃗⃗ 0 with a small sinusoidal field  ℎ⃗  at 100 kHz  and using \nlock-in amplifier detection . The resonance  field 𝐻𝑅 was \nobtained  as a function of the polar and azimuthal angles \n(θH,ϕ𝐻) of the applied magnetic field 𝐻⃗⃗ , as illustrated in Fig. \n1(d), where 𝐻⃗⃗ =𝐻⃗⃗ 0+ℎ⃗  and ℎ≪ 𝐻0. \nThe FMR spectra for TIG(t) films are shown in Figs. \n1 (a, b and c) for thickness es t = 15, 30  and 60 nm respectively. \nThe spectra were measured for 𝐻 applied along three different  \npolar angles: θ𝐻=0° (blue),  θ𝐻≅45° (green ) and θ𝐻=\n90° (red) . The complete dependenc e of 𝐻𝑅, for each sample, \nFig. 1. FMR absorption derivative spectra vs. field scan H for (a) TIG( 15 nm ), (b) TIG( 30 nm ), and (c) TIG( 60 nm ), at room T and 9.5 GHz . \nThe half linewidths ( ∆𝐻) for TIG( 15 nm ) with 𝐻 applied along 𝜃𝐻=0°,50°, and 90° are 112 Oe, 74 Oe, and 72 Oe, r espectively. For TIG(30 \nnm), ∆𝐻 is 82 Oe, 72 Oe, and 65 Oe for 𝜃𝐻=0°,50°, and 90°, respectively. For TIG(60 nm), ∆𝐻 is 72 Oe, 75 Oe, and 61 Oe for 𝜃𝐻=\n0°,45°, and 90°, respectively.  These values were extracted from the fits using the Lorentz function.  (d) Illustration of the FMR experiment \nwhere the magnetization ( 𝑀) under  an applied magnetic field (H) is driven by a microwave . (e), (f) and ( g) show the dependence of the \nresonance field 𝐻𝑅 with 𝜃𝐻 for different thickness of TIG.  The red solid lines are theoretical fits obtained for the FMR condition. \nMagnetization curves are given in reference [44].  \n \n 3  \n as a function of the polar angle ( 0°≤𝜃𝐻≤90°) are shown in \nFigs. 1(e, f and g ). For all samples, 𝐻𝑅 was minimum for 𝜃𝐻=\n0°, confirming that the perpendicular anisotropy field was \nstrong enough to overcome the demagnetization field. While \nthe films with t = 15 nm and 30 nm  exhibited the maximum \nvalue of 𝐻𝑅 for 𝜃𝐻=90° (in-plane), the sample with t = 60 \nnm showed a  maximum 𝐻𝑅at 𝜃𝐻~60°. To explain the \nbehavior of 𝐻𝑅 as a function of the out -of-plane angle  𝜃𝐻, it is \nnecessary  to normalize the FMR data  to compare with the \ntheory described as follows .  \nThe most relevant contributions to the  free magnetic \nenergy  density  𝜖 for GGG(111) / TIG(111)  films , are:  \n                                 𝝐=𝝐𝒁+𝝐𝑪𝑨+𝝐𝑫+𝝐𝑼,                                   (1) \nwhere 𝝐𝒁 is the Zeeman energy  density , 𝝐𝑪𝑨 is the cubic \nanisotropy energy  density  for (111) oriented thin films , 𝝐𝑫 is \nthe demagnetization energy  density , and 𝝐𝑼 is the uniaxial \nenergy  density . Taking into consideration the reference frame \nshown in Fig. 1(d), each energy  density  terms can be written \nas [50]: \n     𝝐𝒁=−𝑴𝑺𝑯(𝒔𝒊𝒏𝜽𝒔𝒊𝒏𝜽𝑯𝒄𝒐𝒔(𝝓−𝝓𝑯)+𝒄𝒐𝒔𝜽𝒄𝒐𝒔𝜽𝑯),        \n(2)           \n                  𝝐𝑪𝑨=𝑲𝟏𝟏𝟐⁄(𝟑−𝟔𝒄𝒐𝒔𝟐𝜽+𝟕𝒄𝒐𝒔𝟒𝜽+\n                                            𝟒 √𝟐𝒄𝒐𝒔𝜽𝒔𝒊𝒏𝟑𝝓𝒔𝒊𝒏𝟑𝜽),                       \n(3) \n𝝐𝑫+𝝐𝑼=𝟐𝝅(𝑴⃗⃗⃗ ∙𝒆̂𝟑)𝟐−𝑲𝟐⊥(𝑴⃗⃗⃗ ∙𝒆̂𝟑𝑴𝑺 ⁄)𝟐  \n                 −𝑲𝟒⊥(𝑴⃗⃗⃗ ∙𝒆̂𝟑𝑴𝑺 ⁄)𝟒,                 (4) \nwhere 𝜽 and 𝝓 are the polar and azimuthal angles of the \nmagnetization  vector   𝑴⃗⃗⃗ , 𝑴𝑺 is the saturation  magnetization ,  \n𝑲𝟏 is the first order cubic anisotropy constant,  and 𝑲𝟐⊥ and 𝑲𝟒⊥ \nare the first and second order uniaxial anisotropy constants.  \nThe uniaxial anisotropy terms come from two sources: growth \ninduced and stress induced anisotropy.  The relation between \nthe resonance field and the excitation frequency 𝝎 can be \nobtained from [51, 52 ]:         (𝝎𝜸⁄)𝟐=𝟏\n𝑴𝟐𝒔𝒊𝒏𝟐𝜽[𝝐𝜽𝜽𝝐𝝓𝝓−(𝝐𝜽𝝓)𝟐],                 (5) \n  \nwhere 𝜸 is the gyromagnetic  ratio.  The subscripts indicate \npartial derivatives with respect to the coordinates, 𝝐𝜽𝜽=\n𝝏𝟐𝝐𝝏𝜽𝟐⁄|𝜽𝟎,𝝓𝟎, 𝝐𝝓𝝓=𝝏𝟐𝝐𝝏𝝓𝟐⁄|𝜽𝟎,𝝓𝟎 and 𝝐𝜽𝝓=\n𝝏𝟐𝝐𝝏𝜽𝝏𝝓⁄|𝜽𝟎,𝝓𝟎, where 𝜽𝟎,𝝓𝟎 are the equilibrium angles of \nthe magnetization determined by the energy  density  minimum \nconditions, 𝝏𝝐𝝏𝜽⁄|𝜽𝟎,𝝓𝟎=𝟎 and 𝝏𝝐𝝏𝝓⁄|𝜽𝟎,𝝓𝟎=𝟎. The \nbest fits to the data obtained with the Eq. (5) are shown in Figs. \n1 (e, f and g) by the solid  red lines. The main physical \nparameters extracted  from the fits , including the effective \nmagnetization 4 𝝅𝑴𝒆𝒇𝒇, are summarized in Table 1 . Here \n4𝝅𝑴𝒆𝒇𝒇= 4𝝅𝑴−𝟐𝑲𝟐⊥/𝑴𝑺 , where the second term is the \nout-of-plane uniaxial anisotropy field 𝑯𝑼𝟐=𝟐𝑲𝟐⊥/𝑴, also \nnamed  as 𝑯⊥. It is important to notice that the large negative \nvalues of  𝑯𝑼𝟐 were  sufficiently strong to saturate the \nmagnetization along the direction perpendicular to the TIG \nfilm’s plane , thus  overcoming the shape anisotropy . We used \nthe saturation magnetization as the nominal value of 𝑴𝑺=\n𝟏𝟒𝟎.𝟎 𝑮. As the thickness of the TIG film increase d, the  \nmagnitude of the  perpendicular magnetic anisotropy field , \n𝑯𝑼𝟐, decrease d due to the relaxation of the induced growth \nstresses  as expected .  \n Table 1.  Physical parameters extracted from the theoretical fits of \nthe FMR  response of the  TIG thin films with thickness 𝑡, performed \nat room 𝑇 and 9.5 GHz . 4𝜋𝑀𝑒𝑓𝑓 is the effective magnetization, H 1C \nis the cubic anisotropy f ield, H U2 and HU4 are the first and second \norder uniaxial anisotropy fields, respectively. H U2 is the out -of-\nplane uniaxial anisotropy field, also named  as 𝐻⊥.    \nTIG film’s thickness  t 15 nm  30 nm  60 nm  \n4𝜋𝑀𝑒𝑓𝑓(G) -979 -799 - 383 \n𝐻1𝐶=2𝐾1𝑀𝑆⁄ (Oe) 31 26 -111 \n𝐻𝑈2=4𝜋𝑀𝑒𝑓𝑓−4𝜋𝑀𝑆 (Oe) -2,739  -2,559  -2,143  \n𝐻𝑈4=2𝐾4⊥𝑀𝑆⁄(Oe) 311 168 432 \n \n \nFig. 2. (a) Ferromagnetic resonance spectra vs. in -plane applied field 𝐻 for a 30 nm -thick TIG film at frequencies ranging from 2 GHz to 14 \nGHz and temperature of 300 K, after normalization by background subtraction. (b) Half linewidth ∆𝐻 versus frequency for T IG(30 nm) at \n300 K. The Gilbert damping parameter 𝛼 was extracted from the linear fitting of the data. (c) Damping 𝛼 versus temperature 𝑇 for TIG films \nwith 30 nm and 60 nm of thickness.  \n \n 4  \n  To obtain  the Gilbert damping  parameter  (𝜶) of the \nTIG thin films, we used the coplanar waveguide technique in \nthe variable temperature insert of a physical property \nmeasurement system (PPMS) . A vector network analyzer \nmeasure d the amplitude of the forward complex transmission \ncoefficients (𝑺𝟐𝟏) as a function of the in -plane magnetic field for different microwave frequencies  (𝒇) and temperatures  (𝑻). \nFigure 2(a) shows the FMR spectra (𝑺𝟐𝟏 versus 𝑯) for \nTIG(30 nm)  corresponding to frequencies ranging from  2 GHz \nto 14 GHz at 300K , with a microwave power of 0 dBm , after \nnormalization by background subtraction . Fitting each FMR \nspectra using the Lorentz function, we were  able to extract the \nhalf linewidth  ∆𝑯 for each  frequency , as shown in Fig. 2(b). \nThen, 𝜶 was estimated  based on  the linear  approximation  \n∆𝑯=∆𝑯𝟎+(𝟒𝝅𝜶𝜸⁄)𝒇, where  ∆𝑯𝟎 reflects the  \ncontribution  of magnetic inhomogeneities , the linear \nfrequency part is caused by the intrinsic Gilbert damping \nmechanism , and 𝜸 is the gyromagnetic ratio [40]. The same \nanalysis was performed for lower temperature  data, and it was \nextended to TIG(60 nm). Due to the weak magnetization of  \nthe thinnest TIG (15 nm ) the coplanar waveguide setup  was \nnot able to  detect its FMR signals. Figure 2(c) shows the \nGilbert damping dependence with 𝑻. At 300 K , 𝜶=𝟎.𝟎𝟏𝟓 \nfor TIG(60 nm) which is in  agree ment  with the values reported \nin the literature  [4, 45 ], and it increases  by 130 %  as 𝑻 goes \ndown to  150 K  [54]. \nNext, this work focused on the generation  of pure \nspin currents  carried by spin waves  in TIG at room 𝑻, followed \nby their propagati on through the interface between  TIG and \nPt, and their spin -to-charge conversion inside the Pt  film. \nInitially , we explored the FMR -driven spin -pumping effect  in \nTIG(60 nm)/Pt(4 nm) , where the coherent magnetization \nprecession of the TIG inject ed a pure spin current 𝑱𝒔 into the \nPt layer, which convert ed as a transverse charge current 𝑱𝒄 by \nmeans  of the inverse spin Hall effect , expressed as 𝑱 𝒄 = \n𝜽𝑺𝑯(𝛔̂  × 𝑱 𝒔), where  𝜽𝑺𝑯 is the  spin Hall angle and 𝜎̂  is the \nspin polarization  [55]. As the FMR was excited  using the \nhomemade spectrometer at 9.5 GHz , a spin pumping voltage \n(𝐕𝐒𝐏) was detected between  the two silver painted electrodes \nFig. 4. Spin Seebeck voltage (V SSE) excited by a thermal gradient in the longitudinal configuration ( 𝛻𝑇∥𝐽 𝑆) at room 𝑇, as shown in (a). (b) \nField scan of V SSE for ∆𝑇=20𝐾 and different field polar angles 𝜃𝐻. (c) Field scan of V SSE for ∆𝑇=12,𝜃𝐻=90° and different azimuthal \nangles 𝜙𝐻. Spin voltage amplitude ∆𝑉𝑆𝑆𝐸 versus (d) 𝜃𝐻, (e) 𝜙𝐻, and (f) ∆𝑇. The solid red lines are theoretical fits of the sine (d), cosine (e) \nand linear (f) dependence of ∆𝑉𝑆𝑆𝐸 with 𝜃𝐻, 𝜙𝐻 and ∆𝑇, respectively . \nFig. 3. Spin pumping voltage (V SP) excited by a FMR microwave of \n9.5 GHz , at room 𝑇, in TIG(60 nm)/Pt(4 nm). (a)  Illustration of the \nspin pumping setup . (b) In-plane field scan of V SP for different \nmicrowave powers. (c) Linear dependence of  the maximum  VSP \nwith the microwave power. ( d) 𝜃𝐻 scan of the  charge current (I SP) \ngenerated by means of the inverse spin Hall effect in the Pt film. ( e) \nIn-plane field s can of the FMR absorption derivative spectrum for 5 \nmW.  \n \n 5  \n placed on the edges of the Pt film, as illustrated in Fig. 3(a). It \nis important to not e that when the magnetization vector was \nperpendicular to the sample ’s plane no V SP was detected. The \nsample TIG(60 nm)/Pt(4 nm) ha d dimension s of 3 x 4 mm2, \nand a resistance between the silver electrodes of 𝟒𝟖  𝛀 at zero \nfield. The 𝑽𝑺𝑷 show ed a peak value of 𝟎.𝟖𝟓 𝛍𝐕 in the \nresonance magnetic field for an incident power of 185 mW , \nand an in -plane dc magnetic field ( 𝛉𝑯=𝟗𝟎°) as shown in Fig. \n3(b). The signal rever sed when the field direction went \nthrough  a 𝟏𝟖𝟎° rotation . The dependence of 𝑽𝑺𝑷 with the \nmicrowave incident power was linear , as shown in Fig. 3(c), \nwhereas the spin pumping charge current ( 𝑰𝑺𝑷=𝑽𝑺𝑷𝑹⁄) had \nthe dependence of 𝑽𝑺𝑷∝𝐬𝐢𝐧𝜽𝑯, showed  in Fig. 3(d), for a \nfixed  microwave power of 100 mW.  The ratio between the \nmicrowave -driven voltage and the microwave power was  \n4µV/W.  \nWe also  excited  pure spin current s via the spin \nSeebeck effect (SSE)  in TIG(60 nm) /Pt(4 nm)  at room 𝑻. SSE \nemerges from  the interplay between the spin and heat currents, \nand it has the potential to harvest and reduce power \nconsumption in spintronic devices [ 16, 18 ]. When a magnetic \nmaterial is subject ed to a temperature gradient, a spin current \nis thermally driven into the adjacent non -magnetic (NM) layer \nby means of the spin -exchange interaction. The spin \naccumulation in the NM layer can be detected by measuring a \ntransversal charge current due to the I SHE. To observe the \nSSE in our samples , the uncovered GGG surface was placed \nover a copper plate, acting as a thermal bath at room 𝑻, while \nthe sample’s  top was in thermal contact with a 𝟐×\n𝟐 𝐦𝐦𝟐 commercial Peltier module through  a thermal paste, \nas illustrate d in Fig. 4(a). The  Peltier module was responsible \nfor creating a controllable temperature gradient across the \nsample . On the other hand, the temperature difference ( ∆𝑻) \nbetween the bottom and top of the sample was measured by a \ndifferential thermocouple. The ISHE voltage  due to the SSE \n(𝑽𝑺𝑺𝑬) was detected between the two silver painted electrodes \nplaced on the e dges of the Pt film . \nThe behaviour  of 𝑽𝑺𝑺𝑬 by sweeping the dc applied \nmagnetic field ( 𝑯), while  ∆𝑻, 𝜽𝑯 and 𝝓𝑯 were  kept fixed  was \ninvestigated . Fixing 𝛟𝑯=𝟎° and varying the magnetic field \nfrom out -of-plane (𝜽𝑯=𝟎°) to in -plane along x -direction \n(𝜽𝑯=𝟗𝟎°), 𝑽𝑺𝑺𝑬 went  from zero to its maximum value of \n5.5 𝝁𝑽 for ∆𝑻=𝟐𝟎 𝑲, as shown in F ig. 4(b). Around zero \nfield, no matter the value of 𝜽𝑯, the TIG’s film magnetization \ntended to rely along its out -of-plane easy axis which zeroes \n𝑽𝑺𝑺𝑬. For in -plane fields ( 𝜽𝑯=𝟗𝟎°) with ∆𝑻=𝟏𝟐 𝑲, 𝑽𝑺𝑺𝑬 \nwas maximum when 𝛟𝑯=𝟎°, and it was zero for 𝛟𝑯=𝟗𝟎°. \nThe reason 𝑽𝑺𝑺𝑬 went to zero for 𝛟𝑯=𝟗𝟎°, may be  \nattributed to  the generated charge flow along the x -direction \nwhile  the silver electrodes were  placed along y -direction , thus \nnot enabling the current detection  (see F ig. 4(c)). The analysis \nof the spin Seebeck amplitude ∆𝑽𝑺𝑺𝑬 versus 𝜽𝑯, 𝛟𝑯 and ∆𝑻 \nshow ed a sine, cosine and linear dependence, respectively as \ncan be seen  in Fig. 4(d)-(e), where the red solid lines are \ntheoretical fits . The Spin Seebeck  coefficient  (SSC)  extracted \nfrom the linear fit of ∆𝑽𝑺𝑺𝑬 vs. ∆𝑻 was 0.54 𝝁𝑽/K. \nIn conclusion , we used the FMR technique to probe \nthe magnetic anisotropies  and the Gilbert damping  parameter  of the  sputtered TIG thin films with perpendicular magnetic \nanisotropy. The results showed higher resonance fields (> 3.5 \nkOe) and broader linewidth s (> 60 Oe) when comparing with \nYIG films  at room 𝑇. Thinner TIG films (t = 15 nm and 30 \nnm) presented  a well -defined PMA; on the other hand, the \neasy axis of thicker  TIG film (60 nm)  showed a de viation of \n30 degrees from normal to the film  plane . By numerically \nadjusting the FMR field dependence with the polar angle , we \nextracted  the effective  magnetization , the cubic  (H1C) and the \nout-of-plane uniaxial anisotropy  (HU2=H⊥) fields for the \nthree TIG films . The thinnest film presented the highest \nintensity for H⊥ as expected, even so H⊥ was strong enough to \novercome the shape anisotropy and g ave place to a \nperpendicular magnetic anisotropy  in all the three thickness of \nTIG films . The Gilbert damping  parameter (𝛼) for TIG(30 \nnm) and TIG(60 nm) films were estimated to be ≈  10-2, by \nanalyzing a set of FMR spectra using the coplanar waveguide \ntechnique at various  microwave frequencies and temperatures. \nAs 𝑇 went down to 150 K the damping increased \nmonotonically  130 % . \nFurthermore , spin waves (magnons) were excited in \nTIG(60 nm)/Pt(4 nm)  heterostructure through the spin \npumping and spin Seebeck effects , at room 𝑇 and 9.5 GHz . \nThe generated pure spin currents carried by the magnons were \nconverted into charge current s once they reached the  Pt film  \nby means of  the inverse spin Hall effect . The charge  currents  \nwere detected as a micro -voltage  measured at the edges of the \nPt film , and they showed sine and cosine dependence  with the \npolar and azimuthal angles , respectively,  of the dc applied \nmagnetic field . This voltage was linearly dependent on the \nmicrowave pow er for the  SPE, and on the temperature \ngradient for the  SSE. These results confirmed  a good spin -\nmixing conductance in the interface TIG/Pt , and an efficient \nconversion of pure spin currents into charge currents inside the \nPt film , which is crucial for the employment of TIG films with \na robust PMA in the development of magnon -based spintronic \ndevices for computing technologies.  \n \nACKNOWLEDGEME NTS  \nThis research is supported in the USA by Army Research \nOffice (ARO W911NF -19-2-0041  and W911NF -19-2-0015 ), \nNSF (DMR 1700137), ONR (N00014 -16-1-2657), in Brazil \nby CAPES (Gilvania  Vilela/POS -DOC -88881.120327/2016 -\n01), FACEPE (APQ -0565 -1.05/14 and APQ -0707 -1.05/14), \nCNPq , UPE (PFA/PROGRAD/UPE 04/2017)  and FAPEMIG \n- Rede de Pesquisa em Materiais 2D and Rede de \nNanomagnetismo , in Chile by Fondo Nacional de Desarrollo \nCientí fico y Tec nológico (FONDECYT) No. 1170723, and  in \nChina by  the National Natural Science Foundation of China  \n(11974025).  \nDATA AVAILABILITY  \nThe data that support the findings of this study are available \nfrom the corresponding author upon reasonable request.  \nREFERENCES   \n 6  \n 1 P. 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Bose National Centre \nfor Basic Sciences, Block JD, Sector-III, Salt Lake, Kolkata 700 106, India \n \n*E-mail: abarman@bose.res.in \n \n \nKeywords: (Thin Film Heterostructures, Interface Properties, Spin Pumping, Spin \nTransparency, Spin-Mixing Conductance, Gilbert Damping, Time-resolved Magneto-optical \nKerr Effect) \n \n \nAbstract \nPure spin current has transfigured the energy-efficient spintronic devices and it has the salient \ncharacteristic of transport of the spin angular momentum. Spin pumping is a potent method to \ngenerate pure spin current and for its increased efficiency high effective spin-mixing \nconductance ( Geff) and interfacial spin transparency ( T) are essential. Here, a giant T is reported \nin Sub/W( t)/Co20Fe60B20(d)/SiO2(2 nm) heterostructures in beta-tungsten (β-W) phase by \nemploying all-optical time-resolved magneto-optical Kerr effect technique. From the variation \nof Gilbert damping with W and CoFeB thicknesses, the spin diffusion length of W and spin-\nmixing conductances are extracted. Subsequently, T is derived as 0.81 ± 0.03 for the β-\nW/CoFeB interface. A sharp variation of Geff and T with W thickness is observed in consonance \nwith the thickness-dependent structural phase transition and resistivity of W. The spin memory \nloss and two-magnon scattering effects are found to have negligible contributions to damping \nmodulation as opposed to spin pumping effect which is reconfirmed from the invariance of \ndamping with Cu spacer layer thickness inserted between W and CoFeB. The observation of \ngiant interfacial spin transparency and its strong dependence on crystal structures of W will be \nimportant for pure spin current based spin-orbitronic devices. \n    \n2 \n 1. Introduction \nThe rapid emergence of spintronics has promised a new paradigm of electronics based on the \nspin degree of freedom either associated with the charge or by itself.[1-3]  This has potential \nadvantages of non-volatility, reduced electrical power consumption, increased data processing \nspeed, and increased integration densities as opposed to its semiconductor counterpart.[4] A \nmajor objective of modern spintronics is to harness pure spin current, which comprises of flow \nof spins without any net flow of charge current.[5, 6] This has the inherent benefit of reduced \nJoule heating and Oersted fields together with the ability to manipulate magnetization . Three \nmajor aspects of spin current are its generation, transport, and functionalization. Pure spin \ncurrent can be generated by spin-Hall effect,[7,8] Rashba-Edelstein effect,[9,10] spin pumping,[11-\n13] electrical injection in a lateral spin valve using a non-local geometry,[14,15] and spin \ncaloritronic effects.[16,17] Among these, spin pumping is an efficient and extensively used \nmethod of spin injection from ferromagnet (FM) into normal metal (NM) where the precessing \nspins from FM transfer spin angular momentum to the conduction electrons of adjacent NM \nlayer in NM/FM heterostructure, which gets dissipated by spin-flip scattering. The efficiency \nof spin pumping is characterized by spin-mixing conductance and spin diffusion length. The \ndissipation of spin current into the NM layer results in loss of spin angular momentum in the \nFM layer leading to an increase in its effective Gilbert damping parameter ( αeff). Thus, spin \npumping controls the magnetization dynamics in NM/FM heterostructures, which is crucial for \ndetermining the switching efficiency of spin-torque based spintronic devices. The enhancement \nin αeff is more prominent in heavy metals (HM) with high spin-orbit coupling (SOC) due to \nstronger interaction between electron spin and lattice. Intense research in the field of spin-\norbitronics has revealed that interface dependent spin transport is highly influenced by the spin \ntransparency, which essentially determines the extent of spin current diffused through the \nNM/FM interface.[18,19]     \n3 \n The highly resistive β-W, which shows a distorted tetragonal phase commonly referred to as \nA15 structure, is well known for exhibiting large spin Hall angle (SHA) (up to ~0.50) [20]  as \ncompared to other transition metal elements such as Pt (0.08) [21]  and β-Ta (0.12).[7] Besides, \nin W/FM heterostructures, W leads to highly stable perpendicular magnetic anisotropy[22]  and \ninterfacial Dzyaloshinskii-Moriya interaction.[23] Another important characteristic associated \nwith W is that it shows a thickness-dependent phase transition in the sub-10 nm thickness \nregime.[24,25] In general, sputter-deposited W films with thickness well below 10 nm are found \nto have β phase with high resistivity, whereas the films with thickness above 10 nm possess \npredominantly α phase (bcc structure) with low resistivity. A small to moderate SHA has been \nreported for the α and mixed (α + β) phase (<0.2) of W.[24] As SHA and effective spin-mixing \nconductance ( Geff) are correlated, one would expect that interfacial spin transparency ( T), which \nis also a function of Geff, should depend on the structural phase of W thin films. Furthermore, \nthe magnitude of the spin-orbit torque (SOT) depends on the efficiency of spin current \ntransmission (i.e. T) across the NM/FM interface. It is worth mentioning that due to high SOC \nstrength, W is a good spin-sink material and also cost-effective in comparison with the widely \nused NM like Pt. On the other hand, CoFeB due to its notable properties like high spin \npolarization, large tunnel magnetoresistance, and low intrinsic Gilbert damping, is used as FM \nelectrode in magnetic tunnel junctions. The presence of Boron at the NM/CoFeB interface \nmakes this system intriguing as some recent studies suggest that a small amount of boron helps \nin achieving a sharp interface and increases the spin polarization, although an excess of it causes \ncontamination of the interface. To this end, determination of T of the technologically important \nW/CoFeB interface and its dependence on the W-crystal phase are extremely important but still \nabsent in the literature.  \nBesides spin pumping, there are different mechanisms like spin memory loss (SML),[26] Rashba \neffect,[10] two-magnon scattering (TMS),[27] and interfacial band hybridization[28] which may \nalso cause loss of spin angular momentum at NM/FM interface, resulting in increase of αeff and   \n4 \n decrease of the spin transmission probability. However, for improved energy efficiency, the \nNM/FM interface in such engineered heterostructures must possess high spin transmission \nprobability. Consequently, it is imperative to get a deeper insight into all the mechanisms \ninvolved in generation and transfer of spin current for optimizing its efficiency. Here, we \ninvestigate the effects of spin pumping on the Gilbert damping in W/CoFeB bilayer system as \na function of W-layer thickness using recently developed all-optical technique, which is free \nfrom delicate micro-fabrication and electrical excitation and detection.[29] This is a local and \nnon-invasive method based on time-resolved magneto-optical Kerr effect (TR-MOKE) \nmagnetometry. Here, the damping is directly extracted from the decaying amplitude of time-\nresolved magnetization precession, which is free from experimental artifacts stemming from \nmultimodal oscillation, sample inhomogeneity, and defects. From the modulation of damping \nwith W layer thickness, we have extracted the intrinsic spin-mixing conductance ( G↑↓) of the \nW/CoFeB interface which excludes the backflow of spin angular momentum and spin diffusion \nlength(𝜆௦ௗ) of W. Furthermore, we have modeled the spin transport using both the ballistic \ntransport model[30, 31] and the model based on spin diffusion theory[32,33]. Subsequently, Geff, \nwhich includes the backflow of spin angular momentum, is estimated from the dependence of \ndamping on the CoFeB layer thicknesses. By using both the spin Hall magnetoresistance \nmodel[34] and spin transfer torque based model utilizing the drift-diffusion approximation[35], \nwe have calculated the T of W/CoFeB interface. The spin Hall magnetoresistance model gives \nlower value of T than the drift-diffusion model, but the former is considered more reliable as \nthe latter ignores the spin backflow. We found a giant value of T exceeding 0.8 in the β phase \nof W, which exhibits a sharp decrease to about 0.6 in the mixed (α+β) phase using spin Hall \nmagnetoresistance model. We have further investigated the other possible interface effects in \nour W/CoFeB system, by incorporating a thin Cu spacer layer of varying thickness between the \nW and CoFeB layers. Negligible modulation of damping with Cu thickness confirms the   \n5 \n dominance of spin pumping generated pure spin current and its transport in the modulation of \ndamping in our system.   \n \n2. Results and Discussion \nFigure 1 (a) shows the grazing incidence x-ray diffraction (GIXRD) patterns of \nSub/W(t)/Co20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures at the glancing angle of 2o. In these \nplots, the peaks corresponding to α and β phase of W are marked. The high-intensity GIXRD \npeak at ∼44.5° and low intensity peak at ∼64° correspond primarily to the β phase (A15 \nstructure) of W (211) and W(222) orientation, respectively. Interestingly, we find these peaks \nto be present for all thicknesses of W, but when t > 5 nm, then an additional peak at ∼40.1° \ncorresponding to α-W with (110) crystal orientation appears. Consequently, we understand that \nfor t ≤ 5 nm, W is primarily in β-phase, while for t > 5 nm a fraction of the α phase appears, \nwhich we refer to as the mixed (α+β) phase of W. These findings are consistent with some \nexisting literature.[24,25]  Some other studies claimed that this transition thickness can be tuned \nby carefully tuning the deposition conditions of the W thin films.[36] The average lattice \nconstants obtained from the β-W peak at 44.5o and α-W peak at 40.1o correspond to about 4.93 \nand 3.15 Å, respectively. By using the Debye-Scherrer formula, we find the average crystallite \nsize in β and α phase of W to be about 14 and 7 nm, respectively. \nIt is well known that the formation of β-W films is characterized by large resistivity due to its \nA-15 structure which is associated with strong electron-phonon scattering, while the α-W \nexhibits comparatively lower resistivity due to weak electron-phonon scattering. We measured \nthe variation of resistivity of W with its thickness across the two different phases, using the \nfour-probe method. The inverse of sheet resistance ( Rs) of the film stack as a function of W \nthickness is plotted in Figure 1(b). A change of the slope is observed beyond 5 nm, which \nindicates a change in the W resistivity. The data have been fitted using the parallel resistors \nmodel[24] (shown in Figure S1  of the Supporting Information). [37] We estimate the average   \n6 \n resistivity of W ( ρW) in β and mixed (α+β) phase to be about 287 ± 19 and 112 ± 14 µΩ.cm, \nrespectively, while the resistivity of CoFeB (ρCoFeB) is found to be 139 ± 16 μΩ.cm. Thus, the \nresistivity results corroborate well with those of the XRD measurement. \nThe AFM image of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) (t = 1, 5 and 10 nm) samples in \nFigure 1(c) revealed the surface topography. We have used WSxM software to process the \nimages.[38] The variation in the average surface roughness of the films with W thickness is listed \nin Table 1. The roughness varies very little when measured at various regions of space of the \nsame sample. The surface roughness in all samples is found to be small irrespective of the \ncrystal phase of W. Due to the small thicknesses of various layers in the heterostructures, the \ninterfacial roughness is expected to show its imprint on the measured topographical roughness. \nWe thus understand that the interfacial roughness in these heterostructures is very small and \nsimilar in all studied samples. Details of AFM characterization is shown in Figure S2  of the \nSupporting Information.[37] \n2.1. Principles behind the modulation of Gilbert damping with layer thickness: \nIn an NM/FM bilayer magnetic damping can have various additional contributions, namely \ntwo-magnon scattering, eddy current, and spin pumping in addition to intrinsic Gilbert damping. \nAmong these, the spin pumping effect is a non-local effect, in which an external excitation \ninduces magnetization precession in the FM layer. The magnetization precession causes a spin \naccumulation at the NM/FM interface. These accumulated spins carry angular momentum to \nthe adjacent NM layer, which acts as a spin sink by absorbing the spin current by spin-flip \nscattering, leading to an enhancement of the Gilbert damping parameter of FM. In 2002, \nTserkovnyak and Brataas theoretically demonstrated the spin pumping induced enhancement \nin Gilbert damping in NM/FM heterostructures using time-dependent adiabatic scattering \ntheory where magnetization dynamics in the presence of spin pumping can be described by a \nmodified Landau-Lifshitz-Gilbert (LLG) equation as: [11-13]   \n7 \n ௗ𝒎\nௗ௧= −𝛾(𝒎×𝑯eff)+𝛼0(𝒎×ௗ𝒎\nௗ௧)+ఊ\nVMೞ𝑰௦                          (1) \nwhere γ is the gyromagnetic ratio,  Is is the total spin current, Heff is the effective magnetic field, \nα0 is intrinsic Gilbert damping constant, V is the volume of ferromagnet and Ms is saturation \nmagnetization of the ferromagnet. As shown in equation (2), Is generally consists of a direct \ncurrent contribution 𝑰𝒔𝟎 which is nonexistent in our case as we do not apply any charge current, \n𝑰𝒔𝒑𝒖𝒎𝒑, i.e. spin current due to pumped spins from the FM to NM and 𝑰𝒔𝒃𝒂𝒄𝒌, i.e. a spin current \nbackflow to the FM reflecting from the NM/substrate interface which is assumed to be a perfect \nreflector. \n𝑰𝒔=𝑰𝒔𝟎+𝑰𝒔pump+𝑰𝒔back                                                    (2) \nHere, 𝑰𝒔𝒃𝒂𝒄𝒌 is determined by the spin diffusion length of the NM layer. Its contribution to \nGilbert damping for most metals with a low impurity concentration is parametrized by a \nbackflow factor β which can be expressed as:[39] \n𝛽=൭2𝜋𝐺↑↓ටఌ\nଷtanhቀ௧\nఒೞቁ൱ିଵ\n                                      (3) \nwhere ε is the material-dependent spin-flip probability, which is the ratio of the spin-conserved \nto spin-flip scattering time. It can be expressed as: [40]                                                                \n 𝜀= (𝜆𝜆௦ௗ⁄)ଶ3⁄                                                                                     (4) \nwhere λel and λsd are the electronic mean free path and spin diffusion length of NM, respectively.  \nThe spin transport through NM/FM interface directly depends on the spin-mixing conductance, \nwhich is of two types: (a) G↑↓, which ignores the contribution of backflow of spin angular \nmomentum, and (b) Geff, which includes the backflow contribution. Spin-mixing conductance  \ndescribes the conductance property of spin channels at the interface between NM and FM. Also, \nspin transport across the interface affects the damping parameter giving rise to αeff of the system   \n8 \n that can be modeled by both ballistic and diffusive transport theory. In the ballistic transport \nmodel, the αeff is fitted with the following simple exponential function:[30,31,39] \n𝐺eff=𝐺↑↓൬1−𝑒ିమ\nഊೞ൰=ସగdM\nఓಳ(𝛼eff−𝛼)                            (5) \n𝛥𝛼=𝛼eff−𝛼=ఓಳீ↑↓൭ଵିషమ\nഊೞ൱\nସdMeff                                         (6) \nHere, the exponential term signifies backflow spin current contribution and a factor of 2 in the \nexponent signifies the distance traversed by the spins inside the NM layer due to reflection from \nthe NM/substrate interface.  \nIn the ballistic approach, the resistivity of NM is not considered while the NM thickness is \nassumed to be less than the mean free path. To include the effect of the charge properties of \nNM on spin transport, the model based on spin diffusion theory is used to describe αeff (t). \nWithin this model, the additional damping due to spin pumping is described as:[32,33,36]  \n            𝐺eff=ீ↑↓\nቆଵାమഐഊೞಸ↑↓\nୡ୭୲୦ቀ௧ఒೞൗቁቇ=ସగdM\nఓಳ(𝛼eff−𝛼)                                         (7)                                                                                       \n                                                                                        \n           ∆𝛼=𝛼eff−𝛼=ఓಳீ↑↓\nସగdMቆଵା మഐഊೞಸ↑↓\nୡ୭୲୦ቀ௧ఒೞൗቁቇ (8)                                                                                            \n     \nwhere ρ is the electrical resistivity of the W layer. Here the term  మఘఒೞீ↑↓\ncothቀ𝑡𝜆௦ௗൗቁ account \nfor the back-flow of pumped spin current into the ferromagnetic layer. \nThe reduction of spin transmission probability implies a lack of electronic band matching, \nintermixing, and disorder at the interface. The spin transparency, T of an NM/FM interface \ntakes into account all such effects that lead to the electrons being reflected from the interface \ninstead of being transmitted during transport. Further, T depends on both intrinsic and extrinsic \ninterfacial factors, such as band-structure mismatch, Fermi velocity, interface imperfections, \netc.[19,39] According to the spin Hall magnetoresistance model, the spin current density that   \n9 \n diffuses into the NM layer is smaller than the actual spin current density generated via the spin \npumping in the FM layer. This model linked T with 𝐺eff  by the following relation:[34,39] \n𝑇=ீeff  tanh൬\nమഊೞ൰\nீeff  coth൬\nഊೞ൰ା\nమഊೞమഐ                              (9) \nThe interfacial spin transparency was also calculated by Pai et al. in the light of damping-like \nand field-like torques utilizing the drift-diffusion approximation. Here, the effects of spin \nbackflow are neglected as it causes a reduction in the spin torque efficiencies. Assuming t ≫ λ \nand a very high value of d, T can be expressed as:[35] \n𝑇=ଶீ↑↓ீಿಾ⁄\nଵାଶீ↑↓ீಿಾ⁄                                                                                                               (10) \nwhere, 𝐺ேெ=\nఘఒೞమ  is the spin conductance of the NM layer. \nIn an NM/FM heterostructure, other than spin pumping, there is a finite probability to have \nsome losses of spin angular momentum due to interfacial depolarization and surface \ninhomogeneities, known as SML and TMS, respectively. In SML, loss of spin angular \nmomentum occurs when the atomic lattice at the interface acts as a spin sink due to the magnetic \nproximity effect or due to the interfacial spin-orbit scattering which could transfer spin \npolarization to the atomic lattice.[26] The TMS arises when a uniform FMR mode is destroyed \nand a degenerate magnon of different wave vector is created.[27] The momentum non-\nconservation is accounted for by considering a pseudo-momentum derived from internal field \ninhomogeneities or secondary scattering. SML and TMS may contribute to the enhancement of \nthe Gilbert damping parameter considerably. Recently TMS is found to be the dominant \ncontribution to damping for Pt-FM heterostructures.[41] In the presence of TMS and SML \neffective Gilbert damping can be approximated as:[41] \n    αeff = α0 + αSP + αSML + αTMS    \n      ∆𝛼=𝛼eff−𝛼= 𝑔𝜇ீeff  ା ீೄಾಽ\nସdM+𝛽்ெௌ𝑑ିଶ                                                      (11) \n   \n10 \n where 𝐺ௌெ is the “effective SML conductance”, and βTMS is a “coefficient of TMS” that \ndepends on both interfacial perpendicular magnetic anisotropy field and the density of magnetic \ndefects at the FM surfaces.  \n2.2. All-optical measurement of magnetization dynamics: \nA schematic of the spin pumping mechanism along with the experimental geometry is shown \nin Figure 2(a). A typical time-resolved Kerr rotation data for the Sub/Co 20Fe60B20(3 nm)/SiO 2(2 \nnm) sample at a bias magnetic field, H = 2.30 kOe is shown in Figure 2(b) which consists of \nthree different temporal regimes. The first regime is called ultrafast demagnetization, where a \nsharp drop in the Kerr rotation (magnetization) of the sample is observed immediately after \nfemtosecond laser excitation. The second regime corresponds to the fast remagnetization where \nmagnetization recovers to equilibrium by spin-lattice interaction. The last regime consists of \nslower relaxation due to heat diffusion from the lattice to the surrounding (substrate) superposed \nwith damped magnetization precession. The red line in Figure 2(b) denotes the bi-exponential \nbackground present in the precessional data. We are mainly interested here in the extraction of \ndecay time from the damped sinusoidal oscillation about an effective magnetic field and its \nmodulation with the thickness of FM and NM layers. We fit the time-resolved precessional data \nusing a damped sinusoidal function given by: \n𝑀(𝑡)=𝑀(0)𝑒ିቀ\nഓቁsin(2π𝑓𝑡+𝜑)                               (12) \nwhere τ is the decay time, φ is the initial phase of oscillation and f  is the precessional frequency. \nThe bias field dependence of precessional frequency can be fitted using the Kittel formula given \nbelow to find the effective saturation magnetization ( Meff): \n𝑓=ఊ\nଶ(𝐻(𝐻+4π𝑀eff))ଵ/ଶ                                                              (13)                       \nwhere γ = gµB/ħ, g is the Landé g-factor and ћ is the reduced Planck’s constant. From the fit, \nMeff and g are determined as fitting parameters. For these film stacks, we obtained effective   \n11 \n magnetization, Meff ≈ 1200 ± 100 emu/cc, and g = 2.0 ± 0.1. The comparison between Meff \nobtained from the magnetization dynamics measurement and Ms from VSM measurement for \nvarious thickness series are presented systematically in Figures S3-S5  of the Supporting \nInformation.[37] For almost all the film stacks investigated in this work, Meff  is found to be close \nto Ms, which indicates that the interface anisotropy is small in these heterostructures. We \nestimate αeff using the expression: [42]  \n𝛼eff=1\nγτ(𝐻+2π𝑀eff)                                           (14) \nwhere τ is the decay time obtained from the fit of the precessional oscillation with equation (12). \nWe have plotted the variation of time-resolved precessional oscillation with the bias magnetic \nfield and the corresponding fast Fourier transform (FFT) power spectra in Figure S6  of the \nSupporting Information.[37] The extracted values of αeff are found to be independent of the \nprecession frequency f. Recent studies show that in presence of extrinsic damping contributions \nlike TMS, αeff should increase with f, while in presence of inhomogeneous anisotropy in the \nsystem αeff should decrease with f.[43] Thus, frequency-independent αeff rules out any such \nextrinsic contributions to damping in our system. \n2.3. Modulation of the Gilbert damping parameter: \nIn Figure 3 (a) we have presented time-resolved precessional dynamics for \nSub/W(t)/Co20Fe60B20(3 nm)/SiO 2(2 nm) samples with 0 ≤ t ≤ 15 nm at H = 2.30 kOe. The \nvalue of α0 for the 3-nm-thick CoFeB layer without the W underlayer is found to be 0.006 ± \n0.0005. The presence of W underlayer causes αeff to vary non monotonically over the whole \nthickness regime as shown by the αeff vs. t plot in Figure 3(b). In the lower thickness regime, \ni.e.  0 ≤ t ≤ 3 nm, Δ α increases sharply by about 90% due to spin pumping but it saturates for t \n≥ 3 nm. However, for t > 5 nm, Δ α drops by about 30% which is most likely related to due to \nthe thickness-dependent phase transition of W.  At first, we have fitted our result for t ≤ 5 nm \nwith equation (6) of the ballistic transport model and determined G↑↓ = (1.46 ± 0.01) × 1015 cm-  \n12 \n 2 and λsd = 1.71 ± 0.10 nm as fitting parameters. Next, we have also fitted our results with \nequation (8) based on spin diffusion theory, where we have obtained G↑↓ = (2.19 ± 0.02) × 1015 \ncm-2 and λsd = 1.78 ± 0.10 nm. The value of G↑↓ using spin diffusion theory is about 28% higher \nthan that of ballistic model while the value of λsd is nearly same in both models. Using values \nfor λel (about 0.45 nm for W) from the literature[44]  and λsd derived from our experimental data, \nwe have determined the spin-flip probability parameter, ε = 2.30 × 10−2 from equation (4). To \nbe considered as an efficient spin sink, a nonmagnetic metal must have ε  ≥ 1.0 × 10-2 and hence \nwe can infer that the W layer acts as an efficient spin sink here.[13] The backflow factor β can \nbe extracted from equation (3). We have quantified the modulation of the backflow factor (Δ β) \nto be about 68% within the experimental thickness regime. \nTo determine the value of 𝐺eff  directly from the experiment, we have measured the time-\nresolved precessional dynamics for Sub/W (4 nm)/Co 20Fe60B20 (d)/SiO2 (2 nm) samples with 1 \nnm ≤ d ≤ 10 nm at H = 2.30 kOe as shown in Figure 4(a). The αeff  is found to increase with the \ninverse of FM layer thickness ( Figure 4(b)). We have fitted our results first with equation (5), \nfrom which we have obtained 𝐺eff  and 𝛼 to be (1.44 ± 0.01) × 1015 cm-2 and 0.006 ± 0.0005, \nrespectively.  \nBy modelling the W thickness dependent modulation of damping of Figure 3(b) using equation \n(5), we have obtained 𝐺eff  of W/CoFeB in β-phase (where ∆𝛼 ≈  0.006) and α+β-mixed phase \n(where ∆𝛼  ≈ 0.004) of W to be (1.44 ± 0.01) × 1015 cm-2 and (1.07 ± 0.01) × 1015 cm-2, \nrespectively. From these, we conclude that β-phase of W has higher conductance of spin \nchannels in comparison to the α+β-mixed phase. The variation of 𝐺eff  with W layer thickness \nis presented in Figure 5(a), which shows that 𝐺eff  increases non monotonically and nearly \nsaturates for t ≥ 3 nm. For t > 5 nm, 𝐺eff  shows a sharp decrease in consonance with the variation \nof αeff.  \nWe have further fitted the variation of αeff  with the inverse of FM layer thickness ( Figure 4(b)) \nusing with equation (11) to isolate the contributions from SML, TMS and spin pumping (SP).   \n13 \n The values of  𝐺ௌெ , and βTMS are found to be (2.45 ± 0.05) × 1013 cm-2 and (1.09 ± 0.02) × 10-\n18 cm2, respectively. 𝐺ௌெ is negligible in comparison with 𝐺eff  which confirms the absence of \nSML contribution in damping. Contribution of TMS to damping modulation ( 𝛽்ெௌ𝑑ଶ) is also \nbelow 2% for all the FM thicknesses. The relative contributions are plotted in Figure 5(b). It is \nclear that spin pumping contribution is highly dominant over the SML and TMS for our studied \nsamples. The value of our 𝐺eff  in β-W/CoFeB is found to be much higher than that obtained for \nβ-Ta/CoFeB[39] measured by all-optical TRMOKE technique as well as various other NM/FM \nheterostructures measured by conventional techniques as listed in Table 2. This provides \nanother confirmation of W being a good spin sink material giving rise to strong spin pumping \neffect. \nWe subsequently investigate the value of T for W/CoFeB interface, which is associated with \nthe spin-mixing conductances of interface, spin diffusion length, and resistivity of NM as \ndenoted in equations (9) and (10). T is an electronic property of a material that depends upon \nelectronic band matching of the two materials on either side of the interface. After determining \nthe resistivity, spin diffusion length and spin-mixing conductances experimentally, we have \ndetermined the value of T which depends strongly on the structural phase of W. Using equation \n(9) based on the spin-Hall magnetoresistance model, Tβ-W and T(α+β)-W are found to be 0.81 ± \n0.03 and 0.60 ± 0.02, respectively. On the other hand, equation (10) of spin transfer torque \nbased model utilizing the drift-diffusion approximation gives Tβ-W and T(α+β)-W to be 0.85 ± 0.03 \nand 0.63 ± 0.02, respectively, which are slightly higher than the values obtained from spin-Hall \nmagnetoresistance model. However, we consider the values of T obtained from the spin-Hall \nmagnetoresistance model to be more accurate as it includes the mandatory contribution of spin \ncurrent backflow from W layer into the CoFeB layer. Nevertheless, our study clearly \ndemonstrates that the value of spin transparency of the W/CoFeB interface is the highest \nreported among the NM/FM heterostructures as listed in Table 2. This high value of T, \ncombined with the high spin Hall angle of β-W makes it an extremely useful material for pure   \n14 \n spin current based spintronic and spin-orbitronic devices. The structural phase dependence of \nT for W also provides a particularly important guideline for choosing the correct thickness and \nphase of W for application in the above devices.  \nFinally, to directly examine the additional possible interfacial effects present in the W/CoFeB \nsystem, we have introduced a copper spacer layer of a few different thicknesses between the W \nand CoFeB layers. Copper has very small SOC and spin-flip scattering parameters and it shows \na very high spin diffusion length. Thus, a thin copper spacer layer should not affect the damping \nof the FM layer due to the spin pumping effect but can influence the other possible interface \neffects. Thus, if other interface effects are substantial in our samples, the introduction of the \ncopper spacer layer would cause a notable modulation of damping with the increase of copper \nspacer layer thickness ( c).[19,39] The time-resolved Kerr rotation data for the Sub/W(4 \nnm)/Cu(c)/Co20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures with 0 ≤ c ≤ 1 nm are presented in \nFigure 6(a) at H = 2.30 kOe and Figure 6(b) shows the plot of αeff as a function of c. The \ninvariance of αeff with c confirms that the interface of Cu/CoFeB is transparent for spin transport \nand possible additional interfacial contribution to damping is negligible, which is in agreement \nwith our modelling as shown in Figure 5(b).  \n \n3. Conclusion \nIn summary, we have systematically investigated the effects of thickness-dependent structural \nphase transition of W in W( t)/CoFeB( d) thin film heterostructures and spin pumping induced \nmodulation of Gilbert damping by using an all-optical time-resolved magneto-optical Kerr \neffect magnetometer. The W film has exhibited structural phase transition from a pure β phase \nto a mixed (α + β) phase for t > 5 nm. Subsequently, β-W phase leads to larger modulation in \neffective damping ( αeff) than (α+β)-W. The spin diffusion length of W is found to be 1.71 ± \n0.10 nm, while the spin pumping induced effective spin-mixing conductance  𝐺eff  is found to be  \n(1.44 ± 0.01) × 1015 cm-2 and (1.07 ± 0.01) × 1015 cm-2 for β and mixed (α+β) phase of W,   \n15 \n respectively. This large difference in 𝐺eff  is attributed to different interface qualities leading \ntowards different interfacial spin-orbit coupling. Furthermore, by analyzing the variation of αeff \nwith CoFeB thickness in W (4 nm)/CoFeB  (d)/SiO2 (2 nm), we have isolated the contributions \nof spin memory loss and two-magnon scattering from spin pumping, which divulges that spin \npumping is the dominant contributor to damping. By modeling our results with the spin Hall \nmagnetoresistance model, we have extracted the interfacial spin transparency ( T) of β-\nW/CoFeB and (α + β)-W/CoFeB as 0.81 ± 0.03 and 0.60 ± 0.02, respectively. This structural \nphase-dependent T value will offer important guidelines for the selection of material phase for \nspintronic applications. Within the framework of ballistic and diffusive spin transport models, \nthe intrinsic spin-mixing conductance ( G↑↓) and spin-diffusion length ( λsd) of β-W are also \ncalculated by studying the enhancement of αeff as a function of β-W thickness. Irrespective of \nthe used model, the value of T for W/CoFeB interface is found to be highest among the NM/FM \ninterfaces, including the popularly used Pt/FM heterostructures. The other possible interface \neffects on the modulation of Gilbert damping are found to be negligible as compared to the spin \npumping effect. Thus, our study helps in developing a deep understanding of the role of W thin \nfilms in NM/FM heterostructures and the ensuing spin-orbit effects. The low intrinsic Gilbert \ndamping parameter, high effective spin-mixing conductance combined with very high interface \nspin transparency and spin Hall angle can make the W/CoFeB system a key material for spin-\norbit torque-based magnetization switching, spin logic and spin-wave devices.  \n \n4. Experimental Section/Methods \n4.1. Sample Preparation \nThin films of Sub/W( t)/Co20Fe60B20(d)/SiO2(2 nm) were deposited by using RF/DC magnetron \nsputtering system on Si (100) wafers coated with 285 nm-thick SiO 2. We varied the W layer \nthickness as t = 0, 0.5, 1, 1.5, 2, 3, 4, 5, 8, 10 and 15 nm and CoFeB layer thickness as d = 1, 2, \n3, 5 and 10 nm. The depositions were performed at an average base pressure of 1.8  × 10-7 Torr   \n16 \n and argon pressure of about 0.5 mTorr at a deposition rate of 0.2 Å/s. Very slow deposition \nrates were chosen for achieving a uniform thickness of the films even at a very thin regime \ndown to sub-nm. The W and CoFeB layers were deposited using average DC voltages of 320 \nand 370 V, respectively, while SiO 2 was deposited using average RF power of 55 watts. All \nother deposition conditions were carefully optimized and kept almost identical for all samples. \nIn another set of samples, we introduced a thin Cu spacer layer in between the CoFeB and W \nlayers and varied its thickness from 0 nm to 1 nm. The Cu layer was deposited at a DC voltage \nof 350 V, argon pressure of 0.5 mTorr and deposition rate of 0.2 Ǻ/s. \n4.2. Characterization \nAtomic force microscopy (AFM) was used to investigate the surface topography and vibrating \nsample magnetometry (VSM) was used to characterize the static magnetic properties of these \nheterostructures. Using a standard four-probe technique the resistivity of the W films was \ndetermined and grazing incidence x-ray diffraction (GIXRD) was used for investigating the \nstructural phase of W. To study the magnetization dynamics, we used a custom-built TR-\nMOKE magnetometer based on a two-color, collinear optical pump-probe technique. Here, the \nsecond harmonic laser pulse (λ = 400 nm, repetition rate = 1 kHz, pulse width >40 fs) of an \namplified femtosecond laser, obtained using a regenerative amplifier system (Libra, Coherent) \nwas used to excite the magnetization dynamics, while the fundamental laser pulse (λ = 800 nm, \nrepetition rate = 1 kHz, pulse width ~40 fs) was used to probe the time-varying polar Kerr \nrotation from the samples. The pump laser beam was slightly defocused to a spot size of about \n300 µm and was obliquely (approximately 30° to the normal on the sample plane) incident on \nthe sample. The probe beam having a spot size of about 100 µm was normally incident on the \nsample, maintaining an excellent spatial overlap with the pump spot to avoid any spurious \ncontribution to the Gilbert damping due to the dissipation of energy of uniform precessional \nmode flowing out of the probed area. A large enough magnetic field was first applied at an \nangle of about 25° to the sample plane to saturate its magnetization. This was followed by a   \n17 \n reduction of the magnetic field to the bias field value ( H = in-plane component of the bias field) \nto ensure that the magnetization remained saturated along the bias field direction. The tilt of \nmagnetization from the sample plane ensured a finite demagnetizing field along the direction \nof the pump pulse, which was modified by the pump pulse to induce a precessional \nmagnetization dynamics in the sample. The pump beam was chopped at 373 Hz frequency and \nthe dynamic Kerr signal in the probe pulse was detected using a lock-in amplifier in a phase-\nsensitive manner. The pump and probe fluences were kept constant at 10 mJ/cm2 and 2 mJ/cm2, \nrespectively, during the measurement. All the experiments were performed under ambient \nconditions at room temperature. \n \nAcknowledgements \n \nAB gratefully acknowledges the financial assistance from the S. N. Bose National Centre for \nBasic Sciences (SNBNCBS), India under Project No. SNB/AB/18-19/211.  SNP, SM and SC \nacknowledge SNBNCBS for senior research fellowship. ArB acknowledges SNBNCBS for \npostdoctoral research associateship. 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Barwal, N. Behera, S. Akansel, P. Svedlindh, S. Chaudhary, \nPhys. Rev. B , 2018, 97, 064420. \n[52] G. Wu, Y. Ren, X. He, Y. Zhang, H. Xue, Z. Ji, Q. Y. Jin, Z. Zhang, Phys. Rev. Appl ., \n2020,  13, 024027. \n[53] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, F. Y. Yang, Phys. Rev. Lett ., \n2014, 112, 197201. \n \n \n    \n22 \n  \n  \n \nFigure 1.  (a) X-ray diffraction patterns measured at 2° grazing angle incidence for different W \nthickness. (b) Variation of inverse sheet resistance with W thickness. (c) AFM images of the \nsamples showing the surface topography.  \n \n  \n23 \n   \nFigure 2. (a) Schematic of experimental geometry and (b) typical TR-MOKE data from \nCo20Fe60B20(3 nm)/SiO 2(2 nm) heterostructure at an applied bias magnetic field of 2.30 kOe. \nThe three important temporal regimes are indicated in the graph. The solid red line shows a \nbiexponential fit to the decaying background of the time-resolved Kerr rotation data. \n  \n24 \n   \nFigure 3. (a) Background subtracted time-resolved Kerr rotation data showing precessional \noscillation for Sub/W( t)/ Co20Fe60B20(3 nm)/SiO 2(2 nm) as function of W thickness at an \napplied bias magnetic field of 2.30 kOe. (b) Experimental result of variation damping with t \n(symbol) fitted with theoretical models (solid and dashed lines) of spin pumping. Two different \nregions corresponding to W crystal phase, namely β and α+β are shown.  \n \n \n \n \n  \n25 \n  \n  \nFigure 4. (a) Background subtracted time-resolved Kerr rotation data showing precessional \noscillation for Sub/W (4 nm)/Co 20Fe60B20 (d)/SiO2 (2 nm) as function of Co 20Fe60B20 thickness \nd at an applied bias magnetic field of 2.30 kOe. (b) Experimental result of variation of damping \nvs 1/d (symbol) fitted with theoretical models (solid and dashed lines).  \n \n  \n26 \n   \nFigure 5.  (a) Variation of effective spin-mixing conductance( 𝐺eff ) with W layer thickness t \n(symbol). The solid line is guide to the eye. (b) Contributions of SP, SML and TMS to the \nmodulation of damping for different Co 20Fe60B20 layer thickness d (symbol). The solid line is \nguide to the eye. 0 2 4 6 8 10039095100 \n  SP\n TMS\n SML\n  Damping (%) \n                d  (nm) 0 2 4 8 12 160.00.51.01.5 \n \n  Geff (1015 cm-2)\nt (nm)(a)\n(b)  \n27 \n   \nFigure 6. (a) Background subtracted time-resolved Kerr rotation data showing precessional \noscillation for Sub/W(4 nm)/Cu( c)/Co20Fe60B20(3 nm)/SiO 2(2 nm) as function of Cu layer \nthickness c at an applied bias magnetic field of 2.30 kOe. (b) Experimental result of variation \nof damping vs c. The dotted line is guide to the eye, showing very little dependence of damping \non Cu layer thickness. \n  \n28 \n Table 1. The average surface roughness values of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) \nsamples obtained using AFM. \n \nTable 2. Comparison of the effective spin-mixing conductance and interfacial spin \ntransparency of the W/CoFeB samples studied here with the important NM/FM interfaces taken \nfrom the literature. \nMaterial  \nInterface Effective Spin-Mixing \nConductance  (×1015 cm-2) Interfacial Spin \nTransparency  \nPt/Py 1.52 [19] 0.25 [19] \nPt/Co 3.96 [19] 0.65 [19] \nPd/CoFe 1.07 [31] N.A. \nPt/FM 0.6-1.2 [35] 0.34-0.67 [35] \nβ-Ta/CoFeB  0.69 [39] 0.50 [39] \nβ-Ta/ CFA 2.90 [40] 0.68 [40] \nPd0.25Pt0.75/Co 9.11 [41] N.A. \nAu0.25Pt0.75/Co 10.73 [41] N.A. \nPd/Co 4.03 [41] N.A. \nPd0.25Pt0.75/FeCoB 3.35 [41] N.A. \nAu0.25Pt0.75/ FeCoB 3.64 [41] N.A. \nGr/Py 5.26 [45] N.A. \nRu/Py 0.24 [46] N.A. \nPt/YIG 0.3-1.2 [47] N.A. \nMoS2/CFA 1.49 [48] 0.46 [48] \nPd/Fe 0.49-1.17 [49] 0.04-0.33 [49] \nPd/Py 1.40 [50] N.A. \nMo/CFA 1.56 [51] N.A. \nMoS2/CoFeB 16.11 [52] N.A. \nTa/YIG 0.54 [53] N.A. \nW/YIG 0.45 [53] N.A. \nCu/YIG 0.16 [53] N.A. \nAg/YIG 0.05 [53] N.A. \nAu/YIG 0.27 [53] N.A. \nβ-W/CoFeB  1.44 (This work)  0.81 (This work)  \nMixed(α+β)-W/CoFeB  1.07 (This work)  0.60 (This work)  \n  \n((N.A. =  Not available)) \n \n t (nm) 0 0.5 1.0 1.5 2 3 5 8 10 15 \nRoughness \n(nm) 0.23 0.21 0.32 0.28 0.25 0.21 0.19 0.29 0.28 0.22 \n   \n29 \n Supporting Information \n \n \nStructural Phase Dependent Giant Interfacial Spin Transparency in W/CoFeB Thin \nFilm Heterostructure \n \nSurya Narayan Panda, Sudip Majumder, Arpan Bhattacharyya, Soma Dutta, Samiran \nChoudhury and Anjan Barman* \n \nDepartment of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre \nfor Basic Sciences, Block JD, Sector-III, Salt Lake, Kolkata 700 106, India \n \nE-mail: abarman@bose.res.in \n \n \nThis file includes: \n1. Determination of resistivity of W and Co 20Fe60B20 layers. \n2. Measurement of surface roughness of the sample using AFM. \n3. Determination of saturation magnetization of the samples from static and dynamic \nmeasurements.  \n4. Variation of effective damping with precessional frequency. \n \n \n \n1. Determination of resistivity of W and CoFeB layers :  \n \nThe variation of sheet resistance ( Rs) of the W( t)/Co20Fe60B20(3 nm) film stack with W layer \nthickness, t is shown in Figure S1 . The data is fitted with a parallel resistor model (Ref. 24 of \nthe article) by the formula given in the inset of the figure. This yields the resistivity of W in its \nβ and (α+β) phase as: 287 ± 19 µΩ.cm and 112 ± 14 µΩ.cm, respectively. On the other hand, \nthe resistivity of Co 20Fe60B20 is found to be 139 ± 16 µΩ.cm. \n   \n30 \n   \n \nFigure S1.  Variation of sheet resistance ( Rs) of the W ( t)/ Co20Fe60B20(3 nm) film stack vs. W \nthickness t used for the determination of resistivity of the W and Co 20Fe60B20 layers. \n \n2. Measurement of surface roughness of the sample using AFM:  \nWe have measured the surface topography of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) thin \nfilms by atomic force microscopy (AFM) in dynamic tapping mode by taking scan over 10 μm \n× 10 μm area. We have analyzed the AFM images using WSxM software. Figures S2 (a) and \nS2(d) show two-dimensional planar AFM images for t = 1 nm and 10 nm, respectively. Figures \nS2(b) and S2(e) show the corresponding three-dimensional AFM images for t = 1 nm and 10 \nnm, respectively. The dotted black lines on both images show the position of the line scans to \nobtain the height variation. Figures S2 (c) and S2(f) show the surface roughness profile along \nthat dotted lines, from which the average roughness ( Ra) is measured as 0.32 ± 0.10  nm and \n0.28 ± 0.12 nm for t = 1 nm and 10 nm, respectively. Topographical roughness is small and \nconstant within the error bar in all samples irrespective of the crystal phase of W. Furthermore, \nsurface roughness varies very little when measured at different regions of same sample. The \ninterfacial roughness is expected to show its imprint on the measured topographical roughness 0 3 10 150200400\n  Rs(Ω)\nt(nm)(ρW)β= 287 µΩ.cm\n(ρW)α+β= 112 µΩ.cm\n= 139 µΩ.cm  \n31 \n due to the small thickness of our thin films. Small and constant surface roughness in these \nheterostructures proves the high quality of the thin films.  \n \n  \nFigure S2.  (a) The two-dimensional AFM image, (b) the three-dimensional AFM image, and \n(c) the line scan profile along the black dotted line for W(1 nm)/ Co 20Fe60B20(3 nm) /SiO 2(2 \nnm) sample. (d) The two-dimensional AFM image, (e) the three-dimensional AFM image, and \n(f) the line scan profile along the black dotted line for W(10 nm)/ Co 20Fe60B20(3 nm) /SiO 2(2 \nnm) sample. \n \n3. Determination of saturation magnetization of the samples from static and dynamic \nmagnetic measurements :  \nWe have measured the in-plane saturation magnetization ( Ms) of all the W( t)/ \nCo20Fe60B20(d)/SiO2(2 nm) samples using vibrating sample magnetometry (VSM). Typical \nmagnetic hysteresis loops (magnetization vs. magnetic field) for W( t)/ Co20Fe60B20(3 \nnm)/SiO 2(2 nm), W(4 nm)/ Co 20Fe60B20(d)/SiO2(2 nm) and W(4 nm)/Cu( c)/ Co20Fe60B20(3 \n  \n32 \n nm)/SiO 2(2 nm) series are plotted in Figures S3 (a), S4(a) and S5(a), respectively. Here, Ms is \ncalculated from the measured magnetic moment divided by the total volume of the Co 20Fe60B20 \nlayer. These films have very small coercive field (~5 Oe).  The effective magnetization Meff of \nthe samples are obtained by fitting the bias magnetic field ( H) dependent precessional frequency \n(f) obtained from the TR-MOKE measurements, with the Kittel formula (equation (13) of the \narticle) (see Figures S3 (b), S4(b) and S5(b)). We have finally plotted the variation of Meff and \nMs with W, Co 20Fe60B20, and Cu thickness in Figures S3 (c), S4(c), and S5(c), respectively. The \nMeff and Ms values are found to be in close proximity with each other, indicating that the \ninterfacial anisotropy is small for all these samples. Since these films were not annealed post-\ndeposition, the interfacial anisotropy stays small and plays only a minor role in modifying the \nmagnetization dynamics for these heterostructures.  \n \n \nFigure S3.     (a) VSM loops for W( t)/ Co20Fe60B20(3 nm)/SiO 2(2 nm). (b) Kittel fit (solid line) \nto experimental data (symbol) of precessional frequency vs. magnetic field for W( t)/ \nCo20Fe60B20(3 nm)/SiO 2( 2 nm) samples. (c) Comparison of variation of Ms from VSM and  Meff \nfrom TR-MOKE as a function of W layer thickness. \n \n \n 0 4 8 12 16500100015002000\n  \nt (nm)Ms (emu/cc)\n500100015002000 Meff (emu/cc)-100001000\n-100001000\n-0.4 0.0 0.4-100001000  \n  \nt =1 nm\nt = 8 nm\n  \n  \nH (kOe)t = 15 nm\n  \n  M (emu/cc)141618\n141618\n1.5 2.0 2.5141618  \n  \nt = 1 nm\nt = 8 nm\nt = 15 nm  \n  f (GHz)\nH (kOe)  \n  (a) (b)\n(c)  \n33 \n  \n \n \nFigure S4.     (a) VSM loops for W(4 nm)/ Co 20Fe60B20(d)/SiO2(2 nm). (b) Kittel fit (solid line) \nto experimental data (symbol) of precessional frequency vs. magnetic field for W(4 nm)/ \nCo20Fe60B20(d)/SiO2(2 nm) samples. (c) Comparison between variation of Ms from VSM and  \nMeff from TR-MOKE as a function of Co 20Fe60B20 layer thickness. \n \n \nFigure S5.     (a) VSM loops for W(4 nm)/Cu( c)/ Co20Fe60B20(3 nm)/SiO 2(2 nm). (b) Kittel fit \n(solid line) to experimental data (symbol) of precessional frequency vs. magnetic field for W(4 \nnm)/ Cu(c)/ Co20Fe60B20(3 nm)/SiO 2(2 nm) samples. (c) Comparison between variation of Ms \nfrom VSM and  Meff from TR-MOKE as a function of Cu layer thickness.  \n \n 0 3 6 9500100015002000\n  \nd (nm)Ms (emu/cc)\n500100015002000 Meff (emu/cc)141618\n141618\n1.5 2.0 2.514161820  \n  \n  \n  \n  \n  f  (GHz)\nH (kOe)d = 10 nmd = 5 nmd = 2 nm\n-100001000\n-100001000\n-0.3 0.0 0.3-100001000  \n  \nd = 5 nmd = 2 nm\nd = 10 nmM (emu/cc)\n  \n  \nH (kOe)  \n  (a) (b)\n(c)\n0.00 0.25 0.50 0.75 1.00500100015002000\n  \nc (nm)Ms (emu/cc)\n500100015002000 Meff (emu/cc)141618\n141618\n1.5 2.0 2.5141618  \n  \n  \n  \n  \n  f (GHz)\nH (kOe)c = 1 nmc = 0.5 nmc = 0 nm\n-100001000\n-100001000\n-0.4 0.0 0.4-100001000\nH (kOe)M (emu/cc)  \n  \nc = 0.5 nmc = 0 nm \nc = 1 nm  \n  \n  \n  (a) (b)\n(c)  \n34 \n 4. Variation of effective damping with precessional frequency:   \nFor all the sample series the time-resolved precessional oscillations have been recorded at \ndifferent bias magnetic field strength. The precessional frequency has been extracted by taking \nthe fast Fourier transform (FFT) of the background-subtracted time-resolved Kerr rotation. \nSubsequently, the time-resolved precessional oscillations have also been fitted with a damped \nsinusoidal function given by equation (12) of the article to extract the decay time τ. The value \nof effective Gilbert damping parameter ( αeff) have then been extracted using equation (14). \nVariation of this αeff with precessional frequency ( f) is plotted to examine the nature of the \ndamping. Here, we have plotted the time-resolved precessional oscillations ( Figure S6( a)), FFT \npower spectra ( Figure S6( b)) and αeff vs. f (Figure S6( c)) for Sub/W(0.5 nm)/Co 20Fe60B20(3 \nnm)/SiO 2(2 nm) sample.  It is clear from this data that damping is frequency independent, which \nrules out the contribution of various extrinsic factors such as two-magnon scattering, \ninhomogeneous anisotropy, eddy current in the damping for our samples.  \n \n \nFigure S6.     (a) Background subtracted time-resolved precessional oscillations at different bias \nmagnetic fields for Sub/W(0.5 nm)/Co 20Fe60B20(3 nm)/SiO 2(2 nm) sample, where symbols \nrepresent the experimental data points and solid lines represent fits using equation (12) of the \narticle. (b) The FFT power spectra of the time-resolved precessional oscillations showing the 0.0 0.3 0.6 0.9 1.2 1.5  \n  \n  \n  \n  \n  \nH = 1.50 kOeH = 1.80 kOeH = 2.10 kOe\n  \n  Kerr Rotation (arb. units)\nTime (ns)H = 2.30 kOe\n0 10 20 30  \n  \n  \n  \n  \n  \n  \n  Power (arb. units)\nf (GHz)12 14 16 180.0000.0080.016\n  \n \nf (GHz)eff\n(c)\n(b) (a)  \n35 \n precessional frequency. (c) Variation of effective damping with precessional frequency is \nshown by symbol and the dotted line is guide to the eye. "    },    {        "title": "2010.00144v1.Quantum_hydrodynamics_of_spin_winding.pdf",        "content": "Quantum hydrodynamics of spin winding\nYaroslav Tserkovnyak,1Ji Zou,1Se Kwon Kim,2and So Takei3\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea\n3Department of Physics, Queens College of the City University of New York, Queens, New York 11367, USA\nAn easy-plane spin winding in a quantum spin chain can be treated as a transport quantity,\nwhich propagates along the chain but has a \fnite lifetime due to phase slips. In a hydrodynamic\nformulation for the winding dynamics, the quantum continuity equation acquires a source term due\nto the transverse vorticity \row. The latter re\rects the phase slips and generally compromises the\nglobal conservation law. A linear-response formalism for the nonlocal winding transport then reduces\nto a Kubo response for the winding \row along the spin chain, in conjunction with the parasitic\nvorticity \row transverse to it. One-dimensional topological hydrodynamics can be recovered when\nthe vorticity \row is asymptotically small. Starting with a microscopic spin-chain formulation, we\nfocus on the asymptotic behavior of the winding transport based on the renormalized sine-Gordon\nequation, incorporating phase slips as well as Gilbert damping. A generic electrical device is proposed\nto manifest this physics. We thus suggest winding conductivity as a tangible concept that can\ncharacterize low-energy dynamics in a broad class of quantum magnets.\nI. INTRODUCTION\nIn addition to e\u000ecient heat transport carried by spin\ndynamics along electrically-insulating spin chains,1there\nhas also been much interest in their transmission of spin\nsignals.2In the case of spin currents polarized along a di-\nrection of axial symmetry, the spin signals can propagate\nballistically or di\u000busively, while generally also undergoing\ndecay due to spin-nonconserving perturbations. Alterna-\ntively, transport based on collective order-parameter dy-\nnamics and rooted in topological conservation laws has\nbeen suggested for potentially more robust propagation\nof signals.3\nThe winding dynamics of planar spins in an easy-plane\n(anti)ferromagnet is one ready example of this. Extend-\ning the natural super\ruid analogy for the SO(2) order\nparameter to the nonequilibrium setting, scenarios for\nspin super\ruidity have been proposed4and experimen-\ntally pursued.5The spontaneously broken U(1) symme-\ntry is replaced here by the axial symmetry (say along the\nzaxis) of the easy-plane spin winding (in the xyplane).\nIf the latter experiences some anisotropies within the xy\nplane, however, the associated SO(2) symmetry gets bro-\nken directly, invalidating spin conservation and possibly\npinning the conjugate phase (i.e., the winding angle) al-\ntogether.\nWhile the spin density \u001azis then no longer acting as\na long-wavelength transport quantity, the winding den-\nsity\u001a/@x'(xbeing the spatial coordinate along the\ntransport channel and 'the azimuthal angle of the order\nparameter in the xyplane) obeys a continuity equation\n(with the associated \rux j/\u0000@t'), irrespective of the\nanisotropies. This is crucially contingent on the abil-\nity to unambiguously de\fne '(x;t) along the channel,\nat all times, which is compromised whenever a vecto-\nrial order parameter traverses one of the poles along the\nhard (z) axis. Such processes could be visualized as vor-\ntices in the (1 + 1)-dimensional space-time, realizing a\nvorticity \row transverse to the xaxis. In analogy to sim-ilar parasitic events in low-dimensional super\ruids and\nsuperconductors,6these can be called phase slips.7,8\nIn this paper, we set out to formulate a rigorous mi-\ncroscopic formalism to address these issues, in regard to\nquantum winding hydrodynamics, at an arbitrary tem-\nperature. Once the formal framework is in place, our fo-\ncus is going to be on the role of anisotropies, phase slips,\nand general magnetic damping, in relation to spatiotem-\nporal transport properties of the spin-winding \rows. In\nparticular, we wish to establish regimes, where the no-\ntion of a winding conductivity can be meaningful both\ntheoretically and experimentally.\nOur discussion is structured as follows. We start, in\nSec. II, by recapping vorticity dynamics in two spatial\ndimensions. The notion of a topological conservation law\nis introduced for a classical theory in Sec. II A, which is\nthen discretized and quantized into an exact quantum\nformulation on a generic spin lattice, in Sec. II B. A simi-\nlar procedure is then attempted for the winding dynamics\nin Sec. III, where the quantum \row of spin winding along\na spin chain gets supplemented with vorticity \row trans-\nverse to it. Here, we develop a quantum Kubo formalism\nfor the winding transport, and establish boundary con-\nditions that could allow us to read it out electrically. In\nSec. IV, a sine-Gordon model is treated systematically,\nin order to study the interplay of the winding \row, phase\nslips, and other sources of dissipation associated with col-\nlective dynamics, both at zero and \fnite temperatures.\nA summary and outlook are o\u000bered in Sec. V.\nII. 2D VORTICITY (HYDRO)DYNAMICS\nA. Classical vorticity dynamics\nA three-component real vector \feld m= (mx;my;mz)\nresiding in 2+1 dimensions, m(r;t), realizes an R2!R3\nmapping, at any given time t. These spatial \feld textures\nare devoid of point defects, as the fundamental homo-arXiv:2010.00144v1  [cond-mat.mes-hall]  30 Sep 20202\ntopy group of the order-parameter space mis trivial:\n\u00191(R3) = 1. Such two-dimensional textures are, fur-\nthermore, all topologically equivalent, having \fxed the\nboundary pro\fle of mon a connected patch of R2, which\nis re\rected in the fact that \u00192(R3) = 1. Despite this,\na smooth vector \feld de\fnes a topological hydrodynam-\nicsgoverned by the continuity equation @\u0016j\u0016= 0 (with\nthe Einstein summation implied over the Greek letters:\n\u0016= 0;1;2!t;x;y ), where9\nj\u0016\u0011\u000f\u0016\u0017\u0018z\u0001@\u0017m\u0002@\u0018m\n2\u0019: (1)\nHere, zis the z-axis unit vector and \u000f\u0016\u0017\u0018is the Levi-\nCivita symbol.\nFor the special case of a rigid texture sliding at a veloc-\nityv, for example: j=\u001av, where\u001a\u0011j0andj= (jx;jy).\nFor another special case of a sharp vortex in a strongly\neasy-plane magnet with the planar order parameter nor-\nmalized to unity, jmj! 1:\u001a\u0019\u000e(r\u0000r0), where r0is\nthe position at which mtilts out of the plane (over an\nappropriate healing length de\fning the size of the core).\nThese examples intuitively suggest a \ruid whose density\nis given by the distribution of vorticity in the system.\nWhile in the extreme easy-plane case, a vortex core car-\nries a quantized topological charge, we do not generally\nassume this special limit.\nThe above conserved quantity j0can be recast as a\n\fctitious \rux\n\u001a=z\u0001r\u0002A\n2\u0019(2)\nassociated with the gauge \feld\nA=mxrmy\u0000myrmx: (3)\nApplying Green's theorem, we then see that the con-\nserved topological charge within a patch \n,\nQ\u0011Z\n\nd2r\u001a=I\n@\ndr\u0001A\n2\u0019=I\n@\nd\u001e\n2\u0019m2\nk; (4)\nis associated with the order-parameter winding around\nits boundary @\n.mkis the \feld's projection onto the\nxyplane (within the order-parameter space) and \u001eis the\nassociated azimuthal angle. This reveals the geometri-\ncal meaning of the conservation law: The charge Qin\nthe bulk can change only in response to a vorticity \row\nthrough the boundary.\nB. Quantum vorticity dynamics\nTo construct a simple quantum theory, which repro-\nduces the above classical hydrodynamics of vorticity in\nthe classical limit of ~!0, let us consider a square lat-\ntice model sketched in Fig. 1. We label each vertex of the\nlattice by two integer indices: \u0010 (along the xaxis) and|\n(along theyaxis). The same indices are used to label thesquare plaquettes, according to their lower left corner, as\nwell as the vertical links going upward and the horizon-\ntal links to the right of the site \u0010 |. Each site contains a\nquantum spin S= (Sx;Sy;Sz), of magnitude S(in units\nof~), characterized by the standard angular-momentum\nalgebra [Sa;Sb] =i\u000fabcSc.\n⇢ı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>x(ı)\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>y(|)\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>jxı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>jyı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>jyı˜|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>jx˜ı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>Sı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>S˜ı˜|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>S˜ı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>Sı˜|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>\nFIG. 1. The quantum spin lattice described by an arbitrary\nHamiltonian H.S\u0010|is the spin operator at site \u0010 |, with index\n\u0010 (|) running along the x(y) axis. ~ \u0010 = \u0010 + 1 and ~ |=|+ 1.\u001a\u0010|\nis the conserved topological charge per plaquette \u0010 |,jx\n\u0010|(jy\n\u0010|) is\nthe \rux per vertical (horizontal) link \u0010 |, which together satisfy\nthe quantum continuity equation (10).\nWe associate a charge density\n\u001a\u0010|\u0011Ax\n\u0010|\u0000Ax\n\u0010~|+Ay\n~ \u0010|\u0000Ay\n\u0010|\n2\u0019a(5)\nto each plaquette, where ais the lattice spacing. Here,\n~ \u0010\u0011\u0010 + 1 and ~|\u0011|+ 1, and\nAx\n\u0010|=z\u0001(S~ \u0010|+S\u0010|)\u0002(S~ \u0010|\u0000S\u0010|)\n4aS2+ H:c:=z\u0001S\u0010|\u0002S~ \u0010|\naS2;\nAy\n\u0010|=z\u0001(S\u0010~|+S\u0010|)\u0002(S\u0010~|\u0000S\u0010|)\n4aS2+ H:c:=z\u0001S\u0010|\u0002S\u0010~|\naS2;\n(6)\nwhich we assign formally to the corresponding horizontal\nand vertical sides of the plaquette, respectively. These\nde\fnitions mimic Eqs. (2) and (3), respectively, and\nshould reproduce them by coarse graining the magnetic\ntextures in the classical limit of S!1 .\nAccording to these de\fnitions,\n\u001a{|=z\u0001c{|\n2\u0019a2;where c{|\u00111\nS2X\nlSl\u0002S~l(7)\nis the vector chirality of the corresponding plaquette,\nwith the sum running over the four vertices labelled by\nl(~lbeing the vertex next to l, in the counterclockwise\ndirection).10We also see [according to Eq. (5)] that\nQ=X\n\u0010|\u001a\u0010| (8)3\nvanishes in the bulk and reduces to the boundary terms,\nwhich we can interpret as the quantum version of the\nvorticity (4). This suggests a conservation law with the\nboundary \ruxes corresponding to the vorticity \row. In-\ndeed, according to the Heisenberg equation of motion\n(for Hamiltonian Hand an arbitrary time-independent\noperatorO),\n@tO\u0011i\n~[H;O]; (9)\nthe quantum vorticity density \u001a\u0010|is seen to satisfy the\ncontinuity equation:\n@t\u001a\u0010|+jx\n~ \u0010|\u0000jx\n\u0010|+jy\n\u0010~|\u0000jy\n\u0010|\na= 0: (10)\nThe \ruxes in the second term are consistent with quan-\ntizing Eq. (1):\njx\n\u0010|=z\u0001(S\u0010~|\u0000S\u0010|)\u0002@t(S\u0010~|+S\u0010|)\n4\u0019aS2+ H:c:; (11)\nand similarly for the other components.\nIt is useful to emphasize that the associated conserva-\ntion law is not rooted in any speci\fc symmetry of the\nsystem. Indeed, the form of the Hamiltonian Hstill re-\nmains arbitrary. The continuity is rather dictated by the\ntopology associated with the vorticity (hydro)dynamics\nin the interior of the system. Speci\fcally, an arbitrary\nlocal deformation of the \feld in the bulk yields the same\nnet vorticity, irrespective of the details of the dynamics.\nIII. 1D WINDING DYNAMICS\nIn contrast to the vorticity \row, winding dynamics\nin, e.g., one-dimensional (1D) super\ruids6or magnets11\nobey the conservation law only approximately. In these\nsystems, the underlying topological invariant relies on\na nonlinear constraint applied to the order parameter,\nwhich ultimately makes the conserved quantity vulnera-\nble to thermal \ructuations. This leads to phase slips,7\nwhich are detrimental to the topological conservation\nlaw.\nThese issues carry over to the quantum regime, where\nquantum phase slips arise due to tunneling.8Supposing\nthese could be neglected, in an appropriate limit, we wish\nto formulate a Kubo approach for the topological quan-\ntum \row in terms of the corresponding current autocor-\nrelator.\nA. Quantum winding dynamics\nLet us illustrate these points by considering winding\ndynamics along a 1D quantum lattice, with an easy-plane\nanisotropy in spin space, which constrains the (ferro- or\nantiferro-)magnetic dynamics to lie close to the xyplane.\nAs we have already mentioned, a coarse-grained classicalhydrodynamics can be formulated in terms of the den-\nsity\u001a=\u0000@x'=\u0019 and \ruxj=@t'=\u0019, where'is the az-\nimuthal angle of the order parameter in the xyplane,11\nsuch that@t\u001a+@xj= 0.\nAllowing for arbitrary (unconstrained) spin dynamics,\nwe now formulate a quantum theory on a lattice through\nthe de\fnitions\n\u001a\u0010=z\u0001S~ \u0010\u0002S\u0010\n\u0019aS2; j\u0010=z\u0001S\u0010\u0002@tS\u0010\n2\u0019S2+ H:c:; (12)\nwhere, as before, @tshould be understood according to\nthe Heisenberg equation of motion (9) (which depends on\na concrete Hamiltonian, to be speci\fed later). Since these\nreduce to the winding density and \rux, in the appropri-\nate coarse-grained classical limit, we may expect them to\napproximately obey the continuity equation (when the\nphase slips can be disregarded). Indeed,\n@t\u001a\u0010+j~ \u0010\u0000j\u0010\na=z\u0001(S~ \u0010\u0000S\u0010)\u0002@t(S~ \u0010+S\u0010)\n2\u0019aS2+ H:c::(13)\nThe term on the right-hand side (RHS), which spoils the\nexact conservation law, can be recognized to be exactly\n(twice) the vorticity \row transverse to the spin chain,\ncf. Eq. (11). If it can be neglected, we would recover\nthe continuity equation and with it the Kubo formula\n(26) that governs the topological \row and the electrical\ntransconductance, to be discussed below.\nB. Boundary conditions\nIn order to place the spin chain (of length L) into a\nmeasurable external circuit, let us suppose it is biased by\nspin torques (polarized along the zaxis)\u001cL(R)at its left\n(right) ends. The (semiclassical) work associated with\nthe left torque is\n\u0001WL=Z\ndt\u001cL@t'=\u0019\u001cLZ\ndtj=\u0019\u001cL\u0001QL;(14)\nwhere \u0001QLis the topological charge transfer into the\nchain through the left end. This translates into the ef-\nfective chemical potential bias at the left end given by\n\u0016L=\u0001WL\n\u0001QL=\u0019\u001cL: (15)\nSimilarly for the right end, we get\n\u0016R=\u0001WR\n\u0001QR=\u0000\u0019\u001cR: (16)\nSuch torques can be induced, for example, by the spin\nHall e\u000bect triggered by an electrical current \rowing trans-\nverse to the chain.4See Fig. 2 below.\nReciprocally to these torques, the precessional dynam-\nics produces a transverse motive force on the electrons\nin the contacts, which can be used to detect the out\row\nof the topological charge through the ends.4,11We will\nreturn to discuss this in more detail in Sec. III D.4\nC. Kubo formula\nWe are now ready to de\fne the bulk impedance for\nthe topological \row, as an intrinsic property of the quan-\ntum magnet. Starting with a continuity equation for the\ncoarse-grained quantum dynamics in the bulk, we have\n@t\u001a+@xj= 0; (17)\nwhere the conserved density and current are obtained\nfrom Eqs. (12), and we neglect the RHS of Eq. (13), i.e.,\nphase slips, for now. We recall that the time derivatives\nare obtained in the Heisenberg picture. If we perturb the\nsystem by a scalar potential \u001e(x;t) that couples linearlyto the topological charge, the Hamiltonian becomes\nH!H+Z\ndx\u001e(x;t)\u001a(x): (18)\nNote that the topological density is even under time re-\nversal, while the \rux is odd (supposing the Hamiltonian\nis time-reversal invariant), so it vanishes in equilibrium,\nwhen\u001e\u00110. For a \fnite time-dependent potential \u001e, on\nthe other hand, the linear response dictates\nj(x;t) =1\n~Z\ndx0dt0G(x\u0000x0;t\u0000t0)\u001e(x0;t0); (19)\nwhere\nG(x\u0000x0;t\u0000t0)\u0011\u0000i\u0012(t\u0000t0)[j(x;t);\u001a(x0;t0)];(20)\naccording to the Kubo formula (with the equilibrium ex-\npectation value implicit on the right-hand side).\nTo invoke the continuity equation, we di\u000berentiate the\nresponse function in time:\n@tG(x\u0000x0;t\u0000t0) =i\u0012(t\u0000t0)[j(x;t);@t0\u001a(x0;t0)]\u0000i\u000e(t\u0000t0)[j(x);\u001a(x0)]\n=\u0000i\u0012(t\u0000t0)[j(x;t);@0\nxj(x0;t0)] +\u000e(t\u0000t0)@0\nxp(x\u0000x0);(21)\nwhere the auxiliary function p(x\u0000x0) is obtained by in-\ntegrating\n@0\nxp(x\u0000x0) =\u0000i[j(x);\u001a(x0)]: (22)\nFourier transforming in time, j(!) =R\ndtei!tj(t) etc., we\n\fnally get\nj(x;!) =i\n~!Z\ndx0&(x\u0000x0;!)\"(x0;!); (23)\nwhere\n&(x\u0000x0;t\u0000t0)\u0011\u0000i\u0012(t\u0000t0)[j(x;t);j(x0;t0)]\n+\u000e(t\u0000t0)p(x\u0000x0)(24)\ninvolves the current autocorrelator and\n\"\u0011\u0000@x\u001e (25)\nis the e\u000bective electric \feld. This gives for the conduc-\ntivity relating j(k;!) to\"(k;!):\n\u001b(k;!) =i\n~!&(k;!); (26)\nhaving also Fourier transformed in real space,R\ndxe\u0000ikx.\nFor a torque-biased spin chain,\n\"=\u0016L\u0000\u0016R\nL=\u0019\u001cL+\u001cR\nL; (27)\nsupposing that the length of the topological transport\nchannelLis long enough, so that the bulk dominatesover the interfacial impedances and focusing on the DC\nlimit.3\nWhile evaluating the Kubo formula, we should in gen-\neral also calculate the phase-slip rate governed by the\nRHS of Eq. (13) (driven by the potential \u001egradient that\ncouples to the winding density \u001a). For the internal con-\nsistency of the hydrodynamic treatment, it must be small\ncompared to the induced winding \rux along the spin\nchain. Writing the phase-slip rate per unit length (i.e.,\nthe vorticity \rux transverse to the chain) as\nj\u001e=\u0014\"; (28)\nwhere\"=\u0000@x\u001eis the e\u000bective \feld that drives the wind-\ning \row, we thus require (taking the k!0 and!!0\nlimit for\u001b)\nL\u001c\u001b\n\u0014; (29)\nfor the validity of the (approximate) conservation law\n(13). At the same time, however, we should not forget\nthatLmust be long enough for us not to concern with\nthe e\u000bective interfacial resistance, if we want the overall\nimpedance to be governed by the bulk.\nD. Electrical transconductance\nIf the winding injection is performed electrically, so\nthat the e\u000bective chemical potential (conjugate to the5\ntopological charge) \u0016=\u0011I, whereIis the applied cur-\nrent, the Onsager reciprocity dictates the backaction mo-\ntive force on the electrons E=\u0011j(which translates di-\nrectly into a measurable voltage).3Putting this together,\nfor a circuit sketched in Fig. 2, we obtain the electrical\ntransconductance\nG=E\nI=\u00112g: (30)\nmediated by the winding \row across the chain. The (lin-\near response) e\u000bective winding conductance here,\ng\u0011j\n\u0016; (31)\nwould be given by \u001b=L in the absence of phase slips, but\nis degraded by them otherwise, as a function of L.12\nL<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>µ=⌘I\n<latexit sha1_base64=\"IEi0mVf0ESla/g4vRm/6o17P7Dc=\">AAABtnicZU9NS8NAFHxbv2r9qnr0UszFU0m0aD0IhV70VsF+YBPLZvtal+5uluymWEL/heBJ/5f/RluDEDKnYebNYybUghvrut+ktLG5tb1T3q3s7R8cHlWPT3omSmKGXRaJKB6E1KDgCruWW4EDHSOVocB+OGuv/P4cY8Mj9WQXGgNJp4pPOKP2V3r2ZXLno6W1h1HVcevuGrUi8TLiQIbOqPrhjyOWSFSWCWrM0HO1DVIaW84ELit+YlBTNqNTHI7nXBtFJZogfVuXzvkplcYsZFgQJbWv+U+JnTSDlCudWFQsH1hMImXNcr3kdoXr/95F0ruse1f1xmPDabWzTWU4g3O4AA9uoAX30IEuMFDwDp/wRZrkhSCZ/p2WSJY5hRyI/gFN2nsP</latexit>I<latexit sha1_base64=\"mEWQgXUx631UH4ZJ5NRVx3GsqXE=\">AAABrXicZU9NT8JAFHyLX4hfVY9ejL14wlaJHzcSLnqDxAIJVLJdHrhhd7vpbolNwy/wqvG3+W8UbEwa5jSZefMyE2nBjfW8b1LZ2Nza3qnu1vb2Dw6PnOOTronThGHAYhEn/YgaFFxhYLkV2NcJUhkJ7EWz1tLvzTExPFbPNtMYSjpVfMIZtb9S52nkuF7dW+F8nfgFcaFAe+R8DccxSyUqywQ1ZuB72oY5TSxnAhe1YWpQUzajUxyM51wbRSWaMH9bdS35OZXGZDJaEyW1r+VPqZ3chzlXOrWoWDmQTWJlzWK15GGJ2//e66R7Xfdv6o1Ow222ik1VOIMLuAQf7qAJj9CGABggvMMHfJIrEpAhefk7rZAicwolkOkP2Gt3hA==</latexit>\nE<latexit sha1_base64=\"7KV0SvEuXic3tf7X+j2zg57pbPQ=\">AAABt3icZU/JTgJBFHyNG+KGevRi5OKJzChBuZEQE4+YyJLAhPQ0b6BDb073EMmEzzDe9Lv8GwUmJhPqVKl69VIVGsGt87wfUtjZ3ds/KB6Wjo5PTs/K5xddq5OYYYdpoeN+SC0KrrDjuBPYNzFSGQrshbPWyu/NMbZcq1e3MBhIOlE84oy6P2kwlNRNGRXp03JUrnhVb43rbeJnpAIZ2qPy53CsWSJROSaotQPfMy5Iaew4E7gsDROLhrIZneBgPOfGKirRBun7unXOT6m0diHDLXFVL/8pcdFjkHJlEoeK5QOLSCtnN0saK9T/e2+T7l3Vv6/WXmqVZivbVIQruIFb8OEBmvAMbegAAw0f8AXfpEFGJCLTzWmBZJlLyIG8/QK3xXwS</latexit>4\n\u0001\u0002\u0003\u0002\u0004\u0005\u0006\u0007\u0004-\b\u0003\u0002\u0004\u0005****\t\n\u000b\f\u0001\u0002\u0003\u0001\u0002\u0004\u0001\u0002\u0005\u0001\u0002\u0006\u0001\u0002\n\u0001\u0002 \u0007\u0002\u0003\u0002\t\n\u000b\f\n\u0001\u0001\n\u0001\u0002 \u0007\u0001\u0002 \u0003\u0001\u0002\u0001\u0002\u0003\u0001\u0001\u0004\u0001\u0005\u00013,41,2(a)\u0001\u0001\u0002\u0001\u0003\u0001(b)(c)\nFIG. 2. Leaky integrate-and-fire functionality of the neuron.(a) The washboard potential. The angular order parameter'of the neuron and its time derivative@t'are analogousto the position and speed of a particle. Inset: The tiltedwashboard potential where local minima become unstable.(b) The firing and non-firing diagram depending on the inputrate!and amplitude⌫0of the incoming train of angularmomentum kicks. Numerics are done with parameters!0=10 GHz and⌘= 5. (c) The time evolution of the orderparameter of the neuron'(t) simulated for samples markedin the diagram. In each simulation, 20 pulses are sent to theneuron until it fires.time, are much less thant0). Thus⌧(t) can be regardedas composed of delta pulses, each elevates the canonicalmomentumMby\u0000M, and, as manifested by Eq. (4),boosts@t'by⌫0=\u0000M/e\u0000.Take the example of an initially stationary statetrapped at'|t=0= 0, a sudden angular momentum kicksets the initial condition@t'|t=0+=⌫0for the relaxationprocess. As a train of topological charges coming in ata rate!, the state'wiggles gently as the e↵ect of eachkick fades, featuring a leaky integration. See Fig.2(c).Only with a su\u0000ciently high input rate and/or large am-plitude, can the signal pulses trigger the neuron to fire—it then overcomes the energy barrier and falls into theneighboring local minimum'=⇡. The dependence ofthe firing or non-firing result on the rate and amplitudeof pulses is plotted in Fig.2(b). When the neuron fires, itexperiences a phase jump of⇡, whereas the phase deep inthe axon remains unchanged. A domain wall with topo-logical charge⇣=+1 is thus created on a time scale oft0,entering the axon with a certain initial velocity, where-after it triggers further actions in the network branches.Another behavior of a biological neuron that we canimitate is the bursting. In a bursting state, a neuron firesrepeatedly in groups separated by quiescent periods [71].For our artificial neuron, a constant torque⌧0(such asa spin transfer torque used in Ref. [45]) e↵ectively tiltsthe washboard potential. This can cause repetitive fir-ing once the torque exceeds a threshold where the lo-cal minima become unstable. On average, the firingrate is proportional to the tilting,r=⌧0/⇡↵eJ. As thephase'winds forward continuously, the neuron repeat-edly switches between the two ground states.Synapse.—A nonvolatile analog memory with an ad-justable weight, as the key element of a synapse, is re-alized in our proposal by an antiferromagnetic nanos-trip (2) holding a metastable winding texture. Wedefine the dimensionless density of topological chargesw=N\u0000/Las the synaptic weight, whereNis the netnumber of topological charges andLis the length of thenanostrip, which is large enough (L\u0000\u0000) to allow usto considerwas an essentially continuous variable. Thenanostrip is a one-dimensional thermodynamic system ofinteracting topological charges, with chemical potentialµ=\u0000F/\u0000N,w h e r eFis its total free energy. The de-pendence of the chemical potentialµon the topological-charge densitywis highly nonlinear. See Fig.3for theplot ofµversuswat zero temperature, calculated for themetastable ground-state textures of the Hamiltonian (2),which are the solutions to the static sine-Gordon equa-tion [68]. The chemical potential determines the leakagerate of topological charges into the dendrite of the post-synaptic neuron when the synapse is activated by a signalfrom the presynaptic neuron. A higher chemical poten-tial results in a stronger synaptic connection.The right end of the synapse [see Fig.1(a)] is initiallysealed by a strong anisotropy—a washboard potential forthe local order parameter\u00002, with a tilting proportionalto the chemical potentialµin the synapse. A topologicalcharge fired by presynaptic neuron arrives from a telo-dendron attpre, and activates the dynamics of\u00002.T h eactivation can be a result of either heating, which softensthe anisotropy, or a kinetic perturbation, which brings\u00002\n\u0001\u0002\u0001\u0002\u0003\u0004\u0002\n\u0005 \u0005\u0002\u0001 \u0005\u0002\u0004\u0005\u0002\u0006 \u0005\u0002\u0007FIG. 3. The nonlinear dependence of the chemical potentialµon the synaptic weightw, extracted from the metastableground-state textures. Inset: the magnetic textures and theircos\u0000(x) for two values ofw. The texture becomes closer toa uniform spiral aswapproaches 1. The asymptote has theanalytical expressionµ=pAK⇡2w.\u0000<latexit sha1_base64=\"Q7hv0XOyK4CymOh9yD3xaeSYsVY=\">AAABsnicZU/LTgJBEOzBF+Jr1aMXIxdPZFeJjxsJF4+YyCPChvQODY7MzE52Zolkwz94M/pn/o2CG5MNdapUdXWqIiOFdb7/zUobm1vbO+Xdyt7+weGRd3zSsXGacGrzWMZJL0JLUmhqO+Ek9UxCqCJJ3WjaXPrdGSVWxPrJzQ2FCidajAVH9yt1BlZMFA69ql/zVzhfJ0FOqpCjNfQ+BqOYp4q04xKt7Qe+cWGGiRNc0qIySC0Z5FOcUH80E8ZqVGTD7G1VuOBnqKydq2hNVOheip9SN74LM6FN6kjzYmA+jrWzi9WS+yVu/nuvk85VLbiu1R/r1UYz31SGM7iASwjgFhrwAC1oA4dXeIdP+GJ19syQ8b/TEsszp1AAkz9WWnna</latexit><latexit sha1_base64=\"AFFCKZ1a2h2fzaNqbvTVGBae2Ts=\">AAABsnicZU/LTgJBEOzBF+Jr1aMXIxdPZFeJjxsJF4+YyCPChvQODY7MzE52Zolkwz94M/pn/o2CG5MNdapUdXWqIiOFdb7/zUobm1vbO+Xdyt7+weGRd3zSsXGacGrzWMZJL0JLUmhqO+Ek9UxCqCJJ3WjaXPrdGSVWxPrJzQ2FCidajAVH9yt1BlM0Bode1a/5K5yvkyAnVcjRGnofg1HMU0XacYnW9gPfuDDDxAkuaVEZpJYM8ilOqD+aCWM1KrJh9rYqXPAzVNbOVbQmKnQvxU+pG9+FmdAmdaR5MTAfx9rZxWrJ/RI3/73XSeeqFlzX6o/1aqOZbyrDGVzAJQRwCw14gBa0gcMrvMMnfLE6e2bI+N9pieWZUyiAyR9RE3nW</latexit>\nFIG. 2. Schematic of a winding \row along a (horizontal)\nspin chain. Transverse charge current Igenerates an e\u000bective\nchemical potential bias \u0016that couples to the winding density\nat the left end. \u0011is a contact-dependent conversion param-\neter, which relates the input current to the out-of-plane spin\nHall torque \u001c=\u0011I=\u0019 acting on the magnetic dynamics in the\nchain.4The injected winding \rux is governed by the winding\nconductivity \u001b, while being dissipated by the transverse vor-\ntex \row/\u0014. The net remaining winding out\row produces\na measurable transverse motive force Eat the right electrical\ncontact.\nIV. SINE-GORDON MODEL\nAs a concrete illustration of this general \feld-\ntheoretic formalism, let us now consider an ideal 1D\nspin chain with an easy-plane collinear order param-\neter parametrized by azimuthal angle '(x) and the\n(canonically-conjugate) out-of-plane spin density s(x).\nOur main focus is on the antiferromagnetic case (which\nwill a\u000bect the phase-slip considerations). The classical\nlow-energy dynamics are generated by the (sine-Gordon)\nHamiltonian density\nH=s2\n2\u001f+A(@x')2\n2+Kcos(2'); (32)along with the Poisson bracket f'(x);s(x0)g=\u000e(x\u0000x0).\nIt is assumed here that the order-parameter con\fguration\nis close to the easy plane, at all times. \u001f,A, andKare\nrespectively the out-of-plane spin susceptibility, order-\nparameter sti\u000bness, and the axial in-plane anisotropy.\nThe Hamiltonian is invariant under time reversal, when\ns!\u0000sand'!'+\u0019. The sign of Kis inconsequential,\nas it can be \ripped by a phase change, '!'+\u0019=2.\nA. Luttinger-liquid mapping\nThis description can be quantized by promoting the\nPoisson bracket to the commutator:\n['(x);s(x0)] =i~\u000e(x\u0000x0); (33)\nmaking the theory formally analogous to a spinless Lut-\ntinger liquid.13s(x) would then correspond to the linear-\nmomentum density \u0005 and the topological density to the\nparticle density @x\u001e=\u0019. Indeed, the (spinless) Luttinger-\nliquid Hamiltonian density is\nH=u\n2\u0014\u0019g\n~\u00052+~\n\u0019g(@x\u001e)2\u0015\n+Kcos(4\u001e); (34)\nwhere [\u001e(x);\u0005(x0)] =i~\u000e(x\u0000x0).uis the speed of sound\nandg > 0 is the interaction parameter ( u!vF, the\nFermi velocity, and g!1, for free electrons; g<1 signals\nelectron repulsion and g>1 attraction). Kparametrizes\numklapp scattering (which requires an appropriate lat-\ntice \flling factor).\nThe corresponding Euclidean Lagrangian density is\nL=~\n2\u0019g\u00141\nu(@\u001c')2+u(@x')2\u0015\n+k\n2(\u0019a)2cos(2');(35)\nabeing a short-distance cuto\u000b. We have rede\fned the\ndisplacement \feld \u001e!'=2 and appropriately rescaled\ng, in order to match the notation of our spin model\n(32). Under the Wilsonian renormalization-group (RG)\nrescaling,13we get the Kosterlitz-Thouless \row equa-\ntions:\ndy\ndl= (2\u0000g)y;dg\ndl=\u0000g3y2=8; (36)\nwherey\u0011k=\u0019~u(which we can take to be positive, with-\nout loss of generality), and we have omitted the nonuni-\nversal cuto\u000b-dependent numerical prefactor on the right-\nhand side of the second equation. The RG \row of ycor-\nresponds simply to the scaling dimension of the cosine\noperator.13The generic reduction in g, under the RG\n\row (36), corresponds to the sti\u000bening of the order pa-\nrameter'due to the cosine term /Kin the Lagrangian\n(35) (which tries to order the \feld ').\nFor our original spin system, Eq. (32), u=p\nA=\u001f,\ng=~=\u0019pA\u001f. We interpret the cuto\u000b as a\u0018p\nA=K?,14\nwhereK?\u001dKis a strong easy-plane anisotropy that\nkeeps spin dynamics close to the xyplane. The bare6\norder-parameter sti\u000bness is A\u0019S2Jaand spin suscep-\ntibility\u001f\u0019~2=4Ja, in the large-spin Heisenberg limit\n(which acquire some corrections due to quantum \ruc-\ntuations when S\u00181). These estimates boil down to:\nu\u0019Ja=~,g\u00192=\u0019S,y\u0018K=K?\u001c1. Going beyond\nthe Heisenberg limit, with a large easy-plane anisotropy,\nwould decrease \u001f, while not similarly a\u000becting A, and\nthus increase the value of g.\nExpanding gclose to its critical value gc= 2,g!2+g,\nwe get13\ndy\ndl=\u0000gy;dg\ndl=\u0000y2; (37)\nwhich parametrize hyperbolic trajectories with g2\u0000y2=\nconst. These equations \row rapidly to a noninteracting\n(gapless) \fxed point g\u0003>0 andy= 0, if 0<jyj< g.\nOtherwise, the \row takes us to a gapped strong coupling,\nwithjyj!1 , where the phase is pinned along the easy\naxis (i.e.,'!0 or\u0019). In this latter case, the elementary\ndynamics could be constructed out of the corresponding\ndomain walls or their composites.13\nB. Quantum phase slips\nIt is important to also make an internal consistency\ncheck to ensure that the original spin system can indeed\nbe described e\u000bectively by the sine-Gordon Lagrangian\n(35) for nonsingular \felds '(i.e., con\fgurations free\nof vortex singularities in the 1 + 1 space-time dimen-\nsions). In the absence of the anisotropy K(or when it\nrenormalizes down to zero due to quantum \ructuations),\nthe easy-plane antiferromagnetic spin chain undergoes a\n(Kosterlitz-Thouless type) spin super\ruid-insulator tran-\nsition atS\u00192 (1=2), for integer (half-integer) spin S,8\nwith larger Splacing us in the (gapless) super\ruid phase.\nIn the latter, the magnetic vortices and antivortices bind,\nand we restore a hydrodynamic easy-plane \u001bmodel.\nThe rate of quantum phase slips, \u0014, in this limit is\ngiven by8\n\u0014/\u001a\u0019q2S\u00003; (38)\nat zero temperature (with a similar scaling with tem-\nperature, at zero winding density \u001a), where the elemen-\ntary vorticity is q= 1 (2) for the integer (half-integer)\nspinS.15The suppression of the phase slips as \u001a!0,\nat low temperatures, gives us an algebraically-large win-\ndow, Eq. (29), for the spin-chain length Lwith conserved\nwinding transport.16\nSummarizing the requirements for the low-energy hy-\ndrodynamic regime described by the e\u000bective long-\nwavelength Hamiltonian\nH=s2\n2\u001f+A(@x')2\n2; (39)\nwith canonically conjugate sand'(and e\u000bective, renor-\nmalized\u001fandA), we needS&1=2 (for the half-integercase, where the leading-order phase slips are mediated by\ndouble vortices, due to destructive interference of single\nvortices,8) whileg\u00182=\u0019S > 2, andy=k=\u0019~u<g\u00002.\nThe generic mean-\feld parameters for a Heisenberg an-\ntiferromagnetic chain17thus place us at a cusp of the\ngapless (hydrodynamic) phase for S= 1=2 (with a clear\ntendency to \row to the gapped phase for a larger spin).\ngcan be decreased beyond the Heisenberg limit when the\neasy-plane anisotropy is large (i.e., comparable to the ex-\nchange interaction), which would suppress the spin sus-\nceptibility\u001f. A more detailed and careful analysis would\nbe needed, in principle, in order to establish the exact\nvalues of the parameters yielding the ballistic transport\nregime in this quantum limit of S\u00191=2. A fermionic\ntreatment in the Wigner-Jordan representation of spin\ndynamics18is one possible approach to that end.\nC. Winding hydrodynamics\nSupposing we wind up with a gapless regime described\nby the Hamiltonian (39) (if the anisotropy Kis weak\nand/or irrelevant), we are ready to formulate a transport\ntheory for the winding density \u001a=\u0000@x'=\u0019. The associ-\nated current operator, j=@t'=\u0019, becomes [cf. Eq. (33)]\nj=i\n\u0019~[H;'] =s\n\u0019\u001f: (40)\nFrom Eq. (22), we thus get\n@0\nxp(x\u0000x0) =\u0000i[j(x);\u001a(x0)] =\u0000~\n\u00192\u001f@0\nx\u000e(x\u0000x0);(41)\nso that\np(x\u0000x0) =~\n\u00192\u001f\u000e(x\u0000x0): (42)\nThe current-current correlator in Eq. (24) vanishes in the\nlimitk!0 (while maintaining a \fnite !),19so that we\nend up with the conductivity (26):\n\u001b(k!0;!) =i\n\u00192\u001f!=iA\n!; (43)\nwhereA= 1=\u00192\u001f.\nRegularizing this result at zero frequency, !!!+i0+,\nwe get Re\u001b=\u0019A\u000e(!). As expected, the static con-\nductivity diverges in the low-frequency limit. In this\ncase, the super\ruid bulk has no impedance and the\nwinding conductance of the entire structure needs to be\ndetermined by carefully considering the interfacial in-\njection physics, which is akin to the Andreev conduc-\ntance of normaljsuperconducting interfaces.20The resul-\ntant winding conductance of a clean \fnite-length spin\nchain should thus be governed by the contact impedance,\nsimilarly to the charge conductance of a ballistic elec-\ntronic wire (whose contact resistance depends both on\nthe details of the contact itself as well as the electron-\nelectron interactions in the wire.21)7\nIf, on the other hand, we add some spin-relaxing impu-\nrities to the Hamiltonian, so that the collective spin den-\nsity de\fning the topological current according to Eq. (40)\nis now allowed to relax, the dynamic spin susceptibility\ncan be obtained at k!0 according to the Bloch phe-\nnomenology:\nds(t)\ndt=\u0000s(t)\u0000s0(t)\n\u001c: (44)\nHeres0(t) =\u001fh(t) is the (instantaneous) equilibrium\nspin density, according to the Zeeman coupling to a mag-\nnetic \feld:H!H\u0000hs. The dynamic spin susceptibility\nis thus:\n\u001fs(!)\u0011s(!)\nh(!)=\u001f\n1\u0000i!\u001c\u0019\u001f(1 +i!\u001c); (45)\nat!\u001c1=\u001c. This, in turn gives us the current (40) self-\ncorrelator, from which we \fnd the low-frequency topo-\nlogical conductivity according to Eq. (24):\n\u001b(!) =i\n!\u0012\n\u0000\u001fs\n(\u0019\u001f)2+1\n\u00192\u001f\u0013\n=\u001c\n\u00192\u001f: (46)\nThe response is Drude-like, with a well-de\fned DC\nlimit and a Drude weight /\u001f\u00001(in the\u001c! 1 ).\nIn the Gilbert damping phenomenology of magnetic\ndynamics,22the spin-pumping rate (i.e., torque \u001c\u000b) into\nthe environment is given by\n\u0000ds\ndt\u0011\u001c\u000b=\u000b@t'=\u000bs\u0000s0\n\u001f; (47)\nwhich can be obtained by supplementing the Heisenberg-\nHamilton dynamics of the planar order parameter with\nthe Rayleigh dissipation function R=\u000b(@t')2=2.23We\nare de\fning the Gilbert constant \u000bhere in units of spin\ndensity. This gives \u001c=\u001f=\u000b for spin-relaxation time,\nresulting in the low-frequency winding conductivity of\n\u001b=1\n\u00192\u000b: (48)\nA larger damping implies lower winding conductivity.\nThe corresponding electrical transconductance (30),\nG/(\u000bL)\u00001, reproduces the spin-super\ruid mediated\ntransconductance (drag) derived in Ref. 4, when L!1 .\nIndeed, in the limit of no axial anisotropy Kin Eq. (32) or\nwhen it renormalizes down by quantum \ructuations, the\nout-of-plane spin sis conserved and follows a super\ruid-\nlike hydrodynamics (with spin \rux /@x'). This pro-\nvides a dual description for the long-range signal propa-\ngation along the spin chain. In general, however, when\nthe spin is not conserved, the winding hydrodynamics\nbased on Eq. (13) establishes a more universal framework\nfor low-frequency long-distance transport in anisotropic\nspin chains. This is illustrated in our next and \fnal ex-\nample.D. Finite temperatures\nIn the opposite, classical limit of S&1 and thusg<2\nin Eq. (36), the phase 'is pinned and the winding dy-\nnamics are gapped. In this case, the zero-temperature\ntransport is exponentially suppressed over a long spin\nchain.18At \fnite temperatures, however, we generally\nanticipate a di\u000busive regime for thermally-activated chi-\nral domain walls,11along with the parasitic thermal\nphase slips.7If the latter is governed by a larger energy\ngap, in comparison to the domain-wall energy (which is\nthe case when K?>K), we can have a meaningful trans-\nport scenario for the conserved winding carried by the\nBrownian motion of domain walls, along a \fnite-length\nspin chain.11\nIn the strong easy-plane limit, when the winding is\ncarried by a classical gas of stable solitonic domain\nwalls (of width \u0015=p\nA=K ) with quantized topologi-\ncal charge\u00061 and mobility M, the corresponding con-\nductivity is simply \u001b= 2nM, wherenis the density of\ndomain walls of a given chirality. The associated dif-\nfusion coe\u000ecient is given by D=kBTM, according to\nthe Einstein-Smoluchowski relation. Within the Gilbert-\ndamping phenomenology,22the mobility of a rigid soli-\nton is given by M\u0018\u0015=\u000b. Since\u001b/n/e\u0000\fE,11while\n\u0014/e\u0000\fF,7whereE= 4p\nAKis the domain-wall energy\nandF= 4pAK?\u001dEis the thermal phase-slip barrier,\nwe can easily satisfy Eq. (29) at low temperatures. In the\nlimit ofK!0 (and/or high temperature, kBT&E), the\ndomain walls coalesce and we reproduce the conductivity\n(48).11\nE. Noise and quantum relaxometry\nIn addition to an electrical measurement of wind-\ning transport, as sketched in Fig. 2, it may be possi-\nble to investigate the associated topological transport\nproperties, such as winding conductivity, using quantum-\nimpurity (such as nitrogen-vacancy) relaxometry.24Sim-\nilarly to the Johnson-Nyquist noise generally associated\nwith charge conductivity, the winding transport is noisy.\nIn particular, the out of the easy-plane spin \ructuations\n(being canonically conjugate to the planar spin preces-\nsion, irrespective of the nature of the spin order), should\nproduce a detectable magnetostatic signal.25We expect\nit to re\rect similar winding transport properties as the\nelectrical setup of Fig. 2 (without the issues pertaining\nto the contacts), in the long-wavelength low-frequency\nlimit of the dynamics. The latter can be controlled by\nthe quantum-impurity positioning and applied magnetic\n\feld (e.g., Zeeman splitting of the nitrogen-vacancy spin\nstates), respectively.268\nV. DISCUSSION\nIn summary, we have constructed a general framework\nto study winding dynamics in spin chains, from the per-\nspective of a transport phenomenon. Motivated by the\nmean-\feld considerations that draw on the notions of\nspin super\rows along the chain and parasitic vorticity\n\rows transverse to it, we developed a fully quantum the-\nory, where both the winding transport and its relaxation\nby phase slips can be treated systematically by \feld-\ntheoretical approaches.\nWe illustrated the general formalism by specializing to\nthe case of antiferromagnetic easy-plane dynamics, whose\nsalient features can be captured by a sine-Gordon model.\nTwo distinct scenarios then arise concerning the winding\n\rows: the spin-super\ruid regime, where the parasitic in-\nplane anisotropy is washed out by quantum (or thermal)\n\ructuations, and the solitonic regime, where chiral do-\nmain walls carry conserved winding density by Brownian\nmotion (at \fnite temperatures). Both of these limits are\naddressed within our general Kubo formalism, reproduc-\ning and complimenting the pertinent special cases known\nin the literature. We see that rather generically, in the\npresence of magnetic damping, the winding \row can ex-\nhibit Drude-like dynamic response. This corresponds to\nan e\u000bectively metallic behavior of the conserved winding\ntransport. The key internal-consistency check for these\n\fndings concerns the transverse vorticity \row, which re-\n\rects phase slips and needs to be smaller than the wind-\ning \row along the spin chain.\nOne of our central motivations for this work is the po-\ntential ability to detect the topological transport, either\nin an electrical device (cf. Fig. 2) or by a nonintrusive\nquantum-impurity relaxometry (cf. Sec. IV E). The \feld-\ntheoretical framework combined with the experimental\ntangibility should open gates for nonelectrical transport-\nbased investigations of correlated magnetic materials. It\nis useful to add, furthermore, that a long-range order ofany kind is neither assumed nor needed for the emergence\nof topological hydrodynamics. Our microscopic quantum\nformulation, which we have explicitly constructed for vor-\nticity and winding \rows, furthermore, does not even rely\non a local ordering or any semiclassical approximations.\nWe have largely left out the contact-impedance consid-\nerations in our device concept sketched in Fig. 2. This\nmay be justi\fed when there a \fnite bulk resistivity for\nthe topological \row. In the opposite, clean limit, the\ntransport physics would, however, generally be domi-\nnated by the contacts and, at low temperatures, poten-\ntially strongly dependent on the many-body e\u000bects away\nfrom the contacts. These aspects are left for future work.\nNo attempt has been made to classify scenarios of topo-\nlogical hydrodynamics for general quantum magnets in\narbitrary dimensions, which also goes beyond our scope\nhere. Quantum skyrmions in two spatial dimensions27\nand hedgehogs in three dimensions28provide other in-\nteresting examples, with skyrmions, like winding, obey-\ning only an approximate continuity equation. We thus\nanticipate rich possibilities for topological hydrodynam-\nics in magnetic materials, with implications for novel\nprobes and device concepts that do not rely on electronic\n(charge) transport.\nACKNOWLEDGMENTS\nWe thank Michael Mulligan for insightful discussions.\nThe work was supported by NSF under Grant No. DMR-\n1742928 (Y.T. and J.Z.). S.K.K. was supported by\nBrain Pool Plus Program through the National Research\nFoundation of Korea funded by the Ministry of Sci-\nence and ICT (Grant No. NRF-2020H1D3A2A03099291)\nand by the National Research Foundation of Korea\nfunded by the Korea Government via the SRC Center\nfor Quantum Coherence in Condensed Matter (Grant No.\nNRF-2016R1A5A1008184). S.T. is supported by CUNY\nResearch Foundation Project #90922-07 10 and PSC-\nCUNY Research Award Program #63515-00 51.\n1X. F. Sun, W. Tao, X. M. Wang, and C. Fan, Phys. Rev.\nLett. 102, 167202 (2009); G. T. Hohensee, R. B. Wilson,\nJ. P. Feser, and D. G. Cahill, Phys. Rev. B 89, 024422\n(2014); A. L. Chernyshev and A. V. Rozhkov, Phys. Rev.\nLett. 116, 017204 (2016); X. Chen, J. Kim, Q. Jia, S. E.\nSullivan, Y. Xu, K. Jarvis, J. Zhou, and L. Shi, Adv. Func.\nMater. 30, 2001637 (2020).\n2F. Meier and D. Loss, Phys. Rev. Lett. 90, 167204 (2003);\nJ. Sirker, R. G. Pereira, and I. A\u000feck, Phys. Rev. B\n83, 035115 (2011); C. Karrasch, D. M. Kennes, and\nF. Heidrich-Meisner, Phys. Rev. B 91, 115130 (2015);\nF. Lange, S. Ejima, T. Shirakawa, S. Yunoki, and\nH. Fehske, Phys. Rev. B 97, 245124 (2018); V. B.\nBulchandani, R. Vasseur, C. Karrasch, and J. E. Moore,\nPhys. Rev. B 97, 045407 (2018); A. Biella, M. Collura,\nD. Rossini, A. De Luca, and L. Mazza, Nat. Comm. 10,4820 (2019).\n3Y. 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Zou, S. K. Kim, and Y. Tserkovnyak, Phys. Rev. B 99,\n180402(R) (2019).\n10By extending this de\fnition of the topological charge den-\nsity in terms of the vector chirality to arbitrarily-shaped\nplaquettes, the theory is naturally generalized to arbitrary\nspin lattices.\n11S. K. Kim, S. Takei, and Y. Tserkovnyak, Phys. Rev. B\n92, 220409(R) (2015).\n12For the simplest scenario with j\u001e/j, the ensuing signal\ndecay would be exponential, which we can see by adding\nthisj\u001eto the RHS of Eq. (17), in the steady state with\n@t\u001a= 0.\n13T. Giamarchi, Quantum Physics in One Dimension (Ox-\nford University Press, Oxford, 2004).\n14This cuto\u000b governs the Landau stability criterion for the\neasy-plane texture.29In the case of easy-plane ferromag-\nnets, the magnon dispersions become nonlinear at wave\nnumbersq\u00181=a. For antiferromagnets, at these wave\nnumbers, the higher-energy magnon branch, with out-of-\nplane N\u0013 eel dynamics, mixes with and modi\fes the sine-\nGordon description of the planar N\u0013 eel \ructuations.\n15In a related context, such topological quantum spin-parity\ne\u000bects were investigated also for chirality tunneling of lo-\ncalized antiferromagmetic domain walls.30.\n16Note that the phase-slip rate \u0014quoted in Eq. (38) was\nevaluated in Ref. 8 with respect to the winding density \u001a\nrather than the e\u000bective \feld \u000fas in our present notation\n[cf. Eq. (28)]. Since in a generic system, they are linearly re-\nlated, in linear response, the basic argument should stand.\nThe key observation here is that the phase slips vanish in\nlinear response in the spin-super\ruid phase.\n17In the case of a ferromagnetic chain, \u001f/1=K?, whichis typically larger than in the antiferromagnetic case, if\nthe easy-plane anisotropy K?is weaker than the exchange\ninteraction. This would reduce g, pushing us deeper into\nthe insulating regime.\n18S. Ho\u000bman, D. Loss, and Y. Tserkovnyak, \\Super\ruid\ntransport in quantum spin chains,\" ArXiv:1810.11470.\n19Y. Tserkovnyak and J. Zou, Phys. Rev. Res. 1, 033071\n(2019).\n20Y. V. Nazarov and Y. M. Blanter, Quantum Transport\n(Cambridge University Press, Cambridge, 2009).\n21D. L. Maslov and M. Stone, Phys. Rev. B 52, R5539\n(1995).\n22T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n23Y. Tserkovnyak and H. Ochoa, Phys. Rev. B 96, 100402(R)\n(2017).\n24C. Du, T. Van der Sar, T. X. Zhou, P. Upadhyaya,\nF. Casola, H. Zhang, M. C. Onbasli, C. A. Ross, R. L.\nWalsworth, Y. Tserkovnyak, and A. Yacoby, Science 357,\n195 (2017); F. Casola, T. van der Sar, and A. Yacoby,\nNat. Rev. Mater. 3, 17088 (2018).\n25Speci\fcally, according to Eq. (40), the winding \rux, /@t'\nis generally proportional to the local spin density. Fluc-\ntuations of the latter, in turn, emit a detectable magne-\ntostatic noise,24which contains direct information about\nthe topological current autocorrelator. According to the\n\ructuation-dissipation theorem,26furthermore, this could\nbe directly related to the (causal) linear response and thus\nthe winding conductivity.\n26B. Flebus and Y. Tserkovnyak, Phys. Rev. Lett. 121,\n187204(R) (2018).\n27H. Ochoa and Y. Tserkovnyak, Inter. J. Mod. Phys. B 33,\n1930005 (2019).\n28J. Zou, S. Zhang, and Y. Tserkovnyak, \\Topolog-\nical transport of decon\fned hedgehogs in magnets,\"\nArXiv:2006.10910.\n29E. B. Sonin, Adv. Phys. 59, 181 (2010).\n30B. A. Ivanov, A. K. Kolezhuk, and V. E. Kireev, Phys.\nRev. B 58, 11514 (1998)."    },    {        "title": "2010.00642v1.Modeling_coupled_spin_and_lattice_dynamics.pdf",        "content": "Modeling coupled spin and lattice dynamics\nMara Strungaru,1Matthew O A Ellis,2Sergiu Ruta,3Oksana Chubykalo-Fesenko,4Richard F L Evans,1and Roy W Chantrell1\n1Department of Physics, University of York, York, United Kingdom\n2Department of Computer Science, University of Sheffield, Sheffield, United Kingdom\n3Department of Physics,University of York, York, United Kingdom\n4Instituto de Ciencia de Materiales de Madrid, CSIC, Madrid, Spain\nA unified model of molecular and atomistic spin dynamics is presented enabling simulations both in micro-\ncanonical and canonical ensembles without the necessity of additional phenomenological spin damping. Trans-\nfer of energy and angular momentum between the lattice and the spin systems is achieved by a coupling term\nbased upon the spin-orbit interaction. The characteristic spectra of the spin and phonon systems are analyzed\nfor different coupling strength and temperatures. The spin spectral density shows magnon modes together with\nthe uncorrelated noise induced by the coupling to the lattice. The effective damping parameter is investigated\nshowing an increase with both coupling strength and temperature. The model paves the way to understanding\nmagnetic relaxation processes beyond the phenomenological approach of the Gilbert damping and the dynamics\nof the energy transfer between lattice and spins.\nI. INTRODUCTION\nWith the emergent field of ultrafast magnetisation\ndynamics1understanding the flow of energy and angular mo-\nmentum between electrons, spins and phonons is crucial for\nthe interpretation of the wide range of observed phenom-\nena2–5. For example, phonons strongly pumped in the THz\nregime by laser excitation can modulate the exchange field\nand manipulate the magnetisation as shown for the magnetic\ninsulator YIG6or in Gd7. The excitation of THz phonons\nleads to a magnetic response with the same frequency in\nGd7, proving the necessity of considering the dynamics of\nboth lattice and spins. Phonon excitations can modulate both\nanisotropy and exchange which can successfully manipulate\n8–10or potentially switch the magnetisation11,12, ultimately\nleading to the development of low-dissipative memories.\nMagnetisation relaxation is typically modeled using the\nphenomenological description of damping proposed by Lan-\ndau and Lifshitz13and later Gilbert14, where the precessional\nequation of motion is augmented by a friction-like term, re-\nsulting in the Landau-Lifshitz-Gilbert (LLG) equation. This\nrepresents the coupling of the magnetic modes (given pri-\nmarily by the localised atomic spin) with the non-magnetic\nmodes (lattice vibrations and electron orbits). The LLG equa-\ntion and its generalisations can be deduced from the quantum-\nmechanical approaches assuming an equilibrium phonon bath\nand the weak coupling of the spin to the bath degrees of\nfreedom15–17. Thus the standard approach works on the sup-\nposition that the time scales between the environmental de-\ngrees of freedom and the magnetic degrees of freedom are\nwell separated and reducing the coupling between the mag-\nnetization and its environment to a single phenomenological\ndamping parameter18,19. In reality, the lattice and magneti-\nsation dynamics have comparable time-scales, where the in-\nteraction between the two subsystems represents a source of\ndamping, hence the necessity of treating spin and lattice dy-\nnamics in a self-consistent way.\nTo investigate these phenomena, and aiming at predictive\npower for the design of competitive ultrafast magnetic nano-\ndevices, advanced frameworks beyond conventional micro-magnetics and atomistic spin dynamics20are needed21. A\ncomplete description of magnetic systems includes the inter-\naction between several degrees of freedom, such as lattice,\nspins and electrons, modeled in a self-consistent simulation\nframework. The characteristic relaxation timescales of elec-\ntrons are much smaller ( \u0019fs) in comparison to spin and lattice\n(100fs\u0000ps), hence magnetisation relaxation processes can be\ndescribed via coupled spin and lattice dynamics, termed Spin-\nLattice Dynamics (SLD)22–29. SLD models can be crucial in\ndisentangling the interplay between these sub-systems.\nSLD models have so far considered either micro-canonical\n(NVE - constant particle number, volume and energy)27,28or\ncanonical (NVT - constant particle number, volume and tem-\nperature) ensembles with two Langevin thermostats connected\nto both lattice and spin subsystems23,30. Damping due to spin-\nlattice interactions only within the canonical ensemble (NVT)\nhas not yet been addressed, but is of interest in future mod-\nelling of magnetic insulators at finite temperature. Here we\nintroduce a SLD model capable of describing both ensem-\nbles. Specifically, our model (i) takes into account the transfer\nof angular momentum from spin to lattice and vice-versa, (ii)\nworks both in a micro-canonical ensemble (constant energy)\nand in a canonical ensemble (constant temperature), (iii) al-\nlows a fixed Curie temperature of the system independent of\nthe spin-lattice coupling strength, (iv) disables uniform trans-\nlational motion of the system and additional constant energy\ndrift, which can be produced by certain spin-lattice coupling\nforms. Furthermore, in this work, the characteristics of the in-\nduced spin-lattice noise, the magnon-phonon induced damp-\ning and the equilibrium properties of the magnetic system has\nbeen systematically investigated.\nThe paper is organised as follows. We start by describ-\ning the computational model of Spin-Lattice Dynamics and\nthe magnetic and mechanical energy terms used in this frame-\nwork (Section II). We then explore the equilibrium properties\nof the system for both microcanonical and canonical simula-\ntions, proving that our model is able to efficiently transfer both\nenergy and angular momentum between the spin and lattice\ndegrees of freedom. In Section III we compute the equilibrium\nmagnetisation as function of temperature for both a dynamic\nand static lattice and we show that the order parameter is notarXiv:2010.00642v1  [cond-mat.mtrl-sci]  1 Oct 20202\ndependent on the details of the thermostat used. In Section IV\nwe analyse the auto-correlation functions and spectral char-\nacteristics of magnon, phonons and the coupling term prov-\ning that the pseudo-dipolar coupling efficiently mediates the\ntransfer of energy from spins to the lattice and vice-versa. We\nthen calculate the temperature and coupling dependence of the\ninduced magnon-phonon damping and we conclude that the\nvalues agree well with damping measured in magnetic insula-\ntors, where the electronic contributions to the damping can be\nneglected ( Section V).\nII. COMPUTATIONAL MODEL\nIn order to model the effects of both lattice and spin dy-\nnamics in magnetic materials an atomistic system is adopted\nwith localised atomic magnetic moments at the atomic coor-\ndinates. Within this framework there are now nine degrees of\nfreedom; atomic magnetic moment (or spin) S, atomic posi-\ntionrand velocity v. The lattice and the magnetic system can\ndirectly interact with each other via the position and spin de-\npendent Hamiltonians. The total Hamiltonian of the system\nconsists of a lattice Hlatand magnetic Hmagparts:\nHtot=Hlat+Hmag: (1)\nThe lattice Hamiltonian includes the classical kinetic and\npairwise inter-atomic potential energies:\nHlat=å\nimiv2\ni\n2+1\n2å\ni;jU(ri j): (2)\nOur model considers a harmonic potential (HP) defined as:\nU(ri j) =(\nV0(ri j\u0000r0\ni j)2=a2\n0ri j<rc\n0 ri j>rc:(3)\nwhere V0has been parametrised for BCC Fe in27anda0=1˚A\nis a dimension scale factor. To be more specific we consider\nthe equilibrium distances r0\ni jcorresponding to a symmetric\nBCC structure. The interaction cut-off is rc=7:8˚A. The pa-\nrameters of the potential are given in Table II. The harmonic\npotential has been used for simplicity, however it can lead to\nrather stiff lattice for a large interaction cutoff.\nAnother choice of the potential used in our model is an an-\nharmonic Morse potential (MP) parameterised in31for BCC\nFe and defined as:\nU(ri j) =(\nD[e\u00002a(ri j\u0000r0)\u00002e\u0000a(ri j\u0000r0)]ri j<rc\n0 ri j>rc(4)\nThe Morse potential approximates well the experimental\nphonon dispersion observed experimentally for BCC Fe32as\nshown in33. The phonon spectra for the choices of potential\nused in this work are given in Section IV. Other nonlinear\nchoices of potential can be calculated via the embedded atom\nmethod34,35.The spin Hamiltonian ( Hmag) used in our simulations con-\nsists of contributions from the exchange interaction, Zeeman\nenergy and a spin-lattice coupling Hamiltonian, given by the\npseudo-dipolar coupling term ( Hc), which we will describe\nlater:\nHmag=\u00001\n2å\ni;jJ(ri j)(Si\u0001Sj)\u0000å\nimiSi\u0001Happ+Hc;(5)\nwhere miis the magnetic moment of atom i,Siis a unit\nvector describing its spin direction and Happis an external ap-\nplied magnetic field. The exchange interactions used in our\nsimulations depend on atomic separation J(ri j). They were\ncalculated from first principle methods for BCC Fe by Ma et\nal23and follow the dependence:\nJ(ri j) =J0\u0012\n1\u0000ri j\nrc\u00133\nQ(rc\u0000ri j); (6)\nwhere rcis the cutoff and Q(rc\u0000ri j)is the Heaviside step\nfunction, which implies no exchange coupling between spins\nsituated at larger distance than rc.\nSeveral previous SLD models suffered from the fact that\nthey did not allow angular momentum transfer between lattice\nand spin systems28. This happened for magnetisation dynam-\nics in the absence of spin thermostat, governed by symmetric\nexchange only, due to total angular momentum conservation.\nTo enable transfer of angular momentum, Perera et al26have\nproposed local anisotropy terms to mimic the spin-orbit cou-\npling phenomenon due to symmetry breaking of the local en-\nvironment. Their approach was successful in thermalising the\nsubsystems, however, single site anisotropy spin terms with\na position dependent coefficients as employed in26can induce\nan artificial collective translational motion of the sample while\nthe system is magnetically saturated, due to the force \u0000¶Htot\n¶riproportional to spin orientation. To avoid large collective mo-\ntion of the atoms in the magnetic saturated state, we consider\na two-site coupling term, commonly known as the pseudo-\ndipolar coupling, described by\nHc=\u0000å\ni;jf(ri j)\u0014\n(Si\u0001ˆri j)(Sj\u0001ˆri j)\u00001\n3Si\u0001Sj\u0015\n: (7)\nThe origin of this term still lies in the spin-orbit interaction,\nappearing from the dynamic crystal field that affects the elec-\ntronic orbitals and spin states. It has been employed previ-\nously in SLD simulations22,27. It was initially proposed by\nAkhiezer36, having the same structure of a dipolar interac-\ntion, however with a distance dependence that falls off rapidly,\nhence the name pseudo-dipolar interaction. The exchange-\nlike term\u00001\n3Si\u0001Sjis necessary in order to preserve the Curie\ntemperature of the system under different coupling strengths\nand to ensure no net anisotropy when the atoms form a sym-\nmetric cubic lattice. For the mechanical forces, the exchange\nlike term eliminates the anisotropic force that leads to a non-\nphysical uniform translation of the system when the mag-\nnetic system is saturated. The magnitude of the interactions3\nis assumed to decay as f(ri j) =CJ0(a0=ri j)4as presented\nin27with Ctaken as a constant, for simplicity measured rel-\native to the exchange interactions and a0=1˚A is a dimen-\nsion scale factor. The constant Ccan be estimated from\nab-initio calculations26, approximated from magneto-elastic\ncoefficients27, or can be chosen to match the relaxation times\nand damping values, as in this work.\nSince the total Hamiltonian now depends on the coupled\nspin and lattice degrees of freedom ( vi,ri,Si), the following\nequations of motion (EOM) need to be solved concurrently to\nobtain the dynamics of our coupled system:\n¶ri\n¶t=vi; (8)\n¶vi\n¶t=\u0000hvi+Fi\nmi; (9)\n¶Si\n¶t=\u0000gSi\u0002Hi; (10)\nFi=\u0000¶Htot\n¶ri+Gi; (11)\nHi=\u00001\nmSm0¶Htot\n¶Si; (12)\nwhere FiandHirepresent the effective force and field, Girep-\nresents the fluctuation term (thermal force) and hrepresents\nthe friction term that controls the dissipation of energy from\nthe lattice into the external thermal reservoir. The strength of\nthe fluctuation term can be calculated by converting the dissi-\npation equations into a Fokker-Planck equation and then cal-\nculating the stationary solution. The thermal force has the\nform of a Gaussian noise:\nhGia(t)i=0; (13)\nhGia(t)Gjb(t0)i=2hkBT\nmidabdi jd(t\u0000t0): (14)\nTo prove that the complete interacting many-body spin-\nlattice framework presented in here is in agreement with the\nfluctuation-dissipation theorem, we have followed the ap-\nproach presented by Chubykalo et al37based on the Onsager\nrelations. Linearising the equation of motion for spins, we find\nthat the kinetic coefficients for the spin system are zero, due\nto the fact that the spin and internal field are thermodynamic\nconjugate variables. Hence, if the noise applied to the lattice\nobeys the fluctuation dissipation theory, the coupled system\nwill obey it as well, due to the precessional form of the equa-\ntion of motion for the spin.\nWe compare the SLD model presented here with other ex-\nisting model that do not take into account the lattice degrees\nof freedom (Atomistic Spin Dynamics - ASD). Particularly,\nin our case we assume a fixed lattice positions. The summary\nof the comparison is presented in Table I. Atomistic spin dy-\nnamics simulations (ASD)18,20,38,39have been widely used to\nstudy finite size effects, ultrafast magnetisation dynamics and\nnumerous other magnetic phenomena. Here the intrinsic spin\ndamping (the Gilbert damping - aG) is phenomenologically\nincluded. In our case since the lattice is fixed it is assumedModel Lattice Lattice Spin Intrinsic Spin\nthermostat thermostat damping\nSLD Dynamic On Off Phonon\ninduced\nASD Fixed Off On Electronic\nmainly\nTABLE I. Summary comparison of the SLD model developed here\nagainst other spin dynamics models.\nQuantity Symbol Value Units\nExchange23J0 0:904 eV\nrc 3:75 ˚A\nHarmonic potential27V0 0:15 eV\nrc 7:8 ˚A\nMorse potential31D 0:4174 eV\na 1:3885 ˚A\nr0 2:845 ˚A\nrc 7:8 ˚A\nMagnetic moment ms 2:22 mB\nCoupling constant C 0:5\nMass m 55:845 u\nLattice constant a 2:87 ˚A\nLattice damping h 0:6 s\u00001\nTABLE II. Parameters used in the spin-lattice model to simulate BCC\nFe.\nto come from electronic contributions. Consequently, only 3\nequations of motion per atom describing the spin dynamics\nare used:\n¶Si\n¶t=\u0000g\n(1+a2\nG)Si\u0002(Hi+aGSi\u0002Hi) (15)\nwith an additional field coming from the coupling to the\nfixed lattice positions. The temperature effects are introduced\nin spin variables by means of a Langevin thermostat. The spin\nthermostat is modeled by augmenting the effective fields by a\nthermal stochastic field ( Hi=xi\u0000¶H=¶Si) and its proper-\nties also follow the fluctuation-dissipation theorem:\nhxia(t)i=0; (16)\nhxia(t)xjb(t0)i=2aGkBT\ngmSda;bdi jd(t\u0000t0): (17)\nThe characteristics of the above presented models are sum-\nmarised in Table I.\nTo integrate the coupled spin and lattice equations of mo-\ntion we used a Suzuki-Trotter decomposition (STD) method40\nknown for its numerical accuracy and stability. The scheme\ncan integrate non-commuting operators, such as is the case of\nspin-lattice models and conserves the energy and space-phase\nvolume. The conservation of energy is necessary when deal-\ning with microcanonical simulations. Considering the gener-\nalized coordinate X=fr;v;Sgthe equations of motion can be4\nre-written using the Liouville operators:\n¶X\n¶t=ˆLX(t)= (ˆLr+ˆLv+ˆLS)X(t): (18)\nThe solution for the Liouville equation is X(t+Dt) =\neLDtX(t). Hence, following the form of this solution and ap-\nplying a Suzuki-Trotter decomposition as in Tsai’s work41,42,\nwe can write the solution as:\nX(t+Dt) =eˆLsDt\n2eˆLvDt\n2eˆLrDteˆLvDt\n2eˆLsDt\n2X(t)+O(Dt3);(19)\nwhere Ls;Lv;Lrare the Liouville operators for the spin, veloc-\nity and position. This update can be abbreviated as (s,v,r,v,s)\nupdate. The velocity and position are updated using a first\norder update, however the spin needs to be updated using a\nCayley transform43,44, due to the fact that the norm of each\nindividual spin needs to be conserved. Thus we have\neˆLvDtvi=vi+Dt\nmiFi; (20)\neˆLrDtri=ri+Dtvi; (21)\neˆLSDtSi=Si+DtHi\u0002Si+Dt2\n2\u0002\n(Hi\u0001Si)Hi\u00001\n2H2\niSi\u0003\n1+1\n4Dt2H2\ni:(22)\nThe spin equations of motions depend directly on the neigh-\nbouring spin orientations (through the effective field) hence\nindividual spins do not commute with each other. We need to\nfurther decompose the spin system ˆLs=åiˆLsi. The following\ndecomposition will be applied for the spin system:\neˆLs(Dt=2)=eˆLs1(Dt=4):::eˆLsN(Dt=2):::eˆLs1(Dt=4)+O(Dt3)(23)\nTests of the accuracy of the integration have been per-\nformed by checking the conservation of energy within the mi-\ncrocanonical ensemble. To ensure that the spin and lattice\nsub-systems have reached equilibrium, we calculate both the\nlattice temperature (from the Equipartition Theorem) and spin\ntemperature45. These are defined as:\nTL=2\n3NkBå\nip2\ni\n2m;TS=åi(Si\u0002Hi)2\n2kBåiSi\u0001Hi: (24)\nIII. SPIN-LATTICE THERMALISATION\nAs an initial test of our model we look at the thermalisation\nprocess within micro-canonical (NVE) and canonical (NVT)\nsimulations for a periodic BCC Fe system of 10\u000210\u000210 unit\ncells. No thermostat is applied directly to the spin system and\nits thermalisation occurs via transfer of energy and angular\nmomentum from the lattice, i.e. via the magnon-phonon inter-\naction. In the case of the NVE simulations, the energy is de-\nposited into the lattice by randomly displacing the atoms from\nan equilibrium BCC structure positions within a 0 :01˚A radius\nsphere and by initialising their velocities with a Boltzmann\nFIG. 1. NVE (top) and NVT (bottom) simulations for a 10 \u000210\u000210\nunit cell BCC Fe system. The spin system is randomly initialised with\na temperature of 1900 K, while the lattice velocities are initialised\nby a Boltzmann distribution at T=300 K. In both cases we obtain\nequilibration of the two subsystems on the ps timescale.\ndistribution at T=300 K. The spin system is initialised ran-\ndomly in the x\u0000yplane with a constant component of mag-\nnetisation of 0.5 in the out of plane ( z) direction. In the case\nof NVT simulations, the lattice is connected to a thermostat\nat a temperature of T=300 K. The parameters used in the\nsimulations are presented in Table II.\nFig. 1 shows the thermalisation process for the two types of\nsimulation. In both cases the spin system has an initial temper-\nature of T=1900 K due to the random initialisation. For the\nNVE simulations, the two subsystems are seen to equilibrate\nat a temperature of T=600 K, this temperature being depen-\ndent on the energy initially deposited into the system. In the\nNVT simulations, the lattice is thermalised at T=300 K fol-\nlowed by the relaxation of the spin towards the same temper-\nature. In both cases we observe that the relaxation of the spin\nsystem happens on a 100 ps timescale, corresponding to typi-\ncal values for spin-orbit relaxation. The corresponding change\nin the magnetisation is emphasized by the green lines in Fig.\n1 showing a transfer of angular momentum between the spin\nand lattice degrees of freedom.\nTo gain a better understanding of properties at thermal equi-\nlibrium within the Spin-Lattice Dynamics model, we have in-\nvestigated the temperature dependence of the magnetic order\nparameter in different frameworks that either enable or dis-\nable lattice dynamics, specifically: SLD or ASD. Tab. I il-\nlustrates the differences between the models. Since reaching\njoint thermal equilibrium depends strongly on the randomness\nalready present in the magnetic system this process is acceler-\nated by starting with a reduced magnetisation of M=MS=0:95\nFIG. 2. Magnetisation versus temperature curves for the SLD model\n(with different choices of lattice potential: MP-Morse Potential, HP-\nHarmonic Potential) and fixed lattice ASD model. The inset zooms\naround the ferromagnetic to paramagnetic phase transition tempera-\nture.\nforT>300 K.\nFig. 2 shows the comparison of the equilibrium magnetisa-\ntion using either the harmonic potential (HP), Morse potential\n(MP) or fixed lattice (ASD) simulations. The magnetisation\nis calculated by averaging for 200 ps after an initial equilibra-\ntion for 800 ps (for SLD type simulations) or 100 ps (for ASD)\nsimulations. We observe that even without a spin thermostat\n(in SLD model) the magnetisation reaches equilibrium via the\nthermal fluctuations of the lattice, proving that both energy\nand angular momentum can be successfully transferred be-\ntween the two sub-systems. Additionally, both the SLD and\nASD methods give the same equilibrium magnetisation over\nthe temperature range considered. This confirms that the equi-\nlibrium quantities are independent of the details of the thermo-\nstat used, in agreement with the fact that both SLD and ASD\nmodels obey the fluctuation-dissipation theorem.\nIn principle, since the strength of the exchange interaction\ndepends on the relative separation between the atoms, any\nthermal expansion of the lattice could potentially modify the\nCurie temperature. However, as highlighted in the inset of\nFig. 2, the same Curie temperature is observed in each model.\nWe attribute this to fact that the thermal lattice expansion is\nsmall in the temperature range considered due to two reasons:\ni) the Curie temperature of the system is well below the melt-\ning point of Fe (\u00191800K) and ii) we model a bulk, constant-\nvolume system with periodic boundary conditions that does\nnot present strong lattice displacements due to surfaces. We\nnote that Evans et al46found a reduction of TCin nanoparticles\ndue to an expansion of atomic separations at the surface that\nconsequently reduces the exchange interactions. For systems\nwith periodic boundary conditions we anticipate fluctuations\nin the exchange parameter due to changes in interatomic spac-\nings to be relatively small. Although the equilibrium proper-\nties are not dependent on the details of the thermostat or thepotential, the magnetisation dynamics could be strongly influ-\nenced by these choices.\nThe strength of the pseudo-dipolar coupling parameters C\ndetermines the timescale of the thermalisation process. Its\nvalue can be parametrised from magneto-elastic simulations\nvia calculations of the anisotropy energy as a function of\nstrain. The magneto-elastic Hamiltonian can be written for a\ncontinuous magnetisation Mand elastic strain tensor eas47,48:\nHm\u0000e=B1\nM2\nSå\niM2\nieii+B2\nM2så\niMiMjei j (25)\nwhere constants B1;B2can be measured experimentally49.\nThe pseudo-dipolar term acts as a local anisotropy, however,\nfor a lattice distorted randomly, this effective anisotropy is av-\neraged out to zero. At the same time under external strain\neffects, an effective anisotropy will arise due to the pseudo-\ndipolar coupling which is the origin of the magneto-elastic\neffects. To calculate the induced magnetic anisotropy energy\n(MAE), the BCC lattice is strained along the zdirection whilst\nfixed in the xyplane. The sample is then uniformly rotated and\nthe energy barrier is evaluated from the angular dependence of\nthe energy. Fig. 3 shows MAE for different strain values and\ncoupling strengths, with the magneto-elastic energy densities\nconstants B1obtained from the linear fit. The values of the ob-\ntained constants B1are larger than the typical values reported\nforBCC FeB1=\u00003:43 MJ m\u00003=\u00006:2415\u000210\u00006eV A\u0000349\nmeasured at T=300 K. Although the obtained magneto-\nelastic coupling constants for BCC Fe are larger than experi-\nmental values, it is important to stress that, as we will see later,\na large coupling is necessary in order to obtain damping pa-\nrameters comparable to the ones known for magnetic insula-\ntors where the main contribution comes from magnon-phonon\nscattering. In reality, in BCC Fe there is an important contribu-\ntion to the effective damping from electronic sources, which if\nconsidered, can lead to the smaller coupling strengths, consis-\ntent in magnitude with experimental magneto-elastic parame-\nters. Indeed, as we will show later, our finding suggests that\nphonon damping is a very small contribution in metallic sys-\ntems such as BCC Fe .\nIV . DYNAMIC PROPERTIES AT THERMAL\nEQUILIBRIUM\nSection III showed that the equilibrium magnetisation does\nnot depend on the details of the thermostat used and a success-\nful transfer of both energy and angular momentum is achieved\nbetween the spin and lattice sub-systems by the introduction\nof a pseudo-dipolar coupling term. In this section, we inves-\ntigate the properties of the magnons, phonons and the cou-\npling term that equilibrates the spin and phonon systems in\nthe absence of a phenomenological spin damping. Two types\nof simulations are presented here: i)magnon and phonon\nspectra calculated along the high symmetry path of a BCC\nlattice and ii)averaged temporal Fourier transform (FT) of\nindividual atoms datasets (spin, velocity, pseudo-dipolar cou-\npling field). The phonon - Fig. 4 and magnon - Fig. 5 spec-6\nFIG. 3. Magnetic anisotropy energy as function of strain for different\ncoupling strengths for T=0K.\ntra are calculated by initially equilibrating the system for 10\nps with a spin thermostat with aG=0:01 and a coupling of\nC=0:5, followed by 10 ps of equilibration in the absence\nof a spin thermostat. For the method i)the correlations are\ncomputed for a runtime of 20 ps after the above thermalisa-\ntion stage. For each point in k-space, the first three maxima\nof the auto-correlation function are plotted for better visual-\nisation. The auto-correlation function is projected onto the\nfrequency space and the average intensity is plotted for dif-\nferent frequencies. The phonon spectra are calculated from\nthe velocity auto-correlation function defined in Fourier space\nas33,50:\nAp(k;w) =Ztf\n0hvp\nk(t)vp\nk(t\u0000t0)ie\u0000iwtdt (26)\nwhere p=x;y;z,tfis the total time and vp\nk(t)is the spatial\nFourier Transform calculated numerically as a discrete Fourier\nTransform:\nvp\nk(t) =å\nivp\nie\u0000ik\u0001ri (27)\nThe same approach is applied for the magnon spectra, us-\ning the dynamical spin structure factor, which is given by\nthe space-time Fourier transform of the spin-spin correlation\nfunction defined as Cmn(r\u0000r0;t\u0000t0) =<Sm(r;t)Sn(r0;t0)>,\nwith m;ngiven by the x,y,z components51:\nSmn(k;w) =å\nr;r0eik\u0001(r\u0000r0)Ztf\n0Cmn(r\u0000r0;t\u0000t0)e\u0000iwtdt(28)\nThe second method ( ii)) to investigate the properties of the\nsystem involves calculating temporal Fourier transform of in-\ndividual atoms datasets, and averaging the Fourier response\nover 1000 atoms of the system. This response represents an\nintegrated response over the k-space. Hence, the projection of\nintensities on the frequency space presented by method i)has\nsimilar features as the spectra presented by method ii). For theresults presented in Fig. 6, a system of 10 \u000210\u000210BCC unit\ncells has been chosen. The system has been equilibrated for\na total time of 20 ps with the method presented in i)and the\nFast Fourier transform (FFT) is computed for the following\n100 ps.\nFig. 4 shows the phonon spectra for a SLD simulations at\nT=300K, C=0:5 for the Morse Potential - Fig. 4(a) and the\nHarmonic Potential - Fig. 4(b) calculated for the high sym-\nmetry path of a BCC system with respect to both energy and\nfrequency units. The interaction cutoff for both Morse and\nHarmonic potential is rc=7:8˚A. The Morse phonon spec-\ntrum agrees well with the spectrum observed experimentally32\nand with the results from33. The projection of the spectra onto\nthe frequency domain shows a peak close to 10.5 THz, due\nto the overlap of multiple phonon branches at that frequency\nand consequently a large spectral density with many k-points\nexcited at this frequency. Moving now to the harmonic poten-\ntial, parameterised as in Ref. 27, we first note that we observe\nthat some of the phonon branches overlap - Fig.4b). Secondly,\nthe projection of intensity onto the frequency domain shows a\nlarge peak at 8.6THz, due to a flat region in the phonon spec-\ntra producing even larger number of k-points in the spectrum\nwhich contribute to this frequency. Finally, the large cutoff\nmakes the Harmonic potential stiffer as all interactions are\ndefined by the same energy, V0, and their equilibrium posi-\ntions corresponding to a BCC structure. This is not the case\nfor the Morse potential which depends exponentially on the\ndifference between the inter-atomic distance and a constant\nequilibrium distance, r0. For a long interaction range, the har-\nmonic approximation will result in a more stiff lattice than the\nMorse parameterisation.\nIn principle, the harmonic potential with a decreased in-\nteraction cutoff and an increased strength could better repro-\nduce the full phonon spectra symmetry for BCC Fe. How-\never, in this work we preferred to use the parameterisation\nexisting in literature27and a large interaction cutoff for sta-\nbility purposes. Although the full symmetry of the BCC Fe\nphonon spectra is not reproduced by this harmonic potential,\nthe phonon energies/frequencies are comparable to the values\nobtained with the Morse potential. Nevertheless, we observed\nthe same equilibrium magnetisation and damping (discussed\nlater) for both potentials, hence the simple harmonic potential\nrepresents a suitable approximation, that has the advantage of\nbeing more computationally efficient.\nFig. 5 shows the magnon spectrum obtained within the SLD\nframework using the Morse potential together with its pro-\njection onto the frequency domain. The results agree very\nwell with previous calculations of magnon spectra28,52. For\nthe harmonic potential the magnon spectrum is found to be\nidentical to that for the Morse potential with only very small\nchanges regarding the projection of intensity onto the fre-\nquency domain. This is in line with our discussion in the pre-\nvious section where the choice of inter-atomic potential had\nlittle effect on the Curie temperature, which is closely linked\nto the magnonic properties. As the harmonic potential is more\ncomputationally efficient than the Morse, we next analyse the\nproperties of the system for a 10 \u000210\u000210 unit cells system\natT=300K with the harmonic potential.7\nFIG. 4. Phonon spectra calculated for a 32 \u000232\u000232 unit cell system at T=300K, C=0:5 for a) Morse potential, b)Harmonic potential. The\nspectra are calculated via method i).\nRight figure includes the projection of the intensity of the spectra onto the frequency domain. Solid lines are the experimental data of\nMinkiewicz et al32. For the Minkiewicz et al data there is only 1 datapoint for the N- Gpath for the second transverse mode which does not\nshow up on the line plots.\nFIG. 5. Magnon spectrum (x component) calculated for a 32 \u000232\u0002\n32 unit cell system at T=300K, C=0:5 for a Morse potential. The\nspectrum is calculated via method i).\nRight figure includes the projection of the intensity of the spectrum\nonto the frequency domain.\nThe power spectral density (auto-correlation in Fourier\nspace) of the magnon, phonons and coupling field at 300 K\nis shown in Fig. 6 computed using method iidetailed previ-\nously. The amplitude of the FFT spectra of velocities and\ncoupling field has been scaled by 0.12 and 0.05 respectively to\nallow for an easier comparison between these quantities. As\nshown in Fig. 6.a) the coupling term presents both magnon\nand phonon characteristics; demonstrating an efficient cou-\npling of the two sub-systems. The large peak observed at\na frequency of 8 :6 THz appears as a consequence of the flat\nphonon spectrum for a Harmonic potential, as observed in the\nspectrum and its projection onto the frequency domain in Fig.\n4.b). Additionally, Fig. 6.a) can give us an insight into the in-duced spin noise within the SLD framework. The background\nof the FFT of the coupling field is flat for the frequencies plot-\nted here, showing that the noise that acts on the spin is uncor-\nrelated. The inset shows a larger frequency domain where it\nis clear that there are no phonon modes for these frequencies,\nand only thermal noise decaying with frequency is visible. At\nthe same time an excitation of spin modes are visible for fre-\nquencies up to ca .100 THz.\nThe characteristics of the coupling field with respect to the\ncoupling strength for a dynamic (SLD) and fixed lattice simu-\nlations (ASD) are presented in Fig. 6(b). The only difference\nbetween the ASD and SLD simulations is given by the pres-\nence of phonons (lattice fluctuations) in the latter. Since the\nlarge peak at 8 :6 THz is due to the lattice vibrations, it is not\npresent in the ASD simulations. The smaller peaks are present\nin both models since they are proper magnonic modes. With\nincreasing coupling the width of the peaks increases suggest-\ning that the magnon-phonon damping has increased. Moving\ntowards the larger frequency regimes, Fig. 6.b) - (inset), we\nobserve that large coupling gives rise to a plateau in the spec-\ntra at around 150 THz, which is present as well for the fixed-\nlattice simulations (ASD). The plateau arises from a weak an-\ntiferromagnetic exchange that appears at large distances due to\nthe competition between the ferromagnetic exchange and the\nantiferromagnetic exchange-like term in the pseudo-dipolar\ncoupling.\nWe have also analysed the characteristics of the magnon\nand phonon spectra for different temperatures- Fig. 7. With8\nFIG. 6. The power spectral density of the auto-correlation function in the frequency domain for magnons, phonons and coupling field for a\nSLD simulations with a Harmonic lattice, calculated by method ii). Panel a) shows the power density of the auto-correlation function of the\nx component of the velocity vx, spin Sxand coupling field Hcx. Panel b) presents the power density of the auto-correlation function for the x\ncomponent of the coupling field for either static (ASD) or dynamic (SLD) lattice. The insets show the high-frequency spectra. For Panel a) the\nvelocity and the coupling field have been multiplied by a factor of 0.12 and 0.05 respectively for easier graphical comparison.\nFIG. 7. The power spectral density of the auto-correlation function in the frequency domain for magnons - Panel a) and phonons - Panel b) for\na SLD simulations with a Harmonic lattice, calculated by method ii), for three distinct temperatures and a coupling constant of C=0:5.\nincreasing temperature, the peaks corresponding to magnons\nshift to smaller frequencies. This is a typical situation known\nas a softening of low-frequency magnon modes due to the in-\nfluence of thermal population, see e.g.53- Panel a). The same\neffect can be observed by calculating the magnon spectra via\nmethod ifor various temperatures. In Panel b), the peak cor-\nresponding to phonons remains almost at the same frequency\nof about 8 :6 THz, as the phonon spectra is not largely affected\nby temperatures up to T=600K. The increase of the effec-\ntive damping (larger broadening) of each magnon mode with\ntemperature is clearly observed.\nV . MACROSOPIC MAGNETISATION DAMPING\nIn this section we evaluate the macroscopic damping pa-\nrameter experienced by magnetisation due to the magnon-\nphonon excitations for a periodic BCC system using our SLD\nmodel. This method for calculating the damping has been\npresented in54–56. The system is first thermalised at a non-\nzero temperature in an external field of Bext=50T applied in\nthezdirection, then the magnetisation is rotated coherently\nthrough an angle of 30\u000e. The system then relaxes back to\nequilibrium allowing the relaxation time to be extracted. The\naveraged zcomponent of magnetisation is then fitted to the\nfunction mz(t) =tanh(agBext(t+t0)=(1+a2))where arep-resents the macrosopic (LLG-like) damping, gthe gyromag-\nnetic ratio and t0a constant related to the initial conditions.\nThe model system consists of 10 \u000210\u000210 unit cells and the\ndamping value obtained from fitting of mz(t)is averaged over\n10 different simulations.\nFig. 8 shows the dependence of the average damping pa-\nrameter together with the values obtained from individual\nsimulations for different temperatures and coupling strengths\nfor two choices of mechanical potential. In our model, the\nspin system is thermalised by the phonon thermostat, hence\nno electronic damping is present. With increasing coupling,\nthe energy and angular momentum transfer is more efficient,\nhence the damping is enhanced. Since the observed value of\ninduced damping is small, calculating the damping at higher\ntemperature is challenging due to the strong thermal fluctua-\ntions that affect the accuracy of the results. Despite the low\ntemperatures simulated here, the obtained damping values (at\nT=50K, a=4:9\u000210\u00005) are of the same order as reported\nfor magnetic insulators such as YIG (1 \u000210\u00004to 1\u000210\u0000657,58\n) as well as in different SLD simulations (3 \u000210\u00005,27). Gener-\nally, the induced damping value depends on the phonon char-\nacteristics and the coupling term, that allows transfer of both\nenergy and angular momentum between the two subsystems.\nFig. 8(a) and (b) compare the calculated damping for the\nMorse and Harmonic potential for two values of the coupling\nstrength. We observe that the values are not greatly affected9\nFIG. 8. Damping parameter extracted from fitting the z component of the magnetisation for two different choices of potential: HP- Harmonic\nPotential (green open squares) and MP-Morse Potential (black open circles) as function of temperature Fig. a), b) and as function of the\ncoupling strength Fig. c), d); Fig a) and b) are calculated for a constant coupling strength of C=0:3,C=0:5 respectively. Fig c) and d)\nare calculated for temperatures of T=100K,T=300Krespectively. The black and green lines represents the average damping parameter\nobtained from the simulations using the Morse and the Harmonic Potentials, respectively.\nby the choice of potential. This arises due to the fact that only\nthe spin modes around Gpoint are excited and for this low k-\nvectors modes the inter-atomic distances between neighbour-\ning atoms do not vary significantly. The extracted damping\nparameter as a function of coupling strength for 100 K and\n300 K is presented in Fig. 8(c) and (d) respectively. The func-\ntional form of the variation is quadratic, in accordance with\nthe form of the coupling term. Measurements of damping in\nmagnetic insulators, such as YIG, show a linear increase in the\ndamping with temperature,58which agrees with the relaxation\nrates calculated by Kasuya and LeCraw59and the relaxation\nrates calculated in the NVE SLD simulations in Ref. 27. How-\never, Kasuya and LeCraw suggest that the relaxation rate can\nvary as Tn, where n=1\u00002 with n=2 corresponding to larger\ntemperature regimes. Nevertheless, the difference between\nthe quadratic temperature variation of the damping observed\nin our simulations and the linear one observed in experiments\nfor YIG can be attributed to the difference in complexity be-\ntween the BCC Fe model and YIG. The difference between\nthe trends may as well suggest that the spin-orbit coupling in\nYIG could be described better by a linear phenomenological\ncoupling term, such as the one used in Refs. 26 and 29, but\nwe note that such forms can lead to a uniform force in the di-\nrection of the magnetisation and so might need further adap-\ntation before being suitable. To test an alternate form of the\ncoupling we have changed the pseudo-dipolar coupling to an\non-site form, specifically Hc=\u0000åi;jf(ri j)((Si\u0001ˆri j)2\u00001\n3S2\ni)\ni.e a N ´eel-like anisotropy term. This leads to much smallerdamping as shown in Fig. 9 ( T=300 K, a=3:3\u000210\u00005,\naveraged over 5 realisations) making it difficult to accurately\ncalculate the temperature dependence of the damping, espe-\ncially for large temperatures. The magnon-phonon damping\ncan clearly have complex behavior depending on the proper-\nties of the system, especially the coupling term, hence no uni-\nversal behaviour of damping as function of temperature can\nbe deduced for spin-lattice models.\nNeglecting the lattice contribution, the temperature depen-\ndence of the macrosopic damping can be mapped onto the\nLandau-Lifshitz-Bloch formalism (LLB)54and theory17and\nASD simulations60have shown it to vary inversely with the\nequilibrium magnetisation. The LLB theory shows that the\nmacrosopic damping is enhanced for large temperatures due\nto thermal spin fluctuations. Using the equilibrium magneti-\nsation it is possible to approximate the variation of damping\nwith temperature produced due to thermal fluctuations within\nthe LLB model. From 100K to 400K the damping calculated\nvia the LLB model increases within the order of 10\u00005, which\nis considerably smaller than the results obtained via the SLD\nmodel. This shows that within the SLD model the temperature\nincrease of the damping parameter is predominantly due to\nmagnon-phonon interaction, and not due to thermal magnon\nscattering, as this process is predominant at larger tempera-\ntures.10\nFIG. 9. Temperature variation of the damping parameter for N ´eel-\nlike on-site coupling, Hc=\u0000åi;jf(ri j)((Si\u0001ˆri j)2\u00001\n3S2\ni). The val-\nues are extracted from mz(t)fittings for 10 realisations;\nVI. CONCLUSIONS AND OUTLOOK\nTo summarise, we have developed a SLD model that is able\nto transfer energy and angular momentum efficiently from\nthe spin to lattice sub-systems and vice-versa via a pseudo-\ndipolar coupling term. Our approached takes the best fea-\ntures from several previously suggested models and general-\nize them which allows modelling in both canonical and mi-\ncrocanonical ensembles. With only the lattice coupled to\na thermal reservoir and not the spin system, we reproduce\nthe temperature dependence of the equilibrium magnetisation,\nwhich agrees with the fact that the spin-lattice model obeys\nthe fluctuation-dissipation theorem. We are able to study the\ndynamic properties such as phonon and spin spectrum and\nmacrosopic damping, showing that the magnetic damping isnot greatly influenced by the choice of potential, however it\nis influenced by the form of the coupling term. This enables\nthe possibility of tailoring the form of the coupling term so it\ncan reproduce experimental dependencies of damping for dif-\nferent materials. In future, the addition of quantum statistics\nfor Spin Lattice Dynamics models61,62may also yield better\nagreement with experimental data.\nThe SLD model developed here opens the possibility of the\ninvestigation of ultrafast dynamics experiments and theoret-\nically studies of the mechanism through which angular mo-\nmentum can be transferred from spin to the lattice at ultrafast\ntimescales. As we have demonstrated that the model works\nwell in the absence of an phenomenological Gilbert damping,\nwhich consists mainly of electronic contributions, the SLD\nmodel can be employed to study magnetic insulators, such\nas YIG, where the principal contribution to damping is via\nmagnon-phonon interactions. Future application of this model\nincludes controlling the magnetisation via THz phonons7\nwhich can lead to non-dissipative switching of the magnetisa-\ntion11,12. With the increased volume of data stored, field-free,\nheat-free switching of magnetic bits could represent the future\nof energy efficient recording media applications. Another pos-\nsible application is more advanced modelling of the ultrafast\nEinstein-de-Haas effect2or phonon-spin transport63.\nVII. ACKNOWLEDGEMENTS\nWe are grateful to Dr. Pui-Wai Ma and Prof. Matt\nProbert for helpful discussions. Financial support of the Ad-\nvanced Storage Research Consortium is gratefully acknowl-\nedged. MOAE gratefully acknowledges support in part from\nEPSRC through grant EP/S009647/1. The spin-lattice simula-\ntions were undertaken on the VIKING cluster, which is a high\nperformance compute facility provided by the University of\nYork. 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Chen3,8,9,1,2,5 \n \n1 WPI Advanced Institute for Material s Research (AIMR), Tohoku University \nSendai 980-8577, Japan \n2 Center for Science and Innovation in Spintronics (CSIS), Tohoku University \nSendai 980-8577, Japan \n3 Purdue Quantum Science and Engineering Ins titute and Department of Physics and Astronomy, \nPurdue University, West Lafayette, Indiana 47907, USA \n4 Frontier Research Institute for Interdiscipli nary Sciences (FRIS), Tohoku University, Sendai \n980-8578, Japan \n5 Center for Spintronics Research Network (CSRN), Tohoku University, Sendai 980-8577, Japan \n6 Graduate School of Environmental Studies, Tohoku University, Sendai 980-8579, Japan \n7 WPI-International Center fo r Materials Nanoarchitectonics (MANA), National Institute for \nMaterial Science, Tsukuba 305-0044, Japan \n8 School of Electrical and Computer Engi neering and Birck N anotechnology Center \nPurdue University, West Lafayette, Indiana 47907, USA \n9Institute of Physics and Astronomy and Villum Center for Hybrid Quantum Materials and \nDevices, Aarhus University, 8000, Aarhus-C, Denmark \n \n           \n \n*idzuchi@tohoku.ac.jp \n#) H. Idzuchi and S. Iihama cont ributed equally to this work.  2Abstract \nSpin transport characteristics of graphene has been extensively  studied so far. The spin transport along c-\naxis is however reported by rather limited number of papers. We  have studied spin transport characteristics \nthrough graphene along c-axis with permalloy(Py)/graphene(Gr)/Pt by gigahertz (GHz) and  terahertz (THz) \nmagnetization dynamics driven by femtosecond laser pulses. The relatively simple sample structure does \nnot require electrodes on the sample. The graphene layer was pr epared by chemical vapor deposition and \ntransferred on Pt film. The quality of graphene layer was chara cterized by Raman microscopy. Time \nresolved magneto-optical Kerr effect is used to characterize gi gahertz magnetization dynamics. \nMagnetization precession is clearly observed both for Pt/Py and  Pt/Gr/Py. The Gilbert damping constant of \nPt/Py was 0.015, indicates spin pumping effect from Py to Pt. T he Gilbert damping constant of Pt/Gr/Py is \nfound to be 0.011, indicates spin injection is blocked by graph ene layer. We have also performed the \nmeasurement of THz emission for Pt/Py and Pt/Gr/Py. While the T Hz emission is clearly observed for Pt/Py, \na strong reduction of THz emission is observed for Pt/Gr/Py. Wi th these two different experiments, and \nhighly anisotropic resistivity of graphite, we conclude that th e vertical spin transport is strongly suppressed \nby the graphene layer.       3Recently, two-dimensional (2D) materials have attracted conside rable attention. Two-dimensional \nmaterials provide handful access on highly crystalline samples,  offering new spintronics research directions. \nSince spin currents can flow in nonmagnetic materials, so far s uch spin transport is widely studied in in-\nplane direction of nonmagnetic 2D materials [1,2,3]. In three-d imensional materials such as Pt, spin \ntransport in out-of-plane direction is often studied with spin Hall effect. The spin current is converted to \ncharge transport with certain geometry: the spin polarization, the flow direction of spin current and the \ndirection of detecti on voltage need to be a ll perpendicular to each other. This geometrical constraint makes \nit difficult to study out-of-plane spin transport through c-axis of 2D material while the spin transport in-\nplane has been relatively well studied. Here, we optically inve stigated spin transport characteristics in c-axis \nof graphene by using gigahertz ( GHz) and terahertz (THz) magnet ization dynamics excited by a \nfemtosecond pulse laser. This makes it more easily to satisfy s uch geometrical conditions as the injector and \ndetector are not required to be electrically connected.  \nFigure 1 represents our sample structure as well as a brief mea surement set up. Here, we employed \nspin pumping and THz emission, both induced by magnetization dy namics excited by a pulsed laser as \ndescribed below. Previously, vertical spin transport in multila yers of graphene have been studied by ferro \nmagnetic resonance. Patra et al fabricated the sample on a co-p laner wave guide and used broad band \nfrequency to characterize spin transport. They found the Gilber t damping is significantly enhanced for Py/Gr \ncompared to Py/Pt where Py stands for permalloy (Ni 80Fe20) [4]. Later Gannett et al studied series of the \nsamples with different thickness of Py to characterize transpor t properties, which shows no detectable \nenhancement for Py/Gr/Cu [5]. While the interface of graphene c an be complicated, in our experiment the \nsample structure is simple (just a multilayer film) and complim entary characteristics are obtained by two \nmethods, which should help reveal t he intrinsic interface spin transport properties.  \n In this study, spin transport was studied on Pt/graphene/Py an d Pt/Py . Pt, graphene, and Py were \nchosen for representative materials for spin Hall effect, 2D ma terial, and soft ferromagnet. Pt film was \nprepared by sputtering with the thickness of 3 nm on silicon su bstrate and glass substrate. The graphene \nfilm was transferred onto Pt in ambient condition. Graphene fil m was prepared on thin copper foil by a \nstandard chemical vapor deposition (CVD) method and transferred  onto the Pt film. Raman microscopy \nwas used to characterize the number of layers in graphene where  the laser wavelength is 532 nm. We have \nobserved clear peaks of D, G, and 2D bands from left to right a s shown in Fig.2a. Particularly from the 2D \npeak, we confirmed the crystallinity of the graphene film and t he film was not folded [6,7]. The Py film and \nMgO capping layer was sputtere d on graphene film with the base pressure of ~ 10-7 T o r r .  T h e  s t a t i c  \nmagnetization process of the film was examined by magneto optic al Kerr effect. For measuring GHz \nmagnetization dynamics induced by femtosecond laser pulse (Fig.  1a), time-resolved magneto-optical Kerr \neffect (TRMOKE) was employed [8]. The wavelength, pulse duratio n, and repetition rate for both the pump \nand probe laser pulses were 800 nm , 120 fs, and 1 kHz, respecti vely. The pump laser beam was irradiated \non the sample from the film normal and the incident angle of th e probe laser beam was ~ 5 degree measured \nfrom the film normal. Kerr rotation angle of the reflected prob e beam was detected via balanced photo- 4detector. The pump laser pulse was modulated by the mechanical chopper with the frequency of 360 Hz \nand then the pump-laser induced change in Kerr rotation angle w as detected by a Lock-in amplifier. A \nmagnetic field was applied with a  10 degree angle measured from  the film normal. The magnetization \nprecession can be excited by the reduction of demagnetizing fie ld due to laser heating. The damping of \nmagnetization precession reflects the transfer of spin into adj acent normal metal layer, referred as spin-\npumping effect [9, 10]. To study spin-transport induced by THz magnetization dynamics, THz time-domain \nspectroscopy was employed [11] (Fig. 1b), in which THz spin-cur rent can be generated by ultrafast \ndemagnetization of Py layer and its angular momentum can be tra nsferred to Pt layer [12,13]. Then, THz \nelectric field can be generated through spin-to-charge conversi on (inverse spin Hall effect) in Pt layer. \nWavelength, pulse duration, repetition rate for the laser pulse  were 800 nm, 120 fs, and 80 MHz, respectively. \nThe femtosecond laser was irradiated from substrate side and th en THz electric field emitted from the film \nside was measured. The THz electric field was detected by elect ro-optic sampling method using a ZnTe \n(110) crystal. All measurements  were conducted at room temperat ure. \n   The spin transport of graphene in vertical direction can be accessed by spin pumping with \nadditional layers. We compare Pt/P y bilayer with Pt/graphene/Py  trilayers to characterize spin transport \nproperties across graphene layer. Figure 2b shows typical TRMOK E signal for Pt/Gr/Py(10nm)/MgO on Si \nsubstrate under the external magnetic field of 10.7 kOe in a di rection tilted by 10 degrees from perpendicular \nto the substrate. We have clearly observed spin precession slow ly decaying over long period right after \ninitial ultrafast dynamics, for the samples of both with and wi thout graphene layer. Those oscillations are \nfitted to the following equations \n𝐴𝐵⋅e x p ሺെ𝑣𝑡 ሻ𝐶⋅e x p ሺെ𝑡 𝜏⁄ሻsinሺ2𝜋𝑓𝑡  𝜙 ሻ \nwhere, A, B, , C, f, , and 0 are signal offset, magnitude of exponential background signal due to recovery \nof magnetization, decay rate, oscillation amplitude, oscillatio n frequency, oscillation life-time, and initial \nphase, respectively. The TRMOKE signals are well fitted by the above equation shown as solid curve in \nFig. 2(b). The f and 1/ values evaluated by fitting with different applied magnetic fi elds are shown in Fig. \n3. f and 1/ can be calculated theoretically  using Landau-Lifshitz Gilbert (LLG) equation as [8,14], \n𝑓ୋ ൌఊ\nଶగඥ𝐻ଵ𝐻ଶ,    ( 1 )  \nଵ\nఛైైృൌଵଶ𝛼𝛾ሺ𝐻ଵ𝐻 ଶሻ,     ( 2 )  \n𝐻ଵൌ𝐻c o s ሺ𝜃െ𝜃 ுሻെ4 𝜋 𝑀 ୣcosଶ𝜃,  (3) \n𝐻ଶൌ𝐻c o s ሺ𝜃െ𝜃 ுሻെ4 𝜋 𝑀 ୣcos 2𝜃 ,  (4) \nwhere H, H (=10 degree in this study), , 4Meff, , and  are external magnetic f ield, field angle, \nmagnetization angle, effective de magnetizing field, gyromagneti c ratio and Gilbert damping constant \nrespectively.  is given by the relation,  =gB/ℏ. The  is determined by the energy minimum condition as,  5𝐻s i n ሺ 𝜃 ுെ𝜃 ሻ2 𝜋 𝑀 ୣsin 2𝜃 ൌ 0 ,  (5) \nThe measured f is well fitted by Eq. (1) with the parameters g = 2.09 (2.06) and 4 Meff = -9.8 (-8.8) kOe for \nPt / Gr / Py (Pt / Py) film. The 1/  calculated using Eq. (2) are sh own in Fig. 3(b) and 3(c). 1/  for Pt / Gr / \nPy / MgO sample (Fig. 3(c)) can be well explained by Eq. (2) wi th  = 0.011. However, 1/  for Pt / Py / \nMgO cannot be explained by Eq. (2), which is due to inhomogeneo us linewidth broadening. Therefore, 1/  \nenhancement due to inhomogeneous linewidth broadening is consid ered as follows, \nଵ\nఛ౪౪ൌଵ\nఛైైృଵଶቚௗሺଶగ ైైృ ሻ\nௗఏಹቚΔ𝜃ு ,    ( 6 )  \nwhere, the first term is identi cal to Eq. (2) and the second te rm is 1/ enhancement due to distribution of H \nwhich may be related to surface roughness of the film [14]. The  black solid and blue dashed curves in Fig. \n3(b) are the calculated results of the first and second terms o f Eq. (6). Hext dependence of 1/  for Pt / Py / \nMgO filmis well explained by the summation of two contributions  in Eq. (6) with the parameters,  = 0.015 \nand H = 0.05 rad [green broken curv e in Fig. 3(b)]. The enhancement of  is due to spin-pumping effect \nat Pt / Py interface associated with dissipation of angular mom entum. This indicates strong suppression of \nspin current with graphene, cons istent with Gannett et al [5]. Previously, it was shown that graphene has \nlong spin diffusion length by means of lateral spin transport w here spin current is flowing in-plane with \nlong spin lifetime probed by Hanle effect [1,15]. Transport alo ng c-direction can be rather different from \nthe one in ab plane. In early studies, graphite crystal shows rather anisotr opic charge transport properties \nand the resistivity of c-axis is reported to larger than the one for ab plane by a factor of 102 to 103 [16]. The \nresistive nature of the graphene along c axis may prevent spin transport.      \nIn the THz method, by irradiating femtosecond laser pulse on th is kind of multilayers, THz electric \nfield can be generated [12, 13]. Ultrafast spin current in nonm agnetic layer can be generated by the ultrafast \ndemagnetization in Py layer, and spin-charge conversion via inv erse spin-Hall effect in Pt layer create \nterahertz charge current and electric field. In our bilayer Pt/ Py, we observed clear THz emission and its \nsignal is inverted with reversed bias magnetic field, as shown in the top panel (a) of Fig. 4. Interestingly, on \ntwo Pt/graphene/Py samples, the THz signal was largely suppress ed [Fig. 4(b) and 4(c)]. This implies strong \nsuppression of spin injection from Py to Pt by graphene monolay er in the terahertz frequency region. The \ninterpretation is qualitatively c onsistent with spin pumping st udy (using ultrafast laser heating and GHz \nmagnetization dynamics). The strong reduction of THz signal is attributed to the strong suppression of spin-\ntransport by inserting graphene monolayer with high resistivity  along the c-axis. Strong reduction of THz \nemission was also reported for Co/ZnO/Pt junctions[17]. With th ese two different characterization methods, \nwe conclude graphene monolayer effectively blocks vertical spin  current. For the second sample of \nPt/graphene/Py (Fig.4c), a small peak appeared around 1 ps. We notice this may or may not be a delayed \nTHz emission, whose precise mechanism (e.g., how it may be rela ted to the graphene barrier, whether it \nmay also be generated by Py itself etc.) is not clear yet at th is stage and open for future study.  \nIn conclusion, we have investigat ed spin injection characterist ics of Py/graphene/Pt by means of  6gigahertz and terahertz magneti zation dynamics driven by a femt osecond laser pulse. Graphene layer was \ngrown by CVD method and the Raman characteristics on Pt showed the characteristics of single layer \ngraphene film. We have clearly observed GHz magnetization prece ssion induced by the laser pulse for the \nsamples of both with and without graphene (Py/Pt). Graphene is observed to give an apparent suppression \nof the damping enhancement due to spin-pumping effect at Py / P t interface, indicating reduction of angular \nmomentum dissipation by graphene monolayer. THz emission induce d by femto-second laser pulse was \nobserved for Py/Pt bilayer, while the THz emission was strongly  suppressed for Py/graphene/Pt, which clear \nindicates graphene blocks spin current in transport along c-axis. Both experiments on spin pumping and \nTHz method can be understood by th e strongly suppressed spin tr ansport across the graphene layer.    \n Data Availability Statements The data that support the findings of this study are available from the corresponding author upon reasonable \nrequest.  \nAcknowledgments We acknowledge the support from AIMR common equipment unit. Thi s work was supported in part by \nAdvanced Institute for Materials R esearch (AIMR) under World Pr emier International Research Center \nInitiative (WPI) of MEXT, Japan, and by AIMR fusion research pr ogram, by the Mazda Foundation, and \nby a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and \nTechnology (MEXT), JSPS KAKENHI (G rant Number 18H 03858, 18H0447 3, 20H04623, and 20K14399). \n  7References  \n[1] N. Tombros et al., Nature  448, 571 (2007). \n[2] Y. P. Liu et al.,  Appl. Phys. Lett.  102, 033105 (2013). \n[3] H. Idzuchi, A. Fert, Y. Otani, Phys. Rev. B  91, 241407 (2015). \n[4] A. K. Patra et al., Appl. Phys. Lett.  101 162407 (2012). \n[5] W. Gannett et al., J. Appl. Phys.  117 213907 (2015). \n[6] L. M. Malard et al., Phys. Rep.  473 51 (2009). \n[7] A. C. Ferrari, D. M. Basko, Nat. Nanotech.  8, 235 (2013). \n[8] S. Iihama et al.,  Phys. Rev. B  89, 174416 (2014). \n[9] Y. Tserkovnyak et al., Phys. Rev. Lett.  88, 117601 (2002). \n[10] S. Mizukami et al., Phys. Rev. B  66, 104413 (2002). \n[11] Y. Sasaki et al., Appl. Phys. Lett.  111, 102401 (2017). \n[12] T. Kampfrath et al., Nat. Nanotech.  8 256 (2013). \n[13] T. Seifert et al., Nat. Photon.  10 485 (2016). \n[14] S. Mizukami et al., Jpn. J. Appl. Phys.  40, 580 (2001). \n[15] D. Khokhriakov et al., Carbon  161, 892 (2020). \n[16] W. Primak and L. H. Fuchs, Phys. Rev.  95 22 (1954). \n[17] G. Li et al., J. Phys. D: Appl. Phys.  51, 134001 (2018). \n    8Figures \n \nFig. 1.  Concept and schematic image of experimental set up of this stu dy. Graphene, spin injector (Py) and detector \n(Pt) are depicted in black hexagon, purple box, and gray box, r espectively. (a) The set up for pump-probe \nmeasurement for magnetization dynamics. The probe beam is sligh tly (~ 5 deg) tilted from the film normal. The \nmagnetic field is applied with a 10 degree angle measured from the film normal. (b) The set up for THz emission. \nThe magnetic field is applied in plane.  \n \n  \nFig.1   9 \nFig. 2. (a) Raman spectroscopy of a graphene layer transferred on to a glass/Pt substrate (glass with Pt sputtered). \nThree main Raman peaks (D, G, and 2D) are clearly observed. Ins et shows the 2D peak, clearly different from bilayer \nor multilayers graphene [6].  (b) Magneto-optical Kerr effect (MOKE ) signal plotted as a function  of pump-probe \ndelay time for Pt/Gr/Py(10nm)/MgO under an external magnetic fi eld of 10.7 kOe. The field is applied with a \n10degree angle measured from the film normal. The measurement s et up is schematically shown in Fig.1a. \n \n \n  \n  \nFig.2   10 \n \nFig. 3. Characterization of GHz magnetization dynamics for multilayers with and without graphene. The \nmeasurement set up is schemati cally shown in Fig.1a. (a) preces sion frequency as a function of the magnetic field. \nThe field is applied with a 10 degree angle measured from the f ilm normal. Closed red squares and open green circles \nindicate data from the Pt/Gr/Py/MgO and Pt/Py/MgO respectively where the thickness of Py is 10 nm for both case. \nThe solid curves are obtained by Eq. (1) with the parameters in  the main text. Inverse lifetime as a function of the \nmagnetic field for (b) Pt / Py / MgO and (c) Pt / Gr / Py / MgO  films. The black solid curves shown in (b) and (c) \ncorrespond to the calculated valu e using LLG eq. (Eq. (2)). The  green dotted and blue broken curves shown in (b) \nare the left hand side and the second term in the right hand si de in Eq. (6), respectively, with the parameters in the \nmain text.      \nFig.3    11  \nFig. 4. Detection of THz electric field generated by femto-second laser  pulse on (a) Pt(3) / Py(2) /MgO , (b) Pt(3) / \nGr / Py(2) / MgO and (c) Pt(3) / Gr / Py(5) / MgO made on glass  substrates. The measurement set up is schematically \nshown in Fig.1b. The numbers in bracket indicate the thickness of the layers in the unit of nanometers. The magnetic \nfield was applied in in-plane direction. Blue solid and red ope n symbols are the signal obtained with opposite \nmagnetic field direction. \n \nFig.4   "    },    {        "title": "2011.05566v1.Reduction_of_back_switching_by_large_damping_ferromagnetic_material.pdf",        "content": "arXiv:2011.05566v1  [cond-mat.mes-hall]  11 Nov 2020Reduction of back switching by large damping ferromagnetic material\nTomohiro Taniguchi1, Yohei Shiokawa2, and Tomoyuki Sasaki2\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nResearch Center for Emerging Computing Technologies, Tsuk uba, Ibaraki 305-8568, Japan,\n2TDK Corporation, Advanced Products Development Center, Ic hikawa, Chiba 272-8558, Japan\nRecent studies on magnetization dynamics induced by spin-o rbit torque have revealed a weak\ndependence of the critical current for magnetization switc hing on the damping constant of a ferro-\nmagnetic free layer. This study, however, reveals that the d amping constant nevertheless plays a key\nrole in magnetization switching induced by spin-orbit torq ue. An undesirable switching, returning\nto an initial state, named as back switching, occurs in a ferr omagnet with an easy axis parallel\nto the current direction. Numerical and theoretical analys es reveal that back switching is strongly\nsuppressed when the damping constant of the ferromagnet is l arge.\nLow-damping ferromagnetic materials has been inves-\ntigated for spintronics applications [1–6]. These materi-\nals are interesting because the critical current for excit-\ning magnetization dynamics by spin-transfer torque[7, 8]\nin two-terminal devices is typically proportional to the\ndampingconstant[9,10], andtherefore, low-dampingfer-\nromagnetic materials help to reduce power consumption.\nThe value of the damping constant in typical ferromag-\nnets, such as Fe, Co, Ni, and their alloys, is on the order\nof 10−3−10−2, even after the effect of spin pumping\n[11, 12] is taken into account [5, 6]. Studies on three-\nterminal devices manipulated by spin-orbit torque [13–\n24], however, have revealed that the dependence of the\ncritical current on the damping constant is weak when\nthe easy axis of the ferromagnet is perpendicular to the\nfilm plane [25, 26] or parallelto the currentdirection [27];\nthese systems are named type Z and type X, respectively,\nin Ref. [19]. Then, a question arises as to what the role\nof the damping constant is in three-terminal devices.\nThe purpose of this study is to investigate the rela-\ntion between the magnetizationstate manipulated by the\nspin-orbit torque and the magnetic damping constant by\nsolving the Landau-Lifshitz-Gilbert (LLG) equation. We\nfocus onthe type-Xsystembecause the switchingmecha-\nnism in the system is not fully understood yet, unlike the\nothersystems[9,10,25,26]. Anundesirablereturnofthe\nmagnetization,calledbackswitchinginthispaper,occurs\nin a ferromagnet, where, although the spin-orbit torque\nbrings the magnetization close to the switched state, it\nreturns to the initial state after turning off the current.\nAs a result, the phase diagramofthe magnetization state\nas a function of the current and external magnetic field\nalternately shows switched and non-switched states. It\nis revealed that the back switching appears as a result of\nthe magnetization precession around the perpendicular\naxis after the current is turned off. The back-switching\nregion is strongly suppressed by using large-damping fer-\nromagnetic materials.\nThe system studied in the work is an in-plane mag-\nnetized ferromagnet placed on a nonmagnetic metal and\nschematically shown in Fig. 1(a). Electric current flow-\ning in the nonmagnet in the xdirection generates pure\nspincurrentthatisinjectedintotheferromagnetwiththez\nxy\njm\nHappl\nj=300 MA cm-2Happl=500 Oemx1\n0\n-1\nTime (ns)0 1(a) (b)\nFIG. 1: (a) Schematic illustration of the system. The spin\nHall effect in the bottom nonmagnet generates spin current\nwith a polarization along the ydirection and excites the mag-\nnetization ( m) dynamics in the top ferromagnet. The easy\naxis of the ferromagnet is parallel to the xaxis. An external\nmagnetic field Happlis applied in the zdirection. (b) Exam-\nple of the time evolution of mxforj= 300 MA cm−2and\nHappl= 500 Oe.\nspin polarization in the ydirection. The spin current ex-\ncitesthespin-transfertorqueactingonthemagnetization\nin the ferromagnet and induces magnetization dynamics.\nThezaxis is perpendicular to the film plane. We denote\nthe unit vectors pointing in the magnetization direction\nin the ferromagnet and along the k-axis (k=x,y,z) as\nmandek, respectively. The magnetization dynamics are\ndescribed by the LLG equation\ndm\ndt=−γm×H−γHsm×(ey×m)+αm×dm\ndt,(1)\nwhereγandαarethe gyromagneticratioandthe Gilbert\ndamping constant, respectively. The magnetic field H\nconsists of the in-plane magnetic anisotropy field HKin\nthexdirection, the demagnetization field −4πMin thez\ndirection, and an external field Happlin thezdirection:\nH=HKmxex+(Happl−4πMmz)ez.(2)\nThe strength of the spin-transfer torque is given by\nHs=/planckover2pi1ϑj\n2eMd, (3)\nwhereϑis the spin Hall angle in the nonmagnet, whereas\nManddare the saturation magnetization and thickness\nof the ferromagnet. The electric current density is de-\nnoted as j. The values of the parameters used in this2\n\"mx.dat\" u 1:2:3 \"mx_relax.dat\" u 1:2:3 (a)\nApplied field, Happl (kOe)Current density, j (MA cm -2 )\n0 1 2 0 1 202004006008001000 1\n0\n-1 mx\nApplied field, Happl (kOe)1\n0\n-1 mx\nS0 S1 S2\njc(b)\nCurrent density, j (MA cm -2 )\n02004006008001000\nFIG. 2: Phase diagrams of mx(a) at a fixed point in the\npresence of current and (b) in a relaxed state after turning o ff\nthe current. The initial state is m0−. The white dotted line in\n(a) represents the theoretical formula of the critical curr ent\ndensity derived in Ref. [27]. Switching regions are distin-\nguished by the labels S n(n= 0,1,2) in (b). The damping\nconstant αis 0.005.\nstudy were taken from typical experiments [13–23] as\nM= 1500 emu c.c.−1,HK= 200 Oe, γ= 1.764×107\nrad Oe−1s−1,ϑ= 0.4, andd= 1.0 nm.\nThe magnetic field His related to the magnetic energy\ndensityEviaE=−M/integraltext\ndm·H,\nE=−MHapplmz−MHK\n2m2\nx+2πM2m2\nz.(4)\nThe energy density Ehas two minima at m0±=\n±/radicalbig\n1−m2\n0zex+m0zez, wherem0z=Happl/(HK+4πM).\nThroughout this paper, the initial state is set to be m0−,\nwhich points in the negative xdirection. Accordingly, we\ncall the other stable state, m0+pointing in the positive x\ndirection, the switched state. Thus, we are interested in\nexperiments where the initial state is reset to m0−dur-\ning each trial ofmagnetization switching. By convention,\nwe will focus on switching by a positive current and field\nregion.\nFigure 1(b) shows an example of the time evolution of\nmxin the presence of current. The damping constant is\nα= 0.005. The results indicate that the magnetization\nsaturates to a fixed point. Figure 2(a) is a phase diagram\nsummarizing the fixed point of mxin the presence of cur-\nrent, where the vertical and horizontal axes represent the\ncurrent density jand the external magnetic field Happl.\nThe magnetization stays near the initial state [ mx≃ −1\nshown in black in Fig. 2(a)] in a relatively small cur-\nrent region, where its boundary is well explained by the\ncritical-currentformula jc[27] shownby the white dotted\nline in Fig. 2(a). On the other hand, the magnetization\nstate above the critical current satisfies mx>0. There-\nfore, one might suppose that the magnetization relaxes\nto the switched state, mx≃+1, after turning off the\ncurrent.\nHowever, the phase diagram after turning off the cur-\nrentshowninFig. 2(b) revealsacomplicateddependence\nof the relaxed state on the current and external magnetic\nfield. Therearestripesdistinguishingtheswitched( m0+,\nyellow) and non-switched ( m0−, black) states. This re-\nsult indicates that the magnetization returns to the ini-tial state under certain conditions. We name this phe-\nnomenon back switching and investigate its relation to\nthe damping constant in the following.\nLet us briefly comment onanotherphenomenon, called\nbackhopping [28–32], to avoid any confusion. Back-\nhopping is a phenomenon in which, after magnetization\nswitchinginafreelayerisachieved,anundesirablereturn\nto the original resistance-state occurs at a high-bias volt-\nage. There are two differences between back switching\nandbackhopping. First, backhoppingintwo-terminalde-\nvices originates from magnetization switching in the ref-\nerence layer. On the other hand, back switching relates\nto the magnetization dynamics in the free layer. Second,\nbackhopping has often been investigated by sweeping the\ncurrent. On the other hand, we reset the initial state of\nthe magnetization at each trial because we are interested\nin designing the operation conditions of memory devices.\nBack switching originates from the fact that the fixed\npoint of the magnetization in the presence of current is\nnot parallel to the easy axis of the magnetization, as dis-\ncussed below, and thus, it will occur in not only type-X\ndevices, but also type-Z devices [25, 26].\nNote that the horizontal stripe in the relatively low\ncurrent region of Fig. 2(b), slightly above jc, was ana-\nlyzed in Ref. [27], and therefore, it will be excluded from\nthe following discussion. We label the other switched\nregions shown by the vertical stripes as the S n-region\n(n= 0,1,2,···); see Fig. 2(b). The role of the integer n\nwill be clarified below.\nFigure 3(a) shows the dynamic trajectory of the mag-\nnetization obtained by numerically solving Eq. (1),\nwherej= 300 MA cm−2andHappl= 500 Oe correspond\nto the S 0region. The red line represents the trajectory\nfrom the initial state to the fixed point in the presence of\nthe current, whereas the blue line shows the relaxation\ndynamics after turning off the current. The fixed point\nsatisfying dm/dt=0in the presence of the current is\nindicated by a yellow circle. Note that the fixed point is\nfar away from the energetically stable states, m0±. This\nis because the spin-transfer torque drives the magnetiza-\ntion in the ydirection. The black solid line in Fig. 3(b)\nshows the time evolution of mxafter turning off the cur-\nrent. Starting from the fixed point located in the region\nofmx>0, the magnetization relaxes to the switched\nstatem0+after showing several precessions around it.\nFigures 3(c) and 3(d), on the other hand, show an exam-\nple of back switching, where the parameter Happl= 750\nOe corresponds to the region sandwiched by the S 0and\nS1-regions. In this case, the magnetization after turning\noff the current shows a precession around the zaxis. As\na result, even though the fixed point in the presence of\nthe current is in the region of mx>0, the magnetization\nreturns to the initial state ( m0−). Let us then move to\nthe next switching region, the S 1-region. The dynamic\ntrajectory, as well as mx, shown in Figs. 3(e) and 3(f),\nindicates that, starting from the fixed point in the region\nofmx>0, the magnetization returns once to the region\nofmx<0 before relaxing to the switched state. Now3\n(a) (b)\nj=300 MA cm-2Happl=500 Oe\nj=300 MA cm-2Happl=750 Oe\nj=300 MA cm-2Happl=1000 Oemz1\n0\n-1\n0-1\n-111\n0mx mymx1\n0\n-1\nTime (ns)0 1m0+m0-\n(e) (f)\nmz1\n0\n-1\n0-1\n-111\n0mx mymx1\n0\n-1\nTime (ns)0 1m0+m0-(c) (d)\nmz1\n0\n-1\n0-1\n-111\n0mx mymx1\n0\n-1\nTime (ns)0 1m0+m0-\nNormalized energy (×10-3) Normalized energy (×10-3) Normalized energy (×10-3)1\n0\n-1\n-2\n-3\n-4\n12\n0\n-1\n-2\n-3\n-4\n1234\n0\n-1\n-2\n-3\n-4\nFIG. 3: (a) Dynamictrajectory in theS 0-region;Happl= 500\nOe, and j= 300 MA cm−2. The red line represents the\ntrajectory from the initial state to the fixed point (yellow\ncircle) in the presence of the current, whereas the blue line\ncorresponds to the relaxation dynamics after turning off the\ncurrent. The symbols m0±indicate the locations of the initial\n(m0−) and switched ( m0+) states. (b) Time evolutions of mx\n(black solid line) and normalized energy density ε(dashed red\nline) after turning off the current. The red triangle indicat es\nthe time at which εbecomes zero. (c) Dynamic trajectory\nand (d) time evolution of mxandεin the region sandwiched\nby the S 0- and S 1-regions; Happl= 750 Oe, and j= 300 MA\ncm−2. (e) Dynamic trajectory and (f) time evolution of mx\nandεin the S 1-region; Happl= 1000 Oe, and j= 300 MA\ncm−2.\nthe meaning of the integer n(n= 0,1,2,···) we used to\ndistinguish the switched regions becomes clear: it repre-\nsents how many times the magnetization returns to the\nregion of mx<0 before relaxing to the switched state.\nThese results imply that the precession around the z\naxis after turning off the current is the origin of back\nswitching. Such a precession is induced by the precession\ntorque,−γm×H. In particular, the precession around\nthezaxis occurs when the energy density at the fixed\npoint of the magnetization in the presence of the current\nis larger than the saddle-point energy given by\nEd=−MH2\nappl\n8πM. (5)In fact, a fixed point of the LLG equation is given by\nmx=HapplHs\nH2s−HK4πM, mz=−HapplHK\nH2s−HK4πM.(6)\nSubstituting Eq. (6) into Eq. (4), the energy density at\nthis fixed point is\nEj=MH2\napplHK\n2(H2s−HK4πM). (7)\nNote that H2\ns−HK4πM >0 because the back switch-\ning appears in the current region above jcforHappl=\n0, which is given by jc= [2eMd/(/planckover2pi1ϑ)]√HK4πM[27].\nTherefore, Eq. (7) is alwayspositive, whereasthe saddle-\npoint energy density given by Eq. (5) is negative. Ac-\ncordingly, the fixed point in the presence of the current is\nin the unstable region corresponding to an energy larger\nthan the saddle-point energy. As a result, after turn-\ning off the current, the magnetization starts to precess\naround the zaxis, as mentioned above. Simultaneously,\nthe magnetization loses energy due to damping torque,\nwhere the dissipated energy density is given by\n∆E(t) =αγM\n1+α2/integraldisplayt\n0dt′/bracketleftBig\nH2−(m·H)2/bracketrightBig\n.(8)\nHere,tis the time after the current is turned off. When\nthe condition,\nEj−∆E(t) =Ed, (9)\nis satisfied, the magnetization relaxes to the nearest sta-\nble state. The point here is that the magnetization al-\nternately comes close to two stable states, m0±, due to\nthe precession around the zaxis before Eq. (9) is sat-\nisfied. As a result, both m0+andm0−can be the final\nstate, which leads to back switching. The conclusions of\nthe discussion are confirmed by the dashed red lines in\nFigs. 3(b), 3(d), and 3(f), where are the time evolutions\nof the energy density. Here, we introduce the normalized\nenergy density,\nε(t) =E(t)−Ed\n4πM2. (10)\nTheenergydensity Eisafunction oftime with the initial\ncondition of E(t= 0) =EjbecauseE, given by Eq. (4),\ndepends on the magnetization direction, and the magne-\ntization changes direction in accordance with the LLG\nequation. The condition given by Eq. (9) is ε(t) = 0\nin terms of the normalized energy density. The times at\nwhichεbecomes zero are indicated by the red triangles\nin the figures. It is clear from Figs. 3(b), 3(d), and 3(f)\nthat the final state of the magnetization is determined by\nwhether mxis positive or negative when εbecomes zero.\nIt is expected that using a large-damping ferromag-\nnetic material will result in a reduction of the back-\nswitching region. Remember that back switching occurs\ndue tothe precessionaroundthe zaxis. When the damp-\ning constant is large, the energy dissipation given by Eq.4\n\"mx_relax.dat\" u 1:2:3 \"mx.dat\" u 1:2:3 mx1\n0\n-1 \nTime (ns)0 1\nNormalized energy (×10 -3 )\nNormalized energy (×10 -3 )\n2\n0\n-2 \n-6 -4 mx1\n0\n-1 \nTime (ns)0 124\n0\n-2 \n-6 -4 \n(c)(a)\nApplied field, Happl (kOe)Current density, j (MA cm -2 )\n0 1 2 0 1 202004006008001000 1\n0\n-1 mx\nApplied field, Happl (kOe)1\n0\n-1 mx(d)(b)\nCurrent density, j (MA cm -2 )\n02004006008001000\nFIG. 4: Time evolutions of mx(black solid lines) and nor-\nmalized energy density ε(red dashed lines) after turning off\nthe current, where j= 300 MA cm−2and (a)Happl= 750 Oe\nand (b) 1000 Oe. The damping constant αis 0.050. The red\ntriangles indicate the time at which εbecomes zero. Phase\ndiagrams of mx(c) at a fixed point in the presence of current\nand (d) in a relaxed state after turning off the current.\n(8) becomes large within a short time t, and the condi-\ntion given by Eq. (9) is immediately satisfied before the\nmagnetizationreturnstothe regionof mx<0bythepre-\ncession around the zaxis. Accordingly, back switching\ndoes not occur. On the other hand, when the damping\nconstant is small, it takes a long time to dissipate the\nenergy to satisfy Eq. (9), during which time the mag-\nnetization shows the precession around the zaxis. As a\nresult, back switching appears.\nTo verify this picture, we evaluated the phase diagram\nof the magnetization state for a relatively large damping\nconstant, α= 0.050, which is ten times large than that\nused above. Figures 4(a) and 4(b) show the time evo-\nlutions of mxandεafter turning off the current, where\nHapplis (a) 750 Oe and (b) 1000 Oe. Contrary to the\ndynamics shown in Figs. 3(d) and 3(f), the magnetiza-tion in Figs. 4(a) and 4(b) does not return to the region\nofmx<0 because the energy dissipates quickly due to\nthe large damping torque. As a result, the magnetiza-\ntion immediately relaxes to the switched state. Figures\n4(c) and 4(d) summarize the magnetization state in the\npresence of current and after turning off the current, re-\nspectively. Comparing Fig. 4(c) with Fig. 2(a), it is\nclear that the phase diagram of the magnetization state\nin the presence of the current is approximately indepen-\ndent of the damping constant. On the other hand, the\nback-switching region is strongly suppressed, as can be\nseen by the comparing Fig. 2(b) and 4(d). This is be-\ncause the large damping torque immediately dissipates\nthe energy from the ferromagnet and leads to a fast re-\nlaxationtothe switched state, due to whichthe S 0-region\ndominates in the phase diagram.\nThe existence of the back switching gives an upper\nlimit of a write-current margin for memory applications,\nand therefore, it restricts the device design and/or ma-\nnipulationconditions. Ontheotherhand, backswitching\nmight be applicable to other devices such as a random\nnumber generator. The analysis in this study provides\nfruitful insights for the development of spintronics appli-\ncations utilizing spin-orbit torque.\nIn conclusion, the phase diagram of the magnetization\nstate in a type-X spin-orbit torque device was calculated\nasafunction ofthe electriccurrentdensity and the exter-\nnal magneticfield. The magnetizationstate afterturning\noff the current has stripe structures alternately showing\nswitched and non-switched states. Such a non-switched\nstate, named back switching here, occurs as a result of\nthe magnetization precession around the perpendicular\naxis after the current is turned off. 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However,\ncountermeasures based on social analyses of the individuals\ncausing flaming are too slow to be effective because of the rapidity\nwith which the influence of online user dynamics propagates.\nA countermeasure technology for the flaming phenomena based\non the oscillation model, which describes online user dynamics,\nhas been proposed; it is an immediate solution as it does not\ndepend on social analyses of individuals. Conventional coun-\ntermeasures based on the oscillation model assume that the\ndamping coefficient is a constant regardless of the eigenfrequency.\nThis assumption is, however, problematic as the damping co-\nefficients are, in general, inherently frequency-dependent; the\ntheory underlying the dependence is being elucidated. This paper\ndiscusses a design method that uses the damping coefficient\nto prevent flaming under general conditions considering the\nfrequency-dependence of the damping coefficient and proposes a\ncountermeasure technology for the flaming phenomena.\nIndex Terms —online flaming, user dynamics, oscillation model,\ndamping coefficient\nI. I NTRODUCTION\nIn recent years, with the spread of social networking sites\nsuch as Twitter and Facebook, users’ activities in online\nsocial networks have come to be closely connected to social\nactivities in the real world, not only in online communities.\nAs a result, the effects of explosive online user dynamics,\nincluding the flaming phenomena, are becoming more serious,\nand countermeasures are needed [1], [2].\nAlthough it is desirable to respond immediately with direct\ncountermeasures to eliminate the factors that cause the flaming\nphenomena, analyzing each event in detail, one by one, will be\ntoo slow to prevent the damage from spreading [3], [4]. Thus\nwe need an engineering framework for flaming countermea-\nsures that does not depend on the details of each individual\nevent. One such framework has been proposed [5]. This is\nbased on the oscillation model on networks [6], [7] which is\nused to describe online user dynamics.\nConventional countermeasures for the flaming phenomena\nhave been discussed under the assumption that the damp-\ning coefficient [5] is a constant, and independent of the\neigenfrequency. However, it is known that regardless of thephenomenon, the damping coefficient generally depends on\nthe eigenfrequency. In fact, the theoretical characteristics of\nthe frequency dependence of the damping coefficient have\nrecently been clarified [8]. Based on these insights, we can\nconsider countermeasures for the flaming phenomena based on\nthe oscillation model, even in general cases where the damping\ncoefficient does depend on the eigenfrequency.\nIn this paper, we introduce a design methodology that allows\nthe damping coefficient to be used to counter the flaming\nphenomena even when the damping coefficient depends on\nthe eigenfrequency; we use it to propose a countermeasure\ntechnology for the flaming phenomena.\nII. O SCILLATION MODEL FOR DESCRIBING ONLINE USER\nDYNAMICS\nLet the Laplacian matrix of the online social network (OSN)\nwithnnodes be L, which is an n\u0002nsquare matrix, and the\nweight of the link from node ito nodej(i!j)bewij. In\naddition, let the eigenvalues of Lbe\u0015\u0016(\u0016= 0;1; :::; n\u00001)\nand the eigenvectors associated with \u0015\u0016bev\u0016. We assume the\neigenvalues are not duplicated.\nThe eigenvalues of Lare generally complex numbers,\nwhose range of existence is given by the largest Gershgorin\ndisk [9] of Las\nDmax\nG(L) =fz2C:jz\u0000dmaxj\u0014dmaxg; (1)\nwheredmax is the maximum weighted out-degree of the\nnetwork. It is known that all the eigenvalues of Llie within\nits largest Gershgorin disk [5].\nThe oscillation model [5], [6] is a minimal model for de-\nscribing user dynamics in OSNs. Let xi(t)be the state of node\n(user)iat timet. Since the influence of interaction between\nusers must propagate through any OSN at a finite speed, its\ndescription by the wave equation should be possible, which\nis the equation for describing the propagation of something at\nfinite speed. For the state vector x(t) :=t(x1(t); :::; xn(t)),\nthe wave equation in the OSN is written as\nd2\ndt2x(t) +\u0000d\ndtx(t) =\u0000Lx(t); (2)arXiv:2011.08117v1  [cs.SI]  16 Nov 2020where \u0000is the matrix expressing the strength of the damping.\nSubstituting the expansion of x(t)byv\u0016as\nx(t) =n\u00001X\n\u0016=0a\u0016(t)v\u0016; (3)\ninto the wave equation (4), yields the equation of motion for\neach oscillation mode a\u0016(t) (\u0016= 0;1; :::; n\u00001)as\nd2\ndt2a\u0016(t) +\r(!\u0016)d\ndta\u0016(t) =\u0000\u0015\u0016a\u0016(t); (4)\nwhere\r(!\u0016)is the damping coefficient; it depends on !\u0016=p\n\u0015\u0016and is expressed as\n\r(!\u0016) :=\r0+\r1\u0015\u0016\nwith the constant \r0and\r1[8]. Note that Re[\r(!\u0016)] =\r0+\n\r1Re[\u0015\u0016]\u00150.\nThe solution of (4) is written as\na\u0016(t) =c+\n\u0016exp\u0014\n\u0000\r(!\u0016)\n2t+ ipr\u0016exp\u0012\ni\u0012\u0016\n2\u0013\nt\u0015\n+c\u0000\n\u0016exp\u0014\n\u0000\r(!\u0016)\n2t\u0000ipr\u0016exp\u0012\ni\u0012\u0016\n2\u0013\nt\u0015\n;(5)\nwherec+\n\u0016andc\u0000\n\u0016are constants that depend on \u0016, andr\u0016and\n\u0012\u0016(\u0000\u0019 < \u0012\u0016\u0014\u0019)are, respectively, the absolute value and\nthe argument of the following complex number:\nr\u0016exp(i\u0012\u0016) :=\u0015\u0016\u0000\u0012\r(!\u0016)\n2\u00132\n=\u0000\u0012\n\u000b+\r(!\u0016)\n2\u00132\n:(6)\nIn the oscillation model, the oscillation energy can be\nconsidered as the strength of the activity of user dynamics [10],\n[11]. Also, the situation in which oscillation energy diverges\nover time is considered to describe explosive user dynamics,\nwhich include the flaming phenomena. Therefore, in order to\nprevent explosive user dynamics, it is necessary to consider the\nconditions under which the oscillation energy does not diverge.\nBy deriving the strength of the damping that satisfies this\ncondition, we can obtain a framework in which the strength of\ndamping can be adjusted to prevents the flaming phenomena.\nThe conventional solutions to the flaming phenomena as-\nsume that the damping coefficient is a constant and indepen-\ndent of eigenfrequency. This corresponds to the special case\nof\r1= 0in (4). Following [5], the condition under which the\noscillation energy does not diverge is given as\n8\u0016;\r0\n2pr\u0016\u0015\f\f\f\fsin\u0012\u0016\n2\f\f\f\f; (7)\nand the value of the damping coefficient required to satisfy\nthis condition is given by\n\r0\u0015p\n2dmax: (8)\nIn the next section, in order to consider countermeasures\nfor the flaming phenomena, we discuss the conditions under\nwhich the oscillation energy does not diverge in the case of\n\r16= 0.III. M ODEL OF EXPLOSIVE USERDYNAMICS\nCONSIDERING FREQUENCY -DEPENDENT DAMPING\nCOEFFICIENT\nSince the oscillation energy is proportional to the square\nof the absolute value of a\u0016(t), we derive the condition under\nwhicha\u0016(t)does not diverge and then the condition under\nwhich flaming does not occur.\nBy decomposing the damping coefficient \r(!\u0016)into real\nand imaginary parts as in\n\r(!\u0016) = (\r0+\r1Re[\u0015\u0016]) + i\r1Im[\u0015\u0016]: (9)\na\u0016(t)is written as\na\u0016(t) =c+\n\u0016exp\u0014\n\u0000\r0+\r1Re[\u0015\u0016]\n2t\u0000pr\u0016sin\u0012\u0012\u0016\n2\u0013\nt\u0015\n\u0002c+\n\u0016exp\u0014\n\u0000i\r1Im[\u0015\u0016]\n2t+ ipr\u0016cos\u0012\u0012\u0016\n2\u0013\nt\u0015\n+c\u0000\n\u0016exp\u0014\n\u0000\r0+\r1Re[\u0015\u0016]\n2t+pr\u0016sin\u0012\u0012\u0016\n2\u0013\nt\u0015\n\u0002c\u0000\n\u0016exp\u0014\n\u0000i\r1Im[\u0015\u0016]\n2t\u0000ipr\u0016cos\u0012\u0012\u0016\n2\u0013\nt\u0015\n:\n(10)\nTo determine if a\u0016(t)diverges or not, we need to check\nwhether the real components of the exponent of the expo-\nnential function in (10) are positive or negative, and if a\u0016(t)\ndiverges, the following condition is satisfied:\n\u0012Re[\r(!\u0016)]\n2pr\u0016+ sin\u0012\u0012\u0016\n2\u0013\u0013\u0012Re[\r(!\u0016)]\n2pr\u0016\u0000sin\u0012\u0012\u0016\n2\u0013\u0013\n<0:\nThis means the one of real components of the exponent is\npositive and the other is negative.\nConsequently, the condition under which the oscillation\nenergy diverges is given by\n9\u0016;\r0+\r1Re[\u0015\u0016]\n2pr\u0016<\f\f\f\fsin\u0012\u0016\n2\f\f\f\f; (11)\nand the condition that the oscillation energy does not diverge\nis obtained as\n8\u0016;\r0+\r1Re[\u0015\u0016]\n2pr\u0016\u0015\f\f\f\fsin\u0012\u0016\n2\f\f\f\f: (12)\nSince the conventional condition (7) that the oscillation energy\ndoes not diverge corresponds to the case of \r1= 0 as per\ncondition (12), the condition (12) for the frequency-dependent\ndamping coefficient is a generalization of the conventional\nresult.\nIV. C OUNTERMEASURE FOR FLAMING PHENOMENA\nGIVEN THE FREQUENCY -DEPENDENT DAMPING\nCOEFFICIENT\nBased on condition (12), i.e., the oscillation energy does\nnot diverge, we consider a design method for the damping\ncoefficient to satisfy (12), and consider a countermeasure\ntechnology for the flaming phenomena by adjusting the value\nof the damping coefficient.A. Adjusting the Damping Coefficient\nAmong parameters \r0and\r1, which determine the strength\nof damping, \r1is the eigenfrequency dependent term. This\nmeans that the value of \r1is a parameter predetermined by the\nstructure of the network. Therefore, \r0is the only parameter\nthat can be manipulated independently of the network struc-\nture. In this framework, even if various values are given as \r1,\nwe can counter the flaming phenomena by adjusting the value\nof\r0. Here, the actual action to adjust the value of \r0includes\ndisseminating other information to attract users’ attention. In\nthe following, we consider the value of \r0necessary to prevent\nthe triggering of explosive user dynamics, and we use its value\nin a flaming countermeasure.\nThe range of eigenvalues of Lis the interior of the largest\nGershgorin disk (including its boundaries) of radius dmax\nwith center (dmax;0). From condition (12), the oscillation\nenergy does not diverge, we can consider the range satisfying\ncondition (12) on the complex plane. Then, if the largest\nGershgorin disk of Llies completely within the range on the\ncomplex plane, the oscillation energy never diverges regardless\nof the network structure.\nIn order to clarify the region on the complex plane in which\ncondition (12) ensures that the oscillation energy does not\ndiverge, the inequality of condition (12) is transformed as\nfollows:\nIm[\u0015\u0016]2\u0014Z\n4\u00004\r0\r1\u00004\r2\n1Re[\u0015\u0016]; (13)\nwhere\nZ=\r4\n0+ 4\r3\n0\r1Re[\u0015\u0016] + 6\r2\n0\r2\n1Re[\u0015\u0016]2\n+ 4\r2\n0X+ 4\r0\r3\n1Re[\u0015\u0016]3+ 8\r0\r1Re[\u0015\u0016]X\n+\r4\n1Re[\u0015\u0016]4+ 4\r2\n1Re[\u0015\u0016]2X;\nandXis the real part of (6).\nThe range of eigenvalues of L, determined by the Gersh-\ngorin theorem, are written as\nIm[\u0015\u0016]2\u0014d2\nmax\u0000(Re[\u0015\u0016]\u0000dmax)2: (14)\nWe compare (14) and the condition (13) that the oscillation\nenergy does not diverge. Because both of them are axially sym-\nmetric on by the real axis, we consider the upper-half plane.\nThe condition that the largest Gershgorin disk is completely\nincluded the range of (13) is expressed as\nd2\nmax\u0000(Re[\u0015\u0016]\u0000dmax)2\u0014Z\n4 (1\u0000\r0\r1\u0000\r2\n1Re[\u0015\u0016]);\nwhich can be transformed to\n\u0000\n\r0\r1+ 1 + 2\r2\n1dmax\u0001\nRe[\u0015\u0016]\n\u0015\u0000(\r2\n0+ 2\r0\r1dmax\u00002dmax); (15)\nby considering Re[\u0015\u0016]\u00150.\nWe consider the conditions for satisfying inequality (15) in\nthe following three cases.\n\u000fif\r0\r1+ 1 + 2\r2\n1dmax= 0,\n\r2\n0+ 2\r0\r1dmax\u00002dmax\u00150: (16)The range of \r0that satisfies the above is as follows from\n\r0\u00150\n\r0\u0015q\n\r2\n1d2max+ 2dmax\u0000\r1dmax: (17)\n\u000fif\r0\r1+ 1 + 2\r2\n1dmax>0,\n\r2\n0+ 2\r0\r1dmax\u00002dmax\n\r0\r1+ 1 + 2\r2\n1dmax\u0015\u0000Re[\u0015\u0016]: (18)\nIn order for this inequality to hold, the numerator needs\nto be non-negative, so we obtain\n\r2\n0+ 2\r0\r1dmax\u00002dmax\u00150: (19)\nConsidering \r0\u00150, the condition of \r0to ensure the\nnon-divergence of oscillation energy for all eigenvalues\nis written as\n\r0\u0015q\n\r2\n1d2max+ 2dmax\u0000\r1dmax: (20)\n\u000fif\r0\r1+ 1 + 2\r2\n1dmax<0,\n\r2\n0+ 2\r0\r1dmax\u00002dmax\n\r0\r1+ 1 + 2\r2\n1dmax\u0014\u0000Re[\u0015\u0016]: (21)\nInequality (21) is transformed to\n(\r0+ 2\r1dmax)2\u00150 (22)\nTherefore, inequality (21) always holds in this case.\nTo summarize the above results, the \r0condition that en-\nsures the oscillation energy does not diverge for all eigenvalues\nis obtained as\n\r0\u0015q\n\r2\n1d2max+ 2dmax\u0000\r1dmax: (23)\nTherefore, given the maximum weighted out-degree of the\nnetwork,dmax, and parameter \r1of the damping coefficient,\nadjusting the value of \r0to satisfy (23) will counter the\nflaming phenomena.\nB. Case Studies\nUsing an example network with dmax= 100 , this section\nconsiders three cases of different values of \r1, the frequency\ndependence of the damping coefficient: \r1= 0,\r1>0or\n\r1<0. In all cases, we confirm that the region of the condition\nthat the oscillation energy does not diverge includes the largest\nGershgorin disk of Lby satisfying the condition (23) of \r0.\nFirst, we confirm the case of complete flaming prevention\nwith\r0=p\n\r2\n1d2max+ 2dmax\u0000\r1dmax, where\r0is the\nminimum value that satisfies condition (23). Figure 1 shows\nthe regions in which the oscillation energy does not diverge\nas given by condition (12), for the cases of \r1= 0:1,\n\r1= 0 and\r1=\u00000:1. These regions are depicted in blue.\nIn addition, the largest Gershgorin disk of Lis depicted in\nred. In all figures, it can be seen that the regions wherein\nthe oscillation energy does not diverge completely include the\nlargest Gershgorin disk, so that no divergence of oscillation\nenergy occurs regardless of the details of the network structure.\nNext, we show the case of incomplete flaming prevention\nby using\r0=p\n\r2\n1d2max+ 2dmax\u0000\r1dmax\u00005, in which\r0Fig. 1. Examples of complete flaming prevention\nFig. 2. Examples of incomplete flaming prevention\nis less than the minimum value that satisfies condition (23).\nFigure 2 shows the regions wherein the oscillation energy\ndoes not diverge as indicated by condition (12), for the cases\nof\r1= 0:1,\r1= 0, and\r1=\u00000:1. In these cases, the\nregions cannot completely enclose the Gershgorin disk. If even\njust one eigenvalue appears outside of the region, the oscil-\nlation energy diverges and the flaming phenomenon occurs.\nTherefore, depending on the position of the eigenvalues of L,\nflaming prevention is not assured.\nV. C ONCLUSION\nIn this paper, we proposed a countermeasure technology for\nthe flaming phenomena based on the oscillation model with the\nfrequency-dependent damping coefficient. The design method\nthat yields the damping coefficients using condition (23) is\na generalized version of the conventional countermeasure\ntechnology for the flaming phenomena. Regardless of the\nvalue of parameter \r1, which is the strength of the frequency\ndependence of damping coefficient, we can prevent explosive\nuser dynamics by setting parameter \r0to be an appropriate\nvalue.\nFurthermore, the required value of \r0can be determined\nfrom justdmax, which is the maximum weighted out-degree\nof the network, and \r1, which is the strength of the frequency-\ndependence of the damping coefficient. One of the methods\nfor increasing the value of \r0in the actual OSNs is that\nto disseminate other information to attract users’ attention is\nmentioned.ACKNOWLEDGEMENT\nThis research was supported by Grant-in-Aid for Scientific\nResearch 19H04096 and 20H04179 from the Japan Society\nfor the Promotion of Science (JSPS).\nREFERENCES\n[1] M. Alonzo and M. Aiken, “Flaming in electronic communication,”\nDecision Support Systems , vol. 36, no. 3, pp. 205–213, 2004.\n[2] Y . Adachi and F. Takeda, “The impact of online flaming on firm value:\nThe evidence from Japan,” IPRC Discussion Paper Series , no. 14, 2014.\n[3] P. Moor, A. Heuvelman and R. Verleur, “Flaming on YouTube,” Com-\nputers in human behavior , vol. 26, no. 6, pp. 1536–1546, 2010.\n[4] S. Tsugawa and H. Ohsaki, “Negative messages spread rapidly and\nwidely on social media,” The 2015 ACM on Conference on Online Social\nNetworks , pp. 151–160, 2015.\n[5] M. Aida, Introduction to Network Dynamics , Morikita Publishing, 2020\n(in Japanese).\n[6] M. Aida, C. Takano and M. Murata, “Oscillation model for describing\nnetwork dynamics caused by asymmetric node interaction,” IEICE\nTransactions on Communications , vol. E101–B, no. 1, pp. 123–136,\n2018.\n[7] M. Aida, C. Takano and M. Murata, “Oscillation model for network\ndynamics caused by asymmetric node interaction based on the sym-\nmetric scaled Laplacian matrix,” The 12th International Conference on\nFoundations of Computer Science (FCS 2016) , pp. 38–44, 2016.\n[8] C. Takano and M. Aida, “Decay characteristics of user dynamics in\nonline social networks,” IEEE Access , vol. 8, pp. 73986–73991, 2020.\n[9] R. Varga, Gershgorin and His Circles , Springer-Verlag, 2004.\n[10] C. Takano and M. Aida, “Revealing of the underlying mechanism of\ndifferent node centralities based on oscillation dynamics on networks,”\nIEICE Transactions on Communications , vol. E101-B, no. 8, pp. 1820–\n1832, 2018.\n[11] C. Takano and M. Aida, “Proposal of new index for describing node\ncentralities based on oscillation dynamics on networks,” 2016 IEEE\nGlobal Communications Conference (GLOBECOM) , Washington, DC,\npp. 1–7, 2016."    },    {        "title": "2011.08181v5.A_Random_Matrix_Theory_Approach_to_Damping_in_Deep_Learning.pdf",        "content": "A Random Matrix Theory Approach to Damping in\nDeep Learning\nDiego Granziol, Huawei AI Theory\nNicholas Baskerville, Bristol University\nAbstract. We conjecture that the inherent difference in generalisation between\nadaptive and non-adaptive gradient methods in deep learning stems from the increased\nestimation noise in the flattest directions of the true loss surface. We demonstrate that\ntypical schedules used for adaptive methods (with low numerical stability or damping\nconstants) serve to bias relative movement towards flat directions relative to sharp\ndirections, effectively amplifying the noise-to-signal ratio and harming generalisation.\nWe further demonstrate that the numerical damping constant used in these methods\ncan be decomposed into a learning rate reduction and linear shrinkage of the estimated\ncurvature matrix. We then demonstrate significant generalisation improvements by\nincreasing the shrinkage coefficient, closing the generalisation gap entirely in both\nlogistic regression and several deep neural network experiments. Extending this\nline further, we develop a novel random matrix theory based damping learner for\nsecond order optimisers inspired by linear shrinkage estimation. We experimentally\ndemonstrate our learner to be very insensitive to the initialised value and to allow\nfor extremely fast convergence in conjunction with continued stable training and\ncompetitive generalisation. We also find that our derived method works well with\nadaptive gradient methods such as Adam.\n1. Introduction\nThe success of deep neural networks across a wide variety of tasks, from speech\nrecognition to image classification, has drawn wide-ranging interest in their optimisation\nand their ability to generalise to unseen data. Optimisation is the process of reducing\nthe value of the neural network loss on its training set. Generalisation refers to the\nloss on an unseen held out test set. The loss of a neural network is a scalar function\nof the network free parameters (the weights of the neural network) that measures how\nwell the network is performing on the data at hand. It can be defined on a single or\nmultiple examples (known as a batch). Loss functions are by definition greater than\nor equal to zero. For example, a neural network’s output may be the probability of a\nparticular image class label being correct. In this case, where we have the true label of\nthe data item, we can use the discrete cross entropy loss, which has roots in information\ntheory [Cover and Thomas, 2012]. The discrete cross entropy loss is zero if and only ifarXiv:2011.08181v5  [stat.ML]  16 Mar 2022A Random Matrix Theory Approach to Damping in Deep Learning 2\nwe predict the true label with probability 1and non zero otherwise. Hence we drive\nthe neural network to learn not only the true class, but to predict the true class with\nprobability 1, i.e. to be confident about the correct predictions. Alternative losses in\nthe regression context could be the well known square loss, along with many others,\nsuch as the Hinge loss [Bishop, 2006].\nDue to enormous dimensionality of the weight space, where billions of parameters\nare the norm, more effective measures than random search must be employed to reduce\nthe loss from a random initialisation of the weights. A very simple but highly effective\nmethod, which underlies the basis of modern machine learning optimisation is stochastic\ngradient descent (SGD). In its simplest form, we simply take the gradient of the neural\nnetwork batch loss and follow the method of steepest descent into a local minima. For\na full discussion on stochastic gradient optimisation and associated convergence proofs\nwe recommend [Nesterov, 2013, Kushner and Yin, 2003]\nThere have been many amendments to stochastic gradient descent, including the\nuse of momentum [Nesterov, 2013]. A particularly fruitful area of optimisation research\nwhich has found its way into Deep Learning are adaptive gradient optimisers . Adaptive\ngradient methods alter the per-parameter learning rate depending on historical gradient\ninformation, which leads to significantly faster convergence of the training loss than\nnon adaptive methods, such as stochastic gradient descent (SGD) with momentum\n[Nesterov, 2013]. Popular examples include Adam [Kingma and Ba, 2014], AdaDelta\n[Zeiler, 2012] and RMSprop [Tieleman and Hinton, 2012].\nHowever, for practical applications the final results on a held-out test set are\nmore important than the training performance. For many image and language\nproblems of interest, the test set performance of adaptive gradient methods is\nsignificantly worse than SGD [Wilson et al., 2017]—a phenomenon that we refer to as\ntheadaptive generalisation gap . As a consequence of this effect, many state-of-the-art\nmodels, especially for image classification datasets such as CIFAR [Yun et al., 2019]\nand ImageNet [Xie et al., 2019, Cubuk et al., 2019], are still trained using SGD with\nmomentum. Although less widely used, another class of adaptive methods which\nsuffer from the same phenomenon [Tornstad, 2020] are stochastic second order methods ,\nwhich seek to alter the learning rate along the eigenvectors of the Hessian of\nthe loss function. KFAC [Martens and Grosse, 2015] uses a Kroenecker factored\napproximation of the Fisher information matrix (which can be seen as a positive definite\napproximation to the Hessian [Martens, 2014]). Other methods use Hessian–vector\nproducts [Dauphin et al., 2014, Martens, 2010] in conjunction with Lanczos methods\nand conjugate gradients [Meurant and Strakoš, 2006]. All second order and adaptive\ngradient methods, are endowed with an extra hyper-parameter called the damping or\nnumerical stability co-efficient respectively. This parameter limits the maximal learning\nrate along the eigenvectors or unit vectors in the parameter space respectively and is\ntypically set to a very small value by practitioners.\nIn this paper we argue that adaptive methods in their typical implementations withA Random Matrix Theory Approach to Damping in Deep Learning 3\nsmalldamping/numericalstabilityco-efficients, areover-confidentintheirupdatesinthe\nflattestdirectionsoftheloss. Weshowthatthisissub-optimalintermsofoptimisingthe\ntrue loss and hence harms generalisation performance. We demonstrate this empirically\nin an online convex example, where we actively perturb the sharp directions, reducing\ngeneralisation without impacting training. We also demonstrate this implicitly for large\nneural networks by altering the damping/stability constant, which we show alters the\neffective learning rate ratio between the sharp and flat directions.\nGiven their widespread adoption, understanding the adaptive generalisation gap\nhas significant implications. In this work we show that altering the numerical stability\nconstant in Adam [Choi et al., 2019] can be interpreted as applying linear shrinkage\nestimation to the noisy Hessian estimate enabling the reduction of the mean squared\nerror [Bun et al., 2016a] in the estimation of the true loss Hessian. We develop this idea\nfurther to produce a novel optimal and highly effective adaptive damping scheme to\nautomatically tune the damping and numerical stability constant for KFAC and Adam\nrespectively.\n2. Background on loss surfaces and generalisation\nWe view a neural network (or any supervised machine learning model) as a prediction\nfunctionh(\u0001;\u0001) :Rdx\u0002RP!Rdy, where RPis the space of parameters of the networks\n(i.e. its weights and biases). A single data point is an element of Rdxand its label,\nwhich could be continuous or discrete, is an element of Rdy. Viewingw2RPas\nparameters of h, we have a parametrised family of functions Rdx!Rdy, namely\nH:=fh(\u0001;w) :w2RPg. To train the network, i.e. optimise the parameters for a\ngiven task, we define a loss function `(y;^y) :Rdy\u0002Rdy!R. The objective is to\nminimise`over a given data distribution. More precisely, let Pdatabe some probability\ndistribution on Rdx\u0002Rdy, thedata distribution . The expectation of the loss over data\ndistribution is known as the true lossz:\nLtrue(w) =Z\n`(h(x;w);y)dPdata(x;y): (1)\nIn practice, given a finite dataset of size N, we only have access to the empirical\nlossorbatch lossx:\nLemp(w) =1\nNNX\ni=1`(h(xi;w);yi); Lbatch(w) =1\nBX\ni2IB`(h(xi;w);yi)(2)\nwhere (xi;yi)are i.i.d. samples from Pdata,IB\u001af1;:::;Nghas cardinality BandB\nis typically much less that N. When optimising deep neural networks in practice, the\nbatch loss is almost always the quantity used as, amongst other reasons, the empirical\nzThe true loss is also often called the Bayes risk .\nxThe empirical loss is also known as the empirical risk .A Random Matrix Theory Approach to Damping in Deep Learning 4\nloss is far too costly to evaluate given the number of training iterations that are required.\nAt the conclusion of some given training procedure, whas been modified to minimise\nas far as possible `onDtrain=f(xi;yi)ji= 1;:::;Ngand so is, in general, statistically\ndependent on the samples in Dtrain, thusLempis no longer an unbiased estimator of\nLtrue. This leads to the possibility of a generalisation gap , i.e.Lemp< Ltrue. The true\nloss is the quantity that is really practically relevant and it can estimated without bias\nusing a held out test setwhich is just another finite sample from Pdataindependent of\nDtrain. Note that the test loss is not free from variance and so a sufficiently large amount\nof the original un-partitioned dataset must be held out so that statistically significant\ndifferences can be observed. The losses, viewed as scalar functions on RPcan be thought\nof as surfaces in RPand our so known as loss surfaces . The objective of learning is to\nfind the lowest possible point on Ltruegiven only Lbatch(orLemp). In practice this is\nachieved stochastic gradient optimisation methods such as stochastic gradient descent\n(SGD) and variants thereof. The gradients required for SGD and optimisation methods\narerLbatchwhere derivatives are with respect the the parameters w2RP. Similarly,\nsecond order methods also make use of the Hessian Hbatch =r2Lbatch. Note that, just\nasLbatchis a random estimate of Ltruecorrupted by some sampling noise, so is rLbatch\na noisy estimate of rLtrueand similarly for the Hessian.\n2.1. Background on adaptive optimisers\nStochastic gradient descent updates weights according to the rule\nwk+1=wk\u0000\u000bkrLbatch (3)\nwherewkare the network parameters after kiterations of SGD and at each iteration\na different batch is used. \u000bk>0is thelearning rate which, in the simplest setting for\nSGD, does not depend on k, but in general can be varied throughout training to achieve\nsuperior optimisation and generalisation. The general for of adaptive optimiser updates\nis\nwk+1=wk\u0000\u000bkB\u00001rLbatch (4)\nwhereBis apre-conditioning matrix . The essential idea of adaptive methods is to use\nthepre-conditioningmatrixtomakethegeometryof LbatchmorefavourabletoSGD.One\napproach is to take Bto be diagonal, which can be thought of as having per-parameter\nlearning rates adapted to the local loss surface geometry. More generally, one might\nseek an approximation Bto the local loss surface Hessian, effectively changing the\nbasis of the update rule to a natural one, with per-direction learning rates. For Adam\n[Kingma and Ba, 2014], the most commonplace adaptive optimiser in the deep learning\ncommunity,Bis given by the diagonal matrix with entriesp\nhg2\nki+\u000f\nhgki. Heregis the\nloss gradient and h\u0001idenotes an empirical exponential moving average or iterations. In\nprinciple there is no reason why a certain parameter gradient should not be zero (or very\nsmall) and hence the inversion of Bcould cause numerical issues. This is the originalA Random Matrix Theory Approach to Damping in Deep Learning 5\nreasongivenby[Kingma and Ba, 2014]forthenumericalstabilitycoefficient \u000f. Similarly\nsoforKFACforwhich B=PP\ni\u0015i\u001ei\u001eT\niwheref\u0015i;\u001eigP\ni=1aretheeigenvalue, eigenvector\npairsofthekroneckerfactoredapproximationtotheHessian. Hencetoeacheigenvaluea\nsmalldampingcoefficient \u000eisadded. Whilstforbothadaptiveandsecondordergradient\nmethods, the numerical stability and damping coefficients are typically treated in the\nliterature as extra nuisance parameters which are required to be non-zero but not of\ngreat theoretical or practical importance, we strongly challenge this view. In this paper\nwe relate these coefficients to the well known linear shrinkage method in statistics. It\nis clear from a random matrix theory perspective, that the sub-sampling of the Hessian\nwill lead to the creation of a noise bulk in its spectrum around the origin, precisely the\nregion where the damping coefficient is most relevant. We show, both experimentally\nand theoretically, that these coefficients should be considered as extremely important\nhyper-parameters whose tuning has a strong impact on generalisation. Furthermore,\nwe provide a novel algorithm for their online estimation, which we find effective in\npreliminary experiments on real networks and datasets.\n3. Previous Work\n0 100 200 300\nEpoch0.00.51.0ErrorSGD Train =0.01\nAdam Train =0.0004\nSGD Val =0.01\nAdam Val =0.0004\n290 2980.3500.375\n(a) SGD quickly outgeneralises Adam\n0 100 200 300\nEpoch0.00.51.0ErrorTrain =0.0004\nTrain =0.0001\nVal =0.0004\nVal =0.0001\n290 2980.000.01 (b) Adam Train/Val for Learning Rates f\u000big\nFigure 1: Adaptive Generalisation Gap and its extent are clearly visible\nwithout regularisation. Train/Val Error on CIFAR- 100using VGG- 16without batch\nnormalisation and weight decay.\n3.1. Learning Rates\nPrevious work has investigated the relationship between generalisation and the ratio\nof learning rates and batch size during SGD optimisation [Jastrzebski et al., 2020], in\naddition to stability analysis [Wu et al., 2017] showing that larger learning rates lead to\nlowerspectralnorms(whichBayesianandminimumdescriptionlengthprinciplessuggest\nlead to better generalisation [Hochreiter and Schmidhuber, 1997]). [Li et al., 2019]\nargue that larger learning rates learn hard to generalise, easy to fit patterns better than\ntheir lower learning rate counterparts and that this forms part of the generalisation\ngap. Whilst we expect increased per iteration weight decay (1\u0000\u000b\r)from a larger\nlearning rate \u000bwithL2regularisation coefficient \rto lead to improved generalisation,\nthis effect remains pertinent even with no regularisation as shown in Fig 1b. What\nfurther remains unclear for adaptive methods is the importance of the global learningA Random Matrix Theory Approach to Damping in Deep Learning 6\nrate, given that each parameter has its own individual learning rate . Additionally, for\na damping/numerical stability coefficient \u000e, the largest possible individual learning rate\nis given by \u000b=\u000e, which for typical setups can be orders of magnitude larger than that of\nSGD. Hence, it is clear that not only the global learning rate is of importance, but also\nthe relative learning rate in different directions, meriting further study.\n3.2. Regularisation\nOne factor hypothesised to contribute to the Adaptive Generalisation Gap is\nthe non-equivalence between traditional weight decay and L2regularisation for\nadaptive [Loshchilov and Hutter, 2018] and second order [Zhang et al., 2018b] methods.\nHowever, as shown in Fig 1a, even when no regularisation is employed, using their\nrespective best settings, the generalisation of SGD strongly outperforms that of Adam.\nThis strongly suggests that SGD inherently generalises better than adaptive methods\nand requires further study. Furthermore a strong understanding as to why weight decay\nstrongly outperforms L2regularisation for Adaptive methods remains elusive and in\nneed of further investigation.\n3.3. Flatness\nThe notion of flatness (or correspondingly, sharpness ) has received considerable\nattention as a property of training loss minima that is predictive of their generalisation.\nAlthough there is no universally accepted definition, flatnessis typically defined through\nproperties of the Hessian (the second derivative of the loss), such as its spectral norm\nor trace. Mathematically when integrating out the product of the maximum likelihood\n(MLE) solution (given by the final weights) with the prior, the posterior is shifted\nrelative to the MLE solution. For sharpminima, the difference in loss for a small shift\nis potentially large, motivating the study of sharpness in the context of generalisation.\nIndeed, the idea of a shiftbetween the training and testing loss surface is prolific in the\nliterature and regularly related to generalisation [He et al., 2019, Izmailov et al., 2018,\nMaddox et al., 2019, Yao et al., 2018, Zhang et al., 2018a, Keskar et al., 2016].\nHowever,thelackofreparameterisationinvarianceoftheHessian[Dinh et al., 2017],\nhas subjected its use for predicting generalisation to criticism [Neyshabur et al., 2017,\nTsuzuku et al., 2019, Rangamani et al., 2019]. Normalized definitions of flatness, have\nbeen introduced [Tsuzuku et al., 2019, Rangamani et al., 2019] in a PAC-Bayesian\nframework, although are not widely available to practitioners and require specialist\nimplementations.\nWhilst [Rangamani et al., 2019] note that empirically Hessian based sharpness\nmeasures correlate with generalisation, we find that \"flatness\" as defined by the\nspectral/Frobenius norm of the Hessian of the loss can give strongly misleading results\nwhen comparing solutions between adaptive and non-adaptive optimisers. As shown\nin Figures 2b and 2c, it is possible to find better generalising solutions with\nadaptive optimisers that are nonetheless significantly “sharper” than those found byA Random Matrix Theory Approach to Damping in Deep Learning 7\nSGD. Furthermore sharpness at a point in weight-space may not be indicative of the\nflatness of the overall basin of attraction. In order to alleviate this, some authors\nhave considered taking random directions in weightspace [Izmailov et al., 2018] for large\ndistances, however it is unclear to what extent 2Dheatmaps are representative in what\nare typically million dimensional spaces.\n3.4. Related Work:\nTo the best of our knowledge, there has been no theoretical work analysing\ntheadaptive generalisation gap , with the notable exception of [Wilson et al., 2017],\nwho consider the poor generalisation performance of adaptive methods to be\ninherent (and show this on a simple example). Practical amendments to improve\ngeneralisation of adaptive methods have included dynamically switching between Adam\nand SGD [Keskar and Socher, 2017], taking the preconditioning matrix in Adam to\nthe power of p2[0;1=2][Chen and Gu, 2018], employing weight decay instead\nofL2regularisation [Zhang et al., 2018b, Loshchilov and Hutter, 2018] and altering\nthe damping or numerical stability constant, typically taken as 10\u00008in Adam\n[Choi et al., 2019]. While empirically effective, these alterations lack a clear theoretical\nmotivation. None of the aforementioned works explain why Adam (or adaptive methods\ningeneral)inherentlygeneraliseworsethanSGD.Instead,theyshowthatswitchingfrom\nAdam to SGD, or making Adam more similar to SGD brings improvements. A clear\nanalysis of how SGD differs from adaptive methods and why this impacts generalisation\nis not provided in the literature and this forms the basis of our paper.\n3.5. Contributions\nWe conjecture that a key driver of the adaptive generalisation gap is the fact that\nadaptive methods fail to account for the greater levels of noise associated with their es-\ntimates of flat directions in the loss landscape . The fundamental principle underpinning\nthis conjecture—that sharp directions contain information from the underlying process\nand that flat directions are largely dominated by noise—is theoretically motivated from\nthe spiked covariance model [Baik and Silverstein, 2004]. This model has been suc-\ncessfully applied in Principal Component Analysis (PCA), covariance matrix estima-\ntion and finance [Bloemendal et al., 2016, Everson and Roberts, 2000, Bun et al., 2017,\nBun et al., 2016b]. We revisit this idea in the context of deep neural network optimisa-\ntion.\nIn particular, we consider a spiked additive signal-plus-noise random matrix model\nfor the batch Hessian of deep neural network loss surfaces. In this model, results from\nrandom matrix theory suggest several practical implications for adaptive optimisation.\nWeuselinearshrinkagetheory[Bun et al., 2016b,Bun et al., 2016a,Bun et al., 2017]to\nilluminatetheroleofdampinginadaptiveoptimisersanduseourinsightstoconstructan\nadaptivedampingschemethatgreatlyacceleratesoptimisation. Wefurtherdemonstrate\nthat typical hyper-parameter settings for adaptive methods produce a systematic biasA Random Matrix Theory Approach to Damping in Deep Learning 8\nin favour flat directions in the loss landscape and that the adaptive generalisation gap\ncan be closed by redressing the balance in favour of sharp directions. To track to bias\ntowards flat vs sharp directions we define the estimated curvature learning rate ratio :\nRest-curv :=\u000bflat\n\u000bsharp(5)\nwhere\u000b\ratand\u000bsharpare the learning rates along the flat and sharp directions,\nrespectively and this ratio encapsulates the noise-to-signal ratio as motivated by our\nconjecture (the terms flatandsharpare defined more precisely in Section 4).\n4. The Spiked Model for the Hessian of the Loss\n4.1. Key Result: Sharp directions from the True Loss surface survive, others wash out\nWe can rewrite the (random) batch hessian Hbatchas the combination of the\n(deterministic) true hessian Htrueplus some fluctuations matrix:\nHbatch(w) =Htrue(w) +X(w): (6)\nIn [Granziol et al., 2020b] the authors consider the difference between the batch and\nempirical Hessian, although this is not of interest for generalisation, the framework\ncan be extended to consider the true Hessian. The authors further show, under the\nassumptions of Lipschitz loss continuity, almost everywhere double differentiable loss\nand that the data are drawn i.i.d from the data generating distribution that the\nelements ofX(w)converge to normal random variables k. Under the assumptions of\nlimited dependence between and limited variation in the variance of the elements of the\nfluctuations matrix, the spectrum of the fluctuations matrix converges to the Wigner\nsemi-circle law [Granziol et al., 2020b, Wigner, 1993], i.e. weakly almost surely\n1\nPPX\ni=1\u000e\u0015i(X)!\u0016SC; (7)\nwhere the \u0015i(X)are the eigenvalues of Xandd\u0016SC(x)/p\n2P2\u0000x2dx. The key\nintuition in this paper is that sharp directions of the true loss surfaces, that is directions\nin which the true Hessian has its largest eigenvalues, are more reliably estimated by\nthe batch loss than are the flat directions (those with small Hessian eigenvalues). This\nintuition is natural in random matrix theory and is supported by results such as the\nfollowing.\nTheorem 1. Letf\u0012igP\ni=1,f\u001egP\ni=1be the orthonormal eigenbasis of the true Hessian\nr2Ltrueand batch Hessian r2Lbatchrespectively. Let also \u0017\u0015:::\u0015\u0017Pbe the eigenvalues\nkNote that although a given batch Hessian is a fixed deterministic property, we are interested in\ngeneric properties of batches drawn at random from the data generating distribution for which we\nmake statements and can hence model the fluctuations matrix as a random matrix.A Random Matrix Theory Approach to Damping in Deep Learning 9\nofr2Ltrue. Assume that \u0017i= 0for alli > r, for some fixed r. Assume that Xis a\ngeneralised Wigner matrix. Then as P!1the following limit holds almost surely\nj\u0012T\ni\u001eij2!8\n<\n:1\u0000P\u001b2\nB\u0017i2ifj\u0017ij>q\nP\nB\u001b;\n0 otherwise;(8)\nwhere\u001bis the sampling noise per Hessian element.\nProof.This is a direct application of a result of [Capitaine and Donati-Martin, 2016]\nwhich is given more explicitly in the case of GOE Wigner matrices by\n[Benaych-Georges and Nadakuditi, 2011]. In particular, we use a scaling of Xsuch that\nthe right edge of the support of its spectral semi-circle is roughly at P1=2B\u00001=2\u001b. The ex-\npression in Section 3.1 of [Benaych-Georges and Nadakuditi, 2011] can then be applied\ntoP\u00001=2Hbatchand re-scaled inp\nPto give the result. Note that the substantiation of\ntheexpressionfrom[Benaych-Georges and Nadakuditi, 2011]inthecaseofquitegeneral\nWigner matrices is given by Theorem 16 of [Capitaine and Donati-Martin, 2016].\nResults like Theorem 1 are available for matrix models other than Wigner, such as\nrotationallyinvariantmodel[Belinschi et al., 2017], andareconjecturedtoholdforquite\ngeneral{models [Benaych-Georges and Nadakuditi, 2011]. We prove in the appendix\nsection Appendix A, under some conditions, convergence of the spectral measure of\nP\u00001=2Xto the semi-circle. This is necessary to obtain (8), but not sufficient. The\ntechnicalities to rigorously prove Theorem 1 without assuming a Wigner matrix for\nXare out of scope for the present work, requiring as they would something like\nan optimal local semi-circle law for X[Erdős and Yau, 2017]. We require only the\ngeneral heuristic principle from random matrix theory encoded in (8), namely that only\nsharp directions retain information from the true loss surface . It is expected that this\nprinciple will hold for a much wider class of random matrices than those for which it has\nbeen rigorously proven. This is acutely important for adaptive methods which rely on\ncurvature estimation, either explicitly for stochastic second order methods or implicitly\nfor adaptive gradient methods.\nThe spectrum of the noise matrix occupies a continuous region that is\nsharp in the asymptotic limit [Bun et al., 2017] known as bulksupported between\n[\u0015\u0000;\u0015+][Bun et al., 2017, Bun et al., 2016b, Bun et al., 2016a] and observed in DNNs\n[Granziol et al., 2019, Papyan, 2018, Sagun et al., 2017]. Within this bulk eigenvectors\nare uniformly distributed on the unit sphere [Benaych-Georges and Nadakuditi, 2011]\nand all information about the original eigenvalue/eigenvector pairs is lost\n[Baik et al., 2005]. Hence from a theoretical perspective it makes no sense to estimate\nthese directions and move along them accordingly. An eigenvalue, \u0015i, corresponds to a\nflatdirection if \u0015i\u0014\u0015+. For finite-size samples and network size, there exists a region\nbeyond the predicted asymptotic support of the noise matrix, called the Tracy–Widom\n{Roughly speaking, models for which a local law can be established [Erdős and Yau, 2017].A Random Matrix Theory Approach to Damping in Deep Learning 10\nEigenvalue \n-\n +\nDensity \n (\n)\nContinuous Bulk\n(a) Hypothetical \u001a(\u0015)\n0.258\n 3.296108\n106\n104\n102\n100\n (b) Val Acc = 94:3, SGD\n6.28\n 58.59108\n106\n104\n102\n100\n (c) Val Acc = 95:1, Adam\nFigure 2: (a) Hypothetical spectral density plot with a sharply supported continuous\nbulk region, a finite size fluctuation shown in blue corresponding to the Tracy-Widom\nregion and three well-separated outliers shown in red. (b,c) VGG- 16Hessian on the\nCIFAR- 10dataset at epoch 300for SGD and Adam respectively. Note the \"sharper\"\nsolution has better validation accuracy.\nregion [Tracy and Widom, 1994, El Karoui et al., 2007], where there may be isolated\neigenvalues which are part of the noise matrix spectrum (also shown in Fig. 2a). The\nwidth of the Tracy–Widom region is very much less than that of the bulk. Anything\nbeyond the Tracy–Widom region \u0015i\u001d\u0015+,\u0015i\u001c\u0015\u0000is considered an outlier and cor-\nresponds to a sharpdirection. Such directions represent underlying structure from the\ndata. The eigenvectors corresponding to these eigenvalues can be shown to lie in a cone\naround their true values [Benaych-Georges and Nadakuditi, 2011] (see Theorem 1). In\nFig. 2b, we show the Hessian of a VGG- 16network at the 300thepoch on CIFAR-100.\nHere, similar to our hypothetical example, we see a continuous region, followed by a\nnumber of eigenvalues which are close to (but not within) the bulk, and finally, several\nclear outliers.\n5. Detailed experimental investigation of Hessian directions\nIn this section we seek to validate our conjecture that movements in the sharp direction\nof the loss landscape are inherently vital to generalisation by studying a convex non-\nstochastic example. For such a landscape there is only a single global minimum\nand hence discussions of bad minima are not pertinent. We implement a second-\norder optimiser based on the Lanczos iterative algorithm [Meurant and Strakoš, 2006]\n(LanczosOPT) against a gradient descent (GD) baseline. We employ a training set\nof1K MNIST [LeCun, 1998] examples using logistic regression and validate on a held\nout test set of 10K examples. Each optimiser is run for 500epochs. The Lanczos\nalgorithm is an iterative algorithm for learning a subset of the eigenvalues/eigenvectors\nof any Hermitian matrix, requiring only matrix–vector products. When the number\nof Lanczos steps, m, is significantly larger than the number of outliers, the outliers\nin the spectrum are estimated effectively [Granziol et al., 2019]. Since the number\nof well-separated outliers from the spectral bulk is at most the number of classes\n[Papyan, 2018] (which is nc= 10for this dataset), we expect the Lanczos algorithm\nto pick out these well-separated outliers when the number of iterations k\u001dnc\n[Granziol et al., 2019, Meurant and Strakoš, 2006] and therefore use k= 50. ToA Random Matrix Theory Approach to Damping in Deep Learning 11\n1 3 10\n0.001\n0.01\n0.1\n1.0\n0.000.020.040.060.080.100.120.14\n(a)\u0001(\u000e;\u0011)Training\n1 3 10\n0.001\n0.01\n0.1\n1.0\n0.000.020.040.060.080.100.120.14 (b)\u0001(\u000e;\u0011)Testing\nFigure 4: Error change with damping/sharp direction perturbation \u000e;\u0011in LanczosOPT,\nrelative to the single best run. Darker regions indicate higher error.\ninvestigate the impact of scaling steps in the Krylov subspace given by the sharpest\ndirections, we consider the update wk+1of the form:\nwk\u0000\u000b\u00121\n\u0011kX\ni=11\n\u0015i+\u000e\u001ei\u001eT\nirL(wk) +PX\ni=k+11\n\u000e\u001ei\u001eT\nirL(wk)\u0013\n(9)\nwhereP= 7850(the number of model parameters) and hence the vast majority of flat\ndirections remain unperturbed. To explore the effect of sharp directions more explicitly,\nwe have introduced perturbations to the optimiser (denoted LOPT [\u0011]), in which we\nreduce the first term in the parenthesis of Equation 9 by a factor of \u0011(we explore\nscaling factors of 3and10). This reduces movement in sharp directions, consequently\nincreases reliance on flat directions (which are left largely unperturbed). For a fixed \u000b,\n\u000econtrols the estimated curvature learning rate ratio.\nExperimental Results: In Fig. 4, where we show in heat map form the difference from\nthebesttrainingandtestingerrorasafunctionof \u000eand\u0011,weobserveevidenceconsistent\nwith our central hypothesis. As we increase Rest-curv(by decreasing the value of \u000efor a\nfixed\u000bvalue of 0:01), the generalisation of the model suffers correspondingly.\nFor each fixed value of \u000e, we see clearly that perturbations of greater magnitude\ncause greater harm to generalisation than training. We also note that for larger values\nof\u000ethe perturbed optimisers suffer more gravely in terms of the effect on both training\nand validation. We show the full training curves in Figure 5. We observe that the\ngeneralisation of all algorithms is worsened by explicit limitation of movement in the\nsharp directions (and an increase of estimated curvature learning rate ratio), however\nfor extremely low damping measures (which are typical in adaptive optimiser settings)A Random Matrix Theory Approach to Damping in Deep Learning 12\n0 100 200 300 400 500\nEpoch0.00.20.40.60.81.0Training ErrorLOPT (=0.001)\nLOPT[3](=0.001)\nLOPT[10] (=0.001)\nLOPT (=0.01)\nLOPT[3](=0.01)\nLOPT[10] (=0.01)\nLOPT (=0.1)\nLOPT[3](=0.1)\nLOPT[10] (=0.1)\nLOPT (=1)\nLOPT[3](=1)\nLOPT[10] (=1)\nGD480 4900.000.050.10\n0 100 200 300 400 500\nEpoch0.20.40.60.81.0Validation Error480 4900.130.150.170.19\nFigure 5: Training/test error of LanczosOPT/Gradient Descent (LOPT/GD) optimisers\nfor logistic regression on the MNIST dataset with fixed learning rate \u000b= 0:01across\ndifferent damping values, \u000e. LOPT [\u0011]denotes a modification to the LOPT algorithm\nthat perturbs a subset of update directions by a factor of \u0011. Best viewed in colour.\nthere is no or very minimal impact in training performance (upper region of Fig. 4 (a)).\nFashion MNIST: We repeat the experimental procedure for the FashionMNIST\ndataset [Xiao et al., 2017], which paints an identical picture (at slightly higher testing\nerror) The full training curves are given in Figure 6.\n0 100 200 300 400 500\nEpoch0.00.20.40.60.81.0Training ErrorLOPT (=0.01)\nLOPT[3](=0.01)\nLOPT[10] (=0.01)\nLOPT (=0.1)\nLOPT[3](=0.1)\nLOPT[10] (=0.1)\nLOPT (=1)\nLOPT[3](=1)\nLOPT[10] (=1)\nGD480 4900.000.050.10\n0 100 200 300 400 500\nEpoch0.20.40.60.81.0Validation Error480 4900.220.250.27\nFigure 6: Training/test error of LanczosOPT/Gradient Descent (LOPT/GD) optimisers\nfor logistic regression on the FashionMNIST dataset with fixed learning rate \u000b= 0:01\nacross different damping values, \u000e. LOPT [\u0011]denotes a modification to the LOPT\nalgorithm that perturbs a subset of update directions by a factor of \u0011. Best viewed\nin colour.\n6. The role of damping\nConsider a general iterative optimiser that seeks to minimise the scalar loss L(w)for a\nset of model parameters w2RP. Recall the k+ 1-th iteration of such an optimiser can\nbe written+as follows:\nwk+1 wk\u0000\u000bkB\u00001rLbatch(wk) (10)\n+Ignoring additional features such as momentum and explicit regularisations.A Random Matrix Theory Approach to Damping in Deep Learning 13\nwhere\u000bkis the global learning rate. For SGD, B=Iwhereas for adaptive methods, B\ntypically comprises some form of approximation to the Hessian i.e. B\u0019r2Lbatch(wk).\nWriting this update in the eigenbasis of the Hessian\u0003r2Lbatch(wk) =PP\ni\u0015i\u001ei\u001eT\ni2\nRP\u0002P, where\u00151\u0015\u00152\u0015\u0001\u0001\u0001\u0015\u0015P\u00150represent the ordered scalar eigenvalues, the\nparameter step takes the form:\nwk+1=wk\u0000PX\ni=1\u000b\n\u0015i+\u000e\u001ei\u001eT\nirLbatch(wk): (11)\nHere,\u000eis a damping (or numerical stability) term. This damping term\n(which is typically grid searched [Dauphin et al., 2014] or adapted during training\n[Martens and Grosse, 2015]) can be interpreted as a trust region [Dauphin et al., 2014]\nthat is required to stop the optimiser moving too far in directions deemed flat ( \u0015i\u00190),\nknown to dominate the spectrum in practice [Granziol et al., 2020b, Papyan, 2018,\nGhorbani et al., 2019], and hence diverging. In the common adaptive optimiser Adam\n[Kingma and Ba, 2014], it is set to 10\u00008. For small values of \u000e,\u000bmust also be small to\navoid optimisation instability, hence global learning rates and damping are coupled in\nadaptive optimisers.\n6.1. Adaptive updates, damping and the estimated curvature learning rate ratio\nThe learning rate in the flattest ( \u0015\u00190) directions is approximately\u000b\n\u000e, which is larger\nthan the learning rate in the sharpest ( \u0015i\u001d\u000e) directions\u000b\n\u000e+\u0015i. This difference in per\ndirectioneffectivelearning ratemakesthebestpossible(damped)trainingloss reduction\nunder the assumption that the loss function can be effectively modelled by a quadratic\n[Martens, 2016]. Crucially, however, it is agnostic to how accurately each eigenvector\ncomponent of the update estimates the true underlying loss surface, which is described\nin Theorem 1. Assuming that the smallest eigenvalue \u0015P\u001c\u000e, we see thatRest-curv =\n1 +\u00151\u0000\u0015P\n\u000e. This is in contrast to SGD where wk+1=wk\u0000PP\ni=1\u000b\u001ei\u001eT\nirLbatch(wk)\nand henceRest-curv = 1. Note that we can ignore the effect of the overlap between the\ngradient and the eigenvectors of the batch Hessian because we can rewrite the SGD\nupdate in the basis of the batch Hessian eigenvectors and hence reduce the problem to\none of the relative learning rates.\nThecrucialpointtonotehereisthatthedifferencein Rest-curvisprimarilycontrolled\nby the damping parameter: smaller values yield a larger Rest-curv, skewing the parameter\nupdates towards flatter directions.\nTo further explore our central conjecture for modern deep learning architectures\n(where a large number of matrix–vector products is infeasible) we employ the KFAC\n[Martens and Grosse, 2015] and Adam [Kingma and Ba, 2014] optimisers on the VGG-\n16[Simonyan and Zisserman, 2014]networkontheCIFAR- 100[Krizhevsky et al., 2009]\ndataset. The VGG-16 allows us to isolate the effect of Rest-curv, as opposed to the effect\n\u0003We assume this to be positive definite or that we are working with a positive definite approximation\nthereof.A Random Matrix Theory Approach to Damping in Deep Learning 14\nof different regularisation implementations for adaptive and non-adaptive methods as\ndiscussed by [Loshchilov and Hutter, 2018, Zhang et al., 2018b].\n6.2. VGG16: a laboratory for adaptive optimisation\nThe deep learning literature contains very many architectural variants of deep neural\nnetworks and a large number of engineering “tricks” which are employed to obtain\nstate of the art results on a great variety of different tasks. The theory supporting\nthe efficacy of such tricks and architectural designs is often wanting and sometimes\nentirely absent. Our primary objective in this work is to illuminate some theoretical\naspects of adaptive optimisers such as appropriate damping and Hessian estimation,\nso we require a simple and clean experimental environment free from, where possible,\ninterferencefromasmanydifferentcompetingeffects. Tothisend, theVGGarchitecture\n[Simonyan and Zisserman, 2014] for computer vision is particularly appropriate. With\n16 layers, the VGG has over 16million parameters and is capable of achieving\ncompetitive test error on a variety of standard computer vision datasets while being\ntrained without batch normalisation [Ioffe and Szegedy, 2015] or weight decay. Indeed,\nfeatures such as weight decay and batch normalisation obscure the effect of learning\nrate and damping, meaning that even quite poor choices can ultimately give reasonable\nresults given sufficient training iterations[Granziol et al., 2020b]. In contrast the VGG\nclearly exposes the effects of learning rate and damping, with training being liable to\nfail completely or diverge if inappropriate values are used. Furthermore as shown in\n[Granziol et al., 2020b] the VGG is highly unstable if too large a learning rate is used.\nThisallowsustoveryexplicitlytestwhetheramendmentsprovidedbytheoryarehelpful\nin certain contexts, such as training stability, as unstable training very quickly leads to\ndivergence.\nLearning Rate Schedule For all experiments unless specified, we use the following\nlearning rate schedule for the learning rate at the t-th epoch:\n\u000bt=8\n>><\n>>:\u000b0; ift\nT\u00140:5\n\u000b0[1\u0000(1\u0000r)(t\nT\u00000:5)\n0:4]if0:5<t\nT\u00140:9\n\u000b0r; otherwise(12)\nwhere\u000b0is the initial learning rate. Tis the total number of epochs budgeted for all\nCIFAR experiments. We set r= 0:01for all experiments.\n6.3. KFAC with VGG-16 on CIFAR-100:\nBy decreasing the global learning rate \u000bwhilst keeping the damping-to-learning-rate\nratio\u0014=\u000e\n\u000bconstant, we increase the estimated curvature learning rate ratio, Rest-curv,\nwhich is determined by\u0015i\n\u0014\u000b+1. As shown in Tab. 1 and in Figure 7 we observe that as we\nincreaseRest-curvthe training performance is effectively unchanged, but generalisationA Random Matrix Theory Approach to Damping in Deep Learning 15\n\u000b\n\u000e0.0004 0.001\n1e-753.1(62.9)\n4e-421.1(64.5)\n1e-39.9(63.5) 20.8(64.4)\n5e-3 9.1(66.2)\n8e-3 2.4(65.8)\u000b\n\u000e0.1 0.001 0.0001\n1e-16.7(65.0)\n1e-2 20.8(64.8)\n1e-3 48.2(62.2)\n3e-4 527.2(60.2)\n1e-4 711.3(56.0)\nTable 1: Spectral norms and generalisation . We report the spectral norm \u00151at\nthe end of training in bold, with corresponding validation accuracy (in parentheses)\nfor learning rate/damping \u000b;\u000eusing Adam (left) and KFAC (right) to train a VGG- 16\nnetwork on CIFAR- 100.\nsuffers ( 35%!37:8%). Whilst decreasing the damping results in poor training for\nlarge learning rates, for very low learning rates the network efficiently trains with a\nlower damping coefficient. Such regimes further increase Rest-curvand we observe that\nthey generalise more poorly. For \u000b= 0:0001dropping the damping coefficient \u000efrom\n0:0003to0:0001drops the generalisation further to 60:2%and then 56%respectively.\nSimilar to logistic regression, for both cases the drop in generalisation is significantly\nlarger than the drop in training accuracy.\n0 50 100 150 200 250 300\nEpoch0.00.20.40.60.81.0Training Error =0.01 =0.1\n =0.01 =0.03\n =0.001 =0.01\n =0.001 =0.003\n =0.0001 =0.001\n =0.0001 =0.0003\n =0.0001 =0.0001\n297 298 2990.0000.0050.0100.015\n0 50 100 150 200 250\nEpoch0.40.60.81.0Validation Error290 2950.360.380.40\nFigure 7: Training/validation error of the KFAC optimiser for VGG- 16on the CIFAR-\n100dataset with various learning rates \u000band damping values, \u000e.\nAdam with VGG-16 on CIFAR-100: We employ Adam with a variety of learning rate\nand damping coefficients with results as shown in Tab. 1 and in Figure 8 and compare\nagainst a baseline SGD with \u000b= 0:01(corresponding to optimal performance). For the\nlargest learning rate with which Adam trains ( \u000b= 0:0004) with the standard damping\ncoefficient\u000e= 10\u00008, we see that Adam under-performs SGD, but that this gap is\nreduced by simply increasing the damping coefficient without harming performance.\nOver-damping decreases the performance. For larger global learning rates enabled by\na significantly larger than default damping parameter, when the damping is set tooA Random Matrix Theory Approach to Damping in Deep Learning 16\n0 50 100 150 200 250 300\nEpoch0.00.20.40.60.81.0Training Error=0.0004 =1e-08\n=0.0004 =0.0004\n=0.0004 =0.001\n=0.001 =0.0003\n=0.001 =0.0006\n=0.001 =0.001\n=0.001 =0.005\n=0.001 =0.0075\nSGD =0.01\n298 2990.0000.002\n0 50 100 150 200 250 300\nEpoch0.40.60.81.0Validation Error2950.340.360.38\nFigure 8: Training/validation error of the Adam optimiser for VGG- 16on the CIFAR-\n100dataset with various learning rates \u000band damping values, \u000e.\nlow, the training is unstable (corresponding to the dotted lines). Nevertheless, many of\nthese curves with poor training out-perform the traditional setting on testing. We find\nthat for larger damping coefficients \u000e= 0:005;0:0075Adam is able to match or even\nbeat the SGD baseline, whilst converging faster. We show that this effect is statistically\nsignificant in Tab. 2. This provides further evidence that for real problems of interest,\nadaptive methods are not worse than their non-adaptive counterparts as argued by\n[Wilson et al., 2017]. We note as shown in Tab. 1, that whilst increasing \u000ealways\nleads to smaller spectral norm, this does not always coincide with better generalisation\nperformance. We extend this experimental setup to include both batch normalisation\n[Ioffe and Szegedy, 2015] and decoupled weight decay [Loshchilov and Hutter, 2018].\nWe use a learning rate of 0:001and a decoupled weight decay of [0;0:25]. For this\nexperiment using a larger damping constant slightly assists training and improves\ngeneralisation, both with and without weight decay.\n0 50 100 150 200 250 300\nEpoch0.00.20.40.60.81.0Training Error=0.001, =1e-08, =0.0\n=0.001, =0.0001, =0.0\n=0.001, =1e-08, =0.25\n=0.001, =0.0001, =0.25\n298 2990.0000.002\n0 50 100 150 200 250 300\nEpoch0.40.60.81.0Validation Error2950.280.300.32\nFigure 9: Training/validation error of the Adam optimiser for VGG- 16BN using Batch\nNormalisation and Decoupled Weight Decay on the CIFAR- 100dataset with various\nlearning rates \u000band damping values, \u000e.\nResNet- 50ImageNet. AsshowninFig.10a,10b, theseprocedureshavepracticalimpact\non large scale problems. Here we show that under a typical 90epoch ImageNet\nsetup [He et al., 2016], with decoupled weight decay 0:01for AdamW and 0:0001\nfor SGD, that by increasing the numerical stability constant \u000ethe generalisation\nperformance can match and even surpass that of SGD, which is considered state-of-\nthe-art and beats AdamW without \u000etuning by a significant margin.A Random Matrix Theory Approach to Damping in Deep Learning 17\n10 20 30 40 50 60 70 80\nEpoch0.20.40.60.8Training ErrorAdamW\nAdamW =0.001\nAdamW =0.005\nSGD\n(a) ResNet- 50Training Error\n10 20 30 40 50 60 70 80\nEpoch0.30.40.50.60.70.8Validation ErrorAdamW\nAdamW =0.001\nAdamW =0.005\nSGD0.240.26 (b) ResNet- 50Testing Error\nFigure 10: (a-b) The influence of \u000eon the generalisation gap. Train/Val curves for\nResNet- 50onImageNet. Thegeneralisationgapiscompletelyclosedwithanappropriate\nchoice of\u000e.\nDataset Classes Model Architecture SGD Adam-D Adam\nCIFAR-100 100 VGG16 65.3 \u00060.6 65.5\u00060.7 61.9\u00060.4\nImageNet 1000 ResNet50 75.7 \u00060.1 76.6\u00060.1 74.04*\nTable 2: Statistical Significance. Comparison of test accuracy across CIFAR 100 (5\nseeds) and ImageNet (3 seeds). Adam-D denotes Adam with increased damping\n(\u000e= 5e\u00003for CIFAR-100, \u000e= 1e\u00004for ImageNet). *Since it is well established that\nVanilla Adam does not generalise well for ImageNet, we do not run this experiment\nfor multiple seeds, we simply report a single seed result for completeness. A more\ncomplete discussion for Adam and its generalisation in vanilla form can be found in\n[Granziol et al., 2020a].\n7. Optimal adaptive damping from random matrix theory\nRecall the scaling applied in the direction of the itheigenvector in (11). We make the\nfollowing observation\n1\n\u0015i+\u000e=1\n\f\u0015i+ (1\u0000\f)\u00011\n\u0014(13)\nwhere\u0014=\f\u00001;\f= (1 +\u000e)\u00001. Hence, using a damping \u000eis formally equivalent to\napplying linear shrinkage with factor \f= (1 +\u000e)\u00001to the estimated Hessian and\nusing a learning rate of \u000b\f. Shrinkage estimators are widely used in finance and\ndata science, with linear shrinkage being a common simple method applied to improve\ncovariance matrix estimation [Ledoit and Wolf, 2004]. The practice of shrinking the\neigenvalues while leaving the eigenvectors unchanged is well-established in the fields\nof sparse component analysis and finance [Bun et al., 2017]. [Bun et al., 2016a] even\nshow that under the free multiplicative noise model, the linear shrinkage estimator,\n~H=\fH+ (1\u0000\f)I= arg minH\u0003jjH\u0003\u0000Htruejj2, gives the minimum error betweenA Random Matrix Theory Approach to Damping in Deep Learning 18\nthe estimator and the true Hessian, and an explicit expression for the optimal \fis\nfound depending only on the dimensionality of the model and the noise variance. In our\noptimisation context, the linear shrinkage estimator is unlikely to be optimal, however\nit has the great advantage of being simple to integrate into existing adaptive optimisers\nand it acts intuitively to reduce the movement of the optimiser in pure-noise directions.\nOur interpretation reveals that the damping parameter should not be viewed as a mere\nnumerical convenience to mollify the effect of very small estimate eigenvalues, but rather\nthat an optimal \u000eshould be expected, representing the best linear approximation to the\ntrue Hessian and an optimal balancing of variance (the empirical Hessian) and bias\n(the identity matrix). This optimal choice of \u000ewill produce an optimiser that more\naccurately descends the directions of the true loss.\nThe linear shrinkage interpretation given by (13) is an elementary algebraic relation\nbut does not by itself establish any meaningful link between damping of adaptive\noptimisers and linear shrinkage estimators. To that end, we return to the random\nmatrix model (6) for the estimated Hessian: Let us write the Hessian as\nHbatch =Htrue+X\nwhereXis a random matrix with EX= 0. Note that this model is entirely general, we\nhave simply defined X=Hbatch\u0000EHbatchandEHbatch =Htrue. We then seek a linear\nshrinkage estimator ~H(\f) =\fHbatch+ (1\u0000\f)Isuch thatE(\f) =P\u00001Tr(~H\u0000Htrue)2\nis minimised. Note that this is the same objective optimised by [Bun et al., 2016a] to\nobtain optimal estimators for various models. In this context, we are not finding the\noptimal estimator for Htruebut rather the optimal linear shrinkage estimator. We have\nE(\f) =1\nPTr [(\f\u00001)Htrue+\fX+ (1\u0000\f)I]2\u00111\nPTr [(\f\u00001)Htrue+Y\f]2\nwhereY\f=\fX+ (1\u0000\f)I.\nA natural assumption in the case of deep learning is that Htrueis low-\nrank, i.e. for P! 1either rank (Htrue) =ris fixed or rank (Htrue) =\no(P). Empirical evidence for this assumption is found in [Granziol et al., 2020b,\nSagun et al., 2016, Sagun et al., 2017, Papyan, 2018, Ghorbani et al., 2019]. In this\ncase the bulk of the spectrum of Y\fis the same as that of (\f\u00001)Htrue+Y\f\n[Benaych-Georges and Nadakuditi, 2011,Capitaine and Donati-Martin, 2016,Belinschi et al., 2017].\nWe will also assume that Xadmits a deterministic limiting spectral measure \u0016Xsuch\nthat\n1\nPPX\nj=1\u000e\u0015(X)i!\u0016 (14)\nweakly almost surely. Say !X(x)dx=d\u0016(x). ThenY\fhas limiting spectral density\n!Y(y) =\f\u00001!X(\f\u00001(y\u00001 +\f)):A Random Matrix Theory Approach to Damping in Deep Learning 19\nThen for large P\nE(\f)\u0019\f\u00001Z\ny2!X(\f\u00001(y\u00001 +\f))dy=Z\n(\fx+ 1\u0000\f)2!X(x)dx\n=\f2\u0016X(x2) + (1\u0000\f)2\nas the centred assumption on Xmeans thatR\nx!X(x)dx= 0.\u0016X(x2)is shorthand forR\nx2!X(x)dx.E(\f)is thus minimised to leading order at \f= (1+\u0016X(x2))\u00001. Recalling\nthat\f\u00001= (1 +\u000e)\u00001, this yields \u000e=\u0016X(x2)i.e. the optimal level of damping at large\nfinitePis approximately\n\u000e=P\u00001TrX2: (15)\nNote that the value (15) is a very natural measure of the Hessian noise variance.\nTherefore if the random matrix model described above is appropriate and the linear\nshrinkage interpretation (13) is meaningful we should expect it to result in close to\noptimal performance of a given adaptive optimiser. The purpose of adaptive optimisers\nis to accelerate training, in part by allowing for larger stable learning rates. As discussed\nthroughout this paper, such optimisation speed often comes at the cost of degraded\ngeneralisation. In this context, ‘optimal performance’ of adaptive optimisers should be\ntaken to mean fast training and good generalisation. As we have discussed above, very\nlarge values of \u000erecover simple non-adaptive SGD, so using (15) we should be able to\nobtain generalisation performance at least as good as SGD and faster optimisation than\nany choice of \u000eincluding the default very small values often used and the larger values\nconsidered in Section 6.\nThe value of (15) can be easily learned by estimating the variance of the Hessian.\nThe Hessian itself cannot be computed exactly, as it is far too large for P\u0015O(107),\nhowever one can compute Hv(and henceH2v) for any vector v, usingr2Lv=\nr(vTrL). The full approach is given in Algorithm 1.\nAlgorithm 1 Algorithm to estimate the Hessian variance\n1:Input:Sample Hessians Hi2RP\u0002P,1\u0014i<N\n2:Output: Hessian Variance \u001b2\n3:v2R1\u0002P\u0018N(0;I)\n4:Initialise\u001b2= 0;i= 0,v v=jjvjj\n5:fori<N do\n6:\u001b2 \u001b2+vTH2\niv\n7:i i+ 1\n8:end for\n9:\u001b2 \u001b2\u0000[vT(1=NPN\nj=1Hj)v]2\n7.1. Experimental Design and Implementation Details\nIn order to test our hypothesis for the derived optimal \u000e(15), we run the classical VGG\nnetwork [Simonyan and Zisserman, 2014] with 16layers on the CIFAR- 100dataset,A Random Matrix Theory Approach to Damping in Deep Learning 20\nwithout weight decay or batch normalisation. This gives us maximal sensitivity to\nthe choice of learning rate and appropriate damping.\nNow in practice the damping coefficient is typically grid searched over several runs\n[Dauphin et al., 2014] or there are heuristics such as the Levenberg–Marquardt to adapt\nthedampingcoefficient[Martens and Grosse, 2015], whichhoweverwefinddoesnotgive\nstable training for the VGG. We hence compare against a fixed set damping value \u000eand\na learned damping value as given by our equation (15). We find that the variance\nof the Hessian (15) at a random point in weight space (such as at initialisation) or\nonce network divergence has occurred is zero, hence the initial starting value cannot\nbe learned as, with a damping of near zero, the network entirely fails to train (no\nchange in training loss from random). This is to be expected, as in this case the local\nquadratic approximation to the loss inherent in adaptive methods breaks down. Hence\nwe initialise the learning algorithm with some starting value \u000e\u0003, which is then updated\nevery 100training iterations using equation (15). Strictly speaking we should update\nevery iteration, but the value of 100is chosen arbitrarily as a computational efficiency.\nSince we are using the variance of the Hessian, which is expensive to compute compared\nto a simple gradient calculation, we do not want to compute this quantity too often if\nit can be helped. We run our experiments on a logarithmic grid search in near factors\nof3. So learning rates and damping rates, either flat or learned are on the grid of\n0:0001;0:0003;0:001:::.\nWe find under this setup that the time taken per epoch against the flat damping\nschedule is only doubled. We get identical results for using a damping gap of 10and\nso do not consider this to be a very relevant hyper-parameter. We further calculate the\nvariance of the Hessian over a sub-sample of 10000examples and do not calculate the\nvariance sample by sample, but over batches of 128to speed up the implementation.\nUndertheassumptionthatthedataisdrawni.i.dfromthedatasetthevarianceissimply\nreduced by a factor (1\nB\u00001\nN)\u00191\nBfor a small batch size. We do not consider the impact\nof using only a sub-sample of the data for estimation, but we expect similar results to\nhold compared to the enitre dataset as long as the sub-sample size S\u001dB. This should\nallow such a method to be used even for very large datasets, such as ImageNet (with\n1-million images), for which a pass of the entire dataset is extremely costly. In theory\nthe sub-sample size and mini-batch size for Hessian variance estimation could be two\nhyper-parameters which are tuned by considering the effect of reduction on training\nset or validation set loss metrics with the trade off for computational cost. We do not\nconduct such analysis here.\nWe also incorporate an exponential moving average into the learned damping with\na co-efficient of 0:7]to increase the stability of the learned damping.\n]This value is not tuned and in fact from our plots it may be advisable to consider higher values for\ngreater stabilityA Random Matrix Theory Approach to Damping in Deep Learning 21\n7.2. Experiment on CIFAR-100 using KFAC to validate the optimal linear shrinkage\nFor large damping values \u000ewe simply revert to SGD with learning rate \u000b=\u000e, so we\nfollow the typical practice of second order methods and use a small learning rate\nand correspondingly small damping coefficient. However as shown in Figure 1 the\ngeneralisation and optimisation are heavily dependent on the global learning rate, with\nlarger learning rates often optimising less well but generalising better and vice versa\nfor smaller learning rates. We hence investigate the impact of our damping learner\non learning rates one order of magnitude apart. Where in the very low learning rate\nregime, we show that our method achieves singificantly improved training stability with\nlow starting damping and fast convergence and for the large learning rate regime that\nwe even exceed the SGD validation set result.\nTraining KFAC with Auto-Damping: We show the results for a global learning rate\nof0:0001in Figure 11a. We see that for the flat damping methods with low values\nof damping, that training becomes unstable and diverges, despite an initially fast\nstart. Higher damped methods converge, but slowly. In stark contrast, our adaptive\ndamping method is relatively insensitive to their chosen initial values. We show here\n\u000e\u0003=\u000b;3\u000b;10\u000band all converge and moreover significantly faster than all flat damping\nmethods. The smaller initial damping coefficients \u000e=\u000b;3\u000bconverge faster than the\nlarger and, interestingly, follow very similar damping trajectories throughout until the\nvery end of training, as shown in Figure 11b.\n20 40 60 80 100\nEpoch0.00.20.40.60.81.0Training Error=0.001\n=0.003\n=0.01\n=0.03\n*=0.0001\n*=0.0003\n*=0.001\n(a) Training Error as a function of Epoch\n50 100\nEpoch104\n103\n102\n101\n100Damping*=0.0001\n*=0.0003\n*=0.001\n (b) Damping per Epoch\nFigure 11: VGG- 16on CIFAR- 100dataset using the KFAC optimiser with \r= 0(no\nweight decay) for a learning rate of \u000b= 0:0001, batch size B= 128and damping set by\n\u000e. For adaptive damping methods the damping is given an initial floor value of \u000e\u0003and\nis then updated using the variance of the Hessian every 100steps.\nGetting Great Generalisation with KFAC and Auto-Damping: WesimilarlytrainKFAC\non the VGG- 16with a larger learning rate of 0:001, in order to achieve betterA Random Matrix Theory Approach to Damping in Deep Learning 22\ngeneralisation. Here we see in Figure 12a that relatively low values of flat damping\nsuch as 0:01and0:03very quickly diverge, wheras a large value of 0:1converges slowly\nto a reasonable test error. The corresponding learned damping curves of 0:01and\n0:03however converge quickly and the 0:03initialised damping curve even beats the\ngeneralisation performance of the large flat damped version and the test result of SGD\non3x as many training epochs.\n20 40 60 80 100\nEpoch0.00.20.40.60.81.0Training Error=0.01\n=0.03\n=0.1\n*=0.01\n*=0.03\n(a) Training Error\n0 20 40 60 80 100\nEpoch0.40.60.81.0Validation Error9798990.340.360.38 (b) Val Error\nFigure 12: VGG- 16on CIFAR- 100dataset using the KFAC optimiser with \r= 0(no\nweight decay) for a learning rate of \u000b= 0:001, batch size B= 128and damping set by\n\u000e. For adaptive damping methods the damping is given an initial floor value of \u000e\u0003and\nis then updated using the variance of the Hessian every 100steps.\n20 40 60 80 100\nEpoch0.00.20.40.60.81.0Training Error=0.03\n=0.1\n*=0.0001\n*=0.03\n(a) Training Error\n0 20 40 60 80 100\nEpoch0.40.60.81.0Validation Error970.350.40 (b) Val Error\n50 100\nEpoch0.00.10.20.30.4Damping*=0.0001\n*=0.03\n (c) Damping\nFigure 13: VGG- 16on CIFAR- 100dataset using the KFAC optimiser with \r= 0(no\nweight decay) for a learning rate of \u000b= 0:0001, batch size B= 128and damping set by\n\u000e. For adaptive damping methods the damping is given an initial floor value of \u000e\u0003and\nis then updated using the variance of the Hessian every 100steps.\nA further look at the value of adaptive damping To elucidate the impact and workings\nof the adaptive damping further, we consider a select set of curves the learning rate\nof\u000b= 0:0001, shown in Figure 13. here we see that starting with an initial damping\nof\u000e=\u000b, the adaptive method reaches a comparable generalisation score to the flat\ndamping of \u000e= 0:03but at a much faster convergence rate. The initial damping of\n\u000e= 0:03converges not quite as quickly but trains and generalises better than its lowerA Random Matrix Theory Approach to Damping in Deep Learning 23\nstarting damping counterpart. Note from Figure 13c that even though the damping\nof this curve reaches \u00190:1that starting with a flat damping of 0:1never achieves a\ncomparable generalisation (or even trains well). This implies as expected that it is\nimportant to adjust damping during training.\n7.3. Adam with Auto-Damping\nGiven that Adam does not employ an obvious curvature matrix, it is curious to\nconsider whether our learned damping estimator can be of practical value for this\noptimiser. As discussed in the previous section, Adam’s implied curvature can be\nconsidered a diagonal approximation to the square root of the gradient covariance.\nThe covariance of the gradients has been investigated to have similarities to the Hessian\n[Jastrzebski et al., 2020] and we further show in Section 8 experimentally that adaptive\nmethods do learn information about the sharpest eigenvalue/eigenvector pairs. However\nthe nature of the square root, derived from the regret bound in [Duchi et al., 2011]\npresents an interesting dillemna. In the case of very very small eigenvalues of B, the\nsquare root actually reduces their impact on the optimisation trajectory, hence it is\nvery plausible that the learned damping could be too harsh (as it is expected to work\noptimially for the eigenvalues of Hand notp\nH). This is actually exactly what we see\nin Figure 14. Whilst an increase in learning rate and damping, along with auto-damping\n50 100 150 200 250 300\nEpoch0.00.20.40.60.81.0Training Error =0.0004 =1e-08\n *=0.001 *=0.0003\n *=0.001 *=0.001\n =0.001 =0.001\n =0.003 *=0.003\n(a) Training Error\n0 50 100 150 200 250 300\nEpoch0.40.60.81.0Validation Error290 2950.340.360.38 (b) Val Error\nFigure 14: VGG- 16on CIFAR- 100dataset using the Adam optimiser with \r= 0(no\nweight decay) for a learning rate of \u000b, batch size B= 128and damping set by \u000e. For\nadaptive damping methods the damping is given an initial floor value of \u000e\u0003and is then\nupdated using the variance of the Hessian every 100steps.\u000b\u0003refers to the use of an\nalternative ramp up schedule where the base learning rate is increased by a factor of 5\nat the start of training before being decreased.\nimproves both the convergence and validation result over the standard baseline (where\nthe damping is kept at the default value and maximal learning rate is found which\nstably trains) the improvements are small and do not make up the gap with SGD. More\nspecifically they are not better than just using a larger learning rate in combination\nwith a larger flat damping, defeating the purpose of learning the damping factor online.\nTo alleviate the effect of overly harsh damping, we consider an alternate learningA Random Matrix Theory Approach to Damping in Deep Learning 24\nrate schedule where the base learning rate is increased by a factor of 5early in training\nand then subsequently decreased. The constant 5is not tuned but simply a place-\nholder to consider a more aggressive learning rate schedule to counter-act the effect of\nthe damping learner. These curves are marked with \u000b\u0003in Figure 14.\nWarm up Learning Rate Schedule For all experiments unless specified, we use the\nfollowing learning rate schedule for the learning rate at the t-th epoch:\n\u000bt=8\n>>>>><\n>>>>>:\u000b0; ift\nT\u00140:1\n\u000b0[1 +(\u0014\u00001)(t\nT\u00000:1)\n0:2;ift\nT\u00140:3\n\u000b0[\u0014\u0000(\u0014\u0000r)(t\nT\u00000:3)\n0:6]if0:3<t\nT\u00140:9\n\u000b0r; otherwise(16)\nwhere\u000b0is the initial learning rate. Tis the total number of epochs budgeted for all\nCIFAR experiments. We set r= 0:01and\u0014= 5.\nWhile this introduces some slight training instability early in training, which could\npotentially be managed by altering the schedule, we find that such a schedule boosts\nthe validation performance, particularly so for auto-damped methods, as shown by the\nblue curve in Figure 14b, which surpasses the generalisation of SGD (shown in Figure\n1).\nTo more clearly expose the combined impact of adaptive damping and this\nalternative learning schedule we consider the variations in Figure 15 for a learning rate\nanddampingbothequalto 0:0001. Hereweseethattheaggressivelearningrateschedule\nwith flat damping diverges, whereas the autodamping stabilises training, allowing for\nconvergence to a solution with excellent generalisation. We see here in Figure 15c that\nthe damping coefficient reacts to this large learning rate increase by increasing its rate\nof damping early, stabilising training. We also show for reference that the typical linear\ndecay schedule, with a larger learning rate and initial damping does not supersede the\nvalidation result of smaller learning rate and flat damping counter-part (it does however\ntrain better). This demonstrates the necessity of an alternative learning rate schedule\nto bring out the value of the adaptive damping. We remark however that optimal\nresults in deep learning almost always require some degree of hand-crafted tuning of\nthe learning rate. Our adaptive damping method is not proposed as a panacea, but\njust an optimal method of setting the damping coefficient. Since changing the damping\ncoefficient effectively changes to geometry of the loss surface, it is entirely reasonable\nthat the learning rate may have to be tweaked to give best results.\n8. Quality of adaptive Hessian approximations\nWe have seen so far the importance of sharp directions in the loss landscape, how\nadaptive methods bias away from movement in sharp directions and towards flat\ndirections and we have seen how this issue can be addressed practically and in aA Random Matrix Theory Approach to Damping in Deep Learning 25\n50 100 150 200 250 300\nEpoch0.00.20.40.60.81.0Training Error *=0.001 *=0.001\n =0.001 =0.001\n *=0.001 =0.001\n =0.003 *=0.003\n(a) Training Error\n0 50 100 150 200 250 300\nEpoch0.40.60.81.0Validation Error290 2950.340.36 (b) Val Error\n100 200 300\nEpoch103\n102\n101\nDamping\n *=0.001 *=0.001\n =0.003 *=0.003\n (c) Damping\nFigure 15: VGG- 16on CIFAR- 100dataset using the Adam optimiser with \r= 0(no\nweight decay) for a learning rate of \u000b, batch size B= 128and damping set by \u000e. For\nadaptive damping methods the damping is given an initial floor value of \u000e\u0003and is then\nupdated using the variance of the Hessian every 100steps.\u000b\u0003refers to the use of an\nalternative ramp up schedule where the base learning rate is increased by a factor of 5\nat the start of training before being decreased.\nprincipled manner using linear shrinkage theory. Behind all of this is the idea that using\nsome approximation to the batch Hessian as a preconditioning matrix for gradients is\nhelpful for optimisation. We have so far focused on mitigating the influence of noise in\nthe batch Hessain compared to the true Hessian. Another natural question to ask is:\nhow closely do adaptive preconditioning matrices approximate the batch Hessian?\nAdam uses the square root of the moving uncentered second moment of the\nper-parameter gradient [Kingma and Ba, 2014] that can be interpreted as a running\ndiagonal approximation to the covariance of gradients, argued to be close to the Hessian\nforDNNs[Zhang et al., 2019,Jastrzebski et al., 2020,Fort et al., 2019,Liu et al., 2019,\nHe and Su, 2019]. Other methods employ Hessian–vector products, using either\nLanczos or conjugate gradient techniques [Meurant and Strakoš, 2006] to approximate\nH\u00001\nbatch(wk)rLbatch(w)[Martens, 2010, Dauphin et al., 2014], which for a number of\nproductsm\u001cPcan be seen as a low rank inverse approximation to Hbatch, or direct\ninversion of Kronecker factored approximations [Martens and Grosse, 2015].\nAre Adam and KFAC similar? We validate the assertion that both Adam and KFAC\napproximate the batch Hessian to differing degrees of accuracy and hence fall under the\nsame class of adaptive methods . The batch loss to second order for any adaptive method\nis given by\nL(w\u0000\u000bB\u00001rL) =L\u0000\u000b(rL)TB\u00001=2\u0012\n1\u0000\u000b\n2B\u00001=2HB\u00001=2\u0013\nB\u00001=2rL;(17)\nwhere we drop wdependence and use the positive definiteness of B. This can derived\nusing a Taylor expansion:\nL(w+\u000ew) =L(w) +\u000ewTrL(w) +1\n2\u000ewTrrL(w)\u000ew (18)A Random Matrix Theory Approach to Damping in Deep Learning 26\nNow substituting \u000ew=\u000bB\u00001rL(w)and re-arranging we have\n\u000eL=L(w+\u000ew)\u0000L(w) =\u000b(rL(w))TB\u00001rL(w)+\u000b2\n2(rL(w))TB\u00001rrL(w)B\u00001rL(w)\nfrom which the result (17) follows immediately. From this expression we can derive\nconditions on the largest possible learning rate \u000bfor which the matrix on the right hand\nside of (17) remains positive definite, namely \u000b < 2(kB\u00001=2HB\u00001=2kop)\u00001. Now let\nf\u0011i;\u001eigP\ni=1be the eigenvalues and normalised eigenvectors of B. We have\nkB\u00001=2HB\u00001=2kop= max\nif\u0011\u00001\nij\u001eT\niH\u001eijg:\nFor SGDB=Iand so the largest stable \u000b= 2(kHkop)\u00001, which we be expect to very\nsmall in practice. Now consider replacing \u0011iby\u0011i+\u000e. The largest stable learning rate\nis bounded above by\n2\u0012\nmax\nij\u001eT\niH\u001eij\n\u0011i+\u000e\u0013\u00001\n(19)\nso we again see that the very smallest \u0011ican very much decrease the largest stable\nlearning rate if \u000eis too small, but that increasing \u000emollifies the effect of the very small\n\u0011i.\nNow set the damping \u000e= 1and note that\nX\ni\u001eT\niH\u001ei= TrUTHUX\nieieT\ni= TrH=)P\u00001kHkopmax\nij\u001eT\niH\u001eij\u0014kHkop\n(20)\nfor some orthogonal matrix U, where (ei)j=\u000eij. So let us write maxij\u001eT\niH\u001eij=\n\u0018kHkopwhere\u00182[P\u00001;1]. However let us view Has a random matrix with eigenvalues\nof typical sizeO(p\nP), so its entries are typically of size O(1)with high probability.\nIf the eigenvectors of Hare in general position yyrelative tof\u001eig, then\u001eiH\u001eiare\nthe just the diagonal entries of Hin a general basis and so with high probability\n\u001eiH\u001ei=kHkop\u0018+o(1)for non-zero \u0018withj\u0018j<1. At the other extreme, if f\u001eig\nis the eigenbasis of H, then\u001eT\niH\u001eiare just the eigenvalues of Hand so the same\nresult holds. We therefore generically expect maxij\u001eiH\u001eijto roughly track the largest\neigenvalue of Hand so the largest stable learning rate is determined by how well the\nlargest eigenvalue/eigenvector pair of His estimated by B. Hence under this setup the\nlargest learning rate achievable is a direct measure of the estimation accuracy of the\nsharpest eigenvalue/eigenvector pairs of the batch Hessian. We run SGD, Adam and\nKFAC on the VGG- 16architecture on the CIFAR- 100dataset with no weight decay for\n300epochs using a linear annealed schedule.\nWe search for the highest stable learning rate \u000balong a logarithmic grid, with\nend points2(0:01;1). All methods incorporate a momentum of \u001a= 0:9. We find\nyyWe expect this e.g. in the case that Bis diagonal (such as in Adam) and Xhas delocalised\neigenvectors.A Random Matrix Theory Approach to Damping in Deep Learning 27\nthat the largest stablerates for SGD, Adam and KFAC are 0:01;0:12;0:32respectively,\nindicating that both KFAC and Adam are significantly better able to estimate sharp\ndirections than curvature-blind SGD. Moreover, the larger stable value of KFAC reflects\nthat KFAC is better able to approximate the sharpest directions than Adam, which is\nrestricted to diagonal forms of B.\n9. Conclusion\nWe show using a spiked random matrix model for the batch loss of deep neural networks\nthat we expect sharp directions of loss surface to retain more information about the\ntrue loss surface compared to flatter directions. For adaptive methods, which attempt\nto minimise an implicit local quadratic of the sampled loss surface, this leads to sub-\noptimal steps with worse generalisation performance. We further investigate the effect\nof damping on the solution sharpness and find that increasing damping always decreases\nthe solution sharpness, linking to prior work in this area. We find that for large neural\nnetworks an increase in damping both assists training and is even able to best the SGD\ntest baseline. An interesting consequence of this finding is that it suggests that damping\nshould be considered an essential hyper-parameter in adaptive gradient methods as it\nalready is in stochastic second order methods. Moreover, our random matrix theory\nmodel motivates a novel interpretation of damping as linear shrinkage estimation of the\nHessian. We establish the validity of this interpretation by using shrinkage estimation\ntheory to derive an optimal adaptive damping scheme which we show experimentally to\ndramatically improve optimisation speed with adaptive methods andcloses the adaptive\ngeneralisation gap.\nOur work leaves open several directions for further investigation and extension.\nMathematically, there is the considerable challenge of determining optimal assumptions\non the network, loss function and data distribution such that the key outlier overlap\nresult in Theorem 1, or sufficiently similar analogues thereof, can be obtained. On\nthe experimental side, we have restricted ourselves to computer vision datasets and a\nsmall number of appropriate standard network architectures. 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Three mechanisms of\nweight decay regularization. arXiv preprint arXiv:1810.12281 .\nAppendix A. Convergence of the noise spectrum to the semi-circle\nAs discussed in the main text, the precise form of the overlap given in Theorem 1\nrequires the Hessian noise matrix to have the semi-circle as its limiting spectral density.\nThis can be established under sensible assumptions and we present the argument in this\nappendix. The key tenet is that there is sufficient (but not full) independence between\ntheelementsofthenoisematrix. Hencefortheadditivemodelthenoisematrixconverges\nto the semi-circle law. We reproduce the technical conditions and key elements in the\nproof of [Granziol et al., 2020b] below for clarity.\nToderiveanalyticresults, weemploytheKolmogorovlimit[Bun et al., 2017], where\nP;B;N!1butP\nB=q>0and to account for dependence beyond the symmetry of the\nnoise matrix elements, we introduce the \u001b-algebras F(i;j), and Lindeberg’s ratio LP(\u001c),\nwhich are defined for any \u001c >0as follows:\nF(i;j):=\u001bf\u000f(w)kl: 1\u0014k\u0014l\u0014P;(k;l)6= (i;j)g;\n1\u0014i\u0014j\u0014P\nLP(\u001c) :=1\nP2PX\ni;j=1Ej\u000f(w)i;jj21(j\u000f(w)i;jj\u0015\u001cp\nP):(A.1)\nWhere\u000f(w)defines the matrix of fluctuations, which denotes the difference matrix\nbetween the true and empirical Hessians i.e.\n\u000f(w) =Htrue(w)\u0000Hemp(w) (A.2)\nRewriting the fluctuation matrix as \u000f(w)\u0011Hbatch(w)\u0000Hemp(w)and assuming\nthe mini-batch to be drawn independently from the dataset, we can infer\n\u000f(w) =\u00121\nB\u00001\nN\u0013BX\nj=1r2`(xj;w;yj)\n\u00001\nNNX\ni=B+1r2`(xi;w;yi)(A.3)A Random Matrix Theory Approach to Damping in Deep Learning 32\nthusE(\u000f(w)j;k) = 0and\nE(\u000f(w)j;k)2=\u0012\n1\nB\u00001\nN\u0013\nVar[r2`(x;w;y)j;k].\nWhereBis the batch size and Nthe total dataset size. The expectation is taken with\nrespecttothedatageneratingdistribution  (x;y). InorderforthevarianceinEquation\nA.3 to exist, the elements of r2`(w;w;y)must obey sufficient moment conditions. This\ncan either be assumed as a technical condition, or alternatively derived under the more\nfamiliar condition of L-Lipschitz continuity, as shown with the following Lemma\nLemma 2. For a Lipschitz-continuous empirical risk gradient and almost everywhere\ntwice differentiable loss function `(h(x;w);y), the elements of the fluctuation matrix\n\u000f(w)j;kare strictly bounded in the range \u0000p\nPL\u0014\u000f(w)j;k\u0014p\nPL. WherePis the\nnumber of model parameters and Lis a constant.\nProof.As the gradient of the empirical risk is LLipschitz continous, as the empirical\nrisk a sum over the samples, the gradient of the batch risk is also Lipschitz continous. As\nthe difference of two Lipschitz functions is also Lipschitz, by the fundamental theorem\nof calculus and the definition of Lipschitz continuity the largest eigenvalue \u0015maxof the\nfluctuation matrix \u000f(w)must be smaller than L. Hence using the Frobenius norm we\ncan upper bound the matrix elements of \u000f(w)\nTr(\u000f(w)2) =PX\nj;k=1\u000f(w)2\nj;k=\u000f(w)2\nj=j0;k=k0\n+PX\nj6=j0;k6=k0\u000f(w)2\nj;k=PX\ni=1\u00152\ni(A.4)\nthus\u000f(w)2\nj=j0;k=k0\u0014PP\ni=1\u00152\ni\u0014PL2and\u0000p\nPL\u0014\u000f(w)j=j0;k=k0\u0014p\nPL.\nAs the domain of the Hessian elements under the data generating distribution is\nbounded, the moments of Equation A.3 are bounded and hence the variance exists. We\ncan even go a step further with the following extra lemma.\nLemma 3. For independent samples drawn from the data generating distribution and\nanL-Lipschitz loss `the difference between the empirical Hessian and Batch Hessian\nconverges element-wise to a zero mean, normal random variable with variance /1\nB\u00001\nN\nfor largeB;N.\nProof.By Lemma 2, the Hessian elements are bounded, hence the moments are\nbounded and using independence of samples and the central limit theorem, (1\nB\u0000\n1\nN)\u00001=2[r2Rtrue(w)\u0000r2Remp(w)]jk\u0000!\na:sN(0;\u001b2\njk).\nTheorem 4. The following technical conditions that in the limit P!1, the limiting\nspectra density of \u000f(w)(w)is given by Wigner’s semi-circle law [Akemann et al., 2011]\ni)1\nP2PP\ni;j=1EjE(\u000f(w)2\ni;jjFi;j)\u0000\u001b2\ni;jj!0,A Random Matrix Theory Approach to Damping in Deep Learning 33\nii)LP(\u001c)!0for any\u001c >0,\niii)1\nPPP\nij1\nPPP\nj=1\u001b2\ni;j\u0000\u001b2\nej!0\niv) max 1\u0014i\u0014P1\nPPP\nj=1\u001b2\ni;j\u0014C\nProof.Lindenberg’s ratio is defined as LP(\u001c) :=1\nP2PP\ni;j=1Ej\u000f(w)i;jj21(j\u000f(w)i;jj \u0015\n\u001cp\nP). By Lemma 3, the tails of the normal distribution decay sufficiently rapidly such\nthatLP(\u001c)!0for any\u001c >0in theP!1limit. Alternatively, using the Frobenius\nidentity and Lipschitz continuityPP\ni;j=1Ej\u000f(w)i;jj21(j\u000f(w)i;jj\u0015\u001cp\nP)\u0014PP\ni;j\u000f(w)2\ni;j=PP\ni\u00152\ni\u0014PL2,LP(\u001c)!0for any\u001c >0.\nBy Lemma 3 we also have E(\u000f(w)i;jjFi;j) = 0. Hence along with conditions (i);(ii);(iii)\nthe matrix\u000f(w)satisfies the conditions in [Götze et al., 2012] and the and the limiting\nspectral density p(\u0015)of\u000f(w)2RP\u0002Pconverges to the semi circle law p(\u0015) =p\n4\u001b2\u000f\u0000\u00152\n2\u0019\u001b2\u000f\n[Götze et al., 2012].\n[Götze et al., 2012] use the condition1\nPPP\ni=1j1\nPPP\nj=1\u001b2\ni;j\u00001j!0, however this simply\nintroduces a simple scaling factor, which is accounted for in condition ii)and the\ncorresponding variance per element of the limiting semi-circle.\nWe note that under the assumption of independence between all the elements of\n\u000f(w)we would have obtained the same result, as long as conditions ii)andiii)were\nobeyed. So in simple words, condition 9i)merely states that the dependence between\nthe elements cannot be too large. For example completely dependent elements have a\nsecondmomentexpectationthatscalesas P2andhencecondition (i)cannotbesatisfied.\nCondition (ii)merely states that there cannot be too much variation in the variances\nper element and condition (iii)that the variances are bounded."    },    {        "title": "2011.14314v1.Cross_sublattice_Spin_Pumping_and_Magnon_Level_Attraction_in_van_der_Waals_Antiferromagnets.pdf",        "content": "Cross-sublattice Spin Pumping and Magnon Level Attraction in van der Waals\nAntiferromagnets\nRoberto E. Troncoso,1Mike A. Lund,2, 1Arne Brataas,1and Akashdeep Kamra1\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Department of Engineering Sciences, University of Agder, 4879 Grimstad, Norway\nWe theoretically study spin pumping from a layered van der Waals antiferromagnet in its canted\nground state into an adjacent normal metal. We \fnd that the resulting dc spin pumping current\nbears contributions along all spin directions. Our analysis allows for detecting intra- and cross-\nsublattice spin-mixing conductances via measuring the two in-plane spin current components. We\nfurther show that sublattice symmetry-breaking Gilbert damping can be realized via interface en-\ngineering and induces a dissipative coupling between the optical and acoustic magnon modes. This\nrealizes magnon level attraction and exceptional points in the system. Furthermore, the dissipative\ncoupling and cross-sublattice spin pumping contrive to produce an unconventional spin current in\nthe out-of-plane direction. Our \fndings provide a route to extract the spin mixing conductance\nmatrix and uncovers the unique opportunities, such as level attraction, o\u000bered by van der Waals\nantiferromagnet-normal metal hybrids.\nIntroduction .{ The dawn of magnetic van der Waals\n(vdW) materials has renewed and invigorated interest\nin low-dimensional phenomena hosted by solid state sys-\ntems [1, 2]. These layered vdWs materials have been\nfound to host various forms of magnetic order [3{8] with\nantiferromagnets (AFs) taking a special place due to their\nvarious unique advantages [9{12]. Among these, a control\nover interfacial exchange coupling to an adjacent metal\nand canted or noncollinear ground state o\u000bers unprece-\ndented pathways to achieve intriguing physics and ap-\nplications [13{19]. The vdW magnets o\u000ber an e\u000bective\ncontrol over both - interface and ground state - due to\ntheir layered structure and relatively weak interlayer an-\ntiferromagnetic exchange [3, 4, 20].\nCapitalizing on these features, magnon-magnon cou-\npling resulting in level repulsion and hybridization has\nrecently been observed in the vdW AF CrCl 3[21{23].\nApproaching the challenge from a di\u000berent direction,\nsimilar magnonic hybridization via their mutual cou-\npling [24{26] has been discovered in carefully chosen plat-\nforms including compensated ferrimagnets [27] and syn-\nthetic AFs [28, 29]. These investigations have, in part,\nbeen driven by the desire to control magnonic systems\nfor quantum information applications [30] and were pre-\nceded by the realization of strong magnon-photon cou-\npling [31, 32]. The latter, being a relatively mature\n\feld, has started to explore dissipative magnon-photon\ncoupling [33, 34] resulting in intriguing phenomena that\nforay into the realm of non-Hermitian physics [35] pro-\nviding a powerful model platform. While not explored\nthus far, such phenomena can also result from dissi-\npative magnon-magnon coupling o\u000bering various advan-\ntages over the magnon-photon platform [36]. Demon-\nstrating these for a vdW AF interfaced with a heavy nor-\nmal metal (NM) forms a key contribution of this work.\nHeterostructures comprising a magnetic insulator in-\nterfaced with a thin NM layer have become basic build-ing blocks in an emerging spin-based paradigm for in-\nformation transport and processing [15, 37{45]. In such\nstructures, magnonic spin in the magnetic insulator can\nbe interfaced with the electronic spin in NM thereby\nallowing their integration with conventional electronics.\nFurthermore, spin generated in NM allows to control,\nand even negate [41], dissipation in the magnetic sys-\ntem via spin transfer torques [46]. Invigorated by re-\ncent breakthroughs, especially employing the magnonic\nspin in AFs [15, 40, 47, 48], an interface-engineering and\ncontrol of spin transfer from AF to NM via magnonic\nspin pumping [13, 14, 49{51] assumes a central role.\nWhile coherently driven spin pumping from an AF into\nNM has recently been observed in its collinear ground\nstate [47, 48], a noncollinear or canted AF should en-\nable unique and novel phenomena emerging from cross-\nsublattice spin pumping [14, 52, 53]. Besides providing\nthe much needed understanding of spin transfer across\nthe AF-NM interface, cross-sublattice pumping may also\no\u000ber a direct probe into the aforementioned dissipative\nand non-Hermitian magnon-magnon coupling phenom-\nena, as shown in this work.\nIn this Letter, we demonstrate heterostructures com-\nprising vdW AFs and a heavy NM to be a unique plat-\nform for observing intriguing phenomena emerging from\ncross-sublattice spin pumping and dissipative magnon-\nmagnon coupling. This niche is enabled by the layered\nstructure of vdW AFs resulting in the possibility of AF-\nNM interface engineering and canted AF ground states\nwith application of relatively small magnetic \felds. We\nshow that in such a non-collinear ground state, the AF\npumps spin into the NM along all three directions on\nexcitation via an rf magnetic \feld. A detection of the\ntwo in-plane spin components via inverse spin Hall e\u000bect\nallows to determine the complete spin mixing conduc-\ntance matrix of the interface. We further \fnd an un-\nconventional out-of-plane spin pumping component thatarXiv:2011.14314v1  [cond-mat.mes-hall]  29 Nov 20202\nx<latexit sha1_base64=\"E+xWb622b2P97o+CO1oWwc/7ors=\">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHYNRo9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmhdlr1K+rFdK1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOjRjQQ=</latexit>\ny<latexit sha1_base64=\"0AzN9pSceEkVyYO5YY8D+4ugxy4=\">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoseiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5qRfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJ+6Lq1aqXzVqlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f6lWNBQ==</latexit>\nz<latexit sha1_base64=\"mBNsSck29HYD+UA8I7CdsBnbA5A=\">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHYNRo9ELx4hkUcCGzI79MLI7OxmZtYECV/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmhdlr1K+rFdK1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOvZjQY=</latexit>\n(a)\n<latexit sha1_base64=\"Ac+cs8EQeHY6Uohp+aTWQJMy8+k=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI97xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE177Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4pXrVzeV0u1myyOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ai0ONUg==</latexit>\n(b)\n<latexit sha1_base64=\"zVpXaXYpCy/5Krr2sSqRENfT7FA=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlh3Jw3iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSSSPU/mR+6pScWaVPwljZkobM1d8TExppPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGllCmuL2VsCFVlBmbTsGG4C2/vEqaFxWvWrm8r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjMiNUw==</latexit>\n(c)\n<latexit sha1_base64=\"0u7bBHPDKBCJbAK7avM20Uz0++4=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI77xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE177Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4pXrVzeV0u1myyOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ajk2NVA==</latexit>\n(d)\n<latexit sha1_base64=\"+qTaDHOrJ165Z1nDA3KKJzLMLOY=\">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BItQL2VXKnosevFY0X5Au5RsNtuGZrNLkhXK0p/gxYMiXv1F3vw3pu0etPXBwOO9GWbm+Yng2jjONyqsrW9sbhW3Szu7e/sH5cOjto5TRVmLxiJWXZ9oJrhkLcONYN1EMRL5gnX88e3M7zwxpXksH80kYV5EhpKHnBJjpYdqcD4oV5yaMwdeJW5OKpCjOSh/9YOYphGThgqidc91EuNlRBlOBZuW+qlmCaFjMmQ9SyWJmPay+alTfGaVAIexsiUNnqu/JzISaT2JfNsZETPSy95M/M/rpSa89jIuk9QwSReLwlRgE+PZ3zjgilEjJpYQqri9FdMRUYQam07JhuAuv7xK2hc1t167vK9XGjd5HEU4gVOoggtX0IA7aEILKAzhGV7hDQn0gt7Rx6K1gPKZY/gD9PkDj9KNVQ==</latexit>\nFIG. 1. (a) Schematic setup for the measurement of pumped\nspin currents in the antiferromagnet (AF)-normal metal (NM)\nstructure. The in-plane spin-polarization of the spin current\ncan be detected by the measurements of inverse spin-Hall\nvoltagesVx\nISHE andVy\nISHE. Possible interface microstructures\nare schematically depicted in (b), (c), and (d). The cross-\nsublattice spin mixing conductance gABvanishes [is \fnite]\nfor the interface depicted in (c) [(b) and (d)].\nresults from a concerted e\u000bect of cross-sublattice spin\npumping and a dissipative coupling resulting from the\nsublattice-symmetry breaking at AF-NM interface. Fur-\nthermore, the ensuing dissipative coupling is found to re-\nsult in magnon-magnon level attraction and coalescence\nobservable via typical magnetic resonance experiments.\nThe system thus constitutes a novel and unique platform\nfor investigating this interplay between magnon level re-\npulsion, attraction, and non-Hermitian physics via in-situ\ndamping matrix engineering by, for example, spin trans-\nfer torques.\nModel .{ We treat the vdW material as a two-sublattice\nmagnet described by the magnetization \felds MAand\nMBthat correspond to the sublattices AandB. We con-\nsider magnetic free-energy density [21] F=\u00160HEMA\u0001\nMB=Ms+\u00160\u0000\nM2\nAz+M2\nBz\u0001\n=2\u0000\u00160H\u0001(MA+MB),\nwithMsthe saturation magnetization of each sublattice.\nThe inter-sublattice exchange coupling parametrized by\nHE>0 favors antiferromagnetic order. In addition, an\nexternal dc magnetic \feld His applied in-plane. The sec-\nond term represents the easy-plane anisotropy. Gilbert\ndamping is accounted for by the viscous Rayleigh dissi-\npation functional [52, 54, 55] via the symmetric matrix\n\u0011\u0010\u00100:R[_MA;_MB] =P\n\u0010\u00100R\nVdr\u0011\u0010\u00100_M\u0010\u0001_M\u00100=2, where\nf\u0010;\u00100g=fA;Bg. The ensuing magnetization dynam-\nics is described by the coupled Landau-Lifshitz-Gilbert\n(LLG) equations,\n_m\u0010=\u0000\u00160\rm\u0010\u0002he\u000b\n\u0010+\u000b\u0010\u00100m\u0010\u0002_m\u00100+\u001c\u0010; (1)\nin terms of the unit vectors m\u0010\u0011M\u0010=Ms. The e\u000bective\felds are given by he\u000b\n\u0010= (1=\u00160)\u0002@F=@M\u0010=H\u0000\nHE\u001bx\n\u0010\u00100m\u00100\u0000Ms(m\u0010\u0001^z)^z, with\u001bxthe Pauli matrix\nand\r > 0 is the gyromagnetic ratio magnitude. The\nGilbert damping parameters are de\fned through \u000b\u0010\u00100\u0011\n\rMs\u0011\u0010\u00100where, in particular, \u000bAB=\u000bBA\u0011\u000bod. Note\nthat sublattice asymmetry in our model is broken only\nby the Gilbert damping [52]. It results from the AF-\nNM interface and spin pumping-mediated losses [49, 52].\nThe magnetization dynamics may be excited by a time-\ndependent magnetic \feld h\u0011\u00160\rhRF(t) that produces\na torque\u001c\u0010=m\u0010\u0002h.\nMagnetization dynamics and magnon modes.{ We now\ninvestigate the magnon modes in the material when the\ntwo sublattice magnetizations are non-collinear in their\nequilibrium con\fguration [Fig. 1(a)]. In the presence of\nan in-plane external magnetic \feld H=H^y, the mag-\nnetic ground state becomes meq\n\u0010=\u0006cos\u001e^x+ sin\u001e^y,\nwhere the non-collinearity is captured by the \fnite an-\ngle\u001ethat satis\fes sin \u001e=H=2HE[see Fig. 1(a)]. The\nmagnetic ground state is invariant under a twofold rota-\ntional operation around the yaxis,C2y, in combination\nwith sublattice exchange A$B, i.e.C2ymeq\nA=meq\nB.\nLinearizing the LLG equations (1), considering m\u0010=\nmeq\n\u0010+\u000em\u0010ei!t, the coupled dynamical equations become,\ni!\u000em+=meq\nA\u0002(A+\u000em++i!\u0001\u0016\u000b\u000em\u0000) +\u001c+;(2a)\ni!\u000em\u0000=meq\nA\u0002(A\u0000\u000em\u0000+i!\u0001\u0016\u000b\u000em+) +\u001c\u0000;(2b)\nwith the two magnetization dynamics or magnon modes\ndescribed by the \felds \u000em\u0006=\u000emA\u0006C2y\u000emB, the\ntorques\u001c\u0006=\u001cA\u0006C2y\u001cBand the operator A\u0006=\n(\u00160\rHE+i!\u0016\u000b)\u0006(\u00160\rHE+i!\u000bod)C2y. Furthermore,\nwe have reformulated the Gilbert damping parameters\nas\u000bAA= \u0016\u000b+ \u0001\u0016\u000band\u000bBB= \u0016\u000b\u0000\u0001\u0016\u000b. Note that\nwhen sublattice symmetry is assumed, i.e., \u000bAA=\u000bBB,\nthe Eqs. (2a) and (2b) become decoupled since \u0001\u0016 \u000b=\n0. In the absence of dissipation, the magnon eigen-\nmodes are captured well by the \felds \u000em\u0006, with the\neigenfrequencies for the so-called optical and acoustic\nmagnon modes being !+=\u00160\rp\n2MsHEcos2\u001eand\n!\u0000=\u00160\rp\n2HE(Ms+ 2HE) sin\u001e, respectively. The\ntwo modes can be excited selectively by a careful choice\nof the rf-\feld h[21]. In general, the excitation of ( \u0006)-\nmodes one at a time, which demands \u001c\u0007= 0 for the\ntorque, imposes the conditions h=\u0006C2yh, respectively.\nSpin pumping .{ We now investigate spin transport\nacross the AF-NM interface resulting from the excitation\nof magnetization dynamics by rf magnetic \feld. The dc\nspin pumping current injected into the adjacent NM is\ngiven by [14]\ne\n~Is=X\n\u0010\u001002fA;Bgg\u0010\u00100hm\u0010\u0002_m\u00100i; (3)\nwhereh\u0001\u0001\u0001i stands for the time-average over the period\nof oscillation. 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The frequency and \feld dependence of the spin-pumping current is represented by the plot of Fj\n\u0006(!), withjthe\ndirections of polarization and damping parameters \u0016 \u000b= 0:05,\u000bod= 0:01 and \u0001\u0016\u000b= 0:005. At panels (a), (b) and (c), we have\ndepicted the spin current with spin-polarization along x,y- andz-direction, respectively, due to coherent excitation of optical\nmodes. Similarly, in panels (d), (e) and (f), we have plotted the polarization components of the spin current when acoustic\nmodes are excited. At the inset of each panel we show Fj\n\u0006, evaluated at the resonant frequencies !=!\u0006. In the inset of panels\n(a), (b), (d) and (e), the curves correspond to di\u000berent \u000bod, while at panels (c) and (f) \u0001\u0016 \u000bis modi\fed. Other parameters\nemployed at the plots were extracted for the vdW antiferromagnet CrCl 3[21].\ndescribe the intra-sublattice spin mixing conductance.\nAn asymmetric interfacial coupling [13{16], resulting in\ngAA6=gBB, occurs when the two magnetic sublattices\nare incommensurately exposed to the NM (see Fig. 1(c)\nfor an example in which gAA= 0). The o\u000b-diagonal\ncross-sublattice conductance satisfy gAB=gBAand are\nnonzero when both the sublattices are (partly) exposed to\nNM [Fig. 1(b) and (d)]. Such interfaces can be achieved\nwith layered magnets [5, 56] (Fig. 1), but are not possi-\nble with synthetic AFs [28]. Our goal here is to estab-\nlish the experimentally detectable spin pumping current\nas a direct probe of the 2 \u00022 spin mixing conductance\nmatrix [Eq. (3)]. In particular, we are interested in es-\ntablishing unique signatures of the cross-sublattice con-\nductancesgAB=gBAthat elude a direct experimental\nobservation thus far.\nIn typical experimental setups (Fig. 1), the spin\npumping current is detected via inverse spin-Hall e\u000bect\n(ISHE) [57{60]. In metals with strong spin-orbit cou-\npling, a nonequilibrium spin current jsinduces a trans-\nverse charge current jc=\u0012SH(2e=~)js\u0002~ \u001b, where~ \u001bde-\nnotes the spin polarization direction and \u0012SHis the spin\nHall angle [57]. Under open circuit conditions, the gen-\nerated charge current is countered by an induced inverse\nspin Hall voltage VISHE proportional to the charge, andthus spin, current. The voltage thus generated is propor-\ntional to the spin current injected into the NM [Eq. (3)]\nand is directly detected in experiments [61].\nIn order to relate spin mixing conductance matrix ele-\nments to experimental observables, we evaluate the spin\npumping current Eq. (3) in the limit \u000bAA\u0019\u000bBB, i.e.,\n\u0001\u0016\u000bsmall. In this perturbative regime, Eqs. (2a) and\n(2b) decouple and the system eigenmodes are the opti-\ncal and acoustic magnon modes. The corresponding spin\npumping currents are evaluated to be\nI\u0006\nz= (gAA\u0000gBB)Fx\n\u0006(!)^x+gABFz\n\u0006(!)^z\n+ (gAA+gBB\u00062gAB)Fy\n\u0006(!)^y:(4)\nThe\u0006index labels optical and acoustic modes, which\nare driven by the rf magnetic \feld with frequency !\nand amplitude h+;\u001e= 2hycos\u001eandh\u0000;\u001e= 2hxsin\u001e,\nrespectively. The expression in Eq. (4) represents a\npure spin current that \rows across the AF-NM inter-\nface (along zaxis). Its various components pertaining to\ndirections in the spin space are proportional to the func-\ntionsFj\n\u0006(!), as detailed in the Supplemental Material\nin [62] (see Eqs. (25)-(27)), which have been obtained to\nthe \frst order in \u0001\u0016 \u000b. The in-plane components Fx;y\n\u0006,\nsatisfying Fy\n\u0006(!) = tan\u001eFx\n\u0006(!), are independent of \u0001\u0016 \u000b.\nHowever, the out-of-plane component of the spin current4\n/Fz\n\u0006(!) scales linearly with \u0001\u0016 \u000b. The resulting \feld-\nand frequency-dependence of the spin current I\u0006\nzcompo-\nnents are plotted in Fig. 2 employing system parameters\nrelevant for the vdW AF CrCl 3[21]. The various spin\ncurrent components are displayed in panels (a), (b) and\n(c) of Fig. 2 for the optical mode, and in panels (d), (e),\nand (f) for the acoustic mode.\nIn contrast with the case of spin pumping via collinear\nmagnets, in which a dc spin current polarized along the\nequilibrium order is generated [47{49, 51, 60, 61], the\ncanted AF under consideration pumps spin with compo-\nnents along all three directions [Eq. (4)]. As detailed in\nthe Supplemental Material [62], Fy;z\n\u0006/sin\u001eimplying\nyandzcomponents of the spin pumping current vanish\nfor\u001e= 0. Our result thus reduces to the existing under-\nstanding of collinear AFs [14, 15]. These additional com-\nponents of the spin pumping current for the canted AF\ntogether with the existence of two independent magnon\nmodes constitute some of the unique features and op-\nportunities o\u000bered by this system. For example, detec-\ntion of the ISHE voltage in two orthogonal directions [as\ndepicted in Fig. 1(a)] enables determination of both in-\nplane spin current components. By detecting these while\nexciting the two magnon modes one at a time, we may\ndetermine the full spin mixing conductance matrix with\nthe cross-sublattice term given by\ngAB=gAA\u0000gBB\n4 tan\u001e\u0012I+\ns;y\nI+s;x\u0000I\u0000\ns;y\nI\u0000s;x\u0013\n; (5)\nwhich assumes the condition gAA6=gBB. This accom-\nplishes a key goal and constitutes a main result of this\npaper.\nFurthermore, existence of the spin current zcompo-\nnent [Eq. (4)] that we \fnd is unconventional and coun-\nterintuitive as the magnon spin is expected to lie in\nthe same plane as the equilibrium sublattice magnetiza-\ntions [24, 25]. However, this component is nonzero only\nwhen \u0001\u0016\u000b6= 0,gAA6=gBB,gAB6= 0, and\u001e6= 0 im-\nplying that it results from a complex interplay of the\nsublattice-symmetry breaking dissipative coupling and\na cross-sublattice interference e\u000bect. Such physics, es-\npecially dissipative coupling [33, 34], appears to go be-\nyond the magnon picture considered thus far and consti-\ntutes another key result of our work. While an out-of-\nplane spin component has not been measured in typical\nspin pumping experiments [57, 60, 61], the recent ther-\nmal drag-mediated detection of such an out-of-plane spin\ncomponent [63] provides one possible method for its di-\nrect observation.\nMagnon level attraction .{ We now discuss the magnon\neigenmodes which become dissipatively coupled [see\nEqs. (2a) and (2b)] due to the sublattice-symmetry\nbreaking Gilbert damping [52], i.e. nonzero \u0001\u0016 \u000b. While a\n\\reactive\" coupling between magnon modes has been ob-\nserved in various systems [21, 22, 27{29], dissipative cou-\npling remains less explored and invokes non-Hermitian\nµ0H[T]\n<latexit sha1_base64=\"BYO018/2BTGPnIVfz+lP5J4t3zQ=\">AAAB+XicbVBNS8NAEN3Ur1q/oh69BIvgqSSi6LHopccK/YImhM120y7d3YTdSbGE/hMvHhTx6j/x5r9x2+agrQ8GHu/NMDMvSjnT4LrfVmljc2t7p7xb2ds/ODyyj086OskUoW2S8ET1IqwpZ5K2gQGnvVRRLCJOu9H4Ye53J1RplsgWTFMaCDyULGYEg5FC2/ZFFrqNvg/0CfLWLAjtqltzF3DWiVeQKirQDO0vf5CQTFAJhGOt+56bQpBjBYxwOqv4maYpJmM8pH1DJRZUB/ni8plzYZSBEyfKlARnof6eyLHQeioi0ykwjPSqNxf/8/oZxHdBzmSaAZVkuSjOuAOJM4/BGTBFCfCpIZgoZm51yAgrTMCEVTEheKsvr5POVc27rt08Xlfr90UcZXSGztEl8tAtqqMGaqI2ImiCntErerNy68V6tz6WrSWrmDlFf2B9/gByiZON</latexit>\n(a)\n<latexit sha1_base64=\"Ac+cs8EQeHY6Uohp+aTWQJMy8+k=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI97xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE177Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4pXrVzeV0u1myyOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ai0ONUg==</latexit>\n(b)\n<latexit sha1_base64=\"zVpXaXYpCy/5Krr2sSqRENfT7FA=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlh3Jw3iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSSSPU/mR+6pScWaVPwljZkobM1d8TExppPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGllCmuL2VsCFVlBmbTsGG4C2/vEqaFxWvWrm8r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjMiNUw==</latexit>\n(c)\n<latexit sha1_base64=\"0u7bBHPDKBCJbAK7avM20Uz0++4=\">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2FXInoMevEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mnGCfkQHkoecUWOlhzI77xVLbsWdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE177Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi4pXrVzeV0u1myyOPJzAKZTBgyuowR3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8Ajk2NVA==</latexit>\n(d)\n<latexit sha1_base64=\"+qTaDHOrJ165Z1nDA3KKJzLMLOY=\">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BItQL2VXKnosevFY0X5Au5RsNtuGZrNLkhXK0p/gxYMiXv1F3vw3pu0etPXBwOO9GWbm+Yng2jjONyqsrW9sbhW3Szu7e/sH5cOjto5TRVmLxiJWXZ9oJrhkLcONYN1EMRL5gnX88e3M7zwxpXksH80kYV5EhpKHnBJjpYdqcD4oV5yaMwdeJW5OKpCjOSh/9YOYphGThgqidc91EuNlRBlOBZuW+qlmCaFjMmQ9SyWJmPay+alTfGaVAIexsiUNnqu/JzISaT2JfNsZETPSy95M/M/rpSa89jIuk9QwSReLwlRgE+PZ3zjgilEjJpYQqri9FdMRUYQam07JhuAuv7xK2hc1t167vK9XGjd5HEU4gVOoggtX0IA7aEILKAzhGV7hDQn0gt7Rx6K1gPKZY/gD9PkDj9KNVQ==</latexit>\nµ0H[T]\n<latexit sha1_base64=\"BYO018/2BTGPnIVfz+lP5J4t3zQ=\">AAAB+XicbVBNS8NAEN3Ur1q/oh69BIvgqSSi6LHopccK/YImhM120y7d3YTdSbGE/hMvHhTx6j/x5r9x2+agrQ8GHu/NMDMvSjnT4LrfVmljc2t7p7xb2ds/ODyyj086OskUoW2S8ET1IqwpZ5K2gQGnvVRRLCJOu9H4Ye53J1RplsgWTFMaCDyULGYEg5FC2/ZFFrqNvg/0CfLWLAjtqltzF3DWiVeQKirQDO0vf5CQTFAJhGOt+56bQpBjBYxwOqv4maYpJmM8pH1DJRZUB/ni8plzYZSBEyfKlARnof6eyLHQeioi0ykwjPSqNxf/8/oZxHdBzmSaAZVkuSjOuAOJM4/BGTBFCfCpIZgoZm51yAgrTMCEVTEheKsvr5POVc27rt08Xlfr90UcZXSGztEl8tAtqqMGaqI2ImiCntErerNy68V6tz6WrSWrmDlFf2B9/gByiZON</latexit>\nµ0H[T]\n<latexit sha1_base64=\"BYO018/2BTGPnIVfz+lP5J4t3zQ=\">AAAB+XicbVBNS8NAEN3Ur1q/oh69BIvgqSSi6LHopccK/YImhM120y7d3YTdSbGE/hMvHhTx6j/x5r9x2+agrQ8GHu/NMDMvSjnT4LrfVmljc2t7p7xb2ds/ODyyj086OskUoW2S8ET1IqwpZ5K2gQGnvVRRLCJOu9H4Ye53J1RplsgWTFMaCDyULGYEg5FC2/ZFFrqNvg/0CfLWLAjtqltzF3DWiVeQKirQDO0vf5CQTFAJhGOt+56bQpBjBYxwOqv4maYpJmM8pH1DJRZUB/ni8plzYZSBEyfKlARnof6eyLHQeioi0ykwjPSqNxf/8/oZxHdBzmSaAZVkuSjOuAOJM4/BGTBFCfCpIZgoZm51yAgrTMCEVTEheKsvr5POVc27rt08Xlfr90UcZXSGztEl8tAtqqMGaqI2ImiCntErerNy68V6tz6WrSWrmDlFf2B9/gByiZON</latexit>Im[\u0000+\u0000\u0000]<latexit sha1_base64=\"dsawMljXjT9cR+hs42pS7ayNkWE=\">AAACB3icbVBNS8NAEN3Ur1q/qh4FCRZBEEoiFT0Wveitgv2AJJbNdtss3WzC7kQsITcv/hUvHhTx6l/w5r9x0/agrQ8GHu/NMDPPjzlTYFnfRmFhcWl5pbhaWlvf2Nwqb++0VJRIQpsk4pHs+FhRzgRtAgNOO7GkOPQ5bfvDy9xv31OpWCRuYRRTL8QDwfqMYNBSt7zvAn2A9DrMHJcE7C49zrqpGwcsr8zrlitW1RrDnCf2lFTQFI1u+cvtRSQJqQDCsVKObcXgpVgCI5xmJTdRNMZkiAfU0VTgkCovHf+RmYda6Zn9SOoSYI7V3xMpDpUahb7uDDEEatbLxf88J4H+uZcyESdABZks6ifchMjMQzF7TFICfKQJJpLpW00SYIkJ6OhKOgR79uV50jqp2rXq6U2tUr+YxlFEe+gAHSEbnaE6ukIN1EQEPaJn9IrejCfjxXg3PiatBWM6s4v+wPj8Af2Wmgg=</latexit>\n0.120.130.140.155.86.26.6\n0.050.100.150.200246810\n0.050.100.150.200.00.51.01.52.0\n0.100.110.120.130.140.150.250.350.450.020.100.180.20.61.0FIG. 3. Eigenfrequencies of the coupled magnon modes are\nplotted as a function of the external dc magnetic \feld. The\nreal (!r) and imaginary ( !i) parts of the frequencies are dis-\nplayed in panels (a) and (c) respectively, for \u0016 \u000b= 0:1 and\n\u000bod= \u0001\u0016\u000b= 0:07. In panel (b), we zoom-in on the level\ncrossing of panel (a). The additional curves shown in pur-\nple and red correspond to the same \u0016 \u000band \u0001\u0016\u000bas (a), but\nwith\u000bod= 0. The level attraction is depicted in gray when\n\u0001\u0016\u000b= 0:07 and \u0016\u000b=\u000bod= 0. (d) Imaginary part of the\nmagnetic susceptibility Im[ \u001f+\n\u001e\u001e]vs.applied magnetic \feld for\nvarious values of \u0001\u0016 \u000b. The susceptibility corresponds to the\noptical mode and has been evaluated at !=!+.\nphysics [35, 64]. Solving Eqs. (2a) and (2b) without an\nexternal rf drive, we obtain the complex eigenmode fre-\nquencies!\u0006=!r\u0006+i!i\u0006.!rand!i, respectively, cap-\nture the energy and inverse lifetime of the magnon modes\nand have been plotted against the external dc magnetic\n\feld in Fig. 3 (a)-(c). Due to the dissipative nature of\nthe coupling, a magnon-magnon level attraction is ob-\nserved. The grey curve in Fig. 3 (b), corresponding to\n\u0016\u000b=\u000bod= 0 and \u0001\u0016 \u000b6= 0, depicts a perfect level coa-\nlescence or mode synchronization [33, 34]. Such values\nfor Gilbert damping matrix require a dc spin transfer\ntorque drive in the NM. An undriven system however\nimposes constraints \u000bAA;\u000bBB>0, i.e. \u0016\u000b > \u0001\u0016\u000band\n\u000bod\u0014p\u000bAA\u000bBB[52]. In this scenario, we \fnd a complex\ninterplay of repulsion and attraction between the two\nmodes. The resulting eigenfrequencies split slightly (see\nFig. 3(a) when \u0016 \u000b= 0:1,\u000bod= 0:07 and \u0001\u0016\u000b= 0:07) while\ncoalescing at a speci\fc point, the so-called exceptional\npoint [65]. Furthermore, Fig. 3(d) depicts the imaginary\npart of dynamic susceptibility, which is directly accessible\nin experiments [21, 27, 66]. A peak in this susceptibility\nprovides an additional experimental signature of the level\nattraction when \u0001\u0016 \u000bis su\u000eciently large. Thus, vdW AFs\nunder consideration constitute a rich platform for realiz-\ning non-Hermitian physics and magnon-magnon level at-\ntraction via AF-NM interface engineering and spin trans-\nfer torques exerted on AF by the NM.\nSummary .{ We have theoretically uncovered unique\nand intriguing cross-sublattice spin pumping and5\nmagnon-magnon level attraction e\u000bects in a model canted\nantiferromagnet. By providing guidance to experiments\nin extracting the interfacial spin mixing conductance ma-\ntrix and key signatures of level attraction, we hope to es-\ntablish van der Waals antiferromagnets interfaced with\na heavy metal layer as a fertile and convenient plat-\nform for realizing and investigating unconventional non-\nHermitian physics.\nNote added: During the manuscript preparation, we\nnoticed a recent related preprint [67] that studies level-\nrepulsion and hybridization of magnonic modes in bulk\nsymmetry-breaking synthetic antiferromagnets. It how-\never does not discuss spin pumping or the dissipative\nmagnon level attraction - the two key novelties of our\nwork.\nThis work was supported by the European Union's\nHorizon 2020 Research and Innovation Programme un-\nder Grant No. DLV-737038 \\TRANSPIRE,\" and the\nResearch Council of Norway through its Centres of Ex-\ncellence funding scheme, Project No. 262633, \\QuSpin\".\n[1] K. S. Burch, D. Mandrus, and J.-G. 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B 82, 214403 (2010).\n[62] SeeSee Supplemental Material at [link] for details of two-\nsubllatice magnetization dynamics and spin pumping..[63] C. O. Avci, E. Rosenberg, M. Huang, J. Bauer, C. A.\nRoss, and G. S. D. Beach, Phys. Rev. Lett. 124, 027701\n(2020).\n[64] B. Flebus, R. A. Duine, and H. M. Hurst, Phys. Rev. B\n102, 180408 (2020).\n[65] Y. Tserkovnyak, Phys. Rev. Research 2, 013031 (2020).\n[66] A. J. Berger, E. R. J. Edwards, H. T. Nembach, A. D.\nKarenowska, M. Weiler, and T. J. Silva, Phys. Rev. B\n97, 094407 (2018).\n[67] J. Lu, M. Li, and W. He, \\Symmetry breaking in-\nduced magnon-magnon coupling in synthetic antifer-\nromagnets,\" (2020), arXiv:2011.01583 [cond-mat.mes-\nhall].7\nSUPPLEMENTAL MATERIAL\nIn this Supplemental Material, we explicitly show the calculation of two-sublattice spin pumping in AF-NM struc-\ntures. The result is applied when the magnetization dynamics is coherently driven by a linearly polarized ac magnetic\n\feld. Basic details of magnon-magnon hybridization are also provided.\nMagnetization dynamics and magnon modes\nMagnetic \ructuations of the two-sublattice magnet results from the linearization of LLG equations. Representing\nthe \ructuations by m\u0010=meq\n\u0010+\u000em\u0010ei!t, with\u0010=fA;Bg, the resulting coupled dynamical equations becomes,\ni!\u000em+=meq\nA\u0002[(\u00160\rHE+i!\u0016\u000b)\u000em++ (\u00160\rHE+i!\u000bod)C2y\u000em+] +i!\u0001\u0016\u000bmeq\nA\u0002\u000em\u0000+\u001c+; (6a)\ni!\u000em\u0000=meq\nA\u0002[(\u00160\rHE+i!\u0016\u000b)\u000em\u0000\u0000(\u00160\rHE+i!\u000bod)C2y\u000em\u0000] +i!\u0001\u0016\u000bmeq\nA\u0002\u000em++\u001c\u0000; (6b)\nwith the torques \u001c\u0006=mA\u0002h\u0006. The Gilbert damping terms are represented by \u000bAB=\u000bBA=\u000bod,\u000bAA= \u0016\u000b+ \u0001\u0016\u000b\nand\u000bBB= \u0016\u000b\u0000\u0001\u0016\u000b. Note that when sublattice symmetry is restored, i.e., \u000bAA=\u000bBB, the modes \u000em+and\u000em\u0000\nbecome decoupled and characterize, optical and acoustic modes respectively. In a compact form the Eqs. (6a) and\n(6b) read,\n\u0012h+\nh\u0000\u0013\n=\u0012M+T\nT\u0003M\u0000\u0013\n|{z}\nM\u0012\n\u000em+\n\u000em\u0000\u0013\n(7)\nwith the ac \felds h+= (h+;\u001e;0)Tandh\u0000= (h\u0000;\u001e;h\u0000;\u0012)T, whereh+;\u001e= 2hycos\u001e,h\u0000;\u001e= 2hxsin\u001eandh\u0000;\u0012= 2hz.\nThe matrices M\u0006andTare de\fned as\nT=\u0012\u0000i!\u0001\u0016\u000b 0\n0i!\u0001\u0016\u000b\u0013\n; (8)\nM+=\u0012\u0000A\u0000i!(\u0016\u000b+\u000bodcos 2\u001e)i!\ni! B +i!(\u0016\u000b\u0000\u000bod)\u0013\n; (9)\nM\u0000=\u0012C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)\u0000i!\n\u0000i!\u0000D\u0000i!(\u0016\u000b+\u000bod)\u0013\n; (10)\nwhere we introduced the following constants A= 2\u00160\rHEcos2\u001eandB=\u00160\rMs,C= 2\u00160\rHEsin2\u001eandD=\n\u00160\r(Ms+ 2HE). It is worth noting that det [ M] = det [ M+] det\u0002\nM\u0000\u0000TM\u00001\n+T\u0003\n\u0019det [M+] det [M\u0000] in the limit of\nsmall \u0001\u0016\u000b. Beyond this approximation, the optical and acoustic magnonic modes are no longer decoupled. Instead,\nthese modes hybridize with \u0001\u0016 \u000bbeing the dissipative coupling that generates level attraction. The magnon-magnon\nhybrid eigenfrequencies were numerically calculated and depicted in Fig. 3.\nAssuming that \u0001\u0016 \u000bis small, we use the standard formula for the inverse of a block matrix to obtain the \felds \u000em\u0006,\n\u000em+=M\u00001\n+h+\u0000M\u00001\n+TM\u00001\n\u0000h\u0000; (11)\n\u000em\u0000=M\u00001\n\u0000h\u0000+M\u00001\n\u0000TM\u00001\n+h+: (12)\nThe \frst term at the right-hand side is the zero order correction in \u0001\u0016 \u000b, where M\u00001\n\u0006corresponds to the symmetric\ndynamic susceptibility matrix. The components are de\fned as\u0002\nM\u00001\n\u0006\u0003\n11=\u001f\u0006\n\u001e\u001e,\u0002\nM\u00001\n\u0006\u0003\n22=\u001f\u0006\n\u0012\u0012and\u0002\nM\u00001\n\u0006\u0003\n12=i\u001f\u0006\n\u001e\u0012,\nwhere\n\u001f+\n\u001e\u001e=B+i(\u0016\u000b\u0000\u000bod)!\n!2\u0000(B+i!(\u0016\u000b\u0000\u000bod)) (A+i!(\u0016\u000b+\u000bodcos 2\u001e)); (13)\n\u001f+\n\u001e\u0012=\u0000!\n!2\u0000(B+i!(\u0016\u000b\u0000\u000bod)) (A+i!(\u0016\u000b+\u000bodcos 2\u001e)); (14)8\nand\n\u001f\u0000\n\u001e\u001e=\u0000D\u0000i!(\u0016\u000b+\u000bod)\n!2\u0000(D+i!(\u0016\u000b+\u000bod)) (C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)); (15)\n\u001f\u0000\n\u0012\u0012=C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)\n!2\u0000(D+i!(\u0016\u000b+\u000bod)) (C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)); (16)\n\u001f\u0000\n\u001e\u0012=!\n!2\u0000(D+i!(\u0016\u000b+\u000bod)) (C+i!(\u0016\u000b\u0000\u000bodcos 2\u001e)): (17)\nThe second contribution in Eqs. (11) and (12) is linear in the damping \u0001\u0016 \u000b, with the 2\u00022 matrices,\nM\u00001\n+TM\u00001\n\u0000=!\u0001\u000b\ndet [M+] det [M\u0000]\u0012i\u0000\nBD+!2\u0001\n(B+C)!\n(A+D)!\u0000i\u0000\nAC+!2\u0001\u0013\n; (18)\nM\u00001\n\u0000TM\u00001\n+=!\u0001\u000b\ndet [M+] det [M\u0000]\u0012i\u0000\nBD+!2\u0001\n(A+D)!\n(B+C)!\u0000i\u0000\nAC+!2\u0001\u0013\n: (19)\nTwo-sublattice Spin Pumping\nIn this section we evaluate the spin pumping in the two-sublattice magnet. Final expressions for the injected spin\ncurrents are found in terms of the dynamical magnetic susceptibility. To start with, let us consider the spin pumping\ncurrent into the normal metal given by Eq. (3). The \ructuations are represented by m\u0010=meq\n\u0010+\u000eM\u0010, with the\nreal-valued \felds de\fned as \u000eMA=\u000eMA;\u0012^z+\u000eMA;\u001e(^z\u0002meq\nA) and\u000eMB=\u000eMB;\u0012^z+\u000eMB;\u001e(^z\u0002meq\nB). We \fnd\nthat the spin pumping current Isbecomes\n4\u0019\n~Is=gAAh\u000eMA\u0002\u000e_MAi+gAB\u0010\nh\u000eMA\u0002\u000e_MBi+h\u000eMB\u0002\u000e_MAi\u0011\n+gBBh\u000eMB\u0002\u000e_MBi: (20)\nNote that since a time-average is involved in the evaluation of previous equation, linear terms in the \ructuations do\nnot contribute. In order to relate each term in Eq. (20) with the dynamical susceptibility, Eqs. (13)-(17), we represent\n\u000eM\u0010by complex-valued \felds as \u000eM\u0010;\u0016= Re\u0002\n\u000em\u0010;\u0016ei!t\u0003\n, where\u0016=f\u0012;\u001eg. Next, we write the \felds \u000em\u0010;\u0016in the\neigenbasis by the following relations \u000emA=B;\u0012 =1\n2(\u000em\u0006;\u0012\u0006\u000em\u0007;\u0012) and\u000emA=B;\u001e =1\n2(\u000em\u0006;\u001e\u0006\u000em\u0007;\u001e). Therefore,\nwe obtain\nh\u000eMA\u0002\u000e_MAi=!\n4\u0000\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n+;\u0012i\u0003\n+ Im\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n+ Im\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n+;\u0012i\u0003\n+ Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\u0001\nmeq\nA;\n(21)\nh\u000eMB\u0002\u000e_MBi=!\n4\u0000\nIm\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n+ Im\u0002\nh\u000em+;\u001e\u000em\u0003\n+;\u0012i\u0003\n\u0000Im\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n+;\u0012i\u0003\n\u0000Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\u0001\nmeq\nB;\n(22)\nh\u000eMA\u0002\u000e_MBi+h\u000eMB\u0002\u000e_MAi=!\n2cos\u001e\u0000\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n\u0000Im\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n+;\u0012i\u0003\u0001^x\n+!\n2sin\u001e\u0000\nIm\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n\u0000Im\u0002\nh\u000em+;\u001e\u000em\u0003\n+;\u0012i\u0003\u0001^y+!\n2sin(2\u001e)Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003^z:(23)\nBelow, we present examples of spin pumping current, evaluating Eqs. (21-23), due to the coherent magnonic\nexcitation by linearly polarized external ac \felds.\nExciting the Optical and Acoustic Modes\nAccording to the selection rules described in the main text, excitation of the optical mode requires a \feld that\nsatisfyh\u0000=0andh+;\u001e6= 0. On the other hand, excitation of the acoustic mode requires h+= 0 andh\u00006= 0. From\nEqs. (11) and (12), we \fnd the \felds \u000em\u0006when the ac magnetic \feld is h+=hy^y(optical mode) and h\u0000=hx^x\n(acoustic mode). To determine the spin current, we evaluate each term given by the Eqs. (23)-(22). Up to linear\norder in the Gilbert damping di\u000berence \u0001\u0016 \u000b, we \fnd\ne\n~I\u0006\ns=!\n4Im\u0002\nh\u000em\u0006;\u001e\u000em\u0003\n\u0006;\u0012i\u0003\n[(gAA\u0000gBB) cos\u001e^x+ ((gAA+gBB)\u00072gAB) sin\u001e^y]\n+!\n2Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003\n\u0006gABsin(2\u001e)^z: (24)9\nThe functions Fj\n\u0006(!) introduced in the main text, are thus de\fned as\nFx\n\u0006(!) =!\n4Im\u0002\nh\u000em\u0006;\u001e\u000em\u0003\n\u0006;\u0012i\u0003\ncos\u001e; (25)\nFy\n\u0006(!) =!\n4Im\u0002\nh\u000em\u0006;\u001e\u000em\u0003\n\u0006;\u0012i\u0003\nsin\u001e; (26)\nFz\n\u0006(!) =!\n2Im\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003\n\u0006sin(2\u001e): (27)\nThe explicit expression for each contribution in Eq. (24) obeys\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n+;\u0012i\u0003\n=!Bh2\n+;\u001e\njdet [M+]j2; (28)\nIm\u0002\nh\u000em\u0000;\u001e\u000em\u0003\n\u0000;\u0012i\u0003\n=!Dh2\n\u0000;\u001e\njdet [M\u0000]j2; (29)\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003\n+= \u0001\u000b!B\u0000\nBD+!2\u0001\u0000\n!2\n\u0000\u0000!2\u0001\nh2\n+;\u001e\njdet [M+]j2jdet [M\u0000]j2; (30)\nIm\u0002\nh\u000em+;\u001e\u000em\u0003\n\u0000;\u001ei\u0003\n\u0000= \u0001\u000b!D\u0000\nBD+!2\u0001\u0000\n!2\u0000!2\n+\u0001\nh2\n\u0000;\u001e\njdet [M+]j2jdet [M\u0000]j2: (31)"    },    {        "title": "2101.02794v2.Mechanisms_behind_large_Gilbert_damping_anisotropies.pdf",        "content": "Mechanisms behind large Gilbert damping anisotropies\nI. P. Miranda1, A. B. Klautau2,∗A. Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4\n1Universidade de São Paulo, Instituto de Física,\nRua do Matão, 1371, 05508-090, São Paulo, SP, Brazil\n2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and\n4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden\n(Dated: November 22, 2021)\nA method with which to calculate the Gilbert damping parameter from a real-space electronic\nstructure method is reported here. The anisotropy of the Gilbert damping with respect to the\nmagnetic moment direction and local chemical environment is calculated for bulk and surfaces\nof Fe 50Co50alloys from first principles electronic structure in a real space formulation. The size\nof the damping anisotropy for Fe 50Co50alloys is demonstrated to be significant. Depending on\ndetails of the simulations, it reaches a maximum-minimum damping ratio as high as 200%. Several\nmicroscopic origins of the strongly enhanced Gilbert damping anisotropy have been examined, where\nin particular interface/surface effects stand out, as do local distortions of the crystal structure.\nAlthough theory does not reproduce the experimentally reported high ratio of 400% [Phys. Rev.\nLett. 122, 117203 (2019)], it nevertheless identifies microscopic mechanisms that can lead to huge\ndamping anisotropies.\nIntroduction: Magnetic damping has a critical impor-\ntanceindeterminingthelifetime,diffusion,transportand\nstability of domain walls, magnetic vortices, skyrmions,\nand any nano-scale complex magnetic configurations [1].\nGiven its high scientific interest, a possibility to obtain\nthis quantity by means of first-principles theory [2] opens\nnew perspectives of finding and optimizing materials for\nspintronic and magnonic devices [3–8]. Among the more\npromising ferromagnets to be used in spintronics devices,\ncobalt-iron alloys demonstrate high potentials due to the\ncombination of ultralow damping with metallic conduc-\ntivity [4, 9].\nRecently, Li et al.[10] reported an observed, gi-\nant anisotropy of the Gilbert damping ( α) in epitaxial\nFe50Co50thin films (with thickness 10 −20nm) reach-\ning maximum-minimum damping ratio values as high as\n400%. TheauthorsofRef. [10]claimedthattheobserved\neffect is likely due to changes in the spin-orbit coupling\n(SOC) influence for different crystalline directions caused\nby short-range orderings that lead to local structural dis-\ntortions. This behaviour differs distinctly from, for ex-\nample, pure bcc Fe [11]. In order to quantitatively pre-\ndict the Gilbert damping, Kambersky’s breathing Fermi\nsurface (BFS) [12] and torque-correlation (TC) [13] mod-\nels are frequently used. These methods have been ex-\nplored for elements and alloys, in bulk form or at sur-\nfaces, mostly via reciprocal-space ab-initio approaches,\nin a collinear or (more recently) in a noncollinear con-\nfiguration [14]. However, considering heterogeneous ma-\nterials, such as alloys with short-range order, and the\npossibility to investigate element specific, non-local con-\ntributions to the damping parameter, there are, to the\nbest of our knowledge, no reports in the literature that\nrely on a real space method.\nIn this Letter, we report on an implementation of ab\ninitiodamping calculations in a real-space linear muffin-tin orbital method, within the atomic sphere approxi-\nmation (RS-LMTO-ASA) [15, 16], with the local spin\ndensity approximation (LSDA) [17] for the exchange-\ncorrelation energy. The implementation is based on the\nBFSandTCmodels, andthemethod(SupplementalMa-\nterial - SM, for details) is applied to investigate the re-\nported, huge damping anisotropy of Fe50Co50(100)/MgO\nfilms [10]. A main result here is the identification of\na microscopic origin of the enhanced Gilbert damping\nanisotropy of Fe50Co50(100) films, and the intrinsic rela-\ntionships to the local geometry of the alloy. Most signifi-\ncantly, wedemonstratethatasurfaceproducesextremely\nlarge damping anisotropies that can be orders of magni-\ntude larger than that of the bulk. We call the attention\nto the fact that this is the first time, as far as we know,\nthat damping values are theoretically obtained in such a\nlocal way.\nResults: We calculated: i)ordered Fe50Co50in theB2\nstructure (hereafter refereed to as B2-FeCo) ii)random\nFe50Co50alloysinbccorbctstructures, wherethevirtual\ncrystal approximation (VCA) was applied; iii)Fe50Co50\nalloys simulated as embedded clusters in a VCA matrix\n(host). In all cases VCA was simulated with an elec-\ntronic concentration corresponding to Fe50Co50. The ii)\nandiii)alloys were considered as in bulk as well as in\nthe (001) surface, with bcc and bct structures (here-\nafter correspondingly refereed as VCA Fe50Co50bcc,\nVCA Fe50Co50bct, VCA Fe50Co50(001) bcc and VCA\nFe50Co50(001) bct). The effect of local tetragonal distor-\ntions was considered with a localc\na= 1.09ratio (SM for\ndetails). All data for cluster based results, were obtained\nfrom an average of several different configurations. The\ntotal damping for a given site iin real-space ( αt, Eqs. S6\nand S7 from SM) can be decomposed in non-local, αij\n(i/negationslash=j), and local (onsite), αonsite(orαii,i=j) contri-\nbutions, each of them described by the tensor elements2\nανµ\nij=g\nmiπ/integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTν\niImˆGijˆTµ\njImˆGji/parenrightBig\nd/epsilon1,(1)\nwheremiis the total magnetic moment localized in the\nreference atomic site i,µ,ν={x,y,z},ˆTis the torque\noperator, and η(/epsilon1) =∂f(/epsilon1)\n∂/epsilon1the derivative of the Fermi\ndistribution. The scalar αijparameter is defined in the\ncollinear regime as αij=1\n2(αxx\nij+αyy\nij).\nTo validate our methodology, the here obtained total\ndamping for several systems (such as bcc Fe, fcc Ni, hcp\nand fcc Co and B2-FeCo) were compared with estab-\nlished values available in the literature (Table S1, SM),\nwhere an overall good agreement can be seen.\nFig. 1 shows the non-local contributions to the damp-\ning for bcc Fe and B2-FeCo. Although the onsite contri-\nbutions are around one order of magnitude larger than\nthe non-local, there are many αijto be added and total\nnet values can become comparable. Bcc Fe and B2-FeCo\nhave very different non-local damping contributions. El-\nement resolved αij, reveal that the summed Fe-Fe in-\nteractions dominate over Co-Co, for distances until 2a\ninB2-FeCo. We observe that αijis quite extended in\nspace for both bcc Fe and B2-FeCo. The different con-\ntributions to the non-local damping, from atoms at equal\ndistance arises from the reduced number of operations in\nthe crystal point group due to the inclusion of SOC in\ncombination with time-reversal symmetry breaking. The\nB2-FeCo arises from replacing every second Fe atom in\nthe bcc structure by a Co atom. It is interesting that this\nreplacement (i.e. the presence of Co in the environment)\nsignificantly changes the non-local contributions for Fe-\nFe pairs , what can more clearly be seen from the Insetin\nFig. 1, where the non-local damping summed over atoms\nat the same relative distance for Fe-Fe pairs in bcc Fe\nandB2-FeCo are shown; the non-local damping of Fe-Fe\npairs are distinctly different for short ranges, while long\nranged (further than ∼2.25 Å) contributions are smaller\nand more isotropic.\nThe damping anisotropy, i.e. the damping change,\nwhen the magnetization is changed from the easy axis\nto a new direction is1\n∆αt=/parenleftBigg\nα[110]\nt\nα[010]\nt−1/parenrightBigg\n×100%, (2)\nwhereα[110]\ntandα[010]\ntare the total damping obtained\nfor magnetization directions along [110]and [010], re-\nspectively. Analogousdefinitionalsoappliesfor ∆αonsite.\n1We note that this definition is different to the maximum-\nminimum damping ratio, defined asα[110]\nt\nα[010]\nt×100%, from Ref.\n[10].We investigated this anisotropy in surfaces and in bulk\nsystems with (and without) tetragonal structural distor-\ntions. Our calculations for VCA Fe50Co50bcc show a\ndamping increase of ∼13%, when changing the magne-\ntization direction from [010]to[110](Table S2 in the\nSM). The smallest damping is found for the easy magne-\ntization axis, [010], which holds the largest orbital mo-\nment (morb) [18]. For VCA Fe50Co50bcc we obtained\na small variation of ∼2%for the onsite contribution\n(α[010]\nonsite = 8.94×10−4andα[110]\nonsite = 8.76×10−4),\nwhat implies that the anisotropy comes mostly from\nthe non-local contributions, particularly from the next-\nnearest neighbours. For comparison, ∆αt∼3%(with\n∆αonsite∼0.4%) in the case of bcc Fe, what corrobo-\nrates the reported [11] small bcc Fe anisotropy at room\ntemperature, andwiththebulkdampinganisotropyrates\n[19].\nWe also inspected the chemical inhomogeneity influ-\nence on the anisotropy, considering the B2-FeCo alloy,\nwhere the weighted average damping (Eq. S7 of SM)\nwas used instead. The B2-FeCo bcc (∼7%) and VCA\nFe50Co50bcc (∼13%) anisotropies are of similar magni-\ntudes. Both B2structure and VCA calculations lead to\ndamping anisotropies which are significantly lower than\nwhatwasobservedintheexperiments, anditseemslikely\nthatthepresenceofdisorderincompositionand/orstruc-\ntural properties of the Fe/Co alloy would be important\nto produce large anisotropy effects on the damping.\n\nα\nij\n\t\n×\n\t\n10\n-4\n−3\n−2\n−1\n0\n1\n2\n3\n4\n\nNormalized\n\t\ndistance\n1.0\n1.5\n2.0\n2.5\n3.0\n\n\t\nB2\n\t\nFe-Co\n\t\nB2\n\t\nFe-Fe\n\t\nB2\n\t\nCo-Co\n\t\nbcc\n\t\nFe-Fe\n\t\n−10\n0\n10\n20\n\n\t\n1.0\n1.5\n2.0\n2.5\n3.0\nFigure 1. (Color online) Non-local damping contributions,\nαij, in (Fe-centered) bulk B2-FeCo and bcc Fe, as a function\nof the normalized distance in lattice constant units a.Inset:\nNon-local contributions from only Fe-Fe pairs summed, for\neach distance, in bcc Fe bulk (empty blue dots) and in the\nB2-FeCo (full red dots). The onsite damping for Fe (Co) in\nB2-FeCo isαFe\nonsite = 1.1×10−3(αCo\nonsite = 0.8×10−3) and for\nbcc Fe it is αFe\nonsite = 1.6×10−3. The magnetization direction\nisz([001]). Lines are guides for the eyes.\nWeanalyzedtheroleoflocaldistortionsbyconsidering3\na hypothetical case of a large, 15%(c\na= 1.15), distortion\non thez-axis of ordered B2-FeCo. We found the largest\ndamping anisotropy ( ∼24%) when comparing the results\nwith magnetization in the [001](α[001]\nt= 10.21×10−3)\nand in the [010](α[010]\nt= 7.76×10−3) directions. This\nconfirms that, indeed, bct-like distortions act in favour of\nthe∆αtenhancement (and therefore, of the maximum-\nminimum damping ratio), but the theoretical data are\nnot large enough to explain the giant value reported ex-\nperimentally [10].\nNevertheless, in the case of an alloy, the local lattice\ndistortions suggested in Ref. [10] are most to likely occur\nin an heterogeneous way [20], with different distortions\nfor different local environments. To inspect this type\nof influence on the theoretical results, we investigated\n(Table S3, SM) clusters containing different atomic con-\nfigurations embedded in a VCA Fe50Co50matrix (with\nFe bulk lattice parameter); distortions were also consid-\nered such that, locally in the clusters,c\na= 1.15(Ta-\nble S4, in the SM). Moreover, in both cases, two types\nof clusters have to be considered: Co-centered and Fe-\ncentered. The αtwas then computed as the sum of the\nlocal and non-local contributions for clusters with a spe-\ncific central (Fe or Co) atom, and the average of Fe-\nand Co-centered clusters was taken. Fe-centered clus-\ntershaveshownlargeranisotropies, onaverage ∼33%for\nthe undistorted (∼74%for the distorted) compared with\n∼8%fortheundistortedCo-centeredclusters( ∼36%for\nthe distorted). Although these results demonstrate the\nimportance of both, local distortions as well as non-local\ncontributions to the damping anisotropy, they are not\nstill able to reproduce the huge observed [10] maximum-\nminimum damping ratio.\nWe further proceed our search for ingredients that\ncould lead to a huge ∆αtby inspecting interface effects,\nwhich are present in thin films, grain boundaries, stack-\ning faults and materials in general. Such interfaces may\ninfluence observed properties, and in order to examine\nif they are relevant also for the reported alloys of Ref.\n[10], we considered these effects explicitly in the calcu-\nlations. As a model interface, we considered a surface,\nwhat is, possibly, the most extreme case. Hence, we per-\nformed a set of αtcalculations for the Fe50Co50(001),\nfirst on the VCA level. Analogous to the respective bulk\nsystems, we found that the onsite contributions to the\ndamping anisotropy are distinct, but they are not the\nmain cause ( ∆αonsite∼18%). However, the lack of in-\nversion symmetry in this case gives a surprisingly large\nenhancement of ∆αt, thus having its major contribution\ncoming from the non-local damping terms, in particular\nfrom the next-nearest neighbours. Interestingly, negative\nnon-local contributions appear when αtis calculated in\nthe[010]direction. These diminish the total damping\n(the onsite contribution being always positive) and gives\nrise to a larger anisotropy, as can be seen by comparisonof the results shown in Table I and Table S5 (in the Sup-\nplemental Material). In this case, the total anisotropy\nwas found to be more than ∼100%(corresponding to a\nmaximum-minimum damping ratio larger than 200%).\nA compilation of the most relevant theoretical results\nobtained here is shown in Fig. 2, together with the ex-\nperimental data and the local density of states (LDOS)\natEFfor each magnetization direction of a typical atom\nin the outermost layer (data shown in yellow). As shown\nin Fig. 2, the angular variation of αthas a fourfold ( C4v)\nsymmetry, with the smallest Gilbert damping occurring\nat 90◦from the reference axis ( [100],θH= 0◦), for both\nsurface and bulk calculations. This pattern, also found\nexperimentally in [10], matches the in-plane bcc crys-\ntallographic symmetry and coincides with other mani-\nfestations of SOC, such as the anisotropic magnetoresis-\ntance [10, 21]. Following the simplified Kambersky’s for-\nmula [13, 22], in which (see SM) α∝n(EF)and, there-\nfore, ∆α∝∆n(EF), we can ascribe part of the large\nanisotropy of the FeCo alloys to the enhanced LDOS dif-\nferences at the Fermi level, evidenced by the close corre-\nlation between ∆n(EF)and∆αtdemonstrated in Fig. 2.\nThus, as a manifestation of interfacial SOC (the so-called\nproximity effect [23]), the existence of ∆αtcan be under-\nstoodintermsofRashba-likeSOC,whichhasbeenshown\nto play an important role on damping anisotropy [24, 25].\nAnalogous to the bulk case, the higher morboccurs where\nthe system presents the smallest αt, and the orbital\nmoment anisotropy matches the ∆αtfourfold symme-\ntry with a 90◦rotation phase (see Fig. S3, SM). Note\nthat a lower damping anisotropy than Co50Fe50(001) is\nfound for a pure Fe(001) bcc surface, where it is ∼49%\n(Table S2, SM), in accordance with Refs. [7, 26], with\na dominant contribution from the onsite damping val-\nues (conductivity-like character on the reciprocal-space\n[19, 27]).\nThe VCA surface calculations on real-space allows to\ninvestigate the layer-by-layer contributions (intra-layer\ndamping calculation), as shown in Table I. We find that\nthemajorcontributiontothedampingsurfaceanisotropy\ncomes from the outermost layer, mainly from the differ-\nence in the minority 3dstates around EF. The deeper\nlayers exhibit an almost oscillatory ∆αtbehavior, simi-\nlar to the oscillation mentioned in Ref. [28] and to the\nFriedel oscillations obtained for magnetic moments. The\ndamping contributions from deeper layers are much less\ninfluenced by the inversion symmetry breaking (at the\nsurface), as expected, and eventually approaches the typ-\nicalbulklimit. Therefore, changesintheelectronicstruc-\ntureconsiderednotonlytheLDOSoftheoutermostlayer\nbut a summation of the LDOS of all layers (including the\ndeeper ones), which produces an almost vanishing differ-\nence between θH= 0◦andθH= 45◦(also approaching\nthe bulk limit). The damping anisotropy arising as a sur-\nface effect agrees with what was observed in the case of\nFe [7] and CoFeB [29] on GaAs(001), where the damping4\n0o45o90o\n135o\n180o\n225o\n270o315o[100][110][010]\n[−110]\n0.10.20.3\nΔn(EF)\n(st./Ry−at.)\n0.0050.0100.015\nαt\nθH\nFigure 2. (Color online) Total damping and LDOS difference\natEF,∆n(EF), as a function of θH, the angle between the\nmagnetizationdirectionandthe [100]-axis. Squares: (redfull)\nVCAFe 50Co50(001)bcc. Triagles: (greenfull)averageover32\nclusters (16 Fe-centered and 16 Co-centered), with bcc struc-\nture at the surface layers (SM) embedded in a VCA medium;\n(gray open) similar calculations, but with a local lattice dis-\ntortion. Circles: (yellow open) ∆n(EF)betweenθH= 0◦and\nthe current angle for a typical atom in the outermost layer\nof VCA Fe 50Co50(001) bcc; (blue full) experimental data [10]\nfor a 10-nm Fe 50Co50/Pt thin film; (purple full) average bulk\nVCA Fe 50Co50bcc; and (brown full) the B2-FeCo bulk. Lines\nare guides for the eyes.\nanisotropy diminishes as the film thickness increases.\nTable I. Total intra-layer damping ( αt×10−3) and anisotropy,\n∆αt(Eq. 2), of a typical (VCA) atom in each Fe 50Co50(001)\nbcc surface layer for magnetization along [010]and[110]di-\nrections. In each line, the sum of all αijin the same layer is\nconsidered. Outermost (layer 1) and deeper layers (2-5).\nLayerαt[010]αt[110] ∆αt\n1 7.00 14.17 +102.4%\n2 1.28 1.16 −9.4%\n3 2.83 3.30 +16.6%\n4 2.18 1.99 −8.7%\n5 2.54 2.53 −0.4%\nWe also studied the impact of bct-like distortions in\nthesurface, initiallybyconsideringtheVCAmodel. Sim-\nilartothebulkcase,tetragonaldistortionsmaybeimpor-\ntant for the damping anisotropy at the surface, e.g. when\nlocal structural defects are present. Therefore, localized\nbct-like distortions of the VCA medium in the surface,\nparticularly involving the most external layer were inves-\ntigated. The structural model was similar to what was\nused for the Fe50Co50bulk, consideringc\na= 1.09(see\nSM). Our calculations show that tetragonal relaxations\naround a typical site in the surface induce a ∆αt∼75%,\nfromα[010]\nt= 8.94×10−3toα[110]\nt= 15.68×10−3. Themain effect of these distortions is an enhancement of the\nabsolute damping values in each direction with respect to\nthe pristine (bcc) system. This is due to an increase on\nαonsite, fromα[010]\nonsite = 7.4×10−3toα[010]\nonsite = 9.5×10−3,\nand fromα[110]\nonsite = 8.7×10−3toα[110]\nonsite = 11.7×10−3;\nthe resulting non-local contributions remains similar to\ntheundistortedcase. Theinfluenceofbct-likedistortions\non the large damping value in the Fe50Co50surface is in\nline with results of Mandal et al.[30], and is related to\nthe transition of minority spin electrons around EF.\nWe then considered explicit 10-atom Fe50Co50clusters\nembedded in a VCA FeCo surface matrix. The results\nfrom these calculations were obtained as an average over\n16 Fe-centered and 16 Co-centered clusters. We con-\nsidered clusters with undistorted bcc crystal structure\n(Fig. 2, yellow open circles) as well as clusters with lo-\ncal tetragonal distortions (Fig. 2, black open circles). As\nshown in Fig. 2 the explicit local tetragonal distortion\ninfluences the damping values ( α[010]\nt= 10.03×10−3and\nα[110]\nt= 14.86×10−3)andtheanisotropy, butnotenough\ntoreproducethehugevaluesreportedintheexperiments.\nA summary of the results obtained for each undis-\ntorted FeCo cluster at the surface is shown in Fig. 3:\nCo-centered clusters in Fig. 3(a) and Fe-centered clusters\nin Fig. 3(b). A large variation of αtvalues is seen from\nclustertocluster, dependingonthespatialdistributionof\natomic species. It is clear that, αtis larger when there is\na larger number of Fe atoms in the surface layer that sur-\nroundsthecentral, referenceclustersite. Thiscorrelation\ncan be seen by the numbers in parenthesis on top of the\nblue symbols (total damping for each of the 16 clusters\nthat were considered) in Fig. 3. We also notice from the\nfigure that the damping in Fe-centered clusters are lower\nthan in Co-centered, and that the [010]magnetization di-\nrection exhibit always lower values. In the Insetof Fig. 3\nthe onsite contributions to the damping, αonsite, and the\nLDOS atEFin the central site of each cluster are shown:\na correlation, where both trends are the same, can be ob-\nserved. The results in Fig. 3 shows that the neighbour-\nhood influences not only the local electronic structure at\nthe reference site (changing n(EF)andαonsite), but also\nmodifies the non-local damping αij, leading to the cal-\nculatedαt. In other words, the local spatial distribution\naffects how the total damping is manifested, something\nwhich is expressed differently among different clusters.\nThis may open up for materials engineering of local and\nnon-local contributions to the damping.\nConclusions: We demonstrate here that real-space\nelectronic structure, based on density functional theory,\nyield a large Gilbert damping anisotropy in Fe50Co50al-\nloys. Theory leads to a large damping anisotropy, when\nthe magnetization changes from the [010]to the [110]di-\nrection, which can be as high as ∼100%(or200%in the\nminimum-maximum damping ratio) when surface calcu-\nlations are considered. This is in particular found for5\n\u0001\u0001\u0002\u0003\u0004\u0005\u0006\u0007\n\u0005\u0004\u0005\b\u0005\u0007\u0005\t\u0005\n\u0001\n\u0001\u0001\n\u000b\f\u0001\r\u000e\u0006\u000f\u0010\u0011\u0002\u0010\u0012\u0010\u0013\u0001\u0002\u0001\u0014\u0005\u0004\u0005\u0015\u0001\u0002\u0001\u0014\u0004\u0004\u0005\u0015\u0001\u0001\u0016\u0016\u0003\u0004\u0005\u0006\u0007\u0005\u0004\u0005\b\u0005\u0007\u0005\n\u0001\u0002\u0003\u0004\u0005\b\u0005\u0007\u0005\t\u0005\n\u0001\b\t\u0017\u0018\u0004\u0005\u0004\b\u0004\t\u0004\u0017\u0001\u0006\u0007\u0007\b\t\n\u000b\n\f\b\u0001\u0002\u0003\u0004\u0005\b\t\n\u000b\n\f\u0006\u0007\u0007\b\t\u000b\u000b\n\f\b\u0001\u0002\u0003\u0004\u0005\b\t\u000b\u000b\n\f\n\u0001\u0001\u0002\u0003\u0001\u0004\u0005\u0006\u0007\n\u0005\b\u0004\u0005\u0004\b\t\u0005\n\u0001\n\u000b\f\r\u0002\u000e\u000f\u0001\u0010\f\u0011\u0012\u000e\u000f\t\u0013\u0014\u0015\u0004\u0005\u0004\t\u0004\u0013\u0004\u0014\u0001\u0016\u0012\u0017\u0001\u0018\u000e\u0006\u0019\u000e\u0010\u0002\u000e\u000f\u000e\u001a\u0001\u0001\u001b\u001b\u0003\u0001\u0004\u0005\u0006\u0007\u0005\b\u0004\u0005\u0004\b\t\u0005\n\u0001\u0002\u0003\u0004\u0005\u0004\u0014\u0004\u0015\t\u0005\t\t\n\u0001\t\u0013\u0014\u0015\u0004\u0005\u0004\t\u0004\u0013\u0004\u0014\u0001(0)(1)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(3)(3)(4)(4)\n(1)(1)(2)(2)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(4)(5)\nFigure 3. Damping for the [010](open circles) and [110](full\ncircles) magnetization directions for distinct types of 10-atom\nFe50Co50bcc clusters, embedded in VCA Fe 50Co50(001) bcc\nand without any distortion around the reference atom (for\nwhichαtandαonsiteare shown). (a) Co-centered and (b)\nFe-centered clusters. The quantity of Fe atoms in the surface\nlayers (near vacuum) are indicated by the numbers in paren-\nthesis and the results have been ordered such that larger val-\nues are to the left in the plots. Insets:αonsitefor the [010]\n(red open circles) and [110](blue filled circles) magnetization\ndirections, and corresponding local density of states, n(EF),\nat the Fermi level (green filled and unfilled triangles) at the\ncentral atom (placed in the outermost layer) for both types\nof clusters. Lines are guides for eyes.\ncontributions from surface atoms in the outermost layer.\nHence the results presented here represents one more ex-\nample, in addition to the well known enhanced surface\norbital moment [31], of the so-called interfacial spin-orbit\ncoupling. This damping anisotropy, which holds a bcc-\nlike fourfold ( C4v) symmetry, has a close relation to the\nLDOS difference of the most external layer at EF(ma-\njorly contributed by the minority dstates), as well as\nto the orbital moment anisotropy with a 90◦phase. As\na distinct example of an interface, we consider explicitly\nthe Fe50Co50cluster description of the alloy. In this case,\nbesides an onsite contribution, we find that the damp-\ning anisotropy is mostly influenced by non-local next-\nnearest-neighbours interactions.\nSeveral Gilbert damping anisotropy origins are also\ndemonstrated here, primarily related to the presence of\ninterfaces, alloy composition and local structural distor-\ntions (as summarized in Table S6, in the SM [32]). Pri-\nmarily we find that: ( i) the presence of Co introduces anenhanced spin-orbit interaction and can locally modify\nthe non-local damping terms; ( ii) the randomness of Co\nin the material, can modestly increase ∆αtas a total ef-\nfect by creating Co-concentrated clusters with enhanced\ndamping; ( iii) at the surface, the spatial distribution of\nFe/Co, increases the damping when more Fe atoms are\npresent in the outermost layer; and ( iv) the existence\nof local, tetragonal distortions, which act in favour (via\nSOC) of the absolute damping enhancement, by modify-\ning theαonsiteof the reference atom, and could locally\nchange the spin relaxation time. Furthermore, in rela-\ntionship to the work in Ref. [10], we show here that bulk\nlike tetragonal distortions, that in Ref. [10] were sug-\ngested to be the key reason behind the observed huge\nanisotropy of the damping, can in fact not explain the\nexperimental data. Such distortions were explicitly con-\nsidered here, using state-of-the-art theory, and we clearly\ndemonstrate that this alone can not account for the ob-\nservations.\nAlthough having a similar trend as the experimen-\ntal results of Ref. [10], we do not reproduce the most\nextreme maximum-minimum ratio reported in the ex-\nperiment,∼400%(or∆αt∼300%). The measured\ndamping does however include effects beyond the intrin-\nsic damping that is calculated from our electronic struc-\nturemethodology. Other mechanismsare knownto influ-\nence the damping parameter, such as contributions from\neddy currents, spin-pumping, and magnon scattering, to\nname a few. Thus it is possible that a significant part\nof the measured anisotropy is caused by other, extrin-\nsic, mechanisms. Despite reasons for differences between\nobservation and experiment on films of Fe50Co50alloys,\nthe advancements presented here provide new insights on\nthe intrinsic damping anisotropy mechanisms, something\nwhich is relevant for the design of new magnetic devices.\nAcknowledgements: H.M.P. and A.B.K. acknowledge\nfinancial support from CAPES, CNPq and FAPESP,\nBrazil. The calculations were performed at the computa-\ntional facilities of the HPC-USP/CENAPAD-UNICAMP\n(Brazil), at the National Laboratory for Scientific Com-\nputing (LNCC/MCTI, Brazil), and at the Swedish Na-\ntional Infrastructure for Computing (SNIC). I.M. ac-\nknowledge financial support from CAPES, Finance Code\n001, process n◦88882.332894/2018-01, and in the Insti-\ntutional Program of Overseas Sandwich Doctorate, pro-\ncess n◦88881.187258/2018-01. O.E. acknowledges sup-\nport from the Knut och Alice Wallenberg (KAW) foun-\ndation, the Swedish research council (VR), the Founda-\ntion for Strategic Research (SSF), the Swedish energy\nagency (Energimyndigheten), eSSENCE, STandUPP,\nand the ERC (synergy grant FASTCORR). D.T. ac-\nknowledges support from the Swedish Research Council\n(VR) through Grant No. 2019-03666.6\n∗aklautau@ufpa.br\n[1] O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik,\nAtomistic Spin Dynamics: Foundations and Applications\n(Oxford University Press, 2017).\n[2] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n[3] A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,\n372 (2012).\n[4] M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J.\nSilva, H. T. Nembach, O. Eriksson, O. Karis, and J. M.\nShaw, Nat. Phys. 12, 839 (2016).\n[5] D. Thonig, Y. Kvashnin, O. Eriksson, and M. Pereiro,\nPhys. Rev. Mater. 2, 013801 (2018).\n[6] Y. Liu, Z. Yuan, R. J. Wesselink, A. A. 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Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4\n1Universidade de São Paulo, Instituto de Física,\nRua do Matão, 1371, 05508-090, São Paulo, SP, Brazil\n2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and\n4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden\n(Dated: November 22, 2021)\nI. Theory\nThe torque-correlation model, first introduced by\nKamberský [1], and later elaborated by Gilmore et al.\n[2], can be considered as both a generalization and an\nextended version of the breathing Fermi surface model,\nwhich relates the damping of the electronic spin orienta-\ntion, with the variation in the Fermi surface when the\nlocal magnetic moment is changed. In this scenario,\nand considering the collinear limit of the magnetic or-\ndering, due to the spin-orbit coupling (SOC), the tilting\nin magnetization ˆmby a small change δˆmgenerates a\nnon-equilibrium population state which relaxes within a\ntimeτtowards the equilibrium. We use an angle θ, to\nrepresent the rotation of the magnetization direction δˆm.\nIftheBlochstatesofthesystemsarecharacterizedbythe\ngenericbandindex natwavevector k(withenergies /epsilon1k,n),\nit is possible to define a tensorfor the damping, that has\nmatrix elements (adopting the isotropic relaxation time\napproximation)\nανµ=gπ\nm/summationdisplay\nn,mdk\n(2π)3η(/epsilon1k,n)/parenleftbigg∂/epsilon1k,n\n∂θ/parenrightbigg\nν/parenleftbigg∂/epsilon1k,m\n∂θ/parenrightbigg\nµτ\n~\n(S1)\nwhich accounts for both intraband ( n=m, conductivity-\nlike) and interband ( n/negationslash=m, resistivity-like) contribu-\ntions [2]. Here µ,νare Cartesian coordinate indices,\nthat will be described in more detail in the discussion\nbelow, while η(/epsilon1k,n) =∂f(/epsilon1)\n∂/epsilon1/vextendsingle/vextendsingle/vextendsingle\n/epsilon1k,nis the derivative of\nthe Fermi distribution, f, with respect to the energy\n/epsilon1, andn,mare band indices. Therefore, the torque-\ncorrelation model correlates the spin damping to vari-\nations of the energy of single-particle states with respect\nto the variation of the spin direction θ, i.e.∂/epsilon1k,n\n∂θ. Us-\ning the Hellmann-Feynmann theorem, which states that\n∂/epsilon1k,n\n∂θ=/angbracketleftψk,n|∂H\n∂θ|ψk,n/angbracketright, and the fact that only the spin-\norbit Hamiltonian Hsochanges with the magnetization\ndirection, the spin-orbit energy variation is given by\n∂/epsilon1k,n(θ)\n∂θ=/angbracketleftψk,n|∂\n∂θ/parenleftbig\neiσ·ˆnθHsoe−iσ·ˆnθ/parenrightbig\n|ψk,n/angbracketright(S2)\nin which σrepresents the Pauli matrices vector, and\nˆnis the direction around which the local moment hasbeen rotated. The expression in Eq. S2 can be eas-\nily transformed into∂/epsilon1k,n(θ)\n∂θ=i/angbracketleftψk,n|[σ·ˆn,Hso]|ψk,n/angbracketright\nand we call ˆT= [σ·ˆn,Hso]thetorqueoperator. In\nview of this, it is straightforward that, in the collinear\ncase in which all spins are aligned to the zdirection,\nσ·ˆn=σµ(µ=x,y,z), originating the simplest {x,y,z}-\ndependent torque operator ˆTµ. Putting together the in-\nformation on Eqs. S1 and S2, and using the fact that\nthe imaginary part of the Greens’ functions can be ex-\npressed, in Lehmann representation, as Im ˆG(/epsilon1±iΛ) =\n−1\nπ/summationtext\nnΛ\n(/epsilon1−/epsilon1n)2+Λ2|n/angbracketright/angbracketleftn|, then it is possible to write in\nreciprocal-space [3]:\nανµ=g\nmπ/integraldisplay /integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTνImˆGˆTµImˆG/parenrightBig\nd/epsilon1dk\n(2π)3.(S3)\nIn a real-space formalism, the Fourier transformation\nof the Green’s function is used to find a very similar ex-\npression emerges for the damping element ανµ\nijrelative to\ntwo atomic sites iandj(at positions riandrj, respec-\ntively) in the material:\nανµ\nij=g\nmiπ/integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTν\niImˆGijˆTµ\njImˆGji/parenrightBig\nd/epsilon1,(S4)\nwhere we defined mi= (morb+mspin)as the total mag-\nnetic moment localized in the reference atomic site iin\nthe pair{i,j}. The electron temperature that enters into\nη(ε)is zero and, consequently, the energy integral is per-\nformed only at the Fermi energy. In this formalism, then,\nthe intraband and interband terms are replaced by onsite\n(i=j) and non-local ( i/negationslash=j) terms. After calculation of\nall components of Eq. S4 in a collinear magnetic back-\nground, we get a tensor of the form\nαij=\nαxxαxyαxz\nαyxαyyαyz\nαzxαzyαzz\n, (S5)\nwhich can be used in the generalized atomistic Landau-\nLifshitz-Gilbert (LLG) equation for the spin-dynamics\nof magnetic moment on site i[4]:∂mi\n∂t=mi×/parenleftBig\n−γBeff\ni+/summationtext\njαij\nmj·∂mj\n∂t/parenrightBig\n. Supposing that all spins are\nparallel to the local zdirection, we can define the scalar2\nαvalue as the average between components αxxandαyy,\nthat is:α=1\n2(αxx+αyy).\nOnce one has calculated the onsite ( αonsite) and the\nnon-local (αij) damping parameters with respect to the\nsite of interest i, the total value, αt, can be defined as\nthe sum of all these α’s:\nαt=/summationdisplay\n{i,j}αij. (S6)\nIn order to obtain the total damping in an heteroge-\nneous atomic system (more than one element type), such\nas Fe50Co50(with explicit Fe/Co atoms), we consider the\nweighted average between the different total local damp-\ning values ( αi\nt), namely:\nαt=1\nMeff/summationdisplay\nimiαi\nt, (S7)\nwheremiis the local magnetic moment at site i, and\nMeff=/summationtext\nimiis the summed total effective magnetiza-\ntion. This equation is based on the fact that, in FMR ex-\nperiments, the magnetic moments are excited in a zone-\ncentered, collective mode (Kittel mode). In the results\npresented here, Eq. S7 was used to calculate αtofB2-\nFeCo, both in bcc and bct structures.\nII. Details of calculations\nThe real-space linear muffin-tin orbital on the atomic-\nsphere approximation (RS-LMTO-ASA) [5] is a well-\nestablished method in the framework of the DFT to de-\nscribetheelectronicstructureofmetallicbulks[6,7], sur-\nfaces [8, 9] and particularly embedded [10] or absorbed\n[11–14] finite cluster systems. The RS-LMTO-ASA is\nbased on the LMTO-ASA formalism [15], and uses the\nrecursion method [16] to solve the eigenvalue problem\ndirectly in real-space. This feature makes the method\nsuitable for the calculation of local properties, since it\ndoes not depend on translational symmetry.\nThe calculations performed here are fully self-\nconsistent, and the spin densities were treated within\nthe local spin-density approximation (LSDA) [17]. In all\ncases, we considered the spin-orbit coupling as a l·sterm\nincluded in each variational step [18–20]. The spin-orbit\nis strictly necessary for the damping calculations due to\nits strong dependence on the torque operators, ˆT. In\nthe recursion method, the continued fractions have been\ntruncated with the Beer-Pettifor terminator [21] after 22\nrecursion levels ( LL= 22). The imaginary part that\ncomes from the terminator was considered as a natural\nchoice for the broadening Λto build the Green’s func-\ntions ˆG(/epsilon1+iΛ), which led to reliable αparameters in\ncomparison with previous results (see Table S1).To account for the Co randomness in the experimental\nFe50Co50films [22], some systems were modeled in terms\nof the virtual crystal approximation (VCA) medium of\nFe50Co50, considering the bcc (or the bct) matrix to have\nthe same number of valence electrons as Fe50Co50(8.5\ne−). However, we also investigated the role of the Co\npresence, as well as the influence of its randomness (or\nordering), by simulating the B2(CsCl) FeCo structure\n(a=aFe). The VCA Fe50Co50andB2-FeCo bulks were\nsimulated by a large matrix containing 8393 atoms in\nreal-space, the first generated by using the Fe bcc lat-\ntice parameter ( aFe= 2.87Å) and the latter using the\noptimized lattice parameter ( a= 2.84Å). Thisachoice\nin VCA Fe50Co50was based on the fact that it is eas-\nier to compare damping results for Fe50Co50alloy and\npure Fe bcc bulk if the lattice parameters are the same,\nand the use of the aFehas shown to produce trustwor-\nthyαtvalues. On the other hand, bct bulk structures\nwithc\na= 1.15(B2-FeCo bct and VCA Fe50Co50bct) are\nbased on even larger matrices containing 49412 atoms.\nThe respective surfaces were simulated by semi-matrices\nof the same kind (4488 and 19700 atoms, respectively),\nconsidering one layer of empty spheres above the outer-\nmost Fe50Co50(or pure Fe) layer, in order to provide a\nbasis for the wave functions in the vacuum and to treat\nthe charge transfers correctly.\nWe notice that the investigations presented here are\nbased on a (001)-oriented Fe50Co50film, in which only a\nsmall lattice relaxation normal to the surface is expected\nto occur (∼0.1%[23]).\nDamping parameters of Fe-centered and Co-centered\nclusters, embedded in an Fe50Co50VCA medium, have\nbeen calculated (explicitly) site by site. In all cases,\nthese defects are treated self-consistently, and the po-\ntential parameters of the remaining sites were fixed at\nbulk/pristine VCA surface values, according to its envi-\nronment. When inside the bulk, we placed the central\n(reference) atom of the cell in a typical site far away\nfrom the faces of the real-space matrix, avoiding any un-\nwanted surface effects. We considered as impurities the\nnearest 14 atoms (first and second nearest neighbours,\nup to 1a) from the central atom, treating also this sites\nself-consistently, in a total of 15 atoms. We calculated\n10 cases with Fe and Co atoms randomly positioned: 5\nwith Fe as the central atom (Fe-centered) and 5 with Co\nas the central atom (Co-centered). An example (namely\ncluster #1 of Tables S3 and S4), of one of these clusters\nembedded in bulk, is represented in Fig. S1(a). As the\nself-consistent clusters have always a total of 15 atoms,\nthe Fe (Co) concentration is about 47% (53%) or vice-\nversa. On the other hand, when inside the surface, we\nplaced the central (reference) atom of the cluster in a\ntypical site of the most external layer (near vacuum),\nsince this has shown to be the layer where the damping\nanisotropy is larger. Therefore, we considered as impu-\nrities the reference atom itself and the nearest 9 atoms3\n(up to 1a), in a total of 10 atoms (and giving a perfect\n50% (50%) concentration). An example of one of these\nclusters embedded in a surface is shown in Fig. S1(b).\n(a)\n(b)\nFigure S1. (Color online) Schematic representation of an ex-\nampleof: (a)Fe-centered15-atomclusterembeddedinaVCA\nFe50Co50bcc bulk medium; (b) Co-centered 10-atom cluster\nembedded in a VCA Fe 50Co50(001) bcc surface medium. Yel-\nlow and blue spheres represent Fe and Co atoms, respectively,\nwhile gray atoms represent the VCA Fe 50Co50sites (8.5 va-\nlence e−). The Fe(Co) concentration in the clusters are: (a)\n53% (47%) and (b) 50% (50%) . The total number of atoms\nincluding the surrounding VCA sites are: (a) 339 and (b) 293.\nThey were all accounted in the sum to obtain αtat the central\n(reference) Fe (a) and Co (b) site.\nTo simulate a bct-like bulk distortion, the 8 first neigh-\nbours of the central atom were stretched in the cdi-\nrection, resulting in ac\na= 1.15ratio. On the other\nhand, when embedded on the Fe50Co50(001) bcc surface,\nthe central (reference) atom is placed in the outermost\nlayer (near vacuum), and we simulate a bct distortion\nby stretching the 4 nearest-neighbours (on the second\nlayer) to reproduce ac\na= 1.09ratio (the maximum per-\ncentage that the atoms, in these conditions, could be\nmoved to form a bct-like defect). In this case, a total\nof 10 atoms (the nearest 9 atoms from the central one\n– up to 1a– and the reference atom) were treated self-\nconsistently, analogous to as shown in Fig. S1(b). As in\nthe case of the pristine bcc Fe50Co50clusters embedded\nin the VCA surface, we considered a total of 32 10-atom\nclusters with different Fe/Co spatial distributions, being\n16 Fe-centered, and 16 Co-centered.III. Comparison with previous results\nTheab-initio calculation of the Gilbert damping, in\nthe collinear limit, is not a new feature in the literature.\nMainly, the reported theoretical damping results are for\nbulk systems [2, 4, 24–28], but, some of them even stud-\nied free surfaces [29]. Therefore, in order to demonstrate\nthe reliability of the on-site and total damping calcula-\ntions implemented here in real-space, a comparison of\nthe presently obtained with previous (experimental and\ntheoretical) results, are shown in Table S1. As can be\nseen, our results show a good agreement with previously\nobtainedαvalues, including some important trends al-\nreadypredictedbefore. Forexample, thereducedGilbert\ndamping of Co hcp with respect to the Co fcc due to\nthe reduction of the density of states at the Fermi level\n[24, 28], (∼10.92states/Ry-atom in the hcp case and\n∼16.14states/Ry-atom in the fcc case).\nIV. Details of the calculated damping values\nThe damping values obtained for the systems studied\nhere are shown in Tables (S2-S5). These data can be use-\nful for the full understanding of the results presented in\nthe main text. For easy reference, in Table S2 the αtof a\ntypical atom in each system (bulk or surface) for different\nspin quantization axes are shown. These data are plot-\nted in Fig. 2 of the main text. The obtained values show\nthat, indeed, for bulk systems the damping anisotropies\nare not so pronounced as in the case of Fe50Co50(001)\nbcc surface.\nAs observed in Table S2, the increase in αtwhen\nchanging from the bcc Fe50Co50(c\na= 1) to the bct\nFe50Co50bulk structure (c\na= 1.15) is qualitatively con-\nsistent to what was obtained by Mandal et al.[33] (from\nαt= 6.6×10−3in the bcc to αt= 17.8×10−3in the\nbct, withc\na= 1.33[33]).\nTables S3 and S4 refer to the damping anisotropies\n(∆αt) for all Fe-centered and Co-centered clusters stud-\nied here, with different approaches: ( i) bcc clusters em-\nbedded in the VCA medium (Table S3) and ( ii) bct-like\nclusters embedded in the VCA medium (Table S4).\nIn comparison with bct-like clusters, we found larger\nabsoluteαtvalues but lower damping anisotropies. In\nall cases, Fe-centered clusters present higher ∆αtper-\ncentages.\nIn Table S5 the onsite damping anisotropies ( ∆αonsite)\nfor each layer of the Fe50Co50(001) bcc surface (\"1\" repre-\nsents the layer closest to vacuum) are shown. In compar-\nison with the total damping anisotropies (Table I of the\nmain text), much lower percentages are found, demon-\nstrating that the damping anisotropy effect comes ma-\njorly from the non-local damping contributions.\nThe most important results concerning the largest\ndampinganisotropiesaresummarizedinTableS6, below.4\nTable S1. Total damping values ( ×10−3) calculated for some bulk and surface systems, and the comparison with previous\nliterature results. The onsite contributions are indicated between parentheses, while the total damping, αt, are indicated\nwithout any symbols. All values were obtained considering the [001]magnetization axis. The VCA was adopted for alloys,\nexcept for the Fe 50Co50bcc in theB2structure (see Eq. S7). Also shown the broadening Λvalue considered in the calculations.\nBulks a(Å) This work Theoretical Experimental Λ(eV)\nFe bcc 2.87 4.2(1.6) 1 .3[2]a/(3.6)[4] 1.9[30]/2.2[31]\nFe70Co30bcc 2.87 2.5(0.7) − 3−5[32]d\nFe50Co50bcc 2.87 3.7(1.0)[VCA]/ 2.3(1.0)[B2]1.0[25]c[VCA]/ 6.6[33] [B2] 2.3[27]\nNi fcc 3.52 27.8(57.7) 23 .7[34]/( 21.6[4])b26.0[31]/24.0[35]\nNi80Fe20(Py) fcc 3.52 9.8(12.1) 3 .9[25]c8.0[30]/5.0[35]\nCo fcc 3.61 [3] 3.2(5.3) 5 .7[28]/(3.9[4])b11.0[30]∼5×10−2\nCo hcp 2.48/4.04 [28] 2.1(6.2) 3 .0[28] 3.7[31]\nCo85Mn15bcc [36] 2.87 [28] 6.2(4.2) 6 .6[28] −\nCo90Fe10fcc 3.56 [37] 3.6(4.2) − 3.0[35]/4.8[37]\nSurfaces a(Å) This work Theoretical Experimental\nFe(001) bcc [110] 2.87 5.8(5.4)e− 7.2[38]h/6.5[39]i\nFe(001) bcc [100] 2.87 3.9(4.4)f∼4[29]g4.2[40]j\nNi(001) fcc 3.52 80.0(129.6)∼10[29]g/12.7[41]m22.1[42]l\nPdFe/Ir(111) [43] fcc 3.84 3.9(2.7)n− −\nPdCo/Ir(111) [44] fcc 3.84 3.2(14.7)o− −\naWith Λ∼2×10−2eV.\nbWith Λ = 5×10−3eV.\ncWith Λ∼1.4×10−4eV.\ndFor a 28%Co concentration, but the results do not significantly change for a 30%Co concentration. Range including results before and\nafter annealing.\neOf a typical atom in the more external surface layer (in contact with vacuum), in the [110] magnetization direction.\nfOf a typical atom in the more external surface layer (in contact with vacuum), in the [100] magnetization direction.\ngFor a (001) bcc surface with thickness of N= 8ML (the same number of slabs as in our calculations), and Λ = 10−2eV.\nhAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001) thin film (sample S2 in Ref. [38]) in the [110] magnetization direction.\niAnisotropic damping obtained for a 1.14 nm Fe/InAs(001) thin film in the in-plane [110] hard magnetization axis.\njFor a 25-nm-thick Fe films grown on MgO(001).\nkFor epitaxial Fe(001) films grown on GaAs(001) and covered by Au, Pd, and Cr capping layers.\nlIntrinsic Gilbert damping for a free 4×[Co(0.2 nm)/Ni(0.6 nm)](111) multilayer. Not the same system as Ni(001), but the nearest system\nfound in literature.\nmFor a Co | Ni multilayer with Ni thickness of 4 ML (fcc stacking).\nnOf a typical atom in the Fe layer.\noOf a typical atom in the Co layer.\nThe alloys with short-range orders (SRO) are described\nas FeCo clusters (with explicit Fe and Co atoms) embed-\nded in the Fe50Co50VCA medium – with and without\nthe bct-like distortion. In this case, the damping is cal-\nculated as a weighted average (Eq. S7). As discussed in\nthe main text, it can be seen from Table S6 that distor-\ntions and disorder can increase the anisotropy but the\nmajor effect comes from the surface. We notice that the\nnumber of clusters considered is limited in the statistical\naverage.\nIV. Kambersky’s simplified formula\nInordertoconnecttheanisotropyoftheGilbertdamp-\ning to features in the electronic structure, we consider in\nthe following Kambersky’s simplified formula for Gilbert\ndamping [47, 48]α=1\nγMs/parenleftBigγ\n2/parenrightBig2\nn(EF)ξ2(g−2)2\nτ.(S8)\nHere,γis the gyromagnetic ratio, n(EF)represents the\nLDOS at the Fermi level, ξis the SOC strength, τis the\nelectron scattering time, Msis the spin magnetic mo-\nment, andgis the spectroscopic g-factor [35, 49]. Note\nthat Eq. S8 demonstrates the direct relation between\nαandn(EF), often discussed in the literature, e.g., in\nRef. [27]. Our first principles calculations have shown\nno significant change in ξ, upon variation of the mag-\nnetization axis, for the FeCo systems ( ξCo= 71.02meV\nandξFe= 53.47meV). Hence, we can soundly relate the\ndamping anisotropy ∆αtto∆n(EF).\nFigure S2 shows how the LDOS difference (per atom)\n∆n(E)between the [010]and [110]magnetization di-\nrections is developed in pure Fe(001) bcc and in VCA\nFe50Co50(001) bcc surfaces, respectively. In both cases,5\nTableS2. Totaldamping( αt×10−3)ofatypicalatomin\neach system for the spin quantization axes [010](θH=\n90◦) and [110](θH= 45◦); also shown for the [001]and\n[111]. Bulk and surface bct systems are simulated with\nc\na= 1.15.\nBulks\nBulk αt[010]αt[110] ∆αt\nFe bcc 4.18 4.31 +3.1%a\nB2-FeCo bcc 2.28 2.44 +7.2%\nB2-FeCo bct 7.76 8.85 +12.4%\nVCA Fe 50Co50bcc 3.70 4.18 +13.0%\nVCA Fe 50Co50bct 4.69 5.10 +8.7%\nαt[010]αt[001] ∆αt\nB2-FeCo bct 7.76 10.21 +24.1%\nVCA Fe 50Co50bct 4.69 5.75 +22.6%\nαt[010]αt[111] ∆αt\nFe bcc 4.18 4.56 +9.1%b\nSurfaces\nSurface αt[010]αt[110] ∆αt\nFe(001) bcc 3.85 5.75 +49.4%\nFe/GaAs(001) bcc [38] 4.7(7) 7.2(7) +53(27)%c\nFe/MgO(001) bcc [45] 3.20(25) 6.15(20) +92(14)%d\nVCA Fe 50Co50(001) bcc 7.00 14.17 +102.4%\nVCA Fe 50Co50(001) bct 15.20 14.80−2.6%\nαt[010]αt[001] ∆αt\nVCA Fe 50Co50(001) bct 15.20 15.56 +2.4%\nVCA Fe 50Co50(001) bcc 7.00 9.85 +40.7%\naMankovsky et al.[24] find a damping anisotropy of ∼12%\nfor bulk Fe bcc at low temperatures ( ∼50K) between\n[010] and [011] magnetization directions. For this result,\nthe definition α=1\n2(αxx+αyy)was used.\nbThis result agrees with Gilmore et al.[46], which find\nthat the total damping of pure Fe bcc presents its higher\nvalue in the [111] crystallographic orientation and the\nlower value in the [001] direction, except for high scatter-\ning rates. Also agrees with Mankovsky et al.[24] results.\ncAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001)\nthin film (sample S2 in Ref. [38]) in the [010] and [110]\nmagnetization directions.\ndFor a Fe(15 nm)/MgO(001) film at T= 4.5K in the high-\nest applied magnetic field, in which only intrinsic contri-\nbutions to the anisotropic damping are left.\nthe chosen layer,denoted as first, is the most external\none (near vacuum). the VCA Fe50Co50(001) bcc we also\ncalculated ∆n(E)for all layers summed (total DOS dif-\nference).\nAs can be seen, although in all cases the quantity\n∆n(E)exhibits some oscillations, differently from what\nwe observe forthe pureFe(001) surface case, at the Fermi\nenergy, there is a non-negligible difference in the minor-\nity spin channel ( 3dstates) for the VCA Fe50Co50(001).\nConsidering the results presented in Table I (main text)\nthe larger contribution to the damping anisotropy comes\nfrom the most external layer. The results by Li et al.\n[22] indicate a small difference (for two magnetization\ndirections) of the total density of states at the FermiTable S3. Total damping anisotropy ( ×10−3) of all stud-\nied Co-centered and Fe-centered bcc clusters for the spin-\nquantization axis [010]and [110], considering the 15-atom\nFeCo cluster together with the VCA medium in the summa-\ntion for total damping.\nCo-centered\nCluster # αt[010]αt[110] ∆αt\n1 10.11 9.65 4.8%\n2 8.09 6.96 16.2%\n3 7.81 7.02 11.3%\n4 7.11 7.02 1.3%\n5 7.48 6.88 8.7%\nAverage 8.12 7.51 8.1%\nFe-centered\nCluster # αt[010]αt[110] ∆αt\n1 2.68 2.03 32.0%\n2 2.49 2.05 21.5%\n3 2.56 1.86 37.6%\n4 2.45 1.79 36.9%\n5 2.76 2.01 37.3%\nAverage 2.59 1.95 32.8%\nTable S4. Total damping anisotropy ( ×10−3) of all stud-\nied Co-centered and Fe-centered bcc clusters for the spin\nquantization axis [010]and [110], with bct-like distortions/parenleftbigc\na= 1.15/parenrightbig\n, considering the 15-atom FeCo cluster together\nwith the VCA medium in the summation for total damping.\nCo-centered\nCluster # αt[010]αt[110] ∆αt\n1 5.85 4.37 33.9%\n2 5.95 4.21 41.3%\n3 5.88 4.35 35.2%\n4 5.90 4.41 33.8%\n5 5.86 4.34 35.0%\nAverage 5.89 4.34 35.7%\nFe-centered\nCluster # αt[010]αt[110] ∆αt\n1 2.36 1.39 69.8%\n2 2.27 1.32 72.0%\n3 2.22 1.26 76.2%\n4 2.25 1.26 78.6%\n5 2.42 1.38 75.4%\nAverage 2.30 1.32 74.2%\nTable S5. Onsite damping ( αonsite×10−3) of a typical atom\nin each layer of the VCA Fe 50Co50(001) bcc for the spin quan-\ntization axis [010]and[110].\nLayerαonsite[010]αonsite[110] ∆αonsite\n1 7.36 8.70 +18.2%\n2 0.63 0.69 +9.5%\n3 1.41 1.44 +2.1%\n4 0.87 0.86−1.1%\n5 0.99 0.97−2.0%6\nTableS6. SummaryofthemainFe 50Co50dampinganisotropy\nresults for: pure ordered ( B2) alloy; pure random (VCA) bulk\nalloy; bcc bulk together with short-range order (SRO) clus-\nters (see Table S3); bulk together with bct-like distorted clus-\nters inside (see Table S4); surface calculations, in the pristine\nmode and with explicit bct-like clusters embedded (surface +\ndistortion). The maximum-minimum ratio according to Ref.\n[22] isα[110]\nt\nα[010]\nt×100%.\nStructure ∆αtMax-min ratio\nOrdered alloy bcc 7.2% 107.2%\nOrdered alloy bct 24.1% 124.1%\nRandom alloy bcc 13% 113%\nRandom alloy bct 22.6% 122.6%\nRandom alloy + SRO 14.9% 114.9%\nRandom alloy + SRO + Distortion 47.2% 147.2%\nSurface (external layer) 102.4% 202.4%\nSurface (ext. layer) + Distortion 75.4% 175.4%\n10-nm Co 50Fe50/Pt [22] (exp.) 281.3% 381.3%\n−2−1 0 1 2\n−0.02 −0.01  0  0.01  0.02Δn(E)[010]−[110]  (states/Ry−atom)\nEnergy (E−EF) (Ry)Fe(001) bcc (first)\nVCA Fe50Co50(001) bcc (first)\nVCA Fe50Co50(001) bcc (all)\nFigure S2. LDOS difference (per atom), ∆n(E), between the\n[010]and[110]magnetization directions, for both spin chan-\nnels (full lines for majority spin and dashed lines for minority\nspin states), in the outermost layer in pure Fe(001) bcc (in\nblack); outermost layer in VCA Fe 50Co50(001) bcc (in blue);\nand all layers summed in VCA Fe 50Co50(001) bcc (in red).\nlevel,N(EF), what the authors claim that could not ex-\nplain the giant maximum-minimum damping ratio ob-\nserved. So, in order to clarify this effect in the VCA\nFe50Co50(001) bcc, ∆n(E)was also calculated for the all\nlayers summed, what is shown in Fig. S2 (in red). This\ndifference is in fact smaller if we consider the DOS of\nthe whole system, with all layers summed. However, if\nwe consider only the most external layer, then the LDOS\nvariation is enhanced. This is consistent with our theo-\nretical conclusions. As we mention in the main text, this\ndo not rule out a role also played by local (tetragonal-\nlike) distortions and other bulk-like factors in the damp-\ning anisotropy.For the outermost layer of Fe(001) bcc, the calculated\nLDOSatEFis∼20.42states/Ry-atominthe[110]direc-\ntion and∼20.48states/Ry-atom in the [010] direction,\nwhich represents a difference of ∼0.3%and agrees with\nthe calculations performed by Chen et al.[38].\nV. Correlation with anisotropic orbital moment\nBesides the close relation exhibited between ∆αt\nand∆n(EF), we also demonstrate the existence of an\nanisotropic orbital moment in the outermost layer, in\nwhich the fourfold symmetry ( C4v) matches the damp-\ning anisotropy with a 90◦phase. Fig. S3 shows this\ncorrelation between ∆αtand∆morbfor two situations:\n(i) for a typical atom in the outermost layer of VCA\nFe50Co50(001) bcc (blue open dots); and ( ii) for a typi-\ncal atom in the VCA Fe50Co50bcc bulk, considering the\nsame ∆morbscale. For case ( i) we find orbital moments\ndifferencesmorethanoneorderofmagnitudehigherthan\ncase (ii).\n0o45o90o\n135o\n180o\n225o\n270o315o[100][110][010]\n[−110]\n0.51.52.5\nΔmorb\n(µB/atom × 10−3)\n0.0050.0100.015\nαt\nθH\nFigure S3. (Color online) Total damping and orbital moment\ndifference, ∆morbas a function of θH, the angle between the\nmagnetizationdirectionandthe [100]-axis. Squares: (redfull)\nVCA Fe 50Co50(001) bcc. Circles: (blue open) morbdifference\nbetweenθH= 90◦and the current angle for a typical atom in\nthe outermost layer of VCA Fe 50Co50(001) bcc; and (yellow\nfull) samemorbdifference but for a typical atom in the VCA\nFe50Co50bcc bulk (in the same scale). Lines are guides for\nthe eyes.\nVI. Contribution from next-nearest-neighbours\nFinally, we show in Fig. S4 the summation of all non-\nlocal damping contributions, αij, for a given normalized\ndistance in the outermost layer of VCA Fe50Co50(001)7\nbcc. As we can see, the next-nearest-neighbours from a\nreference site (normalized distanced\na= 1) have very dis-\ntinctαijcontributions to αtfor the two different mag-\nnetization directions ( [010]and[110]), playing an impor-\ntant role on the final damping anisotropy. We must note,\nhowever, that these neighbours in a (001)-oriented bcc\nsurface are localized in the same layer as the reference\nsite, most affected by the interfacial SOC. Same trend is\nobserved ford\na= 2, however less intense. This is con-\nsistent with our conclusions, about the relevance of the\noutermost layer on ∆αt.\n−3−2−1 0 1 2\n 0.5  1  1.5  2  2.5  3  3.5  4∑αij × 10−3\nNormalized distance[110]\n[010]\nFigure S4. 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Rev. 76, 743 (1949)."    },    {        "title": "2101.07164v3.Topological_electric_driving_of_magnetization_dynamics_in_insulators.pdf",        "content": "Topological electric driving of magnetization dynamics in insulators\nCong Xiao,\u0003Bangguo Xiong,\u0003and Qian Niu\nDepartment of Physics, The University of Texas at Austin, Austin, Texas 78712, USA\nEstablished forms of electromagnetic coupling are usually conservative (in insulators) or dissipa-\ntive (in metals and semiconductors). Here we point out the possibility of nondissipative electric\ndriving of magnetization dynamics, if the valence electronic states have nontrivial topology in the\ncombined space of crystal momentum and magnetization con\fguration. We provide a hybrid in-\nsulator system to demonstrate that the topology-based nonconservative electrical generalized force\nis capable of supporting sustained magnetization motion in the presence of Gilbert damping, with\nquantized and steady energy pumping into magnetization motion from the electric \feld. We also\ngeneralize our results to magnetic textures, and discuss electric \feld induced Dzyaloshinskii-Moriya\ninteraction which can be nonconservative.\nI. INTRODUCTION\nThe study of electrical control of magnetization dy-\nnamics has occupied a large part of solid state research\nfor many decades, which generally falls into two separate\ncategories known as multiferroics [1{3] and spintronics [4]\ndepending on the conductive behavior of the hosting ma-\nterials. The former deals with insulators where electrical\ne\u000bects on magnetization is characterized through the free\nenergy [5], and the resulting torque would be naturally\nconsidered as conservative and unable to drive sustained\nmotion of the magnetization for a static electric \feld. In\nthe latter, one \fnds various current induced magnetic\ntorques in metals and semiconductors [6{8], which can\nprovide a persistent source of energy for sustained mo-\ntion of magnetization, but one has to deal with wasteful\nand prohibitive joule heating in practice.\nMagnetic insulators have recently been utilized to\nachieve low-dissipation magnetization control by combin-\ning the insulator with heavy metals hosting prominent\nspin Hall e\u000bect that injects a spin current into the insu-\nlator [9, 10]. An electric \feld can also directly manipulate\nmagnetization in an insulator without Joule heating by\nmeans of spin-orbit torques mediated by occupied elec-\ntronic states [11{13]. In particular, mesoscopic transport\ntheories proposed the exchange gapped edge states of a\ntwo dimensional topological insulator in hybrid with a\nmagnet as a unique platform for studying the magnetic\nThouless motor [14, 15], which works as the inverse mode\nof the adiabatic charge pumping by a cyclic magnetic\nmotion [16{18] under an applied voltage. By using the\nscattering matrix approach [19], previous works showed\nquantized electrical energy transfer into the magnet if the\nmagnetization accomplishes a cyclic motion [14, 15]. On\nthe other hand, as the Berry curvature in the mixed space\nof crystal momentum and magnetization con\fguration\nunderlies the magnetic Thouless pumping, it would be in-\nteresting to reveal the relation between nonzero electrical\nenergy input into magnetic dynamics and the topological\ncharacteristics in the mixed parameter space [20]. More-\n\u0003These authors contributed equally to this work.over, it has not been shown whether the electric driving\nof sustained magnetic motion, which enables a motor,\ncan be realized in the presence of magnetic damping due\nto coupling of magnetization to other degrees of freedom\nthan electrons.\nIn this study we show the possibility of nondissipative\ndriving of magnetization dynamics with steady energy\npumping by a static electric \feld in insulators. This is\nmotivated by the fact that electric polarization is not al-\nways a single-valued quantity [21, 22], and the adiabatic\ncurrent pumped by cyclic motion of the magnetization\ncan acquire a quantized net amount of energy from a\nstatic electric \feld under certain topological conditions\nof the valence electronic states. We exploit this idea\nin a model system of edge states of a two-dimensional\ntopological insulator gapped by hybrid with a magnetic\nwire, and show explicitly sustained magnetization mo-\ntion when a constant electric \feld is applied to overcome\nGilbert damping.\nOur results can also be generalized to the case of slowly\nvarying magnetic textures. There is a topological current\nbilinear in the gradient and time derivative of the mag-\nnetization density [23], a sort of anomalous Hall current\ninduced by the arti\fcial electric \feld from the time de-\npendent magnetic texture [24{27]. This current provides\na channel of nondissipative drive of the magnetic tex-\nture by an external static electric \feld. In topologically\nnontrivial cases this drive is nonconservative and capa-\nble of delivering a nonzero and quantized amount of en-\nergy when the magnetic texture wraps around the Bloch\nsphere in time. In topologically trivial cases, where the\nelectric polarization induced by magnetization gradient\nis well de\fned, the drive is conservative because it can\nbe identi\fed as originating from a polarization energy\ndensity whose susceptibility to the magnetization gradi-\nent gives the electric \feld induced Dzyaloshinskii-Moriya\ninteraction (DMI) [28{31].\nThe rest of the paper is organized as follows. In Sec. II\nwe focus on the electric-\feld induced generalized force on\na homogeneous magnetization in insulators, and study its\nnonconservative nature which is related to certain topo-\nlogical conditions of the occupied electronic states. These\ngeneral rationales are illustrated in a hybrid insulator in\nSec. III, in which the electric driving of sustained mag-arXiv:2101.07164v3  [cond-mat.mes-hall]  23 Jun 20212\nnetic motion is also demonstrated. Section IV is devoted\nto the electrical generalized force on the magnetization in\ninhomogeneous insulators and its relation to an electric-\n\feld induced DMI. Finally, we concludes the paper in\nSec. V.\nII. ELECTRICAL GENERALIZED FORCE ON\nMAGNETIZATION\nIn the language of analytic mechanics, an generalized\nforce is an amount of work done on the system per\nunit displacement in the dynamical variable (the mag-\nnetization here). Considering a system with a homoge-\nneous magnetization mcoupled to an electronic insula-\ntor, change in the magnetization can in general pump an\nadiabatic electric current\nj=eZ\n[dk] \nkm\u0001_m; (1)\nwhere \n kmis the electronic Bloch-state Berry curvature\nin the parameter space of the magnetization and crystal\nmomentumk(set~= 1 unless otherwise noted), with its\nCartesian components given by \u00002Imh@kiuj@mjui. Here\njuiis the periodic part of the Bloch wave, and the band\nindexnis omitted for simple notation. [ dk]\u0011ddk=(2\u0019)d\nwithdas the spatial dimension, and the summation over\nvalence bands is implied. Through this adiabatic cur-\nrent, an external electric \feld can deliver work on the\nsystem, with the work density \u000ew=E\u0001jdt, which is\nproportional to \u000em=_m\u0001dt. Therefore, we obtain the\nelectrical generalized force density on magnetization as\nEe\nm\u0011\u000ew\n\u000em=eE\u0001Z\n[dk] \nkm: (2)\nThis electrical generalized force is nondissipative be-\ncause of the lack of conduction electrons for joule heat-\ning, and is in fact topological in the sense that it delivers\na quantized amount of energy over a cycle of the mag-\nnetization motion. For simplicity, we \frst consider an\ninsulator with zero Chern numbers in the Brillouin zone\nat each point over the path of m, such that one can take\nak-space periodic gauge to locally de\fne an electronic\npolarizationP=\u0000eR\n[dk]Ak[21], with Ak=huji@kui.\nThen the electrical work density delivered over the cycle\ncan be written in terms of the change of this polarization\nw=I\ndm\u0001Ee\nm=E\u0001\u0001P: (3)\nThis change is quantized in units of \u0001 P=\u0000ea=V0with\nabeing a discrete lattice vector (including the null vec-\ntor) andV0the volume of a unit cell. When this change is\nzero, so that the polarization is globally de\fned, the elec-\ntrical generalized force is conservative in the sense that\nits work can be regarded as a change in the globally well\nde\fned polarization energy density \u0000E\u0001P. When this\nFIG. 1. A ferromagnetic wire (blue bar) hybridizes and gaps\nthe edge states of a two-dimensional topological insulator\n(green region). When the magnetization mmoves around\n(blue circle on the right) on the Bloch sphere, the pumped\nadiabatic current jalong the edge couples to an applied elec-\ntric \feldEto provide energy to overcome Gilbert damping.\nA static magnetic \feld His applied to help preparing the\nsystem into a sustained motion of limit cycle.\nchange is nonzero, the electrical generalized force is non-\nconservative and capable of supporting sustained magne-\ntization motion even in the presence of Gilbert damping\ndue to other dissipative channels.\nSome comments are in order. First, if the electronic in-\nsulator is one dimensional, then the electrical work (per\nunit length) over a cycle of the magnetization reduces to\neEtimes the Chern number over the torus of the com-\nbined space of crystal momentum and the magnetization\n(along its path), corresponding to the quantized number\nof electrons pumped over the cycle. Second, quantization\nof electrical work over the cycle of magnetization also ap-\nplies to insulators with nonzero k-space Chern numbers\nby a simple argument. Although one cannot take a peri-\nodic gauge in k-space, one can always choose a periodic\ngauge over a \fxed one dimensional path of the magneti-\nzation. It is then clear that the electrical work over the\ncycle equals the Brillouin-zone integral of the k-gradient\nof the Berry's phase over the cycle. Topological quantiza-\ntion of this work then follows from the multi-valuedness\nof the Berry's phase. Third, using the Bianchi identity\non Berry curvatures, one can easily show that the electri-\ncal generalized force is curl-free @m\u0002Ee\nm= 0 everywhere\ninm-space, except the singular points where the energy\ngap above the \flled states of the electron system closes.\nWhen can the electrical work on magnetization be\nnonzero? Quantization of its value implies that the elec-\ntrical work is invariant if the path in m-space is deformed\nwithout closing the energy gap. In particular, within a\nsingly connected region where the gap is open, the elec-\ntrical work is zero on all closed paths. This applies for\nexample to the north or south hemispheres of magneti-\nzation in the two dimensional ferromagnetic Dirac model\nstudied in [32], where one can de\fne polarization ener-\ngies separately for each region, although cannot globally\nbecause of gap closing on the equator. Consequently, this\nmodel system cannot provide nonzero electrical work for\nsustained magnetization motion. It is therefore clear that\nthe singular points of gap closing have to be arranged to\nde\fne multiply connected regular regions, where electri-3\ncal work can possibly be nonzero on topologically non-\ntrivial paths.\nIII. A MODEL FOR SUSTAINED\nMAGNETIZATION MOTION\nHere we propose a one dimensional model system,\nwhere the gap closes on the two poles of the magne-\ntization Bloch sphere, and the electrical work per unit\nlength iseEtimes the winding number of the path\naround the poles. The system is constructed by inter-\nfacing a magnetic wire with the topological edge states\nof a two-dimensional topological insulator (Fig. 1). The\nexchange coupling renders the electronic system insulat-\ning by opening a gap in the Dirac spectrum. The relevant\nlow-energy Hamiltonian is\n^h=~vk^\u001by+J^\u001b\u0001m; (4)\nwherevis the Fermi velocity, ^\u001bis the Pauli matrix, and\nJis the coupling constant. The magnetization mis as-\nsumed to have a \fxed magnitude and is parameterized by\nthe polar angle \u0012relative to the yaxis and the azimuthal\nangle\u001eas shown in Fig. 1. The energy gap is open\neverywhere except at the north and south poles of the\nBloch sphere with my=\u0006m(red dots). Assuming that\nthe lower band is \flled and the electric \feld is applied\nalong the magnetic wire (positive xdirection), we can\nevaluate the formula for the electrical generalized force\nto \fnd\nEe=\u0000eE\n2\u0019m^e\u001e\nsin\u0012=\u0000eE@m\u001e\n2\u0019: (5)\nIt is singular at the poles and is a gradient of the multiple-\nvalued azimuthal angle, so that the electrical work den-\nsity over a closed path on the Bloch sphere is quantized\nin terms of the winding number of the path\nI\ndm\u0001Ee=\u0000NteE; (6)\nin line with the aforementioned general topological ar-\nguments. The winding number Ntcounts how many\ntimes the closed path wraps around the yaxis counter-\nclockwise.\nIn such a one dimensional insulator it is also interest-\ning to understand the electrical generalized force from\nthe polarization as Ee=E@mP, where the polarization\nis not single-valued and can only be determined to be\nP=\u0000e\u001e=2\u0019up to an uncertainty quantum \u0000e. Con-\nsistently, the two gap closing poles are singular points of\nthe polarization, and the change of polarization upon a\nclosed path on the Bloch sphere is \u0000eNt.\nWe now proceed to study the dynamics of the mag-\nnetization to see the e\u000bect of this generalized force. In\nthe absence of coupling to the electronic system, we can\nrewrite the Landau-Lifshitz-Gilbert equation of the fer-\nromagnet in the form of \u0000@mG0+_m\u0002\n0\nm\u0000\u00110_m= 0, as\nFIG. 2. Free energy contours in the angular space and typical\nevolution trajectories on the Bloch sphere in the absence (top\npanels) and presence (middle panels) of an electric \feld, and\nin the presence of both electric and magnetic \felds (bottom\npanels). In the last case, a limit cycle emerges.\nbalancing out a conserved force from the free energy G0,\na Lorentz type force from the m-space Berry curvature\n\n0\nm[33], and a frictional force with a scalar damping\ncoe\u000ecient\u00110. We will model the free energy density as\nG0=\u0000K0^m2\nx\u0000Hmywith an easy axis anisotropy and\nan applied static magnetic \feld H. Them-space Berry\ncurvature is given in terms of the gyromagnetic ratio \r0\nas\n0\nm=m=(m2~\r0). The damping coe\u000ecient is related\nto the Gilbert number \u0015as\u0015= (\r0)2\u00110.\nIn the presence of coupling to the electronic system,\nthe equation of motion becomes\nEe\nm\u0000@mG+_m\u0002\nm\u0000\u0011_m= 0; (7)\nwhere the electrical generalized force enters as an ex-\ntra term along with electronic modi\fcations to the other\nterms. The gap opening in the electronic system con-\ntributes a lowering of the free energy Ge=Ke( ^m2\ny\u00001)\nthat we model as a hard-axis anisotropy. The electronic\ncontribution to the m-space Berry curvature is given by\n\ne\nm=R\n[dk] \nm=m=(m2~\re), where \n m=@m\u0002Am\nis derived from Am=huji@mui, and\re= 2\u0019v=J . Fi-4\nnally, we assume that the gap of the electronic system\nremains open during the course of dynamics, so there\nis no electronic contribution to the damping coe\u000ecient\n\u0011=\u00110.\nRepresentative results of the magnetization motion\nare presented in Fig. 2, where we take \re=\r0=\u0019,\nKe=(m=\r0) =K0=(m=\r0) = 1 GHz and \u0011= 0:2=(m\r0).\nShown in the top and middle panels ( H= 0), there are\ntwo types of energy conserved motion in the absence of\ndamping and external \felds, divided by the contour of\nzero energy (the white dashed curve). In the area enclos-\ning the two points of lowest energy, mrotates around the\nxaxis, whereas in the upper and lower areas outside of\nthe zero-energy contour mgoes around the yaxis. This\nsituation is changed in the presence of damping, as shown\nin the top panel, where two points ( \u001e= 0;\u0012= 0:3\u0019)\nand (\u001e= 0;\u0012= 0:7\u0019) outside of the zero-energy contour\nevolve to di\u000berent points of lowest energy. In the middle\npanels, an electric \feld eE=2\u0019= 0:1K0is applied, which\ngives a force in the clockwise (negative \u001e) direction. The\nblue trajectory starting from ( \u001e= 0;\u0012= 0:7\u0019) falls faster\nto the +mxaxis, while the red trajectory starting from\n(\u001e= 0;\u0012= 0:3\u0019) extends for 3 =4 circle before the \f-\nnal decay into the same energy minimum as the other\ntrajectory.\nThe lower panels show the situation where limit cycle\nmotion is found. We found it important to prepare the\nsystem with predominantly around- my-axis energy con-\ntours, so that the non-conservative electrical force can\nbe best utilized. We therefore apply a static magnetic\n\feld inydirection with the magnitude H=K0=mto\nchange the energy landscape. We also switched the di-\nrection of the electric \feld so that the electrical force\ngoes along the directions of the energy contours. We\nfound that all initial points in a wide region, between the\ntwo dashed circles shown on the right of the lower panels\nof Fig. 2, fall into the same limit cycle. For instance,\nthe blue curve starts from ( \u001e= 0;\u0012= 0:4\u0019) and evolves\ninto the right handed limit cycle under an electric \feld\neE=2\u0019=\u00000:1K0. Figure 3 shows how the limit cycle\nmotion is reached in time for two trajectories (blue and\nred) from di\u000berent initial angles, along with one (black)\nthat falls into an energy minimum. On the limit cycle, we\nfound that the energy input from the electrical force bal-\nances out the energy dissipation from the Gilbert damp-\ning,H\ndm\u0001(Ee\nm\u0000\u0011_m) = 0, as can be easily derived from\nthe equation of motion.\nIV. ELECTRICAL DMI FORCE\nSo far we have been concentrating on nondissipative\nelectrical driving on a uniform magnetization. When the\nmagnetization is nonuniform, the electrical generalized\nforce Eq. (2) still applies as a local force density, but\nthere will be additional contributions due to the magne-\ntization gradients. In metals, the electric-current induced\nDMI have been discussed recently [34{36], which is simi-\nFIG. 3. Time dependence of the polar angle for di\u000berent\ninitial conditions, \u001e= 0,\u0012= 0:001\u0019(black),\u0012= 0:4\u0019(red),\n\u0012= 0:8\u0019(blue). Correspondingly on the Bloch sphere shown\nin the inset, the red and blue trajectories evolve into a right-\nhanded limit cycle, whereas the black trajectory evolves into\nthe point of lowest energy.\nlar to the current induced orbital magnetization [37, 38].\nThe intrinsic analog, the electric-\feld induced nondissi-\npative DMI [1, 39], remains elusive in the band picture,\nbut should be well de\fned in insulators as we show now.\nTo \frst order in the gradient, there is an adiabatic\ncurrent pumped by the magnetization dynamics [23] j=\neR\n[dk] \nk[kr]m\u0001_minvolving the second Chern form of\nBerry curvatures \n ks[kr]mj\u0011\nkski\nrimj+\nksri\nmjki+\n\nksmj\nkiri. Through this current, an external electric\n\feld can produce a work density \u000ew=E\u0001jdtpropor-\ntional of\u000em, implying an electrical generalized force lin-\near in the magnetization gradient\nEe\nm=eE\u0001Z\n[dk] \nk[kr]m: (8)\nFor reasons to be discussed later, we will call this an\nelectrical DMI force, although it is nonconservative in\ngeneral and capable of sustained driving of magnetization\ntextures.\nBecause the second Chern form is antisymmetric in the\ncrystal momentum, a nonzero result demands that the\nelectronic system is more than one dimensional. Consider\nfor simplicity a two-dimensional system with the mag-\nnetization gradient in the ydirection (one-dimensional\ndomain wall or a spiral) and an electric \feld applied in\nthe transverse xdirection. The electrical work per unit\ntransverse width over one pumping period may be writ-\nten as\nW=eExNytZ\nT2d2k\n2\u0019Z\nS2d\u0012d\u001e\n2\u0019\nkxky\u0012\u001e=eExNytC2\n(9)\nwhich is topological and quantized in terms of the sec-\nond Chern number C2in the space [40] spanned by the5\nBrillouin zone and the Bloch sphere, and the winding\nnumberNyt=1\n4\u0019R\ndydt^m\u0001(@y^m\u0002@t^m) for the map-\nping ^m(y;t) of theytspace-time onto the Bloch sphere\n[41] ( ^m=m=m). This winding number has previously\nappeared in discussion of quantized electromotive force\ninduced by a moving domain wall [42], the so called ferro-\nJosephson e\u000bect, and the second Chern number may be\nregarded as the quantum measure of the anomalous Hall\nresponse to this emf [43]. The quantized electrical work\nis therefore a result of this quantized Hall current in the\ndirection of the applied electric \feld.\nThe same second Chern number has also been intro-\nduced in study of electric charges carried by magnetic\ntextures such as a skyrmion [30], where it may be un-\nderstood as the quantum measure of charge response to\nthe quantized \rux of arti\fcial magnetic \feld [43]. This is\na sort of Streda dual e\u000bect of the quantum Hall current\nresponse to the arti\fcial electric \feld of the magnetic tex-\nture. This relationship becomes especially clear in the ab-\nsence of spin-orbit coupling, where \n kxky\u0012\u001e= \nkxky\n\u0012\u001e\nandC2reduces to the \frst Chern number in k-space [44]\nwhich characterizes the usual quantum anomalous Hall\ninsulators.\nIn non-Chern insulators where one may choose a pe-\nriodic gauge in k-space, the electrical generalized force\nmay be written as a \feld derivative of the polarization\nenergy, Ee\nm=\u0000\u000emU, with [30, 31]\nU=\u0000Z\ndrE\u0001P=Z\ndrDil@iml; (10)\nwherePis the electric polarization induced by magne-\ntization gradient, including a topological Chern-Simons\npart [23] for which\nDil=e\n2EjZ\n[dk] (Akj\nkiml+Aki\nmlkj+Aml\nkjki):\n(11)\nHowever, this expression for the DMI coe\u000ecient is only\nlocally de\fned because of the gauge dependence of the\nChern-Simons form [45].\nOn the other hand, the electrical DMI force Ee\nm[Eq.\n(8)] is not only gauge invariant and single valued but also\nwell de\fned for Chern insulators. In practice, such a force\nenters directly in determining the static and dynamic be-\nhavior of the magnetic textures. For example, we show\nin the following that the width of a chiral Neel wall may\nbe tuned by such a force, as would normally be antici-\npated from DMI e\u000bects [34, 35]. Speci\fcally, we consider\na chiral Neel wall with easy axis in the zdirection in a\nmodel of the insulating transition metal dichalcogenide\nmonolayer materials with magnetic proximity e\u000bect, and\nshow that its width would be enhanced (decreased) when\nan electric \feld is applied in the x(\u0000x) direction. We\nemploy the model Hamiltonian ^h=^h0+J^\u001b\u0001m, where\n^h0is a six-band tight-binding Hamiltonian suitable forthe low-energy physics in monolayers of AB 2(A = Mo,\nW; B = S, Se, Te), as was detailed in Ref. [46]. Consider\na right-handed up-down Neel-type wall with easy axis in\nFIG. 4. Spin generation due to the electrical generalized force\nin a model of chiral Neel wall of ferromagnetic transition metal\ndichalcogenide monolayer.\nthezdirection, the induced spin is plotted in Fig. 4 un-\nder an electric \feld in the positive xdirection. With the\nlowest two bands \flled, the \frst Chern form contribution\nvanishes. The dominant component, \u000esx, is antisymmet-\nric on the two sides of the domain wall center. Thus the\ntorque exerted on magnetization \u000e\u001c=\u000es\u0002mis in the\npositiveydirection on both sides, increasing the width\nof the domain wall. Apparently, when the electric \feld\nis reversed, the width of the domain wall is decreased.\nV. CONCLUSION\nIn conclusion, we have studied nondissipative electric\ndriving of magnetization motion in uniform and nonuni-\nform magnetic insulators due to nontrivial topologies of\noccupied Bloch states in the combined space of crystal\nmomentum and magnetization con\fguration. The resul-\ntant nonconservative electrical generalized force is capa-\nble of supporting sustained magnetization motion even in\nthe presence of Gilbert damping. A minimal model has\nbeen exploited to show explicitly the limit-cycle behav-\nior of magnetic evolution. For magnetic textures, there is\nan additional nonconservative and nondissipative electri-\ncal generalized force, related to a Chern-Simons DMI for\nnon-Chern insulators in the presence of an electric \feld.\nACKNOWLEDGMENTS\nWe thank Hua Chen, Peng Yan, Yunshan Cao, Liang\nDong and Tianlei Chai for useful discussions. This work\nis supported by NSF (EFMA-1641101) and Welch Foun-\ndation (F-1255).6\n[1] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev.\nLett.95, 057205 (2005).\n[2] M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006).\n[3] Y. Yamasaki, H. Sagayama, T. Goto, M. Matsuura, K.\nHirota, T. Arima, and Y. Tokura, Phys. Rev. Lett. 98,\n147204 (2007).\n[4] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. 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B 88, 085433 (2013)."    },    {        "title": "2102.00909v1.Global_existence_for_semilinear_wave_equations_with_scaling_invariant_damping_in_3_D.pdf",        "content": "arXiv:2102.00909v1  [math.AP]  1 Feb 2021GLOBAL EXISTENCE FOR SEMILINEAR WAVE EQUATIONS WITH\nSCALING INVARIANT DAMPING IN 3-D\nNing-An Lai*1and Yi Zhou2\nAbstract. Global existence for small data Cauchy problem of semilinea r wave equa-\ntions with scaling invariant damping in 3-D is established i n this work, assuming that\nthe data are radial and the constant in front of the damping be longs to r1.5,2q. The\nproof is based on a weighted L2´L2estimate for inhomogeneous wave equation,\nwhich is established by interpolating between energy estim ate and Morawetz type\nestimate.\nKeywords: semilinear wave equation, scaling invariant dam ping, global\nexistence, weighted L2´L2estimate\nMSC2020: 35L71, 35L05, 35B40\n1.Introduction\nThis paper is devoted to studying radial solution of the semilinear wav e equation\nwith scaling invariant damping in 3-D\n(1.1)#\nϕtt´ϕrr´2ϕr\nr`µϕt\nt`2“ |ϕ|p,\nt“0 :ϕ“εϕ0prq,ϕt“εϕ1prq,\nwhereεdenotes the smallness of the initial data. The nonnegative initial dat a come from\nthe energy space and have compact support\n(1.2) supp ϕ 0prq,ϕ1prq Ă trˇˇrď1u.\nThis kind of semilinear wave equation with time dependent variable coeffi cients in\nfront of the damping has been widely studied recently, and the gene ral model is\n(1.3) Φtt´∆Φ`µ\np1`tqβΦt“ |Φ|p.\nAccordingtotheasymptoticbehaviorofthesolutionofthecorres pondinglinearequation,\nwe have four different type dampings, thus\nDate: February 2, 2021.\n1*Corresponding Author: Institute of Nonlinear Analysis an d Department of Mathematics, Lishui\nUniversity, China. Email: ninganlai@lsu.edu.cn\n2School of Mathematical Sciences, Fudan University, Shangh ai, China. and Department of Mathe-\nmatical Sciences, Jinan University, Guangzhou 510632, Chi na.Email: yizhou@fudan.edu.cn\n12 GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPIN G\nβP p´8,´1qoverdampingsolution does not\ndecay to zero\nβP r´1,1q effectivesolution behaves like\nthat of heat equation\nβ“1scaling invariant\nweak dampingthe asymptotic behavior\ndepends on µ\nβP p1,8q scatteringsolution behaves like that\nof wave equation without damping\nWe refer to the works [8, 21, 22, 32, 33, 34] and the conclusive ta ble in [16]. Based on this\nclassification, people try to figure out the critical power( pcpnq,nis the dimension) for\n(1.3) for each case. Here “critical power” denotes the threshold value ofpwhich divides\nthe problem into blow-up and global existence parts. If βă ´1, there is global solution\nfor allpą1, see [11]. If βP r´1,1q, it has been showed that the critical power is exactly\nthe same as Fujita number, i.e., pcpnq “pFpnq “1`2\nn, see [4, 7, 10, 17, 18, 19, 26, 30].\nIfβą1, due to the blow-up results in [14, 28, 31] for 1 ăpăpSpnqandně1 and global\nexistence results in [20] for pąpSpnqandn“3,4, we may conjecture the critical power\nis Strauss exponent, which is the critical power for semilinear wave e quation without\ndamping and the positive root of the quadratic equation\n(1.4) γpp,nq:“2` pn`1qp´ pn´1qp2“0.\nThe case of β“1, which we mean the scaling invariant damping, has attracts more an d\nmore attention, due to the reason that the critical power also dep ends on the size of the\nconstantµin front of the damping. Roughly speaking, if µis relatively large, the critical\npower will be Fujita type while if µis relatively small the critical power will be Strauss\ntype. According to the results in [1, 2, 29] we know the critical powe r ispFpnqat least\nfor\nµě$\n&\n%5{3 forn“1,\n3 forn“2,\nn`2 forně3.\nIfµ“2, it is interesting to see that the equation can be changed into a one without\ndamping by a transformation, and due to [3, 5, 12, 13, 25, 27], we n ow know that the\ncritical power is\npcpnq “maxtpFpnq,pSpn`2qu.\nIfµ‰2, the Strauss type blow-up result was first established in [15], whic h was improved\nby[9]. Wereferthereaderto[16,24]formoredetailedintroduction fortherelatedresults.\nHowever, till the moment, there is no global result for µ‰2 and\n0ăµăn2`n`2\nn`2and p ąpSpn`µq,\nwhich means that in this case the critical power( p“pSpn`µq) is still unfixed. In this\npaper, we will show global existence for (1.1) in R3for\n1.5ďµă2and pSp3`µq ăpď2,\nand this resultwill confirmthe criticalpowerforsome rangeof µinR3. The proofis quite\nelementary, and the key step is to establish a weighted L2´L2estimate by interpolating\nbetween an energy estimate and a Morawetz type estimate.GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPING 3\nRemark 1.1. The similar idea has been used in [13]to show the global existence of\nnon-radial solution for (1.1)withµ“0,pąpSp3qandµ“2,pąpSp5qinR3.\nThe main result is as follows.\nTheorem 1.2. Let1.5ďµă2andpSpn`µq ăpď2. Andεrepresents the smallness\nof the initial data. Assuming the support of the initial data satisfy(1.2). Then there\nexists a positive constant ε0such that if 0ăεăε0, problem (1.1)has global solution.\n2.Weighted L2´L2estimate for inhomogeneous wave equation\nWe first take the transformation\nψpt,rq “ pt`2qµ\n2ϕpt,rq,\nthenψpt,rqsatisfies\n(2.1)$\n’’&\n’’%ψtt´ψrr´2ψr\nr`µp2´µqψ\n4pt`2q2“|ψ|p\npt`2qµpp´1q\n2,\nψp0,rq “2µ\n2εϕ0prq,ψtp0,rq “ε!µ\n22µ\n2´1ϕ0prq `2µ\n2ϕ1prq)\n.\nLet\nu“t`2`r\n2,u“t`2´r\n2\nand\nφpu,uq “ pu´uqψpu`u´2,u´uq,\nthenφsatisfies the following system\n$\n&\n%φu¯u`µp2´µqφ\n4pu`¯uq2“|φ|p\npu´¯uqp´1pu`¯uqµpp´1q\n2,\nt“0 :φ“rψ0,φt“rψ1.\nNext we are going to establish the weighted L2´L2estimate for the following\ninhomogeneous second order partial differential equation\n(2.2)#\nφu¯u`µp2´µqφ\n4pu`¯uq2“Gpu,¯uq,\nt“0 :φ“rψ0,φt“rψ1.\nFirst we show a standard energy estimate for (2.2).\nLemma 2.1 (energy estimate) .Letφpu,¯uqsolve(2.2), and¯Ube a positive constant,\nthen we have\n(2.3)sup\n1\n2ď¯uď¯U˜ż`8\nmax p¯u,2´¯uqφu2du¸1\n2\nÀε´\n}ψ0}2\nH1pR3q` }ψ1}2\nL2pR3q¯1\n2`ż¯U\n1\n2˜ż`8\nmax p¯u,2´¯uqG2du¸1\n2\nd¯u.4 GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPIN G\nProof.Multiplying the equation in (2.2) by φu, we get\nB¯uφu2\n2`µp2´µqBuφ2\n8pu`¯uq2“φuG,\nand then integrating it with respect to uover rmaxp¯u,2´¯uq,`8qone has\nB¯uż`8\nmax p¯u,2´¯uqφu2\n2du`µp2´µqż`8\nmax p¯u,2´¯uqφ2\n4pu`¯uq3\n“ ´ψ2pt,0q\n2`ε\n2rψ0prq `rBrψ0prq `rψ1prqs `ż`8\nmax p¯u,2´¯uqφuGdu.\nIntegrating the above equality with respect to ¯ uover r1\n2,¯Usyields\n(2.4)\nsup\n1\n2ď¯uď¯Uż`8\nmax p¯u,2´¯uqφu2du\nďCε´\n}ψ0}2\nH1pR3q` }ψ1}2\nL2pR3q¯\n`ż¯U\n1\n2˜ż`8\nmax p¯u,2´¯uqφu2du¸1\n2˜ż`8\nmax p¯u,2´¯uqG2du¸1\n2\nd¯u,\nwhere we used the fact 0 ăµp2´µq ă1 and the solution vanishes when ¯ u“1\n2. And\nhence the energy estimate (2.3) follows. /square\nOn the other hand, we may establish a Morawetz type estimate for ( 2.2).\nLemma 2.2 (Morawetz type estimate) .Letφpu,¯uqsolve(2.2), and¯Ube a positive\nconstant, then we have\n(2.5)sup\n1\n2ď¯uď¯U˜ż`8\nmax p¯u,2´¯uqu3pu´¯uqφu2du¸1\n2\nÀε´\n}ψ0}2\nH1pR3q` }ψ1}2\nL2pR3q¯1\n2`ż¯U\n1\n2˜ż`8\nmax p¯u,2´¯uqu3pu´¯uqG2du¸1\n2\nd¯u\nProof.Multiplying the equation in (2.2) with pu´¯uqφuyields\nB¯u„\npu´¯uqφu2\n2\n`φu2\n2`µp2´µqpu´¯uqBuφ2\n8pu`¯uq2“ pu´¯uqφuG,\nintegrating which with respect to uover rmaxp¯u,2´¯uq,`8qwe come to\n(2.6)B¯uż`8\nmax p¯u,2´¯uqu3pu´¯uqφu2\n2du`1\n2ż`8\nmax p¯u,2´¯uqu3φu2du\n´µp2´µqrż`8\nmax p¯u,2´¯uqpBuru3pu´¯uqspu`¯uq ´2u3pu´¯uqq\n8pu`¯uq3φ2dus\n“rpr`2q3\n16rψ0prq `rψ1prq `rBrψ0prqs2`µp2´µq\n256r3pr`2q3ψ2\n0prq\n`ż`8\nmax p¯u,2´¯uqu3pu´¯uqφuGdu.GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPING 5\nA direct calculation shows that\nBuru3pu´¯uqspu`¯uq ´2u3pu´¯uq\n“ p3u2pu´¯uq `u3qpu`¯uq ´2u3pu´¯uq\n“u2tp3r`uqp2u´rq ´2uru\n“u2t2u2´3r2`3uru\n“u2t2u2`3\n4u2´3pr2´ur`u2\n4qu\nď p2`3\n4qu4ď p2`3\n4qupt`2q3,\ntherefore, we obtain from (2.6)\n(2.7)1\n2B¯uż`8\nmax p¯u,2´¯uqu3pu´¯uqφ2\nudu`1\n2ż`8\nmax p¯u,2´¯uqu3φu2du\n´µp2´µq\n2rż`8\nmax p¯u,2´¯uq1\n4p2`3\n4quφ2dus\nÀrpr`2q3\n16rψ0prq `rψ1prq `rBrψ0prqs2`µp2´µq\n256r3pr`2q3ψ2\n0prq\n`ż`8\nmax p¯u,2´¯uqu3pu´¯uqφuGdu.\nNow we claim a Hardy type inequality\n(2.8)ż`8\nmax p¯u,2´¯uqφ2udu ď ´r2pr`2q2\n4ψ2\n0prq `ż`8\nmax p¯u,2´¯uqu3φu2du.\nHence we may get (2.5) by combining (2.7) and claim (2.8), and then inte grating with\nrespect to ¯uover r1\n2,¯Us, since we have\n1\n2´ ´µp2´µq\n21\n4p2`3\n4q ą0\nfor 0 ăµă2. We are left with the proof of claim (2.8). By integration by parts, o ne has\n(2.9)ż`8\nmax p¯u,2´¯uqφ2udu “ ´ˆr`2\n2˙2\nprψ0prqq2´ż`8\nmax p¯u,2´¯uqup2φφuu`φ2qdu,\nwhich implies\n(2.10)2ż`8\nmax p¯u,2´¯uqφ2udu\n“ ´ˆr`2\n2˙2\nprψ0prqq2´2ż`8\nmax p¯u,2´¯uqφφuu2du\nď ´ˆr`2\n2˙2\nprψ0prqq2`2˜ż`8\nmax p¯u,2´¯uqφ2udu¸1\n2˜ż`8\nmax p¯u,2´¯uqu3φ2\nudu¸1\n2\n,\nand this in turn gives (2.8). /square\nInterpolating between (2.3) in Lemma 2.1 and (2.5) in Lemma 2.2, we get6 GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPIN G\nLemma 2.3. Letφpu,¯uqsolve(2.2), and¯Ube a positive constant, then for 0ďsď1,\nthere holds\n(2.11)sup\n1\n2ď¯uď¯U˜ż`8\nmax p¯u,2´¯uqu3spu´¯uqsφu2du¸1\n2\nÀε´\n}ψ0}2\nH1pR3q` }ψ1}2\nL2pR3q¯1\n2`ż¯U\n1\n2˜ż`8\nmax p¯u,2´¯uqu3spu´¯uqsG2du¸1\n2\nd¯u.\n3.Sobolev type inequalities\nIn this section, we shall prove several Sobolev type inequalities. In the following we\ntakes“1\n4`1\n2pand denote\nMpφqp¯uq “˜\n1\n2ż`8\nmax p¯u,2´¯uqu3spu´¯uqsφu2du¸1\n2\n.\nLemma 3.1. For¯uě1, it holds that\n(3.1) sup\nuu2\npφ2pu,¯uq À pMpφqp¯uqq2\nProof.Direct computation implies that\n(3.2)u2\npφ2pu,¯uq “ ´u2\npż`8\nuBλφ2pλ,¯uqdλ\nď2u2\npż`8\nu|φλ||φ|dλ\nď2ż`8\nuλ2\np|φλ||φ|dλ\nď˜ż`8\nmax p¯u,2´¯uqu3spu´¯uqsφu2du¸1\n2\nˆ˜ż`8\nmax p¯u,2´¯uqu5\n2p´3\n4pu´¯uq´sφ2du¸1\n2\n.GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPING 7\nSince 1 ´są0,5\n2p´3\n4ą0, then for ¯ uě1 we have\n(3.3)ż`8\nmax p¯u,2´¯uqu5\n2p´3\n4pu´¯uq´sφ2du“ż`8\n¯uu5\n2p´3\n4pu´¯uq´sφ2du\n“1\n1´sż`8\n¯uu5\n2p´3\n4φ2dpu´¯uq1´s\n“ ´1\n1´sż`8\n¯up5\n2p´3\n4qu5\n2p´7\n4φ2pu´¯uq1´sdu\n´2\n1´sż`8\n¯uu5\n2p´3\n4pu´¯uq1´sφφudu\nď2\n1´sż`8\n¯uu5\n2p´3\n4pu´¯uq1´s|φ||φu|du\nÀˆż`8\n¯uu5\n2p´3\n4φ2pu´¯uq´sdu˙1\n2ˆż`8\n¯uu5\n2p´3\n4pu´¯uq2´sφu2du˙1\n2\n,\nwhich implies\n(3.4)ż`8\nmax p¯u,2´¯uqu5\n2p´3\n4pu´¯uq´sφ2du\nďż`8\n¯uu5\n2p´3\n4pu´¯uq2´sφu2du\nďż`8\n¯uu5\n2p´3\n4`2´2spu´¯uqsφu2du\nÀ pMpφqp¯uqq2,\nwhere we have used the fact that 2 ´sěsand hence\npu´¯uq2´sďu2´2spu´¯uqs,\nand Lemma 3.1 follows. /square\nIn a similar way, we can prove\nLemma 3.2. For¯uď1, it holds that\n(3.5) sup\nuu2\npφ2pu,¯uq À pMpφqp¯uqq2` p1´¯uq3´sψ2\n0p2´2¯uq.8 GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPIN G\nProof.Sinceuě2´¯u, then as in the proof of the last lemma we have\n(3.6)u2\npφ2pu,¯uq “ ´u2\npż`8\nuBλφ2pλ,¯uqdλ\nď2u2\npż`8\nu|φλ||φ|dλ\nď2ż`8\nuλ2\np|φλ||φ|dλ\nďˆż`8\n2´¯uu3spu´¯uqsφu2du˙1\n2\nˆˆż`8\n2´¯uu5\n2p´3\n4pu´¯uq´sφ2du˙1\n2\n.\nSince 1 ´są0,5\n2p´3\n4ą0, then for ¯ uď1 we have\n(3.7)ż`8\n2´¯uu5\n2p´3\n4pu´¯uq´sφ2du\n“1\n1´sż`8\n2´¯uu5\n2p´3\n4φ2dpu´¯uq1´s\nď1\n1´sˆ\np1´¯uq1´sφ2p2´¯u,¯uq ´2ż`8\n2´¯uu5\n2p´3\n4pu´¯uq1´sφφudu˙\nÀˆż`8\n2´¯uu5\n2p´3\n4φ2pu´¯uq´sdu˙1\n2ˆż`8\n2´¯uu5\n2p´3\n4pu´¯uq2´sφu2du˙1\n2\n` p1´¯uq3´sψ2p2´2¯uq,\nwhich implies as in (3.4)\n(3.8)ż`8\n2´¯uu5\n2p´3\n4pu´¯uq´sφ2du\nďż`8\n¯uu5\n2p´3\n4pu´¯uq2´sφu2du` p1´¯uq3´sψ2p2´2¯uq\nÀ pMpφqp¯uqq2` `p1´¯uq3´sψ2p2´2¯uq,\nthen Lemma 3.2 follows from (3.6) and (3.8). /square\n4.Proof of Theorem 1.2\nInthissectionwewillprovethemainresult(Theorem1.2)byglobalite rationmethod.\nFor any ¯ψwith¯φ“r¯ψ, such that sup\n1\n2ď¯uď¯UMp¯φqp¯uq ďM0ε(M0is a positive constant to\nbe determined), we define a map F:¯φÑφsuch thatφsolves\n(4.1)$\n&\n%φu¯u`µp2´µqφ\n4pu´¯uq2“|¯φ|p\npu´¯uqp´1pu`¯uqµpp´1q\n2\nt“0 :φ“εψ0,φt“εψ1.GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPING 9\nWe want to prove that Fmaps the set\nX“ tφ“rψ|ψp0,rq “εψ0prq,ψtp0,rq “εψ1prq,sup\n1{2ď¯uď¯UMpφqp¯uq ďM0εu\nto itself and is a contraction map, thus, for any ¯φ1,¯φ2PX, it holds that\nsup\n1{2ď¯uď¯UMpφ1´φ2qp¯uq ď1\n2sup\n1{2ď¯uď¯UMp¯φ1´¯φ2qp¯uq.\nThen by contraction mapping theorem, Fhas a fixed point, which is our global solution.\nWe only prove that Fmaps the set Xto itself, the contraction can be proved in a similar\nway.\nTakes“1\n4`1\n2p, for¯φPX, by Lemma 2.3 one has\n(4.2)sup\n1\n2ď¯uď¯UMpφqp¯uq Àε`ż¯U\n1ˆż`8\n¯uu3spu´¯uqsG2du˙1\n2\nd¯u\n`ż1\n1\n2ˆż`8\n2´¯uu3spu´¯uqsG2du˙1\n2\nd¯u\nfiε`I`II.\nWe first estimate the integral in IforpSp3`µq ăpď2. By Lemma 3.1 one has\n(4.3)ż`8\n¯uu3spu´¯uqsG2du\n“ż`8\n¯uu3spu´¯uqspu´¯uq´2pp´1qpu`¯uq´µpp´1q|¯φ|2pdu\nďMp¯φq2pp´1qż`8\n¯uu3s´µpp´1qpu´¯uqs´2pp´1q|¯φ|2u´2pp´1q\npdu\n“Mp¯φq2pp´1qż`8\n¯uu´αpu´¯uq´β´1|¯φ|2du,\nwhere forpSp3`µq ăpď2 and 1.5ďµă2\n(4.4)α“ ´3p1\n4`1\n2pq `µpp´1q `2\nppp´1q\n“pµ`2qp2´ pµ`4qp´2\np´2p`21\n4´3\n2p\ną0,\nβ“ ´p1\n4`1\n2pq `2pp´1q ´1ě0,\n2´βą0,\n1´βěs,10 GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPI NG\nwhich leads to\n(4.5)ż`8\n¯uu´αpu´¯uq´1´βφ2du\n“ ´1\nβż`8\n¯uu´αφ2dpu´¯uq´β\n“ ´1\nβż`8\n¯uαu´α´1φ2pu´¯uq´βdu`2\nβż`8\n¯uu´αφ¨φupu´¯uq´βdu\nď2\nβż`8\n¯uu´αφ¨φupu´¯uq´βdu\nÀˆż`8\n¯uu´αpu´¯uq´β´1u2du˙1\n2ˆż`8\n¯uu´αpu´¯uq´β`1φu2du˙1\n2\nÀˆż`8\n¯uu´αpu´¯uq´β´1u2du˙1\n2ˆż`8\n¯uu´α´β`1´4su3spu´¯uqsφu2du˙1\n2\n,\nwhere we used the fact that for 1 ´βěs\npu´¯uq´β`1ď pu´¯uq´β`1´spu´¯uqs\nďu1´β´spu´¯uqs.\nIt is easy to see\nγfiα`β`4s´1\n“µpp´1q `2\nppp´1q `2pp´1q ´2\n“pµ`2qp2´ pµ`4qp´2\np`2\ną2,\nthe by combining\nuě¯u,u´γď¯u´γ,\nwe obtain for pSp3`µq ăpď2 and 1.5ďµă2\n(4.6)ż`8\n¯uu3spu´¯uqsG2du\nÀ¯u´γMp¯φq2pp¯uq.GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPING 11\nWe are going to estimate the integral in IIforpSp3`µq ăpď2. By Lemma 3.2\nwe get\n(4.7)ż`8\n2´¯uu3spu´¯uqsG2du\n“ż`8\n2´¯uu3spu´¯uqspu´¯uq´2pp´1qpu`¯uq´µpp´1q|¯φ|2pdu\nď´\nMp¯φq2pp´1q` p1´¯uqp3´sqpp´1qψ2pp´1q\n0 p2´2¯uq¯\nˆż`8\n2´¯uu3s´µpp´1q´2pp´1q\nppu´¯uqs´2pp´1q|¯φ|2du\nfi´\nMp¯φq2pp´1q` p1´¯uqp3´sqpp´1qψ2pp´1q\n0 p2´2¯uq¯\nˆż`8\n2´¯uu´αpu´¯uq´β´1|¯φ|2du,\nwhereα,βis the same as those in (4.4) and the same conditions are satisfied. Th en we\nmay estimate the last term in the above inequality in a similar way as that of (4.5), and\nthe only difference is that the initial data will appear in this case, thus we have\n(4.8)ż`8\n2´¯uu´αpu´¯uq´1´β¯φ2du\nÀp1´¯uq2´βψ2\n0p2´2¯uq `ż`8\n2´¯uu´αpu´¯uq´1`β¯φ2\nudu\nÀp1´¯uq2´βψ2\n0p2´2¯uq `¯u´γMp¯φq2,\nthis implies by combining (4.7)\n(4.9)ż`8\n2´¯uu3spu´¯uqsG2du\nÀ”\nMp¯φq2pp´1q` p1´¯uqp3´sqpp´1qψ2pp´1q\n0 p2´2¯uqı\nˆ“\n¯u´γMp¯φq2` p1´¯uq2´βψ2\n0p2´2¯uq‰\n.\nPlugging (4.6) and (4.9) into (4.2), finally we get for pSp3`µq ăpď2 and 1.5ď\nµă2\n(4.10) sup\n1\n2ď¯uď¯UMpφqp¯uq ďC0ε`˜C0εp`C1sup\n1\n2ď¯uď¯UMp¯φqpp¯uq,\nwhereC0,˜C0,C1are some positive constants independent of ε, and if we take εď1, then\nthere exists C2ą0 such that\n(4.11) sup\n1\n2ď¯uď¯UMpφqp¯uq ďC2ε`C1sup\n1\n2ď¯uď¯UMp¯φqpp¯uq.\nSetM0“2C2, thenFmapsXtoXprovided that\nC1M0pM0εqp´1ďC2.\nIn a similar way, we can prove Fis a contraction mapping, this finishes the proof of\nTheorem 1.2.12 GLOBAL EXISTENCE OF SEMILINEAR WAVE EQUATIONS WITH DAMPI NG\nAcknowledgement\nNing-An Lai is supported by NSF of Zhejiang Province(LY18A01000 8) and NSFC\n11771194, Yi Zhou is supported by Key Laboratory of Mathematic s for Nonlinear Sci-\nences (Fudan University), Ministry of Education of China, Shangha i Key Laboratory for\nContemporaryAppliedMathematics, SchoolofMathematicalScien ces, FudanUniversity,\nNSFC (11421061), 973 program (2013CB834100) and 111 projec t.\nReferences\n[1] M.D’Abbicco, The threshold of effective damping for semilinear wave equat ions, Math. 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Differential\nEquations, 232(2007), 74-103."    },    {        "title": "2102.03914v4.Spinterface_Induced_Modification_in_Magnetic_Properties_in_Co40Fe40B20_Fullerene_Bilayers.pdf",        "content": "Spinterface Induced Modification in Magnetic\nProperties in Co40Fe40B20/Fullerene Bilayers\nPurbasha Sharangi,†Esita Pandey,†Shaktiranjan Mohanty,†Sagarika Nayak,†\nand Subhankar Bedanta∗,†,‡\n†Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical\nSciences, National Institute of Science Education and Research (NISER), HBNI, P.O.-\nBhimpur Padanpur, Via Jatni, 752050, India\n‡Center for Interdisciplinary Sciences (CIS), National Institute of Science Education and\nResearch (NISER), HBNI, Jatni, 752050 India\nE-mail: sbedanta@niser.ac.in\nAbstract\nOrganic semiconductor/ferromagnetic bilayer thin films can exhibit novel properties\ndue to the formation of spinterface at the interface. Buckminsterfullerene (C 60) has\nbeen shown to exhibit ferromagnetism at the interface when it is placed next to a\nferromagnet (FM) such as Fe or Co. Formation of spinterface occurs due to the orbital\nhybridization and spin polarized charge transfer at the interface. In this work, we have\ndemonstrated that one can enhance the magnetic anisotropy of the low Gilbert damping\nalloy CoFeB thin film by introducing a C60layer. We have shown that anisotropy\nincreases by increasing the thickness of C 60which might be a result of the formation\nof spinterface. However, the magnetic domain structure remains same in the bilayer\nsamples as compared to the reference CoFeB film.\n1arXiv:2102.03914v4  [cond-mat.mtrl-sci]  7 Nov 2021Introduction\nOrganic spintronics has drawn immense research interest in the last few decades due to its\napplications in spin valve, magnetic tunnel junctions etc.1–3In organic spintronics, organic\nsemiconductors (OSCs) (e.g. C 60, Alq 3, ruberene etc.) are used to transport or control spin\npolarized signals.4–8The main advantage of OSCs are their low production cost, light weight,\nflexible and chemically interactive nature. Usually the spin orbit coupling is small in organic\nmaterials (e.g. C 60) as they consist of low Z (atomic number) materials (in particular carbon\n(C)). Moreover, the zero hyperfine interaction in C 60results in a longer spin relaxation\ntime.9–15As a consequence, spin of a carrier weakly interacts in organic environment and\nspin information is maintained for a long time. There are several reports on organic spin\nvalves, organic light emitting diodes (OLED) using C 60as a spacer layer.9–16It has been\nshown that C 60(∼2 nm) can be magnetized when it is placed next to a ferromagnetic (FM)\nlayer.9,17–19d−phybridization at the interface of FM/C 60modifies the density of states\n(DOS) and exhibits room temperature ferromagnetism. Such kind of interface is known as\nspinterface.20It has been shown that the fundamental magnetic properties like magnetic\nmoment, domain structure and magnetic anisotropy can be tuned by depositing C 60on top\nof a Fe, Co or Fe 4N layer.17–19,21Using first-principles calculations Han etal. have shown that\nmagnetic anisotropy energy (MAE) of Fe 4N system is changed from out-of-plane to in-plane\nafter inserting a C 60layer.21Their study indicates a strong d−phybridization between Fe\nand C atoms, which modifies the MAE of the system.21It has been found that ∼2 nm\nof C 60exhibits magnetic moment ∼3µB/cage at the epitaxial Fe/C 60interface.17There\nis a decrement in anisotropy in polycrystalline Fe/C 60system whereas for polycrystalline\nCo/C 60system anisotropy got enhanced. However, to the best of our knowledge no such\nbasic study has been performed on CoFeB system. For spintronic application a low damping\nmaterial is always desired as it directly affects the speed of a device. The main advantage\nof taking CoFeB as a ferromagnet is that it exhibits low Gilbert damping parameter and it\nis amorphous in nature.22It is very important to explore the effect of interface of such a\n2system (CoFeB/OSC) to enrich our fundamental knowledge of spinterface.\nIn this regard, we have prepared CoFeB/C 60bilayer films and compared the magnetic\nproperties to its reference CoFeB film. Also, we have varied the thickness of C 60layer\nto qualitatively define the extent of spinterface and study the modifications in the basic\nmagnetic properties. To study the qualitative nature of the interface, we have performed\nKerr microscopy and ferromagnetic resonance (FMR) measurements.\nMethods\nCoFeB reference film with 5 nm thickness and bilayer (CoFeB/C 60) samples have been de-\nposited on Si (100) substrate in a multi-deposition high vacuum chamber manufactured by\nMantis Deposition Ltd., UK. In the bilayer samples the thickness of CoFeB is fixed to 5 nm\nand the thickness of C 60has been varied between 1.1 to 15 nm. The composition of CoFeB\nconsidered here is 40:40:20. The base pressure in the chamber was 5 ×10−8mbar. CoFeB, C 60\nand MgO layers have been deposited using DC sputtering, thermal evaporation and e-beam\nevaporation techniques, respectively. The samples are named as S1, S2, S3, S4 and S5 for the\nthickness of C 60(tC60) taken as 0, 1.1, 2, 5, 15 nm, respectively. The schematic of the sample\nstructure is shown in Figure 1a (thicknesses not to scale). All the layers were deposited\nwithout breaking the vacuum to avoid oxidation and surface contamination. The deposition\npressure was 5×10−3mbar for CoFeB and 1 ×10−7mbar for C 60and MgO evaporation.\nThe deposition rates for CoFeB and C 60layers were 0.1 and ∼0.1 – 0.15 ˚A/s, respectively.\n2 nm of MgO has been deposited as a capping layer. C 60layer has been deposited normal to\nthe substrate whereas CoFeB plume was at 30◦w.r.t the substrate normal due to chamber’s\nin-built geometry.\nTo understand the growth of each layer and interfaces, cross-sectional TEM has been\nperformed on sample S4 using a high-resolution transmission electron microscope (HRTEM)\n(JEOL F200, operating at 200 kV and equipped with a GATAN oneview CMOS camera).\n3For the compositional analysis we have performed scanning transmission electron microscopy\n- energy dispersive X-ray spectroscopy (STEM - EDX). Selected area electron diffraction\n(SAED) has been performed on sample S4 to investigate the growth of the CoFeB and C 60\nlayers (supplementary information Figure S1). X-ray reflectivity (XRR) has been performed\non all the samples to know the exact thickness and roughness of all the layers (see Figure S2\nand Table S1 in the supplementary information ).\nWe have measured the hysteresis loop and magnetic domain images at room temperature\nby magneto-optic Kerr effect (MOKE) based microscopy manufactured by Evico magnetics\nGmbH, Germany. Longitudinal hysteresis loops are recorded for ±5 mT magnetic field by\nvarying the angle ( φ) between the easy axis (EA) and the applied magnetic field direction.\nTo measure the hysteresis loops along hard axis (HA), we have applied ±17.5 mT magnetic\nfield.\nIn order to determine the magnetic anisotropy constant and observe the anisotropy sym-\nmetry in the samples, angle dependent FMR measurements have been performed at a fre-\nquency of 7 GHz at 5◦interval. During the measurement the sample was kept on the wide\ncoplanar waveguide (CPW) in a flip-chip manner. An in-plane magnetic field (i.e, parallel\nto the sample plane) is applied to the sample (for the detailed measurement configuration,\nrefer to the figure S4 in the supplementary information). Frequency dependent FMR mea-\nsurements have been performed to calculate the Gilbert damping constant( α).\nResults and discussion\nHigh resolution TEM image is shown in Figure 1b and all the layers are marked separately.\nIt shows the amorphous growth of CoFeB and C60(see supplementary information figure S1).\nElement specific mapping has been shown in Figure 1c. Figure 1d shows the STEM image,\nin which the brighter part indicates the layer of the element having high atomic number(Z).\nPresence of Boron(B) is not properly visible as it is a lighter atom. Figure 1e-f represent the\n4Figure 1: (a)Schematic of the sample structure. The thicknesses shown in this schematic is\nnot to scale to the actual thicknesses of the samples. (b) Cross-sectional transmission electron\nmicroscopy (TEM) image of S4. (c) Elemental mapping for individual layers. (d)The region\nof the sample S4 where the STEM-EDX has been performed. (e) EDX line profile for each\nlayer of the sample S4. (f) EDX spectrum of sample S4 showing the presence of different\nelements.\nEDX line profile and EDX spectra, respectively. The position of the Co and Fe peak at the\nsame place indicates the formation of CoFeB alloy. EDX spectra shows the presence of C,\nMg, O, Fe and Co elements in the sample.\nFigure 2a-d show the in-plane angle ( φ) dependent hysteresis loops measured using longi-\ntudinal magneto optic Kerr effect (LMOKE) microscopy at room temperature for the samples\nS1, S2, S4 and S5, respectively. φis defined as the angle between the EA and the applied\nmagnetic field direction. The hysteresis loops along 90◦w.r.t. EA are shown in Figure 2e-h\nfor the samples S1, S2, S4 and S5, respectively. Angle dependent hysteresis loops show that\nthe magnetic HA of the samples is at 90◦w.r.t the EA, which marks the presence of uniaxial\nanisotropy in the system. The easy axis of the anisotropy lies in-plane at an angle of 90◦\nw.r.t the projection of the plume direction. The CoFeB target is at an angle of 30◦w.r.t. the\n5Figure 2: Hysteresis loops measured by magneto optic Kerr effect (MOKE) microscopy at\nroom temperature in longitudinal mode by varying the angle ( φ) between the EA and the\napplied magnetic field direction for the samples (a) S1, (b) S2, (c) S4 and (d) S5. (e) - (h)\nrepresent the hysteresis loops measured along 90◦w.r.t EA for S1, S2, S4 and S5, respectively.\nsubstrate normal due to the in-built geometry of the deposition system. Such kind of oblique\nangle deposition induced uniaxial anisotropy has been reported earlier.19,23–29It should be\nnoticed that there is a change in coercive field ( HC) in the bilayer samples as compared to\nthe single layer reference sample. The values of HCare 1.23, 0.71, 0.91 and 0.81 mT for\nthe samples S1, S2, S4 and S5, respectively. The HCfor the bilayer samples S2 to S5 are\ncomparable. However, the decrease in HCfrom single layer CoFeB to bilayers CoFeB/C 60\ncan be attributed to the formation of a spinterface between the CoFeB and C 60interface. In\nour previous study we have shown that the orbital hybridization at the FM (Fe or Co)/C 60\ninterface promote the change in anisotropy of the system.17–19\nThe square shaped loop along EA indicates the magnetization reversal is happening\nvia domain wall motion whereas, along HA the reversal occurs via coherent rotation. The\nmagnetization reversal is studied as a function of φ. By varying the angle ( φ) w.r.t easy axis\n(0◦), we have recorded the domain images near the HCatφ= 0◦, 30◦, 45◦, 60◦and 90◦.\nFigure 3a-e, Figure 3f-j, Figure 3k-o and Figure 3p-t show the magnetic domain images near\nHCfor the samples S1, S2, S4 and S5, respectively. Branched domains have been observed\n6Figure 3: Domain images near HCfor samples S1, S2, S4 and S5 are shown in (a) - (e), (f)\n- (j), (k) - (o) and (p) – (t), respectively. The scale bars of the domain images for all the\nsamples are same and shown in image (a). The applied field ( H) direction shown in image\n(a) was kept constant for all the measurements and the sample was only rotated to capture\nthe domain images at different φ.\nin all the samples due to the amorphous growth of CoFeB. Domain images captured at\ndifferent applied magnetic fields along EA for samples S1, S2, S4 and S5, are shown in\nFigure S3 in supplementary information. For the samples S2 and S5 the tilt of the domains\nare opposite to S1 and S4. This opposite tilt is due to the anti-clockwise (S2 and S5)\nand clockwise (S1 and S4) rotation of the sample stage w.r.t EA during measurement. It\nis noted that the HCis different between the reference CoFeB and the bilayer samples.\nHowever, the change in domain structure is not significant between the single layer CoFeB\nand the bilayer CoFeB/C 60samples. In our earlier reports it has been shown that the change\nin domain shape and size is significant in other ferromagnetic/OSC systems such as Co/C 60,\nFe/C 60.17–19In epitaxial Fe/C 60system the magnetization reversal process was different\nbetween the reference Fe and the Fe/C 60bilayer systems.17In case of polycrystalline Fe/C 60\n7system the domain size got reduced for the bilyaers as compared to the reference Fe film.18\nHowever, in case of polycrystalline Co/C 60, the domain size increased for the bilayers as\ncompared to the reference Co film.19In this study we considered CoFeB system and the\ndomain shape and size are comparable between the bilayers and the reference sample. The\norigin to this may be investigated theoretically in future work.\nFigure 4:fvsHresand ∆Hvsfplots for S1, S2, S3, S4 and S5 are shown in (a) and (b),\nrespectively. Solid circles represent the experimental data, while the solid lines are the best\nfits to the eqs. 2 and 3.\nWe have further invesigated the magnetization dynamics by performing the frequency\ndependent FMR measurement. The experimental data has been fitted using a Lorentzian\nfunction (eq. 1), where ∆ H,Hres, A1and A 2are linewidth, resonance field, anti-symmetric\nand symmetric components, respectively.30\nFMR signal =A14∆H(H−Hres)\n(4(H−Hres))2+ (∆H)2−A2(∆H)2−4(H−Hres)2\n(4(H−Hres))2+ (∆H)2+offset (1)\nThe plots of fvsHresand ∆Hvsfare shown in Figure 4a and Figure 4b, respectively.\nThe effective damping constant ( α) has been determined by fitting the eqs. 2 and 3:30–32\n8f=γ\n2π/radicalBig\n(HK+Hres)(HK+Hres+ 4πMeff) (2)\nwhere,γ(gyromagnetic ratio) = gµB/¯handg,µB, ¯h,HKare Lande-g factor, Bohr\nmagneton, reduced Planck’s constant and anisotropy field, respectively.\n∆H= ∆H0+4παf\nγ(3)\nwhere, ∆H0is the inhomogeneous line width broadening which depends on the mag-\nnetic inhomogeneity of the sample. αvalues for the samples S1, S2, S3, S4 and S5 are\n0.0095±0.0002, 0.0106±0.0002, 0.0110±0.0002, 0.0124±0.0003 and 0.0169 ±0.0006 , respec-\ntively. It has been observed that αincreases after introducing a C 60layer and it further\nincreased with increasing the C 60thickness. This increase in αmight be due to the interface\nroughness or other effects such as spin pumping at the interface33.\nTo quantify the change in anisotropy in all the samples we have performed in-plane angle\ndependent FMR measurements at a fixed frequency of 7 GHz. Resonance field ( Hres) has\nbeen measured by rotating the sample w.r.t the applied magnetic field in 5◦intervals.\nHresvsφplots have been shown in Figure 5 to calculate the anisotropy constants of the\nsystem. The open circles represent the raw data and the solid lines are the best fits. The\nexperimental data is fitted using Landau-Lifshitz-Gilbert (LLG) equation:34\nf=γ\n2π((H+2K2\nMSCos2φ)(H+ 4πMS+2K2\nMSCos2φ))1/2(4)\nwhere,K2is the in-plane uniaxial anisotropy constant, φis the in-plane angle between the\neasy axis w.r.t the applied magetic field direction and MSis the saturation magnetization.\nTheK2values extracted from the fitting are listed in Table 1. It has been observed\n9Figure 5: Angle dependent resonance field ( Hres) plot for all the five samples to calculate the\nanisotropy constants of the system. The measurement was performed at room temperature\nand a fixed frequency of 7 GHz. Open circles represent the experimental data, while the\nsolid lines are the best fits.\nthat by introducing a C60layer the anisotropy of the system increased. The possible reason\nbehind the enhancement in the magnetic anisotropy is the formation of spinterface at the\nCoFeB/C60interface. The anisotropy increases from 2.9 ×104to 3.1×104erg/cc when the\nC60thickness is varied from 1.1 to 2 nm. With further increase in C60thickness (at 5 nm),\nthe anisotropy become 4.1 ×104erg/cc. There is a small change in the anisotropy (4 .1×104\nto 4.3×104erg/cc) when C60thickness increases from 5 to 15 nm. After a certain thickness of\nTable 1: The value of K2for all the samples extracted from the fitting of LLG equation.\nSampleK2(erg/cc)\nS1 2.4×104\nS2 2.9×104\nS3 3.1×104\nS4 4.1×104\nS5 4.3×104\n10C60layer, the spinterface thickness remains almost constant. However, the exact thickness of\nthe spinterface for amorphous CoFeB/ C60system is not known. In future polarized neutron\nreflectivity (PNR) experiment may be carried out to evaluate the spinterface thickness.\nConclusion\nWe have studied the effect of C 60on the magnetization reversal and the magnetic anisotropy\nof a low damping amorphous CoFeB layer. In comparison to the single layer CoFeB sample\nthe magnetic anisotropy constant has been increased for the CoFeB/C 60bilayer samples. Fur-\nther from the Kerr microscopy measurements it is observed that there is a negligible change\nin the branch domain pattern in the samples. The enhancement in magnetic anisotropy\nmight be the result of d−phybridization between the CoFeB and C 60layer. This study\nreveals that one can enhance the anisotropy of a ferromagnetic CoFeB system by introducing\na C60layer, which can be suitable for future spintronics devices. Further in future, the nature\nof spinterface such as thickness, magnetic moment per atom etc. should be investigated by\nexperimental methods such as polarized neutron reflectometry. The results presented here\nmight bring interest to study similar system theoretically to elucidate the exact nature of\nspinterface and the origin behind it.\nSupporting Information\nSelected area electron diffraction (SAED) on sample S4 is shown in Figure S1. XRR data\nwith the best fits for samples S1 to S5 are shown in Figure S2. MOKE hysteresis loops\nwith corresponding domain images along the EA for samples S1, S2, S4 and S5 are shown\nin Figure S3. The schematic of the FMR measurement set-up and applied field direction is\nshown in Figure S4.\n11Acknowledgement\nWe sincerely thank Dr. Tapas Gosh and Mr. Pushpendra Gupta for helping in TEM imag-\ning. The authors also want to thank Dr. Ashutosh Rath for valuable discussion regarding\nthe SAED images. The authors also acknowledge Department of Atomic Energy, and De-\npartment of Science and Technology - Science and Engineering Research Board, Govt. of\nIndia (DST/EMR/2016/007725) for the financial support.\nReferences\n(1) Naber, W.; Faez, S.; van der Wiel, W. G. Organic Spintronics. J. Phys. D: Appl. Phy.\n2007 ,40, R205.\n(2) Stamps, R. L.; Breitkreutz, S.; ˚Akerman, J.; Chumak, A. V.; Otani, Y.; Bauer, G. E.;\nThiele, J.-U.; Bowen, M.; Majetich, S. A.; Kl¨ aui, M., et al. The 2014 Magnetism\nRoadmap. J. Phys. D: Appl. Phy. 2014 ,47, 333001.\n(3) Kuch, W.; Bernien, M. Controlling the Magnetism of Adsorbed Metal–Organic\nMolecules. J. Phys.: Condens. Matt. 2016 ,29, 023001.\n(4) Dediu, V. A.; Hueso, L. E.; Bergenti, I.; Taliani, C. Spin Routes in Organic Semicon-\nductors. Nat. Mater. 2009 ,8, 707–716.\n(5) Atodiresei, N.; Brede, J.; Lazi´ c, P.; Caciuc, V.; Hoffmann, G.; Wiesendanger, R.;\nBl¨ ugel, S. 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Observation of a Large Spin-Dependent Transport Length in Organic Spin\nValves at Room Temperature. Nat. Commun. 2013 ,4, 1–7.\n(15) Nguyen, T. D.; Wang, F.; Li, X.-G.; Ehrenfreund, E.; Vardeny, Z. V. Spin Diffusion in\nFullerene-Based Devices: Morphology Effect. Phys. Rev. B 2013 ,87, 075205.\n13(16) Liu, H.; Wang, J.; Groesbeck, M.; Pan, X.; Zhang, C.; Vardeny, Z. V. Studies of Spin\nRelated Processes in Fullerene C 60 Devices. J. Mater. Chem. C 2018 ,6, 3621–3627.\n(17) Mallik, S.; Mattauch, S.; Dalai, M. K.; Br¨ uckel, T.; Bedanta, S. Effect of Magnetic\nFullerene on Magnetization Reversal Created at the Fe/C60 interface. Sci. Rep. 2018 ,\n8, 1–9.\n(18) Mallik, S.; Mohd, A. S.; Koutsioubas, A.; Mattauch, S.; Satpati, B.; Br¨ uckel, T.; Be-\ndanta, S. Tuning Spinterface Properties in Iron/Fullerene Thin Films. Nanotechnology\n2019 ,30, 435705.\n(19) Mallik, S.; Sharangi, P.; Sahoo, B.; Mattauch, S.; Br¨ uckel, T.; Bedanta, S. Enhanced\nAnisotropy and Study of Magnetization Reversal in Co/C60 Bilayer Thin Film. Appl.\nPhys. Lett. 2019 ,115, 242405.\n(20) Sanvito, S. Molecular Spintronics: The Rise of Spinterface Science. Nat. Phys 2010 ,6,\n562–564.\n(21) Han, X.; Mi, W.; Wang, X. Spin Polarization and Magnetic Properties at the C60/Fe4\nN (001) Spinterface. J. Mater. Chem. C 2019 ,7, 8325–8334.\n(22) Singh, B. B.; Jena, S. K.; Samanta, M.; Biswas, K.; Satpati, B.; Bedanta, S. Inverse\nSpin Hall Effect in Electron Beam Evaporated Topological Insulator Bi2Se3 Thin Film.\nPhys. Status Solidi RRL 2019 ,13, 1800492.\n(23) Smith, D.; Cohen, M.; Weiss, G. P. Oblique-Incidence Anisotropy in Evaporated\nPermalloy Films. J. Appl. Phys. 1960 ,31, 1755–1762.\n(24) Bubendorff, J.-L.; Zabrocki, S.; Garreau, G.; Hajjar, S.; Jaafar, R.; Berling, D.;\nMehdaoui, A.; Pirri, C.; Gewinner, G. Origin of the Magnetic Anisotropy in Ferro-\nmagnetic Layers Deposited at Oblique Incidence. 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D:\nAppl. Phy. 2019 ,52, 305301.\n(30) Singh, B. B.; Jena, S. K.; Bedanta, S. Study of Spin Pumping in Co Thin Film vis-` a-vis\nSeed and Capping Layers Using Ferromagnetic Resonance Spectroscopy. J. Phys. D:\nAppl. Phys. 2017 ,50, 345001.\n(31) Kittel, C. On the Theory of Ferromagnetic Resonance Absorption. Phys. Rev. 1948 ,\n73, 155.\n(32) Heinrich, B.; Cochran, J.; Hasegawa, R. FMR Linebroadening in Metals due to Two-\nMagnon Scattering. J. Appl. Phys. 1985 ,57, 3690–3692.\n(33) Sharangi, P.; Singh, B. B.; Nayak, S.; Bedanta, S. Spin Pumping and Inverse Spin Hall\nEffect in CoFeB/C60 Bilayers. arXiv preprint arXiv:2106.06829 2021 ,\n15(34) Pan, S.; Seki, T.; Takanashi, K.; Barman, A. Role of the Cr Buffer Layer in the\nThickness-Dependent Ultrafast Magnetization Dynamics of Co2 Fe0.4Mn0.6 Si Heusler\nAlloy Thin Films. Phys. Rev. Appl. 2017 ,7, 064012.\n16Supporting Information\nSpinterface Induced Modification in Magnetic\nProperties in Co40Fe40B20/Fullerene Bilayers\nPurbasha Sharangi,†Esita Pandey,†Shaktiranjan Mohanty,†Sagarika Nayak,†\nand Subhankar Bedanta∗,†,‡\n†Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical\nSciences, National Institute of Science Education and Research (NISER), HBNI, P.O.-\nBhimpur Padanpur, Via Jatni, 752050, India\n‡Center for Interdisciplinary Sciences (CIS), National Institute of Science Education and\nResearch (NISER), HBNI, Jatni, 752050 India\nE-mail: sbedanta@niser.ac.in\nStructural information\nIn order to investigate the growth of deposited layers we have performed selected area electron\ndiffraction (SAED) on sample S4 (i.e., CoFeB(5nm)/C 60(5 nm)/MgO(2nm)). The SAED\nimage (Figure S1) shows the diffuse rings, which confirms the amorphous growth of CoFeB\nand C 60layers. Further, to know the structural information (thikness, roughness) ,we have\nperformed X-ray reflectivity (XRR) measurements on all the samples. We have fitted the\ndata by using GenX software. Figure S2a-e show the XRR data and best fits for samples S1,\nS2, S3, S4 and S5 ,respectively. The extracted thickness ( t) and roughness ( σ) of the layers\nare shown in Table S1.\nS1arXiv:2102.03914v4  [cond-mat.mtrl-sci]  7 Nov 2021Figure S1: SAED image of sample S4.\nFigure S2: XRR data and the best fits for samples S1, S2, S3, S4 and S5 are shown in (a),\n(b),(c), (d) and (e), respectively. The blue open circles are experimental data and the red\nsolid lines represent the best fit using GenX software. The parameters extracted from the\nbest fits are shown in Table S1.\nS2Table S1: Parameters obtained from XRR fits\nSample S1 Sample S2 Sample S3 Sample S4 Sample S5\nLayerst(nm)σ(nm)t(nm)σ(nm)t(nm)σ(nm)t(nm)σ(nm)t(nm)σ(nm)\nCoFeB 5.50 0.91 5.60 0.93 5.50 0.87 5.00 0.81 5.45 0.96\nC60 - - 1.10 0.20 2.00 0.60 5.00 0.52 15.00 1.80\nMgO 1.95 0.61 1.90 0.57 1.80 0.51 2.09 0.76 1.98 0.75\nFigure S3: (a), (b), (c) and (d) show the hysteresis loops for the samples S1, S2, S4 and S5,\nrespectively, along the easy axis (EA). The domain images at different applied fields (marked\ne to x in hysteresis loops) for samples S1, S2, S4 and S5 are shown in (e-i), (j-n), (o-s) and\n(t-x), respectively. The scale bars of the domain images for all the samples are same and\nshown in image (e).\nS3Hysteresis loop and domain imaging\nFigure S3a-d show the hysteresis loops for the samples S1, S2, S4 and S5, respectively, along\nthe easy axis (EA). The domain images are shown in Figure S3e-x are also marked in the\nhysteresis loops. Figure S3e, Figure S3j, Figure S3o and Figure S3t represent the domain\nimages for S1, S2, S4, S5 captured at positive saturation field. Similarly, Figure S3f, Figure\nS3k, Figure S3p, Figure S3u are the domain images near nucleation. Figure S3g, Figure\nS3l, Figure S3q, Figure S3v show the domain images which are captured near coercive field\nand Figure S3h, Figure S3m, Figure S3r, Figure S3w are captured near negative saturation\nfor samples S1, S2, S4 and S5, respectively. Further, domain images captured at negative\nsaturation field are shown in Figure S3i, Figure S3n, Figure S3s, Figure S3x for S1, S2, S4\nand S5, respectively.\nDuring FMR measurement we applied an in-plane magnetic field (i.e, parallel to the\nsample plane). Figure S4 shows the schematic of the FMR measurement set-up and the\napplied field direction.\nFigure S4: Schematic representation of FMR measurement set-up. Sample is placed on the\ncpw in a flip chip manner. The applied field ( Hext) is parallel to the sample plane and\nperpendicular to the rf field ( Hrf).\nS4"    },    {        "title": "2102.07712v2.Magnetodynamic_properties_of_dipole_coupled_1D_magnonic_crystals.pdf",        "content": " \n1 \n Magnetodynamic properties of dipole -coupled 1D magnonic crystals  \nSuraj Singh1*, Xiansi Wang1, Ankit Kumar2, Alireza Qaiumzadeh1, Peter  Svedlindh2, Thomas \nTybell ,3 and Erik Wahlström1 \n1Center for Quantum Spintronics, De partment of Ph ysics, NTNU - Norwegian University of \nScience and Technology, NO -7491 Trondheim, Norway  \n2Department of Materials Sciences and Engineering, Uppsala University, Box 516, SE -75121 \nUppsala, Sweden  \n3Department of Electronic Systems, NTNU - Norwegian University of Science and Technology, \nNO-7491 Trondheim, Norway  \n*Corresponding author e -mail : rsinghsuraj1992@gmail.com  \n \nAbstract  \nMagnonic crystals  are magnetic meta material s, that provide a pr omising way  to manipulate \nmagnetodynamic properties by controlling the geometry of the patterned structures . Here , we \nstudy the  magnetodynamic properties  of 1D magnonic crystals  consist ing of parallel NiFe  strip s \nwith different strip  widths and separations . The strips couple via dipole -dipole interaction s. As \nan alternative to experiments and/or micromagnetic simulations , we investigate the accuracy of  \na simple macrospin model . For the case of simple strips , a model with a single free parameter to \naccount for an overestimation of the out of plane demagn etization of the magnonic lattice  is \ndescribed . By adjusting  this parameter , a good fit with experimental as well as micromagnetic \nresults  is obtained . Moreove r, the  Gilbert damping is found independent of lattice constant  \nhowever  the inhomogeneous linewidth broadening  found to increase with decreasing stripe \nseparation .  \nKeywords:  Permalloy,  Magnonic Crystals, Dipole coupling, Ferromagnetic resonance   \n   \n2 \n 1. Introduction  \nSpin dynamics in nanostructured material s have attracted attention due to  interesting underlying \nphysics and potential for technological applications. Magnonic crystals (MCs)  are a class of \nartificial magnetic media  that offer a promising way to manipulate the magnetodynamic \nproperties  in microwave frequ ency by exploiting the pattern ed geometry1,2. Due to their \ninteresting magnetic properties, MCs find applications in a wide range of magnetic devices such \nas advanced magnetic storage, data processing , and spin logic gates3,4,5. As a consequence, MCs \nhave been studied extensively both theoretically and experimentally in numerous magnetic \nsystem s in order to explore the impact of MCs control parameters on its static and \nmagnetodynamic properties , and for their potential application in novel magnonic devices6-\n8,9,10 ,11,12,13,14. \nAdvances in lithography technique s make it possible to fabricate  nanometer -sized MCs \nwith  narrow spacing s. 1D MCs , with  a periodic magnetic strip pattern along one direction ha ve \nattracted considerable attention due to  their simple geometry , convenient for studying the \nimpact of lattice confinement on the magnetodynamic properties at the nanoscale  7,10,6. In such \nsystem s, the dipolar coupli ng of magnetic strips plays an important role in the magnetodynamic \nproperties. When the strips form a closely packed array, the fundamental mode of individual \nstrips couples via a dynamic dipolar interactio n resulting in formation of  collective spin -wave \nexcitation s6,7,15 ,16,17,18. This i s the result of  the dynamic dipolar magnetic field remov ing the \ndegeneracy between the discrete energy levels of the different magnetic elements.  The collective \ndynamics stemming from the  magnetodynamic  dipolar interaction affect the writing time in \nclosely packed storage media, the synchronization of spin -torque oscillators and most \nimportantly the spin -wave dynamics in MCs19. The spacing between adj acent magnetic elements  \nin such system s is a central  parameter governing the interstrip dipolar coupling . Thus, \ninvestigating the effect of dipolar coupling on the magnetodynamic properties gives valuable \ninformation on the  underlying physics and for poten tial application of MCs in magnonic devices.  \nMost of the previous investigations have been focused on  Brillion Light Scattering  (BLS) \nstud ies of dipolarly coupled 1D MCs, where the interplay of dipolar coupling on collective mode  \n3 \n excitation s and the forma tion of magnonic bandgap s have been studied extensively7,11,16. \nFerromagnetic resonance (FMR) is a sensitive nondestructive technique allowing to study the \nmagn etodynamic properties of the MCs . However, there are only a few report s of FMR stud ies \nof dipole -coupled MCs 20,21, which is especially tru e for  detailed stud ies of important  control \nparameter s such as size and separation of the building blocks making up the MC s and their impact \non the spin dynamics. Also, the impact of dipolar coupling on magnetic damping , which is  \nimportant to  lower the power of magnonic devices , is poorly understood.  In this paper, we \npresent a study of  magnetodynamic properti es of dipolarly coupled 1D MCs by FMR  \nspectroscopy. The 1D MCs consist of parallel Permalloy (Py) strip s prepared using electron beam \nlithography. We report the effect of strip  width  and lattice constant on the resonance field  and \ndescribe a simple macrosp in model that can be used to predict resonance behavior of 1D MCs . \nAlso, the impact  of the MC structure  on the FMR linewidth  has been investigated by broadband \nFMR spectroscopy.  \n2. Experimental Details  \nThe MCs consisted  of Py (Ni 80Fe20) deposited on silicon substrate s by e -beam evaporation . \nElectron beam lithography and lift -off techniques  were  used to fabricate the 1D strip -based  MCs,  \nhaving variable strip width  𝑤 and inter -strip separation  𝑠. The lattice constant of the MCs is  𝜆=\n𝑤+𝑠. Fig. 1 (a) shows the SEM image of a sample described by 𝜆=100 nm and 𝑤=50 nm. \nThe total area of each MC is  11 mm2.  A constant deposition rate was used for all samples to \nensure approximately the same thickness, 𝑑 of 14±3 nm. \nThe magnetodynam ic properties of the MCs were investigated by two complementary \nFMR techniques . Cavity FMR measurements were carried out in a commercial  X-band electron \nparamagnetic resonance (EPR) setup with a fixed microwave frequency of 9.4 GHz (Bruker Bio -\nspin ELEXSYS 500, with a cylindrical TE -011 microwave cavity).  The setup is equipped with a \ngoniometer allowing to rotate the sample 360° in -plane as well as out -of-plane . A schematic of \nthe sample rotation including the magnetization and magnetic field vectors is  shown in fig. 1 (b) . \nA microwave field is applied to the cavity and an applied dc magnetic field is swept to record the \nmicrowave absorption. The measurements were performed with low amplitude modul ation of  \n4 \n static field with lock -in detection to enhance the signal to noise ratio. To extract the resonant \nfield, the measured FMR absorption was fitted to a sum of the derivative of symmetric and \nantisymmetric Lorentzian functions, and the line -shape para meters such as resonant field and \nlinewidth were extracted22. \nFor broadband FMR measurements , a microwave signal generator  FMR setup with a \ncoplanar waveguide (CPW) and  lock-in amplifier  detection technique  was emplo yed. A pair of \nhomemade Helmholtz coils generating a low -frequency (211.5 Hz) and low -amplitude magnetic \nfield (0.25 mT) was used to modulate the microwave signal, which was detected by the lock -in \namplifier.  The FMR spectra  were recorded sweeping  the dc magnetic field at constant microwave \nfrequency. The measurements were taken  at various frequenc ies ranging from 5 to 16 GHz in \nsteps of 0.5 GHz with the dc magnetic field applied parallel  and perpendicular to the magnetic \nstrips of the  MCs23. \n3. Results and Discussion  \n3.1 Resonant field  \nTo investigate  the magnetodynamic properties , we studied  MCs with lattice constants ranging \nfrom 100  nm to 550 nm  with  a fixed width 𝑤=50 nm. In this subsection, all the experiments \nwere done using a 9.4 GHz cavity . The obtained resonant field s as a function of the in-plane \nrotation angle for different  lattice constants  are shown in fig. 2(a). For each sample, two modes \ncan be observed for a magnetic field applied perpendicular to the strips - along the 𝑥-axis (𝜙=\n0°). The two modes shift towards each other as the applied field direction rotates away from 𝜙=\n0°, and at 𝜙= ±15° the modes merge into a single -mode before disappearing.  The two modes \ncorrespond to two equilibrium magnetization directions. No modes are observed for a  wide \nrange of field directions around  the 𝑦-direction i.e., along the strip s. The frequency of the easy -\naxis mode falls outside the detectable range of cavity FMR measurements . The resonant field  \ndecrease s with decreasing  𝜆, wh ich is due to the increasing dipolar  interaction s. We then fix the \nmagnetic field along 𝐇∥𝒙̂ (𝜙=0°) and plot the higher resonant field  versus the lattice constant \n𝜆 in fig. 2(b). The resonant field  increases with increasing 𝜆.  \n5 \n To understand the observed magnetodynamic behaviors, we develop a macrospin \nanalytical model for MCs, and verify its validity  by micromagnetic simulations. Each magnetic \nstrip of the  MCs is considered as a macrospin  with  synchron ized precession of the spin s. The \nmagnetodynamics of the MC is governed by the Landau -Lifshitz -Gilbert (LLG) equation,  \n𝜕𝒎\n𝜕𝑡=−𝛾𝒎×𝑯eff+𝛼𝒎×𝜕𝒎\n𝜕𝑡, \nwhere 𝒎 is the unit vector along the magnetization direction of each strip , 𝛾=𝑔𝜇𝐵\nℏ is the \ngyromagnetic  ratio ,  𝑔 is the La ndé g-factor, µ𝐵 is the Bohr magneton, and ℏ is the reduced \nPlanck’s constant.   There are different values of 𝛾 corresponding to 𝑔=2.00 to 2.17 used in \nliterature24. Here we use 𝛾=176 Grad/s/T corresponding to 𝑔=2.00. The specifi c value of 𝛾 \ndoes not qualitatively  affect the physics we discuss. The term  𝛼 is the Gilbert damping  parameter  \nand 𝑯eff is the  total  effective field , \n𝑯eff=𝐻𝒙̂−𝑀𝑠(𝑁𝑥𝑚𝑥𝒙̂+𝑁𝑦𝑚𝑦𝒚̂+𝑁𝑧𝑚𝑧𝒛̂)+𝑯int, \nwhere 𝐻𝒙̂ is the applied external field along the 𝑥-direction  and −𝑀𝑠(𝑁𝑥𝑚𝑥𝒙̂+𝑁𝑦𝑚𝑦𝒚̂+\n𝑁𝑧𝑚𝑧𝒛̂) is the demagnetization field of the strip with 𝑀𝑆 being the saturation magnetization and \n𝑁𝑥,𝑦,𝑧 the demagnetization components  of each strip which  can be  calculated analytically  from \nthe strip dimensions (width 𝑤, thickness 𝑑 and the length of the strip25). The magneti c damping \nwas neglected when calculating the resonant field  of the modes for the sake of simplicity. 𝑯int is \nthe inter -strip dipolar  interaction  field . To calculate 𝑯int, we consider a strip with two \nneighbo uring strips as illustrated in fig. 2(c) and calculate the dipolar field from the neighbo uring \nstrip s at the center point of the middle  strip . The magnetic charge density on the left and right \nsurfaces (orange surface s in fig. 2(c))  are −𝑀𝑠𝑚𝑥 and 𝑀𝑠𝑚𝑥, respectively . \nThe total field from the left surface (fig.(2c)) can then be estimated by  \n𝐻left=−𝑀𝑠𝑚𝑥\n4𝜋∫∫1\n(𝑥2+𝑦2+𝑧2)−𝑥\n√𝑥2+𝑦2+𝑧2𝑑𝑧𝑑𝑦𝑑\n2\n−𝑑\n2∞\n−∞=arctan𝑑\n2𝑥\n𝜋𝑀𝑠𝑚𝑥, \nSimilarly, for the right surface  \n𝐻right=−arctan𝑑\n2𝑥\n𝜋𝑀𝑠𝑚𝑥,  \n6 \n where  𝑥 is the distance from the center point to the surface.  \nFor the 𝑛th strip, the distance is 𝑥=±𝑤\n2+𝑛𝜆. Thus, the total dipolar field on the center of a \nstrip reads  \n𝐻int𝑥=𝑀𝑠𝑚𝑥\n𝜋∑[arctan𝑑\n(−𝑤+2𝑛𝜆)−arctan𝑑\n(𝑤+2𝑛𝜆)]+∞\n𝑛=−∞, \nSince we have already conside red the shape anisotropy of the center strip, the 𝑛=0 term should \nbe omitted. This term can be absorbed in 𝑁𝑥 (𝑁𝑥′=𝑁𝑥−2\n𝜋∑[arctan𝑑\n(−𝑤+2𝑛𝜆)−+∞\n𝑛=1\narctan𝑑\n(𝑤+2𝑛𝜆)]). On the other hand, the magnetic charge density on top and bottom surfaces \n(blue surfaces) are 𝑀𝑠𝑚𝑧 and −𝑀𝑠𝑚𝑧, respectively. The 𝑥 component cancels out due to the \nMCs symmetry, and the 𝑧 component  at the center is  \n𝐻top𝑧=2𝑀𝑠𝑚𝑧\n4𝜋∫∫1\n(𝑥2+𝑦2+𝑑2/4)−𝑑/2\n√𝑥2+𝑦2+𝑑2/4𝑑𝑥𝑑𝑦𝑛𝜆+𝑤\n2\n𝑛𝜆−𝑤\n2∞\n−∞, \nand the total 𝑧-componet is  \n𝐻int𝑧=−𝑀𝑠𝑚𝑧\n𝜋∑[arctan(𝑤+2𝑛𝜆)\n𝑑−arctan(−𝑤+2𝑛𝜆)\n𝑑]+∞\n𝑛=−∞ \nThis term can be absorbed in 𝑁𝑧 (𝑁𝑧′=𝑁𝑧+2\n𝜋∑[arctan(𝑤+2𝑛𝜆)\n𝑑−arctan(−𝑤+2𝑛𝜆)\n𝑑]+∞\n𝑛=1 ). \nFor all the MCs that we have prepared , 𝑁𝑦<𝑁𝑥′<𝑁𝑧′. Thus, the equilibrium magnetization \ntends to be in-plane and along the 𝑦-direction. When applying the external field along the 𝑥-\ndirection, the equilibrium magnetization is tilted  with respect to the 𝑦-direction  satisfying  𝒎∥\n𝑯eff, so that  \n𝐻−𝑁𝑥′𝑀𝑆cos𝜙𝑚\n−𝑁𝑦𝑀𝑆sin𝜙𝑚=cos𝜙𝑚\nsin𝜙𝑚, \nwhere 𝜙𝑚 is the azimuthal angle of 𝒎. The largest 𝐻 for the above equation to have a solution \n𝜙𝑚=arccos𝐻\n(𝑁𝑥′−𝑁𝑦)𝑀𝑠 is 𝐻=(𝑁𝑥′−𝑁𝑦)𝑀𝑠. Above this value, 𝒎 is saturated along  the 𝑥- \n7 \n direction. To obtain the eigenfrequency of the MC s, we expand 𝒎 around its equilibrium \ndirection, assume the small precessional component to have a harmonic form 𝑒𝑖𝜔𝑡, and keep \nonly  linear  terms . The result is the well -known Smit -Beljers formula26, \n𝜔=\n{    𝛾√(𝑁𝑧′𝑀𝑠−𝑁𝑥′𝑀𝑠)(𝑁𝑦𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻2\n𝑁𝑦𝑀𝑠−𝑁𝑥′𝑀𝑠)       𝐻<(𝑁𝑥′−𝑁𝑦)𝑀𝑠  \n𝛾√(𝑁𝑧′𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻)(𝑁𝑦𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻)                      𝐻≥(𝑁𝑥′−𝑁𝑦)𝑀𝑠 (1) \nwhere 𝜔 = 2𝜋𝑓 is the eigen frequency of the FMR mode . We plot the eigenfrequency versus 𝐻 \nin fig. 2(d) to understand the behavior observed in the f ig. 2(a). The horizontal line at 9.4 GHz \ncorresponds to the frequency of t he cavity, and its intersections with the dispersion curve \ncorre spond  to the two experimentally observed resonant modes.  The low field mode is an \nunsaturated mode around  an intermediate equilibrium magnetization state between the long \nand short axes of the strips . The high field mode is attri buted to the uniform precession in the \nfully satu rated state along  the 𝑥-direction .  \n Concentrating on the saturated mode , the fig. 2(b) depicts the resonance field of the \nmode vs lattice constant . The red solid line is the result from Eq. (1) with 𝑤=50 nm and fitting \nparameter 𝑑=14.0 nm. The result agrees with the experimental data within ±3% error . To be \nmore accurate, and to further verify the validity of the macrospin model , we perform \nmicromagnetic simulation s using  the open -source package Mumax 327. The simulation s were \nperformed using 1×1×𝑑 nm3 meshes. Limited by the computational capacity, the system size \nwas set to be 1024×1024×1 meshes  with periodic boundary conditions.  The exchange \nconstant 𝐴=1.3×10−11 J/m.  For same geometry, the simulation results of the r esonance fields \nare larger than that obtained by the macrospin model.  This fact has also been observed in other \nstudies13. Thus, to fit the experimental data, we have to use a smaller thickness 𝑑=12.5 nm. \nThe simulation results are shown by black squares  in the f ig. 2(b). The reason why results of \nanalytical model based on macrospin approximation differ from the simulations is as follows. T he \nmagnetization throughout each strip is not homogeneous as assumed in the macrospin model.  \nAt the two edges, the precessi onal amplit ude is larger, while near the center the amplitude is \nsmaller  [see inset of the f ig. 2(b) as well as fig. S3(b) of the Supplemental Material ]. The first  \n8 \n consequence is that  the total effective field is also inhomogeneous in the strips28. There is a \ndipolar interaction  field near the center, so a larger external field is necessary to reach the \nresonance than  in the macrospin model. The second consequence is that the demagnetization \nfactor in  the 𝑧-direction  (𝑁𝑧′) is significantly overes timated in the macrospin model, while 𝑁𝑥′ and \n𝑁𝑦 are not affected much. Nevertheless, the macrospin model qualitatively reproduces the \nexperimental results, showing that the main reason of the increasing resonant field is the \ndecrea sing inter -strip dipolar interaction when the separation becomes larger. To compensate \nthe overestimated 𝑁𝑧, we introduce an empirical  dimensionless parameter  𝜂<1 to renormalize  \n𝑁𝑧 in Eq. (1) (for the saturated peak 𝐻≥(𝑁𝑥′−𝑁𝑦)𝑀𝑠), \n𝜔=𝛾√(𝜂𝑁𝑧′𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻)(𝑁𝑦𝑀𝑠−𝑁𝑥′𝑀𝑠+𝐻), (2) \nFor 𝑤=50 nm and 𝑑=12.5 nm obtained from the simulation , we find that 𝜂=0.82 fits the \nsimulation results best , as shown by the green dashed line in  the fig. 2(b). The gray area between \nthe two black dashed lines means the range of 𝐻 from Eq. (2) with 𝑤=50±2 nm and 𝑑=\n12.5±0.3 nm, in which the errors are estimated from the limited precision of lithography and \nthe e -beam evaporation techniques. The experimen tal data are well in the range  indicated by the \ngray area in  the fig. 2(b) .  \nWe also prepared MC s samples with larger strip width, in which the macrospin \napproximation obviously  fails. Fig . 3(a) shows the derivative FMR absorption  of a 𝑤=200 nm, \n𝜆=250 nm sample when rotating the applied field in -plane. We can observe that for any 𝜙 \nangle, there is a t least one resonant mode. When the field is along the 𝑥-direction, there are \nthree resonant modes , instead of two in the previous samples. To under stand  the three modes, \nwe performed micromagnetic simulations for 𝑯∥𝒙̂. After Fourier transform, the result is shown \nin Fig. 3(b). Three peaks can be observed, which is consistent with the experiment. More details \ncan be found in Supplemental Material, si nce the main topic of the paper concerns  the narrow -\nstrip samples.  \n \n  \n9 \n 3.2. Ma gnetic Damping  \nTo investigate the impact of dipolar coupling on magnetic damping,  the recorded  FMR signal was \nfitted to the  sum of  derivative s of the symmetric and antisymmetric Lorentzian functions to \ndetermine the lineshape parameters: resonance field ( Hr) and the full width of half maxima \n(𝛥𝐻)22. The fitting determined linewidth versus frequency data is fitted to the model:  \n𝜇0∆𝐻(𝑓)= 𝜇0∆𝐻0+4𝜋𝛼𝑒𝑓𝑓\n𝛾𝑓 (3) \nwhere 𝛼𝑒𝑓𝑓 is the effective damping parameter  including the Gilbert damping and the eddy \ncurrent contributions29 and ∆𝐻0  is the linewidth at zero frequency also known as  \ninhomogeneous line width  broadening30.  \nThe linewidth broadening was measur ed for the magnetic  field applied parallel and \nperpendicular to the strips.  Table 1  summarizes  the characteristics of the FMR linewidth at \ndifferent frequencies when the field is applied along and perpendicular to the strips ( 𝑦-direction)  \n(spec ific data can be found in Fig. S6 in supplemental materials) . In total , the effective damping \nparameter  𝛼𝑒𝑓𝑓 = 0.0045±0.0005 , and is almost independent of the strip separation. The \ninhomogeneous linewidth broadening decreases with increasing lattice consta nt (or strip \nseparation).  The same trend is obtained when the field is perpendicular to the strips ( 𝑥-direction). \nThe inhomogeneous broadening is small for a reference thin film sample without  the MCs \nstructure, which is within expectation because the thi n film is much more homogeneous .  An \ninteresting observation is that the effective Gilbert damping  is very different for 𝐇∥𝑦̂, 𝐇∥𝑥̂ and \nthin film experiments.  It is noted that different multi -magnon scattering processes31 and metallic \nelectromagnetic effects such as eddy curren ts32 can contribute to such effects.   \n4. Discussions and Summary  \nA lot of effort has been made to understand the dynamics of MCs8,12 -14,31,33,34. For the FMR mode, \none important aspect is to understand the deviation from the macrospin model. Since the \ndimensions of the magnetic strips are usually much l arger than the exchange length, 𝑙𝑒𝑥=√2𝐴\n𝜇0𝑀𝑠2 \n, a deviation is natural  due to an inhomogeneous magnetization profile. As expected, the  \n10 \n micromagnetic simulations shows that, when the mesh size is comparable to strip size, the \nmacrospin model is recover ed. When reducing the mesh size, the resonant field becomes lower, \nuntil the mesh size is as small as  𝑙𝑒𝑥 and the result converges to the “genuine” value. This \nobservation is also applicable when 𝐦 is allowed to be inhomogeneous in the thickness direc tion. \nFor thin strips whose thickness is much smaller than the width, an analytical formula for boundary \nconditions has been derived which can be used to solve the nonuniform magnetization profile35. \nHowever, in our case the thickness and the width are of the same order of magnitude. So, we \nintroduce an empirical parameter 𝜂 which depends only on the bar dimensions, but independent \nof the bar separation, as a correction to the macrospin model. Recen tly, there is a very \ncomprehensive study on the FMR mode of MCs34. Frequency -sweeping FMR was modeled and \nstudied in that paper. Here, we consider field sweeping FMR, and we prov ide an intuitive picture \nfor the deviation and failure of macrospin model. For 50 nm strips, the strip edge and center have \ndifferent oscillation amplitudes but same phase, so the macrospin model is still qualitatively \ncorrect (i.e. emergence of two modes at two equilibrium 𝐦 directions), and the quantitative \nresults can be recovered by introducing a compensation factor. For 200 -nm strips, the oscillations \nat strip edge and center are out -of-phase, so the macrospin model fails and full micromagnetic \nsimula tion is necessary.  \nWe observe a strong anisotropy in the line broadening. Both the interception (the \ninhomogeneous broadening) and the slope (the effective damping) are anisotropic. The effective \ndamping includes the Gilbert damping and dissipation by the eddy current. The Gilbert damping \nis usually isotropic30, but the eddy current is anisotropic because it is related to the geometry of \nthe sample. This is confirmed by our numerical simulation, where we assume an isotropic Gilbert \ndamping and we do not obs erve the anisotropy in the effective damping. Therefore, we attribute \nthe anisotropic effective damping to the eddy current effect29,32. The inhomogeneous line \nbroadening contains the contribution from external sources such as the multi -magnon scattering, \nanisotropy, and scattering due to roughness and defects. These effects can be strongly \nanisotropic31,36, and are not considered in the simulation. So, we suppose that they could be the \nreason for the o bserved anisotropic Δ𝐻0. We also observe that Δ𝐻 is very small for a film but \nshows a decreasing trend when increasing the strip separation. This may relate to the  \n11 \n inhomogeneity of the whole sample. The inhomogeneous broadening is positively related to t he \ninhomogeneity of the sample37. \nIn summary , we have investigated the  magnetodynamic properties of 1D MCs  with different \nlattice constants . The resonant field found to increas e with increasing lattice constant because \nof decrea sing inter -strip dipolar coupling. The experimental results are qualitatively explained by \na macrospin model when the strips are narrow . The accuracy of the macrospin model can be \nquantitatively  improved b y renormalizing the out -of-plane demagnetization factor fitted by \nmicromagnet ic simulation s. Obvious difference in linewidth slopes was found for different field \ndirections.  \n \nAcknowledgements  \nThis work was partly supported by the Research Counc il of Norway through its Centre  of \nExcellence funding scheme, project number 262633, “QuSpin” . S. S. acknowledge s partial funding \nobtained from the Norwegian PhD Network on Nanotechnology for Microsystems, which is \nsponsored by the Research Council of Norway, Di vision for Science, under contract no. \n221860/F40.  The Research Council of Norway is acknowledged for the support to the Norwegian \nMicro - and Nano -Fabrication Facility, NorFab, project number 245963/F50.  Gopal Dutt is \nacknowledged for the AFM measurements.  X. S. W. acknowledges the support from the Natural \nScience  Foundation of China (Grant No. 11804045).  \n   \n12 \n References  \n1 V. V. Kruglyak, S. O. Demokritov, and D. Grundler,  J Phys D Appl Phys 43 (26) (2010).  \n2 S. Neusser and D. Grundler,  Adv Mater 21 (28), 2927 (2009).  \n3 A. V. Chumak, A. A. Serga, and B. Hillebrands,  Nat Commun 5 (2014).  \n4 A. V. Chumak, A. A. Serga, and B.  Hillebrands,  J Phys D Appl Phys 50 (24) (2017).  \n5 A. A. Nikitin, A. B. Ustinov, A. A. Semenov, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, E. \nLahderanta, B. A. Kalinikos, and B. Hillebrands,  Applied Physics Letters 106 (10) (2015).  \n6 M. P. Kostylev, A. A. Stashkevich, and N. A. Sergeeva,  Phys Rev B 69 (6) (2004).  \n7 G. Gubbiotti, S. Tacchi, G. Carlotti, P. 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Crowell,  Phys \nRev B 101 (13), 134430 ( 2020).  \n37 Ernst Schlömann,  Physical Review 182 (2), 632 (1969).  \n   \n14 \n Table 1  The Gilbert damping parameters estimated for MCs  and the reference P y thin film  \n \n \n \n \n \n  Sample  H along y -axis H along x -axis \nα ΔH (Oe)  α ΔH(Oe)  \n𝝀=𝟏𝟎𝟎  nm, 𝒘=𝟓𝟎 nm 0.004  97 0.010  72 \n𝝀=𝟏𝟓𝟎  nm, 𝒘=𝟓𝟎 nm 0.004  58 0.012  35 \n𝝀=𝟐𝟓𝟎  nm, 𝒘=𝟓𝟎 nm 0.005  37 - - \nPy thin-film 0.007  5 0.007  5  \n15 \n Figure captions  \nFig. 1   (a) SEM image of a MC with 𝜆=100 nm and 𝑤=𝑠=50 nm. (b) Schematic diagram of \nthe coordinate system defining geometry of the field.  \nFig. 2   (a) The polar plot of resonant field extracted for samples of different 𝜆. (b) The measured \nhigher resonant fields when the applied field is along x direction for different samples \n(blue dots). The red line is the macrospin model result (Eq. 1) for 𝑑=14.0 nm and 𝑤=\n50 nm. The simulation results for 𝑑=12.5 nm and 𝑤=50 nm are shown as black \nsquares. The green dashed line is effective model (Eq. 2)  for  𝑑=12.5 nm and 𝑤=50 \nnm with empirical coefficient 𝜂=0.82. The gray area indicates the range of resonant \nfield for 𝑑=12.5±0.3 nm and 𝑤=50±2 nm. The inset schematically illustrates th e \namplitude of  magnetization tilting during the precession for the macrospin model and \nthe actual simulation.  (c) Schematics depicting the geometry of magnetic strips used in \nanalytical calculations and micromagnetic simulations. (d) The dispersion curves  for 𝜆=\n100 nm  and 𝜆=550 nm. The dashed horizontal line is the cavity frequency 9.4 GHz  \nFig. 3   (a) Experimental results of angle  (𝜙)-dependent differential FMR absorption for the 𝜆=\n250 nm, 𝑤=200 nm sample at RT. (b) Fourier amplitude of the average magnetization \nestimated by micromagnetic simul ations on  the 𝜆=250 nm, 𝑤=200 nm sample for \nfield along 𝜙=0. Three peaks can be identified, which is consistent with the experiment \n(indicated by the ar rows).  \n   \n16 \n  \n \n \n \n \n \n \n \nFig. 1  \n  \nM \nH \n𝛉𝐌 \n𝜃 \n𝜙 \n𝑦Ԧ \n𝑥Ԧ \n𝑧Ԧ \n (b) \n (a) \n \n17 \n  \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 2  \n  \n(b) \n(c) \n (d) \n(a) \n \n18 \n Fig. 3  \n  \n(a) \n (b)  \n19 \n Supplementary Information  \nMapping the dipolar coupling and magnetodynamic  properties of \ndipole coupled 1D magnonic crystals  \nSuraj Singh1, Xiansi Wang1, Ankit Kumar2, Alireza Qaiumzadeh1, Peter  Svedlindh2, Thomas \nTybell ,3 and Erik Wahlström1 \n1Center for Quantum Spintronics, De partment of Ph ysics, NTNU - Norwegian University of \nScience and Technology, NO -7491 Trondheim, Norway  \n2Department of Materials Sciences and Engineering, Uppsala University, Box 516, SE -75121 \nUppsala, Sweden  \n3Department of Electronic Systems, NTNU - Norwegian University of Science and Technology, \nNO-7491 Trondheim, Norway  \n   \n20 \n Figure S1(a) shows the raw cavity FMR spectra recorded on the magnonic  crystals. Two modes \nwere observed for the magnetic applied perpe ndicular to strips. Fig. S1(b) shows the angle -\ndependent FMR measur ements . The two modes found shifting towards the each other as the \nmagnetic field is rotated away from the angle (φ = 0°). The resona nce field of low field mode \nincreases as a fu nction of angle (φ) whereas the it decreases for the high field mode. The modes \nmerge into a single -mode at around φ= ± 15° and then disappears . \n \n \n \n \n \n \n \n \n \n \n \n  Fig. S1 (a) The FMR lineshape recorded on 𝜆=100 nm and 𝑤=50 nm sample for the magnetic \nfield applied perpendicular to the strips. S2(b) The angle -dependent FMR spectra recorded rotating \nthe magnetic field in the plane of the sample from angle  φ = 0° to 360°.  \n(a) \n (b) \nH \nφ=0° \n H  \n21 \n The micromagnetic simulations performed on the magnonic lattices were used to \ninvestigate  the equilibrium magnetization  profile at the observed modes . Figures. S2(a) and S2(b) \ndepict the equilibrium magnetization profiles of the low -field  mode  and the high -field  mode  \nobserved in 𝜆=100  nm, 𝑤=50 nm sample  for the magnetic field applied along x -axis.  As \ndescribed in the main paper, the high field mode occurs at 𝐻=3.34×103 Oe when the \nmagnetization is homogeneously  magnetized  along the x direction. The low field  occurs at 𝐻=\n1×103 Oe when 𝒎 is oriented about 63°  with respect to the x -axis in the bars .  Although 𝒎 is \nnot uniform over the bar, the variation is not large .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n(a) \n (b) \n(c) \nFig. S2 The equilibrium profile of static magnetization at the low field mode in (a) and high field \nmode in (b) ) for  𝜆=100 nm, 𝑤=50 nm sample . (c) The resonant field  plotted as function \nfrequency for the low field mode .   \n22 \n  \nKnowing the equilibrium magnetization profile, time -dependence of the 𝑚𝑧 component at \nresonance and its behaviors at the edge and the center of the strip  can be plotted, see  figs. S3(a) \nand S 3(b). It can be seen that  the edge and the center precess  in-phase at both peaks . At the first \npeak, the amplitudes at the edge and center are similar, while at the second peak the edge has a \nlarger amplitude than the center.  \n \n  \nFig. S3 The amplitude of the magnetization precession at the low field mode in (a) and at high \nfield mode in (b).   \n(a) \n (b)  \n23 \n The equilibrium magnetization profiles at the three peaks from low field to high field for the  \n𝜆=250 nm, 𝑤=200 nm sample  for the magnetic field applied along x -axis are shown in figs. \nS4(a), S4(b), and S4(c). The magnetic fields are at the resonant peaks shown in Fig. 3(b) in the \nmain text. The magnetization pofile is much different than the 𝜆=100 nm, 𝑤=50 nm sample  \nbecau se the bars are much wider  and the spatial variation of 𝒎 is much more significant in the \ntwo low -field peaks.  The time dependence of 𝑚𝑧 oscillation from the three peaks is shown figs. \nS5(a), S5(b), and S5(c).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nAt the first peak, the edge and the center have a 𝜋-phase difference. Since the energy absorption \nis measured, the energy absorptions at the edge and the center cancel with each other. So \nalthough the oscillation amplitude is large, the peak intensity is small. At the second peak, the \ncenter magnetization is almost along the x direction, but edge magnetization is significantly Fig. S4 The equilibrium magnetization profiles at three peaks observed in 𝜆=250 nm, 𝑤=50 \nnm sample from low to high in (a), (b), and (c) respectively.  \n(a) \n(b) \n(c) \n \n24 \n tilted. The oscillations also have a 𝜋-phase difference, but at the center the amplitude is much \nlarger. Furthermore, the center magnetization has a larger proportion in the whole ba r. So the \nenergy absorption is dominated by the central part, and has the largest intensity. At the third \npeak, the magnetization is fully aligned along the x direction. The oscillations have a 𝜋\n2-phase \ndifference, but the edge has a much larger amplitude . The energy absorption is dominated by \nthe edge part.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFor both  𝜆=100 nm, 𝑤=50 nm and 𝜆=250 nm, 𝑤=200  samples  the amplitudes at the \nedges are larger than that at the center for the saturated peaks. But for other peaks the Fig. S5 The amplitude of the magnetization precession at the three peaks from low to high in (a), \n(b) and (c) respectively.  \n(a) \n (b) \n(c)  \n25 \n comparison is complicated. This can be understood as follows. At equilibrium, the local effective \nfield is paral lel to the local magnetization for each point. The larger magnitude the effective field \nhas, the harder the magnetization precesses. For the saturated peaks, the edges bear larger \ndemagnetization field (because they are closer to the surface magnetic charg es). The \ndemagnetization field is antiparallel to 𝒎, so the total effective field at the edges is smaller than \nthe center. For other non -saturated peaks, the situation is very complicated, so there is no simple \npicture for the observed amplitude difference . \n \nThe details of the measur ements of magnetic damping are summarized in fig . S6. \n \nFig. S6 The linewidth versus frequency data fitted to a straight line for the magnetic field applied \nalong and perpendicular to the magnetic strips for 𝜆 with  𝑤=50 nm samples in 5(a) & 5(b) and \nfor the reference thin -film in fig. 5(c). The filled colored symbols the experimental data points \nwhereas the solid red line shows the fitted data.  \n(b) \n (a) \n(c) "    },    {        "title": "2102.10394v2.Fast_magnetization_reversal_of_a_magnetic_nanoparticle_induced_by_cosine_chirp_microwave_field_pulse.pdf",        "content": "Fast magnetization reversal of a magnetic nanoparticle induced by cosine\nchirp microwave field pulse\nM. T. Islam,1,a)M. A. S. Akanda,1M. A. J. Pikul,1and X. S. Wang2,b)\n1)Physics Discipline, Khulna University, Khulna 9208, Bangladesh\n2)School of Physics and Electronics, Hunan University, Changsha 410082, China\nWe investigate the magnetization reversal of single-domain magnetic nanoparticle driven by the circularly polarized\ncosine chirp microwave pulse (CCMP). The numerical findings, based on the Landau-Lifshitz-Gilbert equation, reveal\nthat the CCMP is by itself capable of driving fast and energy-efficient magnetization reversal. The microwave field am-\nplitude and initial frequency required by a CCMP are much smaller than that of the linear down-chirp microwave pulse.\nThis is achieved as the frequency change of the CCMP closely matches the frequency change of the magnetization pre-\ncession which leads to an efficient stimulated microwave energy absorption (emission) by (from) the magnetic particle\nbefore (after) it crosses over the energy barrier. We further find that the enhancement of easy-plane shape anisotropy\nsignificantly reduces the required microwave amplitude and the initial frequency of CCMP. We also find that there is an\noptimal Gilbert damping for fast magnetization reversal. These findings may provide a pathway to realize the fast and\nlow-cost memory device.\nI. INTRODUCTION\nAchieving fast and energy-efficient magnetization rever-\nsal of high anisotropy materials has drawn much attention\nsince it has potential application in non-volatile data stor-\nage devices1–3and fast data processing4. For high thermal\nstability and low error rate, high anisotropy materials are\nrequired5in device application. But one of the challenging\nissues is to find out the way which can induce the fastest mag-\nnetization reversal with minimal energy consumption. Over\nthe last two decades, many magnetization reversal methods\nhas been investigated, such as by constant magnetic fields6,7,\nby the microwave field of constant frequency, either with or\nwithout a polarized electric current8–10and by spin-transfer\ntorque (STT) or spin-orbit torque (SOT)11–28. However, all\nthe means are suffering from their own limitations. For in-\nstance, in the case of external magnetic field, reversal time\nis longer and has scalability and field localization issues6.\nIn case of the constant microwave field driven magnetiza-\ntion reversal, the large field amplitude and the long rever-\nsal time are emerged as limitations29–31. In the case of the\nSTT-MRAM, the threshold current density is a large and thus,\nJoule heat which may lead the device malfunction durability\nand reliability issues32–38. Moreover, there are several stud-\nies showing magnetization reversal induced by microwaves of\ntime-dependent frequency39–45. In the study39, the magnetiza-\ntion reversal, with the assistance of external field, is obtained\nby a radio-frequency microwave field pulse. Here, a dc ex-\nternal field acts as the main reversal force. In the study40,\nto obtain magnetization reversal, the applied microwave fre-\nquency needs to be the same as the resonance frequency, and\nin the studies42,43, optimal microwave forms are constructed.\nThese microwave forms are difficult to be realized in practice.\nThe study44reports magnetization reversal induced by the mi-\ncrowave pulse, but the pulse is applied such that the magne-\ntization just crosses over the energy barrier, i.e., only positive\na)Electronic mail: torikul@phy.ku.ac.bd\nb)Electronic mail: justicewxs@hnu.edu.cnfrequency range ( +f0to 0) is employed.\nA recent study45has demonstrated that the circularly polar-\nized linear down-chirp microwave pulse (DCMP) (whose fre-\nquency linearly decreases with time from the initial frequency\n+f0to\u0000f0) can drive fast magnetization reversal of uniax-\nial nanoparticles. The working principle of the above model\nis that the DCMP triggers stimulated microwave energy ab-\nsorption (emission) by (from) the magnetization before (after)\ncrossing the energy barrier. However, the efficiency of trig-\ngered microwave energy absorption or emission depends on\nhow closely the frequency of chirp microwave pulse matches\nthe magnetization precession frequency. In DCMP-driven\ncase, the frequency linearly decreases from f0to\u0000f0with\ntime but, in fact, the decrement of magnetization precession\nfrequency is not linear46,47during magnetization reversal . So\nthe frequency of DCMP only roughly matches the magneti-\nzation precession frequency. Thus, the DCMP triggers in-\nefficient energy absorption or emission and the required mi-\ncrowave amplitude is still large.\nTherefore, to achieve more efficient magnetization reversal,\nwe need to find a microwave pulse of proper time-dependent\nfrequency that matches the intrinsic magnetization precession\nfrequency better. In this study, we demonstrate that a cosine\nchirp microwave pulse (CCMP), defined as a microwave pulse\nwhose frequency sweeps in a cosine function with time from\n+f0to\u0000f0in first half-period of the microwave pulse, is ca-\npable of driving the fast and energy efficient magnetization\nreversal. This is because the frequency change of the CCMP\nmatches the nonlinear frequency change of magnetization pre-\ncession better than the DCMP. In addition, this study empha-\nsizes how the shape anisotropy influences the required param-\neters of CCMP and how the Gilbert damping affects the mag-\nnetization reversal. We find that the increase of easy-plane\nshape anisotropy makes the magnetization reversal easier. The\nmaterials with larger damping are better for fast magnetization\nreversal. These investigations might be useful in device appli-\ncations.arXiv:2102.10394v2  [cond-mat.mes-hall]  15 Sep 20212\nmf0\n0-f(b) (a)\nτ t\nFIG. 1. (a) Schematic diagram of the system in which mrepresents\na unit vector of the magnetization. A circularly polarized cosine\n(nonlinear) chirp microwave pulse is applied onto the single domain\nnanoparticle. (b) The frequency sweeping (from +f0to\u0000f0) of a\ncosine chirp microwave pulse.\nII. ANALYTICAL MODEL AND METHOD\nWe consider a square magnetic nanoparticle of area Sand\nthickness dwhose uniaxial easy-axis anisotropy directed in\nthez-axis as shown in FIG. 1(a). The size of the nanopar-\nticle is chosen so that the magnetization is considered as a\nmacrospin represented by the unit vector mwith the mag-\nnetic moment SdM s, where Msis the saturation magnetization\nof the material. The demagnetization field can be approxi-\nmated by a easy-plane shape anisotropy. The shape anisotropy\nfield coefficient is hshape=\u0000m0(Nz\u0000Nx)Ms, and the shape\nanisotropy field is hshape=hshapemzˆz, where NzandNxare\ndemagnetization factors48,49andm0=4p\u000210\u00007N=A2is the\nvacuum magnetic permeability. The strong uniaxial mag-\nnetocrystalline anisotropy hani=hanimzˆzdominates the total\nanisotropy so that the magnetization of the nanoparticle has\ntwo stable states, i.e., mparallel to ˆzand\u0000ˆz.\nThe magnetization dynamics min the presence of circularly\npolarized CCMP is governed by the Landau-Lifshitz-Gilbert\n(LLG) equation50\ndm\ndt=\u0000gm\u0002heff+am\u0002dm\ndt; (1)\nwhere aandgare the dimensionless Gilbert damping con-\nstant and the gyromagnetic ratio, respectively, and heffis the\ntotal effective field which includes the microwave magnetic\nfieldhmw, and the effective anisotropy field hkalong zdirec-\ntion (note that although we consider small nanoparticles that\ncan be treated as macrospins approximately, we still perform\nfull micromagnetic simulations with small meshes and full de-\nmagnetization field to be more accurate).\nThe effective anisotropy field can be expressed in terms of\nuniaxial anisotropy haniand shape anisotropy hshape ashk=\nhani+hshape= [hani\u0000m0(Nz\u0000Nx)Ms]mzˆz. Thus, the resonant\nfrequency of the nanoparticle is obtained from the well-known\nKittel formula\nf0=g\n2p[hani\u0000m0(Nz\u0000Nx)Ms]: (2).\nFor microwave field-driven magnetization reversal from the\nLLG equation, the rate of energy change is expressed as\ndE\ndt=\u0000agjm\u0002heffj2\u0000m\u0001dhmw\ndt: (3)\nThe first term is always negative since damping ais posi-\ntive. The second term can be either positive or negative for\ntime-dependent external microwave field. Therefore, the mi-\ncrowave field pulse can trigger the stimulated energy absorp-\ntion or emission, depending on the angle between the instan-\ntaneous magnetization manddhmw\ndt.\nInitially, because of easy-axis anisotropy, the magnetiza-\ntion prefers to stay in one of the two stable states, \u0006ˆz, cor-\nresponding to two energy minima. The objective of magne-\ntization reversal is to get the magnetization from one stable\nstate to the other. Along the reversal process, the magnetiza-\ntion requires to overcome an energy barrier at mz=0 which\nseparates two stable states. For fast magnetization reversal,\nthe external field is required to supply the necessary energy\nto the magnetization until crossing the energy barrier and, af-\nter crossing the energy barrier, the magnetization releases en-\nergy through damping and =or the external field is required\nto extract (by negative work done) energy from the magne-\ntization. It is mentioned that there is as intrinsic anisotropy\nfieldhanidue to the anisotropy which induces a magnetization\nnatural =resonant frequency proportional to mz. When mag-\nnetization goes from one stable state to another, the magneti-\nzation resonant frequency decreases while the magnetization\nclimbs up and becomes zero momentarily while crossing the\nenergy barrier and then increases with the opposite preces-\nsion direction while it goes down from the barrier. In princi-\nple, for fast and energy-efficient reversal, one requires a chirp\nmicrowave pulse whose frequency always matches the mag-\nnetization precession frequency to ensure the term m\u0001˙hmw\nto be maximal (minimal) before (after) crossing the energy\nbarrier. The study45, employs the DCMP (whose frequency\nlinearly decreases with time) to match the magnetization pre-\ncession frequency roughly. In fact, during the reversal from\nmz= +1 to mz=\u00001 , the decreasing of the resonant fre-\nquency ( while the spin climbs up the energy barrier) and in-\ncreasing of the resonant frequency (while it goes down from\nthe barrier) are not linear46,47. This leads us to consider a\ncosine chirp microwave pulse (CCMP) (a microwave pulse\nwhose frequency decreases non-linearly with time) in order\nto match the change of magnetization precession frequency\nclosely. Thus the CCMP might trigger more efficient stim-\nulated microwave absorptions (emissions) by (from) magne-\ntization before (after) the spin crosses the energy barrier to\ninduce fast and energy-efficient magnetization reversal.\nIn order to substantiate the above mentioned prediction,\nwe apply a circularly polarized cosine down-chirp microwave\npulse in the xyplane of the nanoparticle and solve the LLG\nequation numerically using the MUMAX3 package51. The\ncosine chirp microwave pulse (CCMP) takes the form hmw=\nhmw[cosf(t)ˆx+sinf(t)ˆy], where hmwis the amplitude of the\nmicrowave field and f(t)is the phase. Since the phase f(t)is\n2pf0cos(2pRt)t, where R(in units of GHz) is the controlling3\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s49/s48/s49\n/s46\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s49/s48/s45/s53/s48/s53/s49/s48/s101 /s32/s40/s49/s48/s49/s53\n/s32/s74/s47/s115/s32/s109/s51\n/s41(a) (b)\n(c)\nFIG. 2. Model parameters of nanoparticle of Ms= 106A/m, Hk=\n0:75 T, g= 1:76\u00021011rad/(T\u0001s), and a=0:01. (a) Temporal evo-\nlutions of mzofV= (8\u00028\u00028)nm3driven by the CCMP (with the\nminimal hmw=0:035 T, f0=18:8 GHz and R= 0.32 GHz) (red line)\nand DCMP (with hmw=0:035 T, f0=18:8 GHz and R= 1.53 GHz\n) (blue line). For CCMP case, (b) the corresponding magnetization\nreversal trajectories and (c) temporal evolutions of mz(red lines) and\nthe energy changing rate ˙eof the magnetization against time (blue\nlines).\nparameter, the instantaneous frequency f(t)of CCMP is ob-\ntained as f(t) =1\n2pdf\ndt=f0[cos(2pRt)\u0000(2pRt)sin(2pRt)]\nwhich decreases with time from f0to final\u0000f0at a time\ndependent chirp rate h(t)(in units of ns\u00002) as shown\nin FIG. 1(b). The chirp rate takes the form h(t) =\n\u0000f0h\n(4pR)sin(2pRt)+(2pR)2tcos(2pRt)i\n.\nAccording to the applied CCMP, the second term of right\nhand side of Eq. (3), i.e., the energy changing rate can be\nexpressed as\n˙e=\u0000Hmwsinq(t)sinF(t)\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\n(4)\nwhereF(t)is the angle between mt(the in-plane component\nofmandhmw. Therefore, the microwave field pulse can trig-\nger the stimulated energy absorption (before crossing the en-\nergy barrier) with\u0000F(t)and emission (after crossing the en-\nergy barrier) with F(t).\nThe material parameters of this study are chosen from typ-\nical experiments on microwave-driven magnetization reversal\nasMs=106A=m,hani=0:75 T, g=1:76\u00021011rad=(T\u0001s),\nexchange constant A=13\u000210\u000012J=m and a=0:01. Al-\nthough, the strategy and other findings of this study would\nwork for other materials also. The cell size (2\u00022\u00022)nm3\nis used in this study. We consider the switching time win-\ndow 1 ns at which the magnetization switches/reverses to\nmz=\u00000:9.\nIII. NUMERICAL RESULTS\nWe first investigate the possibility of reversing the magne-\ntization of cubic sample (8\u00028\u00028)nm3by the cosine chirpmicrowave pulse (CCMP). Accordingly, we apply the CCMP\nwith the microwave amplitude hmw=0:045 T, initial fre-\nquency f0=21 GHz and R=1:6 ns\u00001which are same as\nestimated in the study45), to the sample and found that CCMP\ncan drive the fast magnetization reversal. Then, we search the\nminimal hmw,f0, and Rof the CCMP such that the fast re-\nversal is still valid. Interestingly, the CCMP with significantly\nsmaller parameters i.e., hmw=0:035 T, f0=18:8 GHz and\noptimal R=0:32 ns\u00001, is capable of reversing the magnetiza-\ntion efficiently shown by red line in FIG. 2(a).\nThen we intend to show how efficient the CCMP driven\nmagnetization reversal compare to DCMP driven case. For\nfair comparison, we choose the same pulse duration t(as\nshown in Fig. 1, tis the time at which the frequency changes\nfrom +f0to\u0000f0). For the CCMP, we solve cos (2pRt)\u0000\n(2pRt)sin(2pRt) =\u00001, which gives the relation t=1:307\n2pR.\nBut, in case of DCMP, we know that t=2f0\nhand for f0=\n18:8 GHz, the chirp rate becomes h=57:86 ns\u00002and hence\nfind the parameter R(=1=t)= 1.53 ns\u00001. Then we apply the\nDCMP with the hmw=0:035 T, f0=18:8 GHz (which are\nsame as CCMP-driven case), and R=1.53 ns\u00001and found that\nthe magnetization only precesses around the initial state, i.e.,\nthe DCMP is not able to reverse the magnetization as shown\nby blue line in FIG. 2(a). So, it is mentioned that the CCMP\ncan reverse the magnetization with lower energy consumption\nwhich is the desired in device application. To be more explicit,\nthe trajectories of magnetization reversal induced by CCMP\nis shown FIG. 2(b) which shows the magnetization reverses\nswiftly.For further justification of CCMP driven reversal, we\ncalculate the energy changing rate dE=dtrefers to (4) by de-\ntermining the angles F(t)andq(t)and plotted with time in\nFIG. 2(c). The stimulated energy absorption (emission) peaks\nare obtained before (after) crossing the energy barrier as ex-\npected for faster magnetization reversal. This is happened be-\ncause the frequency of the CCMP closely matches the fre-\nquency of magnetization precession frequency, i.e., before\ncrossing the energy barrier, the F(t)remains around\u000090\u000eand\nafter crossing the energy barrier F(t)around 90\u000eto maximize\nthe energy absorption and emission respectively.\nThen, this study emphasizes how shape-anisotropy field\nhshape affects the magnetization switching time, microwave\namplitude hmwand initial frequency f0of CCMP. Since de-\nmagnetization field or shape-anisotropy field hshape should\nhave significant effect on magnetization reversal process as\nit opposes the magnetocrystalline anisotropy field haniwhich\nstabilizes the magnetization along two stable states. Accord-\ningly, to induce the hshape in the sample, we choose square\ncuboid shape samples and, to increase the strength of hshape,\nthe cross-sectional area S=xyis enlarged gradually for the\nfixed thickness d(=z) =8 nm. Specifically, we focus on\nthe samples of S1=10\u000210,S2=12\u000212,S3=14\u000214,\nS4=16\u000216,S5=18\u000218,S6=20\u000220 and S7=22\u000222 nm2\nwith d=8 nm. For the samples of different S, by determining\nthe analytic demagnetization factors NzandNx48,52, the shape\nanisotropy field hshape=m0(Nz\u0000Nx)Msmzˆzare determined.\nThehshape actually opposes the anisotropy field haniand\nhence resonance frequencies f0=g\n2p[hani\u0000m0(Nz\u0000Nx)Ms]4\nTABLE I. Shape anisotropy coefficient, resonant frequency f0, simulated frequency, f0and frequency-band\nCross\u0000sectional area Shape anisotropy coefficient Resonant frequency, Simulated minimal frequency, Simulated frequency-band\nS(nm2) hshape (T) f0(GHz) f0(GHz) (GHz)\nS1 0.09606 18.3 17.8 17.8 \u000019.9\nS2 0.17718 16 14.9 14.9 \u000018.3\nS3 0.2465 14.1 13.7 13.7 \u000015.4\nS4 0.3064 12.4 12.2 12.2 \u000013.8\nS5 0.3588 11 10.7 10.7 \u000012.3\nS6 0.4049 9.6 8.8 8.8 \u000011.5\nS7 0.4459 8.5 7.7 7.4 \u00008.7\ndecreases as shown in the Table I. Then, for the samples of\ndifferent S, with the fixed hmw=0:035 T, the corresponding\nminimal f0and optimal Rof CCMP are determined through\nthe study of the magnetization reversal. The temporal evolu-\ntions of mzfor different Sare shown in FIG. 3(a) and found\nthat for S7\u001522\u000222 nm2, the magnetization smoothly reveres\nwith the shortest time. The switching time tsas a function of\nthe coefficient hshape (corresponding to S) is plotted in FIG.\n3(c). It is observed that, with the increase of hshape orS, the\ntsshows slightly increasing trend but for S7(22\u000222)nm2or\nhshape=0:4459 T , tsdrops to 0.43 ns. For further increment\nofSorhshape,tsremains constant around 0.43 ns which is\nclose to the theoretical limit (0.4 ns) refers to the study37.\nThis is because the hshape reduces the effective anisotropy\nand thus reduces the height of energy barrier (energy differ-\nence between the initial state and saddle point) which is shown\nin the FIG. 3(b). Therefore, after decreasing certain height\nof energy barrier, the magnetization reversal becomes fastest\neven with the same filed amplitude hmw=0:035 T. Due to the\nreduction of height of the energy barrier with the hshape, one\ncan expect that the hmwand f0should also decrease with the\nincrease of the anisotropy coefficient hshape and theses find-\nings are presented subsequently.\nHere we present the effect of the shape anisotropy hshape\non the microwave amplitude hmwand initial frequency f0of\nCCMP. Purposely, for each sample Sorhshape, we numerically\ndetermine (by tuning hmw,f0and optimal R) the minimally re-\nquired hmwandf0of CCMP for the the magnetization reversal\ntime window 1 ns. Interestingly, we find that the fast and effi-\ncient reversal is valid for a wide range of initial frequency f0.\nFIG. 4(a) shows the estimated frequency bands of f0by verti-\ncal dashed lines for different S( for example, f0=19:9\u001817:8\nforhshape=0:096 T) for time window 1 ns. So there is a great\nflexibility in choosing the initial frequency f0which is useful\nin device application. FIG. 3(d) shows how the minimal f0\n(red circles) and hmw(blue square) decrease with the increase\nofhshape. The decay of f0is expected as hshape reduces ef-\nfective anisotropy (refers to Eq.(2)). For more justification, in\nsame FIG. 3(d), theoretically resonant frequency (black solid\nline) and simulated minimal frequency (red circles) as func-\ntion of hshape indicate the agreement. The minimal frequency\nf0always smaller than the theoretical =resonant frequency.\nMoreover, the decreasing trend of hmwwith hshape can be\nattributed by the same reason as the height of the energy bar-\n1\n2\n3\n4\n5\n6\n72\n4\n6\n70\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s53/s49/s48/s49/s53/s50/s48/s32/s102\n/s111\n/s32/s102\n/s111/s32/s40/s84/s104/s101/s111/s114/s101/s116/s105/s99/s97/s108/s41\n/s32/s72\n/s109/s119\n/s104\n/s115/s104/s97/s112/s101/s32/s40/s84 /s41/s102\n/s111/s32/s40/s71/s72/s122/s41\n/s48/s46/s48/s50/s48/s48/s46/s48/s50/s53/s48/s46/s48/s51/s48/s104\n/s109/s119/s32/s40/s84/s41(a) (b)\n(c) (d)\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s52/s53/s54/s55/s56/s57/s116\n/s115/s32/s40/s110/s115/s41\n/s104\n/s115/s104/s97/s112/s101/s32/s40/s84 /s41FIG. 3. (a) Temporal evolution of mzinduced by CCMP (with hmw=\n0:035 T fixed) for different cross-sectional area, S. (b) The energy\nlandscape Ealong the line f=0. The symbols i and s represent the\ninitial state and saddle point. (c) tsas a function of hshape. (d) The\nminimal f0(red dotted) and hmw(blue square) as a function of hshape\nwhile switching time window 1 ns.\n0\n1\n32\n4\n5\n760\n1\n2\n3(a) (b)\nFIG. 4. (a) Minimal switching time tsas a function of estimated fo\nof CCMP with fixed hmwandRcorresponding to different S. (b)\nMinimal tsas a function of Gilbert damping afor different S.\nrier decreases with hshape (refers to FIG. 3(b)). For the larger\nhshape or lower height of energy barrier, the smaller microwave\nfield hmwcan induce the magnetization reversal swiftly.\nThe Gilbert damping parameter, ahas also a crucial effect\non the magnetization magnetization dynamics and hence re-\nversal process and reversal time53–55. In case of CCMP-driven5\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121 /s109\n/s120/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120(a) (b)\nFIG. 5. Magnetization reversal trajectories of biaxial shape (10\u0002\n10\u00028)nm3driven by CCMP for (a) a=0:010. (b) a=0:045.\nmagnetization reversal, smaller (larger) ais preferred while\nthe magnetization climbing the energy barrier (the magnetiza-\ntion goes down to stable states). Therefore, it is meaningful to\nfind the optimal afor samples of different Sat which the re-\nversal is fastest. For fixed hmw=0:035 T, using the optimal f0\nandRcorresponding to S0,S1,S2andS3, we study the CCMP-\ndriven magnetization reversal as a function of afor different\nS. FIG. 4(b) shows the dependence of switching time on the\nGilbert damping for different S. For each S, there is certain\nvalue or range of afor which the switching time is minimal.\nFor instance, the switching time is lowest at a=0:045 for\nthe sample of S1=10\u000210 nm2. To be more clear, one may\nlook at FIG. 5(a) and FIG. 5(b) which show the trajectories\nof magnetization reversal for a=0:01 and a=0:045 respec-\ntively and observed that for a=0:045, the reversal path is\nshorter. This is because, after crossing over the energy bar-\nrier, the larger damping dissipates the magnetization energy\npromptly and thus it leads to faster magnetization reversal.\nThis finding suggests that larger ashows faster magnetiza-\ntion reversal.\nIV. DISCUSSIONS AND CONCLUSIONS\nThis study investigates the CCMP-driven magnetization re-\nversal of a cubic sample at zero temperature limit and found\nthat the CCMP with significantly smaller hmw=0:035 T,\nf0=18:8 GHz and R=0:32 ns\u00001than that of DCMP (i.e.,\nhmw=0:045 T, f0=21 GHz and R= 1.6 ns\u00001) can drive fast\nmagnetization reversal. Since the frequency change of CCMP\nclosely matches the magnetization precession frequency and\nthus it leads fast magnetization reversal with lower energy-\ncost. Then we study the influence of demagnetization field =\nshape anisotropy field hshape, on magnetization reversal pro-\ncess and the optimal parameters of CCMP. Interestingly we\nfind that, with the increase of hshape, the parameters hmwand\nf0of CCMP decreases with increasing hshape.\nSo, we search a set of minimal parameters of CCMP for\nthe fast ( ts\u00181 ns) magnetization reversal of the sample 22 \u0002\n22\u00028 nm3, and estimated as hmw=0:03 T, f0=7:7 GHz\nandR=0:24 ns\u00001which are (significantly smaller than that\nof DCMP) useful for device application. This is happened\nbecause with increase of hshape, the effective anisotropy field\ndecreases and thus the energy barrier (which separates two\nstable states) decreases. In addition, it is observed that thematerials with the larger damping are better for fast magneti-\nzation reversal. There is a recent study56reported that ther-\nmal effect assists the magnetization reversal i.e., thermal ef-\nfect reduces the controlling parameters of chirp microwave\nfield pulse. Thus, it is expected that the parameters of CCMP\nalso might be reduced further at room temperature. To gener-\nate such a cosine down-chirp microwave pulse, several recent\ntechnologies46,47are available. Therefore, the strategy of the\ncosine chirp microwave chirp driven magnetization reversal\nand other findings may lead to realize the fast and low-cost\nmemory device.\nACKNOWLEDGMENTS\nThis work was supported by the Ministry of Education\n(BANBEIS, Grant No. SD2019972). X. S. W. acknowledges\nthe support from the Natural Science Foundation of China\n(NSFC) (Grant No. 11804045) and the Fundamental Research\nFunds for the Central Universities.\nAppendix A: Calculation of ˙e\nIn this Appendix, we show the details of the derivation of ˙e\nin Eq. 4. The rate of change of hmwis\n˙hmw=dhmw\ndt\n=d\ndt(hmw[cosf(t)ˆx+sinf(t)ˆy])\n=hmw[\u0000sinf(t)ˆx+cosf(t)ˆy]df\ndt\n=hmw[\u0000sinf(t)ˆx+cosf(t)ˆy]\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\nThe magnetization is given by\nm=mxˆx+myˆy\n=sinq(t)cosfm(t)ˆx+sinq(t)sinfm(t)ˆy\nwhere q(t)is the polar angle and fm(t)is the azimuthal angle\nof the magnetization m.\nSubstituting mxand˙hmwin Eq. 3, we get,\n˙e=\u0000m\u0001˙hmw\n=hmwsinq(t)[\u0000sinf(t)cosfm(t)+cosf(t)sinfm(t)]\u0001\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\n=hmwsinq(t)sin(f(t)\u0000fm(t))\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\nDefining F(t) =fm(t)\u0000f(t), we have finally\n˙e=hmwsinq(t)sinF(t)\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\nwhereh\nf(t)\nt\u0000d\ndt\u0010\nf(t)\nt\u0011\nti\nrepresents w(t).6\nREFERENCES\n1S. Sun, C. B. Murray, D. Weller, L. Folks, and A. 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The stabili ty estimate consists of the Lipschitz type\ndata discrepancy and the high frequency tail of the source fu nction, where the latter decreases as\nthe upper bound of the frequency increases. The stability al so shows exponential dependence on the\nconstant damping coefficient. The analysis employs Carleman estimates and time decay estimates\nfor the damped plate wave equation to obtain an exact observa bility bound and depends on the\nstudy of the resonance-free region and an upper bound of the r esolvent of the biharmonic operator\nwith respect to the complex wavenumber.\n1.Introduction\nConsider the damped biharmonic plate equation in three dime nsions\n∆2upx,kq ´k2upx,kq ´ikσupx,kq “fpxq, x PR3, (1.1)\nwhereką0 is the wavenumber, σą0 is the damping coefficient, and fPL2pR3qis a assumed to be\na real-valued function with a compact support contained in BR“ txPR3:|x| ăRu, whereRą0 is\na constant. Let BBRbe the boundary of BR. Since the problem is formulated in the open domain,\nthe Sommerfeld radiation condition is imposed usually on uand ∆uto ensure the well-posedness of\nthe problem [17]. This paper is concerned with the inverse so urce problem of determining ffrom\nthe boundary measurements\nupx,kq,∇upx,kq,∆upx,kq,∇∆upx,kq, x P BBR\ncorresponding to the wavenumber kgiven in a finite interval.\nIn general, there is no uniqueness for the inverse source pro blems of the wave equations at a fixed\nfrequency [2,12]. Computationally, a more serious issue is the lack of stability, i.e., a small variation\nof the data might lead to a huge error in the reconstruction. H ence it is crucial to examine the\nstability of the inverse source problems. In [2], the author s initialized the study of the inverse source\nproblem for the Helmholtz equation by using multi-frequenc y data. Since then, it has become an\nactive research topic on the inverse source problems via mul tiple frequency data in order to over-\ncome the non-uniqueness issue and enhance the stability. Th e increasing stability was investigated\nfor the inverse source problems of various wave equations wh ich include the acoustic, elastic, and\nelectromagnetic wave equations [3–6,13,14] and the Helmho ltz equation with attenuation [8]. On\nthe other hand, it has generated sustained interest in the ma thematics community on the boundary\nvalue problems for higher-order elliptic operators [7]. Th e biharmonic operator, which can be en-\ncountered in models originating from elasticity for exampl e, appears as a natural candidate for such\na study [15,16]. Compared with the equations involving the s econd order differential operators, the\nmodel equations with the biharmonic operators are much less studied in the community of inverse\nproblems. We refer to [1,9–11,17] and the references cited t herein on the recovery of the lower-order\ncoefficients by using either the far-field pattern or the Diric hlet-to-Neumann map on the boundary.\n2000Mathematics Subject Classification. 35R30, 31B30.\nKey words and phrases. inverse source problem, the biharmonic operator, the dampe d biharmonic plate equation,\nstability.\n12 P. LI, X. YAO, AND Y. ZHAO\nIn a recent paper [12], the authors demonstrated the increas ing stability for the inverse source prob-\nlem of the biharmonic operator with a zeroth order perturbat ion by using multi-frequency near-field\ndata. The main ingredient of the analysis relies on the study of an eigenvalue problem for the bi-\nharmonic operator with the hinged boundary conditions. But the method is not applicable directly\nto handle the biharmonic operator with a damping coefficient.\nMotivated by [4,8], we use the Fourier transform in time to re duce the inverse source problem\ninto the identification of the initial data for the initial va lue problem of the damped biharmonic\nplate wave equation by lateral Cauchy data. The Carleman est imate is utilized to obtain an exact\nobservability bound for the source function in the framewor k of the initial value problem for the\ncorresponding wave equation, which connects the scatterin g data and the unknown source function\nby taking the inverse Fourier transform. An appropriate rat e of time decay for the damped plate\nwave equation is proved in order to justify the Fourier trans form. Then applying the results in [12]\non the resolvent of the biharmonic operator, we obtain a reso nance-free region of the data with\nrespect to the complex wavenumber and the bound of the analyt ic continuation of the data from the\ngiven data to the higher wavenumber data. By studying the dep endence of analytic continuation\nand of the exact observability bound for the damped plate wav e equation on the damping coefficient,\nwe show the exponential dependence of increasing stability on the damping constant. The stability\nestimate consists of the Lipschitz type of data discrepancy and the high wavenumber tail of the\nsource function. The latter decreases as the wavenumber of t he data increases, which implies that\nthe inverse problem is more stable when the higher wavenumbe r data is used. But the stability\ndeteriorates as the damping constant becomes larger. It sho uld be pointed out that due to the\nexistence of the damping coefficient, we can not obtain a secto rial resonance-free region for the data\nas that in [4,13]. Instead, we choose a rectangular resonanc e-free region as that in [14], which leads\nto a double logarithmic type of the high wavenumber tail for t he estimate.\nThis paper is organized as follows. In section 2, the direct s ource problem is discussed; the\nresolvent is introduced for the elliptic operator, and its r esonance-free region and upper bound are\nobtained. Section 3 is devoted to the stability analysis of t he inverse source problem by using multi-\nfrequency data. In appendix A, we usethe Carleman estimate t o derive an exact observability bound\nwith exponential dependence on the damping coefficient. In ap pendix B, we prove an appropriate\nrate of time decay for the damped plate wave equation to justi fy the Fourier transform.\n2.The direct source problem\nIn this section, we discuss the solution of the direct source problem and study the resolvent of the\nbiharmonic operator with a damping coefficient.\nTheorem 2.1. LetfPL2pR3qwith a compact support. Then there exists a unique solution uof\nSchwartz distribution to (1.1)for every ką0. Moreover, the solution satisfies\n|upx,kq| ďCpk,fqe´cpk,σq|x|\nas|x| Ñ 8,whereCpk,fqandcpk,σqare positive constants depending on k,fandk,σ, respectively.\nProof.Taking the Fourier transform of upx,kqformally with respect to the spatial variable x, we\ndefine\nu˚px,kq “ż\nR3eix¨ξ1\n|ξ|4´k2´ikσˆfpξqdξ, x PR3,\nwhere\nˆfpξq “1\np2πq3ż\nR3fpxqe´ix¨ξdx.\nIt follows from the Plancherel theorem that for each ką0 we have that u˚p¨,kq PH4pR3qand\nsatisfies the equation (1.1) in the sense of Schwartz distrib ution.AN INVERSE SOURCE PROBLEM 3\nDenote\nGpx,kq “ż\nR3eix¨ξ1\n|ξ|4´k2´ikσdξ.\nBy a direct calculation we can write u˚px,kqas\nu˚px,kq “ pG˚fqpxq “1\n2κ2ż\nR3´eiκ|x´y|\n4π|x´y|´e´κ|x´y|\n4π|x´y|¯\nfpyqdy, (2.1)\nwhereκ“ pk2`ikσq1\n4such that ℜκą0 andℑκą0. Since fhas a compact support, we obtain\nfrom (2.1) that the solution u˚px,kqsatisfies the estimate\n|u˚px,kq| ďCpk,fqe´cpk,σq|x|\nas|x| Ñ 8, whereCpk,fqandcpk,σqare positive constants dependingon k,fandk,σ, respectively.\nBydirectcalculations, wemayalsoshowthat ∇u˚and∆u˚havesimilarexponential decayestimates.\nNext is show the uniqueness. Let ˜ u˚px,kqbe another Schwartz distributional solution to (1.1).\nClearly we have\np∆2´k2´ikσqpu˚´˜u˚q “0.\nTaking the Fourier transform on both sides of the above equat ion yields\np|ξ|4´k2´ikσqp{u˚´˜˚uqpξq “0.\nNotice that for ką0 we have |ξ|4´k2´ikσ‰0 for all ξPR3. Taking the generalized inverse\nFourier transform gives u˚´˜u˚“0, which proves the uniqueness. /square\nTo study the resolvent we let\nu˚px,κq:“upx,kq, κ “ pk2`ikσq1\n4,\nwhereℜκą0 andℑκą0. By (1.1), u˚satisfies\n∆2u˚´κ4u˚“f.\nDenote by R“ tzPC:pδ,`8q ˆ p´ d,dquthe infinite rectangular slab, where δis any positive\nconstant and d!1. ForkPR, denote the resolvent\nRpkq:“ p∆2´k2´ikσq´1.\nThen we have Rpκq “ p∆2´κ4q´1. Hereafter, the notation aÀbstands for aďCb,whereCą0\nis a generic constant which may change step by step in the proo fs.\nLemma 2.2. For each kPRandρPC8\n0pBRqthe resolvent operator Rpkqis analytic and has the\nfollowing estimate:\n}ρRpkqρ}L2pBRqÑHjpBRqÀ |k|j\n2e2Rpσ`1q|k|1\n2, j “0,1,2,3,4.\nProof.It is clear to note that for a sufficiently small d, the set tpk2`ikσq1\n4:kPRubelongs to the\nfirst quadrant. Consequently, pk2`ikσq1\n4is analytic with respect to kPR. By [12, Theorem 2.1],\nthe resolvent Rpκqis analytic in Czt0uand the following estimate holds:\n}ρRpκqρ}L2pBRqÑHjpBRqÀ |κ|´2xκyjpe2Rpℑκq´`e2Rpℜκq´q, j “0,1,2,3,4,(2.2)\nwherex´:“maxt´x,0uand xκy “ p1` |κ|2q1{2. On the other hand, letting k“k1`ik2, we have\nfrom a direct calculation that\nk2`ikσ“k2\n1´k2\n2´k2σ` p2k1k2`k1σqi.\nIt is easy to see that if dis sufficiently small, which gives that |k2|is sufficiently small, there is a\npositive lower bound for |k2`ikσ|withkPRand then |κ| ącfor some positive constant c. The\nproof is completed by replacing κwith pk2`ikσq1\n4in (2.2). /square4 P. LI, X. YAO, AND Y. ZHAO\n3.The inverse source problem\nIn this section, we address the inverse source problem of the damped biharmonic plate equation\nand present an increasing stability estimate by using multi -frequency scattering data.\nDenote\n}upx,kq}2\nBBR:“ż\nBBR´\npk4`k2q|upx,kq|2`k2|∇upx,kq|2\n` pk2`1q|∆upx,kq|2` |∇∆upx,kq|2¯\ndspxq.\nThe following lemma provides a relation between the unknown source function and the boundary\nmeasurements. Hereafter, by Remark B.3, we assume that fPHnpBRqwhereně4.\nLemma 3.1. Letube the solution to the direct scattering problem (1.1). Then\n}f}2\nL2pBRqÀ2eCσ2ż`8\n0}upx,kq}2\nBBRdk.\nProof.Consider the initial value problem for the damped biharmoni c plate wave equation\n#\nB2\ntUpx,tq `∆2Upx,tq `σBtUpx,tq “0, px,tq PBRˆ p0,`8q,\nUpx,0q “0,BtUpx,0q “fpxq, x PBR.(3.1)\nWe define Upx,tq “0 whentă0 and denote UTpx,tq “Upx,tqχr0,Tsptqand\nxUTpx,kq “żT\n0Upx,tqeiktdt.\nBy the decay estimate (B.2) we have that Upx,tq PL2\ntp0,`8qand lim TÑ8UTpx,tq “Upx,tqin\nL2\ntpRquniformly for all xPR3. It follows from the Plancherel Theorem that xUTalso converges in\nL2\nkpRqto a function u˚px,kq PL2\nkpRquniformly for all xPR3, which implies that u˚px,kqis the\nFourier transform of Upx,tq.\nDenoteby x¨,¨yandStheusualscalarinnerproductof L2pR3qandthespaceofSchwartzfunctions,\nrespectively. We take u˚px,kqas a Schwartz distribution such that u˚px,kqpϕq “ xu˚,ϕyfor each\nϕPS. In what follows, we show that u˚px,kqsatisfies the equation (1.1) in the sense of Schwartz\ndistribution.\nFirst we multiply both sides of the wave equation (3.1) by a Sc hwartz function ϕand take inte-\ngration over R3. Using the wave equation (3.1) and the integration by parts w ith respect to the t\nvariable over r0,TsforTą0, we obtain\n0“żT\n0xB2\ntU`∆2U`σBtU,ϕyeiktdt\n“eikTxBtUpx,Tq,ϕy ´ikeikTxUpx,Tq,ϕy `σeikTxUpx,Tq,ϕy\n´ xBtUpx,0q,ϕy `AżT\n0p∆2U´k2U´ikσU qeiktdt,ϕE\n. (3.2)\nItfollowsfromthedecay estimate(B.2)that |BtUpx,tq|,|Upx,tq| À p1`tq´3\n4uniformlyforall xPR3,\nwhich give\nlim\nTÑ8eikTxBtUpx,Tq,ϕy “lim\nTÑ8ikeikTxUpx,Tq,ϕy “lim\nTÑ8σeikTxUpx,Tq,ϕy “0.AN INVERSE SOURCE PROBLEM 5\nOn the other hand, we have from the integration by parts that\nAżT\n0p∆2U´k2U´ikσU qeiktdt,ϕE\n“AżT\n0Udt,∆2ϕE\n`AżT\n0p´k2U´ikσU qeiktdt,ϕE\n. (3.3)\nSince lim TÑ`8xUTpx,kq “u˚px,kqinL2\nkpRquniformly for xPR3, we can choose a positive sequence\ntTnu8\nn“1such that lim nÑ8Tn“ `8and lim nÑ8yUTnpx,kq “u˚px,kqpointwisely for a.e. kPRand\nuniformly for all xPR3. Define a sequence of Schwartz distributions tDnu8\nn“1ĂS1as follows\nDnpϕq:“ xyUTn,ϕy, ϕ PS.\nSince lim nÑ8yUTnpx,kq “u˚px,kqfor a.e.kPRand uniformly for all xPR3, we have\nlim\nnÑ8Dnpϕq “ xu˚,ϕy.\nConsequently, replacing TbyTnin (3.3) and letting nÑ 8, we get\nlim\nnÑ8´@żTn\n0Udt,∆2ϕD\n`@żTn\n0p´k2U´ikσU qeiktdt,ϕD¯\n“u˚p∆2ϕq ´k2u˚pϕq ´ikσu ˚pϕq\n“ p∆2´k2´ikσqu˚pϕq,\nwhich further implies by (3.2) that\np∆2´k2´ikσqu˚pϕq “ xf,ϕy\nfor every ϕPS. Then u˚px,kqis a solution to the equation (1.1) as a Schwartz distributio n.\nFurthermore, it follows from the uniqueness of the direct pr oblem that we obtain u˚px,kq “upx,kq,\nwhich gives that upx,kqis the Fourier transform of Upx,tq.\nBy Theorem B.1, we have the estimates\n|B2\ntU|,|BtU|,|Bt∇U|,|Bt∆U|,|∆U|,|∇∆U| À p1`tq´3\n4.\nMoreover, they are continuous and belong to L2\ntpRquniformly for all xPR3. Similarly, we may show\nthat\nyB2\ntU“ ´k2u,yBtU“iku, {Bt∇U“ik∇u,\n{Bt∆U“ik∆u,y∆U“∆u,{∇∆U“∇∆u.\nIt follows from Plancherel’s theorem thatż`8\n0´\n|B2\ntU|2` |BtU|2` |Bt∇U|2` |Bt∆U|2` |∆U|2` |∇∆U|2¯\ndt\n“ż`8\n´8´\n|k2u|2` |ku|2` |k∇u|2` |k∆u|2` |∆u|2` |∇∆u|2¯\ndk. (3.4)\nBy (3.4) and the exact observability bounds (A.1), we obtain\n}f}2\nL2pBRqÀeCσ2ż`8\n´8ż\nBBR´\npk4`k2q|upx,kq|2`k2|∇upx,kq|2\n` pk2`1q|∆upx,kq|2` |∇∆upx,kq|2¯\ndspxqdk\nÀeCσ2ż8\n´8}upx,kq}2\nBBRdk.6 P. LI, X. YAO, AND Y. ZHAO\nSincefpxqis real-valued, we have upx,kq “upx,´kqforkPRand then\nż8\n´8}upx,kq}2\nBBRdk“2ż8\n0}upx,kq}2\nBBRdk,\nwhich completes the proof. /square\nLetδbe a positive constant and define\nIpkq “żk\nδ}upx,ωq}2\nBBRdspxqdω.\nThe following lemma gives a link between the values of an anal ytical function for small and large\narguments (cf. [14, Lemma A.1]).\nLemma 3.2. Letppzqbe analytic in the infinite rectangular slab\nR“ tzPC:pδ,`8q ˆ p´ d,dqu,\nwhereδis a positive constant, and continuous in Rsatisfying#\n|ppzq| ďǫ1, z P pδ,Ks,\n|ppzq| ďM, z PR,\nwhereδ,K,ǫ1andMare positive constants. Then there exists a function µpzqwithzP pK,`8q\nsatisfying\nµpzq ě64ad\n3π2pa2`4d2qeπ\n2dpa\n2´zq,\nwherea“K´δ, such that\n|ppzq| ďMǫµpzq@zP pK,`8q.\nLemma 3.3. Letfbe a real-valued function and }f}L2pBRqďQ. Then there exist positive constants\ndandδ,Ksatisfying 0ăδăK, which do not depend on fandQ, such that\n|Ipkq| ÀQ2e4Rpσ`2qκǫ2µpkq\n1 @kP pK,`8q\nand\nǫ2\n1“żK\nδż\nBBR}upx,kq}2\nBBRdspxqdk, µ pkq ě64ad\n3π2pa2`4d2qeπ\n2dpa\n2´kq,\nwherea“K´δ.\nProof.Let\nI1pkq “żk\nδż\nBBR´\npω4`ω2qupx,ωqupx,´ωq `ω2∇upx,ωq ¨∇upx,´ωq\n` pω2`1q∆upx,ωq∆upx,´ωq `∇∆upx,ωq ¨∇∆upx,´ωq¯\ndspxqdω,\nwherekPR. Following similar arguments as those in the proof of Lemma 2 .2, we may show that\nRp´kqis also analytic for kPR. Sincefis real-valued, we have upx,kq “upx,´kqforkPR, which\ngives\nI1pkq “Ipkq, k ą0.\nIt follows from Lemma 2.2 that\n|I1pkq| ÀQ2eCσ2e4Rpσ`1q|k|, k PR,\nwhich gives\ne´4Rpσ`2q|k||I1pkq| ÀQ2eCσ2, k PR.AN INVERSE SOURCE PROBLEM 7\nAn application of Lemma 3.2 leads to\nˇˇe´4Rpσ`2q|k|IpkqˇˇÀQ2ǫ2µpkq@kP pK,`8q,\nwhere\nµpkq ě64ad\n3π2pa2`4d2qeπ\n2dpa\n2´kq,\nwhich completes the proof. /square\nHere we state a simple uniqueness result for the inverse sour ce problem.\nTheorem 3.4. LetfPL2pBRqandIĂR`be an open interval. Then the source function fcan\nbe uniquely determined by the multi-frequency Cauchy data tupx,kq,∇upx,kq,∆upx,kq,∇∆upx,kq:\nxP BBR,kPIu.\nProof.Letupx,kq “∇upx,kq “∆upx,kq “∇∆upx,kq “0 for all xP BBRandkPI. It suffices to\nprove that fpxq “0. By Lemma 2.2, upx,kqis analytic in the infinite slab Rfor anyδą0, which\nimplies that upx,kq “∆upx,kq “0 for all kPR`. We conclude from Lemma 3.1 that f“0./square\nThe following result concerns the estimate of upx,kqfor high wavenumbers.\nLemma 3.5. LetfPHnpBRqand }f}HnpBRqďQ. Then the following estimate holds:\nż8\ns}upx,kq}2\nBBRdkÀ1\nsn´3}f}2\nHnpBRq.\nProof.Recall the identity\nż8\ns}upx,kq}2\nBBRdk“ż8\nsż\nBBR´\npk4`k2q|upx,kq|2`k2|∇upx,kq|2\n` pk2`1q|∆upx,kq|2` |∇∆upx,kq|2¯\ndspxqdk. (3.5)\nUsing the decomposition\nRpκq “ p∆2´κ4q´1“1\n2κ2“\np´∆´κ2q´1´ p´∆`κ2q´1‰\n,\nwe obtain\nupxq “ż\nBR1\n2κ2´eiκ|x´y|\n4π|x´y|´e´κ|x´y|\n4π|x´y|¯\nfpyqdy, x P BBR.\nFor instance, we consider one of the integrals on the right-h and side of (3.5)\nJ:“ż8\nsk4|upx,kq|dk\n“ż8\nsk4ˇˇˇż\nBR1\n2κ2´eiκ|x´y|\n4π|x´y|´e´κ|x´y|\n4π|x´y|¯\nfpyqdyˇˇˇ2\ndk.\nUsing the spherical coordinates r“ |x´y|originated at y, we have\nJ“1\n8πż8\nsż\nBBRk2ˇˇˇż2π\n0dθżπ\n0sinϕdϕż8\n0peiκr´e´κrqfrdrˇˇˇ2\ndspxqdk.\nBy the integration by parts and noting xP BBRand supp fĂBˆRĂBRfor some ˆRăR, we obtain\nJ“1\n4πż8\nsż\nBBRk2ˇˇˇż2π\n0dθżπ\n0sinϕdϕż2R\nR´ˆR´eiκr\npiκqn´e´κr\np´κqn¯Bnpfrq\nBrndrˇˇˇ2\ndspxqdk.8 P. LI, X. YAO, AND Y. ZHAO\nSincexP BBRand |κ| ěk1{2forką0, we get from direction calculations that\nJÀ }f}2\nHnpBRqż8\nsk2´ndkÀ1\nsn´3}f}2\nHnpBRq.\nTheotherintegrals ontheright-handsideof (3.5)can beest imated similarly. Thedetails areomitted\nfor brevity. /square\nDefine a real-valued function space\nCQ“ tfPHnpBRq:ně4,}f}HnpBRqďQ,suppfĂBˆRĂBR, f:BRÑRu,\nwhereˆRăR. Now we are in the position to present the main result of this p aper.\nTheorem 3.6. Letupx,κqbe the solution of the scattering problem (1.1)corresponding to the source\nfPCQ. Then for ǫsufficiently small, the following estimate holds:\n}f}2\nL2pBRqÀeCσ2´\nǫ2`Q2\nK1\n2pn´3qpln|lnǫ|q1\n2pn´3q¯\n, (3.6)\nwhere\nǫ:“żK\n0}upx,kq}2\nBBRdk“żδ\n0}upx,kq}2\nBBRdk`ǫ2\n1.\nProof.We can assume that ǫďe´1, otherwise the estimate is obvious.\nFirst, we link the data Ipkqfor large wavenumber ksatisfying kďLwith the given data ǫ1of\nsmall wavenumber by using the analytic continuation in Lemm a 3.3, where Lis some large positive\ninteger to be determined later. It follows from Lemma 3.3 tha t\nIpkq ÀQ2ec|κ|ǫµpκq\n1\nÀQ2exptcκ´c2a\na2`c3ec1pa\n2´κq|lnǫ1|u\nÀQ2expt´c2a\na2`c3ec1pa\n2´κq|lnǫ1|p1´c4κpa2`c3q\naec1pκ´a\n2q|lnǫ1|´1qu\nÀQ2expt´c2a\na2`c3ec1pa\n2´Lq|lnǫ1|p1´c4Lpa2`c3q\naec1pL´a\n2q|lnǫ1|´1qu\nÀQ2expt´b0e´c1L|lnǫ1|p1´b1Lec1L|lnǫ1|´1qu,\nwherec,ci,i“1,2 andb0,b1are constants. Let\nL“#”\n1\n2c1ln|lnǫ1|ı\n, k ď1\n2c1ln|lnǫ1|,\nk, k ą1\n2c1ln|lnǫ1|.\nIfKď1\n2c1ln|lnǫ1|, we obtain for sufficiently small ǫ1that\nIpkq ÀQ2expt´b0e´c1L|lnǫ1|p1´b1Lec1L|lnǫ1|´1qu\nÀQ2expt´1\n2b0e´c1L|lnǫ1|u.\nNotinge´xďp2n`3q!\nx2n`3forxą0, we have\nIpLq ÀQ2ep2n`3qc1L|lnǫ1|´p2n`3q.AN INVERSE SOURCE PROBLEM 9\nTakingL“1\n2c1ln|lnǫ1|, combining the above estimates, Lemma 3.1 and Lemma 3.5, we g et\n}f}2\nL2pBRqÀeCσ2´\nǫ2`IpLq `ż8\nLż\nBBR}upx,kq}2\nBBRdk¯\nÀeCσ2´\nǫ2`Q2ep2n`3qc1L|lnǫ1|´p2n`3q`Q2\nLn´3¯\nÀeCσ2´\nǫ2`Q2´\n|lnǫ1|2n`3\n2|lnǫ1|´p2n`3q` pln|lnǫ1|q3´n¯ ¯\nÀeCσ2´\nǫ2`Q2´\n|lnǫ1|´2n`3\n2` pln|lnǫ1|q3´n¯ ¯\nÀeCσ2´\nǫ2`Q2pln|lnǫ1|q3´n¯\nÀeCσ2´\nǫ2`Q2\nK1\n2pn´3qpln|lnǫ1|q1\n2pn´3q¯\nÀeCσ2´\nǫ2`Q2\nK1\n2pn´3qpln|lnǫ|q1\n2pn´3q¯\n,\nwhere we have used |lnǫ1|1{2ěln|lnǫ1|for sufficiently small ǫ1and ln |lnǫ1| ěln|lnǫ|.\nIfKą1\n2c1ln|lnǫ1|, we have from Lemma 3.5 that\n}f}2\nL2pBRqÀeCσ2´\nǫ2`ż8\nKż\nBBR}upx,kq}2\nBBRdk¯\nÀeCσ2´\nǫ2`Q2\nKn´3¯\nÀeCσ2´\nǫ2`Q2\nK1\n2pn´3qpln|lnǫ|q1\n2pn´3q¯\n,\nwhich completes the proof. /square\nIt can be observed that for a fixed damping coefficient σ, the stability (3.6) consists of two parts:\nthe data discrepancy and the high frequency tail. The former is of the Lipschitz type. The latter\ndecreases as Kincreases which makes the problem have an almost Lipschitz s tability. But the\nstability deteriorates exponentially as the damping coeffic ientσincreases.\nAppendix A.An exact observability bound\nConsider the initial value problem for the damped biharmoni c plate wave equation\n#\nB2\ntUpx,tq `∆2Upx,tq `σBtUpx,tq “0, px,tq PBRˆ p0,`8q,\nUpx,0q “0,BtUpx,0q “fpxq, x PBR.(A.1)\nThe following theorem presents an exact observability boun d for the above equation. The proof\nfollows closely from that in [8, Theorem 3.1].\nTheorem A.1. Let the observation time 4p2R`1q ăTă5p2R`1q. Then there exists a constant\nCdepending on the domain BRsuch that\n}f}2\nL2pBRqďCeCσ2`\n}B2\ntU}2\nL2pBBRˆp0,Tqq` }BtU}2\nL2pBBRˆp0,Tqq` }Bt∇U}2\nL2pBBRˆp0,Tqq\n` }Bt∆U}2\nL2pBBRˆp0,Tqq` }∆U}2\nL2pBBRˆp0,Tqq` }∇∆U}2\nL2pBBRˆp0,Tqq˘\n.(A.2)10 P. LI, X. YAO, AND Y. ZHAO\nBefore showing the proof, we introduce the energies\nEptq “1\n2ż\nΩ`\n|BtUpx,tq|2` |∆Upx,tq|2` |Upx,tq|2˘\ndx,\nE0ptq “1\n2ż\nΩ`\n|BtUpx,tq|2` |∆Upx,tq|2˘\ndx,\nand denote\nF2“ż\nBΩˆpt1,t2q`\n|B2\ntUpx,tq|2` |BtUpx,tq|2` |Bt∇Upx,tq|2\n` |Bt∆Upx,tq|2` |∆Upx,tq|2` |∇∆Upx,tq|2˘\ndspxqdt.\nLemma A.2. LetUbe a solution of the damped biharmonic plate wave equation (A.1)with the\ninitial value fPH1pBRq,suppf ĂBR. Let0ďt1ăt2ďTand1ď2σ. Then the following\nestimates holds:\nEpt2q ďe4pt2´t1q2p2Ept1q `F2q, (A.3)\nEpt2q ďep2σ`4pt2´t1qqpt2´t1qpEpt2q `F2q. (A.4)\nProof.Multiplying both sides of (A.1) by pBtUqeθtand integrating over Ω ˆ pt1,t2qgive\nż\nΩˆppt1,t2q´1\n2BtpBtUq2`∆2UBtU`σpBtUq2¯\neθtdxdt“0.\nUsing the integration ∆2UBtUby parts over Ω and noting ∆ UBtp∆Uq “1\n2Bt|∆U|2, we obtain\nżt2\nt1pBtE0ptqqeθtdt`ż\nΩˆpt1,t2qσpBtUq2eθtdxdt\n`ż\nBΩˆpt1,t2qpBνp∆UqBtU´∆UBtpBνUqqeθtdspxqdt“0.\nHence,\nE0pt2qeθt2´E0pt1qeθt1“ż\nΩˆpt1,t2q´θ\n2ppBtUq2` |∆U|2q ´σpBtUq2¯\neθtdxdt\n´ż\nBΩˆpt1,t2qpBνp∆UqBtU´∆UBtpBνUqqeθtdspxqdt“0.\nLettingθ“0, using Schwartz’s inequality, and noting σą0, we get\nE0pt2q ďE0pt1q `ż\nΩˆpt1,t2qp´σqpBtUq2dxdt\n`1\n2ż\nBΩˆpt1,t2q´\npBtUq2` pBtpBνUqq2¯\ndspxqdt\n`1\n2ż\nBΩˆpt1,t2q´\np∆Uq2` pBνp∆Uqq2¯\ndspxqdt\nďE0pt1q `F2.AN INVERSE SOURCE PROBLEM 11\nSimilarly, letting θ“2σ, we derive\nE0pt1qe2σt1ďE0pt2qe2σt2`ż\nΩˆpt2,t1q´σp∆Uq2dxdt\n`1\n2ż\nBΩˆpt1,t2q´\npBtUq2` pBtpBνUqq2¯\ne2σtdspxqdt\n`1\n2ż\nBΩˆpt1,t2q´\np∆Uq2` pBνp∆Uqq2¯\ne2σtdspxqdt\nďE0pt2qe2σt2`1\n2ż\nBΩˆpt1,t2q´\npBtUq2` pBtpBνUqq2¯\ne2σtdspxqdt\n`1\n2ż\nBΩˆpt1,t2q´\np∆Uq2` pBνp∆Uqq2¯\ne2σtdspxqdt.\nwhich gives\nE0pt1q ďe2σpt2´t1qpE0pt2q `F2q.\nThe proof is completed by following similar arguments as tho se in [8, Lemma 3.2]. /square\nNow we return to the proof of Theorem A.2\nProof of Theorem A.2. Letϕpx,tq “ |x´a|2´θ2pt´T\n2q2, where dist pa,Ωq “1,θ“1\n2. Using the\nCarleman-type estimate in [18], we obtain\nτ6ż\nQ|U|2e2τϕdxdt`τ3ż\nQ|BtU|2e2τϕdxdt`τż\nQ|∆U|2e2τϕdxdt\nÀż\nQppB2\nt`∆2qUq2e2τϕdxdt\n`ż\nBQτ6p|Bν∆U|2` |Bt∆U|2` |B2\ntUq|2qe2τϕdspxqdt. (A.5)\nIt is easy to see that 1 ´θ2ε2\n0ďϕon Ω ˆ t|t´T\n2| ăε0ufor some positive εă1. Then we have from\nA.4 that\nτ6ż\nQ|U|2e2τϕdxdt`τ3ż\nQ|BtU|2e2τϕdxdt`τż\nQ|∆U|2e2τϕdxdt\něτ6ż\nΩˆpT\n2´ε0,T\n2`ε0q|U|2e2τp1´θ2ε2\n0qdxdt`τ3ż\nΩˆpT\n2´ε0,T\n2`ε0q|BtU|2e2τp1´θ2ε2\n0qdxdt\n`τż\nΩˆpT\n2´ε0,T\n2`ε0q|∆U|2e2τp1´θ2ε2\n0qdxdt\něτe2τp1´θ2ε2\n0qż\nΩˆpT\n2´ε0,T\n2`ε0qEptqdt\něτe2τp1´θ2ε2\n0qε0p2e´p2σ`4TqTEp0q ´F2q. (A.6)\nMoreover, it follows from (A.4) and ϕď p2R`1q2´θ2T2{4 on Ω ˆ p0,Tqthat\nτ6ż\nQ|U|2e2τϕdxdt`τ3ż\nQ|BtU|2e2τϕdxdt`τż\nQ|∆U|2e2τϕdxdt\nďτ6e2τpp2R`1q2´θ2T2{4qpEp0q `EpTqq\nďτ6e2τpp2R`1q2´θ2T2{4qppe4T2`1qEp0q `e4T2F2q.12 P. LI, X. YAO, AND Y. ZHAO\nBy (A.5) and (A.6), we obtain\nτe2τp1´θ2ε2\n0qε0e´p2σ`1`4TqTEp0q\n`τ6ż\nQ|U|2e2τϕdxdt`τ3ż\nQ|BtU|2e2τϕdxdt`τż\nQ|∆U|2e2τϕdxdt\nď´\nσ2ż\nQ|BtU|2e2τϕdxdt`ż\nBQτ6p|Bν∆U|2` |Bt∆U|2` |B2\ntUq|2qe2τϕdspxqdt\n` pτe2τp1´θ2ε2\n0q`τ6e2τpp2R`1q2´θ2T2{4qe4T2qF2`τ6e2τpp2R`1q2´θ2T2{4qe4T2Ep0q¯\n.(A.7)\nChoosing τsufficiently large, we may remove the first integral on the righ t hand side of (A.7). We\nalso choose T2“4p2R`1q2\nθ2`4ε2\n0andτ“ p2σ`8TqT`lnp2pε0q´1Cq `Cσ2. Noting τ5e´τď5!, we\nhave\nτ5e2τpp2R`1q2´θ2T2{4´1`θ2ε2\n0q`p2σ`8TqT“τ5e´2τ`p2σ`8TqT\nď5!e´τ`p2σ`8TqTďε0\n2C.\nIn addition, since Tď5p2R`1q, it follows that\nτ5e2τpp2R`1q2´1`θ2ε2\n0`p2σ`4TqTqďτ5e2pp2σ`8TqT`Cσ2`Cqp2R`1q2`p2σ`4TqTďCeCσ2.\nUsing the above inequality and the inequality ϕă p2R`1q2onQand dividing both sides in (A.7)\nby the factor of Ep0qon the left hand side, we obtain\nEp0q ďCeCσ2F2.\nSincefis supported in Ω, there holds }U}L2pBBRˆp0,TqqďC}BtU}L2pBBRˆp0,Tqq, which completes the\nproof. /square\nAppendix B.A decay estimate\nWe prove a decay estimate for the solution of the initial valu e problem of the damped plate wave\nequation\n#\nB2\ntUpx,tq `∆2Upx,tq `σBtUpx,tq “0, px,tq PR3ˆ p0,`8q,\nUpx,0q “0,BtUpx,0q “fpxq, x PR3,(B.1)\nwherefpxq PL1pR3q XHspR3q. By the Fourier transform, the solution Upx,tqof (B.1) is given as\nUpx,tq “F´1pmσpt,ξqˆfpξqqpxq,\nwhereF´1denotes the inverse Fourier transform,\nmσpt,ξq “e´σ\n2t\na\nσ2´4|ξ|4´\ne1\n2t?\nσ2´4|ξ|4´e´1\n2t?\nσ2´4|ξ|4¯\n,\nandˆfpξqis the Fourier transform of f, i.e.,\nˆfpξq “1\np2πq3ż\nR3e´ix¨ξfpxqdx.\nLeta\nσ2´4|ξ|4“ia\n4|ξ|4´σ2when |ξ|4ąσ2\n4. Then we have\nmσpt,ξq “$\n’&\n’%e´σ\n2tsinh pt\n2?\nσ2´4|ξ|4q?\nσ2´4|ξ|4, |ξ|4ăσ2\n4,\ne´σ\n2tsinpt\n2?\n4|ξ|4´σ2q?\n4|ξ|4´σ2, |ξ|4ąσ2\n4.AN INVERSE SOURCE PROBLEM 13\nIt is clear to note from the representation of mσpt,ξqthat the solution Upx,tqdepends on both of\nthe low and high frequency of ξ. In fact, the solution Upx,tqbehaves as a “parabolic type” of e´t∆2f\nfor the low frequency part, while for the high frequency part it behaves like a “dispersive type” of\neit∆2f.\nTheorem B.1. LetUpx,tqbe the solution of (B.1). ThenUpx,tqsatisfies the decay estimate\nsupxPR3|Bα\nxBj\ntUpx,tq| À p1`tq´3`|α|\n4}f}L1pR3q`e´ct}f}HspR3q, (B.2)\nwherejPN,αis a multi-index vector in N3such that Bα“ Bα1x1Bα2x2Bα3x3,są2j` |α| ´1\n2andcą0\nis some positive constant. In particular, for |α| “s“0, the following estimate holds:\nsupxPR3|Upx,tq| À p1`tq´3\n4p}f}L1pR3q` }f}L2pR3qq. (B.3)\nRemark B.2. The estimate (B.3)provides a time decay of the order Opp1`tq´3\n4qforUpx,tq\nuniformly for all xPR3, which gives\nsup\nxPR3ż8\n0|Upx,tq|2dtÀż8\n0p1`tq´3{2dtă `8.\nHence, let Upx,tq “0whentă0, thenUpx,tqhas a Fourier transform ˆUpx,kq PL2pRqfor each\nxPR3. Moreover, the following Plancherel equality holds:\nż`8\n0|Upx,tq|2dt“ż`8\n´8|ˆUpx,kq|2dk.\nRemark B.3. To study the inverse source problem, it suffices to assume that fPH4pR3q. In this\ncase, it follows from the above theorem that both B2\ntUpx,tqand∆2Upx,tqare continuous functions.\nMoreover, we have from (B.2)that the following estimate holds:\nsupxPR3|Bj\ntUpx,tq| À p1`tq´3\n4}f}L1pR3q`e´ct}f}HspR3q, j “1,2,\nsupxPR3|Bα\nxUpx,tq| À p1`tq´3`|α|\n4}f}L1pR3q`e´ct}f}HspR3q,|α| ď4.\nProof.Without loss of generality, we may assume that σ“1, and then\nmσpt,ξq “e´1\n2t\na\n1´4|ξ|4´\ne1\n2t?\n1´4|ξ|4´e´1\n2t?\n1´4|ξ|4¯\n.\nFirst we prove (B.2) for j“0. Choose χPC8\n0pR3qsuch that supp χĂBp0,1\n2qandχpξq “1 for\n|ξ| ď1\n4. Let\nUpx,tq “F´1pmpt,ξqχpξqˆfq `F´1pmpt,ξqp1´χpξqqˆfq\n:“U1px,tq `U2px,tq.\nForU1px,tq, sincea\n1´4|ξ|4ď1´2|ξ|4when 0 ď |ξ| ď1\n2, we have for |ξ| ď1\n2that\nmpt,ξq “1a\n1´4|ξ|4e´t\n2p1˘?\n1´4|ξ|4qď2e´t|ξ|4, t ě0.\nFor each xPR3we have\nBαU1px,tq “ż\nR3eix¨ξpiξqαmpt,ξqχpξqˆfpξqdξ,\nwhich gives\nsup\nxPR3|Bα\nxU1px,tq| ďż\n|ξ|ď1\n2|ξ|αe´t|ξ|4|ˆfpξq|dξÀ }ˆf}L8pR3qż\n|ξ|ď1\n2|ξ|αe´t|ξ|4dξ.14 P. LI, X. YAO, AND Y. ZHAO\nSince\nż\n|ξ|ď1\n2|ξ|αe´t|ξ|4dξď#\nC, 0ďtď1,\nt´3`|α|\n4, t ě1,\nand }ˆf}L8pR3qď }f}L1pR3q, we obtain\nsup\nxPR3|Bα\nxU1px,tq| À p1`tq´3`|α|\n4|f}L1pR3q@αPN3. (B.4)\nTo estimate U2px,tq, noting\np1´∆qp\n2U2px,tq “ż\nR3eix¨ξp1` |ξ|2qp\n2mpt,ξqp1´χpξqqˆfpξqdξ,\nwe have from Plancherel’s theorem that\nż\nR3|p1´∆qp\n2U2px,tq|2dx“ż\nR3p1` |ξ|2qp|mpt,ξqp1´χpξqqˆfpξq|2dξ. (B.5)\nIt holds that\n|mpt,ξq| ď$\n’’’’&\n’’’’%te´t\n2p1´?\n1´4|ξ|4qˇˇˇ1´e´t?\n1´4|ξ|4\nt?\n1´4|ξ|4ˇˇˇÀe´t\n8,1\n2ă |ξ| ď?\n2\n2,\n1\n2e´t\n2sint\n2?\n4|ξ|4´1\nt\n2?\n4|ξ|4´1Àe´t\n8,?\n2\n2ă |ξ| ď1,\ne´t\n2?\n4|ξ|4´1|sint\n2a\n4|ξ|4´1| ďe´t\n2?\n4|ξ|4´1, |ξ| ą1.\nHence, when |ξ| ě1\n2we have\n|p1` |ξ|2qmpt,ξq| Àe´t\n8.\nIt follows from (B.5) that\n}U2px,tq}2\nHppR3qďż\n|ξ|ě1\n2|p1` |ξ|2qp\n2mpt,ξqˆfpξq|2dξ\nďe´t\n4ż\nR3|p1` |ξ|2q´1`p\n2ˆfpξq|2dξ“e´t\n4}f}2\nHp´2pR3q.\nOn the other hand, by Sobolev’s theorem, we have for pą3\n2that\nsup\nxPR3|U2px,tq| ď }U2p¨,tq}HppR3qÀe´t\n8}f}Hp´2pR3q.\nMore generally, for any αPN3it holds that\np1´∆qp\n2Bα\nxU2px,tq “F´1pp1` |ξ|2qp\n2mpt,ξqp1´χpξqqyBαfq,\nwhich leads to\nsup\nxPR3|Bα\nxU2px,tq| Àe´t\n8}Bαf}Hp´2pR3qÀe´t\n8}f}HspR3q. (B.6)\nHeres“p´2` |α| ą |α| ´1\n2by choosing pą3\n2. Combining the estimate (B.4) with (B.6) yields\n(B.2) for j“0.\nNext we consider the general case with j‰0. Noting\nBj\ntUpx,tq “ż\nR3eix¨ξBj\ntmpt,ξqˆfpξqdξ,AN INVERSE SOURCE PROBLEM 15\nwe obtain from direct calculations that\nBj\ntmpt,ξq “ Bj´e´1\n2t\na\n1´4|ξ|4´\ne1\n2t?\n1´4|ξ|4´e´1\n2t?\n1´4|ξ|4¯¯\n“jÿ\nl“02´jpa\n1´4|ξ|4ql´1e´t\n2´\ne1\n2t?\n1´4|ξ|4` p´1ql`1e´1\n2t?\n1´4|ξ|4¯\n:“jÿ\nl“0mlpt,ξq.\nHence we can write Bj\ntUpx,tqas\nBj\ntUpx,tq “jÿ\nl“0ż\nR3eix¨ξmlpt,ξqˆfpξqdξ:“jÿ\nl“0Wlpx,tq. (B.7)\nFor each 0 ďlďj, j‰0, using similar arguments for the case j“0 we obtain\nsup\nxPR3|Bα\nxWlpx,tq| ď p1`tq´3`|α|\n4}f}L1pR3q`e´t\n8}f}HspR3q (B.8)\nforsą2l` |α| ´1\n2. Combining (B.7) and (B.8), we obtain the general estimate ( B.2). /square\nRemark B.4. For the damped biharmonic plate wave equation, besides the d ecay estimate (B.2), we\ncan deduce other decay estimates of the Lp-Lqtype and time-space estimates by more sophisticated\nanalysis for the Fourier multiplier mpt,ξq. For example, it can be proved that\n}Upx,tq}LqpR3qÀ p1`tq´3\n4p1\np´1\nqq}f}LppR3q`e´ct}f}Wq,spR3q,\nwhere1ăpďqă `8andsě3p1\nq´1\n2q ´2. We hope to present the proofs of these Lp-Lqestimates\nand their applications elsewhere.\nAcknowledgement\nWe would like to thank Prof. Masahiro Yamamoto for providing the reference [18] on Carleman\nestimates of the Kirchhoff plate equation. The research of PL is supported in part by the NSF grant\nDMS-1912704. The research of XY is supported in part by NSFC ( No. 11771165). The research of\nYZ is supported in part by NSFC (No. 12001222).\nReferences\n[1] T. Aktosun and V. Papanicolaou, Time evolution of the sca ttering data for a fourth-order linear differential\noperator, Inverse Problems, 24 (2008), 055013.\n[2] G. Bao, J. Lin, and F. Triki, A multi-frequency inverse so urce problem, J. Differential Equations, 249 (2010),\n3443–3465.\n[3] G. Bao, P. Li, and Y. Zhao, Stability for the inverse sourc e problems in elastic and electromagnetic waves, J.\nMath. Pures Appl., 134 (2020), 122–178.\n[4] J. Cheng, V. Isakov, and S. Lu, Increasing stability in th e inverse source problem with many frequencies, J.\nDifferential Equations, 260 (2016), 4786–4804.\n[5] M. Entekhabi and V. Isakov, On increasing stability in th e two dimensional inverse source scattering problem\nwith many frequencies, Inverse Problems, 34 (2018), 055005 .\n[6] M. Entekhabi and V. Isakov, Increasing stability in acou stic and elastic inverse source problems, SIAM J. Appl.\nMath., 52 (2020), 5232–5256.\n[7] F. Gazzola, H.-C. Grunau, and G. Sweers, Polyharmonic bo undary value problems, Springer-Verlag Berlin Hei-\ndelberg, 2010.\n[8] V. Isakov and S. Lu, Increasing stability in the inverse s ource problem with attenuation and many frequencies,\nSIAM J. Appl. Math., 78 (2018), 1–18.\n[9] K. Iwasaki, Scattering theory for 4th order differential operators: I-II, Japan. J. Math., 14 (1988), 1–96.16 P. LI, X. YAO, AND Y. ZHAO\n[10] K. Krupchyk, M. Lassas, and G. Uhlmann, Inverse boundar y value problems for the perturbed poly-harmonic\noperator, Trans. Amer. Math. Soc., 366 (2014), 95–112.\n[11] M. Lassas, K. Krupchyk and G. Uhlmann, Determining a firs t order perturbation of the biharmonic operator by\npartial boundary measurements, J. Funct. Anal., 262 (2012) , 1781–1801.\n[12] P. Li, X. Yao, and Y. Zhao, Stability for an inverse sourc e problem of the biharmonic operator, arXiv:2102.04631.\n[13] P. Li and G. Yuan, Increasing stability for the inverse s ource scattering problem with multifrequencies, Inverse\nProblems Imag., 11 (2017), 745–759.\n[14] P. Li, J. Zhai, and Y. Zhao, Stability for the acoustic in verse source problem in inhomogeneous media, SIAM J.\nAppl. Math., to appear.\n[15] N.V. Movchan, R.C. McPhedran, A.B. Movchan, and C.G. Po ulton, Wave scattering by platonic grating stacks,\nProc. R. Soc. A, 465 (2009), 3383–3400.\n[16] J. Rousseau and L. Robbiano, Spectral inequality and re solvent estimate for the bi-harmonic operator, J. Eur.\nMath. Soc., 22 (2020), 1003–1094.\n[17] T. Tyni and V. Serov, Scattering problems for perturbat ions of the multidimensional biharmonic operator, Inverse\nProblems and Imaging, 12 (2018), 205–227.\n[18] G. Yuan and M. Yamamoto, Lipschitz stability in inverse problems for a Kirchhoff plate equation, Asymptotic\nAnalysis, 53 (2007), 29–60.\nDepartment of Mathematics, Purdue University, West Lafaye tte, Indiana 47907, USA\nEmail address :lipeijun@math.purdue.edu\nSchool of Mathematics and Statistics, China Central Normal University, Wuhan, Hubei, China\nEmail address :yaoxiaohua@mail.ccnu.edu.cn\nSchool of Mathematics and Statistics, China Central Normal University, Wuhan, Hubei, China\nEmail address :zhaoyueccnu@163.com"    },    {        "title": "2103.03885v1.Universal_spin_wave_damping_in_magnetic_Weyl_semimetals.pdf",        "content": "arXiv:2103.03885v1  [cond-mat.str-el]  5 Mar 2021Universal spin wave damping in magnetic Weyl semimetals\nPredrag Nikoli´ c1,2\n1Department of Physics and Astronomy, George Mason Universi ty, Fairfax, VA 22030, USA and\n2Institute for Quantum Matter at Johns Hopkins University, B altimore, MD 21218, USA\n(Dated: March 9, 2021)\nWe analyze the decay of spin waves into Stoner excitations in magnetic Weyl semimetals. The\nlifetime of a mode is found to have a universal dependence on i ts frequency and momentum, and\non a few parameters that characterize the relativistic Weyl spectrum. At the same time, Gilbert\ndamping by Weyl electrons is absent. The decay rate of spin wa ves is calculated perturbatively\nusing the s-d model of itinerant Weyl or Dirac electrons coup led to local moments. We show that\nmany details of the Weyl spectrum, such as the momentum-spac e locations, dispersions and sizes\nof the Weyl Fermi pockets, can be deduced indirectly by probi ng the spin waves of local moments\nusing inelastic neutron scattering.\nI. INTRODUCTION\nWeyl semimetals are condensed matter realizations\nof massless fermions with a chiral relativistic three-\ndimensional spectrum1–3. Topologically protected gap-\nless Fermi “arc” states on the system boundaries, and\nunconventional transport properties such as the intrinsic\nanomalous Hall effect, set Weyl semimetals apart from\nother weakly interacting conductors. One way to ob-\ntain a Weyl spectrum involves breaking the time-reversal\nsymmetry in a material that has Dirac quasiparticles.\nThe presence of magnetization, for example, will remove\nthe spin degeneracy of a Dirac node by splitting it into\na dipole of opposite-chirality Weyl nodes in momentum\nspace. Magnetism then becomes intimately related to\nthe presence of Weyl electrons. Alternatively, Weyl spec-\ntrum of itinerant electrons can be created by a broken\ninversion symmetry, e.g. due to the crystal structure,\nand then coupled to magnetism if the material possesses\nadditional local moments or undergoes a spin density\nwave instability. Some of these theoretical scenarios are\nslowly finding their actualization in experimentally stud-\nied magnetic Weyl semimetals4–13.\nHere we analyze an important imprint of Weyl elec-\ntrons on the magnetic dynamics – the damping of spin\nwaves via particle-hole (Stoner) excitations. This basic\ninteractioneffect revealsthe definingfeatures ofthe Weyl\nspectrum, relativity and chirality. We will show that the\nlifetime of spin waves exhibits a universal dependence on\nthe modefrequency andmomentum whichcanbe used to\nextract detailed properties of the underlying Weyl elec-\ntrons. By measuring the mode lifetime throughout the\nfirst Brillouin zone, it is possible to discern the locations\nof the Weyl nodes in momentum space, their relative chi-\nralities, slope of the energy versusmomentum dispersion,\nand the size of the Fermi pockets on the Weyl nodes.\nThe spin wave lifetime is obtained from the width of the\nscattering intensity peaks in inelastic neutron scattering\nexperiments, provided that a sufficient energy resolution\nis available and other sources of decoherence (thermal\nbroadening, disorder, phonons) do not mask the elec-\ntronic source.\nEven though neutron scattering is a powerful Green’sfunction probe, its ability to detect fermionic quasipar-\nticles is normally ruined by the incoherent continuum\nof excitations that can absorb an angular momentum\nquantum. Interestingly, this problem is reduced in Weyl\nsemimetals14, and fortunately it is also possible to in-\ndirectly characterize the quasiparticles via collective ex-\ncitations. The latter has been achieved in the neutron\nstudies of samarium hexaboride (SmB 6)15,16, where the\nmeasured dispersion of a “spin exciton” has revealed a\nnon-trivial topology of the underlying electronic quasi-\nparticles. An energygap protects the exciton’s coherence\nin SmB 6, but the gaplessquasiparticlesin Weyl semimet-\nals will generally induce ubiquitous damping of collective\nmodes. Such a damping can in fact reveal the existence\nand properties of chiral fermionic quasiparticles. The\nWeyl electron characterization through damping could\npotentially overcome various issues that plague other ap-\nproaches, such as correlation effects in the case of band-\nstructure calculations, limited resolution in the case of\nARPES, sensitivity to conventional bands (that coexist\nwith Weyl nodes) in transport measurements, etc.\nClosely related to the physics we pursue here is the ex-\ntensively studied damping in metallic ferromagnets17–29.\nStoner excitations provide a mechanism for the decay\nof spin waves, and also typically give rise to Gilbert\ndamping30– the dissipated precession of uniform mag-\nnetization in an external magnetic field. Many works\nhave been devoted to the calculation of Gilbert damp-\ning since it is possible to measure it by ferromagnetic\nresonance31,32and time-resolved magneto-optical Kerr\neffect33,34. A careful consideration of the relativistic\nelectron dynamics has revealed that Gilbert damping\noriginates in the spin-orbit coupling and depends on\nthe electrons’ mass25. In the case of massless Weyl\nelectrons, we show here that Gilbert damping is ab-\nsent. However, spin waves unavoidably decay via Stoner\nexcitations35–39,41,42, and their damping features “non-\nreciprocity” – different polarization modes that carry\nthe same momentum have different damping rates. This\naccompanies the non-dissipative aspects of chiral spin-\nmomentumlocking44,45. Spinwave“non-reciprocity”has\nbeen anticipated in spiral magnets46, magnetic interfaces\nwith a Dzyaloshinskii-Moriya interaction derived from2\nthe Rashba spin-orbit coupling43,47–52, and observed in\nseveral experiments53–58. In the context of magnetic\nWeyl semimetals, initial theoretical studies have been fo-\ncused on the domain wall dynamics59,60.\nThe rest of this paper is organized as follows. Section\nII presentsthe approachand the main results ofthe anal-\nysis, focusing on the observable physical characteristics\nof the spin wave damping by Weyl electrons. Section III\nis devoted to the technical development of the damping\ntheory. It contains separate derivations of the dissipative\ntermsintheeffectivespinaction(IIIA),spinwavedamp-\ning (IIIB), and Gilbert damping from the semiclassical\nfield equation (IIIC). The last section IV summarizesthe\nconclusions and discusses the broader applicability and\nlimitations of the damping theory.\nII. SUMMARY OF THE RESULTS\nIn this paper, we work with the s-d model of Weyl\nelectrons coupled to local moments. We perturbatively\ncalculate the dissipative non-Hermitian parts of the mo-\nments’effectiveaction,whichdeterminetherate γofspin\nwavedamping. γalsodependsonthe magneticorderand\nthe wave’s propagation direction relative to the magneti-\nzation, but it is always controlled by the components of\nthe universal damping rate tensor given by\nγab\nmn(q) =a3J2\nKΩ2\n128πSv3fab\nmn/parenleftbigg|Ω|\nvq,|Ω|\n2|µ|;sign(µ,Ω)/parenrightbigg\n(1)\nfor ferromagnetic local moments of spin magnitude S.\nThe upper indices a,b∈ {x,y,z}refer to spin projec-\ntions. The universal scaling functions fab\nmnare dimen-\nsionless, the factor a3is the unit-cell volume of the local\nmoment’s lattice, JKis the Kondo or Hund coupling en-\nergy scale, vandµare the Fermi velocity and Fermi\nenergy of the Weyl electrons respectively, and Ω is the\nreal spin wave frequency (we use the units /planckover2pi1= 1). The\nspin wave momentum qin this expression is measured\nrelative to the difference ∆ Q=Qm−Qnbetween the\nwavevectors Qm,Qnof any two Weyl nodes in the first\nBrillouin zone. Coherent collective excitations that span\nthe entire first Brillouin zone can be used to separately\naddress many pairs of Weyl nodes – by tuning the total\nwavevector ∆ Q+qto the vicinity of ∆ Q. Representa-\ntive functions fab\nmnfor the Weyl nodes with finite Fermi\nsurfaces are plotted in Figures 1 and 2\nWemakeanalyticalprogressandgainvaluablephysical\ninsight through several idealizations: all Weyl nodes are\nassumed to be identical, sphericallysymmetric and living\nat the same node energy. Their chiralities χm=±1 and\nlocations Qmare arbitrary (as long as the total chirality\nin the first Brillouin zone vanishes). Under these condi-\ntions, only three tensor components of γabare finite and\nindependent, γ/bardbl/bardbl,γ⊥⊥andγ⊥⊥′. Here and throughout\nthe paper ∝bardblindicates the spin direction parallel to the\nmode’s wavevector q, and⊥,⊥′are the spin directions(a)\n(b)\nFIG. 1. The plots of functions (a) f⊥⊥and (b) f/bardbl/bardblfor the\ndamping rates of transverse and longitudinal spin waves re-\nspectively, contributed by the Fermi surfaces on a particul ar\npair of Weyl nodes. Solid red lines are for the same-chiralit y\nnodes, and the dashed blue lines are for the opposite-chiral ity\nnodes.|Ω|= 1.4|µ|was assumed in this example.\nFIG. 2. The plots of selected universal functions fabfeatured\nin the damping rate γ∼Ω2f(vq/|Ω|;xµ). The functions\nare parametrized by xµ= 2|µ/Ω|, with finer dashes corre-\nsponding to larger Weyl Fermi pockets (solid lines refer to\nthe Fermi level that crosses the Weyl nodes). Shown func-\ntionsincludetransverse( ⊥⊥)andchiral( ⊥⊥′)dampingchan-\nnels shaped by electron scattering between equal-chiralit y (+)\nandopposite-chirality ( −)Weyl nodes. Longitudinal channels\n(∝bardbl∝bardbl) are similar to the shown transverse channels, compare\nwith Fig.1.3\n(a)\n (b)\nFIG. 3. Examples of the damping rate map in momentum\nspace for (a) µ∝negationslash= 0 and (b) µ= 0 (with and without a Fermi\nsurface of Weylelectrons respectively). Brightness depic ts the\nrateγ(q) of spin wave damping, and the red crosshair shows\nthe reference ∆ Qfor the local wavevector q= 0. These are\nqz= 0 slices through the full 3D map. Observing patterns\nof this kind in the full Brillouin zone scan will indicate the\nWeyl-electron origin of damping and reveal the complete set\nof ∆Q=Qm−Qnwavevectors from which the individual\nnode wavevectors Qmcan be deduced (assuming, for exam-\nple,/summationtext\nmQm= 0). The bright outer ring, which shrinks and\ncloses when 2 |µ|<|Ω|, originates in the inter-band electron\nscattering and gains strength from the rapidly growing Weyl\nelectron density of states. Note that various details in the se\nmaps, such as the anisotropy and ring sizes, will generally\ndepend on the concrete spin-wave dispersion Ω( q+∆Q), po-\nlarization, type and orientation of magnetic order, as well as\nthe chiralities and symmetries of the Weyl nodes.\nwhich areperpendicular to qand eachother. The full ex-\npression for damping rates is presented in Section IIIB;\nin Weyl ferromagnets, it becomes\nγmn=γ⊥⊥\nmn±γ⊥⊥′\nmn (2)\nfor the two polarizations of spin waves propagating along\nthe magnetization direction.\nThe essential utility of the universal damping comes\nfrom its qualitative features that reflect the relativistic\nnature of Weyl electrons. If the Fermi energy µlies away\nfrom the energy of the Weyl nodes, Fermi surfaces will\nform. Then, the spin wave damping rate is expected to\nexhibitasetofminimumsandmaximumsasafunctionof\nthefrequencyΩandmomentum q. Thelocationsofthese\nextremums depend on the parameters that characterize\nthe Weyl nodes: Fermi velocity v, chemical potential µ\nand even their relative chiralities χmχn=±1. Fig.3\ndemonstrates how the locations Qmof Weyl nodes can\nbe extracted from the full Brillouin zone map of the spin\nwave’s damping rate γ(q). Once the wavevectors Qmare\nknown, Fig.4 illustrates how the observation of enough\nextremums enables indirect measurements of the Weyl\nelectron spectra on multiple Weyl nodes. The presence\nof Weyl Fermi pockets also introduces spin-momentum\nlocking into the damping rates ( γ⊥⊥′\nmn∝ne}ationslash= 0), but only on\nthe pairs of Weyl nodes with opposite chiralities. As a\nvqΩ\n02\u00012 \u0000\u0002\n\u0003Ω=vq\nΩ=vq-2\u0004\nFIG. 4. A density plot of the collective mode damping rate\nγ(q,Ω) induced by Weyl electrons. Thin solid green lines in-\ndicateγ= 0, and the thin dashed green line indicates the\nlocal maximum of γ. The thick dashed yellow line represents\nthe dispersion Ω( q+ ∆Q) of a hypothetical spin-wave exci-\ntation (note that the origin of the plot corresponds to the\nmomentum difference ∆ Qof two Weyl nodes in the first Bril-\nlouin zone). The spin-wave’s damping rate will exhibit loca l\nminimums and maximums at the shown red points, which are\ncharacteristic for the relativistic spectrum of Weyl elect rons.\nResolving two of these points is enough for the determinatio n\nof the Weyl Fermi velocity vand the chemical potential µof\nthe Weyl nodes addressed via ∆ Q. Resolving three points al-\nlows an independent verification that Weyl nodes are indeed\nresponsible for the damping. The two-parameter scaling of\nthe damping rate (1) across a range of energies is the most\ngeneral signature of Weyl electrons, and can be used to verif y\nthe Weyl-electron origin of damping even if the visible spin\nwave dispersion does not cross any of the shown characterist ic\npoints.\nconsequence, the two spin wave modes that carry oppo-\nsite spin currents at the same wavevector qhavedifferent\npeak widths in inelastic neutron scattering.\nThe above qualitative features of damping disappear if\nthe Fermi energy sits exactly at the Weyl nodes. How-\never, the damping rate then becomes a universalfunction\nofa single parameter |Ω|/vq. This kind of scalingis a sig-\nnature of the relativistic Weyl electrons – it is caused by\n“inter-band” transitions in which an electron below the\nWeyl nodeisexcited toastate abovethe Weyl node. The\nplots of universal functions fab\nmnthat appear in Eq. 1 at\nµ= 0 are shown in Fig.2.\nThe magnitude of the damping rate depends\non the Kondo/Hund scale JKwhich may not be\nknown. However, the spin wave damping caused\nby Weyl electrons is always related to the effec-\ntive strength Jof the Weyl-electron-induced Ruder-\nman–Kittel–Kasuya–Yosida(RKKY)interactionsamong4\nqqJKJK\nFIG. 5. The Feynman diagram for two-spin interactions.\nThick external lines represent local moment fields and thin\nlines represent Weyl electron propagators. The two-spin co u-\nplings include Heisenberg, Kitaev and Dzyaloshinskii-Mor iya\ninteractions, but the Weyl-electron origin of spin dynamic s\nalso creates a dissipation channel in which spin waves decay\ninto electron-hole pairs.\nthe local moments45:\nγ\nJ∼1\n(aΛ)3/parenleftBigq\nΛ/parenrightBig2\n×/parenleftbiggΩ\nvq/parenrightbigg2\n, J∼vΛ/parenleftbigga3Λ2JK\nv/parenrightbigg2\n.\n(3)\nHere, Λisthemomentumcut-offforthelinearWeylspec-\ntrum,|q|<Λ. SinceaΛ<1 and the characteristic fea-\ntures of the universal damping appear near |Ω| ∼vq, the\ndamping rates are generally comparable to the energy\nscaleJof the induced RKKY interactions. For example,\nthe RKKY energy scale in the magnetic Weyl semimetal\nNdAlSi13can be crudely estimated as J∼1 meV. Even\nif the damping rate is more than an order of magnitude\nbelow this value of J, it should be detectable with high\nresolutionneutron instruments (a spin echospectrometer\ncan achieve energy resolution below 10 µeV).\nIII. DISSIPATION BY WEYL ELECTRONS\nHere we calculate the Gaussian dissipative part of the\neffective action for local moments which arises due to\ntheir coupling to itinerant Weyl electrons. The non-\ndissipative part of this action, computed in Ref.45, cap-\ntures the induced RKKY interactions among the lo-\ncal moments: Heisenberg, Kitaev and Dzyaloshinskii-\nMoriya. All Gaussian terms δnaΓabδnbof the action ob-\ntain from a single two-point Feynman diagram which in-\nvolves momentum integration of a singular function; the\nprincipal part of this integral yields the interactions, and\nthe contribution of its pole singularity amounts to dissi-\npation. We will focus only on the latter, following the\nprocedure from Ref.45.\nThe essential dynamics of local moments ˆnicoupled to\nconduction electrons ψiis given by the Hamiltonian:\nH0=Hn+/summationdisplay\nkǫkψ†\nkψk+JK/summationdisplay\niˆniψ†\niσψi.(4)\nBoth the local moments and electrons live on a lattice\nwhose sites are labeled by i, but we will immediately\ntake the continuum limit. The basic two-spin correla-\ntions∝an}bracketle{tˆna\niˆnb\nj∝an}bracketri}htare contained in the second-order Feynman\ndiagram shown in Fig.5:\nΓab\nmn(q) =iJ2\nK\n2/integraldisplayd4k\n(2π)4tr/bracketleftBig\nGm/parenleftBig\nk−q\n2/parenrightBig\nσaGn/parenleftBig\nk+q\n2/parenrightBig\nσb/bracketrightBig\n(5)The Weyl electron Green’s functions\nGn(ω,k) =/bracketleftBig\nω−Hn(k)+isign(ǫn(k))0+/bracketrightBig−1\n(6)\nare treated as spinor matrices and refer to the low-energy\nelectronic states near any Weyl node nwhose wavevector\nin the first Brillouin zone is Qn; the wavevector kis a\n“small” displacement |k|<Λ fromQn, where Λ is the\nmomentum cut-off for the linear Weyl dispersion. These\nlow-energy electrons are described by the Hamiltonian\nHn(k+Qn) =vχnσk−µ , (7)\nwhereµis the chemical potential that determines the\nWeyl Fermi pocket character and size, vis the Fermi\nvelocity, and χn=±1 is the Weyl node chirality. We\nassume for simplicity that all Weyl nodes are spherically\nsymmetric, share the same node energy, chemical poten-\ntial and Fermi velocity, but have arbitrary wavevectors\nQnand chiralities χn=±1 (as long as the chiralities of\nall nodes in the first Brillouin zone add up to zero). By\nthis construction, the expression (5) is associated with a\npairm,nof Weyl nodes, and qis a “small” wavevector\nmeasured relative to Qm−Qn.\nWe will carry out all calculations with the formal as-\nsumption that no external or effective magnetic field is\nexerted on electrons. Realistically, however, we are inter-\nested in magnetic Weyl semimetals whose local moments\nmay carry a non-zero net magnetization ˆn0that presents\nitself as an effective magnetic field B=−JKˆn0to elec-\ntrons. This is of no concern because the correction of the\nspectrum (7) amounts merely to a shift of the wavevector\nk→k−B/vχn. Hence, an effective magnetic field only\nalters the locations Qnof the Weyl nodes in momentum\nspace, which are arbitrary in our formalism.\nThe full effective action matrix Γ for local moments\ntakes contributions from all Weyl node pairs:\nΓ(Q,Ω) =/summationdisplay\nm,nΓmn(Q−Qm+Qn,Ω).(8)\nIn this sense, it is possible to experimentally address a\nparticular pair of Weyl nodes, or a set of pairs, by prob-\ningthe momentumspaceinthe vicinityof Q∼Qm−Qn.\nThe dissipative part of Γ mnwill contain information\nabout the addressed Weyl nodes.\nA. Calculation of the dissipative terms in the\neffective spin Lagrangian\nThe calculationof(5) is lengthy, sowe will onlyoutline\nits key steps. The trace has been evaluated before45, and\nthe frequency integration yields:5\nΓab\nmn(q) =−J2\nK\n2/integraldisplayd3k\n(2π)3/bracketleftBigg\nXab(Ω,q;vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle+Ω\n2−µ,k)\n2vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ−vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/parenrightbig\nΩ+vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−vsχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle+i0+F(sχn,χm)\n−Xab(Ω,q;−vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle+Ω\n2−µ,k)\n2vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ+vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/parenrightbig\nΩ−vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−vsχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle+i0+F(sχn,−χm)\n+Xab(Ω,q;vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω\n2−µ,k)\n2vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ−vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/parenrightbig\nvχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω−vsχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−i0+F(χn,sχm)\n−Xab(Ω,q;−vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω\n2−µ,k)\n2vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ+vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/parenrightbig\n−vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω−vsχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−i0+F(−χn,sχm)\nHere,θ(x) is the step function, and two more functions,\nXab(Ω,q;ω,k) andF(s+,s−) are introduced to simplify\nnotation. The function Xab(Ω,q;ω,k) obtains from the\nnumerator of the trace in (5). Introducing the Kronecker\nsymbolδaband the Levi-Civita symbol ǫabc, we have:\nXab(Ω,q;ω,k) =/bracketleftbigg\n(ω+µ)2−Ω2\n4/bracketrightbigg\nδab\n+v2χmχn/bracketleftbigg\n2/parenleftbigg\nkakb−qaqb\n4/parenrightbigg\n−δab/parenleftbigg\nkckc−qcqc\n4/parenrightbigg/bracketrightbigg\n+ivǫabc/bracketleftbigg\nχm/parenleftbigg\nω+Ω\n2+µ/parenrightbigg/parenleftbigg\nkc−qc\n2/parenrightbigg\n−χn/parenleftbigg\nω−Ω\n2+µ/parenrightbigg/parenleftbigg\nkc+qc\n2/parenrightbigg/bracketrightbigg\n. (9)\nThe function F(s+,s−) withs+,s−=±1 keeps track of\nthe infinitesimal imaginary terms in the denominators of\nGreen’s functions:\nF(s+,s−) = sign/parenleftBig\nvs+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle−µ/parenrightBig\n−sign/parenleftBig\nvs−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n=θ/parenleftbigg\n|qk|−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigµ\nv/parenrightBig2\n−k2−q2\n4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n×/bracketleftbigg/parenleftbigg\nsign(µ)+s++s−\n2/parenrightbigg\nsign(qk)+s+−s−\n2/bracketrightbigg\n+(s+−s−)θ/parenleftbigg\nk2+q2\n4−|qk|−/parenleftBigµ\nv/parenrightBig2/parenrightbigg\n.(10)\nAt this point, we use the relationship\n1\nx±i0+=P1\nx∓iπδ(x) (11)\nto isolate the dissipative processes that curb the x→0\nresonances. Dropping all terms that involve the principal\npartP, we get:\n/tildewideΓab\nmn(q) =iπJ2\nK\n8v2/summationdisplay\nsm,snsmsn/integraldisplayd3k\n(2π)3F′(snχn,smχm)\nχmχn/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle\n×Xab/parenleftbigg\nΩ,q;vsmχm/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle+Ω\n2−µ,k/parenrightbigg\n(12)\n×δ/parenleftBig\nΩ+vsmχm/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle−vsnχn/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n×/bracketleftBig\nθ/parenleftBig\nµ−vsmχm/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n−θ/parenleftBig\nµ−vsnχn/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig/bracketrightBigWe introduced F′= sign(F)(1−δF,0), and the sum goes\noversm,sn=±1. All chirality factors χm,χn=±1\nthat appear outside of Xabare clearly eliminated by the\nsummation over sm,sn, so it will be convenient do define\ns−=smχm=±1 ands+=snχn=±1. The Dirac\nδ-function in (12) imposes:\ns+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle−s−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle=Ω\nv. (13)\nThis pins the magnitude of the wavevector kto\nk=|Ω|\n2v/radicalBigg\nΩ2−v2q2\nΩ2−v2q2cos2θ, (14)\nassuming qk=qkcosθ, and further requires satisfying\none of these two conditions:\n|Ω|>vq ∧s±=±sign(Ω)\n|Ω|<vq|cosθ| ∧s+=s−= sign(Ωcos θ)\nThe wavevector k= (k,θ,φ) integration in (12) is now\nconveniently performed in the spherical coordinate sys-\ntemreferencedtotheexternalwavevector q. Theintegral\noverk=|k|is immediately solved due to the Dirac δ-\nfunction and we merely need to replace the occurrences\nof|k|with (14). The integral over φaffects only the\nquantities (9) leading to/integraltext\ndφXab= 2πv2/tildewideXabwith the\nfollowing non-zero components:\n/tildewideX/bardbl/bardbl=s+s−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle (15)\n+χmχn/bracketleftbigg\nk2(2cos2θ−1)−q2\n4/bracketrightbigg\n/tildewideX⊥⊥=s+s−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle\n+χmχn/parenleftbigg\n−k2cos2θ+q2\n4/parenrightbigg\n/tildewideX⊥⊥′=iǫ/bardbl⊥⊥′/bracketleftbigg\n−/parenleftBig\nχms+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle+χns−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBigq\n2\n+/parenleftBig\nχms+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle−χns−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\nkcosθ/bracketrightbigg\n.\nHere and onward, the upper spin indices denote direc-\ntions∝bardblparallel to q, and two mutually perpendicular6\ndirections ⊥,⊥′which are also perpendicular to q. Note\nthatǫ/bardbl⊥⊥′implements a chiral “right-hand-rule” rela-\ntionship between the three spin directions. The integral\noverθis finite and conveniently evaluated numerically.\nAt the end, we arrive at:\n/tildewideΓab\nmn(q) =iJ2\nKΩ2\n128πv3fab\nmn/parenleftbigg|Ω|\nvq,|Ω|\n2|µ|;sign(µ,Ω)/parenrightbigg\n(16)\nwhere the dimensionless functions f=α+βhave con-\ntributions from intra-band αand inter-band βelectronscattering. Note that the inter-band processes require\ntransferringan electron between the two states whoseen-\nergies have opposite signs, and thus can occur only when\n|Ω|>2|µ|. Defining\nλ=vq\n|Ω|, x=2|µ|\n|Ω|, κ=/radicalBigg\n1−λ2\n1−λ2ξ2(17)\nwith|ξ|=|cosθ|, we have:\nα⊥⊥\nmn=1/integraldisplay\n0dξθ/parenleftbig\n2κλξ−|x2−κ2−λ2|/parenrightbig\nκ2(18)\n×/bracketleftBigg/parenleftBigg\n1−χmχn−κ2ξ2+λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(1−λ)+/parenleftBigg\n1+χmχn−κ2ξ2+λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(λξ−1)/bracketrightBigg\nα/bardbl/bardbl\nmn=1/integraldisplay\n0dξθ/parenleftbig\n2κλξ−|x2−κ2−λ2|/parenrightbig\nκ2\n×/bracketleftBigg/parenleftBigg\n1−χmχnκ2(2ξ2−1)−λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(1−λ)+/parenleftBigg\n1+χmχnκ2(2ξ2−1)−λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(λξ−1)/bracketrightBigg\nα⊥⊥′\nmn=−iǫ/bardbl⊥⊥′1/integraldisplay\n0dξθ/parenleftbig\n2κλξ−|x2−κ2−λ2|/parenrightbig\nκ2\n×/summationdisplay\ns=±1(χm+χn)sign(µ)+s(χm−χn)sign(Ω)\n2/radicalbig\nκ2+λ2−2sκλξ(κξ−sλ)/bracketleftBig\nθ(1−λ)−sθ(λξ−1)/bracketrightBig\n,\nand\nβ⊥⊥\nmn=1/integraldisplay\n−1dξθ/parenleftbig\nκ2+λ2−2κλ|ξ|−x2/parenrightbig\nκ2\n1−χmχn−κ2ξ2+λ2\n/radicalBig\n(κ2+λ2)2−(2κλξ)2\nθ(1−λ) (19)\nβ/bardbl/bardbl\nmn=1/integraldisplay\n−1dξθ/parenleftbig\nκ2+λ2−2κλ|ξ|−x2/parenrightbig\nκ2\n1−χmχnκ2(2ξ2−1)−λ2\n/radicalBig\n(κ2+λ2)2−(2κλξ)2\nθ(1−λ)\nβ⊥⊥′\nmn=−iǫ/bardbl⊥⊥′1/integraldisplay\n−1dξθ/parenleftbig\nκ2+λ2−2κλ|ξ|−x2/parenrightbig\nκ2/summationdisplay\ns=±1s(χm−χn)sign(Ω)\n2/radicalbig\nκ2+λ2−2sκλ|ξ|(κ|ξ|−sλ)θ(1−λ).\nThe functions fabhave the same characteristics in all\nspin channels a,b∈ {⊥⊥,∝bardbl∝bardbl,⊥⊥′}. Their plots in Fig-\nures 1, 2, 4 illustrate that fabvanish for |Ω|<v|q|−2|µ|,\n2|µ|−v|q|>|Ω|>v|q|and|Ω|=v|q|. Thedissipationat|Ω|>max(2|µ|,v|q) is dominated by the collective mode\ndecay into “high energy” particle-hole pairs which are\nexcited across the Weyl node. Outside of this frequency-\nmomentum region, the decay occurs by generating “low7\nenergy” particle-hole pairs across the Fermi surface on\nthe Weyl node. This “low energy” channel is weaker,\nbut has several features that clearly reveal the relativis-\ntic properties of the Weyl spectrum. Fig. 4 shows how\ntheminimumsandmaximumsofacollectivemodedamp-\ning rate can be used to characterize the Fermi surface of\nWeyl electrons.\nB. Spin wave damping\nThe actual damping rate of collective excitations gen-\nerally obtains from a mixture of spin channels. Consider\nthe spin waves with wavevectors ∆ Q+qin the vicinity\nof the momentum-space separation ∆ Q=Qm−Qnbe-\ntween two particular Weyl nodes. Let −SΩab\n0(q) be the\nintrinsic part of the effective Lagrangian density δLeff\nfor the local moment fluctuations δn, excluding the spin\nBerry phase SΩδab(Sis the spin magnitude of local\nmoments). This can contain any exchange interactions\nof the localized electrons and crystal field anisotropies.\nThe Lagrangian density terms induced by the itinerant\nWeyl electrons are all contained in the Γabtensor (5).\nThe principal part of (5) yields a variety of induced\nRKKY interactions45, while its dissipative components\n/tildewideΓabare collected in (16). The presence of magnetic or-\nder in the ground state further affects the dynamics of\nspin waves because the small spin fluctuations δnof low-\nenergy modes must be orthogonal to the local spins ˆn.\nThis can be incorporated into the general analysis44, but\nwe will simplify the discussion here by considering only a\nferromagnetic ground state ˆn(r) =ˆn0. The spectrum of\ndamped spin waves is extracted from the Gaussian part\nof the Lagrangian density in momentum space\nδLeff= (δna)∗/bracketleftBig\nSΩδab−SΩab\n0(q)+a3Γab(Ω,q)/bracketrightBig\nδnb(20)\nThe factor of a unit-cell volume a3converts the energy\ndensity Γabto the energy per lattice unit-cell, and the\nfactor of1\n2in the Berry phase term Ω is appropriate for\nthe local moments with spin S=1\n2. Introducing\ngab= Ωab\n0−a3\nSΓab(21)\nto simplify notation, the spin wave modes obtain by di-\nagonalizing PMP, wherePab=δab−ˆna\n0ˆnb\n0projects-out\nthe high-energy amplitude fluctuations (keeps δn⊥ˆn0)\nand\nMab= Ωδab−g⊥⊥(δab−ˆqaˆqb)−g/bardbl/bardblˆqaˆqb−g⊥⊥′ǫabcˆqc\nis the matrix embedded in (20). An arbitrary choice of\nthe background magnetization ˆn0=ˆzreveals two polar-\nization modes δn= (δnx,δny) atq=qˆq\nδn±∝/parenleftBigg\ng/bardbl/bardbl−g⊥⊥\n2(ˆq2\nx−ˆq2\ny)±δǫ\n(g/bardbl/bardbl−g⊥⊥)ˆqxˆqy−g⊥⊥′ˆqz/parenrightBigg\n(22)with energies\nΩ±=g⊥⊥\n0+g/bardbl/bardbl−g⊥⊥\n2(1−ˆq2\nz)±δǫ(23)\nwhereδǫ=1\n2/radicalbig\n(g/bardbl/bardbl−g⊥⊥)2(1−ˆq2z)2−(2g⊥⊥′ˆqz)2.\nThese polarizations are generally elliptical, but become\ncircularδn∝(±i,1) with Ω ±=g⊥⊥∓ig⊥⊥′for the\nmodes that propagate along the magnetization direction\n(q∝bardblˆn0), and linear δn+∝ˆq,δn−∝ˆn0׈qwith\nΩ+=g/bardbl/bardbl, Ω−=g⊥⊥respectively for the modes that\npropagate in the plane perpendicular to the magneti-\nzation (q⊥ˆn0). The character and non-degeneracy\nof the two polarization modes is the hallmark of the\nRKKY interactions induced through the spin-orbit cou-\npling: Dzyaloshinskii-Moriya(DM) in the caseof circular\npolarizations, and Kitaev in the case of linear polariza-\ntions.\nThe equation (23) has to be solved self-consistently\nsince the components of the gabtensor on its right-hand\nside depend on frequency, but the revealed form of its\nsolutions ensures all of the spin wave properties that\nwe discuss. The two circular polarizations at the same\nwavevector q∝bardblˆn0carry opposite spin currents\nja\ni=−iqiǫabc(δnb)∗δnc∝ ∓|g⊥⊥′|2qiδaz,(24)\nso their energy difference Ω ±=g⊥⊥∓ig⊥⊥′due to the\nDM interaction implies spin-momentum locking. Note\nthat the DM interactions appears as gab\nDM∝ǫabc(iqc), so\nit does shift the spin wave energy. The dissipative com-\nponents/tildewidegab∝/tildewideΓabofgabimpart an imaginary part on\nthe pole frequency Ω, which corresponds to the damping\nrate. The signs of both /tildewideΓ⊥⊥,/tildewideΓ/bardbl/bardbl(f⊥⊥,f/bardbl/bardbl>0) in-\ndeed correspond to damping and not an instability, and\nthe chiral contributions are not large enough to overturn\nthis at any Ω. The chiral dissipative part extracted from\n(16) is real,/tildewidegab\nDM∝ǫabcqc, and hence introduces differ-\nent damping rates for the two circular spin waves. These\nqualitative conclusions hold for the elliptical modes as\nwell.\nC. The absence of uniform precession damping\nThe universal dependence of (16) on |Ω|/vqintroduces\nanon-analyticbehavioratΩ ,q→0inthedampingterms\n/tildewideLof the spin Lagrangian density. Therefore, one cannot\nstrictly expand /tildewideLin powers of Ω ,qto represent the dis-\nsipation as a result of local processes. /tildewideLcan be approx-\nimated by an expansion only in special limits. Suppose\nthe spin waves have dispersion |Ω|=uqat low ener-\ngies (in the vicinity of ∆ Q=Qm−Qn→0 for intra-\nnode scattering m=n). If the spin wave velocity uis\nsmaller than the Weyl electrons’ velocity v, then a suf-\nficiently large qpushes the spin waves into the regime\n|Ω|< vq−2|µ|where/tildewideΓab= 0 in (16) and the damping\nis absent (see Fig.2). Alternatively, if u≫v, then the8\nspin waves are in the regime |Ω| ≫vqand their damp-\ning at energies |Ω|>2|µ|is approximately characterized\nby the dominant local terms /tildewideΓ/bardbl/bardbl,/tildewideΓ⊥⊥∼i(AΩ2+Bq2)\nand a smaller chiral term /tildewideΓ⊥⊥′∼DqΩ. Together with\nthe non-dissipative Hermitian terms χ−1\n0, the electron-\ninduced part of the local moments’ effective Lagrangian\ndensity (20) contains\nΓab|Ω|≫vq− −−−− →1\n2/bracketleftBig\n(χ−1\n0)ab+i(AabΩ2+Babq2)+DǫabcqcΩ/bracketrightBig\n(25)\nwithAab=A⊥⊥(δab−qaqb/q2)+A/bardbl/bardblqaqb/q2andlikewise\nforBab. By construction (5), Γ ≡1\n2χ−1is the inverse\ntime-ordered correlation function\n∝an}bracketle{tδsa(q,Ω)δsb(q′,Ω′)∝an}bracketri}ht=iχab(q,Ω)δ(q+q′)δ(Ω+Ω′)\nfor the small fluctuations δsof the Weyl electron spins\naway from their equilibrium magnetization. We will con-\nsider only the simplest case of a collinear ferromagnet\nin the following analysis. The equilibrium state will be\ngiven by the uniform magnetization of local moments ˆn0\nand electrons ∝an}bracketle{ts0∝an}bracketri}ht ∝bardblˆn0.\nA semiclassical representation of the local moment dy-\nnamics is given by the field equation for ˆn. The pres-\nence of non-Hermitian damping terms in the effective ac-\ntion for local moments prevents us from deriving the field\nequation by considering the stationary action condition.\nInstead, we can use linear response theory to learn about\nthe semiclassical dynamics. The retarded electrons’ spin\ncorrelation function\nχR(q,Ω) =/braceleftBigg\nχ(q,Ω),Ω>0\nχ†(q,Ω),Ω<0(26)\nis readily obtained from (25)\n(χ−1\nR)ab|Ω|≫vq− −−−− →(χ−1\n0)ab(27)\n+sign(Ω)/bracketleftBig\ni(AabΩ2+Babq2)+DǫabcqcΩ/bracketrightBig\n,\nand then the response of electron spins to the local mo-\nment field is\n∝an}bracketle{tδsa(r,t)∝an}bracketri}ht=JK\na3/integraldisplay\ndt′d3r′χab\nR(r−r′,t−t′)δnb(r′,t′).\n(28)\nThis follows from the Kondo interaction JKin (4) be-\ntween the “perturbation” field nand the responding\nelectrons spin s=ψ†σψon a lattice site (the unit-cell\nvolumea3effectively converts the integration over coor-\ndinates to a summation over lattice sites). Note that\nχab\nR(q,Ω) = (χab\nR)∗(−q,−Ω) is established globally in\nmomentumspace(notnecessarilyintheimmediatevicin-\nity of the Weyl node wavevector ∆ Q)61, so that its in-\nverse Fourier transform χab\nR(r,t) is real. The thermody-\nnamic potential for local moments is simply\nF[ˆn] =JK∝an}bracketle{ts∝an}bracketri}htˆn. (29)The local moment dynamics is driven by an effective\n“magnetic” field in units of energy\nHeff(r,t) =−δF[ˆn]\nδˆn(r,t)=−JK∝an}bracketle{ts(r,t)∝an}bracketri}ht(30)\nTaking into account the Berry phase of local moments\nyields the usual semiclassical field equation\n∂ˆn\n∂t=ˆn×Heff. (31)\nwith\nHa\neff(r,t)≈ −JKˆna\n0−J2\nK\na3/integraldisplay\ndδtd3δrχab\nR(δr,δt) (32)\n×δˆnb(r+δr,t+δt)\n≈ −JKˆna\n0−J2\nK\na3/integraldisplay\ndδtd3δrχab\nR(δr,δt)\n×/bracketleftbigg\n1+δr∇+δt∂\n∂t+···/bracketrightbigg\nδˆnb(r,t)\nThis is seen to generate Gilbert damping which dissi-\npates the precession of uniform magnetization in typical\nferromagnets\n∂ˆn\n∂t=ˆn×Heff=···+ˆn×αG∂ˆn\n∂t(33)\nwith the damping tensor\nαab\nG=−J2\nK\na3/integraldisplay\ndδtd3δrχab\nR(δr,δt)δt (34)\n=iJ2\nK\na3/integraldisplaydΩ\n2π/integraldisplay\ndδte−iΩδt∂χab\nR(0,Ω)\n∂Ω\n=−J2\nK\na3∂Imχab\nR\n∂Ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n(q,Ω)=0.\nThe real part of χR(q,Ω) generally does not contribute\nbecause it is an even function of Ω at q= 0 (even though\nit diverges for gapless spin waves when Ω →0). In the\ncase of damping induced by Weyl electrons, the imagi-\nnary part of χRbecomes zero when 2 |µ|−vq>|Ω| ≥vq,\nfollowing the behavior of the time-ordered χ−1≡Γab\nthat was discussed earlier (see Fig.4). Therefore, χRis\nreal in the limit Ω ,q→0 and the decay of spin waves\ninto Stoner excitations of the Weyl electrons does not\ngenerate Gilbert damping.\nThe complete equation of motion for local moments\ncanbe extractedfrom(31)and(32), but thenon-analytic\nfrequency dependence of the dissipative terms in (27) in-\ntroduces (via its Fourier transform) non-local relation-\nships between the fields ˆn(t) at different times t. If one\nwere to ignore this issue, or approximate the non-local\neffect by couplings over small time intervals, then a local\nfield equation would be obtained from the expansion in-\ndicated in (32). We will not pursue this here any further.9\nIV. CONCLUSIONS AND DISCUSSION\nWe analyzed the dynamics of local magnetic moments\ncoupled to itinerant Weyl electrons, and focused on the\ndissipation of spin waves via the continuum of Stoner\nparticle-holeexcitations. Wedescribedthisdissipationat\nthe level of the effective Lagrangian of local moments, or\nequivalently the spin-spin correlation function (dynamic\nsusceptibility). For the spin waves at wavevector∆ Q+q\nand frequency Ω in the vicinity of the momentum differ-\nence∆Q=Qm−QnbetweentwoWeylnodes,thedamp-\ning rate is proportional to Ω2and a universal function of\n|Ω|/v|q|wherevis the Weyl electron (Fermi) velocity.\nThe presence of Fermi pockets with chemical potential µ\nintroduces additional dependence of the damping rate on\n|Ω/µ|. If the Weyl nodes are well-separated in momen-\ntum space, then there is no cross-talk between them in\nthe damping rates and the momentum-space locations of\nthe Weyl nodes can be discerned from the wavevectors\nat which the spin wave dissipation is locally maximized.\nThe Weyl-electron origin of dissipation can be experi-\nmentally verified by the universal relativistic properties\nof damping over a range of mode frequencies and mo-\nmenta, while various parameters of the Weyl spectrum\ncan be extracted from the momentum space locations\nof the characteristic damping features (e.g. local maxi-\nmums and points where damping vanishes). The damp-\ning rates involving Weyl electrons also generally exhibit\n“non-reciprocity”or chirality– the modes ofdifferent po-\nlarizationsthat propagateatthe samemomentum qhave\ndifferent lifetimes. We presented a procedure to obtain\nthe field equation for the semi-classical dynamics of the\nlocalmomentmagnetizationfield, andfoundthatthedis-\nsipation on Weyl electrons does not give rise to Gilbert\ndamping.\nOne important conclusion of this study is that the\nspin wave damping rate reveals the relativistic nature\nof Weyl electrons – both through its universal depen-\ndence on |Ω|/v|q|and the places in momentum space\nwhere it vanishes. We computed the damping rate asso-\nciated with Stoner excitations, but similar results should\nhold for zero-spin particle-hole excitations as well. Then,\notherkindsofcollectivemodescoupledtoWeylelectrons,\ne.g. the phonons of the crystal or a charge density wave,\nshouldexhibitsimilaruniversalityintheirdampingrates.\nThiswouldbeinterestingtoexploreinthefuturesincein-\nelastic neutron scattering is sensitive to phonons as well.\nThe developed theory is very general within its limi-\ntations. It makes no assumptions about the Weyl node\nlocations, so it applies to Diracsemimetals aswell (where\nthe opposite-chirality Weyl nodes coexist at the samewavevectors). It also makes no assumptions about the\nmagnetic order, so it holds for ferromagnets, antiferro-\nmagnets and paramagnets, with or without local spin\nanisotropy. In this regard, however, the damping rates\nof spin waves are affected by the nature of magnetic or-\nder; we demonstrated the calculations only in the fer-\nromagnetic (and implicitly also the paramagnetic) case.\nAnalytical progress was made by simplifying the model\nto spherically symmetric Weyl nodes that all live at the\nsame energy. This is the main limitation of the current\ntheory, although many implications of realistic model ex-\ntensions can be readily anticipated. Energy differences\nbetween the nodes are easily included by associating dif-\nferent chemical potentials to the nodes, while a small\nWeyl node anisotropy is expected to introduce a simi-\nlar anisotropy in the induced dynamics and dissipation\nof local moments. It is possible that type-II Weyl nodes\nfalloutsideofthistheory’sdomain, sotheirexplorationis\nleft for future study. We also did not consider corrections\ndue to finite temperature and disorder.\nTheusefulnessofthistheoryfortheexperimentalchar-\nacterization of magnetic Weyl semimetals is guarantied\nin principle, but depends on several factors in reality.\nThe needed level of detail is not easy to achieve in the\nmeasurements of spin wave spectra. It requires at least\nvery clean samples, low temperatures, as well as a suffi-\nciently high energy resolution and adequate statistics to\nresolvewithlowerrorbarsthe energy/momentumdepen-\ndence of the inelastic neutron scattering. These aspects\nof measurements can always be improved, but there are\nalso material-related constraints: phonons, for example,\nmust not coexist with spin waves at the same momenta\nand frequencies. Still, some regions of the first Brillouin\nzone should expose the electronic damping mechanism\nand enable the proposed experimental characterization\nof magnetic Weyl semimetals. On the purely theoretical\nfront, the present study was concerned with a basic but\nintricate and important aspect of interaction physics in\na topological system. It plays a role in piecing together\na broader picture of magnetic correlated topological ma-\nterials, which can host non-trivial anisotropic magnetic\norders13, chiral magnetic states and excitations44, and\npossibly even exotic spin liquids62.\nV. ACKNOWLEDGMENTS\nI am very grateful for insightfull discussions with\nJonathanGaudet and Collin Broholm. 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Nikoli´ c, Physical Review B 101, 115144 (2020)."    },    {        "title": "2103.05871v3.Anisotropic_superconducting_spin_transport_at_magnetic_interfaces.pdf",        "content": "Anisotropic superconducting spin transport at magnetic interfaces\nYuya Ominato1, Ai Yamakage2, and Mamoru Matsuo1;3;4;5\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan and\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: October 18, 2022)\nWe present a theoretical investigation of anisotropic superconducting spin transport at a magnetic\ninterface between a p-wave superconductor and a ferromagnetic insulator. Our formulation describes\nthe ferromagnetic resonance modulations due to spin current generation depending on spin-triplet\nCooper pair, including the frequency shift and enhanced Gilbert damping, in a uni\fed manner. We\n\fnd that the Cooper pair symmetry is detectable from the qualitative behavior of the ferromagnetic\nresonance modulation. Our theory paves the way toward anisotropic superconducting spintronics.\nIntroduction.| Use of spin-triplet Cooper pairs as car-\nriers for spin currents in the emergent \feld of super-\nconducting spintronics is challenging1,2. Previous stud-\nies have demonstrated spin transport mediated by spin-\ntriplet Cooper pairs that formed at the s-wave supercon-\nductor (SC)/ferromagnet interfaces of Josephson junc-\ntions. The spin-singlet pairs in SCs are converted into\nspin-triplet pairs in half-metallic CrO 23. However, pre-\nvious studies on spin-triplet pairs at magnetic interfaces\nhave been limited to cases induced by the proximity ef-\nfect.\nOne promising candidate material system for investi-\ngation of spin-triplet currents to enable more active use\nof spin-triplet pairs is the p-wave SC/ferromagnetic in-\nsulator (FI) bilayer thin \flm system4,5. Tunneling of the\nspins is driven by the magnetization dynamics excited\nby ferromagnetic resonance (FMR) in the ferromagnetic\nmaterial via interfacial exchange coupling between the\nmagnetization in the FI and the electron spins in the\np-wave SC, and a spin-triplet current is expected to be\ngenerated. Furthermore, as a backaction of spin injec-\ntion, both the FMR frequency and the Gilbert damping\nof the FI should be modulated6{8. Although similar sce-\nnarios have already been studied vigorously in s-wave\nSC/ferromagnet systems, most previous studies have fo-\ncused on the Gilbert damping modulation due to spin\ninjection9{22. To gain an in-depth understanding of the\nspin-triplet transport mechanisms, the FMR modulation\nprocesses, including both the frequency shift and the en-\nhanced Gilbert damping, should be formulated micro-\nscopically in a systematic manner.\nDetermination of the pairing symmetry of the spin-\ntripletp-wave SCs within the same framework is also\ndesirable. Despite many years of research based on sev-\neral experimental techniques that detect the pairing sym-\nmetry, including nuclear magnetic resonance23, polar-\nized neutron scattering24{26, and muon-spin resonance\ntechniques27, there are few established candidate systems\nfor spin-triplet SCs28{32. The FMR modulation has been\nobserved in various nanoscale magnetic multilayers. Ac-\ncordingly, the technique is widely used to investigate a\n(c) FMR modulation due to the coupling between\n     spin-triplet Cooper pair and magnetization\nH\nH0H0+ δH\n(b) Spin-triplet Cooper pair\n(i) Chiral p-wave (ii) Helical p-wave\nα+ δα\nH\nH0αH0H0+ δHFISC\nFIxz\nY, yθ(a) System\nθZ\nXS-HFISC\nFIG. 1. Mechanism of FMR modulation due to anisotropic\nsuperconducting spin transport at magnetic interfaces. (a)\nPrecession axis located on the x-zplane, where the angle\nbetween the precession axis and the zaxis is\u0012(where 0\u0014\u0012\u0014\n\u0019=2). (b) Two types of spin-triplet Cooper pairs considered\nin this work. (c) FMR signal modulation in the SC/FI bilayer\nsystem compared with the signal in the FI monolayer.\nspin transport property in a variety of nanoscale thin\n\flm systems because it is highly sensitive. Thus one can\nexpect that the FMR measurements in p-wave SC/FI bi-\nlayer systems provide useful information about pairing\nsymmetry.\nIn this Letter, we investigate anisotropic superconduct-\ning spin transport at the magnetic interfaces of hybrid\nsystems composed of p-wave SC/FI thin \flms theoret-\nically, as illustrated in Fig. 1(a). The two-dimensional\nbulk SC is placed on the FI, where the FMR occurs. The\nprecession axis is rotated by an angle \u0012from the direc-\ntion perpendicular to the interface. Here, we use two\ncoordinate systems: ( x;y;z ) and (X;Y;Z ). Thezaxis is\nperpendicular to the interface and the xandyaxes arearXiv:2103.05871v3  [cond-mat.supr-con]  15 Oct 20222\nalong the interface. The ( X;Y;Z ) coordinate is obtained\nby rotating the angle \u0012around the yaxis, so that the\nprecession axis and the Zaxis are parallel. Figure 1(b)\nshows a schematic image of the spin-triplet Cooper pairs\nfor the chiral and helical p-wave SCs considered in this\nwork. Figure 1(c) shows a schematic image of the FMR\nsignal in the FI monolayer and the SC/FI bilayer. The\nFMR frequency and linewidth in the SC/FI bilayer are\nboth modulated because of the spin transfer occurring at\nthe interface.\nUsing the nonequilibrium Green's function method,\nwe formulate the FMR modulations due to the back\naction of the spin-triplet transport process systemati-\ncally. The main advantage of using the nonequilibrium\nGreen's function is dealing with both a spectral function\nand a nonequilibrium distribution function. Indeed, the\ninterface spin current is given by the expression using\nthe nonequilibrium distribution function, which shows\nthat the interface spin current by the spin pumping and\nthe enhanced Gilbert damping are proportional to each\nother. Furthermore, as an advantage of \feld theoretical\ntreatment, the frequency shift and the enhanced Gilbert\ndamping are both described in a uni\fed manner. Addi-\ntionally, it is shown that the symmetry of the spin-triplet\npairs can be extracted from the FMR modulations. The\nresults presented here o\u000ber a pathway toward develop-\nment of anisotropic superconducting spintronics.\nModel Hamiltonian.| The FMR modulation due to the\nSC adjacent to the FI is calculated microscopically using\nthe spin tunneling Hamiltonian method9{11,33{38. The\ne\u000bect of the SC on the FI is treated as a perturbation\nand suppression of ferromagnetism with the onset of su-\nperconductivity is assumed to be negligible, which is con-\nsistent with the results of spin pumping experiments in\nmagnetic multilayer thin \flms. The details of the model\nHamiltonians and the formulations are described in the\nSupplemental Material39. In the main text, we focus on\ngiving an overview of the model Hamiltonians and the\nformulations.\nThe total Hamiltonian H(t) comprises three terms\nH(t) =HFI(t) +HSC+Hex: (1)\nThe \frst term HFI(t) describes the bulk FI,\nHFI(t) =X\nk~!kby\nkbk\u0000h+\nac(t)by\nk=0\u0000h\u0000\nac(t)bk=0;(2)\nwhereby\nkandbkdenote the creation and annihilation op-\nerators of magnons with the wave vector k= (kx;ky;kz),\nrespectively. We assume the parabolic dispersion ~!k=\nDk2\u0000~\rH, where\r(<0) is the electron gyromagnetic ra-\ntio. The coupling between the microwave radiation and\nthe magnons is given by h\u0006\nac(t) = ~\rhacp\nSN=2e\u0007i!t,\nwherehacand!are the amplitude and the frequency of\nthe microwave radiation, respectively. Sis the magni-\ntude of the localized spin and Nis the number of sites\nin the FI. Note that the precession axis for the localized\nspin is \fxed along the Zaxis [see Fig. 1(a)].The second term HSCdescribes the two-dimensional\nbulk SCs,\nHSC=1\n2X\nkcy\nkHBdGck; (3)\nwhere we use the four-component notations\ncy\nk= (cy\nk\";cy\nk#;c\u0000k\";c\u0000k#); (4)\nck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T: (5)\nHere,cy\nksandcksdenote creation and annihilation op-\nerators, respectively, of electrons with the wave vector\nk= (kx;ky) and thezcomponent of the spin s=\";#.\nThe Bogoliubov-de Gennes Hamiltonian HBdGis a 4\u00024\nmatrix given by\nHBdG=\u0012\n\u0018k\u001b0\u0001k\n\u0000\u0001\u0003\n\u0000k\u0000\u0018k\u001b0\u0013\n; (6)\nwhere\u0018krepresents the energy of the electrons as mea-\nsured from their chemical potential, \u001b0is a 2\u00022 unit\nmatrix, and the pairing potential \u0001 kis also a 2\u00022 ma-\ntrix. We consider three pairing potential types, including\nthe spin-singlet s-wave pairing \u0001 k= \u0001i\u001byand two spin-\ntripletp-wave pairings \u0001 k= (dk\u0001\u001b)i\u001by, where their d\nvectors are given by\ndk=(\n\u0001(0;0;ei\u001ek) : Chiral p\u0000wave\n\u0001(\u0000sin\u001ek;cos\u001ek;0) : Helical p\u0000wave(7)\nwhere\u001ek= arctan(ky=kx) is an azimuth angle. The\nphenomenological form of the gap function is assumed\n\u0001 = 1:76kBTctanh\u0010\n1:74p\nTc=T\u00001\u0011\n; (8)\nwithTcthe superconducting transition temperature. By\ndiagonalizing HBdG, the quasiparticle energy is given by\nEk=p\n\u00182\nk+ \u00012for all SCs considered here. There-\nfore, one cannot distinguish them by the energy spectrum\nalone, and they are simple models suitable for studying\nthe di\u000berence of the magnetic responses due to the pair-\ning symmetry40.\nThe third term Hexrepresents the proximity exchange\ncoupling that occurs at the interface, which describes the\nspin transfer between the SC and the FI10,33,\nHex=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+ h:c:\u0001\n; (9)\nwhereJq;kis the matrix element for the spin transfer pro-\ncesses,\u001b\u0006\nq= (\u001bX\nq\u0006i\u001bY\nq)=2 represent the spin-\rip opera-\ntors for the electron spins in the SCs, and S\u0000\n\u0000k=p\n2Sby\nk\nandS+\nk=p\n2Sbkrepresent the Fourier component of the\nlocalized spin in the FI. Note that the precession axis is\nalong theZaxis, so that the Zcomponent of the spin\nis injected into the SC when the FMR occurs. Using3\nthe creation and annihilation operators of electrons and\nmagnons,Hexis written as\nHex=X\nq;k;k0;s;s0\u0010p\n2SJq;k\u001b+\nss0cy\nk0sck0+qs0by\n\u0000k+ h:c:\u0011\n:\n(10)\nFrom the above expression, one can see that Hexde-\nscribes electron scattering processes with magnon emis-\nsion and absorption.\nModulation of FMR.| The FMR modulation can be\nread from the retarded component of the magnon Green's\nfunction33, which is given by\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!); (11)\nwhere the Gilbert damping constant \u000bis introduced\nphenomenologically41{43. In the second-order perturba-\ntion calculation with respect to the matrix element Jq;k,\nthe self-energy caused by proximity exchange coupling is\ngiven by\n\u0006R\nk(!) =\u0000X\nqjJq;kj2\u001fR\nq(!); (12)\nwhere the dynamic spin susceptibility of the SCs is de-\n\fned as\n\u001fR\nq(!) :=Z\ndtei(!+i0)ti\n~\u0012(t)h[\u001b+\nq(t);\u001b\u0000\n\u0000q(0)]i:(13)\nThe pole of GR\nk(!) indicates the FMR modulation, i.e.,\nthe shift of resonance frequency and the enhancement of\nthe Gilbert damping. By solving the equation\n!\u0000!k=0\u0000(2S=~)Re\u0006R\nk=0(!) = 0; (14)\nat a \fxed microwave frequency !, one obtains the mag-\nnetic \feld at which the FMR occurs. The imaginary part\nof the self-energy gives the enhancement of the Gilbert\ndamping. Consequently, the frequency shift and the en-\nhanced Gilbert damping are given by\n\u000eH=2S\n\r~Re\u0006R\nk=0(!); \u000e\u000b =\u00002S\n~!Im\u0006R\nk=0(!):(15)\nFrom the above equations and Eq. (12), one can see that\nthe FMR modulation provides information about both\nthe interface coupling properties and the dynamic spin\nsusceptibility of the SCs.\nThe form of matrix element Jq;k=0depends on the\ndetails of the interface. In this work, we assume the\ninterface with uncorrelated roughness. jJq;k=0j2is given\nby\njJq;k=0j2=J2\n1\nN\u000eq;0+J2\n2l2\nNA; (16)\nwhere the \frst and second terms describe averaged uni-\nform contribution and uncorrelated roughness contribu-\ntion, respectively39.J1andJ2correspond to the meanvalue and variance, respectively. Ais the area of the in-\nterface, which is equal to the system size of the SC. lis\nan atomic scale length. Using Eq. (16), the self-energy\nfor the uniform magnon mode is given by\n\u0006R\nk=0(!) =\u0000J2\n1\nN\u001fR\nuni(!)\u0000J2\n2l2\nNA\u001fR\nloc(!); (17)\nwhere the uniform and local spin susceptibilities are de-\n\fned as\n\u001fR\nuni(!) := lim\njqj!0\u001fR\nq(!); \u001fR\nloc(!) :=X\nq\u001fR\nq(!):(18)\nThe self-energy \u0006R\nk=0(!) consists of two terms originating\nfrom the uniform and roughness contributions, so that\nboth\u001fR\nuni(!) and\u001fR\nloc(!) contribute to \u000eHand\u000e\u000b.\nHere, we discuss the FI thickness dependence on the\nFMR modulation44. From Eqs. (15), and (17), one can\nsee that the FMR modulation is inversely proportional\nto the FI thickness ( /A=N ) because\u001fR\nuni(!)/Aand\n\u001fR\nloc(!)/A2. This is consistent with the experiments on\nthe spin pumping in Y 3Fe5O12=Pt heterostructures45. In\norder to observe the FMR modulation experimentally, it\nis necessary to prepare a sample that is su\u000eciently thin,\ne.g., typically, the thickness of several tens of nanometers.\nNumerical results.| In the following, we consider a \rat\ninterface where J2= 0, so that the behavior of the FMR\nmodulation is determined by \u001fR\nuni(!). The roughness\ncontribution proportional to \u001fR\nloc(!) is discussed later.\nFigure 2 shows the frequency shift \u000eHand the enhanced\nGilbert damping \u000e\u000bas a function of temperature and fre-\nquency. Here, we set \u0012= 0 and \u0000=kBTc= 0:05, where \u0000\nis a constant level broadening of the quasiparticle intro-\nduced phenomenologically39.\nFirst, we explain the qualitative properties of \u000eHand\n\u000e\u000bfor the chiral p-wave SC. In the low frequency re-\ngion, where ~!=kBTc\u00141,\u000eHis \fnite and remains al-\nmost independent of !near the zero temperature and\n\u000e\u000bdecreases and becomes exponentially small with the\ndecrease of the temperature. In the high frequency re-\ngion, where ~!=kBTc\u00151, a resonance peak occurs at\n~!= 2\u0001 for both \u000eHand\u000e\u000b. The qualitative proper-\nties of\u000eHand\u000e\u000bfor the helical p-wave SC are the same\nas those of the chiral p-wave SC.\nNext, we explain the qualitative properties of \u000eHand\n\u000e\u000bfor thes-wave SC. In the low frequency region, where\n~!=kBTc\u00141, both\u000eHand\u000e\u000bdecrease and become\nexponentially small with the decrease of the temperature.\nIn the high frequency region, where ~!=kBTc\u00151, both\n\u000eHand\u000e\u000bvanish.\nThep-wave SCs show two characteristic properties\nthat thes-wave SC does not show: a \fnite \u000eHatT= 0\nand a resonance peak of \u000eHand\u000e\u000b. These properties\ncan be understood by the analogy between SCs and band\ninsulators as follows. The uniform dynamic spin suscepti-\nbility consists of contributions from intraband transitions\nwithin particle (hole) bands and interband transitions\nbetween particles and holes. In the low temperature or4\n(a) (b)Γ/kBTc=0.05 Chiral p-wave\n(c) (d)Γ/kBTc=0.05 Helical p-wave\n(e) (f)Γ/kBTc=0.05 s-waveT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n42\n1\n0\nT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nFIG. 2. The frequency shift \u000eHand the enhanced Gilbert\ndamping\u000e\u000bas a function of temperature and frequency nor-\nmalized by the characteristic values \u000eH1=\u0000SJ2\n1DF=(N\r~)\nand\u000e\u000b1=SJ2\n1DF=(NkBTc) in the normal state. DF(/A)\nis the density of states at the Fermi level in the normal state.\nWe set\u0012= 0 and \u0000=kBTc= 0:05. The sign of \u000eHcorresponds\nto the sign of Re \u001fR\nuni(!), which can be positive and negative\nat low and high frequencies, respectively. In contrast, \u000e\u000bis\npositive at any frequency.\nhigh frequency region, the intraband contribution is neg-\nligible and the interband contribution is dominant. In\nthe case of the s-wave SC, the interband transitions are\nforbidden because the Hamiltonian and the spin operator\ncommute. As a result, there is no spin response in the\nlow-temperature or high-frequency regions. In contrast,\nthe Hamiltonian for the p-wave SCs and the spin operator\ndo not commute. Therefore, \u000eHhas a \fnite value near-\nzero temperature due to the interband contribution. In\naddition, a resonance peak occurs when ~!= 2\u0001 because\nthe density of states diverges at the band edge E=\u0006\u0001.\nA detailed proof of the above statement is given in the\nSupplemental Material39.\nThe angle dependences of \u000eHand\u000e\u000bare distinct for\nchiral and helical p-wave SCs, as shown in Fig. 3. In\nboth cases, we set ~!=kBTc= 3:0 as the typical values\nat high frequencies, where the main contribution of the\nuniform spin susceptibility is the interband transitions.\nIn the chiral p-wave SC,\u000eHand\u000e\u000btend to decrease and\nare halved at a \fxed temperature when \u0012increases from\n0 to\u0019=2. Conversely, in the helical p-wave SC, the qual-\n0.4\nT/Tc1.0 0.2 0.08\n4\n0\n0.8 0.6δH/δH1\n2\n-210\n60.4\nT/Tc1.0 0.2 0.08\n4\n0\n0.8 0.6δH/δH1\n2\n-210\n6\nHelical p-waveChiral p-wave\nθ=0\nπ/4\nπ/2\nθ=0π/4π/2\n0.4\nT/Tc1.0 0.2 0.02\n0\n0.8 0.6δα/δα1\n13θ=0\nπ/4\nπ/2\nθ=0π/4π/20.4\nT/Tc1.0 0.2 0.02\n0\n0.8 0.6δα/δα1\n13Γ/kBTc=0.05, hω/kBTc=3.0\n(a) (b)\n(c) (d)Γ/kBTc=0.05, hω/kBTc=3.0FIG. 3. Frequency shift and the enhanced Gilbert damping\nas a function of temperature at angles of \u0012= 0;\u0019=4;\u0019=2. The\nupper and lower panels show the characteristics for the chiral\nand helical p-wave SCs, respectively.\nitative behavior shows the opposite trend. \u000eHand\u000e\u000b\nboth tend to increase and become 1 :5 times larger at a\n\fxed temperature when \u0012increases from 0 to \u0019=2. In\nfact, the angle dependences are approximately obtained\nto be/1 +cos2\u0012and 1 +(sin2\u0012)=2 for chiral and helical\np-wave SCs, respectively39. Therefore, the spin con\fg-\nuration of the Cooper pair can be detected from the \u0012\ndependence data for the FMR modulation.\nThe FMR modulation properties of the three SCs are\nsummarized in Table I. All SCs considered here can be\ndistinguished based on three properties: the frequency\nshift in the low temperature limit, the presence of their\nresonance peak, and their \u0012dependence. For the s-wave\nSC,\u000eHbecomes exponentially small in T!0, while for\nthep-wave SCs, \u000eHis \fnite in T!0. For the s-wave\nSC,\u000eHand\u000e\u000bshow no resonance and no \u0012dependence,\nwhile for the chiral and helical p-wave SCs, both \u000eHand\n\u000e\u000bexhibit a resonance at ~!= 2\u0001 and a \u0012dependence.\nIn addition, these two p-wave SCs can be distinguished\nfrom their\u0012dependences of \u000eHand\u000e\u000b, which are char-\nacterized by @\u0012(\u000eH) and@\u0012(\u000e\u000b), respectively. Here, it\nshould be emphasized that the pairing symmetry can be\ncharacterized by the sign of @\u0012(\u000eH) and@\u0012(\u000e\u000b). These\nproperties are summarized in the Table I.\nSpin-triplet current generation.| The relationship be-\ntween the enhanced Gilbert damping discussed above5\nand the spin-triplet current generation must also be dis-\ncussed. The enhancement of the Gilbert damping is\nknown to originate from the spin current generation at\nthe magnetic interface6,33. The interface spin current in-\nduced by FMRhISiSPis given by39\nhISiSP=N(~\rhac)2\n2\u000b\u0002\n\u0000ImGR\nk=0(!)\u0003\n\u000e\u000b: (19)\nOne can see that hISiSPand\u000e\u000bare proportional to each\nother. In our setup, the enhanced Gilbert damping \u000e\u000b\nwill lead to the generation of both the Cooper pair spin-\ntriplet current and the quasiparticle spin current. Since\nthe angular dependence of \u000e\u000bre\rects the direction of\nthe Cooper pair spins, it is expected that the spin-triplet\ncurrent can be controlled by varying the magnetization\ndirection of the FI.\nDiscussion.| We have considered a \rat SC/FI inter-\nface. In the presence of roughness, the correction term\nproportional to \u001fR\nloc(!) contributes to the FMR mod-\nulation, as shown in Eq. (17). In the rough limit,\nJ2\n1\u001cJ2\n2,\u001fR\nloc(!) dominates to make the FMR modu-\nlation isotropic, due to the angle average by summation\noverq. Namely, the anisotropy peculiar to p-wave SC\nis smeared by the roughness. The detailed behavior of\n\u001fR\nloc(!) is shown in the Supplemental Material39. This\nresult implies that it is crucial to control the interface\nroughness. In principle, the roughness of the interface\ncan be observed using transmission electron microscopy\nof interfaces46{48and it is possible to detect whether the\ninterface of the sample is \rat or rough. More detailed\nspectroscopy can be obtained from the FMR modulation\nby using a \rat interface.\nOur results show that the pairing symmetry can be\ndetected by the sign of @\u0012(\u000eH) and@\u0012(\u000e\u000b) around the\nin-plane magnetic \feld ( \u0012\u0018\u0019=2), where the vortices are\nnegligible. When the external magnetic \feld has a large\nout-of-plane component, the vortex formation may cause\nproblems in observing the angular dependence. The qual-\nitative behavior is expected to change when the out-of-\nplane magnetic \feld approaches the upper critical \feld\n(H\u0018Hc2\u00181T). This is because the coherence length\nof the Cooper pair and the distance between the vor-\ntices can become comparable. Indeed, it has been exper-\nimentally reported that the vortex formation suppresses\nthe characteristic properties in the spin pumping into\nSCs20. Therefore, the out-of-plane magnetic \feld should\nbe as small as possible when FMR measurements are per-\nformed for H\u0018Hc2.\nTABLE I. FMR modulation properties for the \rat SC/FI in-\nterface where J16= 0 andJ2= 0.\nPairing symmetry s Chiral Helical\n\u000eHin the limit of T!0 0 \fnite \fnite\nResonance peak of \u000eH,\u000e\u000b { X X\n@\u0012(\u000eH),@\u0012(\u000e\u000b) 0 negative positiveRecent experiments have reported that UTe 2is a can-\ndidate material for spin-triplet p-wave SCs31, which has\nattracted a great deal of attention. Various experi-\nments, including spectroscopic measurements, are now\nin progress to investigate the pairing symmetry of UTe 2,\nand indicated that the superconducting transition tem-\nperature is about 1K \u001830 GHz. Therefore, the resonance\ncondition ~!= 2\u0001 shown above is accessible to recent\nbroadband FMR measurements.\nIn addition, experiments on spin pumping into d-wave\nSCs have recently been reported49and a theoretical in-\nvestigation of the enhancement of the Gilbert damp-\ning in ad-wave SC/FI bilayer system has recently been\npresented50. Thus anisotropic superconducting spintron-\nics can be expected to develop as a new research direc-\ntion.\nWe should emphasize two important aspects of the\nFMR method presented here: the spectroscopic probe\nmethod for the p-wave SC thin \flms and the versa-\ntile spin injection method. First, the FMR measure-\nment procedure can provide a new spin-sensitive mea-\nsurement method that will complement other measure-\nment methods to enable a breakthrough in the discovery\nof spin-triplet SCs. Second, the FMR method represents\na promising way to generate spin-triplet currents in p-\nwave SC thin \flms.\nConclusions.| We have investigated the anisotropic\nsuperconducting spin transport at magnetic interfaces\ncomposed of a p-wave SC and an FI based on a micro-\nscopic model Hamiltonian. The FMR signal in these p-\nwave SC/FI bilayer systems is modulated via spin trans-\nfer at the interface, which generates spin-triplet currents.\nWe have shown that the pairing symmetry of the SCs\ncan be extracted from the FMR modulation character-\nistics. Our approach provides a unique way to explore\nanisotropic superconducting spintronics, which will be\nuseful for application to emerging device technologies.\nNote added.| After the submission of this manuscript,\nwe became aware of a closely related work, where a way\nto convert spin-triplet currents to magnon spin currents\nin SC/FI bilayer systems is discussed51.\nWe thank R. Ohshima, M. Shiraishi, H. Chudo, G.\nOkano, K. Yamanoi, and Y. Nozaki for helpful discus-\nsions. This work was supported by the Priority Pro-\ngram of the Chinese Academy of Sciences under Grant\nNo. XDB28000000, and by JSPS KAKENHI under\nGrants Nos. JP20K03835, JP20H04635, JP20H01863,\nJP21H04565, and JP21H01800.6\nSUPPLEMENTAL MATERIAL\nI. MODEL HAMILTONIAN\nIn this section, we describe the derivation and details of the model Hamiltonian used in the main text.\nA. Ferromagnetic Heisenberg model\nThe ferromagnetic Heisenberg model with the transverse AC magnetic \feld due to the microwave radiation is given\nby\nHFI(t) =\u0000JX\nhi;jiSi\u0001Sj+~\rHX\njSZ\nj\u0000~\rhacX\nj\u0000\nSX\njcos!t\u0000SY\njsin!t\u0001\n; (S.1)\nwhereJ >0 is the exchange coupling constant, hi;jirepresents summation over all nearest-neighbor sites, Sjis the\nlocalized spin at site jin the ferromagnetic insulator (FI), \r(<0) is the gyromagnetic ratio, His a static magnetic\n\feld,hacis an amplitude of an transverse oscillating magnetic \feld due to the microwave radiation with a frequency\n!. The rotated coordinates ( X;Y;Z ) are shown in Fig. 1(a).\nIt is convenient to introduce the boson creation and annihilation operators in order to formulate the problem in\nterms of the quantum \feld theory. In the current problem, we perturbatively treat the excitation of the FI. In this\ncase, the Holstein-Primako\u000b transformation is useful, where the localized spin can be described using boson creation\nand annihilation operators bj;by\njin Hilbert space constrained to 2 S+ 1 dimensions. The spin operators are written as\nS+\nj=SX\nj+iSY\nj=\u0010\n2S\u0000by\njbj\u00111=2\nbj; (S.2)\nS\u0000\nj=SX\nj\u0000iSY\nj=by\nj\u0010\n2S\u0000by\njbj\u00111=2\n; (S.3)\nSZ\nj=S\u0000by\njbj; (S.4)\nwhere we require [ bi;by\nj] =\u000ei;j;in order that the S+\nj,S\u0000\nj, andSZ\njsatisfy the commutation relation of angular\nmomentum. The deviation of SZ\njfrom its ground-state value Sis quanti\fed by the boson particle number.\nWe consider low-energy excitation in the FI, where the deviation of SZ\njfrom the ground state is small hby\njbji=S\u001c1.\nThe ladder operators S\u0006\njare approximated as\nS+\nj\u0019(2S)1=2bj; (S.5)\nS\u0000\nj\u0019(2S)1=2by\nj; (S.6)\nwhich is called spin-wave approximation. Here, we de\fne the magnon operators\nbk=1p\nNX\nje\u0000ik\u0001rjbj; (S.7)\nby\nk=1p\nNX\njeik\u0001rjby\nj; (S.8)\nwhereNis the number of sites and k= (kx;ky;kz). The inverse transformation is then given by\nbj=1p\nNX\nkeik\u0001rjbk; (S.9)\nby\nj=1p\nNX\nke\u0000ik\u0001rjby\nk: (S.10)\nThe magnon operators satisfy [ bk;by\nk0] =\u000ek;k0and describe the quantized collective excitations. Using the spin-wave\napproximation and the magnon operators, the Hamiltonian HFI(t) is written as\nHFI(t)\u0019X\nk~!kby\nkbk\u0000h+\nac(t)by\nk=0\u0000h\u0000\nac(t)bk=0; (S.11)7\nwhere ~!k=Dk2\u0000~\rHwithD= 2JSa2and the lattice constant a,h\u0006\nac(t) =~\rhacp\nSN=2e\u0007i!t, and constant\nterms are omitted.\nB. BCS Hamiltonian\nWe derive a mean-\feld Hamiltonian, which describes a bulk superconductor (SC), and we diagonalize the mean-\feld\nHamiltonian with the Bogoliubov transformation. At the end of this section, the spin density operators of the SC are\nwritten in terms of the Bogoliubov quasiparticle creation and annihilation operators.\nWe start with the e\u000bective Hamiltonian in momentum space\nHSC=X\nk;s\u0018kcy\nkscks+1\n2X\nk;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4; (S.12)\nwhere\u0018kis the band energy measured relative to the chemical potential, and cy\nksandcksare the creation and\nannihilation operators of electrons with the wave vector k= (kx;ky) and thezcomponent of the spin s=\";#. The\nmatrix elements satisfy\nVs1;s2;s3;s4(k;k0) =\u0000Vs2;s1;s3;s4(\u0000k;k0); (S.13)\nVs1;s2;s3;s4(k;k0) =\u0000Vs1;s2;s4;s3(k;\u0000k0); (S.14)\nbecause of the anticommutation relation of fermions, and\nVs1;s2;s3;s4(k;k0) =V\u0003\ns4;s3;s2;s1(k0;k); (S.15)\nbecause of the Hermitianity of the Hamiltonian. We consider a mean-\feld, which is called a pair potential\n\u0001k;ss0=\u0000X\nk0;s3;s4Vs0;s;s 3;s4(k;k0)hck0s3c\u0000k0s4i; (S.16)\nand its conjugate\n\u0001\u0003\n\u0000k;ss0=X\nk0;s1;s2Vs1;s2;s0;s(k0;k)hcy\n\u0000k0s1cy\nk0s2i: (S.17)\nHere, we consider a mean-\feld approximation where the interaction term is replaced as follows\ncy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4!cy\n\u0000ks1cy\nks2hck0s3c\u0000k0s4i+hcy\n\u0000ks1cy\nks2ick0s3c\u0000k0s4\u0000hcy\n\u0000ks1cy\nks2ihck0s3c\u0000k0s4i; (S.18)\nso that the interaction term is rewritten as\nX\nk;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4!X\nk;s1;s2h\n\u0001k;s1s2cy\nks1cy\n\u0000ks2\u0000\u0001\u0003\n\u0000k;s1s2c\u0000ks1cks2i\n; (S.19)\nwhere an constant term is omitted. Consequently, we derive a mean-\feld Hamiltonian\nHSC=X\nk;s\u0018kcy\nkscks+1\n2X\nk;s1;s2\u0002\n\u0001k;s1s2cy\nks1cy\n\u0000ks2\u0000\u0001\u0003\n\u0000k;s1s2c\u0000ks1cks2\u0003\n: (S.20)\nUsing a four-component notation\ncy\nk= (cy\nk\";cy\nk#;c\u0000k\";c\u0000k#); (S.21)\nck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T; (S.22)\nthe mean-\feld Hamiltonian is written as\nHSC=1\n2X\nkcy\nkHBdGck: (S.23)8\nHBdGis the 4\u00024 matrix\nHBdG= \n\u0018k\u001b0\u0001k\n\u0000\u0001\u0003\n\u0000k\u0000\u0018k\u001b0!\n; (S.24)\nwhere\u001b0is the 2\u00022 unit matrix and \u0001 kis the 2\u00022 matrix given as\n\u0001k= \n\u0001k;\"\"\u0001k;\"#\n\u0001k;#\"\u0001k;##!\n: (S.25)\nIn principle, the pair potential is obtained by solving the gap equation self-consistently for an explicit form of the\nmatrix elements Vs1;s2;s3;s4(k;k0). In this work, we do not solve the gap equation, but instead assume an explicit\nform of the pair potential and perform calculations using a phenomenological gap function. For the singlet pairing,\nthe pair potential is given by\n\u0001k= ki\u001by; (S.26)\nwith an even function  k= \u0000k. For ans-wave SC, the pair potential is given by\n\u0001k= \u0001 \n0 1\n\u00001 0!\n: (S.27)\nFor the triplet pairing, the pair potential is given by\n\u0001k= [dk\u0001\u001b]i\u001by; (S.28)\nwith an odd vectorial function dk=\u0000d\u0000k. For a chiral p-wave SC and a helical p-wave SC,dkis given by\ndk=(\n\u0001(0;0;ei\u001ek) : chiral p\u0000wave\n\u0001(\u0000sin\u001ek;cos\u001ek;0) : helical p\u0000wave(S.29)\nwith\u001ek= arctan(ky=kx), so that the pair potential is given by\n\u0001k=8\n>>>><\n>>>>:\u0001 \n0ei\u001ek\nei\u001ek0!\n: chiralp\u0000wave\n\u0001 \nie\u0000i\u001ek0\n0iei\u001ek!\n: helicalp\u0000wave(S.30)\nThe phenomenological gap function is given by\n\u0001 = 1:76kBTctanh\u0010\n1:74p\nTc=T\u00001\u0011\n: (S.31)\nThe Bogoliubov transformation to diagonalize HBdGis given by\nUk= \nukvk\nv\u0003\n\u0000ku\u0003\n\u0000k!\n; (S.32)\nUy\nk= \nuk\u0000vk\n\u0000v\u0003\n\u0000ku\u0003\n\u0000k!\n; (S.33)\nwith the 2\u00022 matricesukandvkgiven by\nuk=s\n1\n2\u0012\n1 +\u0018k\nEk\u0013\n\u001b0; (S.34)\nvk=\u0000s\n1\n2\u0012\n1\u0000\u0018k\nEk\u0013\u0001k\n\u0001; (S.35)9\nwhereEkis the eigenenergy\nEk=q\n\u00182\nk+ \u00012: (S.36)\nUsing the Bogoliubov transformation Uk, the 4\u00024 matrixHBdGis diagonalized as\nUy\nkHBdGUk=0\nBBB@Ek0 0 0\n0Ek0 0\n0 0\u0000Ek0\n0 0 0\u0000Ek1\nCCCA: (S.37)\nThe excitation of HSCis described by the creation and annihilation operators of the Bogoliubov quasiparticles \r(y)\nk\n\ry\nk= (\ry\nk\";\ry\nk#;\r\u0000k\";\r\u0000k#); (S.38)\n\rk= (\rk\";\rk#;\ry\n\u0000k\";\ry\n\u0000k#)T; (S.39)\nwhere they are obtained by the Bogoliubov transformation\n\rk=Uy\nkck; (S.40)\n\ry\nk=cy\nkUk: (S.41)\nThe spin density operators \u001ba(r) (a=x;y;z ) is de\fned as\n\u001ba(r) :=1\nAX\nk;k0;s;s0e\u0000i(k\u0000k0)\u0001r\u001ba\nss0cy\nksck0s0; (S.42)\nwhereAis the area of the system. \u001ba(r) (a=x;y;z ) is expanded in Fourier series\n\u001ba(r) =1\nAX\nqeiq\u0001r\u001ba\nq; (S.43)\nand the Fourier coe\u000ecient is given by\n\u001ba\nq=Z\ndre\u0000iq\u0001r\u001ba(r) =X\nk;s;s0\u001ba\nss0cy\nksck+qs0: (S.44)\nUsing the Bogoliubov transformation Uk, the above expression is rewritten as\n\u001ba\nq=X\nk;s;s0\"\u0010\nsa(1)\nk;k+q\u0011\ns;s0\ry\nks\rk+qs0+\u0010\nsa(2)\nk;k+q\u0011\ns;s0\r\u0000ks\ry\n\u0000k\u0000qs0+\u0010\nsa(3)\nk;k+q\u0011\ns;s0\ry\nks\ry\n\u0000k\u0000qs0+\u0010\nsa(4)\nk;k+q\u0011\ns;s0\r\u0000ks\rk+qs0#\n;\n(S.45)\nwith the 2\u00022 matricessa(i)\nk;k+qgiven by\nsa(1)\nk;k+q=uy\nk\u001bauk+q; (S.46)\nsa(2)\nk;k+q=vy\nk\u001bavk+q; (S.47)\nsa(3)\nk;k+q=uy\nk\u001bavk+q; (S.48)\nsa(4)\nk;k+q=vy\nk\u001bauk+q: (S.49)\nThe \frst and second terms describe the intraband transition from particle-to-particle and from hole-to-hole, respec-\ntively. The third and fourth terms describe the interband transition from hole-to-particle and from particle-to-hole,\nrespectively.10\nC. Proximity exchange coupling at interface\nWe start with a model for the proximity exchange coupling given by\nHex=Z\ndrX\njJ(r;rj)\u001b(r)\u0001Sj: (S.50)\nWe rewrite the above expression in the real space into the expression in the wave space. The proximity exchange\ncoupling is rewritten as\nHex=Z\ndrX\njJ(r;rj)1\nAp\nNX\nq;kei(q\u0001r+k\u0001rj)\u0000\n\u001b+\nqS\u0000\nk+\u001b\u0000\nqS+\nk\u0001\n+Z\ndrX\njJ(r;rj)\u001bZ(r)SZ\nj; (S.51)\nwhere the Fourier series are given by\n\u001b(r) =1\nAX\nqeiq\u0001r\u001bq; (S.52)\nSj=1p\nNX\nkeik\u0001rjSk; (S.53)\nwith the area of the SC, A, and the number of sites in the FI, N, and the ladder operators are given by\n\u001b\u0006=1\n2(\u001bX\u0006i\u001bY); (S.54)\nS\u0006=SX\u0006iSY: (S.55)\nThe matrix element is given by\nJq;k=1\nAp\nNZ\ndrX\njJ(r;rj)ei(q\u0001r+k\u0001rj): (S.56)\nConsequently, the exchange coupling which we use in the main text is derived as\nHex=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+J\u0003\nq;k\u001b\u0000\n\u0000qS+\n\u0000k\u0001\n; (S.57)\nwhere we use a relation J\u0000q;\u0000k=J\u0003\nq;k, and we omit the last term\nZ\ndrX\njJ(r;rj)\u001bZ(r)SZ\nj; (S.58)\nin order to focus on the spin transfer at the interface. For the uniform magnon mode jkj= 0, the matrix element is\ngiven by\nJq;k=0=1\nAp\nNZ\ndrX\njJ(r;rj)eiq\u0001r: (S.59)\nII. TIME DEPENDENT QUANTUM AVERAGE\nIn this section, we show that the ferromagnetic resonance (FMR) frequency and linewidth are read from the\nmagnon Green's function. We consider the Hamiltonian H(t) composed of the unperturbed Hamiltonian H0and the\nperturbation V(t)\nH(t) =H0+V(t): (S.60)\nThe time-dependent quantum average of a physical quantity Ois calculated as\nhO(t)i=hSy(t;\u00001)~O(t)S(t;\u00001)i; (S.61)11\nwhere ~O(t) is the interaction picture and the S matrix S(t;t0) is given by\nS(t;t0) =Texp Zt\nt0dt0~V(t0)\ni~!\n: (S.62)\nThe time-dependent quantum average hO(t)iis written as\nhO(t)i=hOieq+\u000ehO(t)i; (S.63)\nwherehOieq= Tr (\u001aeqO) is the equilibrium value and \u000ehO(t)iis deviation from the equilibrium. When the perturbation\nis written as V(t) =\u0000AF(t), the \frst order perturbation calculation gives\n\u000ehO(t)i=\u0000Zt\n\u00001dt01\ni~h[~O(t);~A(t0)]iF(t0)\n=\u0000Z1\n\u00001dt0GR(t0)F(t\u0000t0); (S.64)\nwhere we de\fne the retarded Green's function\nGR(t) =1\ni~\u0012(t)h[~O(t);~A(0)]i: (S.65)\nWhen the external force is written as F(t) =Fe\u0000i(!+i0)t,\u000ehO(t)iis written as\n\u000ehO(t)i=\u0000Fe\u0000i(!+i0)tZ1\n\u00001dt0ei(!+i0)t0GR(t0)\n=\u0000Fe\u0000i!tGR(!): (S.66)\nUsing the above formula, the dynamics of \u000ehS+\nk=0(t)iis written as\n\u000ehS+\nk=0(t)i=\u0000~\rhacp\nN\n2e\u0000i!tGR\nk=0(!); (S.67)\nwhereGR\nk(!) is the Fourier transform of the retarded component of the magnon Green's function GR\nk(t). They are\nde\fned as\nGR\nk(t) :=1\ni~\u0012(t)h[S+\nk(t);S\u0000\n\u0000k(0)]i; (S.68)\nGR\nk(!) :=Z1\n\u00001dt0ei(!+i0)t0GR\nk(t0): (S.69)\nFrom Eq. (S.67), one can see that the FMR frequency and linewidth are read from GR\nk(!).\nIII. MAGNON GREEN'S FUNCTION\nIn this section, we perform perturbative calculation for the magnon Green's function. We treat the proximity\nexchange coupling as a perturbation. The Hamiltonian is written as\nH=H0+V; (S.70)\nwhereH0is the unperturbed Hamiltonian\nH0=X\nk~!kby\nkbk+X\nk;sEk\ry\nks\rks; (S.71)\nandVis the perturbation\nV=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+ h:c:\u0001\n: (S.72)12\n(e) Vertex(b) Keldysh contour\n(d) Self-energytime(a) Magnon Green’s function (c) Dyson equation\nk/uni2032 +qs/uni2032 −k\n−k/uni2032 −qs/uni2032 /uni03C3+\nqS−\nk= + + +−k −k −k−k/uni2032 s k/uni2032 s k/uni2032 s\n−k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 sGk(/uni03C4,/uni03C4/uni2032 )= = + /uni03A3\n/uni03A3 = + + +\nk/uni2032 +qs/uni2032 −k/uni2032 −qs/uni2032 −k/uni2032 s k/uni2032 s k/uni2032 s\n−k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 s\nBogoliubov quasiparticle: Magnon:\nFIG. 4. (a) The Feynman diagram for the magnon Green's function. (b) Keldysh contour to perform perturbative calculations.\n(c) The Feynman diagram for the Dyson equation. (d) The self-energy within the second-order perturbation is given by the\ndynamic spin susceptibility of the SCs. (e) The Feynman diagrams for the vertex \u001b+\nqS\u0000\nk, which represent scattering of a\nBogoliubov quasiparticle with magnon emission. The solid and wavy lines represent a Bogoliubov quasiparticle and a magnon,\nrespectively.\nWe de\fne the magnon Green's function\nGk(\u001c;\u001c0) :=1\ni~hTCS+\nk(\u001c)S\u0000\n\u0000k(\u001c0)i; (S.73)\nwhereTCis the time-ordering operator on the Keldysh contour (see Figs. 4(a) and (b)). To perform the perturbative\ncalculation, we introduce interaction picture. The perturbation is written as\n~V(t) =X\nq;k\u0010\nJq;k~\u001b+\nq(t)~S\u0000\nk(t) + h:c:\u0011\n: (S.74)\nThe magnon Green's function is given by\nGk(\u001c;\u001c0) =1\ni~hTCSC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c0)iconn; (S.75)\nwhereh\u0001\u0001\u0001i connmeans the connected diagrams and the S matrix is given by\nSC=TCexp Z\nCd\u001c~V(\u001c)\ni~!\n: (S.76)\nThe above expressions lead to the Dyson equation (see Fig. 4(c))\nGk(\u001c;\u001c0) =G(0)\nk(\u001c;\u001c0) +Z\nCd\u001c1Z\nCd\u001c2G(0)\nk(\u001c;\u001c1)\u0006k(\u001c1;\u001c2)Gk(\u001c2;\u001c0); (S.77)\nwhereG(0)\nk(\u001c;\u001c0) is the unperturbed magnon Green's function\nG(0)\nk(\u001c;\u001c0) =1\ni~hTC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c0)i; (S.78)\nand \u0006 k(\u001c1;\u001c2) is the self-energy. Within the second-order perturbation, the self-energy is given by (see Fig. 4(d))\n\u0006k(\u001c;\u001c0) =1\ni~X\nqjJq;kj2hTC~\u001b+\nq(\u001c)~\u001b\u0000\n\u0000q(\u001c0)i: (S.79)\nThe Feynman diagram for the vertex is shown in Fig. 4(e). Substituting the ladder operators expressed in terms of13\n\r(y)\nks, the self-energy is written as\n\u0006k(\u001c;\u001c0) =\u0000i~X\nqjJq;kj2X\nk0;s;s0\"\u0012\f\f\f(s+(1)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(1)\nk0;k0+q)s;s0(s\u0000(2)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ngk0;s(\u001c0;\u001c)gk0+q;s0(\u001c;\u001c0)\n+\u0012\f\f\f(s+(2)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(2)\nk0;k0+q)s;s0(s\u0000(1)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ng\u0000k0;s(\u001c;\u001c0)g\u0000k0\u0000q;s0(\u001c0;\u001c)\n\u0000\u0012\f\f\f(s+(3)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(3)\nk0;k0+q)s;s0(s\u0000(3)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ngk0;s(\u001c0;\u001c)g\u0000k0\u0000q;s0(\u001c0;\u001c)\n\u0000\u0012\f\f\f(s+(4)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(4)\nk0;k0+q)s;s0(s\u0000(4)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ng\u0000k0;s(\u001c;\u001c0)gk0+q;s0(\u001c;\u001c0)#\n;\n(S.80)\nwhere the quasiparticle Green's function is de\fned as\ngk;s(\u001c;\u001c0) :=1\ni~hTC~\rks(\u001c)~\ry\nks(\u001c0)i: (S.81)\nThe \frst and second terms give the intraband contribution, and the third and fourth terms give the interband\ncontribution. Evaluating the Dyson equation, the retarded component of the magnon Green's function is given by\nGR\nk(!) =1\nh\nG(0)R\nk(!)i\u00001\n\u0000\u0006R\nk(!); (S.82)\nwhere the unperturbed Green's function is written as\nG(0)R\nk(!) =2S=~\n!\u0000!k+i\u000b!: (S.83)\nHere, we introduce the phenomenological dimensionless damping parameter \u000b. Using Eq.(S.83), the retarded Green's\nfunction is written as\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!): (S.84)\nFrom the above expression, the frequency shift at a \fxed !is given by\n\u000eH=2S\n\r~Re\u0006R\nk(!); (S.85)\nand the enhanced Gilbert damping is given by\n\u000e\u000b=\u00002S\n~!Im\u0006R\nk(!): (S.86)\nThe Fourier transform of the self-energy is given as\n\u0006R\nk(!) =Z\ndtei(!+i0)t\u0006R\nk(t) =\u0000X\nqjJq;kj2\u001fR\nq(!); (S.87)\nwhere the dynamic spin susceptibility of the SC is de\fned as\n\u001fR\nq(!) :=Z\ndtei(!+i0)ti\n~\u0012(t)h[~\u001b+\nq(t);~\u001b\u0000\n\u0000q(0)]i: (S.88)\nEvaluating the self-energy Eq. (S.87), one can obtain the information of the FMR modulation, \u000eHand\u000e\u000b. Using the\nsystem's symmetry, the dynamic spin susceptibility \u001fR\nq(!) can be written as\n\u001fR\nq(!) = cos2\u0012\u001fxx\nq(!) +\u001fyy\nq(!) + sin2\u0012\u001fzz\nq(!); (S.89)\nwhich means that both \u000eHand\u000e\u000bshow a dependence on \u0012when the dynamic spin susceptibility is anisotropic.14\nIV. SPIN CURRENT AT THE INTERFACE\nIn this section, we derive the general expression of spin current at the interface. We treat the tunneling Hamiltonian\nas a perturbation and the other terms as the unperturbed Hamiltonian\nH(t) =H0(t) +Hex; (S.90)\nH0(t) =HFI(t) +HSC: (S.91)\nThe operator of spin current \rowing from the SC to the FI at the interface is de\fned by\nIS:=\u0000~\n2_\u001bZ\ntot=\u0000~\n21\ni~[\u001bZ\ntot;Hex] =i\n2[\u001bZ\ntot;Hex]; (S.92)\nwhere\u001bZ\ntotis given by\n\u001bZ\ntot=Z\ndr\u001bZ(r): (S.93)\nCalculating the commutation relation, we obtain the following expression\nIS=iX\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk\u0000h:c:\u0001\n: (S.94)\nThe time-dependent quantum average of ISis written as\nhIS(t)i= Re2\n42iX\nq;kJq;kh\u001b+\nq(t)S\u0000\nk(t)i3\n5; (S.95)\nwhereh\u0001\u0001\u0001i = Tr[\u001a0\u0001\u0001\u0001] denotes the statistical average with an initial density matrix \u001a0. In order to develop the\nperturbation expansion, we introduce the interaction picture\nhIS(\u001c1;\u001c2)i= Re2\n42iX\nq;kJq;khTCSC~\u001b+\nq(\u001c1)~S\u0000\nk(\u001c2)i3\n5: (S.96)\nSCand ~O(t) are given by\nSC=TCexp Z\nCd\u001c~Hex(\u001c)\ni~!\n; (S.97)\nand\n~O(t) =Uy\n0(t;t0)OU0(t;t0); (S.98)\nwhere\nU0(t;t0) =Texp\u0012Zt\nt0dt0H0(t0)\ni~\u0013\n: (S.99)\nExpandingSCas\nSC\u00191 +Z\nCd\u001cTC~Hex(\u001c)\ni~; (S.100)\nthe spin current is given by\nhIS(\u001c1;\u001c2)i=X\nq;kjJq;kj2Re\"\n2\n~Z\nCd\u001chTC~\u001b+\nq(\u001c1)~\u001b\u0000\n\u0000q(\u001c)ihTC~S+\n\u0000k(\u001c)~S\u0000\nk(\u001c2)i#\n: (S.101)15\nUsing the contour ordered Green's functions\n\u001fq(\u001c1;\u001c) =\u00001\ni~hTC~\u001b+\nq(\u001c1)~\u001b\u0000\n\u0000q(\u001c)i; (S.102)\nGk(\u001c;\u001c2) =1\ni~hTC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c2)i; (S.103)\nthe above equation is rewritten as\nhIS(\u001c1;\u001c2)i=X\nq;kjJq;kj2Re\"\n2~Z\nCd\u001c\u001fq(\u001c1;\u001c)G\u0000k(\u001c;\u001c2)#\n: (S.104)\nWe put\u001c2on the forward contour and \u001c1on the backward contour to describe spin transfer at the interface in\nappropriate time order. Assuming a steady state, the spin current is written as\nhISi= 2~X\nq;kjJq;kj2Re\"Z1\n\u00001d!0\n2\u0019\u0010\n\u001fR\nq(!0)G<\n\u0000k(!0) +\u001f<\nq(!0)GA\n\u0000k(!0)\u0011#\n: (S.105)\nWe introduce the distribution functions as\n\u001f<\nq(!) =fSC\nq(!)\u0002\n2iIm\u001fR\nq(!)\u0003\n; (S.106)\nG<\nk(!) =fFI\nk(!)\u0002\n2iImGR\nk(!)\u0003\n: (S.107)\nThe formula of the spin current at the interface is derived as\nhISi= 4~X\nq;kjJq;kj2Z1\n\u00001d!0\n2\u0019Im\u001fR\nq(!0)\u0002\n\u0000ImGR\n\u0000k(!0)\u0003\u0002\nfFI\n\u0000k(!0)\u0000fSC\nq(!0)\u0003\n: (S.108)\nWhen both the SC and the FI are in equilibrium, the di\u000berence of the distribution functions is zero (i.e. fFI\n\u0000k(!0)\u0000\nfSC\nq(!0) = 0), so that no spin current is generated. Under the microwave irradiation, the distribution function of\nthe FI deviates from equilibrium, which generates the interface spin current. Performing a second-order perturbation\ncalculation, the deviation of the distribution function of the FI, \u000efFI\n\u0000k(!0), is given by\n\u000efFI\n\u0000k(!0) =2\u0019NS (\rhac=2)2\n\u000b!0\u000ek;0\u000e(!0\u0000!): (S.109)\nConsequently, the interface spin current is written as\nhISiSP= 4~X\nq;kjJq;kj2Z1\n\u00001d!0\n2\u0019Im\u001fR\nq(!0)\u0002\n\u0000ImGR\n\u0000k(!0)\u0003\n\u000efFI\n\u0000k(!0): (S.110)\nFinally, one can show that the spin current is proportional to the enhanced Gilbert damping\nhISiSP= 4~NS(\rhac=2)2\n\u000b!\u0002\n\u0000ImGR\nk=0(!)\u0003X\nqjJq;k=0j2Im\u001fR\nq(!);\n=N(~\rhac)2\n2\u000b\u0002\n\u0000ImGR\nk=0(!)\u0003\n\u000e\u000b: (S.111)\nV. MODEL FOR INTERFACE CONFIGURATIONS\nIn order to calculate Eq. (S.87), one needs to set up an explicit expression for jJq;k=0j2. We consider an interface\nwith uncorrelated roughness. To model this interface, we assume that J(r;rj) satis\fes\nDX\njJ(r;rj)E\nave=J1; (S.112)\nDX\nj;j0J(r;rj)J(r0;rj0)E\nave=J2\n1+J2\n2l2\u000e(r\u0000r0); (S.113)16\nwhereh\u0001\u0001\u0001i avemeans interface con\fguration average. The spatially averaged J(r;rj) is given by a constant J1as\nshown in Eq. (S.112). Equation (S.113) means that the interface roughness is uncorrelated and J2\n2l2is a variance.\nJ1andJ2are coupling constants with dimension of energy, and are independent of the system size. lis introduced\nbecause the Hamiltonian of the SCs is treated as a continuum model. Performing the interface con\fguration average,\nand using Eq. (S.112) and (S.113), one can obtain the expression for jJq;k=0j2in the main text.\nVI. DYNAMIC SPIN SUSCEPTIBILITY OF SC\nEvaluating the retarded component of the self-energy Eq. (S.80), the dynamic spin susceptibility of the SC is given\nby\n\u001fR\nq(!) =\u0000Z1\n\u00001dEf(E)X\n\u0015;k(\nM\u0015;\u0015(a)\nk;k+q\u0014\n\u00001\n\u0019ImgR\n\u0015;k(E)gR\n\u0015;k+q(E+~!)\u00001\n\u0019ImgR\n\u0015;k+q(E)gA\n\u0015;k(E\u0000~!)\u0015\n+M\u0015;\u0000\u0015(a)\nk;k+q\u0014\n\u00001\n\u0019ImgR\n\u0015;k(E)gR\n\u0000\u0015;k+q(E+~!)\u00001\n\u0019ImgR\n\u0000\u0015;k+q(E)gA\n\u0015;k(E\u0000~!)\u0015)\n;\n(S.114)\nwhereM\u0015;\u00150(a)\nk;k+qwitha=s;c;andhare given by\nM\u0015;\u00150(s)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q+\u00012\n4\u0015Ek\u00150Ek+q; (S.115)\nM\u0015;\u00150(c)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q\u0000\u00012e\u0000i(\u001ek\u0000\u001ek+q)\n4\u0015Ek\u00150Ek+qcos2\u0012; (S.116)\nM\u0015;\u00150(h)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q\u0000\u00012sin\u001eksin\u001ek+q\n4\u0015Ek\u00150Ek+qsin2\u0012: (S.117)\n\u0015;\u00150=\u0006give a sign, and a=s;c;andhcorrespond to matrix elements for s-wave, chiral p-wave, and helical p-wave\nSCs, respectively. In Eq. (S.114), the terms multiplied by M\u0015;\u0015(a)\nk;k+qdescribe the intraband transition processes, i.e.,\ntransition processes from particle to particle and from hole to hole, and the terms multiplied by M\u0015;\u0000\u0015(a)\nk;k+qdescribe\nthe interband transition processes, i.e., transition processes from particle to hole and vice versa. The retarded and\nadvanced Green's functions of the quasiparticles gR=A\n\u0015;k(E) are given by\ngR\n\u0015;k(E) =1\nE\u0000\u0015Ek+i\u0000; (S.118)\ngA\n\u0015;k(E) =1\nE\u0000\u0015Ek\u0000i\u0000; (S.119)\nwhere \u0000 is a constant level broadening introduced phenomenologically. \u0000 is introduced to incorporate the intraband\ncontribution in the calculation of the uniform spin susceptibility. The details are explained in the next section.\nThe sum over kis replaced by the integral near the Fermi energy\nX\nkF(k)!DFZ1\n0dEDs(E)X\n\u0011=\u0006F\u0011(E); (S.120)\nX\nkF(k) sin2\u001ek!DFZ1\n0dEDs(E)X\n\u0011=\u00061\n2F\u0011(E); (S.121)\nwhereDFis the density of states near the Fermi energy in the normal state and Ds(E) is the density of states of\nquasiparticles\nDs(E) =jEjp\nE2\u0000\u00012\u0012(jEj\u0000\u0001): (S.122)\nF\u0011(E) means to assign \u0011p\nE2\u0000\u00012to\u0018contained in F(k).17\nVII. UNIFORM SPIN SUSCEPTIBILITY\nIn this section, we explain three properties related to the calculation of the uniform spin susceptibility. First, the\nmatrix element's properties are explained, which is essential to understand the qualitative di\u000berence between spin-\nsinglets-wave and spin-triplet p-wave SCs. Second, the reason to introduce the constant level broadening \u0000. Third,\nthe analytical expression for the uniform spin susceptibility of the p-wave SCs is given.\nPerforming the angular integral and replacing the sum over kby theEintegral, the matrix elements are replaced\nby\nM\u0015;\u00150(s)\nk;k!1 +\u0015\u00150\n4\u0015\u00150; (S.123)\nM\u0015;\u00150(c)\nk;k!(1 +\u0015\u00150)E2\u0000(1 + cos2\u0012)\u00012\n4\u0015\u00150E2; (S.124)\nM\u0015;\u00150(h)\nk;k!(1 +\u0015\u00150)E2\u0000(1 +1\n2sin2\u0012)\u00012\n4\u0015\u00150E2: (S.125)\nHere, the \frst-order terms in \u0018kare omitted because they vanish in the Eintegral. From the above expressions, the\nintraband matrix elements become \fnite for all SCs considered here, while the interband matrix elements vanish in\nthes-wave SC and becomes \fnite in the p-wave SCs. The above properties of the intraband and interband matrix\nelements can be understood using the commutation relation between the Hamiltonian and the spin operators. We\nintroduce the BdG form of the spin operators \u001ba\nBdG(a=x;y;z ) as below\n\u001ba\nBdG= \n\u001ba0\n0\u0000(\u001ba)T!\n: (S.126)\nThe commutation relation of HBdGand\u001ba\nBdGis given by\n[HBdG;\u001ba\nBdG] = 0 :s\u0000wave; (S.127)\n[HBdG;\u001ba\nBdG]6= 0 :p\u0000wave: (S.128)\nEquation (S.127) means that both the Hamiltonian and the spin operator are diagonalized simultaneously, so that the\nmatrix elements of the spin operator between a particle and a hole with the same wave-number vanish. This is because\nthes-wave SC is spin singlet. Therefore, the interband matrix elements vanishes in the s-wave SC. In contrast, in\nthep-wave SCs, the commutation relation between the Hamiltonian and the spin operator is \fnite as shown in Eq.\n(S.128), so that the matrix elements of the spin operator between a particle and a hole with the same wave-number\nis \fnite. This is because the p-wave SCs are spin triplet. As a result, the interband matrix elements are \fnite.\nHere, we explain the reason to introduce the constant level broadening \u0000 for gR=A\n\u0015;k(E). The intraband and interband\ntransitions are schematically shown in Fig. 5. The quasiparticles are scattered due to the magnon emission or\nabsorption. The scattering process conserves the wave-number. Consequently, in the case of the intraband transition,\nthe transition process is forbidden when \u0000 = 0. In order to incorporate the intraband processes, one needs to introduce\n\u0000, otherwise the intraband contribution vanishes, which can be directly shown by calculating Eq. (S.114).\nWhen \u0000 = 0, the uniform spin susceptibility for the chiral p-wave SCs is given by\nRe\u001fR\nuni(!) =2DFZ1\n\u0001dEEp\nE2\u0000\u00012(1 + cos2\u0012)\u00012\n4E2(f(E)\u0000f(\u0000E))\u00121\n2E+~!+1\n2E\u0000~!\u0013\n; (S.129)\nand\nIm\u001fR\nuni(!) =2\u0019DFj~!=2jp\n(~!=2)2\u0000\u00012(1 + cos2\u0012)\u00012\n(~!)2(f(\u0000~!=2)\u0000f(~!=2)): (S.130)\nFrom the above expressions, one can show that both the real part and imaginary part of the uniform spin susceptibility\ndiverge at ~!= 2\u0001, leading a resonance peak. The expressions for the helical p-wave SC can be obtained by replacing\ncos2\u0012with1\n2sin2\u0012. Therefore, \u0012dependence of \u001fR\nuni(!) explained in the main text is obtained from the above\nexpressions.18\nΓintra\ninter\nk+Ek\n−EkE E\nSpectral\nfunction\n2/uni0394/uni210F/uni03C9\n/uni210F/uni03C9Γ\nFIG. 5. Schematic image of intraband transition and interband transitions. The intraband transition gives contribution to the\nuniform spin susceptibility when the excitation energy is comparable to or smaller than the level broadening, ~!.\u0000. The\ninterband contribution is dominant when the excitation energy is comparable to the superconucting gap, ~!\u00192\u0001.\nVIII. LOCAL SPIN SUSCEPTIBILITY\nPerforming the angular integral and replacing the sum over k;qby theE;E0integral, the matrix elements are\nreplaced by\nM\u0015\u00150(s)\nk;q!1\n4+\u00012\n4\u0015E\u00150E0; (S.131)\nM\u0015\u00150(c)\nk;q!1\n4; (S.132)\nM\u0015\u00150(h)\nk;q!1\n4: (S.133)\nThe matrix elements for the chiral and helical p-wave SCs are identical. From the above expressions, one can see\nthat the interband contribution in the s-wave SC is suppressed. Unlike the uniform spin susceptibility, the intraband\ncontribution for the local spin susceptibility is \fnite even when \u0000 = 0. This is because the transition processes\nconsidered here leads to momentum transfer and the intraband transition is not forbidden. Therefore, we calculate\nthe local spin susceptibility at \u0000 = 0. The local spin susceptibility for the s-wave SC is given by\n\u001fR\nloc(!) =\u0000D2\nFZ1\n\u00001dEZ1\n\u00001dE0Ds(E)Ds(E0)\u0012\n1 +\u00012\nEE0\u0013f(E)\u0000f(E0)\nE\u0000E0+~!+i0; (S.134)\nand the local spin susceptibility for the p-wave SCs is given by\n\u001fR\nloc(!) =\u0000D2\nFZ1\n\u00001dEZ1\n\u00001dE0Ds(E)Ds(E0)f(E)\u0000f(E0)\nE\u0000E0+~!+i0: (S.135)\nIX. FMR MODULATION: ROUGH INTERFACE\nIn this section, we show the numerical results and summarize the characteristic properties of the FMR modulation\nfor the rough interface limit. In the following calculations, we set J1= 0 and assume that only \u001fR\nloc(!) contributes to\n\u000eHand\u000e\u000b.\nFigures 6 show (a) \u000eHand (b)\u000e\u000bfor the chiral and helical p-wave SCs as a function of frequency and temperature.\n\u000eHis \fnite inT!0 and has a resonance peak at ~!= 2\u0001.\u000e\u000bexhibits a coherence peak just below the transition\ntemperature in the su\u000eciently low frequency region, where ~!=kBTc\u001c1.\u000e\u000bdrops abruptly at ~!= 2\u0001.\u000e\u000bis\nalmost independent of both frequency and temperature when ~!>2\u0001.\nFigures 6 show (c) \u000eHand (d)\u000e\u000bfor thes-wave SC as a function of frequency and temperature. In the low\nfrequency region, where ~!=kBTc\u00141,\u000eHat a \fxed frequency decreases by about thirty percent with the decrease of\nthe temperature, and \u000eHis \fnite inT!0. As the frequency increases, \u000eHis almost independent of the temperature.\n\u000e\u000bshows a coherence peak just below the transition temperature in the su\u000eciently low frequency, where ~!=kBTc\u001c1.19\nThe coherence peak in the s-wave SC is larger than the corresponding coherence peak in the p-wave SCs. \u000e\u000bhas a\nkink structure at ~!= 2\u0001.\nNote that the cuto\u000b energy Ecwas introduced here to cause the integral for Re \u001fR\nloc(!) to converge. Although\nRe\u001fR\nloc(!) is approximately proportional to Ec, the qualitative properties explained above are independent of Ec.\nThe FMR modulation properties of the three SCs are summarized in Table II. In the case of the rough interface\nlimit, the pairing symmetry can be detected from either the absence or the existence of the resonance peak of \u000eH. The\npairing symmetry may also be detected from the properties of \u000e\u000b, the height of the coherence peak, and the structure\nat~!= 2\u0001. When compared with the resonance peak for \u000eH, however, the properties of \u000e\u000bare too ambiguous to\nallow the pairing symmetry to be distinguished clearly.\n(c) (d)s-wave(a) (b)Chiral & Helical p-wave\nT/T\nchω/kBTcδH/δH2\n0.00.51.00\n2\n410\n8\n6\nT/T\nchω/kBTcδα/δα2\n0.00.51.00\n2\n42\n1\n0\nT/T\nchω/kBTcδH/δH2\n0.00.51.00\n2\n410\n8\n6\nT/T\nchω/kBTcδα/δα2\n0.00.51.00\n2\n42\n1\n0\nFIG. 6. (a) The frequency shift and (b) the enhanced Gilbert damping as a function of both frequency and temperature for the\np-wave SCs. (c) The frequency shift and (d) the enhanced Gilbert damping as a function of both frequency and temperature\nfor thes-wave SC. The terms \u000eH2and\u000e\u000b2are given by \u000eH2=\u00002\u0019SJ2\n2l2D2\nFkBTc=(NA\r ~) and\u000e\u000b2= 2\u0019SJ2\n2l2D2\nF=(NA),\nwhere they are characteristic values in the normal state. The cuto\u000b energy is set to be Ec=kBTc= 10.\nTABLE II. 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Linder,\nPhysical Review Letters 127, 207001 (2021)."    },    {        "title": "2103.07008v1.Magnetoelastic_Gilbert_damping_in_magnetostrictive_Fe___0_7__Ga___0_3___thin_films.pdf",        "content": "Magnetoelastic Gilbert damping in magnetostrictive Fe 0.7Ga 0.3thin films\nW. K. Peria,1X. Wang,2H. Yu,2S. Lee,2, 3I. Takeuchi,2and P. A. Crowell1,∗\n1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA\n2Department of Materials Science and Engineering,\nUniversity of Maryland, College Park, Maryland 20742, USA\n3Department of Physics, Pukyong National University, Busan 48513, South Korea\nWe report an enhanced magnetoelastic contribution to the Gilbert damping in highly magne-\ntostrictive Fe 0.7Ga0.3thin films. This effect is mitigated for perpendicular-to-plane fields, leading\nto a large anisotropy of the Gilbert damping in all of the films (up to a factor of 10 at room tem-\nperature). These claims are supported by broadband measurements of the ferromagnetic resonance\nlinewidths over a range of temperatures (5 to 400 K), which serve to elucidate the effect of both the\nmagnetostriction andphonon relaxation on the magnetoelastic Gilbert damping.\nAmong the primary considerations in the design of\nspintronics devices is Gilbert damping. However, a full\nunderstanding of the mechanisms which cause damping\nof magnetization dynamics in ferromagnets remains elu-\nsive. Reports of anisotropy in the Gilbert damping have\nproven to be useful tools in the understanding of the un-\nderlying mechanisms involved [1–3], but there is much\nthat is yet unclear. Studies of the temperature depen-\ndence also promise to be a uniquely powerful tool for a\ncomplete physical understanding [4, 5], however, there\nare few such reports in existence.\nRecently, it has been shown that spins can be co-\nherently coupled over large distances ( ∼1 mm) using\nmagnon-phonon coupling [6–8]. It is also well known\nthat magnetization dynamics can be excited elastically\nthrough this phenomenon [9], but its effect on Gilbert\ndamping has been largely confined to theoretical calcu-\nlations [10–13] and lacks clear experimental validation.\nFurthermore, most studies have focused on yttrium iron\ngarnet (YIG), which is weakly magnetostrictive.\nIn this Letter, we observe a large and anisotropic mag-\nnetoelastic contribution to the Gilbert damping in highly\nmagnetostrictive Fe 0.7Ga0.3films through broadband\nmeasurements of the ferromagnetic resonance (FMR)\nlinewidths over a wide range of temperatures. The\nperpendicular-to-plane linewidths exhibit a relatively low\nminimum in the Gilbert damping of approximately 0.004,\nsimilar to that of bcc Fe [14]. At room temperature, the\nGilbert damping is as large as a factor of 10 greater with\nfield applied in plane relative to out of plane. In fact, for\nany given sample and temperature, the anisotropy is, at\nminimum, about a factor of 2. We argue this is due to\na mitigation of the magnetoelastic contribution for per-\npendicular magnetization, arising from finite-thickness\nboundary conditions and weak elastic coupling to the\nsubstrate. The nonmonotonic temperature dependence\nof the Gilbert damping also shows the competing effects\nof the magnetostriction, which increases at low tempera-\nture, and the phonon viscosity, which generally decreases\nat low temperature.\nThe Fe 0.7Ga0.3films studied in this letter were de-\nposited on SiO 2/Si wafers at room temperature by dcmagnetron sputtering of an Fe 0.7Ga0.3target. The base\npressure of the deposition chamber was 5 ×10−8torr,\nand the working pressure was kept at 5 ×10−3torr\nwith Ar gas. The composition of the Fe 0.7Ga0.3films\nwas quantitatively analyzed by energy dispersive spec-\ntroscopy (EDS). Films were grown with thicknesses of\n21 nm, 33 nm, 57 nm, and 70 nm (the 21 nm, 57 nm,\nand 70 nm belong to the same growth series). An addi-\ntional 33 nm film was grown at 200◦C. The 33 nm room\ntemperature deposition was etched using an ion mill to\nobtain films with thicknesses of 17 nm and 26 nm. The\nthicknesses of the films were measured using x-ray reflec-\ntometry (see Supplemental Material).\nThe FMR linewidths were measured using a setup in-\nvolving a coplanar waveguide and modulation of the ap-\nplied magnetic field for lock-in detection as described in\nRef. [15]. Measurements were done with the field applied\nin the plane (IP) and perpendicular to the plane (PP) of\nthe film. The sample temperature was varied from 5 K\nto 400 K for both IP and PP configurations [16] with\nmicrowave excitation frequencies up to 52 GHz. The res-\nonance fields and linewidths were isotropic in the plane,\nand the absence of in-plane magnetic anisotropy was ver-\nified with vibrating sample magnetometry (see Supple-\nmental Material). This is also consistent with the abun-\ndance of grain boundaries observed with atomic force mi-\ncroscopy (AFM). In analyzing the FMR linewidths, we\nconsider three contributions: Gilbert damping 4 παf/γ\n(αis the Gilbert damping coefficient, fis the microwave\nfrequency, and γis the gyromagnetic ratio), inhomo-\ngeneous broadening ∆ H0, and two-magnon scattering\n∆HTMS (for IP fields). Eddy current damping and ra-\ndiative damping contributions [17] are neglected because\nwe expect them to be small ( <10−4) for these films.\nLinewidths of the 70 nm film at 300 K for both con-\nfigurations of the applied field are shown in Fig. 1(a),\nand the IP linewidths with individual contributions to\nthe linewidth plotted separately in Fig. 1(b). We fit the\nIP linewidths using a model of two-magnon scattering\nbased on granular defects [15, 18, 19]. The fit for the\n70 nm film is shown in Fig. 1(b), along with the two-\nmagnon contribution alone given by the magenta curve.arXiv:2103.07008v1  [cond-mat.mtrl-sci]  11 Mar 20212\n01 02030405005001000150020000500100015002000/s61508\nH0/s61508HTMS/s61508H (Oe)F\nrequency (GHz)H in-plane/s61508\nHGilbertT = 300 K/s61537\n = 0.0390 /s61617 0.0005/s61560\n = 17 nm(a)(\nb)/s61508 H (Oe)T  = 300 K/s61537\nIP = 0.0390 /s61617 0.0005/s61537\nPP = 0.0035 /s61617 0.0001/s61508HIP/s61508\nHPP70 nm film\nFIG. 1. (a) FMR linewidths for IP (black squares) and\nPP (red circles) configurations for the 70 nm film. The IP\nlinewidths are fit to a model of two-magnon scattering and\nthe PP linewidths are fit using the standard Gilbert damping\nmodel. (b) Total linewidth (solid black), Gilbert linewidth\n(dotted blue), two-magnon scattering linewidth (dashed ma-\ngenta), and inhomogeneous broadening (dashed/dotted red)\nfor the 70 nm film with IP field.\nThe fit parameters are the Gilbert damping α(indicated\non the figure) and the RMS inhomogeneity field H/prime. The\ndefect correlation length ξis fixed to 17 nm based on the\nstructural coherence length obtained with x-ray diffrac-\ntion (XRD), which agrees well with the average grain di-\nameter observed with AFM (see Supplemental Material).\nFurthermore, the high-frequency slope of the linewidths\napproaches that of the Gilbert damping since the two-\nmagnon linewidth becomes constant at high frequencies\n[see Fig. 1(b)].\nWe now compare the IP and PP linewidths of the\n70 nm film shown in Fig. 1(a). The two-magnon scat-\ntering mechanism is inactive with the magnetization per-\npendicular to the plane [20], and so the PP linewidths are\nfit linearly to extract the Gilbert damping. We obtain a\nvalue of 0.0035±0.0001 for PP fields and 0 .039±0.0005\nfor IP fields, corresponding to an anisotropy larger than\na factor of 10. Li et al. [3] recently reported a large\nanisotropy (∼factor of 4) in epitaxial Co 50Fe50thin films.\nFirst we discuss the dependence of the PP Gilbert\ndampingαPPon temperature for all of the films, shown\nin Fig. 2. We observe a significant temperature depen-\n01 002 003 004 00024685\n7 nm 17 nm2\n6 nm3\n3 nm (RT dep)/s61537 (×10-3)T\nemperature (K)21 nm 3\n3 nm (200 °C dep)H\n perpendicular-to-plane70 nmFIG. 2. Gilbert damping αfor PP field shown as a func-\ntion of temperature for the 17 nm (orange), 21 nm (blue),\n26 nm (green), 33 nm room temperature deposition (ma-\ngenta), 33 nm 200◦C deposition (gold), 57 nm (red), and\n70 nm (black) Fe 0.7Ga0.3films.\ndence in all cases (with the exception of the 33 nm room\ntemperature deposition), characterized by a maximum\nat around 50 K. Then, at the lowest temperatures (5 to\n10 K),αPPapproaches the same value for all of the films\n(/similarequal0.004).\nNow we turn to the temperature dependence of the\nIP Gilbert damping αIPshown in Fig. 3. The values\nobtained here were obtained by fitting the linewidths lin-\nearly, but excluding the low-frequency points ( <∼20 GHz)\nsince the two-magnon scattering becomes constant at\nhigh frequencies [21]. Here we note, upon comparison\nwith Fig. 2, that a large anisotropy of the Gilbert damp-\ning exists for all of the samples. In the 70 nm film, for\ninstance,αIPis more than a factor of 10 larger than αPP\nat 300 K. In the temperature dependence of αIP, we ob-\nserve behavior which is similar to that seen in αPP(Fig.\n2), namely, a maximum at around 50 K (with the excep-\ntion of the 21 nm film). Here, however, αIPdoes not\napproach a common value at the lowest temperatures in\nall of the samples as it does in the PP case.\nThe IP Gilbert damping is larger than the PP Gilbert\ndamping for all of the samples over the entire range of\ntemperatures measured. This anisotropy of the Gilbert\ndamping—along with the nonmonotonic temperature\ndependence—in all seven samples implies a contribution\nto the Gilbert damping in addition to Kambersk´ y damp-\ning. We have verified that the orientation of FeGa(110)3\nplanes is completely random with XRD for the 33 nm\n(both depositions) and 70 nm films (see Supplemen-\ntal Material), and it is therefore not possible that the\nanisotropy is due to Kambersk´ y damping. Interface\nanisotropy has reportedly led to anisotropic Kambersk´ y\ndamping in ultrathin ( ∼1 nm) films of Fe [2], but this\nis highly unlikely in our case due to the relatively large\nthicknesses of the films. In addition, the fact that the\ndamping anisotropy shows no clear correlation with film\nthickness furthers the case that intrinsic effects, which\ntend to show a larger anisotropy in thinner films [2],\ncannot be the cause. The longitudinal resistivity ρxxof\nthe 33 nm (both depositions) and 70 nm films (see Sup-\nplemental Material) shows very weak temperature de-\npendence. In the Kambersk´ y model, the temperature\ndependence of the damping is primarily determined by\nthe electron momentum relaxation time τ, and we would\ntherefore not expect the Kambersk´ y damping to show\na significant temperature dependence for samples where\nthe residual resistivity ratio is approximately unity. It is\nplausible that the Kambersk´ y damping would still show\na temperature dependence in situations where the spin\npolarization is a strong function of temperature, due to\nchanges in the amount of interband spin-flip scattering.\nThis kind of damping, however, would be expected to\ndecrease at low temperature [22, 23]. The temperature\ndependence we observe for both αPPandαIPis therefore\ninconsistent with Kambersk´ y’s model, and the similarity\nbetween the two cases in this regard suggests that the\nenhanced Gilbert damping has a common cause that is\nmitigated in the PP configuration.\nIt has been proposed that magnetoelastic coupling\ncan lead to Gilbertlike magnetization damping through\nphonon relaxation processes [10, 12, 24]. Similar treat-\nments calculate the magnetoelastic energy loss through\ninteraction with the thermal population of phonons\n[11, 25]. The Kambersk´ y mechanism is often assumed to\nbe the dominant Gilbert damping mechanism in metal-\nlic samples, so magnetoelastic Gilbert damping is usually\nstudied in magnetic insulators, particularly yttrium iron\ngarnet (YIG). There is the possibility, however, for the\nmagnetoelastic damping to dominate in metallic samples\nwhere the magnetostriction is large, such as in Fe-Ga al-\nloys. Later we will discuss how magnetoelastic damping\ncan be mitigated in thin films by orienting the magneti-\nzation perpendicular to the plane, and how the degree to\nwhich it is mitigated depends on the boundary conditions\nof the film.\nHere we outline a theory of magnetoelastic damping,\nwhich relies on the damping of magnetoelastic modes\nthrough phonon relaxation mechanisms. Figure 4 illus-\ntrates the flow of energy through such a process. Analyt-\nically, the procedure is to equate the steady-state heating\nrate due to Gilbert damping to the heating rate due to\ncrystal viscosity, and solve for the Gilbert damping α\nin terms of the crystal shear viscosity ηand the mag-\n01 002 003 004 000123453\n3 nm (200 °C dep)17 nm2\n6 nm/s61537 (×10-2)T\nemperature (K)33 nm (RT dep)57 nm2\n1 nmH in-plane7\n0 nmFIG. 3. Gilbert damping αfor IP field shown as a func-\ntion of temperature for the 17 nm (orange), 21 nm (blue),\n26 nm (green), 33 nm room temperature deposition (ma-\ngenta), 33 nm 200◦C deposition (gold), 57 nm (red), and\n70 nm (black) Fe 0.7Ga0.3films.\nnetostrictive coefficients λhkl. Shear strain uijresult-\ning from the magnetoelastic interaction can be expressed\nasuij=λ111mimj[26], where mi≡Mi/Msare the\nreduced magnetizations. The leading-order shears thus\nhave equations of motion given by ˙ uiz=λ111˙mi, where\ni=xory, andzis the direction of the static magnetiza-\ntion so that mz≈1. Longitudinal modes are quadratic\nin the dynamical component of the magnetization [24]\nand so will be neglected in this analysis.\nThe heating rate due to Gilbert damping can be writ-\nten as ˙Qα=Ms\nγα( ˙m2\nx+ ˙m2\ny), and the heating rate due to\nthe damping of phonon modes as ˙Qη= 4η( ˙u2\nxz+ ˙u2\nyz) =\n4ηλ2\n111( ˙m2\nx+ ˙m2\ny) [12], with the factor of 4 accounting\nfor the symmetry of the strain tensor. Equating the two,\nand solving for α(henceforward referred to as αme), we\nobtain\nαme=4γ\nMsηλ2\n111. (1)\nWe will restrict our attention to the case of isotropic mag-\nnetostriction, and set λ111=λ.\nIn order to use Eq. 1 to estimate αmein our films,\nwe first estimate the shear viscosity, given for transverse\nphonons with frequency ωand relaxation time τas [27]\nη=2ρc2\nt\nω2τ, (2)4\n(b)\nu(t)\nphonon pumpingM(t) H0\nM(t)\nu(t)(a)\nkph\ndH0\nFIG. 4. (a) Depiction of magnetoelastic damping process\nfor magnetization in plane and (b) perpendicular to plane,\nwhere M(t) is the magnetization vector and u(t) is the lattice\ndisplacement. In panel (b), the magnon-phonon conversion\nprocess is suppressed when d < π/k ph, wheredis the film\nthickness and kphis the transverse phonon wavenumber at\nthe FMR frequency.\nwhereρis the mass density and ctis the transverse\nspeed of sound. Using ω/2π= 10 GHz, τ= 10−11s,\nandct= 2.5 km/s, we obtain η≈2.3 Pa s. (The\nestimate of the phonon relaxation time is based on a\nphonon mean free path of the order of the grain size:\n∼10 nm.) Furthermore, the magnetostriction of an equiv-\nalent sample has been measured to be ∼100 ppm at room\ntemperature [28]. Then, with γ/2π= 29 GHz/T and\nMs= 1123 emu/cc (extracted from FMR data taken at\n300 K), we estimate αme≈0.016. This estimate gives us\nimmediate cause to suspect that magnetoelastic Gilbert\ndamping is significant (or even dominant) in these films.\nWe now discuss why the magnetoelastic damping can\nbe much weaker for PP magnetization in sufficiently thin\nfilms. We will start by assuming that there is no coupling\nbetween the film and substrate, and later we will relax\nthis assumption. In this case the only phonons excited\nby the magnetization, to leading-order in the magneti-\nzations and strains, are transverse modes propagating in\nthe direction of the static magnetization [24]. One may\nassume that the minimum allowable phonon wavenum-\nber is given by π/d, wheredis the film thickness, since\nthis corresponds to the minimum wavenumber for a sub-\nstrate having much lower acoustic impedance than the\nfilm (requiring the phonons to have antinodes at the in-\nterfaces) [13]. (We also assume an easy-axis magnetic\nanisotropy at the interfaces, so that the dynamical mag-\nnetizations have antinodes at the interfaces.) We expect\nthen that the magnetoelastic damping will be suppressed\nfor cases where the phonon wavelength, at the frequencyof the precessing magnetization, is greater than twice the\nfilm thickness [see Fig. 4(b)]. Thus, in sufficiently thin\nfilms (with weakly-coupled substrates), the magnetoelas-\ntic damping process can be suppressed when the mag-\nnetization is perpendicular to the plane. However, the\nmagnetoelastic damping can be active (albeit mitigated)\nwhen there is nonnegligible or “intermediate” coupling\nto the substrate.\nBefore moving on, we briefly note the implications of\nEq. (1) for the temperature dependence of the Gilbert\ndamping. On the basis of the magnetostriction alone,\nαmewould be expected to increase monotonically as tem-\nperature is decreased ( λhas been shown to increase by\nnearly a factor of 2 from room temperature to 4 K in\nbulk samples with similar compositions [29]). However,\nthe viscosity ηwould be expected to decrease at low tem-\nperature, leading to the possibility of a local maximum\ninαme. In polycrystalline samples where the grain size\nis smaller than the phonon wavelength, viscous damping\nof phonons due to thermal conduction caused by stress\ninhomogeneities can be significant [27, 30]. (In our case\nthe phonon wavelengths are ∼100 nm and the grain\nsizes are∼10 nm.) This effect scales with temperature\nasη∼Tα2\nT/Cχ [30], where αTis the thermal expansion\ncoefficient, Cis the specific heat at constant volume, and\nχis the compressibility. At higher temperatures, αTand\nCwill approach constant values, and χwill always de-\npend weakly on temperature. We therefore expect that\nthe viscosity is approximately linear in T. In this case,\nαmeis maximized where λ2(T) has an inflection point.\nWe proceed to explain our data in terms of the mecha-\nnism described above, turning our attention again to the\nPP Gilbert damping for all of the films shown in Fig.\n2. We previously argued that the magnetoelastic damp-\ning mechanism will be suppressed for the case where the\nacoustic impedances of the film and substrate are mis-\nmatched. However, the clear dependence on tempera-\nture, which we have already shown is inconsistent with\nKambersk´ y damping, appears to be consistent with the\nmagnetoelastic damping mechanism. We estimate that\nthe acoustic impedance of the film (defined as the product\nof mass density ρand transverse speed of sound ct[13])\nis about a factor of 2 larger than the substrate. This sug-\ngests that the elastic coupling between the film and sub-\nstrate, albeit weak, may be nonnegligible. Furthermore,\nexperiments with YIG/GGG heterostructures (where the\nacoustic match is good) have demonstrated magnetic ex-\ncitation of phononic standing waves that have boundary\nconditions dictated by the combined thickness of the film\nand substrate, rather than the film thickness alone (i.e.,\nthe wavelengths are much larger than the film thickness)\n[6, 31]. In this case, the Gilbert damping may contain\nsome contribution from the magnetoelastic mechanism.\nA final point is that αPPapproaches/similarequal0.004 at 5 to\n10 K for all of the films. Both the magnetostriction and\nthe viscosity are quantities which could have significant5\n01 0020030040002468/s61537me (×10-2)T\nemperature (K)Ref. [29]2\n1 nm70 nm5\n7 nm0\n.000.250.500.751.00/s61548\n2(T)//s615482(0)010020030001/s61544 (arb. units)T\n (K)\nFIG. 5. Magnetoelastic Gilbert damping αmefor the 21 nm\n(blue), 57 nm (red), and 70 nm (black) films (left ordinate)\nandλ2(T)/λ2(0) from Clark et al. [29] (magenta; right or-\ndinate) shown as a function of temperature. Inset shows the\nratio ofαmeandλ2(T)/λ2(0), labeled as η(T), along with lin-\near fits for the 21 nm (blue), 57 nm (red), and 70 nm (black)\nfilms.\nvariation between samples, leading to variations in αme.\nHowever, the viscosity becomes small at low temperature,\nwhich means that the Gilbert damping will approach the\nKambersk´ y “limit,” a property that is determined by the\nelectronic structure, implying that the Kambersk´ y damp-\ning is/similarequal0.004 in these films and that it is the primary\ncontribution to the Gilbert damping near T= 0.\nNow we revisit the IP Gilbert damping shown in Fig.\n3. In this configuration, there is a strong temperature\ndependence of the Gilbert damping similar to that of\nthe PP case, again implying the presence of magnetoe-\nlastic damping. However, the overall magnitude is much\nhigher. That is because in this case arbitrarily long wave-\nlength phonons can be excited regardless of the thick-\nness of the film. Although we cannot directly measure\nthe magnetostriction as a function of temperature, we\nestimate the scaling behavior of λby interpolating the\ndata in Ref. [29] taken for bulk samples of similar com-\nposition. In order to demonstrate that αIPscales with\ntemperature as expected from the model, we have plot-\nted the quantities αmeandλ2(T)/λ2(0) as functions of\ntemperature in Fig. 5—where we define the quantity\nαme≡αIP−0.004—for the 21 nm, 57 nm, and 70 nm\nfilms (which are part of the same growth). The corre-\nlation between the two quantities is not completely con-\nvincing. There is, however, an additional temperature\ndependence in αmebesidesλ2(T), namely, the viscosity\nη(T). The inset of Fig. 5 shows the ratio of αmeand\nλ2(T), which [from Eq. (1)] is proportional to η(T). The\nlinear fits provide strong evidence that the mechanism\nbehind the viscosity is indeed the thermal conduction\nprocess that we have argued is approximately linear inT. It is noteworthy that the maximum in αme(∼50 to\n75 K for all of the samples) coincides approximately with\nthe inflection point in λ2(T). This was a consquence of\nour assumption that η(T) should be roughly linear. We\nalso obtain a significant value for the zero-temperature\nviscosity, which is around 25 % of the value at 300 K. This\nis likely due to boundary-scattering processes which will\npreventαmefrom going to zero at low temperatures, par-\nticularly for in-plane magnetization where αmeis much\nlarger than 0.004 (our estimate for the Kambersk´ y damp-\ning). For the PP case, αmeis much smaller due to limita-\ntions on the wavelengths of phonons that can be excited,\nso the Gilbert damping of all the samples approaches the\nKambersk´ y limit of 0.004 near zero temperature. We also\nfound that η(T) was linear for the 33 nm (200◦C depo-\nsition) film, but had a more complicated dependence on\nTfor the 17 nm, 26 nm, and 33 nm (room temperature\ndeposition) films (the latter three being notably of the\nsame growth). The viscosity near zero temperature is\nwithin roughly a factor of 2 for all seven of the samples,\nhowever.\nFinally, we propose that this mechanism may be re-\nsponsible for a Gilbert damping anisotropy of similar\nmagnitude reported in Ref. [3], observed in an epitax-\nial Co 0.5Fe0.5thin film. The authors attributed the\nanisotropy to the Kambersk´ y mechanism [22, 23, 32, 33],\narising from tetragonal distortions of the lattice. The\nmagnetostriction is known to be highly anisotropic in\nbulk Co 0.5Fe0.5,viz.,λ100= 150 ppm and λ111= 30 ppm\n[34]. We therefore expect that the Gilbert damping aris-\ning from the mechanism we have described may be much\nlarger for M/bardbl(110) than M/bardbl(100), which is precisely\nwhat the authors observed.\nIn summary, we observe large and anisotropic magne-\ntoelastic Gilbert damping in Fe 0.7Ga0.3polycrystalline\nthin films (thicknesses ranging from 17 to 70 nm). At\n300 K, the damping coefficient is more than a factor of\n10 larger for field in plane than it is for field perpendicu-\nlar to the plane in the 70 nm film. The large anisotropy\nis caused by a mitigation of the magnetoelastic effect for\nperpendicular-to-plane fields due to a dependence on the\nelastic coupling of the film to the substrate, which in our\ncase is weak. Finally, there is a nonmonotonic tempera-\nture dependence of the Gilbert damping, which we show\nis consistent with our model.\nWe acknowledge Rohit Pant and Dyland Kirsch for as-\nsistance with thin film deposition and characterization.\nThis work was supported by SMART, a center funded\nby nCORE, a Semiconductor Research Corporation pro-\ngram sponsored by NIST. Parts of this work were carried\nout in the Characterization Facility, University of Min-\nnesota, which receives partial support from NSF through\nthe MRSEC program, and the Minnesota Nano Cen-\nter, which is supported by NSF through the National\nNano Coordinated Infrastructure Network, Award Num-\nber NNCI - 1542202.6\n∗Author to whom correspondence should be addressed:\ncrowell@umn.edu\n[1] K. Gilmore, M. D. Stiles, J. Seib, D. Steiauf, and\nM. F¨ ahnle, Anisotropic damping of the magnetization\ndynamics in Ni, Co, and Fe, Phys. Rev. B 81, 174414\n(2010).\n[2] L. Chen, S. Mankovsky, S. Wimmer, M. A. W. Schoen,\nH. S. 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Ferromagnetic resonance linewidths of 70 nm film 2\nS3. X-ray reflectivity 2\nS4. X-ray diffraction 3\nS5. Atomic force microscopy 3\nS6. Vibrating sample magnetometry 3\nS7. Longitudinal resistivity 3\nReferences 4\nS1. MAGNETIZATION DYNAMICS\nThe treatment of magnetization dynamics begins with the Landau-Lifshitz-Gilbert equation of motion\ndM\ndt=−γM×Heff+α\nMsM×dM\ndt(S1)\nwhere the relaxation is characterized by the Gilbert damping parameter α. Upon linearizing this equation in the\ndynamic component of the magnetization, one obtains for the ac magnetic susceptibility of the uniform q= 0 mode\nχac(q= 0,ω)∝αω/γ\n(H−HFMR )2+ (αω/γ )2(S2)\nso that the field-swept full-width-at-half-maximum linewidth is given by ∆ HFWHM = 2αω/γ . Therefore, the Gilbert\ndamping parameter αis obtained by measuring ∆ HFWHM as a function of ω.\nRelaxation of the uniform mode can include mechanisms which are not described by Gilbert damping. The most\ncommon of these is inhomogeneous broadening, which results from inhomogeneities in the system and is constant\nas a function of frequency. Another mechanism is two-magnon scattering, which is also extrinsic in nature. Two-\nmagnon scattering originates from the negative group velocity at low qof the backward volume mode magnons for\nin-plane magnetization. The negative group velocity is due to a lowering of the magnetostatic surface charge energy\nfor increasing q. The existence of negative group velocity at low qleads to the appearance of a mode at nonzero q\nthat is degenerate with the uniform mode. Two-magnon scattering refers to the scattering of the uniform mode to\nthe nonuniform degenerate mode.\nMuch work has been done on the treatment of two-magnon scattering [S1–S3], and here we will simply give an\nexpression for the contribution of two-magnon scattering to the field-swept linewidth\n∆HTMS =γ2ξ2H/prime2\ndω/dH/integraldisplay\nd2qΛ0q1\n(1 + (qξ)2)3/21\nπωα\n(ωα)2+ (ω−ωFMR )2(S3)\nwithξthe defect correlation length, H/primethe RMS inhomogeneity field, and Λ 0qthe magnon-magnon coupling. In\ngeneral, this leads to a nonlinear dependence of the linewidth on frequency. Eq. S3 is used to fit the IP linewidths.arXiv:2103.07008v1  [cond-mat.mtrl-sci]  11 Mar 20212\n0102030405005001000150020002500(b) \n5 K \n50 K \n150 K \n225 K \n300 K \n400 K/s61508H (Oe)F\nrequency (GHz)H in-plane\n010203040500100200300400500 5 K \n50 K \n150 K \n225 K \n300 K \n400 K/s61508H (Oe)F\nrequency (GHz)H perpendicular-to-plane(a)\nFIG. S1. FMR linewidths of the 70 nm film with field PP (a) and IP (b) for sample temperatures of 5 K (blue), 50 K (gold),\n150 K (black), 225 K (magenta), 300 K (red), and 400 K (orange). The solid lines are linear fits in both panels. In (b), the\nvertical dashed line indicates the lower bound of the points included in the fit.\n12345610-1100101102103104105106(c)Intensity (arb. units)/s61553\n/2/s61553 (degrees)d = 56.6 nm\n12345610-1100101102103104105106(b)Intensity (arb. units)/s61553\n/2/s61553 (degrees)200 °C depositiond\n = 33.1 nm \n123410-1100101102103104105106(a)R\nT depositiond\n = 33.2 nm Intensity (arb. units)/s61553\n/2/s61553 (degrees)\nFIG. S2. X-ray reflectivity data (black) overlaid with fits (red) for the (a) 33 nm (room temperature deposition), (b) 33 nm\n(200◦C deposition), and (c) 57 nm films. Thicknesses dobtained from the fits are indicated on the figure.\nS2. FERROMAGNETIC RESONANCE LINEWIDTHS OF 70 nm FILM\nThe field-swept FMR linewidths of the 70 nm film are shown in Fig. S1 for field PP and IP. For the case of field\nIP, the data above 23 GHz were fit linearly to obtain the Gilbert damping. (This value varied between different\nsamples since the characteristic roll-off frequency depends on both defect lengthscale and film thickness, but remained\nin the range 20 to 25 GHz.) It is safe to do this provided there are no inhomogeneities at lengthscales smaller than a\nfew nm, which could cause the two-magnon scattering contribution to the linewidth to roll off at higher frequencies.\nWe believe that defects at such small lengthscales are highly unlikely given the characterization performed on these\nsamples.\nS3. X-RAY REFLECTIVITY\nIn Fig. S2 we show x-ray reflectivity measurements at grazing incidence for 33 nm (room temperature and 200◦C\ndepositions) and 57 nm films. The measurements were taken using a Rigaku SmartLab diffractometer. The thicknesses\ndyielded by the fits of the data are indicated on the figure.3\n4243444546036912F\ne0.7Ga0.3(110)(a)I\nntensity (arb. units)/s61553\n/2/s61553 (degrees)33 nm (RT deposition)2\n/s61553c= 44.14 /s61617 0.03 °F\nWHM = 0.78 /s61617 0.09 °\n4243444546036912F\ne0.7Ga0.3(110)(b)I\nntensity (arb. units)/s61553\n/2/s61553 (degrees)33 nm (200 °C  deposition)2\n/s61553c = 44.07 /s61617 0.05 °F\nWHM = 1.18 /s61617 0.17 °\n4243444546036912(c)F\ne0.7Ga0.3(110)Intensity (arb. units)/s61553\n/2/s61553 (degrees)2/s61553c = 44.36 /s61617 0.02 °F\nWHM = 0.50 /s61617 0.07 °70 nm\nFIG. S3. X-ray diffraction symmetric θ/2θscans for (a) 33 nm room temperature deposition, (b) 33 nm 200◦C deposition,\nand (c) 70 nm films. Full width at half maxima (FWHM) and 2 θcenter positions are indicated on the figure.\nS4. X-RAY DIFFRACTION\nX-ray diffraction (XRD) measurements were performed in order to determine both the degree of orientation and\nthe structural coherence length of the films.\nSymmetric θ/2θscans were taken with a Rigaku Smartlab diffractometer using Cu Kα 1(λ= 1.54˚A) radiation.\nThe data for both samples are shown in Fig. S3. The grain size was estimated using the Scherrer formula for spherical\ngrains [S4] as 13 nm, 9 nm, and 17 nm for the 33 nm (room temperature deposition), 33 nm (200◦C deposition), and\n70 nm films respectively.\nTwo-dimensional images were collected with a Bruker D8 Discover diffractometer using Co Kα 1(λ= 1.79˚A)\nradiation. Detector images showing the “ring” corresponding to the Fe 0.7Ga0.3(110) peak in four different samples\nare shown in Fig. S4. The ring indicates that the Fe 0.7Ga0.3(110) planes are randomly oriented over the range of the\ndetector, which we take to be evidence that there is no texture over a macroscopic scale in these samples. Furthermore,\nthe films were grown directly on top of amorphous SiO 2layers, so we do not expect an epitaxial relationship between\nthe film and substrate. The Fe 0.7Ga0.3(110) peaks were the only measurable Bragg peaks since the structure factor\nis highest for this case.\nS5. ATOMIC FORCE MICROSCOPY\nAtomic force microscopy data are shown in Fig. S5 for the 33 nm (room temperature and 200◦C depositions),\n57 nm, and 70 nm films. The field-of-view is 250 nm for the 33 nm films and 500 nm for the 57 nm and 70 nm films.\nThe root-mean-square (RMS) roughness of the sample surfaces is 0.7 nm, 0.4 nm, 1.5 nm, and 1.3 nm for the 33 nm\n(room temperature deposition), 33 nm (200◦C deposition), 57 nm, and 70 nm films, respectively .\nS6. VIBRATING SAMPLE MAGNETOMETRY\nVibrating sample magnetometry (VSM) data for the 33 nm (room temperature and 200◦C depositions) and 70 nm\nfilms are shown in Fig. S6. The magnetic field was applied in 3 different directions, with no discernible difference in the\nhysteresis loops. We conclude that there is no in-plane magnetocrystalline anisotropy over macroscopic lengthscales,\nwhich is consistent with the FMR measurements.\nS7. LONGITUDINAL RESISTIVITY\nLongitudinal resistivity ρxxwas measured as a function of temperature for the 33 nm (room temperature and 200◦C\ndepositions) and 70 nm films (Fig. S7) by patterning Hall bars and performing 4-wire resistance measurements.4\n65 60 55 50 45 40 35\n2/s61553(°)\n0100200300400Intensity\n(arb.units)\n65 60 55 50 45 40 35\n2/s61553(°)\n0100200300400Intensity\n(arb.units)\n65 60 55 50 45 40 35\n2/s61553(°)\n020406080Intensity\n(arb.units)\n65 60 55 50 45 40 35\n2/s61553(°)\n020406080Intensity\n(arb.units)(b) (a)\n(c) (d)q\nxyz\nq\nxyz33 nm (RT) 33 nm (200 °C) \n57 nm 70 nm\nFIG. S4. Two-dimensional detector images of the Fe 0.7Ga0.3(110) peak for (a) 33 nm (room temperature deposition), (b)\n33 nm (200◦C deposition), (c) 57 nm, and (d) 70 nm films. The total scattering angle is 2 θand is shown on the abscissa. The\nmeasurement is conducted such that the symmetric configuration corresponds to the center of the detector, which is to say\nthat the incident radiation is at an angle ω/similarequal26◦relative to the sample surface. In panel (a), the effect of moving vertically\nfrom the center of the detector on the scattering vector qis shown ( qis canted into the y-zplane).\n[S1] R. Arias and D. L. Mills, Extrinsic contributions to the ferromagnetic resonance response of ultrathin films, Phys. Rev. B\n60, 7395 (1999).\n[S2] R. D. McMichael and P. Krivosik, Classical Model of Extrinsic Ferromagnetic Resonance Linewidth in Ultrathin Films,\nIEEE Trans. Magn. 40, 2 (2004).\n[S3] P. Krivosik, N. Mo, S. Kalarickal, and C. E. Patton, Hamiltonian formalism for two magnon scattering microwave relaxation:\nTheory and applications, J. Appl. Phys. 101, 083901 (2007).5\n-2-1012Height(nm)(a)\n-2-1012Height(nm)(b)\n(c) (d)\n-2-1012Height(nm)\n-2-1012Height(nm)100 nm\n100 nm100 nm\n100 nm33 nm (RT)\n57 nm33 nm (200 °C) \n70 nm\nFIG. S5. Atomic force microscopy for (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), (c) 57 nm,\nand (d) 70 nm films. RMS roughnesses are (a) 0.7 nm, (b) 0.4 nm, (c) 1.5 nm, and (d) 1.3 nm.\n[S4] M. Birkholz, Thin Film Analysis by X-Ray Scattering (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2006).6\n-2000-1000010002000-101M/MsF\nield (Oe) H || Si[100] \nH || Si[110] \nH || Si[010]70 nm(c)\n-200-1000100200-101 \nH || Si[100] \nH || Si[110] \nH || Si[010]M/MsF\nield (Oe)33 nm (\n200 °C)(b)\n-50-2502550-101 \nH || Si[100] \nH || Si[110] \nH || Si[010]M/Ms F\nield (Oe)33 nm (RT)(a)\nFIG. S6. Vibrating sample magnetometry of (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), and\n(c) 70 nm films for H/bardblSi[100] (black), H/bardblSi[110] (red), and H/bardblSi[010] (blue).\n0100200300400020040060080010007\n0 nm(c)/s61554 xx (µΩ cm)T\nemperature (K)\n0100200300400050100150200250/s61554xx (µΩ cm)T\nemperature (K)33 nm (200 °C deposition)(b)\n0100200300400050100150200250(a)/s61554xx (µΩ cm)T\nemperature (K)33 nm (RT deposition)\nFIG. S7. Longitudinal resistivity ρxxas a function of temperature for the (a) 33 nm (room temperature deposition), (b) 33 nm\n(200◦C deposition), and (c) 70 nm films."    }]